Geometric and Algebraic Topological Meth

Geometric and Rlgebrctic Topologicol Methods in Quantum Mechanics Geometric one) fllgebroic Topologicol Methods in Qu...

Author: Giovanni Giachetta | Luigi Mangiarotti | G. Sardanashvily

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Geometric and Rlgebrctic Topologicol Methods in Quantum Mechanics

Geometric one) fllgebroic Topologicol Methods in Quantum Mechanics

Giovanni Giachetta & Luigi Mangiarotti University of Camerino, Italy

Gennadi Sardanashvily Moscow State University, Russia

\fc World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

GEOMETRIC AND ALGEBRAIC TOPOLOGICAL METHODS IN QUANTUM MECHANICS Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 891-256-129-3

ISBN 981-256-129-3

Printed in Singapore.

Preface

Contemporary quantum mechanics meets an explosion of different types of quantization. Some of these quantization techniques (geometric quantization, deformation quantization, BRST quantization, noncommutative geometry, quantum groups, etc.) call into play advanced geometry and algebraic topology. These techniques possess the following main peculiarities. • Quantum theory deals with infinite-dimensional manifolds and fibre bundles as a rule. • Geometry in quantum theory speaks mainly the algebraic language of rings, modules, sheaves and categories. • Geometric and algebraic topological methods can lead to nonequivalent quantizations of a classical system corresponding to different values of topological invariants. Geometry and topology are by no means the primary scope of our book, but they provide the most effective contemporary schemes of quantization. At the same time, we present in a compact way all the necessary up to date mathematical tools to be used in studying quantum problems. Our book addresses to a wide audience of theoreticians and mathematicians, and aims to be a guide to advanced geometric and algebraic topological methods in quantum theory. Leading the reader to these frontiers, we hope to show that geometry and topology underlie many ideas in modern quantum physics. The interested reader is referred to extensive Bibliography spanning mostly the last decade. Many references we quote are duplicated in E-print arXiv (http://xxx.lanl.gov). With respect to mathematical prerequisites, the reader is expected to be familiar with the basics of differential geometry of fibre bundles. For the sake of convenience, a few relevant mathematical topics are compiled in Appendixes. V

Contents

Preface

v

Introduction

1

1.

2.

Commutative geometry

17

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

17 23 27 31 39 56 59 70 85

....

Classical Hamiltonian systems 2.1 2.2 2.3 2.4 2.5 2.6 2.7

3.

Commutative algebra Differential operators on modules and rings Connections on modules and rings Homology and cohomology of complexes Homology and cohomology of groups and algebras Differential calculus over a commutative ring Sheaf cohomology Local-ringed spaces Algebraic varieties

91

Geometry and cohomology of Poisson manifolds Geometry and cohomology of symplectic foliations Hamiltonian systems Hamiltonian time-dependent mechanics Constrained Hamiltonian systems Geometry and cohomology of Kahler manifolds Appendix. Poisson manifolds and groupoids

Algebraic quantization 3.1

....

91 110 115 136 157 172 189 195

GNS construction I. C*-algebras of quantum systems . . . 195 vii

viii

Geometric and Algebraic Topological Methods in Quantum Mechanics

3.2 3.3 3.4 3.5 3.6 3.7 3.8 4.

5.

209 217 224 229 234 238 249

Geometry of algebraic quantization

257

4.1 4.2 4.3 4.4 4.5 4.6 4.7

257 271 274 278 282 286 290

Banach and Hilbert manifolds Dequantization Berezin's quantization Hilbert and C*-algebra bundles Connections on Hilbert and C*-algebra bundles Example. Instantwise quantization Example. Berry connection

Geometric quantization 5.1 5.2 5.3 5.4 5.5 5.6 5.7

6.

GNS construction II. Locally compact groups Coherent states GNS construction III. Groupoids Example. Algebras of infinite qubit systems GNS construction IV. Unbounded operators Example. Infinite canonical commutation relations . . . . Automorphisms of quantum systems

Leafwize geometric quantization Example. Quantum completely integrable systems Quantization of time-dependent mechanics Example. Non-adiabatic holonomy operators Geometric quantization of constrained systems Example. Quantum relativistic mechanics Geometric quantization of holomorphic manifolds

295 ....

295 306 312 324 332 335 342

Supergeometry

347

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11

347 352 358 366 382 385 388 392 401 423 426

Graded tensor calculus Graded differential calculus and connections Geometry of graded manifolds Lagrangian formalism on graded manifolds Lagrangian supermechanics Graded Poisson manifolds Hamiltonian supermechanics BRST complex of constrained systems Appendix. Supermanifolds Appendix. Graded principal bundles Appendix. The Ne'eman-Quillen superconnection

Contents

7.

8.

9.

10.

ix

Deformation quantization

433

7.1 7.2 7.3 7.4 7.5 7.6

433 444 450 459 472 475

Gerstenhaber's deformation of algebras Star-product Fedosov's deformation quantization Kontsevich's deformation quantization Deformation quantization and operads Appendix. Monoidal categories and operads

Non-commutative geometry

483

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10

484 486 492 498 503 507 509 512 514 518

Modules over C*-algebras Non-commutative differential calculus Differential operators in non-commutative geometry . . . . Connections in non-commutative geometry Connes' non-commutative geometry Landsman's quantization via groupoids Appendix. if-Theory of Banach algebras Appendix. The Morita equivalence of C*-algebras Appendix. Cyclic cohomology Appendix. KK-Theory

Geometry of quantum groups

523

9.1 9.2 9.3

523 530 535

Quantum groups Differential calculus over Hopf algebras Quantum principal bundles

Appendixes 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12

Categories Hopf algebras Groupoids and Lie algebroids Algebraic Morita equivalence Measures on non-compact spaces Fibre bundles I. Geometry and connections Fibre bundles II. Higher and infinite order jets Fibre bundles III. Lagrangian formalism Fibre bundles IV. Hamiltonian formalism Fibre bundles V. Characteristic classes /f-Theory of vector bundles Elliptic complexes and the index theorem

541 541 546 553 565 569 586 611 618 626 633 648 650

x

Geometric and Algebraic Topological Methods in Quantum Mechanics

Bibliography

661

Index

683

Introduction

Geometry of classical mechanics and field theory is mainly differential geometry of finite-dimensional smooth manifolds, fibre bundles and Lie groups. The key point why geometry plays a prominent role in classical field theory lies in the fact that it enables one to deal with invariantly defined objects. Gauge theory has shown clearly that this is a basic physical principle. At first, a pseudo-Riemannian metric has been identified to a gravitational field in the framework of Einstein's General Relativity. In 60-70th, one has observed that connections on a principal bundle provide the mathematical model of classical gauge potentials [120; 284; 442]. Furthermore, since the characteristic classes of principal bundles are expressed in terms of the gauge strengths, one can also describe the topological phenomena in classical gauge models [142]. Spontaneous symmetry breaking and Higgs fields have been explained in terms of reduced G-structures [341]. A gravitational field seen as a pseudo-Riemannian metric exemplifies such a Higgs field [230]. In a general setting, differential geometry of smooth fibre bundles gives the adequate mathematical formulation of classical field theory, where fields are represented by sections of fibre bundles and their dynamics is phrased in terms of jet manifolds [169]. Autonomous classical mechanics speaks the geometric language of symplectic and Poisson manifolds [l; 279; 426]. Non-relativistic time-dependent mechanics can be formulated as a particular field theory on fibre bundles over R [294]. At the same time, the standard mathematical language of quantum mechanics and perturbative field theory, except gravitation theory, has been long far from geometry. In the last twenty years, the incremental development of new physical ideas in quantum theory (including super- and BRST symmetries, geometric and deformation quantization, topological field the1

2

Geometric and Algebraic Topological Methods in Quantum Mechanics

ory, anomalies, non-commutativity, strings and branes) has called into play advanced geometric techniques, based on the deep interplay between algebra, geometry and topology. Let us briefly survey some peculiarities of geometric and algebraic topological methods in quantum mechanics. Let us recall that, in the framework of algebraic quantization, one associates to a classical system a certain (e.g., von Neumann, C*-, canonical commutation or anticommutation relation) algebra whose different representations are studied. Quantization techniques under discussion introduce something new. Namely, they can provide non-equivalent quantizations of a classical system corresponding to different values of some topological and differential invariants. For instance, a symplectic manifold X admits a set of non-equivalent star-products indexed by elements of the cohomology group i?2(X)[[/i]] [206; 340]. Thus, one may associate to a classical system different underlying quantum models. Of course, there is a question whether this ambiguity is of physical or only mathematical nature. Prom the mathematical viewpoint, one may propose that any quantization should be a functor between classical and quantum categories (e.g., some subcategory of Poisson manifolds on the classical side and a subcategory of C*-algebras on the quantum side) [271]. From the physical point of view, dequantization becomes important. There are several examples of sui generis dequantizations. For instance, Berezin's quantization [145] in fact is dequantization. One can also think of well-known Gelfand's map as being dequantization of a commutative C*-algebra A by the algebra of continuous complex functions vanishing at infinity on the spectrum of A. This dequantization has been generalized to non-commutative unital C*-algebras [105; 239]. The concept of the strict C*-algebraic deformation quantization implies an appropriate dequantization when h —> 0 [269; 372]. In Connes' non-commutative geometry, dequantization of the spectral triple in the case of a commutative algebra C°°(X) is performed in order to restart the original differential geometry of a spin manifold X [107; 368]. I. Let us start with familiar differential geometry. There are the following reasons why this geometry contributes to quantum theory. (i) Most of the quantum models comes from quantization of the original

Introduction

3

classical systems and, therefore, inherits their differential geometric properties. First of all, this is the case of canonical quantization which replaces the Poisson bracket {/, / ' } of smooth functions with the bracket [/, /'] of Hermitian operators in a Hilbert space such that Dirac's condition

[f,T] = -ih{fJ!) holds. Let us mention Berezin-Toeplitz quantization [47; 145; 365] and geometric quantization [141; 401; 426; 438] of symplectic, Poisson and Kahler manifolds. (ii) Many quantum systems are considered on a smooth manifold equipped with some background geometry. As a consequence, quantum operators are often represented by differential operators which act in a pre-Hilbert space of smooth functions. A familiar example is the Schrodinger equation. The Kontsevich deformation quantization is based on the quasi-isomorphism of the differential graded Lie algebra of multivector fields (endowed with the Schouten-Nijenhuis bracket and the zero differential) to that of polydifferential operators (provided with the Gerstenhaber bracket and the modified Hochschild differential) [219; 255]. (iii) In some quantum models, differential geometry is called into play as a technical tool. For instance, a suitable [/(l)-principal connection is used in order to construct the operators / in the framework of geometric quantization. Another example is Fedosov's deformation quantization where a symplectic connection plays a similar role [149]. Let us note that this application has stimulated the study of symplectic connections [165]. (iv) Geometric constructions in quantum models often generalize the classical ones, and they are build in a similar way. For example, connections on principal superbundles [21], graded principal bundles [405], and quantum principal bundles [293] are defined by means of the corresponding oneforms in the same manner as connections on smooth principal bundles with structure finite-dimensional Lie groups. II. In quantum models, one deals with infinite-dimensional smooth Banach and Hilbert manifolds and (locally trivial) Hilbert and C*-algebra bundles. The definition of smooth Banach (and Hilbert) manifolds follows that of finite-dimensional smooth manifolds in general, but infinite-dimensional

4

Geometric and Algebraic Topological Methods in Quantum Mechanics

Banach manifolds are not locally compact, and they need not be paracompact [273; 422]. In particular, a Banach manifold admits the differentiable partition of unity if and only if its model space does. It is essential that Hilbert manifolds (but not, e.g., nuclear manifolds) satisfy the inverse function theorem and, therefore, locally trivial Hilbert bundles are denned. However, they need not be bundles with a structure group. (i) Infinite-dimensional Kahler manifolds provide an important example of Hilbert manifolds [327]. In particular, the projective Hilbert space of complex rays in a Hilbert space E is such a Kahler manifold. This is the space the pure states of a C*-algebra A associated to the same irreducible representation n of A in a Hilbert space E [129]. Therefore, it plays a prominent role in many quantum models. For instance, it has been suggested to consider a loop in the projective Hilbert space, instead of a parameter space, in order to describe Berry's phase [7; 43]. We have already mentioned the dequantization procedure which represents a unital C*-algebra by a Poisson algebra of complex smooth functions on a projective Hilbert space [105]. (ii) Sections of a Hilbert bundle over a smooth finite-dimensional manifold X make up a particular locally trivial continuous field of Hilbert spaces in [129]. Conversely, one can think of any locally trivial continuous field of Hilbert spaces or C*-algebras as being the module of sections of a topological fibre bundle. Given a Hilbert space E, let B C B{E) be some C*-algebra of bounded operators in E. The following fact reflects the nonequivalence of Schrodinger and Heisenberg quantum pictures. There is the obstruction to the existence of associated (topological) Hilbert and C*algebra bundles £ —» X and B —+ X with the typical fibres E and B, respectively. Firstly, transition functions of £ define those of B, but the latter need not be continuous, unless B is the algebra of compact operators in E. Secondly, transition functions of B need not give rise to transition functions of £. This obstruction is characterized by the Dixmier-Douady class of B in the Cech cohomology group H3(X, Z). There is the similar obstruction to the [/(l)-extension of structure groups of principal bundles [73; 86]. One also meets the Dixmier-Douady class as the obstruction to a bundle gerbe being trivial [58; 87]. (iii) There is a problem of the definition of a connection on C*-algebra bundles which comes from the fact that a C*-algebra (e.g., any commutative C*-algebra) need not admit non-zero bounded derivations. An unbounded derivation of a C*-algebra A obeying certain conditions is an infinitesimal generator of a strongly (but not uniformly) continuous one-parameter group

Introduction

5

of automorphisms of A [62]. Therefore, one may introduce a connection on a C*-algebra bundle in terms of parallel transport curves and operators, but not their infinitesimal generators [15]. Moreover, a representation of A does not imply necessarily a unitary representation of its strongly (not uniformly) continuous one-parameter group of automorphisms. In contrast, connections on a Hilbert bundle over a smooth manifold can be defined both as particular first order differential operators on the module of its sections [296] and a parallel displacement along paths lifted from the base [228]. (iv) Instantwise geometric quantization of time-dependent mechanics is phrased in terms of Hilbert bundles over R [174; 401]. Holonomy operators in a Hilbert bundle with a structure finite-dimensional Lie group are well known to describe the non-Abelian geometric phase phenomena [44]. At present, holonomy operators in Hilbert bundles attract special attention in connection with quantum computation and control theory [159; 181; 349]. III. Geometry in quantum systems speaks mainly the algebraic language of rings, modules and sheaves due to the fact that the basic ingredients in the differential calculus and differential geometry on smooth manifolds (except non-linear differential operators) can be restarted in a pure algebraic way. (i) Any smooth real manifold X is homeomorphic to the real spectrum of the M-ring C°°(X) of smooth real functions on X provided with the Gelfand topology [17; 233]. Furthermore, the sheaf Cg of germs of / G C°°(X) on this topological space fixes a unique smooth manifold structure on X such that it is the sheaf of smooth functions on X. The pair (X, C^) exemplifies a local-ringed space. A sheaf SRon a topological space X is said to be a local-ringed space if its stalk 9lx at each point x £ X is a local commutative ring [414], One can associate to any commutative ring A the particular local-ringed space, called an affine scheme, on the spectrum Spec .4. of A endowed with the Zariski topology [421]. Furthermore, one can assign the following algebraic variety to any commutative finitely generated /C-ring A over an algebraically closed field K. Given a ring fC[x] of polynomials with coefficients in /C, let us consider the epimorphism 4> : K\x\ —» A defined by the equalities
6

Geometric and Algebraic Topological Methods in Quantum Mechanics

Every affine variety V in turn yields the affine scheme Spec/Cy such that there is one-to-one correspondence between the points of Spec/Cv and the irreducible subvarieties of V. For instance, complex algebraic varieties have a structure of complex analytic manifolds. (ii) Given a (connected) compact topological space X and the ring C°(X) of continuous complex functions on X, the well-known Serre-Swan theorem [409] states that a C°(X)-module is finitely generated projective if and only if it is isomorphic to the module of sections of some (topological) vector bundle over X. Moreover, this isomorphism is a categorial equivalence [237], and its variant takes place if X is locally compact [369]. If X is a compact smooth manifold, there is the similar isomorphism of a finitely generated projective C°°(X)-modules on X to the modules of sections of some smooth vector bundle over X [430], and this is also true if X is not necessarily compact. A variant of the Serre-Swan theorem for Hilbert modules over non-commutative C*-algebras holds [239]. (iii) Let K. be a commutative ring, A a commutative /C-ring, and P, Q some ^-modules. The /C-linear Q-valued differential operators on P can be defined [202; 233; 261]. The representative objects of the functors Q —> Diff S(P, Q) are the jet modules J"P of P. Using the first order jet module JlP, one also restarts the notion of a connection on an ^-module P [260; 296]. Such a connection assigns to each derivation r £ dA of a /C-ring .4 a first order P-valued differential operator VT on P obeying the Leibniz rule V r (ap)=r(o)p + aVT(p). For instance, if P is a C°° (X)-module of sections of a smooth vector bundle Y —> X, we come to the familiar notions of a linear differential operator on Y, the jets of sections of Y —» X and a linear connection on Y —» X. Similarly, connections on local-ringed spaces are introduced [296]. In supergeometry, connections on graded modules over a graded commutative ring and graded local-ringed spaces are defined [2l]. In non-commutative geometry, different definitions of a differential operator on modules over a non-commutative ring have been suggested [50; 136; 286]. Roughly speaking, the difficulty lies in the fact that, if d is a derivation of a non-commutative /C-ring A, the product ad, a € A, need not be so. There are also different definitions of a connection on modules over a non-commutative ring [137; 267]. (iv) Let K. be a commutative ring, A a (commutative or noncommutative) /C-ring, and Z(A) the center of A. Derivations of A make up a Lie /C-algebra 5.4. Let us consider the Chevalley-Eilenberg com-

Introduction

7

plex of /C-multilinear morphisms of DA to A, seen as a DA-module [160; 426]. Its subcomplex O*($A, d) of Z (^-multilinear morphisms is a differential graded algebra, called the Chevalley-Eilenberg differential calculus over A. It contains the minimal differential calculus O*A generated by elements da, a € A. If ^4. is the R-ring C°°(X) of smooth real functions on a smooth manifold X, the module QC°°(X) of its derivations is the Lie algebra of vector fields on X and the Chevalley-Eilenberg differential calculus over C°°(X) is exactly the algebra of exterior forms on a manifold X where the Chevalley-Eilenberg coboundary operator d coincides with the exterior differential, i.e., O*(X>C°°(X),d) is the familiar de Rham complex. In a general setting, one therefore can think of elements of the ChevalleyEilenberg differential calculus Ok(QA, d) over an algebra .4 as being differential forms over A. Similarly, the Chevalley-Eilenberg differential calculus over a graded commutative ring is constructed [160]. IV. As was mentioned above, homology and cohomology of spaces and algebraic structures often play a role of sui generis hidden quantization parameters which can characterize non-equivalent quantizations. (i) First of all, let us mention the abstract de Rham theorem [220] and, as its corollary, the homomorphism H*(X,Z)-+H*(X) of the Cech cohomology of a smooth manifold X to the de Rham cohomology of exterior forms on X. For instance, the Chern classes c* € H2l(X, Z) of a [/(n)-principal bundle P —» X are represented by the de Rham cohomology classes of certain characteristic exterior forms V-2.%(FA) on X expressed into the strength two-form FA of a principal connection A on F - » I [142]. The Chern class c-i of a complex line bundle plays a prominent role in many quantization schemes, e.g., geometric quantization. The well-known index theorem establishes the equality of the index of an elliptic operator on a fibre bundle to its topological index expressed in terms of the characteristic forms of the Chern character, Todd and Euler classes. Let us note that the classical index theorem deals with linear elliptic operators on compact manifolds. They are Fredholm operators. In order to generalize the index theorem to non-compact manifolds, one either imposes conditions sufficient to force operators to be the Fredholm ones or

8

Geometric and Algebraic Topological Methods in Quantum Mechanics

considers the operators which are no longer Fredholm, but their index can be interpreted as a real number by some kind of averaging procedure [375]. (ii) Geometric quantization of a symplectic manifold (X, Cl) is affected by the following ambiguity. Firstly, the equivalence classes of admissible connections on a prequantization bundle (whose curvature obeys the prequantization condition R = ifl) are indexed by the set of homomorphisms of the homotopy group TT\(X) of X to U(l) [257; 312]. Secondly, there are non-equivalent bundles of half-forms over X in general and, consequently, the non-equivalent quantization bundles exist [141]. This ambiguity leads to non-equivalent quantizations. (iii) The cohomology analysis gives a rather complete picture of deformation quantization of symplectic manifolds. Let K, be a commutative ring and K.[[h]\ the ring of formal series in a real parameter h. Let us recall that, given an associative (resp. Lie) algebra A over a commutative ring /C, its Gerstenhaber deformation [166] is an associative (resp. Lie) /C[[/i]]-algebra Ah such that Ah/hAh = A. The multiplication in Ah reads oo

a*b = aob+

Y^ hrCr(a, b) r=l

where o is the original associative (resp. Lie) product and Cr are 2-cochains of the Hochschild (resp. Chevalley-Eilenberg) complex of A. The obstruction to the existence of a deformation of A lies in the third Hochschild (resp. Chevalley-Eilenberg) cohomology group. Let A = C°°(X) be the ring of complex smooth functions on a smooth manifold X. One considers its associative deformations Ah where the cochains Cr are bidifferential operators of finite order. The multidifferential cochains make up a subcomplex of the Hochschild complex of A, and its cohomology equals the space of multi-vector fields on X [433]. If
Introduction

9

of Poisson deformations of the Poisson bracket on a symplectic manifold X are parameterized by i?2(X)[[/t]]. A star-product on a Poisson manifold is defined as an associative deformation of C°°(X) such that C\ (/,/') — C\ (/',/) is the Poisson bracket. The existence of a star-product on an arbitrary symplectic manifold has been proved in [125], and this is true for any regular Poisson manifold [150; 302]. Moreover, any star-product on a symplectic manifold is equivalent to Fedosov's one, and its equivalence classes are parameterized by H2(X)[[h}] [206; 340]. (iv) Let us also mention BRST cohomology, called into play in order to describe constrained symplectic systems [156; 217; 259]. Let (Z, fl) be a symplectic manifold endowed with a Hamiltonian action of a Lie group G, J the corresponding momentum mapping of Z to the Lie coalgebra g* of G, and TV = J~ 1 (0) a regular constraint surface. The classical BRST complex is defined as the bicomplex Bn'm =

AQ*®Ag®C°°(Z),

where the n- and m-gradings are the ghost and antighost degrees, respectively. The differential S . B*'* -> B*+h* is the coboundary operator of the Chevalley-Eilenberg cohomology of the Lie algebra g of G with coefficients in the g-module g ® C°°(Z), while d:B*'* -> B*'*'1 is the Koszul boundary operator. The algebra B is provided with the graded Poisson bracket [,], and there exists an element 9 of B, called the BRST charge, such that [0,0] = 0 and D = [Q,.] = 5 + d up to extra terms of non-zero ghost number is the nilpotent classical BRST operator. The BRST cohomology is defined as the cohomology of this classical BRST operator. The BRST complex has been built for constrained Poisson systems [245] and time-dependent Hamiltonian systems with Lagrangian constraints [295] as an extension of the Koszul-Tate complex of constraints through introduction of ghosts. Quantum BRST cohomology has been studied in the framework of geometric [419] and deformation [49] quantization. V. Contemporary quantum models appeal to a number of new algebraic structures and the associated geometric techniques.

10

Geometric and Algebraic Topological Methods in Quantum Mechanics

(i) For instance, SUSY models deal with graded manifolds and different types of supermanifolds, namely, H°°-, G°°-, GH°°-, G-supermanifolds over (finite) Grassmann algebras, R°°- and .R-supermanifolds over ArensMichael algebras of Grassmann origin and the corresponding types of DeWitt supermanifolds [21; 22; 69]. Their geometries are phrased in terms of graded local-ringed spaces. Let us note that one usually considers supervector bundles over G-supermanifolds. Firstly, the category of these supervector bundles is equivalent to the category of locally free sheaves of finite rank (in contrast, e.g., with Gff°°-supermanifolds). Secondly, derivations of the structure sheaf of a G-supermanifold constitute a locally free sheaf (this is not the case, e.g., of G°°-supermanifolds). Moreover, this sheaf is again a structure sheaf of some G-superbundle (in contrast with graded manifolds). At the same time, most of the quantum models uses graded manifolds. They are not supermanifolds, though there is the correspondence between graded manifolds and DeWitt if00-supermanifolds. By virtue of the well-known Batchelor theorem, the structure ring of any graded manifold with a body manifold Z is isomorphic to the graded ring AE of sections of some exterior bundle AE* —> Z. In physical models, this isomorphism holds fixed from the beginning as a rule and, in fact, by geometry of a graded manifold is meant the geometry of the graded ring AE- For instance, the familiar differential calculus in graded exterior forms is the graded Chevalley-Eilenberg differential calculus over such a ring. (ii) Non-commutative geometry is mainly developed as a generalization of the calculus in commutative rings of smooth functions [107; 194; 267]. In a general setting, any non-commutative /C-ring A over a commutative ring /C can be called into play. One can consider the above mentioned Chevalley-Eilenberg differential calculus O*A over A, differential operators and connections on A-modules (but not their jets). If the derivation K.module T)A is a finite projective module with respect to the center of A, one can treat the triple (A, (3A, O*A) as a non-commutative space. For instance, this is the case of the matrix geometry, where A is the algebra of finite matrices, and of the quantum phase space, where A is a finitedimensional algebra of canonical commutation relations. Non-commutative field theory also can be treated in this manner [133; 359], though the bracket of space coordinates [xfl,x1/} = ie^

in this theory is also restarted from Moyal's star-product xM * xv [99; 133].

Introduction

11

A different linear coordinate product [x»,xl'] = i<$'xx comes from Connes' non-commutative geometry [195]. In Connes' non-commutative geometry, the more deep analogy to the case of commutative smooth function rings leads to the notion of a spectral triple (A, E, V) [107; 109]. It is given by an involutive subalgebra A C B(E) of bounded operators on a Hilbert space E and an (unbounded) selfadjoint operator V in E such that the resolvent (V - A)"1, A € C \ E is a compact operator and [D,A] C B(E). Furthermore, one assigns to elements u> = aqdai • • • dak of the universal differential calculus (O*A, d) over A the operators 7r(w) = a 0 [P,oi] •••[!>,ofc] in E. This however fails to be a representation of the differential algebra O* A because 7r(o>) = 0 does not imply ir(dw) = 0. The appropriate quotient O*A 3 4> -> 1
together with the differential d[u>] := [dw] overcomes this difficulty, though 7r([w]) is not an operator in E. Let us note that other variants of spectral data, besides a spectral triple, are also discussed [158]. The algebra C°°{X) and the Dirac operator P on a compact manifold X exemplifies Connes' commutative geometry [108; 368]. Spectral triples have been studied for non-commutative tori, the Moyal deformations of E", non-commutative spheres 2-, 3- and 4-spheres [95; 110], and quantum Heisenberg manifolds

N.

(iii) Formalism of groupoids provides the above mentioned categorial C*-algebraic deformation quantization of some class of Poisson manifolds [270; 271]. A groupoid is a small category whose morphisms are invertible [287; 367]. For instance, given an action of a group G on a set X on the right, the product © = X x G is brought into the action groupoid where: • a pair ((x,g), (x',g')) is composable if and only if x' — xg, • the inversion (a;,^)" 1 = (xg,g~x), • the partial multiplication (x,g)(xg,g') — (x,gg'), • the range r{(x,g)) = (x, 1G), • the domain l((x,g)) :— (xg, 1Q). The unit space 0° = r(<8) = l(<5) of this groupoid is naturally identified to X. Any group bundle Y —> X (e.g., a vector bundle) is a groupoid

12

Geometric and Algebraic Topological Methods in Quantum Mechanics

whose elements make up composable pairs if and only if they belong to the same fibre, and whose unit space is the set of unit elements of fibres of Y —> X. Let 21 —> ©° be an Abelian group bundle over the unit space ©° of a groupoid ©. The pair (©,21) together with a homomorphism © —> Iso2l is called the ©-module bundle. One can associate to any ©-module bundle a cochain complex C*(©,21). Let 21 be a 6-module bundle in groups [/(I). The key point is that, similarly to the case of a locally compact group [129], one can associate a C*-algebra C*(0,u) to any locally compact groupoid © provided with a Haar system by means of the choice of a two-cocycle a G C2(©,21) [367]. The algebras C*(<S,a) and C*(©, (xy,x,y) is a Lagrangian submanifold of the symplectic manifold (© x ©x 6 , f l e f i 6 f i ) [27]. A Poisson manifold P is called integrable if there exists a symplectic groupoid ©(P) over P. It is unique up to an isomorphism. Integrable Poisson manifolds subject to a certain class morphisms (isomorphism classes of regular dual pairs) make up a suitable category Poisson [270]. Since the groupoid ©(P) is I- and /-simple connected, one considers the category LG of Lie groupoids possessing this property. Any Lie groupoid yields an associated Lie algebroid A(<S) which is the restriction to ©° of the vertical tangent bundle of the fibration r : © -> ©° [287]. The key point is that, similarly to the dual of a Lie algebra, the dual A*(<8) of A(<5) is a Poisson manifold. Then the assignment © i-> A*(&) is a functor from LG to Poisson [271]. Let LPoisson denote its image. One can show that T : L*(©) e-> C*(©)

13

Introduction

is a functor from the category LPoisson to the above mentioned category of C*-algebras [271]. It is a desired functorial quantization. This functor is equivariant under the Morita equivalence of Poisson manifolds in LG [444] and that of C*-algebras [371]. Furthermore, the functorial quantization A*(<&) — I > C*(6) is amplified into the above mentioned strict quantization of C*(<&) by an appropriate continuous field of C*-algebras over R [27l]. Connes' tangent groupoid provides an example of such strict quantization [269]. (iv) Hopf algebras and, in particular, quantum groups make a contribution to many quantum theories [97; 249; 292; 293]. At the same time, the development of differential calculus and differential geometry over these algebras has met difficulties. Given a (complex or real) Hopf algebra H = (H, m, A, e, 5), one introduces the first order differential calculus (henceforth FODC) (fi1,^) over H just as for a non-commutative algebra. It is said to be left-covariant if Q1 possesses the structure of a left iJ-comodule A; : n1 -> H 9} such that Ai{adb) = A(o)(Id ®d)A(b),

a,beH,

[249]. By virtue of Woronowicz's theorem [440], left-covariant FODCs are classified by right ideals Tl— {x £ Kere : S(xi)dx2 = 0, x = y ^ x \ ®x 2 ) of H contained in the kernel of its counit e. The linear subspace T - {X e H* : X (l) = 0, X (K) = 0} of the dual H* is the quantum (enveloping) Lie algebra (quantum tangent space [214]) associated to the left-covariant FODC (Ql,d) (see [286] for a general construction of the enveloping algebra for a non-commutative FODC). A problem lies in the definition of vector fields as a sum a%Ui of invariant vector fields Ui{a) = aiXi{.a2) [13] because they satisfy the deformed Leibniz rule deduced from the formula Xi(ab) = Xi(a)e(6) + $3/ij(a)x,-(&), i

where {xi} is a basis for T and /y are complex linear forms on H. One can model the vector fields obeying such a Leibniz rule by the so called Cartan

14

Geometric and Algebraic Topological Methods in Quantum Mechanics

pairs [50]. These are elements u of the right H-du&l fi1 together with the morphisms u: H 3 a >->= u(da) £ H

which obey the relations (bu)(a) — bu(a),

u(ba) = u(b)a + (ub)(a).

Another problem of geometry of Hopf algebras is the notion of a quantum principal bundle [75; 82; 293]. In the case of Lie groups, there are two equivalent definitions of a smooth principal bundle, which is both a set of trivial bundles glued together by means of transition functions and a bundle provided with the canonical action of a structure group on the right. In the case of quantum groups, these two notions of a principal bundle are not matched, unless the base is a smooth manifold [139; 355]. • The first definition of a quantum principal bundle repeats the classical one and makes use of the notion of a trivial quantum bundle, a covering of a quantum space (e.g., by a family of non-intersecting closed ideals), and its reconstruction from local pieces [76] which however is not always possible [81]. • The second definition of a quantum principal bundle is algebraic [74; 293]. Let H be a Hopf algebra and V a right K-comodule algebra with respect to the coaction /3 : V —» V H. Let : (3(p) = p ® l }

M = {p£V

be its invariant subalgebra. The triple (P,H,f3) is called a quantum principal bundle if the map ver : V ® V 9 (p ® q) ^ p/3{q) £ ? ® W M

M

is a linear isomorphisms. This condition, called the Hopf-Galois condition, is a key point of this algebraic definition of a quantum principal bundle. By some reasons, one can think of it as being a sui generis local trivialization. (v) Finally, one of the main point of Tamarkin's proof of the formality theorem in deformation quantization is that, for any algebra A over a field of characteristic zero, its Hochschild cochain complex and its Hochschild cohomology are algebras over the same operad [219; 411]. This observation has been the starting point of 'operad renaissance' [253; 297]. Monoidal categories provide numerous examples of algebras for

Introduction

15

operads. Furthermore, homotopy monoidal categories lead to the notion of a homotopy monoidal algebra for an operad. In a general setting, one considers homotopy algebras and weakened algebraic structures where, e.g., a product operation is associative up to homotopy [276]. Their well-known examples are A^-spaces and Aoo-algebras [403]. At the same time, the formality theorem is also applied to quantization of several algebraic geometric structures such as algebraic varieties [255; 450].

Chapter 1

Commutative geometry

In comparison with classical mechanics and field theory phrased in terms of smooth finite-dimensional manifolds, quantum theory speaks the algebraic language adapted to describing systems of infinite degrees of freedom. Geometric techniques are involved in quantum theory due to the fact that the differential calculus over an arbitrary ring can be denned. Their relation to the familiar differential geometry of smooth manifolds is based on the fact that any manifold can be characterized in full by a certain algebraic construction and, furthermore, there is the categorial equivalence between the vector bundles over a smooth manifold and the finite projective modules over the ring of smooth real functions on this manifold.

1.1

Commutative algebra

In this Section, the relevant basics on modules over commutative algebras is summarized [272; 288]. An algebra A is an additive group which is additionally provided with distributive multiplication. All algebras throughout the book are associative, unless they are Lie algebras. A ring is a unital algebra, i.e., it contains a unit element 1. Unless otherwise stated, we assume that 1 ^ 0 , i.e., a ring does not reduce to the zero element. One says that A is a division algebra if it has no a divisor of zero, i.e., ab = 0, a,b £ A, implies either a — 0 or b = 0. Non-zero elements of a ring form a multiplicative monoid. If this multiplicative monoid is a multiplicative group, one says that the ring has a multiplicative inverse. A ring A has a multiplicative inverse if and only if it is a division algebra. A field is a commutative ring whose non-zero elements make up a multiplicative group. A subset I of an algebra A is called a left (resp. right) ideal if it is a 17

18

Geometric and Algebraic Topological Methods in Quantum Mechanics

subgroup of the additive group A and ab e l (resp. 6a € 1) for all a E A, b e l If J is both a left and right ideal, it is called a two-sided ideal. An ideal is a subalgebra, but a proper ideal (i.e., 1 ^ A) of a ring is not a subring because it does not contain a unit element. Let A be a commutative ring. Of course, its ideals are two-sided. Its proper ideal is said to be maximal if it does not belong to another proper ideal. A commutative ring A is called local if it has a unique maximal ideal. This ideal consists of all non-invertible elements, of A. A proper twosided ideal I of a commutative ring is called prime if db £ 1 implies either a £ J or b £ 1. Any maximal two-sided ideal is prime. Given a two-sided ideal 1 c A, the additive factor group A/1 is an algebra, called the factor algebra. If A is a ring, then A/1 is so. If J is a prime ideal, the factor ring A/1 has no divisor of zero, and it is a field if J is a maximal ideal. Remark 1.1.1. We will refer to the following particular construction in the sequel. Let K be a commutative ring and S its multiplicative subset which, by definition, is a monoid with respect to multiplication in K. Let us say that two pairs (a,s) and (a',s'), a,a' £ /C, s,s' £ S, are equivalent if there exists an element s" £ S such that s"{s'a - so,') = 0. We abbreviate with a/s the equivalence classes of (a, s). The set S~1IC of these equivalence classes is a ring with respect to the operations s/a + s'/a' := (s'a + sa')/(ss'), (a/s) • (a'/s') := (aa')/(ss'). There is a homomorphism $s : K 3^ a/1 £ S^IC

(1.1.1)

such that any element of $s(S) is invertible in S^1^. If a ring K has no divisor of zero and S does not contain a zero element, then $ s (1.1.1) is a monomorphism. In particular, if 5 is the set of non-zero elements of K-, the ring S~1fC is a field, called the field of quotients of the fraction field of /C. If K. is field, its fraction field coincides with K. • Given an algebra A, an additive group P is said to be a left (resp. right) A-module if it is provided with distributive multiplication A x P —> P by elements of A such that (ab)p = a(bp) (resp. (ab)p = b(ap)) for all a, b € A and p £ P. If A is a ring, one additionally assumes that lp = p = pi for

19

Chapter 1 Commutative Geometry

all p € P. Left and right module structures are usually written by means of left and right multiplications (a, p) H-» ap and (a, p) >—> pa, respectively. If P is both a left module over an algebra A and a right module over an algebra A', it is called an (A — .4')-bimodule (an .4-bimodule if A — .4'). If A is a commutative algebra, an (.4 — .4)-bimodule P is said to be commutative if ap = pa for all a € A and p £ P. Any left or right module over a commutative algebra A can be brought into a commutative bimodule. Therefore, unless otherwise stated, any module over a commutative algebra A is called an .4-module (see Section 8.1). A module over a field is called a vector space. If an algebra A is a module over a ring K., it is said to be a IC-algebra. Any algebra can be seen as a Z-algebra. Remark 1.1.2. Any AC-algebra A can be extended to a unital algebra A by the adjunction of the identity 1 to A. The algebra A, called the unital extension of A, is defined as the direct sum of ^-modules K © A provided with the multiplication (Ai,ai)(A2,a2) = (AiA2,Aia2 + A 2 ai+aia 2 ),

Ai,A 2 £/C,

ai,a2eA

Elements of A can be written as (A, a) = Al + a, A € /C, a G A. Let us note that, if A is a unital algebra, the identity 1^ in A fails to be that in A. In this case, the algebra A is isomorphic to the product of A and the algebra K,(l — 1A)D In this Chapter (except Sections 1.5C), all associative algebras are assumed to be commutative, unless they are graded. The following are standard constructions of new modules from old ones. • The direct sum Pi © P 2 of ,4-modules Pi and P 2 is the additive group Pi x P 2 provided with the .4-module structure a(Pi,P2) = (api,ap2),

Pi,2 G P l i 2 ,

a & A.

Let {Pi} ie / be a set of modules. Their direct sum ©P* consists of elements (..., pi,...) of the Cartesian product n Pi s u c n that pi ^ 0 at most for a finite number of indices i € I. • The tensor product P ® Q of ^-modules P and Q is an additive group which is generated by elements p® q, p € P, q 6 Q, obeying the relations {p + p') ® q = P ® q + p' ® q, p ® (q + q') = p®q+p®q'', p e P, q € Q, a € A, pa (8) q = p aq,

20

Geometric and Algebraic Topological Methods in Quantum Mechanics

(see Remark 10.4.1), and it is provided with the .4-module structure a(p ® q) = (ap) ®q = p® (qa) = (p q)a.

If the ring A is treated as an .4-module, the tensor product A ®^ Q is canonically isomorphic to Q via the assignment A ®A

QBa®q<->aq£Q.

• Given a submodule Q of an .4-module P, the quotient P/Q of the additive group P with respect to its subgroup Q is also provided with an «4-module structure. It is called a factor module. • The set Horn ,4 (P, Q) of .4-linear morphisms of an .4-module P to an .4-module Q is naturally an .4-module. The .4-module P* = Horn ^(P, A) is called the dual of an .4-module P. There is a natural monomorphism P-» P**. An .4-module P is called free if it has a basis, i.e., a linearly independent subset I C P spanning P such that each element of P has a unique expression as a linear combination of elements of / with a finite number of non-zero coefficients from an algebra A. Any vector space is free. Any module is isomorphic to a quotient of a free module. A module is said to be finitely generated (or of finite rank) if it is a quotient of a free module with a finite basis. One says that a module P is protective if it is a direct summand of a free module, i.e., there exists a module Q such that P®Q is a free module. A module P is projective if and only if P = pS where 5 is a free module and p is a projector of S, i.e., p 2 = p. If P is a projective module of finite rank over a ring, then its dual P* is so, and P** is isomorphic to P. THEOREM

1.1.1. Any projective module over a local ring is free.

•

Now we focus on exact sequences, direct and inverse limits of modules [288; 303]. A composition of module morphisms P - U Q -^->T is said to be exact at Q if Ker j = Im i. A composition of module morphisms

O^P -UQ -Ur^o

(1.1.2)

21

Chapter 1 Commutative Geometry

is called a short exact sequence if it is exact at all the terms P, Q, and T. This condition implies that: (i) i is a monomorphism, (ii) Ker j = Imi, and (iii) j is an epimorphism onto the quotient T — Q/P. THEOREM 1.1.2. Given an exact sequence of modules (1.1.2) and another ,4-module R, the sequence of modules

0->EomA{T,R) i^RomA{Q,R)

^Eom{P,R)

(1.1.3)

is exact at the first and second terms, i.e., j * is a monomorphism, but i* HI need not be an epimorphism. One says that the exact sequence (1.1.2) is split if there exists a monomorphism s :T —> Q such that j o s = IdT or, equivalently,

Q = i{P) ® s{T) ^P®T. The exact sequence (1.1.2) is always split if T is a projective module. A directed set 7 is a set with an order relation < which satisfies the following three conditions: (i) i < i, for all i € I; (ii) if i < j and j < k, then i < k; (iii) for any i,j € / , there exists k £ I such that i < k and j < k. It may happen that i ^ j , but i < j and j < i simultaneously. A family of modules {P{\i^i (over the same algebra), indexed by a directed set / , is called a direct system if, for any pair i < j , there exists a morphism r* : Pi —> Pj such that r\ = I d P i ,

r) or{ = ri,

i<j<

k.

A direct system of modules admits a direct limit. This is a module P^ together with morphisms r ^ : Pi —> P^ such that r ^ = r£, o rj for all i < j . The module P^ consists of elements of the direct sum ©Pj modulo the identification of elements of Pi with their images in Pj for all i < j . An example of a direct system is a direct sequence Po —»Pi ^ • • • P / M . . . ,

J = N.

(1.1.4)

It should be noted that direct limits also exist in the categories of commutative algebras and rings, but not in categories whose objects are non-Abelian groups.

22

Geometric and Algebraic Topological Methods in Quantum Mechanics

THEOREM 1.1.3. Direct limits commute with direct sums and tensor products of modules. Namely, let {Pi} and {Qi} be two direct systems of modules over the same algebra which are indexed by the same directed set / , and let P^ and Q^ be their direct limits. Then the direct limits of the direct systems {Pi © Qi} and {Pt Qi} are P^ © Qoo and Poo ® Qoo, respectively. •

A morphism of a direct system {Pi,rlj}i to a direct system {Qi>,plj,}i> consists of an order preserving map / : / — » / ' and morphisms Fj : Pj —> Qf(i) which obey the compatibility conditions 0S^)oFi = Fior). If PQQ and Qoo are limits of these direct systems, there exists a unique morphism F^ : P ^ —> Qoo such that

p£>oFi=F00ori0. Moreover, direct limits preserve monomorphisms and epimorphisms. To be precise, if all Ft : Pi —> Q/(») axe monomorphisms or epimorphisms, so is $00 : Poo —* Qoo- As a consequence, the following holds. THEOREM

1.1.4. Let short exact sequences 0-^Pi

^Qi

^Ti^O

(1.1.5)

for all i £ I define a short exact sequence of direct systems of modules {P,}/, {Qi}i, and {Tj}/ which are indexed by the same directed set / . Then there exists a short exact sequence of their direct limits O^Poo ^ Q o o ^ T o o ^ O .

(1.1.6)

• In particular, the direct limit of factor modules Qi/Pi is the factor module Qoo/Poo- By virtue of Theorem 1.1.3, if all the exact sequences (1.1.5) are split, the exact sequence (1.1.6) is well. Example 1.1.3. Let P be an ^-module. We denote P®k =®P. Let us consider the direct system of ^-modules with respect to monomorphisms A -^(A®P)

— > - - - ( ^ © P © - - - © P 0 f e ) —>••• •

Chapter 1 Commutative Geometry

23

Its direct limit ®P = A® P <$>••• ®P®k®---

(1.1.7)

is an N-graded ,4-algebra with respect to the tensor product . It is called the tensor algebra of a module P. Its quotient with respect to the ideal generated by elements pp'+p'' ®p, p,p' e P, is an N-graded commutative algebra, called the exterior algebra of a module P. • We restrict our consideration of inverse systems of modules to inverse sequences

P° «— P1 <

p* *£1... .

(1.1.8)

Its inductive limit (the inverse limit) is a module P°° together with morphisms 7if° : P°° -> Pi such that 7if° = irj o itf for all i < j . It consists of elements (... ,p*,...), pl € Pl, of the Cartesian product f] Pl such that p1 = K{ (p3) for all i < j . THEOREM 1.1.5. Inductive limits preserve monomorphisms, but not epimorphisms. If a sequence

Q-^Pi

^Q{

-51*T*,

ieN,

of inverse systems of modules {P1}, {Q1} and {T1} is exact, so is the sequence of the inductive limits poo

/f»oo

0_>p°° *-^>Q°° ?—>T°°.

n In contrast with direct limits, the inductive ones exist in the category of groups which are not necessarily commutative. Example 1.1.4. Let {Pi} be a direct sequence of modules. Given another module Q, the modules Hom(Pj,<3) make up an inverse system such that • its inductive limit is isomorphic to Horn (Poo, Q)1.2

Differential operators on modules and rings

This Section addresses the notion of a (linear) differential operator on a module over a commutative ring [202; 233; 26l].

24

Geometric and Algebraic Topological Methods in Quantum Mechanics

Let K be a commutative ring and .A a commutative /C-ring. Let P and Q be Amodules. The /C-module Homjc(P, Q) oi /C-module homomorphisms $ : P —> Q can be endowed with the two different Amodule structures (afc)(p) := a$(p),

($ • o)(p) := $(ap),

o£i,

p£?.

(1.2.1)

For the sake of convenience, we will refer to the second one as the .A*-module structure. Let us put <Sa$ := a$ - $ • a,

(1-2.2)

a £ A.

1.2.1. An element A <£ Hom;c(-P, Q) is called a Q-valued differential operator of order s on P if

DEFINITION

$ao°---°5aA

=°

for any tuple of s + 1 elements ao,..., as of A

•

The set Diff S (P, Q) of these operators inherits the A- and .A*-module structures (1.2.1). Of course, an s-order differential operator is also of (s + l)-order. In particular, zero order differential operators obey the condition <5aA(p) = oA(p) - A(ap) = 0,

a € A,

p € P,

and, consequently, they coincide with ,4-module morphisms P —> Q. A first order differential operator A satisfies the condition 5boSaA(p)

= baA(p)-bA(ap)-aA(bp)

a,b G A (1.2.3)

+ A{abp)=0,

The following fact reduces the study of Q-valued differential operators on an Amodule P to that of Q-valued differential operators on the ring A PROPOSITION

1.2.2. Let us consider the Amodule morphism /i s :Diff,(A<3)->Q,

M A ) = A(1).

(1-2-4)

Any Q-valued s-order differential operator A G Diff s (P, Q) on P uniquely factorizes A : P -^Diff S (AQ) - ^ Q

(1-2-5)

through the morphism hs (1.2.4) and some homomorphism fA:P-Diffs(AQ),

(Up)(a) = A(op),

aeA

(1-2-6)

25

Chapter 1 Commutative Geometry

of the Amodule P to the .A*-module Diff s(.4,<3) [261]. The assignment A H-> fA defines the isomorphism Diffs(P,Q) = EomA_A.(P,Ditt8{A,Q)).

(1.2.7)

n Let P = A. Any zero order Q-valued differential operator A on A is defined by its value A(l). Then there is an isomorphism Diff o(-4, Q) = Q via the association Q9QH4 A,eDiff o (AQ), where A, is given by the equality A,(l) = q. A first order Q-valued differential operator A on A fulfils the condition a, b e A.

A(ab) = bA(a) + aA(b) - 6aA(l),

It is called a Q-valued derivation of A if A(l) = 0, i.e., the Leibniz rule A(ab) = A(a)b + aA(b),

a,b € A,

(1.2.8)

holds. One obtains at once that any first order differential operator on A falls into the sum A(a) = aA(l) + [A(a) - aA(l)] of the zero order differential operator aA(l) and the derivation A(a) — aA(l). If d is a derivation of A, then ad is well for any a € A- Hence, derivations of A constitute an .A-module V(A, Q), called the derivation module. There is the ,4-module decomposition Difi1(A,Q) = Q®V(A,Q).

(1.2.9)

Remark 1.2.1. Let us recall that, given a (non-commutative) /C-algebra A and an .4-bimodule Q, by a Q-valued derivation of A is meant a Kmodule morphism u : A —> Q which obeys the Leibniz rule u(ab) =u(a)b + au(b),

a,b G A.

(1.2.10)

It should be emphasized that this derivation rule differs from that (6.2.3) of graded derivations. A Q-valued derivation u of A is called inner if there O exists an element q £ Q such that u(a) = qa — aq.

26

Geometric and Algebraic Topological Methods in Quantum Mechanics

If Q = A, the derivation module $A of A is also a Lie algebra over the ring K. with respect to the Lie bracket [ti,!i'] = i i o t i ' - u ' o t i ,

u, u'£ A.

(1.2.11)

Accordingly, the decomposition (1.2.9) takes the form (1.2.12)

DiSi(A) = A®dA.

An s-order differential operator on a module P is represented by a zero order differential operator on the module of s-order jets of P as follows. Given an .A-module P, let us consider the tensor product A ®JC P of ^-modules A and P . We put 5b(a®p) := {ba)®p-a®(bp),

p € P,

a,b £ A.

(1.2.13)

Let us denote by /xfc+1 the submodule of A®K. P generated by elements of the type S*0

O-.-OSbk(a®p).

The k-order jet module Jk{P) of a module P is defined as the quotient of the ^-module A <8>JC P by /i fe+1 . We denote its elements c ®k p. In particular, the first order jet module JX{P) consists of elements c®\p modulo the relations Sa o 5b(l ®ip) = ab®xp-b

i (op) - a ®i (bp) + 1 ®i (abp) = 0. (1.2.14)

The /C-module Jk (P) is endowed with the A- and .A*-module structures b(a ®k p) := ba ®k p,

b» (a®kP) •= a®k(bp).

(1.2.15)

There exists the module morphism Jk :P3p^l®kPe

Jk(P)

(1.2.16)

of the .A-module P to the ,4*-module Jk(P) such that Jk{P), seen as an ,4-module, is generated by elements Jkp, p 6 P. Due to the natural monomorphisms /j,r —> fis for all r > s, there are .4-inodule epimorphisms of jet modules 7rj+1 : Ji+1(P)

- • J*(P).

(1-2.17)

In particular, Trl:J1(P)Ba®ip^ap£P.

(1.2.18)

27

Chapter 1 Commutative Geometry

The above mentioned relation between differential operators on modules and jets of modules is stated by the following theorem [261]. THEOREM 1.2.3. Any Q-valued differential operator A of order k on an ,4-module P factorizes uniquely

A : P A j f e ( P ) —+Q through the morphism Jk (1.2.16) and some ,4-module homomorphism fA :

Jk(P) -> Q.

•

The proof is based on the fact that the morphism Jk (1.2.16) is a fc-order i7 (P)-valued differential operator on P. Let us denote fc

J: P3pi->l®p£A®P. Then, for any f e Horn .4 (.4 ® P,Q), we obtain 6btfoJ)(p) = f(6b(l®p)). The correspondence A >—> fA defines an .A-module isomorphism DiBa(P,Q)= Horn A(J'(P),Q). 1.3

(1.2.19)

Connections on modules and rings

We employ the jets of modules in previous Section in order to introduce connections on modules and commutative rings [296]. Let us consider the jet modules Js = JS{A) of the ring A itself. In particular, the first order jet module J1 consists of the elements a <S>\ b, a,b £ A, subject to the relations ab ! 1 - b i a - a <S>i b + 1 ®i (ab) = 0. The A- and ^.'-module structures (1.2.15) on J1 read c(a ! b) := (ca) i 6,

c • (a <E>i b) := a ®i (cb) = (a ® i b)c.

Besides the monomorphism J1 : A^a^->l®la£Jl (\.2.1(^), there is the .4-module monomorphism ii : A 3 a h^> a ®! 1 £ Jx.

(1.3.1)

28

Geometric and Algebraic Topological Methods in Quantum Mechanics

With these monomorphisms, we have the canonical A-module splitting (1.3.2)

J1=ii(A)®O1, oJ x (6) = a <8>i b = ab ®x 1 + a(l ®i b - b i 1),

where the Amodule O1 is generated by the elements 1 ®i & — &(g>i 1 for all 6 € A Let us consider the corresponding Amodule epimorphism /i1:J13l®i6i-+ligi1&-fc®1lGC1

(1.3.3)

and the composition (1.3.4)

d1 = h1oJ1:Aab^l®lb-b®1l€O1,

which is a /C-module morphism. This is a C^-valued derivation of the Jt-ring A which obeys the Leibniz rule d1 (ab) = 1 ®i ab - ab ®i 1 + a ®i b - a <8>! b = a d ^ + ( d 1 ^ . It follows from the relation (1.3.1) that adx6 = (d1b)a for all a, 6 € A Thus, seen as an ,4-module, O1 is generated by the elements dla for all o e A Let O1* = Horn .4 ( 0 \ .4) be the dual of the A-module O1. In view of the splittings (1.2.12) and (1.3.2), the isomorphism (1.2.19) reduces to the duality relation (1.3.5)

DA=OU, QA 9 u <-> 4>u e O1*,

(f>u(d1a) := u(a),

a€A

(1.3.6)

In a more direct way (see Proposition 8.2.1 below), the isomorphism (1.3.5) is derived from the facts that C 1 is generated by elements dla, a E A, and that (dla) is a derivation of A for any £ O1*. However, the morphism oi

_>

oi.*

=

0 ^»

need not be an isomorphism. Let us define the modules Ok, k = 2,..., as the exterior products of the Amodule O1. There are the higher degree generalizations hk

:ji(Ok-1)^Ok,

dk = hk o J 1 : Ok-1 -+ Ok

(1.3.7)

of the morphisms (1.3.3) and (1.3.4). The operators (1.3.7) are nilpotent, i.e., dk o dk~l = 0. They form the cochain complex 0-»/C -^A

^ O

1

^...Qkd<^...

.

(1.3.8)

29

Chapter 1 Commutative Geometry

Let us return to the first order jet module J1(P) of an .A-module P. It is isomorphic to the tensor product J1(P)=J1®P,

(a®1bp)*->{a®1b)®p.

(1.3.9)

Then the isomorphism (1.3.2) leads to the splitting J\P)

= {A®OX)®P

= {A®P)®

{O1 ® P),

(1.3.10)

a (g>i bp <-» (ab + ad}{b)) ®p. Applying the epimorphism ?r^ (1.2.18) to this splitting, one obtains the short exact sequence of A- and A' -modules 0 — > O 1 ® P - » J\P)

^ P —>0,

(1.3.11)

(a ®! b — ab ®i 1) ® p —> (c (2)! 1 + a i b — ab ®i 1) p —» cp. This exact sequence is canonically split by the ^.'-module morphism P3 ap^ \®ap = a®p + dl(a)®pe Jl{P)However, it need not be split by an ,4-module morphism, unless P is a projective .4-module. DEFINITION 1.3.1. A connection on an ^4-module P is defined as an Amodule morphism T:P^J\P),

T(ap) = aT(p),

(1.3.12)

which splits the exact sequence (1.3.11) or, equivalently, the exact sequence 0-*O1<8>P-+(A®O1)®P-*P-^0.

(1.3.13)

• If a splitting T (1.3.12) exists, it reads J1p = T(p) + V(p),

(1.3.14)

where V is the complementary morphism V : P -> O1 ® P,

V(p) = l ® i p - r ( p ) .

(1.3.15)

Though this complementary morphism in fact is a covariant differential on the module P, it is traditionally called a connection on a module. It satisfies the Leibniz rule V(ap) =d1a<8>p + aV(p),

(1.3.16)

30

Geometric and Algebraic Topological Methods in Quantum Mechanics

i.e., V is an (O1 ® P)-valued first order differential operator on P. Thus, we come to the equivalent definition of a connection [260]. DEFINITION 1.3.2. A connection on an ^.-module P is a /C-module mor• phism V (1.3.15) which obeys the Leibniz rule (1.3.16). The morphism V (1.3.15) can be extended naturally to the morphism V : O1 ® P -> O2 <8> P.

Then we have the morphism iJ = V2 :P^O2®P,

(1.3.17)

called the curvature of the connection V on a module P. In view of the isomorphism (1.3.5), any connection in Definition 1.3.2 determines a connection in the following sense. DEFINITION 1.3.3. A connection on an A-module P is an ^-module morphism ViA3ut-^VueDifii(P,P)

(1.3.18)

such that the first order differential operators V u obey the Leibniz rule V u (ap) = u(a)p + aVu(p),

a € A,

p £ P.

(1.3.19)

• 1

Definitions 1.3.2 and 1.3.3 are equivalent if O = DA*. The curvature of the connection (1.3.18) is defined as a zero order differential operator R{u, u') = [V«, Vu.) - V[UiU/]

(1.3.20)

on the module P for all u, u' € DA. Let P be a commutative .A-ring and DP the derivation module of P as a AT-ring. Definition 1.3.3 is modified as follows. DEFINITION 1.3.4. A connection on an ,4-ring P is an .4-module morphism 5 i 9 u H V u £ DP,

(1.3.21)

which is a connection on P as an ^-module, i.e., obeys the Leinbniz rule • (1.3.19).

31

Chapter 1 Commutative Geometry

Two such connections V u and V^ differ from each other in a derivation of the .A-ring P, i.e., which vanishes on A C P. The curvature of the connection (1.3.21) is given by the formula (1.3.20). 1.4

Homology and cohomology of complexes

This Section summarizes the relevant basics on homology and cohomology of complexes of modules over a commutative ring [288; 303]. Let K. be a commutative ring. A sequence 0^B0 £-2?! <^...Bpa£±I...

(1.4.1)

of /C-modules Bp and homomorphisms dp is said to be a chain complex if dp o dp+i = 0,

p G N,

i.e., Im9 p + i C Ker9 p . The homomorphisms dp are called boundary operators. Elements of the module Bp are said to be p-chains, while elements of its submodules Kerdp C Bp and imd p + i c Ker9 p are called p-cycles and p-boundaries, respectively. The p-th homology group of the chain complex B* (1.4.1) is defined as the factor module Hp(B,) =

Keidp/lmdlH.i.

It is a /C-module. In particular, we have HQ{B.) = BQ/Imdi. A chain complex (1.4.1) is exact at a term Bp if HP(B*) = 0. It is an exact sequence if all homology groups are trivial. A chain complex (1.4.1) is called acyclic if its homology groups Hp>o are trivial. A chain complex 5» is acyclic if there exists a homotopy operator h. This is defined as a set of module morphisms hp:Bp^Bp+1,

peN,

such that h p _! odp + dp+l o h p = IdB p ,

p&N+.

It follows that, if dpbp = 0, then bp = dp+i(hpbp), and Hp>0(Bt,) = 0.

32

Geometric and Algebraic Topological Methods in Quantum Mechanics

A chain complex (1.4.1) is said to be a chain resolution of a module B if it is acyclic and H0(B*) = B. This complex defines the exact sequence 0 <—B <— Bo &-Bx &-...Bp9^---.

(1.4.2)

Any module B admits a chain resolution. Indeed, B is a quotient QO/BQ of some free module Qo, where BQ is also a quotient Q\/B\ of a free module Qi, and so on. The following are the standard constructions of new chain complexes from old ones. • Given chain complexes (£», d*) and (B^d*), their direct sum B1t®B'il is a chain complex of modules (B.®B'Jp = Bp®B'p with respect to the boundary operators

df(bp + b'p):=dpbp + dpb'p. • Given a subcomplex (C*,d*) of a chain complex (B*,d*), the factor complex B+/C* is defined as the chain complex of factor modules Bp/Cp provided with the boundary operators dp[bp] := [dpbp],

where \bp] € Bp/Cp denotes the coset of an element bp. • Given chain complexes (B*,9») and (B»,9«), their tensor product -B* <8> B!, is the chain complex of modules ( B , ® B't)p =

© Bk B'r k+r=p

with respect to the boundary operators 9® (Bfc ® B'r) := (dkbk) ® b'r + (-l)kbk

® (d'rbr)-

A chain morphism of chain complexes 7 : £* -> SI

(1.4.3)

is denned as a family of degree-preserving /C-module homomorphisms 7p

: i? p -> s ; ,

peN,

which commute with the boundary operators, i.e., 9

P+I°7P+I

=lp°dp+i.

Chapter 1 Commutative Geometry

33

It follows that if bp £ Bp is a cycle or a boundary, then 7P(6P) G B'p is well. Therefore, the chain morphism of complexes (1.4.3) yields the induced homomorphism of their homology groups

[ 7 ], : H.(B.)

-» H.(B't),

[7]([6]) := [7(6)],

(1-4.4)

where [b] denotes the homology class of 6 € B». Let 7 , 7 ' : £» —> B't be two different chain morphisms of the same chain complexes. By a chain homotopy h is meant a family of /C-module homomorphisms

hp:Bp^

B'p+1,

peN,

of degree +1 such that

d'p+i °hP + hp-i o dp = 7 P - 7p.

If a chain homotopy exists, the chain morphisms 7 and 7' are called homotopic. The difference 7 — 7' of homotopic chain morphisms sends cycles onto boundaries, i.e., these morphisms induce the same homomorphisms [7], and [7']* (1.4.4) of homology groups. In particular, a chain morphism 7 (1.4.3) is said to be a homotopy equivalence if there exists a chain morphism £ : B'% —» B* such that the compositions £ ° 7 and 70£ are homotopic to the identity morphisms of the chain complexes B* and B'^, respectively. Chain complexes connected by a homotopy equivalence are called homotopic. Their homology groups are isomorphic. Let us consider a short sequence of chain complexes

0 -» C, - ^ B . -$-> F . -> 0,

(1.4.5)

34

Geometric and Algebraic Topological Methods in Quantum Mechanics

represented by the commutative diagram 0

0

I

I

dp C p - i <—

• • • <—

7P-1

I d -Dp-1

• • • <

••• <

i'p-l

I

I

B

<

I

CP-I

Cp <

7P

<

Op

I

CP *

0

•• •

£p *

I 0

It is called an exact sequence if all columns of this diagram are exact sequences of modules, i.e., 7 is a chain monomorphism and £ is a chain epimorphism onto the factor complex F* = 5*/C*. One says that the exact sequence (1.4.5) is split if there exists a set s* of degree-preserving monomorphisms sp : Fp —> Bp such that d£ = Cp-i o dp3 o sp,

CpOSp = ldFp,

peN+.

Then Bp^Cp® Fp for all p € N. A splitting s* is called a chain splitting if it is a chain morphism, i.e., s p _! o d% = d^ o sp

or, equivalently, if 5 , is isomorphic to the direct sum of chain complexes C* © Ft. Then we have H,(B.) = H.(C.)®H,(F,). 1.4.1. The short exact sequence of chain complexes (1.4.5) yields the long exact sequence of their homology groups THEOREM

0 <—H0(F*) ^H0(B*)

t ^ f f o ( C . ) ^^(F*)

«—ffp(F.) &^HP{B.)

[

T

&-•••

^HP(C*) ^HP+1(F*)

<

(1.4.6) ,

35

Chapter 1 Commutative Geometry

where [7]* and [£]„ axe the induced homomorphisms (1.4.4), and r» is called the joint homomorphism. O Let us consider a direct sequence of chain complexes Bl —> B\ — • • • B* % Bi+1 — » • • • .

(1.4.7)

Theorem 1.1.4 leads to the following important result [303]. THEOREM 1.4.2. The direct sequence (1.4.7) admits a direct limit 5 J ° which is a chain complex, whose homology groups H*(B^) are the direct limit of the direct system of homology groups H,(B°)

— > f f . ( B j ) — » • • • H*(B*)h^>]

H.(B!t+1)

— ....

D This statement is also true for a direct system of chain complexes indexed by an arbitrary directed set. However, the similar assertion for an inverse system of chain complexes fails because the inductive limit of epimorphisms need not be an epimorphism. Let us turn to cochain complexes. A sequence 0->B°

-^B1

^•••Bp

-£*•••

(1.4.8)

of modules Bp and their homomorphisms 5P is said to be a cochain complex (henceforth, simply, a complex) if p+i o <JP = 0,

p £ N,

i.e., Im

36

Geometric and Algebraic Topological Methods in Quantum Mechanics

Example 1.4.1. Given a chain complex JB» (1.4.1), let Cp = B* be the K.duals of Bp. Let us define the /C-module homomorphisms Sp : Cp —> C p + 1 as 5pcp := (f o 9 p + 1 : Bp+1 -*K,

tf>

e Cp.

(1.4.9)

It is readily observed that Sp+1 o8p = 0. Then {Cp, 5P} is the dual complex of the chain complex £?». Let us note that, if the chain complex 5» is exact, • the dual complex need not be so (see Theorem 1.1.2). A complex (B*, 6*) is called acyclic if its cohomology groups HP>O(B*) are trivial. It is acyclic if there exists a homotopy operator h, defined as a set of module morphisms h P+i

. BP+I

^BPt

p e

N(

such that hp+1o6p

+ Sp-1ohp

= IdBp,

peN+.

Indeed, if 5pbp = 0, then bp = 5p-l{\ipbp), and HP>O(B*) = 0. A complex (B*, 6*) is said to be a resolution of a module B if it is acyclic and H°{B*) = B. The following are the standard constructions of new complexes from old ones. • Given complexes (B^, 5*) and (B^S^), their direct sum B± © J3| is a complex of modules (B*x ® B*)p = Bp ® BP with respect to the coboundary operators

6l(bp + bp2):=5pbp + 5PbP. • Given a subcomplex (C*, 5*) of a complex (B*,S*), the factor complex B*/C* is defined as a complex of factor modules Bp/Cp provided with the coboundary operators 5P[bP] : = [<W], where [bP] G Bp/Cp denotes the coset of the element bp. • Given complexes (5J, <5J) and (B^S^), their tensor product B\ B% is a complex of modules

(Bl ® B^)p =

© £f ® 5£

fc+r=p

Chapter 1 Commutative Geometry

37

with respect to the coboundary operators

a|(fl? ® Br2) := (tf bf) ® &£ + (-l)fc&? ® (*$&$). A cochain morphism of complexes 7 : 5j* -> B 2

(1.4.10)

is defined as a family of degree-preserving homomorphisms Y : B\ -> B p ,

peN,

which commute with the coboundary operators, i.e., ^O7"

=y+1o^,

peN.

It follows that if V e £?p is a cocycle or a coboundary, then 7p(bp) £ B p is so. Therefore, the cochain morphism of complexes (1.4.10) yields an induced homomorphism of their cohomology groups [7]* : E*{B\) - H*{B*2).

(1.4.11)

Let 7,7' : B\ —> B2 be two different cochain morphisms. By their cochain homotopy h is meant a family of homomorphisms

hP:Bl^Bl~\

peN+,

such that $P-I ohP + hp+16p = 7 P - ip. If a cochain homotopy exists, the cochain morphisms 7 and 7' are called homotopic. Homotopic cochain morphisms 7 and 7' induce the same homomorphisms [7]* and [7']* (1.4.11) of cohomology groups. One says that the cochain morphism 7 (1.4.10) is a homotopy equivalence if there exists a cochain morphism £ : B2 —* B\ such that the compositions C ° 7 and 7 o £ axe homotopic to the identity morphisms of the complexes B* and B2, respectively. Complexes connected by a homotopy equivalence are called homotopic, and their cohomology groups are isomorphic. Let us consider a short sequence of complexes 0 -^ C* ^UB* -^-> F* -> 0,

(1.4.12)

38

Geometric and Algebraic Topological Methods in Quantum Mechanics

represented by the commutative diagram 0

0

.._ U J._... Tp

...

y

I

BP

7p+l

BP+1

- ^

CP I ...

»

I

Cp+i

pp i f U

> •• •

I >...

pp+i

I

I

0

0

It is said to be exact if all columns of this diagram are exact, i.e., 7 is a cochain monomorphism and £ is a cochain epimorphism onto the quotient F* = B*/C*. The following assertions are similar to Theorems 1.4.1 and 1.4.2. 1.4.3. The short exact sequence of complexes (1.4.12) yields the long exact sequence of their cohomology groups THEOREM

0->H°(C*)

l

^H°(B*)

[

^H°(F*)

—>HP(C*) ^ > F P ( B * )

^H\C*)

1

^>HP(F*)

—>•••

^HV+1{C*)

(1.4.13) —••••.

• THEOREM

1.4.4. A direct sequence of complexes B*

_ » B J —»• • • J3J 7 ^B* k + 1 -*..•

(1.4.14)

admits a direct limit B^ which is a complex whose cohomology ,ff*(.B£o) is a direct limit of the direct sequence of cohomology groups H*(B*0)

-*H*{Bl)

-^...H*(B*k)b^>]H*(B*k+1)

—....

This statement is also true for a direct system of complexes indexed by an arbitrary directed set. •

39

Chapter 1 Commutative Geometry

1.5

Homology and cohomology of groups and algebras

Subsections: A. Homology and cohomology of groups, 39; B. The Koszul complex, 44; C. Hochschild cohomology, 49; D. Chevalley-Eilenberg cohomology, 53. We briefly sketch homology and cohomology of some algebraic systems needed in the sequel. These are homology and cohomology of groups, homology of the Koszul complex, Hochschild cohomology, ChevalleyEilenberg cohomology.

A. Homology and cohomology of groups Homology and cohomology of groups demonstrate the standard techniques of constructing homology and cohomology of algebraic systems [288]. Given a set Z, one can introduce a chain complex as follows. Let Zk be a A;+l

free Z-module whose basis is the Cartesian product x Z. In particular, Zo is a free Z-module whose basis is Z. Let us define Z-linear homomorphisms d0 : Zo 3mi(zl0) i-> Ylmi

eZ

'

m

»£z.

(1.5.1)

i

3k+1

: Zk+1 -+ Z*,

ke N, fe+1

dk+i(z0, • • •, z/M-i) = ^2(~'i-y(zo,---,Zj,...,zk+i), j=o

(1.5.2)

where the caret "denotes omission. It is readily observed that dkodk+\ = 0 for all k € N. Thus, we obtain the chain complex 0 <— L <— Zo <— Z\ <— • • • Zjt <

•,

(1.5.3)

called the standard chain complex of a set Z. The chain complex (1.5.3) admits the homotopy operator h z ( l ) = (z),

hz(z0,...,zk)

= (z,z0,...,zk),

(1.5.4)

where z is some fixed element of Z. Consequently, the chain complex (1.5.3) is exact at all the terms Zk, k E N. Moreover, this complex is also exact at the term Z since the boundary operator do (1.5.1) is an epimorphism. Hence, it provides a resolution of a ring Z by free Z-modules. Let Z = G be a group and G* the chain complex O^z<^-Go ^ - d <

Gp$±i---

(1.5.5)

40

Geometric and Algebraic Topological Methods in Quantum Mechanics

(1.5.3). The Z-module Go becomes a ring with respect to the multiplication (mrgr)(nkgk) = mrnk{grgk),

gr,gk&G.

mr,nk£i,

It is called the group ring ZG. Accordingly, the set Gk, k > 0, is brought into a free left ZG-module by letting g{go,---,gk)

= (ggo,---,ggk),

g&G.

(1.5.6)

Its basis consists of (k + l)-tuples (1, gi,..., gk). For the sake of simplicity, left ZG-modules are usually called G-modules. It is readily observed that the boundary operators do (1.5.1) and dk>o (1.5.2) are G-module morphisms, where Z is treated as a trivial G-module. Therefore, the chain complex (1.5.5) is a resolution of Z = HQ(G*) by free G-modules. At the same time, the homotopy operator (1.5.4) fails to be a G-module morphism. Let us consider an isomorphic chain complex G* of G-modules which exhibits useful features as follows. Its terms Gk>o arefi"eeG-modules having the fc-tuples [gi,. • • ,gk] as their bases, while Go = ZG consists of elements mrgr[). The boundary operators of the chain complex G* are given by the formula dk>o[gi,---,gk]

=9i\92,---,9k\

+

(1.5.7)

1

X ^ - ) ^ ! ' • • ^9j9j+u • • • ,5fc] + (-l)*bi> • • -,9k-i], 3

while do(g[ ]) = 1 is a G-module morphism from ZG to Z, regarded as a trivial G-module. It is readily observed that the chain complexes G* and G» are isomorphic via the association [ 9 i , - - - , 9 k ] >-> ( 1 . 5 1 , 5 1 0 2 , • • • , 3 i * • - 9 k ) -

In particular, the homotopy operator for the chain complex G* reads h(l) = [ ] ,

h ( g [ ] ) = [g},

h(g\g1,...,gk]) = \g,glt...,gk].

Let A be a left G-module. Given the chain complex G*, let us consider the cochain complex whose terms Gk=BomG(Gk,A) are left G-modules of G-module morphisms fk : Gk —> A. These morphisms can also b e seen as A-valued functions fk(gi,---,gk) of A; arguments from

41

Chapter 1 Commutative Geometry

the group G. Similarly to the formula (1.4.9), the coboundary operators of the complex Q* are denned as (Skfk)(9u

• • .,9k+i) = fk(dk+i[9i,

k

• • • ,
9if {92, • • • ,9k+i) + '%2(-l) f (9i,-

(1-5.8)

• -,9j9j+i,- • -,9k+i)

j

+(~l)k+1fk(9i,...,9k),

k

e

K

In particular, the module G° is isomorphic to A via the association ^ f l n j j e G°,

f°a([ ]) = a.

For instance, we have S°foa(9) = ga-a, l 1

(1.5.9)

geG,

1

1

S f (9i,92)=9if (92)-f (9i92)

1

+ f (9i),

2 2

$ f (9i,92,93) = 9if{92,93) - f{9192,93) + 2

f {9i,929z)-

(1-5.10) (1.5.11)

f'(91,92)•

Cohomology H*(G;A) of the complex Q* is called cohomology of the group G with coefficients in the G-module A. In particular, the expression (1.5.9) shows that H°{G,A) = Kei5° is isomorphic to the additive subgroup of G-invariant elements of A, treated as a trivial G-module. Example 1.5.1. Let G be a finite group of order m. There is a G-module morphism h:Qk->Qk-\

k£N+,

given by the formula (hfk)(9i, • • • ,9k-i) = (-l)fc £/ f e (3i, • • -,9k-i,9).

(1.5.12)

geG

A simple calculation shows that ho6k + 5k-1oh = mLdgk.

(1.5.13)

It follows that, if fk is a cocycle, then mfk is a coboundary. Consequently, the cohomology groups Hk>0(G, A) consist of cyclic elements of order m.

42

Geometric and Algebraic Topological Methods in Quantum Mechanics

In particular, let us suppose that, for any element a of the module A, the element m~1a is defined. Then m" 1 /! is a homotopy operator, and Hk>0(G,A) = 0. For instance, Hk>o{G,Z)=0, where Z is treated as a trivial G-module. We refer the reader to [288] for cohomology of cyclic and • free groups. By cohomology of a group G with coefficients in a G-module A is also meant the cohomology HQ(G, A) of the subcomplex QQ of the complex Q* whose fc-cochains are .A-valued functions of k arguments from G which vanish whenever one of the arguments is equal to 1. It is easily verified

that skgk c gk+1.

Let us show that the cohomology group HQ(G,A) classifies the extensions of the group G by an additive group A. Such an extension is defined as a sequence (1.5.14)

O-*A^W-^G->1

of group homomorphisms, where i is a monomorphism onto a normal subgroup of W and TT is an epimorphism onto the factor group G = W/A, i.e., Imi = 7r~ 1 (l). By analogy with sequences of additive groups, the sequence (1.5.14) is said to be exact. For the sake of simplicity, we will identify A with its image i(A) c W, and write the group operation in W in the additive form. Any extension (1.5.14) yields a homomorphism of G to the group of automorphisms of A as follows. Let w(g) be representatives in W of elements g £ G. Then any element w G W is uniquely written in the form w = aw + w(g), aw € A. Let us consider the automorphism <j)g : a H-> ga = w(g) + a — w(g),

a G A.

(1.5.15)

Certainly, this automorphism depends only on an element g G G, but not on its representative in W. The association g >—> g defines a desired homomorphism 0:G->AutA

(1.5.16)

This homomorphism <j> makes A to a G-module denoted by A^. Conversely, the homomorphism <j> (1.5.16) corresponds to some extension (1.5.14) of the group G. Among these extensions, there is the semidirect product W = Ax^G with the group operation (a, g) + (a', g') = (a + ga', gg'),

ga' = g{a').

43

Chapter 1 Commutative Geometry

THEOREM 1.5.1. There is one-to-one correspondence between the classes of isomorphic extensions (1.5.14) of a group G by an Abelian group A associated to the same homomorphism (1.5.16) and the elements of the • cohomology group H^(G,A^) [288].

Outline of proof. Given representatives w(gi) and w(g2) of elements <72 € G, their product is obviously a representative of the element gig2 £ G, i.e., (1.5.17)

w(gi)+w(g2) =w(gi92) + f (91,92),

where f(gi,g2) £ A is called the extension factor. Since w(l) = 0, we have / ( s i . i ) = /(i,ftj) = O,

si.saeG.

Therefore, the extension factor f(gi,g2) can be seen as a two-cochain of the complex QQ. Let us rewrite the expression (1.5.15) for ga as the commutation condition (1.5.18)

w(g) + a = ga + w(g).

By virtue of the formulae (1.5.17) and (1.5.18), the product in W of elements ai +w(gi) and a2 + w(g2) reads {ax + iw(ffi)) + (o2 + w(g2)) = ai + gia2 + f(gi,g2) + w(gig2). Using this formula, let us write the products [w(gi) + w(g2)] + w(g3) = f(gx, g2) + f(gi92,9z)

+w(gig2g3),

w(gi) + [w(g2) + w(g3)} = gif(g2,93) + f (91,9293) +w(gi9293)Then the associativity condition leads to the relation 9if(92,93) + f(9i, 5253) = f(9i,92) + f (9i92,93)A comparison of this relation with the expression (1.5.11) shows that the extension factor f(gi,g2) is a two-cocycle of the complex QQ. Of course, / depends on the choice of representatives w(g). It is easily verified that the extension factor / ' corresponding to representatives w'(g) = w(g) + u(g) differs from / in the two-coboundary 51u (1.5.10). This states the result. QED

44

Geometric and Algebraic Topological Methods in Quantum Mechanics

In particular, the semidirect product A x^ G corresponds to 0 of the cohomology group Hl(G,A$).

B. The Koszul complex The Koszul complex is a finite chain complex of the alternating product of a free module [272]. Let A be a commutative ring. For a fixed tuple {a\,..., ar) of elements of .4, the Koszul complex K* = K* {a\,..., ar) of rank r is denned as follows. We put: • Ko = A; • K\ is a free .4-module of rank r whose basis is denoted by { e i , . . . , e r }; P

• Kp = A K\, p = 2 , . . . r, is a free ,4-module with the basis {e^ A • • • A ejp},

h

< •••

In particular, Kr is a free .4-module of rank 1 with the basis element ei A . . . A e r . The boundary operators of the Koszul complex are (1.5.19)

d1(ei)=ai, dp>i(e<1 A - - - A e i p ) = 1

(1.5.20) A • • • A ei3. A • • • A e ip .

^2(-iy- aijeil o

A direct verification shows that di o di+i = 0, i — 1,... , r — 1. Thus, we have a finite chain complex 0^«4^*ri<

£-tfp-" ?^Kr

*—0.

(1.5.21)

Let H*(K*) denote its cohomology. Prom the expression (1.5.19), we obtain at once that H0(K*) =

A/lTnd1=A/I,

where 1 C A is the ideal generated by the elements a\,...,ar. The Koszul complex is independent of the choice of the tuple to the following extent.

(ai,...,ar)

1.5.2. Let (a[,..., a'r) be a different tuple of elements of the ring A such that PROPOSITION

45

Chapter 1 Commutative Geometry

and det(c].) = 1. Then the Koszul complexes K*(a,i,... ,aT) and K*(a[,... ,a'r) are isomorphic due to the chain morphism e'k = c*kej. • In particular, the Koszul complexes if*(ai,..., ar) and K*(a[,...,a'r) are isomorphic if (a\,...,ar) is a permutation of (a[,...,a'r). The forthcoming theorem enables one to use the induction procedure. THEOREM

1.5.3. There is a natural chain isomorphism K*(ai,...,

ar) = K*{a{)

®---®

K*(ar),

where K*(a,i) is the chain complex 0 ^— A ^-K^cii)

(1.5.22)

<—0.

• More generally, let Q be an ,4-module. The Koszul complex of Q is defined as K*[Q) = K.(au ..., ar; Q) := Q ® Kt, together with the boundary operators 9»(9®e)=g(g>a,(e),

q£Q,

e G K*,

where d*(e) is given by the expressions (1.5.19) - (1.5.20). In particular, we have K0{Q) = Q®A = Q, and the boundary operator d\ is di {q ® ej) = qa,i = cnq,

q€Q-

Thus, the Koszul complex K* (Q) of Q reads 0 <— Q <— Q <8) ATi • • • <— Q®KP--- <— Q®KT *— 0.

(1.5.23)

We abbreviate with H*(ai,... ar; Q) the homology of this complex. The first and last homology groups can be obtained directly from the definition of boundary operators. They are H0(ai,...ar;Q)=Q/QI, Hr(a1,...ar;Q)=Kerdr

= {qeQ

: q! = 0}.

46

Geometric and Algebraic Topological Methods in Quantum Mechanics

By virtue of Theorem 1.5.3, the Koszul complexes of an ,4-module Q obey the iteration formula Kt(alt

...,ar;Q)

, a r _ i ; Q) K*(ar).

= K*{cn,...

(1.5.24)

Therefore, let us study the tensor product C* <S> K* (a) of an arbitrary chain complex (C*,9f) of ,4-modules with the Koszul complex K*(a) (1.5.22). The chain complex C* ® K* (a) consists of the chain modules (C, ® K,(a))p

= Cp © (Cp_i ® tfi(a)),

together with the boundary operators

9?[Cp®(Cp-1®K1(a))] = d%Cp © {^L^p-i ® K^a) +

(-lf-'Cp-^K^a)}.

A glance at this expression shows that there is the chain monomorphism C* —> C* ® K* (a) so that we have the short exact sequence of complexes 0 _» c , —»C, ® X*(a) —»(C, ® K.(a))/C. -» 0,

(1.5.25)

represented by the commutative diagram 0

0

• • • <—

I

op_i

1

• • • •— C p _i 0 (Cp_2 ® ifi(o))

•• • •—

I I

Cp-2 ® ^ i ( a )

0

<—

88 ^ "^-

I

op

I

<— • • •

Cp® {Cp-i ® /iTi(a)) <

I I

<

0

(1.5.26) Obviously, the exact sequence (1.5.25) admits a splitting, but this is not a chain morphism. A glance at the diagram (1.5.26) shows that the factor complex (C* ®

47

Chapter 1 Commutative Geometry

K{a))/C* is precisely C* <8> -Jsfi(a) with a one unit degree shift, i.e., ((C. ® K.{a))/C.)p>0

= C p -i ® ^ i ( a ) ,

((C.<8>lf.(a))/C.)o = O, where the boundary operators dp_i ® Id do not act on the module Ki(a). Hence, there are isomorphisms of homology groups HP>O{(C. ®K*{a))/C*) = H p _ i ( C ) . With these isomorphisms, the exact sequence (1.4.6) of homology groups induced by the short exact sequence of chain complexes (1.5.25) takes the form 0 ±—H0(C* ® K*(a)) *— H0(C.) 2-Ho(C.) ffi(C. ® /f.(o)) *

<—

(1.5.27)

HP(C, ® ^.(o)) <— £rp(C.) ^±i flp(C.)

<—H p + 1 (C,®ii:.(a)) <

.

1.5.4. The joint homomorphism r p + i in the exact sequence D (1.5.27) is a multiplication by (—l)pa in the .A-module HP(C*).

PROPOSITION

Indeed, one can write (C* <8> K*(a))p = CP® (Cp_i ® Xi(o)) =* Cp © C p _i, where ^(Cp.cp-O = (9pccp + ( - l r - ^ C p - i . a p l i C p - ! ) .

(1.5.28)

Then the proof of Theorem 1.4.1 shows that the elements (—l)p~1acp_i modulo boundaries constitute the range of the joint homomorphism r p . Furthermore, let (c p ,c p _i) be a cycle. Then we obtain from (1.5.28) that (ac p ,ac p _0 = c ^ ( O , ( - l ) % ) is a boundary. This fact states the following. PROPOSITION 1.5.5. A multiplication by a annihilates the homology groups of the chain complex C* 0 K(a). • Let us return to the iteration formula (1.5.24) and put C* = K(ai,..., a r _i; Q). Then the exact sequence of homology groups (1.5.27)

48

Geometric and Algebraic Topological Methods in Quantum Mechanics

takes the form 0<-Ho(ai,...ar;Q) Hi(ai,...ar;Q) T

<—#o(ai, • • • ar-i; Q) 2-Ho(a±,.. .o r _i;Q) <— < Hp(ai,...ar;Q) <—H p (ai,...a r -i\Q)

^-Hp(a1,...ar^1;Q)

<—#p+i(ai,...

a r ; Q) <

,

where the joint homomorphisms r p are multiplications by (—l)p~1ar. Proposition 1.5.5 states that the element ar € A annihilates all homology groups Hp(ai,. ..ar;Q) of the Koszul complex K*(ai,...,ar;Q) (1.5.23). Moreover, since the Koszul complexes for different permutations of the tuple (oi,... ,ar) are isomorphic, any element of the ideal X generated by a\,...,ar annihilates homology groups Hp(a\,... ar; Q). In particular, if X = A, the Koszul complex Kt(ai,..., aT; Q) is exact. This result can be improved as follows. One says that a tuple of elements (oi,...,flr)of A is regular in an ,4-module Q if: (i)Q/(ai,...,or)Q^0, (ii) a\ is not a divisor of 0 in Q (i.e., aiq — 0 implies q = 0), (iii) for k = 2,... r, an element a^ is not a divisor of 0 in Q/(ai,...,o fe _i)Q. THEOREM 1.5.6. [272]. If a\,..., a r is a regular tuple in Q, the homology groups Hp>o(ai,.. .ar;Q) of the Koszul complex K*(ai,...,ar;Q) are trivial, i.e., the Koszul complex (1.5.23) is acyclic. The corresponding exact sequence is

0 ^Q/XQ

^Q

•_ J R- 1 (Q)... <—KP{Q)---

^Kr(Q)

This is a finite free resolution of the module Q/XQ.

<—0. (1.5.29) •

In particular, let Q = A. Since K*{a\,... ,ar;A) is isomorphic to K*(a,i, •. •, ar)t Theorem 1.5.6 leads to the following important result. 1.5.7. Let ai,... ,ar be a regular tuple in a commutative ring A- Then the Koszul complex /ST»(ai,... ,a r ) (1.5.21) provides the free resolution THEOREM

0 «— A/I <— A*—Ki--of the quotient A/I.

*—Kp--' ±—Kr <— 0.

(1.5.30) •

49

Chapter 1 Commutative Geometry

C. Hochschild cohomology Let K. be a commutative ring and A a AC-ring which need not be commutative. Hochschild cohomology is the cohomology of a /C-ring A with coefficients in a .4-bimodule. Throughout this Section, by <8> is meant the tensor product of modules over /C. Let Bk(A), k 6 N + , be an .4-bimodule whose basis is the tensor product of AC-modules ®A, i.e., it consists of elements [«i ® • • • <8> afc], a* € A. The Bo (A) is denned as an .4-bimodule of rank 1 whose basis element is denoted by [ ]. One considers the chain complex B*(A) with respect to boundary operators defined as the .4-bimodule morphisms dk>o[ai <8> • • • ® ak] = ai[a2 ® • • • ® ak] +

(1.5.31)

fc-i

^2(-l)j[ai

• • • ® ajdj+i <8> • • • (8> ofc] + (-l) f c [ai <8> • • • ® afe^i]afe-

In particular, we have 9 0 : B 0 (X) 3 a{ }a' i-> ao' e A 9i(a[ai]a') = aai[ }a' — a[ ]a\a!. This chain complex admits the homotopy operator given by the right ^-module morphisms h : a[ai ® • • • (g> a*] H-> [a ai ® • • • ® at],

fc

€ N.

(1.5.32)

For instance, we have h o (a[ ]a') = [a]a'. It follows that the chain complex B* (^) is acyclic. The corresponding exact sequence reads 0 •— A £ - J50(>l) <

Bfc(^) 84±1 • •. .

(1.5.33)

It is readily observed that the homotopy operator h (1.5.32), completed by the map

is the homotopy operator for the chain complex (1.5.33). Therefore, this chain complex is exact at all terms Bk(A), k S N. Moreover, it is exact

50

Geometric and Algebraic Topological Methods in Quantum Mechanics

at A since A has a unit element and, therefore, do is a /C-module epimorphism. Hence, the chain complex (1.5.33) is a resolution of the ring A by .4-bimodules. Remark 1.5.2. Given the canonical monomorphism of K, —> A, one also constructs the chain complex of ,4-bimodules C0(A) = B0(A),

Ck>0(A) = ®{A/K)

(1.5.34)

whose boundary operators take the form (1.5.31). The chain complexes JB» (A) and C* (.4) are proved to be homotopic [288]. • Remark 1.5.3. Given a (non-commutative) /C-ring A, let us consider the chain complex 5* (.4, .4.) (called the Hochschild chain complex) whose k+l

members Bk(A, A), k £ N, are the tensor products ® A of ^-modules .4. They are /C-modules generated by the elements ao ® • • • a^, aj e A. The boundary operators of this chain complex reads dk>o(ao ® • • • ® ak) = fc-i ^ ( - l ) J ' o o ® • • • ® ajCij+i ® • • • ® a.k + j=o (-l)haka0

(1.5.35)

<8> • • • (8»afc_i.

For instance, BQ(A,A) = A, and we put do = 0. The homology H+{A,A) of the chain complex B* (A, A) are called the Hochschild homology of the /C-ring A. • Let us turn now to Hochschild cohomology. Let Q be an .4-bimodule. Given the chain complex Bi,(A), let us consider the cochain complex 0 -» B°(A, Q) f^B\A,Q)-^---

Bk(A, Q) ^

•••,

(1.5.36)

whose terms are the ,4-bimodules Bk(A,Q) = RomK(Bk(A),Q). It is called the Hochschild complex. Their elements can be seen as Q-valued ,ak) on A. The coboundary operators /C-multilinear functions fk{a\,..,

51

Chapter 1 Commutative Geometry

read (Skfk)(ai,..

.,ak+1)

= fk(dk+i[ai

® • • • ® ak+1)) =

ai/fc(fl2, • • •, ajt+i) + ^ T ( - l ) J / f c ( a i . • • .,a,jaj+i,.. i +(-l)fc+1/fc(ai,---,Ofc)ofc+i,

(1.5.37) .,ak+i)

fceN,

(see Example 1.4.1). In particular, the module B°(A,Q) is isomorphic to Q via the association /°([])=«.

Q3q~f°£B°(A,Q), For instance, we have 5°f°{a)=aq-qa,

(1.5.38)

a £ A,

1

1

S W . a a ) = a i / (a 2 ) - / W 2 ) + / (ai)o2, 2 2

2

2

« / (oi, oa, as) = ai/ (a 2 , a3) - / (a 1 a 2 , o3) + oi,a 2 ) a 3 € A / 2 (oi,a 2 a 3 ) - / 2 ( o i , a 2 ) a 3 )

(1-5.39) (1.5.40)

Cohomology ff*(^,Q) of the complex B*(A,Q) (1.5.36) is called the Hochschild cohomology of a /C-ring .4 with coefficients in an ,4-module Q. The Hochschild cohomology coincides with cohomology of the normalized subcomplex C*(A, Q) of the Hochschild complex (1.5.36) whose terms are the .4-bimodules

Ck(A,Q)

=HomA(Ck(A),Q),

where C*(A) is the chain complex (1.5.34). Its elements are represented by Q-valued /C-multilinear functions fk(ai,..., ak) on A which vanish whenever one of its arguments belongs to K. C A. The Hochschild complex B*(A, Q) (1.5.36) is functorial in Q, i.e., every •4-bimodule morphism Q —> P yields a cochain /C-module morphism

B*(A,Q)^B*(A,P) and, consequently, a homomorphism of Hochschild cohomology groups

H*(A,Q)^H*(A,P) [167]. In particular, an A-bimodule morphism $ : Q —» P is called K.allowable if there exists a ^-module morphism A : P —> Q such that $ o A o

52

Geometric and Algebraic Topological Methods in Quantum Mechanics

$ = $ . This condition is always satisfied if K. is a field. Each short exact sequence 0 -» Q' —»Q —> Q" -» 0 of /C-allowable morphisms yields the long exact sequence • • . # * ( . * , Q') —>ff*(A H k + \ A , Q ' )

—••••.

The Hochschild cohomology modules of low degrees have the following natural interpretation. (i) Since H°(A, Q) = Ker 6° = Q, it follows from the expression (1.5.38) that H° (A, Q) is isomorphic to the center ZQ of the .A-bimodule Q. (ii) A glance at the expression (1.5.39) shows that any normalized onecocycle Z 1 is a Q-valued derivation of the /C-ring A and vice versa. In particular, one-coboundaries 5°/® are inner Q-valued derivations up to the sign minus. Therefore, H1(A,Q) is sometimes called the module of outer derivations of A. (iii) Similarly to the case of cohomology of groups, the cohomology module H2(A,Q) classifies the non-equivalent extensions of a /C-ring A by .4-bimodules Q given by the exact sequences

0-><2^#—>.A->0, where B is a A^-ring [l 67]. An important instance of the foregoing occurs when Q is A itself seen as an ,4-bimodule. In this case, the Hochschild complex B*(A, A) is provided with the following two products [167]. (i) The first one is the cup-product f

w

/'(oii • • •, am+n) = / ( o i , • • •. am)f'(am+i,...,

am+n),

(1.5.41)

which obeys the equality

Sm+n(f ^ /') = Smf - / ' + (-1)"7 - 5nf. Hence, it induces the cup-product [/] - [/'] := [/ - /']

(1-5-42)

of the Hochschild cohomology classes [/],[/'] € H*(A, A). This product makes H*(A, A) into a graded commutative algebra, i.e., the relation

lf}-[f') = (-l)im'llf'}-lf]

Chapter 1 Commutative Geometry

53

holds, where | / | denotes the degree of cohomology classes. (ii) The composition product of / G Bm(A,A) and / ' € Bn(A,A) is defined by the formula /o/'(oll...,am+n_1)=

(1.5.43)

m

^ ( _ l ) ( * - i ) ( » - i ) y ( a i ) . . .^ai-i,f{au

.. .,an+i-i),an+i,...

,a n + m _i).

It obeys the relation

s(f ° n = (-i)"-1*/ of'+fosf+(~i)"(/' - / - (-ir n / - /'). It is easily justified that, if / and / ' are cocycles, the bracket .[/, fT = f ° / ' - (-I)*" 1 - 1 '*"- 1 )/' o /

(1.5.44)

is also a cocycle. Consequently, the bracket (1.5.44) induces the bracket of the Hochschild cohomology classes

[[/M/']]° :=[[/, m

(1-5-45)

called the Gerstenhaber bracket. This bracket makes H* (A, A) into a graded Lie algebra where elements [/] € Hm(A, A) are provided with the graded degree m — 1. The cup-product (1.5.42) and the Gerstenhaber bracket (1.5.45) in the Hochschild cohomology H*(A, A) satisfy the relations making H* (A, A) into a Gerstenhaber algebra. In Section 8.9, we consider cyclic cohomology which is cohomology of a certain subcomplex of the complex (1.5.36) when Q = A* is the dual of the /C-module A. D. Chevalley—Eilenberg cohomology One associates the Chevalley-Eilenberg complex to a Lie algebra. In Section 1.6, the Chevalley-Eilenberg complex of the Lie algebra of derivations of a commutative ring provides the differential calculus over this ring. In this Section, JJ denotes a Lie algebra (not necessarily finitedimensional) over a commutative ring K.. Let 0 act on a /C-module P on the left by endomorphisms gxPB {e,p)^ep£P, [e, s']p = (e o e' - e' o e)p,

e.e'Gfl.

54

Geometric and Algebraic Topological Methods in Quantum Mechanics

One says that P is a g-module. A ^-multilinear skew-symmetric map ck : x g -> P is called a P-valued fc-cochain on the Lie algebra g. These cochains form a g-module Ck\g;P}. Let us put C°[g;P] = P. We obtain the cochain complex 0 - > P -^CMfljP] ^•••Ck[g;P]

il*...,

(1.5.46)

with respect to the Chevalley-Eilenberg coboundary operators k k k

^ ( - 1 ) ^ ^ ( 6 0 , . . . , e i , . ..,ek)

6 c (e0,...,ek)=

+

(1.5.47)

t=0

{~l)i+jck{{£i,ej\,e0,...,eu...,ej,...,ek),

^ i
where the caret~denotes omission [160]. The complex (1.5.46) is called the Chevalley-Eilenberg complex with coefficients in a module P. It is finite if the Lie algebra g is finite-dimensional. For instance, (1.5.48)

5°p(eo)=soP, tfVfo.ei)

1

=e O c (£i) - e i c ^ o ) -^([eo.ei])-

(1-5-49)

Cohomology H*(g;P) of the complex C*[g;P] is called the ChevalleyEilenberg cohomology of the Lie algebra g with coefficients in the module P. The following are two standard variants of the Chevalley-Eilenberg complex. (i) Let P = g be regarded as a g-module with respect to the adjoint representation s:e'h->

[e, e'}

, e, e' G 0.

We abbreviate with C*[g] the Chevalley-Eilenberg complex of g-valued cochains on g. Cohomology H*(g) of this complex is called the ChevalleyEilenberg cohomology or, simply, the cohomology of a Lie algebra g. In particular, C°[g] = g, while C 1 ^] consists of endomorphisms of the Lie algebra g. Accordingly, the coboundary operators (1.5.48) and (1.5.49)

55

Chapter 1 Commutative Geometry

read (1.5.50)

6°e(eo) = {eo,e}, 1

= [eo.c^ei)] - {euc\eQ)\

5 c\e0,e1)

- ^([eo.ei]).

(1.5.51)

1

A glance at the expression (1.5.51) shows that a one-cocycle c on g obeys the relation c1([eo,ei]) = [c1(eo),ei] + [eo,c1(ei)] and, thus, it is a derivation of the Lie algebra g. Accordingly, any onecoboundary (1.5.50) is an inner derivation of g up to the sign minus. Therefore, one can think of the cohomology H1 (g) as being the set of outer derivations of g. One can show that the cohomology H2(g) indexes non-equivalent infinitesimal deformations of the Lie algebra g [160]. (ii) Let P — K and g : K. —* 0. Then the Chevalley-Eilenberg complex C*[g;/C] is the exterior algebra Ag* of the dual Lie algebra g*. The Chevalley-Eilenberg coboundary operators (1.5.47) on this algebra read k

^ ( - l ) i + : ' ^ ( [ e ^ e ^ e o , •.. ,ei,. •. , £ j , . . • ,ek).

k k

5 c (e0,...,ek)=

(1.5.52) In particular, <50c°(£0) = 0, 1 1

c° € K, 1

i c (e o ,ei) = -c ([eo,ei]), 2 2

6 c (e0,ei,e2)

c1 € g*,

2

2

(1.5.53) 2

= -c ([e 0 ,ei],e2) + c ([e0,£2],ei) - c ([£i,e2],£o)-

Cohomology H*(g; K.) of the complex C*[g; K] are called the ChevalleyEilenberg cohomology with coefficients in the trivial representation. It is provided with the cup-product H-[c'] = [cA4

(1.5.54)

where [c] denotes the cohomology class of a cocycle c. This product makes H*(g;fC) into a graded commutative algebra. For instance, let g be the right Lie algebra of a finite-dimensional real Lie group G. Then the relation (1.5.53) is the well-known Maurer-Cartan equation. Written with respect to the basis Si for g and the dual basis 9l for g*, this equation reads Sek = -^j0iA9j.

(1.5.55)

56

Geometric and Algebraic Topological Methods in Quantum Mechanics

There is a monomorphism of the complex C*[g; R] onto the subcomplex of right-invariant exterior forms of the de Rham complex (1.8.9) of exterior forms on G. This monomorphism induces an isomorphism of the ChevalleyEilenberg cohomology /P(g; K) to the de Rham cohomology of G [160]. For instance, if G is semisimple, then H1(S;R) = H°(g;R) = 0.

1.6

Differential calculus over a commutative ring

In a general setting, the de Rham complex is defined as a cochain complex which is also a differential graded algebra. By a gradation throughout this Section is meant the N-gradation. A graded algebrafi*over a commutative ring K. is denned as a direct sum k

of /C-modules Clk, provided with an associative multiplication law a • ft, a,/3 £ Q*, such that a • /3 £ fiH+l'3!, where \a\ denotes the degree of an element a € fl'"'. In particular, it follows that Q° is a (non-commutative) /C-algebra A, while Qk>0 are ,4-bimodules and fi* is an (.4 — ,4)-algebra. A graded algebra is said to be graded commutative if a-p = (-l)^mP-a,

a,/3eQ*.

A graded algebra Q.* is called a differential graded algebra if it is a cochain complex of /C-modules 0->/C -^A

^Q1

^..-Qk

-*-»...

(1.6.1)

with respect to a coboundary operator S which obeys the graded Leibniz rule 5(a-/3) = 8a-0+(-l)WafS0.

(1.6.2)

In particular, <5 : A —> Q1 is a f^-valued derivation of a /C-algebra A. The cochain complex (1.6.1) is the above mentioned de Rham complex of the differential graded algebra (fi*, S). This algebra is also said to be a differential calculus over A. Cohomology H*(Q*) of the complex (1.6.1)

57

Chapter 1 Commutative Geometry

is called the de Rham cohomology of a differential graded algebra. It is a graded algebra with respect to the cup-product (1.6.3)

[a}^[(3}:=[a-/3],

where [a] denotes the de Rham cohomology class of elements a £ ft*. A morphism 7 between two differential graded algebras (ft*,<5) and (ft'*,<5') is defined as a cochain morphism, i.e., 7 o 6 = 7 o 8'. It yields the corresponding morphism of the de Rham cohomology groups of these algebras. One considers the minimal differential graded subalgebra ft* .4 of the differential graded algebra ft* which contains A. Seen as an (A—,4)-algebra, it is generated by the elements 5a, a £ A, and consists of monomials a = aoSai • • • Sak,

&i S A,

whose product obeys the juxtaposition rule (0,060,1) • (bo8bi) = ao$(fli&o) • <^i ~~ flo^i^o • 5bi

in accordance with the equality (1.6.2). The differential graded algebra (ft*.4,5) is called the minimal differential calculus over A. Let us show that any commutative /C-ring A defines a differential calculus. As was mentioned above, the derivation module DA of A is also a Lie /C-algebra. Let us consider the extended Chevalley-Eilenberg complex (1.6.4)

O^JC ^C*{QA;A]

of the Lie algebra 5.4. with coefficients in the ring A, regarded as a QAmodule. This complex contains a subcomplex 0*[t).A] of ,4-multilinear skew-symmetric maps (1.6.5)

k :xt>A->A with respect to the Chevalley-Eilenberg coboundary operator k

d(j)(u0,...,

life) = ^2(-iyui{4>(uQ,

...,Ui,...,

uk)) +

i=0

^ ( - l ) t + J 0 ( [ u i , U j ] , U O , ...,Ui,...,Uj,..., i<j

Uk).

(1.6.6)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

Indeed, a direct verification shows that if is an ,4-multilinear map, so is d. In particular, a e A,

(da)(u) = u(a),

u£DA,

(d)(uo, m) = u o (0(ui)) - ui(
^ O

1

(1.6.7) (1-6.8)

^ ] ,

C^M] = Horn

A(0A,A).

It follows that d(l) = 0 and d is a O^O^J-valued derivation of A. The graded module 0*[Ov4] is provided with the structure of a graded ,4-algebra with respect to the product 0A0'(iti,...,u r + s ) =

(1.6.9)

s g n l ™ 1 - ^ ^ , . . . ,uir)c/>'(uh,..

J2

.,ujs),

il<---
£Or [*A],

f € O « [VA],

uk G DA,

where sgn;;; is the sign of a permutation. This product obeys the relations d( A ') = d(4>) A'+ (-l)W4>

M f

A d('), cp, 4? € O*[l>A], (1.6.10)

= (-1)WI*V'A^.

(1.6.11)

By virtue of the first one, (O*[T)A],d) is a differential graded /C-algebra, called the Chevalley-Eilenberg differential calculus over a /C-ring A. The relation (1.6.11) shows that O*[0^4] is a graded commutative algebra. The minimal Chevalley-Eilenberg differential calculus O*A over a ring A consists of the monomials aodai A • • • A dak,

en £ A.

Its complex 0 - • K —> A - ^ OlA -±-> • • • OkA - ^ • • •

(1.6.12)

is exactly the cochain complex (1.3.8). Indeed, comparing the equalities (1.3.6) and (1.6.7) shows that dl = d on the .A-module O1A = Ox
Chapter 1 Commutative Geometry

59

algebra with respect to the cup-product (1.6.3) induced by the exterior product A of elements of O*A so that M-[^]:=[^A^], [ 0 ] - [ ^ ] = (-l)l[*]IH*']l[0']-[0]. 1.7

(1.6.13)

Sheaf cohomology

We now address algebraic systems on topological spaces. They are described in terms of sheaves. In this Section, the relevant material on sheaves of modules over a commutative ring is summarized. We follow the terminology of [64; 220]. A sheaf on a topological space X is a topological fibre bundle ir : S —> X in modules over a commutative ring K, where the surjection n is a local homeomorphism and fibres Sx, x € X, called the stalks, are provided with the discrete topology. Global sections of a sheaf S make up a /C-module S(X), called the structure module of S. Any sheaf is generated by a presheaf. A presheaf S{u} on a topological space X is defined if a module Su over a commutative ring K. is assigned to every open subset U C X (S^ = 0) and if, for any pair of open subsets V C U, there exists the restriction morphism ry : Su —» SV such that r%=IdSu,

r^

= r^rv,

W C V C U.

Every presheaf S{uy on a topological space X yields a sheaf on X whose stalk Sx at a point x G X is the direct limit of the modules Su, x £ U, with respect to the restriction morphisms ry. It means that, for each open neighborhood U of a point x, every element s e Su determines an element sx £ Sx, called the germ of s at x. Two elements s 6 Su and s' € Sy belong to the same germ at x if and only if there exists an open neighborhood W C U n V of x such that r^s = r^s'. Example 1.7.1. Let C°Vy be the presheaf of continuous real functions on a topological space X. Two such functions s and s' define the same germ sx if they coincide on an open neighborhood of x. Hence, we obtain the sheaf C^- of continuous functions on X. Similarly, the sheaf Cg of smooth functions on a smooth manifold X is defined. Let us also mention the presheaf of real functions which are constant on connected open subsets of X. It generates the constant sheaf on X denoted by R. •

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Geometric and Algebraic Topological Methods in Quantum Mechanics

Two different presheaves may generate the same sheaf. Conversely, every sheaf 5 defines a presheaf S({U}) of modules S(U) of its local sections. It is called the canonical presheaf of the sheaf 5. Global sections of 5 make up the structure module S(X) of 5. If a sheaf 5 is constructed from a presheaf S{uy, there are natural module morphisms SuBs^

s(U) e 5(17),

s{x) := sx,

xeU,

which are neither monomorphisms nor epimorphisms in general. For instance, it may happen that a non-zero presheaf defines a zero sheaf. The sheaf generated by the canonical presheaf of a sheaf 5 coincides with S. A direct sum and a tensor product of presheaves (as families of modules) and sheaves (as fibre bundles in modules) are naturally defined. By virtue of Theorem 1.1.3, a direct sum (resp. a tensor product) of presheaves generates a direct sum (resp. a tensor product) of the corresponding sheaves. Remark 1.7.2. In the terminology of [414], a sheaf is introduced as a presheaf which satisfies the following additional axioms. (51) Suppose that U C X is an open subset and {Ua} is its open cover. If s, s' £ 5(7 obey the condition

4As) = 4js>) for all Ua, then s = s'. (52) Let U and {Ua} be as in previous item. Suppose that we are given a family of presheaf elements {sa € Sua} such that r

u2nux(sa) = rt/*ni/A(s<0

for all Ua, U\. Then there exists a presheaf element s G Su such that Canonical presheaves are in one-to-one correspondence with presheaves obeying these axioms. For instance, the presheaves of continuous, smooth and locally constant functions in Example 1.7.1 satisfy the axioms (SI) -

(S2).

•

Remark 1.7.3. The notion of a sheaf can be extended to sets, but not to non-commutative groups. One can consider a presheaf of such groups, but it generates a sheaf of sets because a direct limit of non-commutative groups need not be a group. •

Chapter 1 Commutative Geometry

61

There is a useful construction of a sheaf on a topological space X from local sheaves on open subsets which make up a cover of X. 1.7.1. Let {U^} be an open cover of a topological space X and S( a sheaf on U^ for every U^. Let us suppose that, if U^ n U^ •£ 0, there is a sheaf isomorphism

PROPOSITION

Pa '• si lc/cnc/4—»• S( lu^nu^

and, for every triple (U$, Ug, UL), these isomorphisms fulfil the cocycle condition Pit o/JoO^lt/cnt^niO = P€i(Sf1|ucnt;4nyJThen there exists a sheaf S on X together with the sheaf isomorphisms <; : S\u( —> SQ such t h a t

D A morphism of a presheaf S{u} to a presheaf Sry. on the same topological space X is denned as a set of module morphisms ju : Su —> S'v which commute with restriction morphisms. A morphism of presheaves yields a morphism of sheaves generated by these presheaves. This is a bundle morphism over X such that j x : Sx -* S'x is the direct limit of morphisms 7t/, x S U. Conversely, any morphism of sheaves 5 —* S" on a topological space X yields a morphism of canonical presheaves of local sections of these sheaves. Let Horn (S\u,S'\u) be the commutative group of sheaf morphisms S\u —*• S'\u for any open subset U C X. These groups are assembled into a presheaf, and define the sheaf Horn (S,S') on X. There is a monomorphism Horn (S, S')(U) -> Horn (S{U),S'(U)),

(1.7.1)

which need not be an isomorphism. By virtue of Theorem 1.1.4, if a presheaf morphism is a monomorphism or an epimorphism, so is the corresponding sheaf morphism. Furthermore, the following holds. THEOREM

1.7.2. A short exact sequence 0 - S'{u} -> S{u} - S'(u} -v 0

(1.7.2)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

of presheaves on the same topological space yields the short exact sequence of sheaves generated by these presheaves 0 - S' -» S -» S" -* 0,

(1.7.3)

where the factor sheaf S" = S/S' is isomorphic to that generated by the factor presheaf S"u} = s{u}ls'{uy If the exact sequence of presheaves (1.7.2) is split, i.e., C

b

{uy

r^j Qf

= b

rr*

nil

{u} ® b{u}>

the corresponding splitting S S 5 ' © S" of the exact sequence of sheaves (1.7.3) holds.

•

The converse is more intricate. A sheaf morphism induces a morphism of the corresponding canonical presheaves. If S —> S' is a monomorphism, S{{U}) —* S'({U}) is also a monomorphism. However, if S —> S" is an epimorphism, S({U}) —» S'({U}) need not be so. Therefore, the short exact sequence (1.7.3) of sheaves yields the exact sequence of the canonical presheaves 0 -> S'({U}) -» S({U}) -» 5"({C/}),

(1.7.4)

where .Sdf/}) —» 5"'({f/}) is not necessarily an epimorphism. At the same time, there is the short exact sequence of presheaves 0 -> S'({U}) - S({U}) -* 5 ^ } - 0,

(1.7.5)

where the factor presheaf S'(U}=S({U})/S'({U}) generates the factor sheaf S" = S/S', but need not be its canonical presheaf. THEOREM

1.7.3. Let the exact sequence of sheaves (1.7.3) be split. Then S({U})^S'({U})®S"({U}),

and the canonical presheaves make up the short exact sequence 0 ^ S\{U}) - S({U}) -> S"({U}) -> 0.

(1.7.6)

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Chapter 1 Commutative Geometry

• Let us turn now to sheaf cohomology. We follow its definition in [220]. In the case of paracompact topological spaces, it coincides with a different definition of sheaf cohomology based on the canonical flabby resolution (see Remark 1.7.6 below). Let S{u} be a presheaf of modules on a topological space X, and let il = {Ui}iei be an open cover of X. One constructs a cochain complex where a p-cochain is defined as a function sp which associates an element sp(i0,...,ip)

E

Suion-nuip

to each (p+ l)-tuple (io, • • •, iv) of indices in I. These p-cochains are assembled into a module Cp(il, S{V}). Let us introduce the coboundary operator 6p:Cp(ii,S{u})->Cp+1(U,S{u}), P+i

= ]T(-l)fer^Sp(io,...,Ifc)...,W)-

p p

5 s (i0,...,iP+i)

fe=O

w = u i a n . . . n uip+1,

wk = uion---nuikn---n

(1.7-7)

uip+1.

One can easily check that 5P+1 o 5P = 0. Thus, we obtain the cochain complex of modules 0->C°(U,5{l/})

^•••Cp(ii,S{u})

^Cp+1(lX,S{u])

—

••• . ( 1 . 7 . 8 )

Its cohomology groups Hp(tt; S{u})

= Ker <5P/Im 5p~l

are modules. Of course, they depend on an open cover il of the topological space X. Remark 1.7.4. Throughout the book, only proper covers are considered, i.e., Ui y^ Uj if i ^ j . A cover il' is said to be a refinement of a cover il if, for each U' e il', there exists U e il such that U' CU. D Let il' be a refinement of the cover il. Then there is a morphism of cohomology groups

Let us take the direct limit of cohomology groups iJ*(il; S^yy) with respect to these morphisms, where il runs through all open covers of X. This limit

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Geometric and Algebraic Topological Methods in Quantum Mechanics

H*(X; S{uy) is called the cohomology of X with coefficients in the presheaf S{u}Let S be a sheaf on a topological space X. Cohomology of X with coefficients in S or, simply, sheaf cohomology of X is defined as cohomology H*(X;S) = H*(X]S({U})) with coefficients in the canonical presheaf S({U}) of the sheaf S. In this case, a p-cochain sp 6 Cp(ii, S({U})) is a collection A — \b [IQ, . . . , lp)}

of local sections sp(io, • • •, ip) of the sheaf S over Ui0 D • • • PI Uip for each (p + l)-tuple (Ui0,..., Uiv) of elements of the cover it. The coboundary operator (1.7.7) reads P+i

5Psp(i0,...,

iP+i) - ^2(-l)ksP(i0,...

,? f c ) ..., iP+i)\uion-nuip+1

• (1-7.9)

fe=0

For instance, 5°so(i,j) = [so(j)-s°(i)}\UinUj, 1

<5V(i,j,fc) = [s (j,fc) - s\i,k)

(1.7.10) + s\i,j)}\UinUjnUk.

(1.7.11)

A glance at the expression (1.7.10) shows that a zero-cocycle is a collection s = {s(i)}j of local sections of the sheaf S over Ui e il such that s(i) — s{j) on Ui n Uj. It follows from the axiom (S2) in Remark 1.7.2 that s is a global section of the sheaf 5, while each s(i) is its restriction s ^ to Ui. Consequently, the cohomology group H°(U,S({U})) is isomorphic to the structure module S(X) of global sections of the sheaf S. A one-cocycle is a collection {s(i, j)} of local sections of the sheaf S over overlaps Ui (~l Uj which satisfy the cocycle condition [s(j,k) - s(i,k) + s(i,j)}\UinUjnuk

= 0.

(1-7.12)

If X is a paracompact space, the study of its sheaf cohomology is essentially simplified due to the following fact [220]. THEOREM 1.7.4. Cohomology of a paracompact space X with coefficients in a sheaf S coincides with cohomology of X with coefficients in any presheaf • generating the sheaf 5*.

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Chapter 1 Commutative Geometry

Remark 1.7.5. We follow the definition of a paracompact topological space in [220] as a Hausdorff space such that any its open cover admits a locallyfiniteopen refinement, i.e., any point has an open neighborhood which intersects only a finite number of elements of this refinement. A topological space X is paracompact if and only if any cover {U^} of X admits a subordinate partition of unity {/^}, i.e.: (i) /^ are real positive continuous functions on X; (ii) supp/^ Ct/j; (iii) each point x € X has an open neighborhood which intersects only a finite number of the sets supp / j ; (iv) J2 fax) = 1 for all x € X. D The key point of the analysis of sheaf cohomology is that short exact sequences of presheaves and sheaves yield long exact sequences of sheaf cohomology groups. Let S{t/} and S',y, be presheaves on the same topological space X. It is readily observed that, given an open cover 11 of X, any morphism S{uj —> S'ry-i yields a cochain morphism of complexes C*(ll,S{u})^C*(il,S'{u}) and the corresponding morphism H'(U,S{U})^H'{U,S'{V}) of cohomology groups of these complexes. Passing to the direct limit through all refinements of il, we come to a morphism of the cohomology groups H*(X,S{U})->H*(X,S'{U}) of X with coefficients in the presheaves S[u} and S'^y In particular, any sheaf morphism S —> S' yields a morphism of canonical presheaves S({U}) —» S'({U}) and the corresponding cohomology morphism H*{X,S)^H*(X,S'). By virtue of Theorems 1.4.3 and 1.4.4, every short exact sequence 0 - S'{u} —»S {u} —> S'(U} - 0

(1.7.13)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

of presheaves on the same topological space X and the corresponding exact sequence of complexes (1.7.8) yield the long exact sequence O^H°(X;S'{U}) H^XiSfa) P

H (X;S'(U})

-^H°(X;S{U}) -^•••H*>{X]S'{U)) -^H

P+1

(X;S'{U})

—> H°(X;S'{U}) — -^W{X-S{U})

(1.7.14)

—>

—> • • •

of the cohomology groups of X with coefficients in these presheaves. This result however is not extended to an exact sequence of sheaves, unless X is a paracompact space. Let 0-»S" —*S —>S" -»0

(1.7.15)

be a short exact sequence of sheaves on X. It yields the short exact sequence of presheaves (1.7.5) where the presheaf 5" a } generates the sheaf S". If X is paracompact, H*(X;S'(U}) = H*(X;S") in accordance with Theorem 1.7.4, and we have the exact sequence of sheaf cohomology Q-*H°(X;S') -^H°(X;S) ^H°{X;S") —* P P H\X;S') —^•••H {X;S') -—>H (X;S) —> HP(X;S") -^Hp+1{X;S') —

(1.7.16)

Let us point out the following isomorphism between sheaf cohomology and singular (Chech and Alexandery) cohomology of a paracompact space [64; 402]. THEOREM 1.7.5. The sheaf cohomology H*(X;Z) (resp. H*(X;Q), H*(X;R)) of a paracompact topological space X with coefficients in the constant sheaf Z (resp. Q, M) is isomorphic to the singular cohomology of X with coefficients in the ring Z (resp. Q, R). •

Since singular cohomology is a topological invariant (i.e., homotopic topological spaces have the same singular cohomology) [402], the sheaf cohomology groups H*(X;Z), H*(X;Q), H*(X\R) of a paracompact space are also topological invariants. Let us turn now to the abstract de Rham theorem which provides a powerful tool of studying algebraic systems on paracompact spaces.

Chapter 1 Commutative Geometry

67

Let us consider an exact sequence of sheaves 0 - > 5 -^->50 - ^ S i - ^ • • • S p - ^ • • • .

(1.7.17)

It is said to be a resolution of the sheaf S if each sheaf Sp>o is acyclic, i.e., its cohomology groups Hk>0(X; Sp) vanish. Any exact sequence of sheaves (1.7.17) yields the sequence of their structure modules 0-*S(X)

^So(X)

^StiX)

^•••SP(X)

^ • • -

(1.7.18)

which is always exact at terms S(X) and So(X) (see the exact sequence (1.7.4)). The sequence (1.7.18) is a cochain complex because hl+1 oh% = 0. If X is a paracompact space and the exact sequence (1.7.17) is a resolution of S, the abstract de Rham theorem establishes an isomorphism of cohomology of the complex (1.7.18) to cohomology of X with coefficients in the sheaf 5 as follows [220]. 1.7.6. Given a resolution (1.7.17) of a sheaf 5 on a paracompact topological space X and the induced complex (1.7.18), there are isomorphisms THEOREM

H°(X;S)=Kerh°,

H9(X;S) = Kei hqjlmh^1,

q>0. (1.7.19)

• We will also refer to the following minor modification of Theorem 1.7.6 [173; 410]. THEOREM O^S

1.7.7. Let -^So

£+Si

^ • • •

h

^ S

p

^Sp+i,

p>l,

(1.7.20)

be an exact sequence of sheaves on a paracompact topological space X, where the sheaves Sq, 0 < q < p, are acyclic, and let 0_5(X)

^So(X)

^Sx(X)

!^L...h^Sp(X)

!^SP+1(X)

(1.7.21) be the corresponding cochain complex of structure modules of these sheaves. Then the isomorphisms (1.7.19) hold for 0 < q < p. • Any sheaf on a topological space admits the canonical resolution by flabby sheaves as follows.

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Geometric and Algebraic Topological Methods in Quantum Mechanics

A sheaf S on a topological space X is called flabby (or flasque in the terminology of [414]), if the restriction morphism S(X) —> S(U) is an epimorphism for any open U C X, i.e., if any local section of the sheaf S can be extended to a global section. A flabby sheaf is acyclic. Indeed, given an arbitrary cover it of X, let us consider the complex C*(il, S({U})) (1.7.8) for its canonical presheaf S({U}). Since 5 is flabby, one can define a morphism

h : C(il, S({U})) -> C-HU, S({U})), p

hs {i0,...,

p > 0,

p

tp_i) := j * s (i 0 ,..., tp-i.j),

(1.7.22)

where Uj is a fixed element of the cover 11 and j*sp is an extension of sp(io,..., ip-i,j) onto £/j0 n • • • n {/ip_i• A direct verification shows that h (1.7.22) is a homotopy operator for the complex C*(il, S({U})) and, consequently, HP>0(il,S({U})) = 0. Given an arbitrary sheaf 5 on a topological space X, let SF({U}) denote the presheaf of all (not-necessarily continuous) sections of the sheaf S. It generates a sheaf SF on X, and coincides with the canonical presheaf of this sheaf. There are the natural monomorphisms S({U}) —> Sp({U}) and S —* Sp. It is readily observed that the sheaf S^ is flabby. Let us take the quotient Sp/S and construct the flabby sheaf

SF = (Sp/S)F. Continuing the procedure, we obtain the exact sequence of sheaves 0 - > 5 —»Sj. -^Sp

—>•••,

(1.7.23)

which is a resolution of S1 since all sheaves are flabby and, consequently, acyclic. It is called the canonical flabby resolution of the sheaf S. The exact sequence of sheaves (1.7.23) yields the complex of structure modules of these sheaves 0 -+ S(X) — > S°p(X)

—•+ SF(X)

—>....

(1.7.24)

If X is paracompact, the cohomology of X with coefficients in the sheaf S coincides with cohomology of the complex (1.7.24) by virtue of Theorem 1.7.6. Remark 1.7.6. An important peculiarity of flabby sheaves is that a short exact sequence of flabby sheaves on an arbitrary topological space provides the short exact sequence of their structure modules. Therefore, there is a

Chapter 1 Commutative Geometry

69

different definition of sheaf cohomology. Cohomology of a topological space X with coefficients in a sheaf S is defined directly as cohomology of the complex (1.7.24) [64]. For a paracompact space, this definition coincides with above mentioned one due to Theorem 1.7.6. • In the sequel, we also refer to a fine resolution of sheaves, i.e., a resolution by fine sheaves. A sheaf S on a paracompact space X is called fine if, for each locally finite open cover it = {f/j}ig/ of X, there exists a system {hi} of endomorphisms hi : S —» S such that: (i) there is a closed subset Vi C Hi and hi(Sx) = 0 if x $ Vi, (ii) ^2 hi is the identity map of S. A fine sheaf on a paracompact space iei

is acyclic. Indeed, given an arbitrary locally finite cover 11 = {Ui}i^i of X and a p-cochain sp, let us define the (p — l)-cochain hsp(i0,...,

ip_i) = J2

rsP i

( > *o>• • • > V-i)

(1.7.25)

where i*sp, by definition, is equal to hiSp on the set Ui n Ui0 n • • • Uip_1 and to 0 outside this set. Then the morphism h (1.7.25) is a homotopy operator. There are the following important examples of fine sheaves. 1.7.8. Let X be a paracompact topological space which admits a partition of unity performed by elements of the structure module 2l(X) of some sheaf 21 of real functions on X. Then any sheaf S of 21modules on X, including 21 itself, is fine. O PROPOSITION

Outline of proof. Let il = {Ui]i^i be a locally finite open cover of X and {fi} the subordinate partition of unity. For any open set U C X, we put hi(s) = fiS,

s G S(U).

This defines an endomorphism hi of the canonical presheaf <S'({f7}) and, consequently, an endomorphism of the sheaf 5. It is readily observed that QED these endomorphisms hi satisfy the definition of a fine sheaf. In particular, the sheaf C*x of continuous functions on a paracompact topological space is fine, and so is any sheaf of C^-modules. A smooth manifold X admits a partition of unity performed by smooth real functions.

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Geometric and Algebraic Topological Methods in Quantum Mechanics

It follows that the sheaf Cx of smooth real functions on X is fine, and so is any sheaf of Cjf-modules, e.g., the sheaves of sections of smooth vector bundles over X. We complete our exposition of sheaf cohomology with the following useful theorem [21].

1.7.9. Let / : X —» X' be a continuous map and S a sheaf on X. Let either / be a closed immersion or every point x' £ X' have a base of open neighborhoods {[/} such that the sheaves S |/-i([/) are acyclic. Then • the cohomology groups H*(X; S) and H*(X'; /»5) are isomorphic. THEOREM

1.8

Local-ringed spaces

Local-ringed spaces are sheafs of local rings. For instance, smooth manifolds, represented by sheaves of real smooth functions, make up a subcategory of the category of local-ringed spaces. A sheaf D\ on a topological space X is said to be a ringed space if its stalk 9ix at each point x G X is a commutative ring [414]. A ringed space is often denoted by a pair (X, 9^) of a topological space X and a sheaf 91 of rings on X which are called the body and the structure sheaf of a ringed space, respectively. A ringed space is said to be a local-ringed space (a geometric space in the terminology of [414]) if it is a sheaf of local rings. For instance, the sheaf C^ of continuous real functions on a topological space X is a local-ringed space. Its stalk C°, x E X, contains the unique maximal ideal of germs of functions vanishing at x. Let A be an arbitrary commutative ring and I its prime ideal. Then A\ X is a multiplicative subset of A (see Remark 1.1.1) and (A\ X)~^A is a local ring. Using this fact, one can associate to any commutative ring A a local-ringed space as follows. Let SpeCv4 be the set of prime ideals of the ring A. It is called the spectrum of A. Let us assign to each ideal X of A the set

V(X) = {xeSpecA : XCx}.

(1.8.1)

Chapter 1 Commutative Geometry

71

These sets possess the properties ^({0}) = Spec A

n v(ii) = v( 53 ii),

V(A) = 0,

v(i) u v(x') = v(ii').

i

In view of these properties, one can regard the sets (1.8.1) as closed sets of some topology on the spectrum Spec A It is called the Zariski topology. A base for this topology consists of the sets U{a) = {x£ Spec A : a $ x } = Spec A \ V{Aa)

(1.8.2)

as a runs through A The set of closed points is nothing but the set Specm A of all maximal ideals of A. Endowed with the relative Zariski topology, it is called the maximal spectrum of A A ring morphism C, : A —» A' induces the continuous map C" : Spec A 3 I ' H C V ) G Spec A

(1.8.3)

In particular, let 7r : A —> A/X be the quotient morphism with respect to an ideal X of A. Then (^ (1.8.3) is a homeomorphism of Spec(.4/Z) onto the closed subspace V(I) C Spec A Given a commutative ring A and its spectrum Spec A, one can define a sheaf 21 on Spec A whose stalk at a point x € Spec .4 is the local ring 21* = (A \ x)~xA. The ring 2l(Spec A) of global sections of the sheaf 21 is exactly the ring A itself. The local ringed space (Spec A, 21) is called an affine scheme [397; 421]. A local-ringed space (X, KH) which is locally isomorphic to an affine scheme is called a scheme. Let us recall the following standard notions. (i) If a scheme (X, SH) has an affine open cover {£/, = Spec A } such that every A% is a Noetherian ring (i.e., any ideal of A is finitely generated), (X, 9t) is said to be locally Noetherian. A locally Noetherian scheme is called Noetherian if its body X is quasi-compact. (ii) A scheme (X, £H) is called reduced if the stalk of 9^ at each point of X has no nilpotent elements. (iii) A scheme (X, £R) is said to be irreducible if its body X is not a union of two proper closed subsets. (iv) A scheme is called integral if it is reduced and irreducible. Morphisms of local-ringed spaces are defined to be particular morphisms of sheaves on different topological spaces as follows.

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Geometric and Algebraic Topological Methods in Quantum Mechanics

Let ip : X —> X ' be a continuous map. Given a sheaf S on X, its direct image y>»S on X' is generated by the presheaf of assignments X' DU' ^

S(ip-\U'))

for any open subset U' C X'. Conversely, given a sheaf 5 ' on X', its inverse image ip*S' on X is denned as the pull-back onto X of the topological fibre bundle 5" over X', i.e., X' be a closed subspace of X'. Then i»S is a unique sheaf on X' such that i*S\x = 5,

t , 5 | X ' \ x = 0.

Indeed, i f x ' s X c X', then i*S(U') = S(U' n X) for any open neighborhood U of this point. If x' $ X, there exists its neighborhood U' such that U' H X is empty, i.e., i*S(U') = 0. The sheaf i*S is called the trivial • extension of the sheaf S. By a morphism of ringed spaces (X, 9^) —» (X', 9V) is meant a pair (ip, $) of a continuous map ip : X —> X' and a sheaf morphism $ : 9t' —>
(1.8.4)

(X, JH) is said to be a scheme over (X', 91'). A morphism

X x X X'

is closed. In this case, one says that (X, JH) is separated over X'. A scheme (X, 9t) is called separated if it is a separated over Spec Z. All afflne schemes are separated.

Chapter 1 Commutative Geometry

73

A morphism of schemes ( X , 9 i ) - » ( * ' = Spec .4,21) is said to be locally of finite type (resp. of finite type) if (X, 9t) has an open affine cover (resp. a finite open affine cover) {Ui = Spec.4j} such that each Ai is a finitely generated ,4-algebra. A general morphism of schemes tp (1.8.4) is said to be locally of finite type (resp. of finite type) if there is an open affine cover {V{\ of X' such that every restriction of

(iR(tf')). Given a local-ringed space (X, 91), a sheaf P on X is called a sheaf o/9lmodules if every stalk Px, x £ X, is an 9tx-module or, equivalently, if P(U) is an 9t(t/)-module for any open subset U C X. A sheaf of 9l-modules P is said to be locally free if there exists an open neighborhood U of every point x G X such that P(U) is a free 9t(f7)-module. If all these free modules are of finite rank (resp. of the same finite rank), one says that P is of finite type (resp. of constant rank). The structure module of a locally free sheaf is called a locally free module.

The following is a generalization of Proposition 1.7.8 [220]. 1.8.1. Let X be a paracompact space which admits a partition of unity by elements of the structure module S(X) of some sheaf S of real functions on X. Let P be a sheaf of 5-modules. Then P is fine and, consequently, acyclic. • PROPOSITION

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Geometric and Algebraic Topological Methods in Quantum Mechanics

Let (X, SH) be a ringed space. A sheaf of 9^-modules S is said to be quasi-coherent if for each point x of X there exists a neighborhood U of x and an exact sequence M —*N —• S| l / -»o, where M and N are free 9t|(y-modules. Let 5 be a locally free sheaf of 9t-modules of finite type. It is said to be of finite presentation if, locally, there exists an exact sequence 9tm —>9tn —>5->0, where m and n are positive integers (which need not be globally constant). A locally free sheaf of finite type is called coherent if the kernel of any homomorphism 9t|j} —* S\u, where n is an arbitrary positive integer and U is an open set, is of finite type. Obviously, if S is coherent, then it is of finite presentation, and is quasi-coherent. If 91 itself is coherent as a sheaf of 9t-modules, then it is called the coherent sheaf of rings. In this case, every sheaf of Dt-modules of finite presentation is coherent. For instance, the structure sheaf of a locally Noetherian scheme is a coherent sheaf of rings. Let (X = Spec A, 9^ = 21) be an affine scheme. Then every quasicoherent sheaf S of 2l-modules on X is generated by its global sections. The correspondence S —> S(X) defines an equivalence between the category of quasi-coherent sheaves on X and the category of .4-modules. If .4 is Noetherian, then the coherent sheaves and the finite ^.-modules correspond to each other under this equivalence. Let (X,fR) be a separated scheme and il = {[/*} an affine open cover of X. For each quasi-coherent sheaf S of 9^-modules, the cohomology H*(X;S) of X with coefficients in S is canonically isomorphic to the cohomology H*(iX;S(U)). One defines the cohomological dimension cd(X) of a scheme (X,SR.) as the largest integer r such that Hm(X;S) ^ 0 for a quasi-coherent sheaf of fK-modules on X. For instance, if X is an affine scheme, then cd(X) = 0. The converse is true under the assumption that X is Noetherian. The sheaf Cg of smooth real functions on a smooth manifold X provides an important example of a local-ringed spaces. Remark 1.8.3. Throughout the book, smooth manifolds are finitedimensional real manifolds, though infinite-dimensional Banach and Hubert manifolds are also smooth. A smooth real manifold is customarily assumed

Chapter 1 Commutative Geometry

75

to be Hausdorff and second-countable (i.e., it has a countable base for topology) . Consequently, it is a locally compact space which is a union of a countable number of compact subsets, a separable space (i.e., it has a countable dense subset), a paracompact and completely regular space. Being paracompact, a smooth manifold admits a partition of unity by smooth real functions. One also can show that, given two disjoint closed subsets N and N' of a smooth manifold X, there exists a smooth function / on X such that /|AT = 0 a n d /|AT' = 1. Unless otherwise stated, manifolds are assumed to be connected and, consequently, arcwise connected. We follow the notion of a manifold without boundary. D Similarly to the sheaf C^ of continuous functions, the stalk C£° of the sheaf Cj^P at a point x € X has a unique maximal ideal of germs of smooth functions vanishing at x. Though the sheaf Cjj? is defined on a topological space X, it fixes a unique smooth manifold structure on X as follows. 1.8.2. Let X be a paracompact topological space and (X,9\) a local-ringed space. Let X admit an open cover {Ui} such that the sheaf 9K restricted to each Ui is isomorphic to the local-ringed space (R",C^,). Then X is an n-dimensional smooth manifold together with a natural isomorphism of local-ringed spaces (X,9i) and (X,Cj^). • THEOREM

One can think of this result as being an alternative definition of smooth real manifolds in terms of local-ringed spaces. A smooth real manifold X is also algebraically reproduced as a certain subspace of the spectrum of the K-ring C°°(X) of smooth real functions on X as follows [17]. Let A be a commutative R-ring and Specm.4 its maximal spectrum. The real spectrum of A is the subspace SpecR.4 C Specm A of the maximal ideals X such that the quotients A/I are isomorphic to R. It is endowed with the relative Zariski topology. There is the bijection between the set of R-algebra morphisms of A to the field R and the real spectrum of A, namely, HomR(.4,R) 3

Ker e SpecRA SpecR.4 9 x H-> irx e HomR(.4,R),

nx : A -> A/x = R.

Any element a € A induces a real function fa : Spec R «4 3 x )—> nx(a)

on the real spectrum SpecRA This function need not be continuous with respect to the Zariski topology, but one can provide SpecR.4 with another

76

Geometric and Algebraic Topological Methods in Quantum Mechanics

topology, called the Gel'fand topology, which is the coarsest topology which makes all such functions continuous. If A = C°°(X), the Zariski and Gel'fand topologies coincide. 1.8.3. Given the R-ring C°°(X) of smooth real functions on a manifold X, let fix denote the maximal ideal of functions vanishing at a point x e X. Then there is a homeomorphism THEOREM

Xx-X3x^

fixe SpecRC°°(X).

(1.8.5)

a Let X and X' be two smooth manifolds. Any smooth map 7 : X —> X' induces the R-ring morphism 7* : C°°{X') -» C°°{X) which associates to a function / on X' the pull-back function 7*/ = / o 7 on X. Conversely, each R-ring morphism C : C°°{X') -» C°°(X) yields the continuous map C*1 (1-8.3) which sends SpecKC°°(X) C SpecC°°(X) to SpecRC°°(X') C SpecC°°{X') so that the induced map Xx'C* o xx •• X -» X' is smooth. Thus, there is one-to-one correspondence between smooth manifold morphisms X -> X' and the R-ring morphisms C°°(X') -> C°°(X). Remark 1.8.4. Let X x X' be a manifold product. The ring C°° (X x X') is constructed from the rings C°°(X) and C°°(X') as follows. Whenever referring to a topology on the ring C°°(X), we will mean the topology of compact convergence for all derivatives [374]. The C°°(X) is a Frechet ring with respect to this topology, i.e., a complete metrizable locally convex topological vector space. There is an isomorphism of Frechet rings

C^POSC^PO *£ C°°(X x X'),

(1.8.6)

where the left-hand side, called the topological tensor product, is the completion of C°°{X) ® C°°(X') with respect to Grothendieck's topology, defined as follows. If Ei and E2 are locally convex topological vector spaces, Grothendieck's topology is the finest locally convex topology on Ei <&EQ. such that the canonical mapping of E\ x E2 to Ei ® E2 is continuous [374]. It is

77

Chapter 1 Commutative Geometry

also called the 7r-topology in contrast with the coarser e-topology on E±®E2 [356; 417]. Furthermore, for any two open subsets U C X and V C X', let us consider the topological tensor product of rings C0O([/)(g)C°o(C/'). These Due to tensor products define a locally ringed space (X x X',Cx®Cx,). the isomorphism (1.8.6) written for all U C X and U' C X', we obtain the sheaf isomorphism /TOO Zi./-«X)

.TOO

/ I n 71

(1.S.7J

OX®OX, -CXxX,.

D Since a smooth manifold admits a partition of unity by smooth functions, it follows from Proposition 1.8.1 that any sheaf of C^-modules on X is fine and, consequently, acyclic. For instance, let Y —> X be a smooth (finite-dimensional) vector bundle. The germs of its sections make up a sheaf of C^-modules, called the structure sheaf Sy of a vector bundle Y —> X. The sheaf Sy is fine. In particular, all sheafs Ox, k G N+, of germs of exterior forms on X is fine. These sheaves constitute the de Rham complex —>K — >u

x

— > Ux

— > • • • Ux

—>••• .

(l.o. a)

The corresponding complex of structure modules of these sheaves is the de Rham complex 0 - • R —* C°°{X) -1+ Ol{X) -iU • • • Ok(X) - ^ • • •

(1.8.9)

of exterior forms on a manifold X. Its cohomology is called the de Rham cohomology H*(X) of X. Due to the Poincare lemma, the complex (1.8.8) is exact and, thereby, is a fine resolution of the constant sheaf R on a manifold. Then a corollary of Theorem 1.7.6 is the classical de Rham theorem. THEOREM

1.8.4. There is the isomorphism Hk(X)=Hk(X;R)

(1.8.10)

of the de Rham cohomology H*(X) of a manifold X to cohomology of X with coefficients in the constant sheaf M.. Q Remark 1.8.5. The de Rham cohomology groups H*(X) of a compact manifold X are finite-dimensional real vector spaces so that dimHh(X) is

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Geometric and Algebraic Topological Methods in Quantum Mechanics

the k-th Betty number of X, and X(X)=

J2dimHk(X) k

is the Euler characteristic of X. Remark 1.8.6. sheaves

•

Let us consider the short exact sequence of constant 0 - > Z —»R —>I7(1)-»O,

(1.8.11)

where C/(l) = M/Z is the circle group of complex numbers of unit modulus. This exact sequence yields the long exact sequence of the sheaf cohomology groups O ^ Z —>R —>tf(l) -^Hl{X;Z) HP{X;Z)

^HP(X;R)

—> • • •

-^H\X;R)

—*HP(X;U(1))

—>HP+1(X;Z)

—*••-,

where H°(X;Z)=Z,

H°(X;R)=R

and H°(X; U(l)) = U{\). This exact sequence defines the homomorphism H*(X;Z)^H*(X;R)

(1.8.12)

of cohomology with coefficients in the constant sheaf Z to that with coefficients in R. Combining the isomorphism (1.8.10) and the homomorphism (1.8.12) leads to the cohomology homomorphism H*(X\Z)->ir(X).

(1.8.13)

Its kernel contains all cyclic elements of cohomology groups Hk{X\ Z). • Given a vector bundle Y —> X, the structure module of the sheaf Sy coincides with the structure module Y(X) of global sections of Y —» X. The Serve-Swan theorem, shows that these modules exhaust all projective modules of finite rank over C°°(X). As was mentioned in Introduction, this theorem has been proved in the case of a compact manifold X. However, its proof can be generalized to an arbitrary smooth manifold due to Theorem 10.6.2.

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Chapter 1 Commutative Geometry

1.8.5. Let X be a smooth manifold. A C°°(X)-module P is isomorphic to the structure module of a smooth vector bundle over X if and only if it is a projective module of finite rank. • THEOREM

Outline of proof. Let {(^,V'c)>/9Cc}' £» C = 1> • • • i^> D e a finite atlas of a smooth vector bundle Y —* X of fibre dimension m. Given a smooth partition of unity {/^} subordinate to the cover {U(}, let us set

It is readily observed that {£|} is also a partition of unity subordinate to {U^}. Every element s £ Y(X) defines an element of a free C°°(X)-module of rank m + k as follows. Let us put s% = ip£ o s\u( • It fulfils the relation a

€=Z>«'csC-

(1-8-14)

C

There are a module monomorphism F : Y(X) 3 s » (llSl,...,

lksk) G "©* C°°(X)

and a module epimorphism *: m © f c C°°(Jf) B (tu...

,tk) ^ (su ... ,sk) €Y(X),

3i=

Y^PH^i3

In view of the relation (1.8.14), 9oF

=

IdY{X),

i.e., Y(X) is a projective module of finite rank. The converse assertion is QED proved similarly to that in [430], Theorem 1.8.5 states the above mentioned categorial equivalence between the vector bundles over a smooth manifold X and projective modules of finite rank over the ring C°°{X) of smooth real functions on X. The following are corollaries of this equivalence • The structure module Y*(X) of the dual Y* —> X of a vector bundle Y -» X is the C°°(X)-dual Y(X)* of the structure module Y(X) of Y -> X. • Any exact sequence of vector bundles 0 -> y —>Y' —>Y" -> 0

(1.8.15)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

over the same base X yields the exact sequence 0 -» Y(X)

—* Y'(X)

—• Y"(X) -> 0

(1.8.16)

of their structure modules, and vice versa. In accordance with Theorem 10.6.5, the exact sequence (1.8.15) is always split. Every its splitting defines that of the exact sequence (1.8.16), and vice versa. • For instance, the derivation module of the M-ring C°°(X) coincides with the C°°(X)-module Ti(X) of vector fields on X, i.e., with the structure module of the tangent bundle TX of X. Hence, it is a projective C°°(X)-module of finite rank. It is the C°°(X)-dual TX(X) = O1{X)* of the structure module O1(X) of the cotangent bundle T*X of X which is the module of differential one-forms on X and, conversely, O1(X) = Tx{X)*. It follows that the Chevalley-Eilenberg differential calculus over the R-ring C°°(X) is exactly the differential graded algebra (O*(X),d) of exterior forms on X, where the Chevalley-Eilenberg coboundary operator d (1.6.6) coincides with the exterior differential. Accordingly, the de Rham complex (1.6.12) of the R-ring C°°(X) is the de Rham complex (1.8.9) of exterior forms on X. Moreover, one can show that (O*(X), d) is a minimal differential calculus, i.e., the C°° (X)-modu\e O1(X) is generated by elements df, f € C°°(X). Indeed, using the notation in the proof of Theorem 1.8.5, one can write

O\X) B <$> = £Z|0 = J^lfacb? = ^2(l^d(l^)

-

l^^dk),

(1.8.17) where (xM) are local coordinates on [/{ and /^xM and 1% are functions on X. Remark 1.8.7. Let us note that the above mentioned ChevalleyEilenberg differential calculus over the M-ring CCO(X) is a subcomplex of the Chevalley-Eilenberg complex of the Lie algebra T\{X) with coefficients in C°°(X). It consists of skew-symmetric morphisms of T\(X) to C°°(X) which are not only R-multilinear, but C°°(X)-multilinear. The ChevalleyEilenberg cohomology of smooth vector fields with coefficients in a trivial representation and in spaces of smooth tensor fields has been studied in detail [160] • • Let Y —> X be a vector bundle and Y(X) its structure module. The r-order jet manifold JrY of Y —> X consists of the equivalence classes j^s, x £ X, of sections s of Y —> X which are identified by the r + 1 terms of their Taylor series at points x £ X (see Section 10.7). Since Y —> X is a

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Chapter 1 Commutative Geometry

vector bundle, so is the jet bundle JrY —» X. Its structure module JrY(X) of the C°°(X)-module Y(X) is exactly the r-order jet module Jr(Y(X)) in Section 1.2 [261]. As a consequence, the notion of a connection on the structure module Y(X) is equivalent to the standard geometric notion of a connection on a vector bundle Y —» X [296]. Indeed, connection on a fibre bundle Y —> X is defined as a global section F of the affine jet bundle JXY -> y . If V —> X is a vector bundle, there exists the exact sequence 0->T*X®Y—>JlY

—>y->0

(1.8.18)

x over X which is split by F. Conversely, any slitting of this exact sequence yields a connection Y —> X. The exact sequence of vector bundles (1.8.18) induces the exact sequence of their structure modules

o-»o 1 (x)®y(x) -->JlY{x) —>y(x)->o.

(1.8.19)

Then any connection F on a vector bundle Y —> X defines a splitting of the exact sequence (1.8.19) which, by Definition 1.3.1, is a connection on the C°°(X)-module Y(X), and vice versa. Let now P be an arbitrary C°°(X)-module. One can reformulate Definitions 1.3.2 and 1.3.3 of a connection on P as follows. DEFINITION 1.8.6. A connection on a C°°(X)-module P is a C°°(X)module morphism

V:P-*O1{X)<»P,

(1.8.20)

which satisfies the Leibniz rule V ( / p ) = # ® p + /V(p),

f€C°°(X),

pGP. D

DEFINITION 1.8.7. A connection on a C°°(X)-module P associates to any vector field r e 7i{X) on X a first order differential operator V r on P which obeys the Leibniz rule

V T (/p) = (Tj4f)p + / V r p .

(1.8.21)

• Since O1(X) = Ti(X)*, Definitions 1.8.6 and 1.8.7 are equivalent.

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Let us note that a connection on an arbitrary C°° (X)-module need not exist, unless it is a projective or locally free module (see Theorem 1.8.11 below). The curvature of a connection V in Definitions 1.8.6 and 1.8.7 is defined as the zero-order differential operator iJ(T,T') = [V T ,V T .]-V [T , T /]

(1.8.22)

on a module P for all vector fields T,T' e TX{X) on X (cf. (10.6.64)). Let us turn now to the notion of a connection on a local-ringed space. Let (X,d\) be a local-ringed space and ty a sheaf of 9"l-modules on X. For any open subset U C X, let us consider the jet module Jx{ty{U)) of the module ty{U). It consists of the elements of £H(C/) ® ^P(E7) modulo the pointwise relations (1.2.14). Hence, there is the restriction morphism

jlmu)) -

j\y(v))

for any open subsets V C U, and the jet modules ^(^(U)) constitute a presheaf. This presheaf defines the sheaf 3ls$ of jets of ty (or simply the jet sheaf). The jet sheaf O^SH of the sheaf SH of local rings is introduced in a similar way. Since the relations (1.2.14) and (1.3.1) on the ring W.(U) and modules ty{U), J71(?J(C/)), v71(fH([/)) are pointwise relations for any open subset U C X, they commute with the restriction morphisms. Therefore, the direct limits of the quotients modulo these relations exist [303]. Then we have the sheaf O 1 ^ of one-forms over the sheaf Dt, the sheaf isomorphism 0;1(q3) = (fH©C»1fH)®^, and the exact sequences of sheaves 0-*£> 1 9lq3^a 1 (q3)-+q3-+0, O - * © 1 ^ ® ^ - ^ (9S © O 1 ^)

0.

(1.8.23) (1.8.24)

They reflect the quotient (1.3.3), the isomorphism (1.3.10) and the exact sequences of modules (1.3.11), (1.3.13), respectively. Remark 1.8.8. It should be emphasized that, because of the inequality (1.7.1), the duality relation (1.3.5) is not extended to the sheaves 59^ and O 1 ^ in general, unless VfR and Ol are locally free sheaves of finite rank. If 93 is a locally free sheaf of finite rank, so is CJ1^D Following Definitions 1.3.1, 1.3.2 of a connection on modules, we come to the following notion of a connection on sheaves.

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Chapter 1 Commutative Geometry

DEFINITION 1.8.8. Given a local-ringed space {X,9\) and a sheaf *p of Dt-modules on X, a connection on a sheaf ^3 is denned as a splitting of the exact sequence (1.8.23) or, equivalently, the exact sequence (1.8.24). •

Theorem 1.7.3 leads to the following compatibility of the notion of a connection on sheaves with that of a connection on modules. 1.8.9. If there exists a connection on a sheaf ?p in Definition 1.8.8, then there exists a connection on a module *P(t/) for any open subset U C X. Conversely, if for any open subsets V c U
• X be a vector bundle. Every linear connection Example 1.8.9. Let Y —> F on Y —> X defines a connection on the structure module Y(X) such that the restriction T\u is a connection on the module Y(U) for any open subset U C X. Then we have a connection on the structure sheaf Yx- Conversely, a connection on the structure sheaf Yx defines a connection on the module • Y(X) and, consequently, a connection on the vector bundle Y —» X. As an immediate consequence of Proposition 1.8.9, we find that the exact sequence of sheaves (1.8.24) is split if and only if there exists a sheaf morphism V :

«p,

(1.8.25)

satisfying the Leibniz rule V(/s) = d / ® s + /V(s),

f€A(U),

s£V(U),

for any open subset U € X. It leads to the following equivalent definition of a connection on sheaves in the spirit of Definition 1.3.2. DEFINITION

sheaf q3.

1.8.10. The sheaf morphism (1.8.25) is a connection on the

•

Similarly to the case of connections on modules, the curvature of the connection (1.8.25) on a sheaf ty is given by the expression R = V 2 :

dCp>q{Z) c Cp+2'g-1(Z) © Cp+1'q{Z) © Cp'q+1(Z) © C p - l l 9 + 2 (Z).

PROPOSITION 2.6.4. [250]. An almost complex structure J on a manifold Z is a complex (i.e., torsionless) structure if and only if the following equivalent conditions hold: (i) the distribution TZ1'0 is involutive, (ii) the distribution TZ 0 ' 1 is involutive, (iii) dCx-°{Z) C C2-°(Z) ®C1-\Z) and dC^x{Z) c CX<\Z) ©C°-2(Z). D

The condition (iii) in Proposition 2.6.4 implies the relation dCp'q(Z)

C Cp+1'q(Z)

© CP'9+1(Z),

p,q =

0,...,m.

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Geometric and Algebraic Topological Methods in Quantum Mechanics

As a consequence, the exterior differential d on C*{Z) can be split into a sum d = d + d of holomorphic and antiholomorphic differentials d = i ( d - {-l)p+qUdJ)

: Cp'q(Z) -> C p+1 ' 9 (2'),

3 = l(d+(-l)P+HJdJ)

: Cp'g{Z) -> C P ' 9+1 (Z)

such that

J o 9 = —i9o J,

Jo d = ido J,

d(i> = d~(jj,

d = d~$.

With the differentials 9 and 9, the graded commutative algebra C*(Z) becomes a bicomplex. Let us recall the following notions. • A smooth complex function / on a complex manifold (Z, J) is called holomorphic (resp. antiholomorphic) if df = 0 (resp. df = 0). Accordingly, an exterior (p, 0)-form (resp. a (0, (?)-forin) 0 on a complex manifold (Z, J) is called holomorphic (resp. antiholomorphic) if c?(/> = 0 (resp. dcf> = 0). • A TZ1'°-valued (resp. TZ1'°-valued) vector field i ) o n a complex manifold (Z, J) is said to be holomorphic (resp. antiholomorphic) if $J
*=H^-w)' a

_ 1(

d

, .

dzx = dRe zx + idlm zx,

d

(26 i3)

'-

\ dzx = dRe zx - idlm zx.

(2.6.14)

Chapter 2 Classical Hamiltonian Systems

179

The following relations hold: J(%) = - i % ,

J(dx) = idx, x

x

J(dz ) = idz , d»\dzv = 6£,

J{dzx) = -idzx, dp\dzv = 8%,

d»\dzv = djj\dzv = 0.

The tuples {d\}, {^} play the role of holonomic bases for holomorphic and antiholomorphic tangent bundles TZ1'0 and TZ0'1, while {dzx}, {dzx} are the dual bases for holomorphic and antiholomorphic cotangent bundles TZIQ and TZ$t\, respectively. Relative to these bases, the tangent-valued form J reads J = i(dzx dx - dzx ® % ) .

(2.6.15)

Accordingly, holomorphic and antiholomorphic differentials read d{(j))=dzxAdx,

d() = dzxAdx-.

(2.6.16)

In particular, a holomorphic (resp. antiholomorphic) vector field on Z takes the coordinate form $ = tixd\ (resp. i? = i?A%), where dx are holomorphic (resp. antiholomorphic) local functions on Z. Accordingly, a holomorphic (resp. antiholomorphic) exterior form is an element of C*'°(Z) (resp. C°'*(Z)) with holomorphic (resp. antiholomorphic) coefficients.

C. Hermitian manifolds A Riemannian metric g on an almost complex manifold (Z, J) is called Hermitian if it is J-invariant, i.e., g(v,v')=g(Jv,Jv'),

v,v'eTzZ,

z G Z.

(2.6.17)

Clearly, a Hermitian metric yields a Hermitian scalar product on each tangent space TZZ, z € Z. An almost complex (resp. complex) manifold, provided with a Hermitian metric, is called an almost Hermitian (resp. Hermitian) manifold. 2.6.5. Any almost complex manifold (Z, J) admits a Hermitian metric, i.e., it is an almost Hermitian manifold [250]. D THEOREM

Indeed, the existence of an almost complex structure on a manifold Z implies the corresponding contraction of the structure group GL(2m,]R) of the linear frame bundle LZ to the subgroup GL(m, C) (2.6.2) and, consequently, to the unitary group U(m). This means that the structure group

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of LZ is reduced to the orthogonal group O(2m) D U(m). In accordance with Theorem 10.10.5, this contraction defines a Riemannian metric g on Z. We have monomorphisms LgZ -» LJZ -» LZ of the corresponding reduced subbundles L9Z and L J Z . It follows that L9Z consists of all #-orthonormal and J-invariant frames in TZ. Hence, g is a Hermitian metric with respect to a given almost complex structure J on Z. Moreover, there is one-to-one correspondence between such metrics and the global sections of the quotient bundle LJZ/U{m)

-> Z.

Given an almost Hermitian manifold (Z,J,g), a Hermitian metric g defines the non-degenerate two-form (i.e., an almost symplectic form) Q(0,0')=g(JW),

V,#'eCTzZ,

zeZ,

(2.6.18)

on Z. It is J-invariant, i.e., 9,{v, v') = Sl{Jv, Jv'),

v, v' e TZZ,

ze Z,

(2.6.19)

and is called the fundamental form of the Hermitian metric g. An almost complex structure, a metric and an almost symplectic form, obeying the compatibility conditions (2.6.17) - (2.6.19), are three main "players" blending to form an almost Hermitian structure on a manifold Z. Any one of them is entirely determined by the remaining two. A linear connection on an almost Hermitian manifold (Z,J,g,Q) is called Hermitian if it preserves any two of the forms J, g, Cl and, consequently, the remaining one. It follows that Hermitian connections on an almost Hermitian manifold (Z, J, g, fi) differ from each other in a torsion, and they are classified by the form of their torsion tensor [163; 196]. 2.6.6. A Hermitian manifold admits a unique Hermitian connection whose torsion tensor is totally skew-symmetric [163]. It is called a KT-connection or a Bismut connection. • THEOREM

D. Kahler manifolds Given an almost Hermitian manifold (Z,J,g), a Hermitian metric g is called a Kahler metric if its fundamental form Cl (2.6.18) is closed, i.e., is

Chapter 2 Classical Hamiltonian Systems

181

a symplectic form. A closed fundamental form is called the Kdhler form. Accordingly, an almost Hermitian manifold (resp. a Hermitian manifold) with a closed fundamental form is said to be an almost Kahler manifold (resp. a Kahler manifold). In view of the forthcoming theorem, one can start with a symplectic form on a manifold in order to make it into an almost Kahler manifold [311; 415]. THEOREM 2.6.7. Any symplectic manifold (Z, Q) admits a compatible almost complex structure J and, thus, it is an almost Kahler manifold with • a Kahler metric [199]. At the same time, the existence of a Kahler structure imposes additional topological restrictions on a symplectic manifold (Z, Q) (see Proposition 2.6.10 below). 2.6.8. An almost Hermitian manifold (Z, J,g, Cl) is a Kahler manifold if and only if the Levi-Civita connection of the Hermitian metric g is a Hermitian connection [250]. • THEOREM

Given a Kahler manifold (Z,J,g,£l), the Kahler metric g is uniquely extended to the complex tangent bundle CTZ of Z so that: • g@,ti') = s(i9,tf') for all vectors ti,& G CTZZ, z £ Z (the Hermitian condition); • g(•&,•&) > 0 for all non-vanishing complex tangent vectors i? G CTZ; • g('&,'&') = 0 if #, •&' € CTZZ, z £ Z, are simultaneously holomorphic or antiholomorphic tangent vectors. Accordingly, the Kahler form Q is extended to CTZ so that • Q(d, i?') = fi(j?, i?') for all §, •&' G CTZZ, z G Z; • fi(tf,tf') = 0 if •&,•&' e CTZZ, z e Z, are simultaneously holomorphic or antiholomorphic; • dd = 0. Since the Kahler form Q, is closed, it takes the local form Q = iddF,

(2.6.20)

where F is a real function on Z. It is called the potential function for the Kahler structure on Z. Remark 2.6.2. If Z is a non-compact simply connected manifold, local Kahler potentials can be glued into a global one. If Z is not simply connected, a Kahler potential exists at least on an open subset U obtained

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from Z by deleting a (real) submanifold of lower (real) dimension [145]. Alternatively, one may pass to the universal covering manifold over Z. • Let the Kahler manifold (Z,J,g,Q,) be equipped with complex coordinates (zx), A = l , . . . , m . We will follow the notation (2.6.13) - (2.6.14). Let the complex tangent and cotangent bundles of Z be provided with the bases {d\,d\} and {dzx,dzx}, respectively. Relative to these bases, the Kahler metric g reads 9=

gapdzaVdz/3-

The Hermitian condition of g implies that (2.6.21)

9a0 =9*0 = 90oiSince the Kahler form fi = iga0-dza A dzp is closed, components ga^ of g also obey the equalities dx9a0 = dagX0,

(2-6.22)

9X9a0 = ^9ax-

In particular, the Kahler metric takes the local form (2.6.23)

9cT0 = da^F,

where F is the potential function in the expression (2.6.20). In view of the relations (2.6.21) - (2.6.22), the Levi-Civita connection of a Kahler metric g has the following non-vanishing components:

Its curvature tensor (10.6.67) reads %//3 = ^ V

= -9aVd^9^

+ Sf^dtffntdxgsj,

Then the Ricci tensor can be brought into the form %/j = % / / 3 = -g^dpd^g^ + g£Vg^d0g^g^ -^^(lndet^p)).

=

(2.6.24)

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Chapter 2 Classical Hamiltonian Systems

E. Cohomology of Kahler manifolds Let (Z, J, g, Cl) be a Kahler manifold of complex dimension m. The Hodge star operator with respect to the Kahler metric g and the corresponding Hodge inner product are extended to complex exterior forms on Z as follows: * : Cp'q{Z) -> Cm-p'm-g(Z),

** =

(-l)p+g,

(
(2.6.25)

where
are the adjoints of the holomorphic and antiholomorphic differentials d and d with respect to the Hodge inner product (2.6.25). The operators d* and d are nilpotent and satisfy the relations

a* o d * + o* o e* = o,

d o a * + a * o d = o,

d * o a + a o a * = o.

Let us construct the complex Laplacian a = dod* + d* od.

(2.6.26)

One can show that, in the case of a Kahler metric g, the Laplacian (2.6.26) commutes with the conjugation 4> —* , i-e-,

and • = ^A,

(2.6.27)

where A is the standard Laplacian. Thus, for any fixed number p £ N, we obtain the elliptic Dolbeault complex Cp>*(Z) = {Cp'"(Z),d,d\D}, O^UP

i ^ c P l 0 ( Z ) -^CP'\Z)

(2.6.28) - ^ • • • C P ' « ( Z ) - ! + . . . , (2.6.29)

where IAV is the complex space of holomorphic p-forms on Z. The corresponding complex of sheaves n

, 7/p

v/"p'°

^rP'1

v

rP'i

.pP'i+1

d

-

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is proved to be a fine resolution of the sheaf Uvz of holomorphic p-forms on Z [220]. Hence, by virtue of Theorem 1.7.6, the cohomology group Hq(Z;U^) of Z with coefficients in the sheaf Uvz is isomorphic to the g-th cohomology Hp>q(Z) group of the complex (2.6.29). Namely, HP'°(Z) = Up and Hp>q{Z), q > 0, is the complex space of 9-closed (p, g)-forms module the 9-exact ones. Applied to the Dolbeault complex (2.6.28), the Hodge theorem 10.12.3 implies that, with respect to the inner product (2.6.25), the space CPA{Z) °f (Pi
®~5*Cp'q+1(Z) ®B™(Z),

(2.6.30)

where BPi9(Z) C KerD is the space of harmonic (p,q)-forms. It follows that <j> € Bp>q(Z) if and only if d = d*<j> = 0 , and that Hp'q(Z) = Bp'q(Z). Hence, the cohomology groups Hp'q{Z), called the Dolbeault cohomology, are finite-dimensional. Then h™ := dim Hp'q{Z)

(2.6.31)

are said to be the Hodge numbers. In particular, hp'q — 0 if p > m or q > m. With the Hodge numbers (2.6.31), the index (10.12.15) of the Dolbeault complex (2.6.28) reads r(Cp'*) = ^ ( - 1 ) 9 ^ . 9 .

It is important that the Betti numbers, the Euler characteristic and the signature of a compact Kahler manifold are expressed into the Hodge numbers (see the formulae (2.6.34), (2.6.35) and Theorem 2.6.11 below). By means of integration Cm'm(Z)5<j>^

I 4>&
one can define the bilinear form

fizfa, 4>') = j A ', z

G Cp'q(Z),

cf>' 6 C m - " ' m - q ( Z ) ,

Chapter 2 Classical Hamiltonian Systems

185

which depends only on cohomology classes of complex smooth forms and leads to the Poincare duality h? —• <j> yields an anti-isomorphism from Bp'q(Z) to Bq'p(Z), and the Hodge numbers obey the relations (2.6.32)

hP,g = hg,P_

Furthermore, the equality (2.6.27) leads to the cohomology isomorphism F r (Z;C)=

Bp'q(Z).

^

(2.6.33)

p+q=r

Therefore, the r-th Betti number br of Z satisfies

br=

Yl

hP q

'-

(2.6.34)

p+q=r

It should be emphasized that the formulae (2.6.32) - (2.6.34) fail for nonKahler manifolds, in general. For instance, the Kahler form Q, is a harmonic (1, l)-form. Its exterior fc-product £lk, k — 2 , . . . , m, is a non-zero harmonic (k, fc)-form. Hence, the spaces Bk
Betti numbers.

2.6.9. A compact Kahler manifold has non-vanishing even •

As was mentioned above, this property is also satisfied for compact symplectic manifolds. The following assertion provides the so called Thurston's example of symplectic manifolds with no Kahler structure [71; 415] (see, e.g., [112] for other examples). PROPOSITION

fold are even.

2.6.10. The odd Betti numbers of a compact Kahler mani•

As an immediate consequence of the formula (2.6.34), one obtains that

X = Y,(-l)p+qhp'q

(2.6.35)

p. q

is the Euler characteristic of a compact Kahler manifold Z. The following result requires the more sophisticated analysis [220].

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Geometric and Algebraic Topological Methods in Quantum Mechanics

THEOREM

2.6.11. The signature (the index) of a compact Kahler manifold

is

p,q

• In the proof of this Theorem, the following important construction is utilized. Given the Kahler form Cl, let us introduce the operators (2.6.36)

L+ : CP'9{Z) BAne Cp+1'q+1(Z), L~ : CP'"(Z) 9 i - *(O A *<j>) G C ^ 1 ' 9 " 1 ^ ) ,

(2.6.37) (2.6.38)

L = [L+,L-}.

If Z is compact, the operators L+ and L~ are mutually adjoint relative to the Hodge inner product (2.6.25). One can also show that L (2.6.38) is the scalar operator L(j> = {V + q-n)(j>,

cPeC^iZ),

[154]. It follows at once that the operators (2.6.36) - (2.6.38) obey the commutation relations [L,L+] = 2L+,

[L,L~] = -2L-

and generate the Lie algebra s/(2,C). The relations [L+,d]=0, [L-,d*} = 0,

[L+,d}=0,

[L+,d*]=id,

[L-,T] = 0,

[L-,d] = id\

[L+,d*] = -id, [L~,d\ = -i&¥

also hold. It follows that the operators (2.6.36) - (2.6.38) commute with the Laplacian A. Therefore, they yield the homomorphisms

L+ L~

:Bp'q(Z)-*Bp+i-q+1(Z), :Bp'q{Z)^Bp-l'q-l{Z)

of spaces of harmonic forms and, consequently, of the Dolbeault cohomology groups Hp'*(Z) = B'«(Z). An important class of harmonic (1, l)-forms on a compact Kahler manifold (Z, J, g, Q) is related to the Ricci tensor of different Kahler metrics on

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Chapter 2 Classical Hamiltonian Systems

Z. Given the Ricci tensor (2.6.24) of a Kahler metric g, one can introduce the harmonic (1, l)-form R = YR\l3dz!3

Ad X

z

= - j-^(lndet(5 M T7)),

(2.6.39)

called the Ricci form. In accordance with well-known theorem [100], the cohomology class of this form depends only on the complex structure J on Z and equals the first Chern class c\(Z) of Z. Moreover, this result has been improved as follows [80; 449]. 2.6.12. Given a Kahler manifold {Z, J, g, 0), let ^ be a harmonic (1, l)-form which represents the first Chern class c\(Z) of (Z, J). Then one can find a J-compatible Kahler metric g' on (Z, J) such that is its Ricci form (2.6.39). Furthermore, there exists a unique Kahler metric g' whose O Kahler form has the same de Rham cohomology class as that of £1. THEOREM

For instance, any Kahler manifold (Z, J, g, ft) of the vanishing first Chern class (called a Calabi-Yau manifold) admits a Ricci-flat Kahler metric g' whose Kahler form Q.' belongs to the same de Rham cohomology class as that of O [84; 449]. Usually, most interest centers on Calabi-Yau manifolds of three complex dimension (see [11; 28]).

F. Hyper-Kahler manifolds Three almost complex structures (resp. complex structures) Ji, Ji and J3 on a manifold Z form an almost hypercomplex structure (resp. a hypercomplex structure) if J\ = J2J3 = —J%J2This relation implies the relations Ji = eijkJjJk,

(2.6.40)

i,j,k = l , 2 , 3 ,

where e is the totally skew-symmetric Levi-Civita tensor with the component ei23 = 1. Hence, there is the two-sphere of complex structures

{a\Ji + a-zJi + azJz : aj + 02 + ag = 1}. We will use the notation dkcf> : = (-1)1*1 Jkd(Jk), dk = -(d-idk),

dk = -(d + idk),

k =

1,2,3.

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Geometric and Algebraic Topological Methods in Quantum Mechanics

Theorem 2.6.3 on the existence of a torsionless connection on a complex manifold is extended to a hypercomplex manifold as follows [19; 196]. 2.6.13. An almost hypercomplex structure (Jk) on a manifold Z is a hypercomplex structure if and only if there exists a torsionless affine connection which preserves all Jk- It is called the Obata connection. • THEOREM

When g is a Riemannian metric on the hypercomplex manifold (Z,Jk) such that it is a Hermitian with respect to each complex structure Jk, the tuple (Z,Jk,g) is called a hyper-Hermitian manifold. Thus, a hyperHermitian manifold Z is a triple of Hermitian structures (Jk,g) on Z such that the complex structures Jk satisfy the relation (2.6.40). A linear connection on a hyper-Hermitian manifold (Z, Jk,g) is called a hyper-Hermitian connection if it is Hermitian with respect to all the complex structures Jk, i.e., it is a metric connection for g and preserves Jk, k = 1,2,3. A hyper-Hermitian connection is said to be hyper-Kdhler if its torsion tensor is totally skew-symmetric. Clearly, a hyper-Kahler connection is a KT-connection with respect to all the Hermitian structures (Jfc, <;). Therefore, it is also called the HKT-connection. THEOREM 2.6.14. [196]. A hyper-Hermitian manifold (Z,Jk,g) admits a hyper-Kahler connection if an only if the fundamental forms fife, associated to Hermitian structures (Jfc,g), fulfils the equivalent conditions: (i) diOi = d.2^2 = d3rt3, (ii) di(V.2 + in3) = 0 , (iii) 9 i ( f i 2 - t f i 3 ) = 0 , (iv) ditlj = —26ijT — eijkdQk, where T is a three-form denned by the skew-symmetric torsion tensor. If a hyper-Kahler connection exists, it is unique. •

If a hyper-Hermitian manifold admits a hyper-Kahler connection, it is said to be an HKT-manifold. It is readily justified from the equivalent conditions in Theorem 2.6.14 that a hyper-Kahler connection is symmetric (i.e., it coincides with the Levi-Civita connection of the metric g and the Obata connection in Theorem 2.6.13) if and only if fife, k = 1,2,3, are Kahler forms. Then (Z, Jk,g) is a hyper-Kdhler manifold. Let (Z,Jk,g,£lk) be a compact hyper-Kahler manifold. For each k = 1,2,3, one can introduce the operators L^ and Lj~ by the formulae (2.6.36) - (2.6.37), but with Qk playing the role of fi. Let us put

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Chapter 2 Classical Hamiltonian Systems

Then the following relations hold [154; 432]: [L+ ,LJ}= 2eijkKk + SijL, [Ku Kj] = eijkKk,

[L, L+) = 2L+,

[Kit L+] = eijkL+,

[L, L~] = -2Lr,

[Kit Lj] =

eijkL^,

with all the other brackets equal zero. These are the commutation relations of the Lie algebra so(4,1). Indeed, let Ipq, p, q = 1 , . . . , 5, be the standard basis for the Lie algebra so(5). One can reproduce the above commutation relations by letting

L'l = -(/i5 + Ha),

L~ - - ( i j 5 - ila),

L = 2iJ45,

Kk — tujhj.

We refer the reader to [383] for cohomology of hyper-Kahler manifolds.

2.7

Appendix. Poisson manifolds and groupoids

In this Section, we describe the category Poisson of integrable Poisson manifolds [85; 268; 426; 444], which will be quantized in Section 8.6. We start with the notion of a complete Poisson morphism. Given Poisson manifolds (Pi, W\) and (P21W2), let g : P\ —> Pi be a Poisson morphism. Then, given a function / 6 C°°{P-i) and its Hamiltonian vector field $f (2.1.13), we have the Hamiltonian vector field -de*f on P\ whose trajectories project onto those of flf. A Poisson morphism g is called complete if the Hamiltonian vector field i9 e -/ is complete whenever the Hamiltonian vector field i?/ is complete. The injection of every symplectic leaf in a Poisson Example 2.7.1. manifold is a complete Poisson morphism. More generally, the range of a complete Poisson morphism is a union of symplectic leaves. D Let (P, w) be a Poisson manifold. The symbol P~ further stands for the Poisson manifold (P,—w). Given a Poisson manifold P, a left (resp. right) symplectic P-module is defined as a symplectic manifold S together with a complete Poisson morphism g : S —» P (resp. S —> P~). Just as in algebra (see Section 10.4), one can define the tensor product of right and left symplectic P-modules g : 5 —> P and g' : S' —> P~. Namely, let us consider the fibre product SSx S1 = {(x,y)eSxS'

: g(x) = g'(y)}

(2.7.1)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

(see the notation (10.4.6)), which is the inverse image of the diagonal of P x P~ under the Poisson morphism {g,g') -.SxS'^PxP-. If either g or g' is a surjective submersion, the fibre product (2.7.1) is a coisotropic smooth submanifold of S x S' and, accordingly, a presymplectic manifold (see Example 2.1.5). Then one can define the tensor product S*S' of right and left symplectic P-modules as the quotient of the presymplectic manifold (2.7.1) by its characteristic foliation. In accordance with Proposition 2.1.5, this tensor product is a symplectic manifold if the characteristic foliation of the presymplectic manifold (2.7.1) is simple. Given Poisson manifolds Pi and P 2 , a symplectic (Pi, P2)-bimodule is denned as a symplectic manifold S and a pair of maps (2.7.2)

Pl^S^P2

which make S both into a left Pi-module and a right P2-module, and satisfy the commutation condition {elh, £2/2} = 0,

/1 e C^iPi),

h e C°°(P2),

where {.,.} is the Poisson bracket on S. Two symplectic (Pi, P2)-bimodules Pl^S^P2,

P1^S'^P2

are said to be isomorphic if there exists a symplectomorphism (j> : S —> 5' such that gi = g\ o (f>. Remark 2.7.2. Let us note that, in [27l], symplectic bimodules, without • assumption of completeness, are called dual pairs. Given symplectic bimodules Pl

^

S

^P

2

,

P2 ^ S ' - ^ P 3 ,

(2.7.3)

one can construct their tensor product S * S' over P 2 . As was mentioned above, it however need not be a symplectic (Pi — P3)-bimodule, unless some extra "regularity" conditions hold. In order to formulate these conditions, let us relate Poisson manifolds with Lie groupoids. A Lie groupoid (25 is called a symplectic groupoid if it is provided with a symplectic formfisuch that the multiplication graph M = {(x,y,xy)

: (x,y) G 0 2 } C <25 x (5 x <2T

(2.7.4)

Chapter 2 Classical Hamiltonian Systems

191

is a Lagrangian submanifold of the product 0 x 0 x <5~ provided with the symplectic form fi © 0, 9 fi. Example 2.7.3. The cotangent bundle T*X of a smooth manifold X endowed with the canonical symplectic form (2.1.3) exemplifies a symplectic groupoid with respect to the fibrewise addition. More generally [426], the cotangent bundle T*(0) of any Lie groupoid © is a symplectic groupoid over the dual V*<8 of the Lie algebroid V<& —> 0° associated to 0 (see Section 10.3). D A symplectic groupoid (0,fi) possesses the following properties. • Its unit space 0° is a Lagrangian submanifold of 0. • The inversion map x —> x~l, x & <&, is the anti-symplectic involution • The fibres 0 U and <5U, u 6 0°, are symplectic orthogonal of one another. • At each point x € 0, we have Ker(Tlx) = {0r.f(x) Ker {Trx) = {^f{x)

: / G C°°(0 0 )} : / e C°°((S0)}.

• The unit space 0° admits a unique Poisson structure such that r : (5 -> (5°,

I : 0 -> <5°~

are Poisson maps. In view of the last property, a Poisson manifold P is called integrable if there exists a symplectic groupoid (0, fi) (called the integration of P) over

0° = P.

THEOREM

2.7.1. If 0 is a symplectic groupoid over a Poisson manifold P,

then r : 0 -> 0° = P is a complete submersion. Conversely, if there exists a symplectic P-module S —> P which is a submersion, then P is an integrable Poisson manifold. Moreover, there exists a symplectic groupoid, integrating P, whose i-fibres are connected and simply-connected, and this groupoid, denoted 0(P), is unique up to isomorphisms. D

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Geometric and Algebraic Topological Methods in Quantum Mechanics

Let © be a symplectic groupoid acting on a symplectic manifold S on the left with a moment g : S —> ©° (see Section 10.4). Its action © x S-+ S 00

is called symplectic if the graph of this action is a Lagrangian submanifold of <8xSxS~. Let P be an integrable Poisson manifold and © a symplectic groupoid over P. Given a left symplectic action of © on a symplectic manifold S, the moment g : S —» P is a complete Poisson morphism defining a symplectic module over P. Conversely, a symplectic P-module g : S —* P implies a left symplectic action of the ^-connected and Z-simply connected symplectic groupoid ©(P) in Theorem 2.7.1 on S. Consequently, there is one-to-one correspondence between P-modules and left symplectic actions of ©(P). Accordingly, given integrable Poisson manifolds Pi and P2 and a symplectic bimodule (2.7.2), we have a left symplectic action of the symplectic groupoid 6(Pi) and a right symplectic action of the symplectic groupoid ©(P2) on S. Moreover, these actions mutually commute and, consequently, 5 is a symplectic (<5(Pi) — 0(P2))-bibundle (see Section 10.4). A symplectic bimodule (2.7.2) is called regular if the associated symplectic (©(Pi) - 0(P2))-bibundle is left principal. 2.7.2. A symplectic bimodule (2.7.2) is regular if and only if £1 is a submersion, p2 is a. surjective submersion with connected and simplyconnected fibres, and the gi- and £>2-fibres are symplectic orthogonal of one another. •

PROPOSITION

The following theorem states that the tensor product of regular symplectic bimodules is well denned [268]. 2.7.3. The tensor product S*S' of regular symplectic bimodules (2.7.3) is a regular symplectic (Pi — P3)-bimodule which coincides with the tensor product S*S' (10.4.9) of S and S' as principal symplectic (©(Pi) ©(P2))- and (6(P 2 ) - ©(P3))-bibundles. • THEOREM

In particular, it follows that the tensor product of regular symplectic bimodules is associative up to isomorphisms, while the regular symplectic bimodule P^<5(P) -» P

Chapter 2 Classical Hamiltonian Systems

193

associated to the (0(P)-(9(P))-bibundle S = <5(P) plays the role of a twosided unit. As a consequence, the following category of Poisson manifolds is denned. 2.7.4. The category Poisson has integrable Poisson manifolds as objects and classes of isomorphic regular symplectic bimodules as morphisms. •

DEFINITION

Accordingly, the notion of the Morita equivalence of groupoids leads to the following definition of the Morita equivalence of integrable symplectic manifolds in the sense of [444]. Namely, integrable symplectic manifolds Pi and Pi are Morita equivalent if and only if they are isomorphic object in the category Poisson, i.e., their integrations ©(Pi) and <3(P2) in Theorem 2.7.1 are Morita equivalent symplectic groupoids.

Chapter 3

Algebraic quantization

Algebraic quantum theory follows the hypothesis that a quantum system can be characterized by Hermitian elements of a C*-algebra A and positive forms f on A treated as mean values of quantum observables (see Remark 3.1.1 below). In accordance with the Gelfand-Naimark-Segal (henceforth GNS) construction, any positive form on a C*-algebra A determines its cyclic representation by bounded operators in a Hilbert space. This Chapter addresses some modifications of this GNS construction. 3.1

GNS construction I. C*-algebras of quantum systems

Subsections: A. Involutive algebras, 195; B. Hilbert spaces, 198; C. Countably Hilbert spaces and nuclear spaces, 201; D. Operators in Hilbert spaces, 202; E. Representations of involutive algebras, 204; F. The GNS representation, 206. We start with a brief exposition of the conventional GNS representation of C*-algebras [129] A. Involutive algebras A complex algebra A is called involutive, if it is provided with an involution * such that {a*y=a,

{a + Xby =a*+Jb*,

(ab)*=b*a*,

a,b e A,

XeC.

Let us recall the standard terminology. An element a £ A is normal if aa* = a*a, and it is Hermitian or self-adjoint (see Section 3.6) if a* = a. If 195

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Geometric and Algebraic Topological Methods in Quantum Mechanics

A is a unital algebra, a normal element such that aa* = a*a = 1

is called unitary. Remark 3.1.1. A product of two Hermitian elements need not be Hermitian, unless they mutually commute. As a consequence, Hermitian elements fail to make up an associative algebra. Therefore, one considers Jordan algebras in order to describe quantum observables [3; 144]. Hermitian elements of an involutive algebra constitute a Jordan algebra with respect to the symmetrized product a- a' = -{aa' + a'a).

• An involutive algebra A is called a normed algebra (resp. a Banach algebra) if it is a normed (resp. complete normed) vector space whose norm ||.|| obeys the multiplicative conditions ||ab|| < ||a||||6||,

K | | = ||a||,

a,beA.

An involutive Banach algebra A is called a C* -algebra if \\a\\2 = \\a*a\\

for all a € A. If A is a unital C"*-algebra, then ||1|| = 1. A C*-algebra is provided with the normed topology, i.e., it is a topological involutive algebra. Let X be a closed two-sided ideal of a C*-algebra A. Then X is selfadjoint, i.e., X* = X. Endowed with the quotient norm, the quotient A/X is a C*-algebra. Remark 3.1.2. It should be emphasized that by a morphism of normed algebras is customarily meant a morphism of the underlying involutive algebras, without any condition on the norms and continuity. At the same time, an isomorphism of normed algebras means always an isometric morphism. Any morphism of C*-algebras is automatically continuous due to the property ||0(a)|| < ||a||,

aeA.

(3.1.1)

•

197

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Any involutive algebra A can be extended to a unital algebra A = C®A by the adjunction of the identity 1 to A (see Remark 1.1.2). The unital extension of A is also an involutive algebra with respect to the operation (Al + a)* = (Al + a*),

AeC,

a&A.

If A is a normed involutive algebra, a norm on A is extended to A, but not uniquely. If A is a C*-algebra, a norm on A is uniquely prolonged to the norm ||Al + a|| := sup ||Aa'+aa'|| l|a'll• 0 and \\auL — a|| —> 0 for every a £ A. It should be noted that the existence of an approximate identity is an essential condition for many results. In particular, the GNS representation is relevant to involutive Banach algebras with an approximate identity. However, there is no loss of generality if we restrict our study of the GNS representation to C*-algebras because any involutive Banach algebra A with an approximate identity defines the enveloping C*-algebra A* such that there is one-to-one correspondence between the representations of A • and those of A1 (see Remark 3.2.1 below). Remark 3.1.4. Unless otherwise stated, by the tensor product A® A' of C*-algebras A and A' is meant their minimal (or spatial tensor) product. This is the C*-algebra denned as the completion of the tensor product of involutive algebras A and A' with respect to the minimal norm which obeys the condition ||aa'|| = ||a|| ||a'||.

a £ A,

a' G A.

For instance, if A and A' are operator algebras in Hilbert spaces E and E', this norm is exactly the operator norm (3.1.14) of operators in the tensor product E <8> E' of Hilbert spaces E and E'. In general, there are several ways of completing the algebraic tensor product of C*-algebras in order to • obtain a C*-algebra [382].

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B. Hilbert spaces An important example of C*-algebras is the algebra B(E) of bounded (and, equivalently, continuous) operators in a Hilbert space E. Every closed involutive subalgebra of B(E) is a C*-algebra and, conversely, every C*algebra is isomorphic to a C*-algebra of this type (see Theorem 3.1.2 below). Let us recall the basic facts on pre-Hilbert and Hilbert spaces [56]. A Hermitian form on a complex vector space E is defined as a sesquilinear form (.|.) such that (e\e') =J^\e),

(Ae|e'> = <e|Ae') = X{e\e'),

e,e' G E,

A e C.

Remark 3.1.5. There exists another convention where (e|Ae') = A(e|e') (see Section 8.1). • A Hermitian form (.|.) is said to be positive if (e|e) > 0 for all e € E. Throughout the book, all Hermitian forms are assumed to be positive. A Hermitian form is called non-degenerate if the equality (e|e) = 0 implies e = 0. A complex vector space endowed with a (positive) Hermitian form is called a pre-Hilbert space. Morphisms of pre-Hilbert spaces, by definition, are isometric. A Hermitian form provides E with the topology determined by the seminorm ||e|| = {e\e)ll\

(3.1.2)

Hence, a pre-Hilbert space is Hausdorff if and only if the Hermitian form (.|.) is non-degenerate, i.e., the seminorm (3.1.2) is a norm. In this case, the Hermitian form (.|.) is called a scalar product. A family {e^}/ of elements of a pre-Hilbert space E is called orthonormal if its members are mutually orthogonal and ||ej|| = 1 for all i £ I. Given an element e S E, there exists at most a countable set of elements ej of an orthonormal family such that (e|ej) ^ 0 and

E^}

2

< Ikll2.

A family {e^}/ is called total if it spans a dense subset of E or, equivalently, if the condition (e|ej) = 0 for all i £ I implies e = 0. A total orthonormal family in a Hausdorff pre-Hilbert space is called a basis for E. Given a basis

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199

{ei}i, any element e £ E admits the decomposition

e=5>| e i > e i , i€l

||e|| 2 =£|<e| ei >| 2 . iel

A basis for a pre-Hilbert space need not exist. 3.1.1. Every Hausdorff pre-Hilbert space, satisfying the first axiom of countability (e.g., if it is second-countable), has a countable orthonormal basis. D PROPOSITION

Remark 3.1.6. The notion of a basis for a pre-Hilbert space differs from that of an algebraic basis for a vector space. • A Hilbert space is denned as a complete Hausdorff pre-Hilbert space. Any Hausdorff pre-Hilbert space can be completed to a Hilbert space so that its basis, if any, is also a basis for its completion. Every Hilbert space has a basis, and any orthonormal family in a Hilbert space can be extended to its basis. All bases for a Hilbert space have the same cardinal number, called the Hilbert dimension. Moreover, given two bases for a Hilbert space, there is its isomorphism sending these bases to each other. A Hilbert space has a countable basis if and only if it is separable. Then it is called a separable Hilbert space. A separable Hilbert space is second-countable. Remark 3.1.7. Unless otherwise stated, by a Hilbert space is meant a complex Hilbert space. A complex Hilbert space (E, {.].)) seen as a real vector space ER is provided with a real scalar product (e,e') := \{(e\e') + (e'|e» = Re <e|e'),

(3.1.3)

which makes EK into a real Hilbert space. It is also a real Banach space. Conversely, the complexification E = C® V of a real Hilbert space (V, (.,.)) is a complex Hilbert space with respect to the Hermitian form (e^ie^+ie'z)

:= (e1;e[) + i{(e2,e[) - {eue'2)) + (e2,e'2).

(3.1.4) D

The following are the standard constructions of new Hilbert spaces from old ones.

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• Let (EL, (.|.)BO be a set of Hilbert spaces and Yl EL denote the direct sum of vector spaces EL. For any two elements e = (el) and e' = (e") of ^ E ' , the sum

(e|e')e := £ < e V V

C3-1-5)

is finite, and defines a non-degenerate Hermitian form on ^2,El. The completion ®EL of X3 El with respect to this form is a Hilbert space, called the Hilbert sum of {EL}. This is a subspace of the Cartesian product Yl EL which consists of the elements e = (eL) such that

^ H e 1 s ' H is defined as the completion of the tensor product of vector spaces E and H with respect to the scalar product

K M ® = 5>ile2>is Wf>*. i,p

ioi = ^ e i ® f t i ,

u>2 = ^ e f ®/if,

e[,e$ e E,

h\,h% G H.

Let {ej} and {/ij} be bases for E and H, respectively. Then {e^ ® hj} is a basis for E ® H. • Let £" be the topological dual of a Hilbert space E. Then the assignment e w e ( e ' ) := (e'|e),

e,e'e E,

(3.1.6)

defines an antilinear bijection of I? onto JB', i.e., Ae = Ae. The dual E' of a Hilbert space is a Hilbert space provided with the scalar product (e\e')' := (e'\e)

(3.1.7)

such that the morphism (3.1.6) is isometric. The E' is called the dual Hilbert space, and is usually denoted by E. A Hilbert space E and its dual E' seen as real Hilbert and Banach spaces are isomorphic to each other.

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Chapter 3 Algebraic Quantization

C. Countably Hilbert spaces and nuclear spaces Physical applications of Hilbert spaces are limited by the fact that the dual of a Hilbert space E is anti-isomorphic to E. The construction of a rigged Hilbert space describes the dual pairs (E,E') where E' is larger than E [164]. For instance, let us mention the space of smooth functions of rapid decrease and its dual space of tempered distributions (generalized functions) [4l]. Let a complex vector space E have a countable set of non-degenerate Hermitian forms (.|.)fe, k G N + , such that <e|e)i<-"<(e|e)fc<"for all e £ E. The family of norms ||.|U = <.|.)i/2,

(3.1.8)

k€N+,

yields a Hausdorff topology on E (see Section 10.5A). The space E is called a countably Hilbert space if it is complete with respect to this topology [164]. For instance, every Hilbert space is a countably Hilbert space where all Hermitian forms (.|.)fc coincide. Let Ek denote the completion of £ with respect to the norm ||.||jt (3.1.8). There is the chain of injections (3.1.9)

E1DE2D---EkD--together with the homeomorphism E = nEk-

The dual spaces form the

increasing chain E[ cE'2C---cE'kC---

,

(3.1.10)

and E' = UE'h. k

fe

The dual E' of E can be provided with the weak* and strong topologies. One can show that a countably Hilbert space is reflexive. Given a countably Hilbert space E and m
EnDEBet-^eeEcEm to the continuous map of En onto a dense subset of Em. A countably Hilbert space E is called a nuclear space if, for any m, there exists n such

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that T™ is a nuclear map, i.e., i l

where: (i) {e n} and {e^} are bases for the Hilbert spaces En and Em, respectively, (ii) A, > 0, (iii) the series Yl^i converges [164]. An important property of nuclear spaces is that they are perfect, i.e., every bounded closed set in a nuclear space is compact. It follows immediately that a Banach (and Hilbert) space is not nuclear, unless it is finite-dimensional. Since a nuclear space is perfect, it is separable, and the weak* and strong topologies (and, consequently, all topologies of uniform convergence) on a nuclear space E and its dual E' coincide. Let E be a nuclear space, provided with still another non-degenerate Hermitian form (.|.) which is separately continuous, i.e., continuous with respect to each argument. It follows that there exist numbers M and m such that (e|e) < M||e|| m ,

e G E.

(3.1.11)

Let E denote the completion of E with respect to this form. There are the injections (3.1.12)

EcEcE',

where E is a dense subset of E and E is a dense subset of E', equipped with the weak* topology. The triple (3.1.12) is called the rigged Hilbert space. Furthermore, bearing in mind the chain of Hilbert spaces (3.1.9) and that of their duals (3.1.10), one can convert the triple (3.1.12) into the chain of spaces (3.1.13)

Ec---cEkC---E1dEcE'1c---(ZE'k(Z---cE'.

Real countably Hilbert spaces, nuclear spaces and rigged Hilbert spaces are similarly described.

D. Operators in Hilbert spaces Unless otherwise stated (see Section 3.6), we deal with bounded operators a £ B(E) in a Hilbert space E. They are provided with the operator norm \\a\\ :=

sup ||ae|| B ,

l|e||B = l

a € B(E).

(3.1.14)

Chapter 3 Algebraic Quantization

203

This norm makes the involutive algebra B(E) of bounded operators in a Hilbert space E into a C*-algebra. The corresponding topology on B{E) is called the normed operator topology. One also provides B(E) with the strong and weak operator topologies, determined by the families of seminorms {pe(a) = \\ae\\, e £ E}, {pe,e.(a) = |(oe|e')|,

e.e'efi),

respectively. The normed operator topology is finer than the strong one which, in turn, is finer than the weak operator topology. The strong and weak operator topologies on the subgroup U(E) C B(E) of unitary operators coincide with each other. Remark 3.1.8. It should be emphasized that B(E) fails to be a topological algebra with respect to strong and weak operator topologies. Nevertheless, the involution in B(E) is also continuous with respect to the weak operator topology, while the operations B{E) 9 a i-> aa' G B(E), B(E) B o H a ' a e B{E), where a' is a fixed element of B(E), are continuous with respect to all the above mentioned operator topologies. D Remark 3.1.9. Let N be a subset of B(E). The commutant N' of N is the set of elements of B(E) which commute with all elements of N. It is a subalgebra of B{E). Let N" = (N1)' denote the bicommutant. Clearly, N C N". An involutive subalgebra B of B(E) is called a von Neumann algebra if B = B". This property holds if and only if B is strongly (or, equivalently, weakly) closed in B(E) [129]. For instance, B(E) is a von Neumann algebra. Since a strongly (weakly) closed subalgebra of B{E) is also closed with respect to the normed operator topology on B(E), any von Neumann algebra is a C*-algebra. D Remark 3.1.10. An operator in a Hilbert space E is called completely continuous if it is compact, i.e., it sends any bounded set into a set whose closure is compact. An operator a € B(E) is completely continuous if and

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Geometric and Algebraic Topological Methods in Quantum Mechanics

only if it can be represented by the series oo

a{e)=J2Xk(e\ek)ek,

(3.1.15)

k=i

where e^ are elements of a basis for E and Afe are positive numbers which tend to zero as k —> oo. For instance, every degenerate operator (i.e., an operator of finite rank which sends E onto its finite-dimensional subspace) is completely continuous. Moreover, the set T(E) of completely continuous operators in E is the completion of the set of degenerate operators with respect to the operator norm (3.1.14). Every completely continuous operator can be written as a = UT, where U is a unitary operator and T is a positive completely continuous operator, i.e., (Te\e) > 0 for all e £ E. A completely continuous operator a is called the Hilbert-Schmidt operator if the series

IMIHS : =

z2^l' k

called the Hilbert-Schmidt norm, converges. Hilbert-Schmidt operators make up an involutive Banach algebra with respect to this norm, and it is a two-sided ideal of the algebra B(E). A completely continuous operator a in a Hilbert space E is called a nuclear operator if the series

IMITY

:= 5Z A f e ' k

called the trace norm, converges. Nuclear operators make up an involutive Banach algebra with respect to this norm, and it is a two-sided ideal of the algebra B(E). Any nuclear operator is a Hilbert-Schmidt one. Moreover, the product of arbitrary two Hilbert-Schmidt operators is a nuclear operator, and every nuclear operator is of this type. D

E. Representations of involutive algebras In this Section, we restrict our consideration to representations of involutive algebras by bounded operators in Hilbert spaces [129; 351]. It is a morphism IT of an involutive algebra A to the algebra B(E) of bounded operators in a Hilbert space E, called the carrier space of n. Representations throughout are assumed to be non-degenerate, i.e., there is no element e / 0

Chapter 3 Algebraic Quantization

205

of E such that Ae = 0 or, equivalently, AE is dense in E. A representation 7r of an involutive algebra A is uniquely prolonged to a representation n of the unital extension A of A. THEOREM

tation.

3.1.2. If A is a C*-algebra, there exists its isomorphic represenD

Two representations TTI and TT2 of an involutive algebra A in Hilbert spaces Ei and E E% such that ^2(0-)

= 7O7Ti(a) 0 7 " ' ,

a e A

Let {TT1} be a family of representations of an involutive algebra A in Hilbert spaces El. If the set of numbers ||vri'C«.) II is bounded for each a € A, one can construct the continuous linear operator n(a) in the Hilbert sum (BE1 which induces 7r'(a) in each EL. For instance, this is the case of a C*-algebra A due to the property (3.1.1). Then n is a representation of A in ®El, called the Hilbert sum of representations nx. Given a representation •n of an involutive algebra A in a Hilbert space E, an element 0 € E is said to be a cyclic vector for n if the closure of ir(A)9 is equal to E. Accordingly, TT is called a cyclic representation. 3.1.3. Every representation of an involutive algebra A is a Hilbert sum of cyclic representations. •

THEOREM

A representation TV of an involutive algebra A in a Hilbert space E is called topologically irreducible if the following equivalent conditions hold: • the only closed subspaces of E invariant under n(A) are 0 and E; • the commutant of tr(A) in B{E) is the set of scalar operators; • every non-zero element of E is a cyclic vector for TT. Let us recall that irreducibility of n in the algebraic sense means that the only subspaces of E invariant under ir(A) are 0 and E. If A is a C*-algebra, the notions of topologically and algebraically irreducible representations are equivalent. Therefore, we will further speak on irreducible representations of a C*-algebra without the above mentioned qualification. An algebraically irreducible representation n of an involutive algebra A is characterized by its kernel Ker -K C A. This is a two-sided ideal, called primitive. The assignment ASJTM

Ker?r € Prim(A)

(3.1.16)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

defines the canonical surjection of the set A of the equivalence classes of algebraically irreducible representations of an involutive algebra A onto the set Prim(>l) of primitive ideals of A. It follows that algebraically irreducible representations with different kernels are necessarily inequivalent. The set Prim(^4) is equipped with the so called Jacobson topology (see [129] for detail). This topology is not Hausdorff, but obeys the Frechet axiom, i.e., for any two distinct points of Prim(>l), there is a neighborhood of one of the points which does not contain the other. Then the set A is endowed with the coarsest topology such that the surjection (3.1.16) is continuous. It is called the spectrum of an involutive algebra A. PROPOSITION 3.1.4. If the spectrum A satisfies the Frechet axiom (e.g., A is Hausdorff), the map A —> Pnm(A) is a homeomorphism, i.e., algebraically irreducible representations with the same kernel are equivalent. D

3.1.5. If an involutive algebra A is unital, Prim(A) and A are quasi-compact , i.e., they satisfy the Borel-Lebesgue axiom, but need not be Hausdorff. O PROPOSITION

PROPOSITION 3.1.6. The spectrum A of a C"*-algebra A is a locally quasicompact space. •

If A is a C*-algebra, the topology of its spectrum A can be described as follows. Let {at} be a dense set of elements of A and Zt the set of n € A such that ||7r(at)|| > 1. Then ZL constitute a base for the topology on A. Example 3.1.11. A C*-algebra is said to be elementary if it is isomorphic to the algebra T(E) C B(E) of compact operators in some Hilbert space E. Every non-trivial irreducible representation of an elementary C* algebra A = T(E) is equivalent to its isomorphic representation by compact operators in E [129]. Hence, the spectrum of an elementary algebra is a singleton set. • F. The GNS representation Let / be a complex form on an involutive algebra A. It is called positive if f(a*a) > 0 for all a £ A. Given a positive form / , the Hermitian form (a\b):=f{b*a),

a,b G A,

(3.1.17)

Chapter 3 Algebraic Quantization

207

makes A into a pre-Hilbert space. In particular, the relation \f(b*a)\2
a,b G A,

(3.1.18)

holds. If A is a normed involutive algebra, positive continuous forms on A are provided with the norm

Il/H = sup |/(a)|, ||a|| = l

aeA.

One says that / is a state of A if ||/|| = 1. Positive forms on a C*-algebra are continuous. Conversely, a continuous form / on an unital C*-algebra is positive if and only if / ( I ) = ||/||. In particular, it is a state if and only if /(1) = 1. For instance, let A be an involutive algebra, n its representation in a Hilbert space E, and 0 an element of E. Then the map u8 • a -> (7r(a)0|0>

(3.1.19)

is a positive form on A. It is called the vector form determined by n and 9. This vector form is a state if the vector 0 is normalized. Let w$1 and uj02 be two vector forms on A determined by representations TTI in E\ and TT2 in £'2. If u>0j = ui02, there exists a unique isomorphism of E\ to Ei which sends TK\ to 1x2 and 9\ G E\ to 02 € £2The following theorem states that, conversely, any positive form on a C*-algebra equals a vector form determined by some representation of A called the GNS representation [129]. THEOREM 3.1.7. Let / be a positive form on a C*-algebra A. It is extended to a unique positive form J o n the unital extension A of A such that / ( I ) = ||/||. Let Nf be a left ideal of A consisting of those elements aeA such that f{a*a) = 0. The quotient A/Nf is a Hausdorff pre-Hilbert space with respect to the Hermitian form obtained from f(b*a) (3.1.17) by passage to the quotient. We abbreviate with Ef the completion of A/Nf and with 0/ the canonical image of 1 G A in A/Nf C Ef. For each aeA, let r(a) be the operator in A/Nf obtained from the left multiplication by a in A by passage to the quotient. Then the following holds. (i) Each r(a) has a unique extension to an operator nf(a) in the Hilbert space Ef. (ii) The map a H-+ 7T/(a) is a representation of A in Ef. (iii) The representation iTf admits a cyclic vector Of. (iv) f(a) = (Tr(a)6f\6f) for each aeA. •

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The representation 7iy and the cyclic vector Of in Theorem 3.1.7 are said to be determined by the form / , and the form / equals the vector form determined by 717 and Of. Conversely, given a representation n of A in a Hilbert space E and a cyclic vector 0 for n, let w be the vector form on A determined by n and 0. Let 7rw and 0w be the representation in Eu and the vector of Eu determined by w in accordance with Theorem 3.1.7. Then there is a unique isomorphism of E to Eu which sends n to n^ and 0 to 0^. Example 3.1.12. In particular, any cyclic representation of a C*-algebra A is a summand of the universal representation ®fftf of A, where / runs through all positive forms on A. • It may happen that different positive forms on a C*-algebra determine the same representation as follows. 3.1.8. (i) Let A be a C*-algebra and / a positive form on A which determines a representation 717 of A and its cyclic vector Of. Then for any b e A, the positive form a —> f(b*ab) on A determines the same representation 717. (ii) Conversely, any vector form / ' on A determined by the representation 717 is the limit a —> F(b*abi), where {6j} is a convergent sequence with respect to the normed topology on A. • PROPOSITION

Now let us specify positive forms on a C*-algebra A which determine its irreducible representations. A positive form / ' on an involutive algebra A is said to be dominated by a positive form / if / — / ' is a positive form. A non-zero positive form / on an involutive algebra A is called pure if every positive form / ' on A which is dominated by / reads A/, 0 < A < 1. THEOREM 3.1.9. The representation of 717 of a C*-algebra A determined by a positive form / on A is irreducible if and only if / is a pure form [129]

•

In particular, any vector form determined by a vector of a carrier space of an irreducible representation is a pure form. Therefore, it may happen that different pure forms determine the same irreducible representation. THEOREM 3.1.10. (i) Pure states f\ and / 2 of a C*-algebra A yield equivalent representations of A if and only if there exists an unitary element U

Chapter 3 Algebraic Quantization

209

of the unital extension A of A such that f2(a) = fi(U*aU),

aeA.

(ii) Conversely, let 7r be an irreducible representation of a C*-algebra A in a Hilbert space E. Given two different elements 6\ and #2 of E (they are cyclic for 7r), the vector forms on A determined by (TT, #i) and (ft, 82) are equal if and only if there exists A G C, |A| = 1, such that #i = A02(iii) There is one-to-one correspondence between the pure states of a C*-algebra A associated to the same irreducible representation n of A in a Hilbert space E and the one-dimensional complex subspaces of E. It follows that these states constitute the projective Hilbert space PE in Example 4.1.5 below. • Let P(A) denote the set of pure states of a C*-algebra A. Theorem 3.1.10 implies a surjection P(A) —> A, where A is the spectrum of A. This surjection is a bijection if and only if any irreducible representation of A is one-dimensional, i.e., A is a commutative C*-algebra. In this case, A is the C*-algebra of continuous complex functions vanishing at infinity on A, while a pure state on A is a Dirac measure ex, x € A, on A, i.e., ex(a) — a(x) for all aEA. Being a subset of the topological dual A' of the Banach space A, the set P(A) is provided with the normed topology. However, one usually refers to P(A) equipped with the weak* topology. In this case, the canonical surjection P(A) —> A is continuous and open [129]. 3.2

GNS construction II. Locally compact groups

Given a locally compact group G provided with a Haar measure, the space LQ(G) of the equivalence classes of complex integrable functions on G is an involutive Banach algebra with an approximate identity. There is oneto-one correspondence between the representations of this algebra and the strongly continuous unitary representations of a group G. Thus, one can employ the GNS construction in order to describe these representations of G [129]. Let a left Haar measure dg on G hold fixed, and by an integrability condition throughout is meant the dg-integrability. A uniformly (resp. strongly) continuous unitary representation of a locally compact group G in a Hilbert space E is a continuous homeomorphism

210

Geometric and Algebraic Topological Methods in Quantum Mechanics

IT of G to the group U(E) C B(E), called the unitary group, of unitary operators in E provided with the normed (resp. strong) operator topology. A uniformly continuous representation is strongly continuous. However, the uniform continuity of a representation is rather rigorous condition. For instance, a uniformly continuous irreducible unitary representation of a connected locally compact real Lie group is necessarily finite-dimensional. Therefore, one usually studies strongly continuous representations of locally compact groups. In this case, any element £ of a carrier Hilbert space E yields the continuous map G B g .-> TT(<7)£ € E.

Since strong and weak operator topologies on the unitary group U(E) coincide, we have a bounded continuous complex function
(3-2.1)

on G for any fixed elements £,77 G E. It is called a coefficient of the representation IT. There is the obvious equality
= VvAd'1)-

The Banach space LQ(G) of the equivalence classes of integrable complex functions on G is provided with the structure of an involutive Banach algebra with respect to the contraction /1 * fy (10.5.20) and the involution

/(ff)-/•(«/) = A( S - 1 )7(F T ), where A is the modular function of G. It is called the group algebra of G. The map / >-* f(9)dg defines an isometric monomorphism of LQ(G) to the Banach algebra M1(G, C) of bounded complex measures on G provided with the involution [j,* = fj,-1. Unless otherwise stated, L^(G) will be identified with its image in M1(G,C). In particular, the group algebra L^(G) admits an approximate identity which converges to the Dirac measure £1 G M1(G, C). The group algebra L^(G) is not a C*-algebra. Its Remark 3.2.1. enveloping C*-algebra C*(G) is called the C* -algebra of a locally compact group G. Let us recall the notion of an enveloping C*-algebra [129]. Let A

Chapter 3 Algebraic Quantization

211

be an involutive Banach algebra A with an approximate identity, and let P(A) be the set of pure states of A. For each a e A, we put \\a\\'= sup f(aa*)1/2, feP(A)

a € A.

(3.2.2)

It is a seminorm on A such that ||a||' < ||a||. If A is a C*-algebra, \\a\\' = \\a\\ due to the relation (3.1.1) and the existence of an isomorphic representation of A. Let I denote the kernel of ||.||'. It consists of o € A such that ||o||' = 0. Then the completion A* of the factor algebra A/1 with respect to the quotient of the seminorm (3.2.2), is a C*-algebra, called the enveloping C*-algebra of A. There is the canonical morphism r : A —> A*. Clearly, A = A* if A is a C*-algebra. The enveloping C*-algebra A* possesses the following important properties. (i) If 7T is a representation of A, there is exactly one representation TT* of A* such that vr = n^ o r. Moreover, the map 7r —> TT* is a bijection of the set of representations of A onto the set of representations of A^. (ii) A representation vr of A is non-degenerate (resp. topologically irreducible) if and only if TT* is non-degenerate (resp. irreducible). (iii) If / is a continuous positive form on A, there exists exactly one positive form p on A* such that / = / * O T . Moreover, ||/t|| = ||/||. The map / —> p is a bijection of the set of continuous positive forms on A onto the set of positive forms on A+. • Unitary representations of a locally compact group G and representations of the group algebra L^(G) are related as follows [129]. Let 7T be a (strongly continuous) unitary representation of G in a Hilbert space E. Given a bounded positive measure /i on G, let us consider the integrals

Vt.vit1) = /

(""(^MM

of the coefficient functions
(3.2.3)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

The assignment /z —> 7r(/x) provides a representation of the involutive Banach algebra Ml(G, C) in E. Its restriction * L ( / ) = J*(g)f(g)dg

€ B(£?)

(3.2.4)

to Lp(G) is non-degenerate. One says that the representations (3.2.3) of M1(G, C) and (3.2.4) of L^(G) are determined by the unitary representation 7T of G. Conversely, let nL be a representation of the involutive Banach algebra Z>c(G) in a Hilbert space E. There is a monomorphism g i-> eff of the group G onto the subgroup of Dirac measures eg, g € G, of the algebra M1(G, C). Let {ut(
*(/) = ji>{g)f(g)dg

(3.2.5)

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Chapter 3 Algebraic Quantization

by an element tp of the space Lg3 (G) of the equivalence classes of integrable complex functions on G of infinite degree. However, the function ip should satisfy the following additional condition in order that the form (3.2.5) to be positive. A continuous complex function ip on G is called positive-definite, if X^teJ19%)CiCj > 0 for any finite set g\,...,gm of elements of G and any complex numbers C\, • • • > Cm- In particular, if m = 2 and g\ = 1, we obtain

i>{g~l) = W),

M(s)
sec,

i.e., ip(l) is bounded. 3.2.1. The continuous form (3.2.5) on LQ(G) is positive if and only if -0 € L^? (G) locally almost everywhere equals a continuous positivedefinite function. • LEMMA

Then cyclic representations of the group algebra L^(G) and the unitary cyclic representations of a locally compact group G are defined by continuous positive-definite functions on G in accordance with the following theorem. 3.2.2. Let 7ty be a representation of LXC{G) in a Hilbert space Eif, and 6^ a cyclic vector for n^, which are determined by the form (3.2.5). Then the associated unitary representation n^ of G in E^ is characterized by the relation THEOREM

1>(9) = «9)h\h)-

(3-2.6)

Conversely, a complex function ip on G is continuous positive-definite if and only if there exists a unitary representation TT,/, of G and a cyclic vector 8$ for 7ty such that the equality (3.2.6) holds. • Example 3.2.2. Let a group G acts on a Hausdorff topological space Z on the left. Let fi be a quasi-invariant measure on Z, i.e., j(g)fi = hg\i where hg is the Radon-Nikodym derivative in Theorem 10.5.1. Then there is the representation G3g:f^

U(g)f,

(U(g)f)(z) = hy\z)f(gz)

(3.2.7)

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of G in the Hilbert space LQ(Z,/J.) of the equivalence classes of square /xintegrable complex functions on Z [121]. It is a unitary representation due to the equality II/IIM

= / I/(*)IM(*) = / \f{g{z)\2Kg{z)) =

(3.2.8)

Jhg(z)\f(9(z)\2ti(z) = \\n(g)f\\l. A group G can be equipped with the coarsest topology such that the representation (3.2.7) is strongly continuous. For instance, let Z = G be a locally compact group, and let /J, = dg be a left Haar measure. Then the representation (3.2.7) comes to the left-regular representation (Tl(g)f){q) = f{g-1q),

f G L2C(G),

q G G,

(3.2.9)

of G in the Hilbert space LQ(G) of the equivalence classes of square integrable complex functions on G. Let us note that the above mentioned coarsest topology on G is coarser then the original one, i.e., the representation (3.2.9) is strongly continuous. • Let us consider unitary representations of a locally compact group G which are contained in its left-regular representation (3.2.9). In accordance with the expression (3.2.3), the corresponding representation II(/i) of the group algebra L^(G) in L^.(G) reads (H(h)f)(q) = J h(g)(Il(g)f)(q)dg = J h(g)f(g-1q)dg = (h * /)(«/). (3.2.10) Let G be a unimodular group. There is the following criterion that its unitary representation is contained in the left-regular representation. 3.2.3. If a continuous positive-definite function tp on a unimodular locally compact group G is square integrable, then the representation TT,/, of G determined by ip is contained in the left-regular representation II (3.2.9) of G. Conversely, let TT be a cyclic unitary representation of G which is contained in II, and let 6 be a cyclic vector for n. Then continuous positive-definite function (n(g)9\6) on G is square integrable. El PROPOSITION

The representation 7ty in Proposition 3.2.3 is constructed as follows. Given a square integrable continuous positive-definite function ip on G,

Chapter 3 Algebraic Quantization

215

there exists a positive-definite function 6 G L^(G) such that •# = 0*6 = 6*9* = 9* *T. This is a cyclic vector for 7ty. The coefficients (3.2.1) of the representation 7T,/, read are orthogonal in L$.(G), while the functions ip and ip' fulfil the relations / ip'(9M9)dg = 0,

V * $ = 0-

(3.2.11)

An irreducible unitary representation n of a locally compact group G in a Hilbert space £?„. is called square integrable if there exists a vector £ € En such that the coefficient <^£ of n is square integrable over G. This vector is called admissible. If G is a unimodular group, a square integrable representation of G is contained in its left-regular representation in accordance with Proposition 3.2.3. One can show that all coefficients ^i7? of this representation are square integrable over G [129]. Furthermore, there exists a unique constant dv (0 < d^ < oo) such that / U9)h',r,'(9)dg = d-1{m')'W),

^

*&',„'=<£ W)0£M

(3-2.12)

(3.2.13)

for any £,»7,£V G En. The constant d^ is called the formal dimension of the representation ir. However, its value depends on the choice of the Haar measure on G. If G is compact and dg is the Haar measure of total mass 1, then dn is the (finite) dimension of IT in the usual sense. If -n and n' are inequivalent square integrable representations of G, their coefficients
Jz,v(g)e,n'(g)dg = o,

(3.2.14)

Lv * fa'tf = 0-

(3-2.15)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

Example 3.2.3. Square integrable representations need not exist. For instance, let G be a commutative group. Its irreducible unitary representation is a character X, i-e., a non-zero homomorphism of G to C such that |x| 2 = 1. If G is compact, each character x of G is square integrable, and so is the corresponding irreducible unitary representation of G. If G is noncompact, it has no square integrable representations. D Now, let G be a connected locally compact (i.e.,finite-dimensional)real Lie group. Any unitary representation of G yields a representation of its right Lie algebra g as follows. A finite-dimensional unitary representation of G in a Hilbert space E is analytic, and the Lie algebra g is represented by bounded operators in E. If IT is an infinite-dimensional (strongly continuous) unitary representation of G in a Hilbert space E, a representation of the Lie algebra g fails to be defined everywhere on E in general. To construct the carrier space of g, let us consider the space /C°°(G,C) C L^(G) of smooth complex functions on G of compact support and the vectors ef=nL(f)e

= J n(g)f(g)edg,

e e E,

f G /C°°(G, C),

(3.2.16)

where nL is the representation (3.2.4) of the group algebra L^(G) [223]. The vectors e/ (3.2.16) exemplify smooth vectors of the representation TT because, for any rj € E, the coefficients (pef,r](g) of vr are smooth functions on G. The vectors ef (3.2.16) for all e £ E and / € /C°°(G,C) constitute a dense vector subspace E^ of E. Let ua be a right-invariant vector field on G corresponding to an element o 6 j . Then the assignment ^ ( a ) : ef -* KL(ua\df)e

(3.2.17)

provides a representation of the Lie algebra g in E^. In particular, let n be a (topologically) irreducible unitary representation of the locally compact Lie group G, and let e = £ be a cyclic vector for 7T. The morphism W^1 (3.3.20) below yields a monomorphism of the space /C°°(G, C) to Eoo C E, and TTOQ (3.2.17) is an irreducible representation of the Lie algebra g in WT1(/COC(G,C)). Given another cyclic vector £'€ J B\^- 1 (/C OO (G,C)) for 7T, we have a different irreducible representation of g. It follows that the representation 7roo (3.2.17) of the Lie algebra g in E ^ is reducible in general.

Chapter 3 Algebraic Quantization

217

Let 7T be a square integrable representation of G. Using the decomposition (3.3.21) below, one can show that EQQ contains all smooth vectors for TT. It is called the Garding space. 3.3

Coherent states

Using unitary square integrable representations of a locally compact group, one conies to the notion of coherent states, which play a prominent role in many quantum models [4; 353; 452]. We start with the more general construction. Let E be a Hilbert space. A family Sz = {£z} of non-zero vectors of E is said to be a reproducing system if it is indexed by points of a locally compact topological space Z, and it determines the decomposition

V = J(v\Z*)tzti*)

(3-3.1)

z of any element 77 of E with respect to some positive measure /x on Z. The integrand in (3.3.1) is brought into the form FZV = (Zzl&PzV, where Pz is the projector onto the vector £z. Hence, the equality (3.3.1) can be rewritten as a resolution of the identity JFZII(Z) = 1.

(3.3.2)

z Remark 3.3.1. In Dirac's bra-ket notation, the integrand in the resolution • of the identity (3.3.2) reads Fz = %){(,z\. A reproducing system need not exist. In a generic case, one deals with a separable Hilbert space E and a positive bounded operator a in E which admits a resolution

JFzii{z) = a z

(3.3.3)

with respect to a positive measure /x on some locally compact topological space Z such that Fz are compact operators in E. The operator a in (3.3.3) need not be invertible, i.e., the domain of a~1 is not dense in E.

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Geometric and Algebraic Topological Methods in Quantum Mechanics

If the operator a is invertible (but a" 1 need not be bounded), one says that (E,Fz,a) is the reproducing triple. For instance, this is the general case of covariant coherent states below. If the operator a" 1 is bounded and all operators Fz are of the same rank n, one can construct a reproducing system of vectors {/*}, i = 1 , . . . , n, of the Hilbert space E which provides a resolution of the identity. Then, similarly to coherent states below, one also call /* the coherent states [4]. In particular, we throughout are in the case of n = 1. Let us consider a complex function (3.3.4)

K(z,z') = (^zl)

on the Cartesian product Z x Z. It is called a reproducing kernel because of the relation K(z, z') = jK(z, z")K(z", z')n(z"), z

z, z',z" € Z,

(3.3.5)

derived from the decomposition (3.3.1). By the very definition of K(z,z') (3.3.4), we also have K{z,z)>0,

K(z,z') = K{z',z),

z,z'eZ.

(3.3.6)

Let us consider the morphism W:E^Ll(Z,fi),

(Wr,)(z) = (r]\^),

r, G E,

(3.3.7)

of E to the Hilbert space L$.(Z, fi) of the equivalence classes of square integrable complex functions on Z. This is an isometric monomorphism of Hilbert spaces whose range Ew consists of the equivalence classes of square integrable functions f on Z satisfying the equality

f(z)= jK(z',z)f(z')fi(z'). z

(3.3.8)

A glance at this equality shows that the functions K(z,z') = WZz(z'),

*eZ,

constitute a reproducing system in the Hilbert space Ew- Namely, the corresponding decomposition (3.3.1) reads

/(*)= / Jf(z")K(z",z>Mz") K(z',z)Kz'). z Lz

219

Chapter 3 Algebraic Quantization

The inverse W" 1 : Ew -> E to the morphism W (3.3.7) reads

z Then one can assign the operator

J=W~1ofoW=

jf{z)Fzix{z) z

on the Hilbert space E to any element / G Lg^Z,/z) (see Remark 10.5.6), e.g., to any bounded continuous function on Z. Conversely, let a £ B(E) be a bounded operator in the Hilbert space E. One can associate to a a complex function fa(z,z') = (a£z\Sz')

(3-3.9)

/«(*) = {a£x\Zz)

(3-3.10)

on Z x Z. Its restriction

to the diagonal Z in Z x Z is a, mean value function. By virtue of the decomposition (3.3.1), we obtain (fa * fb)(z) := J fa(z', z)fb(z, *>(*') = fab(z), z

a, b € B(E).

(3.3.11)

Let us note that this formula is also true for unbounded operators a and b if Sz c D(a),

Sz C D(b),

bSz C D{a).

A unitary square integrable representation of a locally compact group G exemplifies a reproducing system. Here, G is not necessarily a unimodular group, and not all coefficients of its square integrable representation are square integrable functions on G [4; 20l]. Let 7r be a square integrable representation of G and £ an admissible vector, i.e.,

j M9)m)\2d9 < oo.

(3.3.12)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

It is readily observed that every vector n(g)£, g e G, is admissible. Hence, the set 5(TT) of admissible vectors is G-invariant and dense in the carrier space E. If G is a unimodular group, then 5(71") = E. 3.3.1. If TT is a square integrable representation of G, there exists a unique closed positive invertible operator C on S(TT) C E such that PROPOSITION

Ji,r,(9)
V,

i G E.

(3.3.13)

• It follows that coefficients £>v of a square integrable representation are square integrable functions if one of the vectors £ and r\ is admissible. If G is a unimodular group, then C = Al, A > 0, and vice versa. In this case, the relations (3.3.13) come to the relations (3.2.12). The relations (3.2.13) are generalized to an arbitrary locally compact group as follows. Let £ be an admissible vector of a square integrable representation TT of G such that

J\
(3-3.14)

i.e., (C£|C£) = 1. Then one can easily justify that the function K(g, g') = (n(g)^(g')0 = ^Mg^(g)

(3.3.15)

is a reproducing kernel on G, i.e., it obeys the relations K(g,g) >0,

g^G,

K(g,g') = jK(g,g")K(g",g')dg".

(3.3.16)

In particular, if G is a unimodular group, then (£|£) = dn and the relation (3.3.16) for the reproducing kernel (3.3.15) follows from (3.2.13). Thus, we come to the conventional notion of a coherent state. Let 7T be a square integrable representation of a locally compact group G, and let £ G E be an admissible vector obeying the condition (3.3.14). Let us consider the orbit St = {tg=n(g)£:geG}

(3.3.17)

of the group G in E. Its elements £g are called coherent states. They possess the following important properties.

221

Chapter 3 Algebraic Quantization

(i) Since •n is an irreducible representation, coherent states (3.3.17) constitute a total family, but this family is not orthonormal in general. (ii) Coherent states define the reproducing kernel (3.3.15) on the group G. (iii) Let us consider the map Wc-E^LliG),

(W(r,)(g)=J^)

= (r,\7r(g)0,

V e E.

(3.3.18)

By virtue of the relations (3.3.13) and (3.3.14), it is a monomorphism of Hilbert spaces whose range E^ is dense in LQ(G). Moreover, the map W% (3.3.18) is the intertwining operator between •K and the left-regular representation II of G, i.e., Wt«(g) = TI(g)Wz for all g €G. (iv) It follows that a square integrable representation TT of a locally compact group G is contained in its left-regular representation. Using the relations (3.3.13) and (3.3.14), one can show that the space E$ consists of elements / of L^ (G) obeying the condition

f(g) = jK(g',g)f(g')dg',

(3.3.19)

where K{g,g') is the reproducing kernel (3.3.15). (v) The inverse morphism E^ —> E reads

W[1(f)=Jf(g)£gdg,

/ e %

(3.3.20)

In particular, we obtain the decomposition

V = j(W^)(g)^dg

= J(v\Zg)tgdg

(3.3.21)

of any element r) e E with respect to coherent states (cf. the decomposition (3.3.1)). Thus, coherent states constitute a reproducing system. There is the following serious inconsistency of the above construction of coherent states. Let us consider the integral (3.3.12) and suppose that H% is the subgroup of G which keeps £ invariant up to a phase multiplier, i.e., TT(/I)£

= eia(-h\,

h G Hv

(3.3.22)

Clearly, the integrand in (3.3.12) does not depend on g, but only on a coset q € G/H^. Consequently, the finiteness of the integral (3.3.12) forces

222

Geometric and Algebraic Topological Methods in Quantum Mechanics

the subgroup H^ to be compact. In order to avoid this restriction, one introduces the notion of covariant coherent states [4]. Let G be a locally compact group and n its strongly continuous irreducible unitary (not necessarily square integrable) representation in a Hilbert space E. Let H be a closed subgroup of G. We suppose that: • G/H is endowed with a positive G-invariant measure fi (this is the case of a compact group H [23]), • there exists a /i-measurable (not necessarily global) section a of the principal fibre bundle G —> G/H, • there exists a vector £ £ E such that the integral

j IM
V£E,

(3.3.23)

G/H

is finite and aa is a bounded positive invertible operator in E (though a" 1 need not be bounded). Let £ be normalized by (£|aCT£) = 1- Then the family of covariant coherent states is denned as Sa = {&(,) = 7r(orfa))e, q e G/H}.

(3.3.24)

The following summarizes the main properties of covariant coherent states. (i) The family 5CT (3.3.24) is total in E. (ii) Let us introduce the function Ka(q,q') = (t;aiq)\a-1Zaiq,))

(3.3.25)

on the Cartesian product G/H x G/H. There is a monomorphism

Wt:E-^Ll{G/H,n),

(W^)te) = fol&(,)>, V G E.

(3.3.26)

Its range E^ consists of the equivalence classes of square integrable functions / on G/H which satisfy the relation

/(
similar to the condition (3.3.19) in the case of coherent states.

Chapter 3 Algebraic Quantization

223

(iii) The inverse morphism E% —>• E reads

G/H

(cf. (3.3.20)). In particular, we have

G/H

G/H

or, equivalently,

aaV = I (»?|&(,)>&(,)M9)-

(3-3.27)

G/H

(iv) The integrand in (3.3.27) can be rewritten as

where Pa(q) is the projector onto the vector ^ ^ . Then the equality (3.3.27) takes the form

aa=

J Fa{q)ii{q).

(3.3.28)

G/H

In particular, if G/H is compact, aa is a Hilbert-Schmidt operator. (v) The range E^ of the monomorphism W^ (3.3.26) is complete with respect to the norm

(f\f), = (ftW^a-'W^f)

(3.3.29)

so that the morphism W^ of E onto E^ is isometric. The above mentioned results simplify somewhat if the operator a^1 is bounded. Then Sa is called a frame.. This terminology is borrowed from the theory of non-orthogonal expansions [122J. The further simplification occurs when Sa is a tight frame, i.e., aa — Al, A > 0. This is also the case of generalized coherent states when H = H^ (3.3.22) [353]. If aa = 1, the

224

Geometric and Algebraic Topological Methods in Quantum Mechanics

function Ka(q,q') (3.3.25) fulfils the relation K{q,q)a>0,

qeG/H,

tf(g,<7%=W^)CT, * W ) * = J K(q,q")aK(q",q%Li(q") G/H

and, thus, defines a reproducing kernel on G/H. The norm (3.3.29) coincides with the standard norm on the space L^(G/H, /x) so that Wj (3.3.26) is an isometric monomorphism. The equality (3.3.28) provides a resolution of the identity J Fa{q)n{q) = 1

(3.3.30)

G/H

(cf. (3.3.2)). Thus, generalized coherent states constitute a reproducing system. Of course, the above mentioned construction of covariant coherent states depends on the choice of a particular section a of the principal bundle G —» G/H. However, there is a certain degree of "section-independence" in the method. In particular, if a provides the admissibility condition (3.3.23), the similar condition holds for any translated section <Jg(q) = 9v(9~1q),

g^G.

If two sections a and a' fulfil the admissibility conditions (3.3.23) for a given vector £, there is one-to-one correspondence between the corresponding families of covariant coherent states Sa and Sai carried out by bounded operators £
=Tqt.a>(q).

One is also free to specify a vector £ of the carrier space E in order to obtain a family of covariant coherent states. In many quantum models, there exists a vector £ such that the covariant coherent states produced from £ appear to be minimal uncertainty states. 3.4

GNS construction III. Groupoids

Generalizing the case of a locally compact group, one can associate a C*algebra to a locally compact groupoid equipped with a left Haar system as

Chapter 3 Algebraic Quantization

225

follows [367]. Let a topological groupoid (5 (see Section 10.3) be locally compact. Let /C(<5, C) be the space of continuous complex functions of compact support on (55 endowed with the inductive limit topology. A left Haar system for a groupoid (25 is a family of measures {/xu,u £ <50} on (25 indexed by points of the unit space (25° of (25 such that: • the support of the measure /xu is (25U; • for any function / £ /C((9, C), the function /^e^uHnu(f)

ec

(3.4.1)

is continuous; • for any x € 6 and / G £(<25,C),

/ f(xy)vi(x)(y) = / f(y)vr{x)(y)One can show that, if 0 is a locally compact groupoid with a left Haar system, then the map tC(<5,C)3f^U£tC(®°,C) is continuous and the surjection r : (25 —> <S° is open. Example 3.4.1. Let G be a locally compact group which acts continuously on a locally compact space S. Let us consider the action groupoid 5 x G in Example 10.3.2. It is a locally compact groupoid with the left Haar system {fis, s € S}, where ns = es /J,G is the product of the Dirac D measure es on S and the left Haar measure /^G on G. Example 3.4.2. Let © be a locally compact group bundle in Example 10.3.3. If 0 admits a left Haar system, this system is defined up to the multiplication /J,'U = h(u)nu by an arbitrary positive continuous function h in <8°. In particular, let G be a locally compact group. The set SQ of subgroups of G becomes a compact Hausdorff space when equipped with Fell's topology [l5l]. Let us define the topological groupoid <S = {(H,g) : H€SG,g£H}cSGxG

(3.4.2)

provided with the relative topology and the following groupoid structure:

.{H,g)-l:={H,g-1); • a pair ((H,g), (H',g')) is composable if and only if H = H';

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Geometric and Algebraic Topological Methods in Quantum Mechanics

.(H,g)(H,g>) = (H,gg>). The groupoid (3.4.2) is a locally compact group bundle over SQ called the bundle of subgroups of a group G. It admits the left Haar system {/J,JI, H G SG} where /J,H the left Haar measure on H. • Example 3.4.3. A locally compact groupoid 0 is called r-discrete if its unit space is an open subset. This groupoid possesses the following properties. • For any u £ 0°, the subspaces 0 U and 0 U are discrete. • If © admits a left Haar system, one can choose it to be /Uu({u}) := 1 for all u € 0°. Then the invariance condition results in fj,u({x}) = 1 for all • The multiplication 0 2 —> 0 and the map r : 0 —> 0° are local homeomorphisms. • Let 0 be a locally compact groupoid with a left Haar system {fiu}- Let us denote fiu = [i~l the inverse measure. The measures fiu, u € (5°, are assembled into a right Haar system. Given a measure A on the unit space 0°, let us introduce the measure

v=fiiu\{u)

(3.4.3)

on <8 by means of the condition

K / ) = //M(«)M«).

/e/C(0,C),

where /M is the function (3.4.1). A measure A on 0° is said to be quasiinvariant (resp. invariant) if the measure v (3.4.3) is equivalent (resp. equal) to the inverse measure V-i

= j f\{u).

Example 3.4.4. If 0 = G is a locally compact group, then 0° = {1} and the left Haar system for 0 reduces to the left Haar measure on G. A quasi-invariant measure on {1} is exemplified by the Dirac measure, and the measure v (3.4.3) on G coincides with the left Haar measure. • Example 3.4.5. Let us consider the action groupoid 0 = S x G in Example 10.3.2 with the left Haar system in Example 3.4.1. Let A be a

Chapter 3 Algebraic Quantization

111

measure on 5. Then the measure v (3.4.3) on (25 is the product A no, where fxc is the left Haar measure on G. Let G acts transitively on S, i.e., 5 is an orbit of its element s. Then a quasi-invariant measure on S is exemplified by the Dirac measure e3. • Let us turn now to the notion of a C*-algebra of a groupoid [367]. Let © be a locally compact groupoid with a left Haar system {fJ,u}- Let (6,21) be a <5-module bundle where 2t = <S°x[/(l), and let a be a two-cocycle of the complex (10.3.6). The space /C((S, C) is a topological involutive algebra with respect to the operations

{f*9){x) = Jf(xy)g{y-1)a(xy,y-1)fM{x)(y),

(3.4.4)

r(x)=f(x-i)
(3.4.5)

Let us denote this algebra by C(<8,cr). Of course, this algebra depends on the choice of a two-cocycle a. One however can show that, if cocycles a and a' belongs to the same cohomology class, the involutive algebras C(<S,a) and C(<8,a') are isomorphic. The involutive algebra C(<8,a) becomes a normed algebra, called the convolution algebra, if one provides the space /C((5,C) with the norm

||/||w=max(||/|| r , H/IIO,

||/|| P = sup f\f\nu,

/G/C(0,C),

(3.4.6)

Il/U, = sup f \ f \ n \

The corresponding normed topology on /C((25,C) is finer than the inductive limit topology. By a representation n of the convolution algebra C(<25,
lk(/)ll < ll/llw for a l l / e / C ( 0 , C ) . A representation of the convolution algebra C(<S,a) can be constructed as follows. Let the unit space 0° be provided with a quasi-invariant measure

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Geometric and Algebraic Topological Methods in Quantum Mechanics

A. Let H —> (5° be a group bundle whose fibres are separable Hilbert spaces. Let us consider a morphism

of the groupoid 0 to the frame groupoid Iso H in Example 10.3.4 such that: • ^(x) is represented by some morphism

• the equality

V(x)V(y) =
on<252; • the equality

^(a;)- 1

=a(x,x-1)V(x-1)

holds almost everywhere on <S with respect to the measure v (3.4.3); • the map

is ^-measurable for any pair of A-measurable sections 4> and 4>' of the group bundle H -> <5°. The morphism $ is called the a-representation of the groupoid 0 in H. Two cr-representations ($!,H,\) and ( # ' , # ' , A') of (5 are said to be equivalent if there exists an isomorphism $ of if to H' over (8° such that (*'o$)(a;) = ($o^)(x)

almost everywhere on 0 with respect to the measure v (3.4.3). Let L2(H, A) be the Hilbert space of sections of H —> (5° which are square A-integrable. Then a representation of the convolution algebra C(<8,a) in L2(H,X) is given by the expression 7T*(/)0(«) = j f(x)(V(x)(ci> o l)(x)A-V2v(x),

(3-4.7)

Chapter 3 Algebraic Quantization

229

where A is the modular function of the measure v. Different arepresentations \? and \&' of the groupoid 6 obviously yield different representations 7T* and 7r

and W of <& are equivalent, the representations TT* and Try of C(0,a) are also equivalent. Let us provide the convolution algebra C(<&,a) with a different norm | | / | | = sup ||7r(/)||, where TT runs through all bounded representations of the algebra C(<S,a). The completion C*(<8,cr) of C(<3,cr) with respect to this norm is a C*algebra, called the convolution C* -algebra of the groupoid <8. Let A be a positive measure on the unit space (25° of a locally compact groupoid 0. Then \{f\@o), f £ C(<5,a), is a positive continuous form on C(<3,cr) equipped with the inductive limit topology. If <S is an r-discrete groupoid and A a measure of total mass 1, this form is extended to a state on the C*-algebra C*(<8,a) of 6. 3.5

Example. Algebras of infinite qubit systems

Let Q be the two-dimensional complex space C2 equipped with the standard positive non-degenerate Hermitian form (.|.)2- Let Mi be the algebra of complex 2 x 2-matrices seen as a C*-algebra. A system of m qubits is usually described by the Hilbert space m

and the C* -algebra m

Am = ®M 2 , called the input algebra, which coincides with the algebra B{Em) of bounded operators in Em [240]. One can straightforwardly generalize this description to an infinite set S of qubits by analogy with a spin lattice [144]. Its algebra As admits non-equivalent irreducible representations. If S = Z + , there is one-to-one correspondence between the representations of As and those of an algebra of canonical commutation relations. Given a system of m qubits, one also considers an algebra of complex m

functions on the set x Z2. It is regarded as the output algebra of a qubit system [240]. There is a natural monomorphism of this algebra to the

230

Geometric and Algebraic Topological Methods in Quantum Mechanics

input one Am. Using the technique of groupoids in previous Section, one can generalize these algebras to infinite qubit systems. We start with the input algebra, and follow the construction of infinite tensor products of Hilbert spaces and C*-algebras in [144]. Let {Qs,s £ S} be a set of two-dimensional Hilbert spaces Qs = C 2 . Let x Qs be s the complex vector space whose elements are finite linear combinations of elements {qs} of the Cartesian product J\QS of the sets Qs- The tensor s product Q3 of complex vector spaces Qs is the quotient of x Qs with s s respect to the vector subspace generated by the elements of the form: • {is} + {q's} — Ws } , where qr + q'r = q" for some element r £ S and q3 = q's = q'J for all the others, • {qs} ~ ^W3}> A € C, where qr = Xq'r for some element r £ S and qs — q's for all the others. Given an element

8 = {e.}e]lQ. s e

such that all 0S ^ 0, let us denote ® Qs the subspace of (g>Qs spanned by s the vectors ®qs where qs ^ 6S only for a finite number of elements s G S. It is called the 9-tensor product of vector spaces Qs, s £ S. Let us choose a family 6 = {0S} of normalized elements 8S £ Qs, i.e., all \63\ = 1. Then ®eQs is a pre-Hilbert space with respect to the positive non-degenerate Hermitian form s€S

Its completion Qes is a Hilbert space whose orthonormal basis consists of the elements

eir = ®qs,

r£S,

i = 1,2,

such that qsitr = 6S and qr = ei, where {e^} is an orthonormal basis for Q. Let now {A3,s € 5} be a set of unital C*-algebras As = Mi. These algebras are provided with the operator norm H I = (AoAo + AiAi + A2A2 + A 3 A3) 1/2 ,

a = i\ol

+

^ A ^ , t=l,2,3

where a1 are the Pauli matrices. Given the family { l s } , let us construct the {l s }-tensor product ®AS of vector spaces As. It is a normed involutive

Chapter 3 Algebraic Quantization

231

algebra with respect to the operations (<8>as)(®a's) = <8>(aaa's),

(<8>as)* = a*

and t h e norm

I ® a J = JJlKUs

Its completion As is a C*-algebra. Then the following holds [144]. PROPOSITION 3.5.1. Given a family 9 = {6S} of normalized elements 9S E Qs, the natural representation of the involutive algebra ®AS in the pre-Hilbert space ®BQ3 is extended to the representation of the C*-algebra As in the Hilbert space Qes such that As = B(Qes) is the algebra of all bounded operators in Q8S. D PROPOSITION 3.5.2. Given two families 9 = {9S} and 9' = {9's} of normalized elements, the representations of the C*-algebra As in the Hilbert spaces Qs and Qs are equivalent if and only if

£||<6U60|-l|
ses

D

For instance, if S = Zfc, we come to a spin lattice. Let us turn now to the output algebra. One can associate to a system of qubits {Qs,s £ S} the following groupoid. Let Z2 = {1, p : p 2 = 1} be the smallest Coxeter group. Let us consider the set X = Z2S of Zivalued functions on 5. It is a set 2s of all subsets of S, and it can be brought into a Boolean algebra. Let G C X be a subset of functions which equal p G Z2 at most finitely many points of S. Both X and G are commutative Coxeter groups with respect to the pointwise multiplication. Given the action of G on X on the right, the product <8 = X x G is the action groupoid whose unit space 0° is naturally identified with X (see Example 10.3.2). Since G acts freely on X, the action groupoid (25 is principal, i.e., the map (r, I ) : 0 - 4 0 ° x 0 °

232

Geometric and Algebraic Topological Methods in Quantum Mechanics

is an injection. Provided with the discrete topology, X is a locally compact space. Then (5 is a locally compact groupoid. Since (25° C (25 is obviously an open subset, (25 is an r-discrete groupoid. Let us note that the action groupoid (25 is isomorphic to the following one. Let us say that functions x, y € X are equivalent x ~ y if and only if they differ at most finitely many points of S. Let (25' c X x X be the graphic of this equivalence relation, i.e., it consists of pairs (x,y) of equivalent functions. One can provide & with the following groupoid structure: • a pair ((x, y), (y', z)) is composable if and only if y = y', • {x,y)~l -=(y,x), • {z,y)(y,z) = (x,z), • r : ( i , j/) i-> (x,x), I : (x,y) H-> (y,y). The unit space (25/0 of this groupoid is naturally identified with X. The isomorphism of groupoids (25 and (25' is given by the assignment <8B(x,g)^(x,xg)e&.

In particular, 0 * / 0 if and only if x ~ y. Let /C(<25, C) be the space of complex functions on (25 of compact support provided with the inductive limit topology. Since (55 is a discrete space, any function on 0 is continuous. A left Haar system for the groupoid (25 is a family of measures {fj,x, x e X} on 6 indexed by points of the unit space <S° = X such that: • the support of the measure /xx is <25X, • for any (x,g) € (25 and any / € K.(<8,C), we have

J f((x,g)(y,g'))fixg(y,g') = j ' f(v,g')nx(v,g'). Since 0 ° is a discrete space, the function (3.4.1) is always continuous. As was mentioned in Example 3.4.1, a left Haar system for the action groupoid (25 is given by the measures Mx = £x x

MG,

where ex is the Dirac measure on X with support at x G X and HG is the left Haar measure on the locally compact group G. We have

/ f(v,g)i*x{y,g) = / f(x,g)nG(g) =

J

J

^2f(x,g),

g€G

where the sum is finite since / € £(<25,C) are of compact support.

233

Chapter 3 Algebraic Quantization

With this left Haar system, the space /C(<8,C) is brought into the following convolution algebra C(<5). Since the t/(l)-extension of G is trivial up to equivalence [288], the cohomology group H2(<&, 21) is trivial (see Theorem 1.5.1 and Remark 10.3.7), and algebraic operations (3.4.4) - (3.4.5) in /C(<5,C) read

(f*f)(x,9)= ^/(z.^O/W 1 .^" 1 ).

(3.5.1)

g'eG

(3.5.2)

r{x,g) = f((x,g)-i). This algebra is provided with the norm (3.4.6):

Il/H = max(sup Y2\f(x,g)\, sup £ l/O^^fiT1)!)X

9

9

In particular, let us consider a subspace Ac C C(<&) of functions f(x,g) = f(g) independent of x £ X. The algebraic operations (3.5.1) - (3.5.2) on these functions take the form

(/ * /')(
rig) = 'W1) = W)-

g'eG

Thus, AQ is exactly the group algebra of the locally compact group G provided with the norm

11/11 = £ 1/(5)1g

There is the monomorphism of this algebra to the algebra ®AS as follows. Let us assign to each element g £ G the element "g = ®as G <8>AS, where aa = 1 if g(s) = 1 and as = cr1 if g(s) = p. Then the above mentioned monomorphism is given by the association

/(s)^ £/(
Thus, one can think of Ac and, consequently, C(&) as being a generalization of the output algebra in [240]. The monomorphism Ac —> ®AS provides the representations of the involutive algebra Ac in Hilbert spaces Qss. One can construct representations of the whole convolution algebra C(<S) as follows. Given the Hilbert space Q6S, let us consider the product € = X x Q%

234

Geometric and Algebraic Topological Methods in Quantum Mechanics

seen as a group bundle in the Abelian groups Q% over X. It is a groupoid such that • a pair

{{x,v),(x',v')),

v,v'eQes,

x,x'£X,

is composable if and only if x' = x, • {x^)-1 :=(x,-v), • (x,v)(x,v') = (x,v + v'), • r((x,v)) = l((x,v)) =x. Its unit space is X, and its fibres €x are isomorphic to Q6S. Let the discrete topological space X be provided with the measure A such that

/e/C(A\C).

ff(x)X(x)=^2f(x), J

x

Let Iso <£ be the set of all isomorphisms 7j:€ w -»Cx,

x,ye<5°.

It is the frame groupoid in Example (10.3.4) whose unit space <£° is naturally i > Id of the group bundle <£ —> X. Then we come to the representation n (3.4.7) of the convolution algebra C((5) in L2(<£,fix) by the formula

(7r(/)0)(z) = £ / ( ( * , s))^,?). 9

The algebra C(<3) can be provided with the operator norm ||/|| = ||7r(/)|| with respect to this representation. Its completion relative to this norm is the C*-algebra of the action groupoid <S.

3.6

GNS construction IV. Unbounded operators

There are algebras (e.g., of quantum fields) whose representations in Hilbert spaces need not be normed. Therefore, the generalization of the conven-

Chapter 3 Algebraic Quantization

235

tional GNS representation of C*-algebras to some classes of unnormed topological involutive algebras has been studied. By an operator in a Hilbert (or Banach) space E is meant a linear morphism a of a dense subspace D(a) of E to E. The D(a) is called a domain of an operator a. One says that an operator b on D(b) is an extension of an operator a on D(a) if D(a) C D(b) and &|.D(a) = a. For the sake of brevity, we will write a C b. An operator a is said to be bounded on D(a) if there exists a real number r such that ||oe||
e G £>(a).

If otherwise, it is called unbounded. Any bounded operator on a domain Z)(a) is uniquely extended to a bounded operator everywhere on E. Therefore, by bounded operators in E are usually meant bounded (continuous) operators defined everywhere on E. An operator a on a domain D(a) is called closed if the condition that a sequence {e{\ C D(a) converges to e G E and that the sequence {aej} does to e' G E implies that e £ -D(G) and e' = ae. Of course, any operator defined everywhere E is closed. An operator a on a domain D(a) is called closable if it can be extended to a closed operator. The closure of a closable operator a is defined as the minimal closed extension of a. Operators a and b in E are called adjoint if (ae|e'> = (e\be'), e € £>(a),

e' G D(b).

Any operator a has a maximal adjoint operator a*, which is closed. Of course, a C a** and b* c a* if a C b. An operator a is called symmetric if it is adjoint to itself, i.e., a C a*. Hence, a symmetric operator is closable. One can obtain the following chain of extensions of a symmetric operator: a C a C a** C a* = a* = a***. In particular, if a is a symmetric operator, so are a and a**. At the same time, the maximal adjoint operator a* of a symmetric operator a need not be symmetric. A symmetric operator a is called self-adjoint if a = a*, and it is called essentially self-adjoint if a = a* = a*. It should be emphasized that a symmetric operator a is sometimes called essentially self-adjoint if a** = a*. We here follow the terminology of [360]. If a is a closed operator, the both notions coincide. For bounded operators, the notions of symmetric, self-adjoint and essentially self-adjoint operators coincide.

236

Geometric and Algebraic Topological Methods in Quantum Mechanics

Let E be a Hilbert space. The pair (B, D) of a dense subspace D of E and a unital algebra B of (unbounded) operators in E is called the Op*-algebra (O*-algebra in the terminology of [394]) on the domain D if, whenever b £ B, we have: (i) D(b) = D and bD C D, (ii) D C £>(&*), (iii) 6* |D C B [222; 360]. The algebra B is provided with the involution b\-+ b+ = b*\o, and its elements are closable. A representation TT(A) of an involutive algebra A in a Hilbert space E is an Op*-algebra if there exists a dense subspace D(ir) C E such that D(n) = D(n(a)) for all a £ A and this representation is Hermitian, i.e., 7r(a*) C vr(a)* for all a € A. In this case, one also considers the representations W:a->W(a):=ir(a)\D(^,

D(W) = Q D(7r(a)),

7T* : a -» 7T*(a) := 7r(a*)*|D(7r.);

£ ( 0 = f| £>(ir(a)'), a£A

7T** : a -»7r**(a) := 7r*(aT| D(7r .. );

£ ( O = f ) i?(7r'(a)'),

called the closure of a representation TT, an adjoint representation and a second adjoint representation, respectively. There are the representation extensions 7T C 7f C 7T** C 7T*,

where TTI C TT2 means -D(TTI) C £)(TT2). The representations n and ?r** are Hermitian, while TT* = TT* = TT***. A Hermitian representation n(A) is said to be closed if vr = ?f, and it is self-adjoint if vr = TT*. Herewith, a representation 7T(J4) is closed (resp. self-adjoint) if one of operators ir(A) is closed (resp. self-adjoint). The representation domain D(ir) is endowed with the graph-topology. It is generated by the neighborhoods of the origin U(M,e) = {x€D(n)

: £

||7r(a)z|| < e},

where M is a finite subset of elements of A. All operators of TT(A) are continuous with respect to this topology. Let us note that the graph-topology

Chapter 3 Algebraic Quantization

237

is finer than the relative topology on -D(TT) C E, unless all operators ir(a), a G A, are bounded [394]. Let N denote the closure of a subset N C D(TT) with respect to the graph-topology. An element 6 e D(ir) is called strongly cyclic (cyclic in the terminology of [394]) if

D(TT) C R I ) 0 f . Then the GNS representation Theorem 3.1.7 can be generalized as follows [222; 394]. THEOREM 3.6.1. Let A be a unital topological involutive algebra and / a positive continuous form on A such that / ( I ) = 1 (i.e., / is a state). There exists a strongly cyclic Hermitian representation {^f,0f) of A such that

4>{a) = (7r(a)0*|fy),

a e A.

a We point out the particular class of nuclear barreled "-algebras. Let A be a locally convex topological involutive algebra whose topology is denned by a set of multiplicative seminorms pt which satisfy the condition pL(a*a)=pL(a)2,

a & A.

It is called a b*-algebra. A unital 6*-algebra as like as a C*-algebra is regular and symmetric, i.e., any element (1 + a*a), a € A, is invertible and, moreover, (1 -{-a*^^1 is bounded [5; 227]. The 6*-algebras are related to C*-algebras as follows. PROPOSITION 3.6.2. Any 6*-algebra is the Hausdorff projective limit of a • family of C*-algebras, and vice versa [227].

In particular, every C*-algebra A is a barreled 6*-algebra, i.e., every absorbing balanced closed subset is a neighborhood of the origin of A. Let us additionally assume that A is a nuclear algebra (i.e., a nuclear space). Then we have the following variant of the GNS representation theorem [227] 3.6.3. Let A be a unital nuclear barreled 6*-algebra and / a positive form on A. There exists a unique (up to unitary equivalence) cyclic representation 717 of A in a Hilbert space by operators on a common invariant domain D. This domain can be topologized to conform a rigged THEOREM

238

Geometric and Algebraic Topological Methods in Quantum Mechanics

Hilbert space such that all the operators representing A are continuous on D. D The following is an example of a nuclear barreled b*-algebra which is very familiar from quantum field theory. Let Q be a nuclear space. Let us consider the direct limit §Q = C © Q © Q § Q © - - . Q ® n © . . .

(3.6.1)

of the vector spaces ®~nQ = C © Q © Q®Q • • • © Q®n, where ® is the topological tensor product with respect to the Grothendieck's topology (which coincides with the e-topology on the tensor product of nuclear spaces [356]). The space (3.6.1) is provided with the inductive limit topology, the finest topology such that the morphisms ®~nQ —» ®Q are continuous and, moreover, are imbeddings [417]. A convex subset V of ®Q is a neighborhood of the origin in this topology if and only if V D ®~~nQ is so in ®~ Q. Furthermore, one can show that ®Q is a unital nuclear barreled LF-algebra [31]. The LF-property implies that a linear form / on ®Q is continuous if and only if the restriction of / to each §~"Q is so [417]. If a continuous conjugation * is defined on Q, the algebra ®Q is involutive with respect to the operation *(9i®"-®g n ) = 9n®-"9i

(3-6-2)

on Q®" extended by continuity and linearity to Q®n. One can show that ®Q is a b*-algebra as follows. Since Q is a nuclear space, there is a family ||.||fe, k £ N + , of continuous norms on Q. Let Qk denote the completion of Q with respect to the norm ||.||fe. Then one can show that the tensor algebra ®Qk is a C*-algebra and that ®Q (3.6.1) is the projective limit of these C*-algebras with respect to morphisms ®Qk+i —> ®Qk [227]. In quantum field theory, one usually choose Q the space of functions of rapid decrease or the Schwartz space. 3.7

Example. Infinite canonical commutation relations

The canonical commutation relations (henceforth the CCR) are of central importance in many quantum models as the traditional method of canonical

Chapter 3 Algebraic Quantization

239

quantization. A remarkable result about the CCR for finite degrees of freedom is the Stone-von Neumann uniqueness theorem which states that all irreducible representations of the CCR for n degrees of freedom are unitarily equivalent [354]. On the contrary, the CCR for infinite degrees of freedom admit infinitely many inequivalent irreducible representations (see [157] for a survey). One can find the comprehensive description of representations of the CCR modelled over an infinite-dimensional nuclear space Q in [164]. Let Q be a real nuclear space provided with a non-degenerate separately continuous Hermitian form (.|.). This Hermitian form makes Q into a Hausdorff pre-Hilbert space. A nuclear space Q, the completion Q of the pre-Hilbert space Q, and the (topological) dual Q' of Q make up the rigged Hilbert space Q C Q C Q' (3.1.12). Given a real nuclear space Q together with a non-degenerate separately continuous Hermitian form (.|.), let us consider the group G(Q) of triples 9 = (9i)<72,A) of elements q\, qi of Q and complex numbers A of unit modulus which are subject to multiplications (9i»92, W n g * A') = (qi +
(3-7.1)

It is a Lie group whose group space is a nuclear manifold modelled over W = Q®Q®R.

(3.7.2)

Let us denote T(q) = (q,0,0),

P(q) = (0,q,0).

Then the multiplication law (3.7.1) takes the form T(q)T(q') = T(q + q'), P(q)P(q') = P(q + q'),

(3.7.3)

P(q)T(q')=exp[i(q\q')}T(q')P(q). Written in this form, G(Q) is called the nuclear Weyl CCR group. The complexified Lie algebra of the nuclear Lie group G(Q) is the Heisenberg CCR algebra Q{Q). It is generated by the elements <j>{q), n(q), q G Q, and / which obey the Heisenberg CCR commutation relations [(q),I] = [n(q), I] = 0, \-K{q),4>{q')] = -i(q\q')I.

\cj>(q), <j>(q')} = [*(
240

Geometric and Algebraic Topological Methods in Quantum Mechanics

There is the exponential map T{q) = exp[^( 9 )],

P(q) = exp[t7r(g)].

Due to the relation (3.1.11), the normed topology on the pre-Hilbert space Q defined by the Hermitian form (.|.) is coarser than the nuclear space topology. The latter is metric, separable and, consequently, secondcountable. Hence, the pre-Hilbert space Q is also second-countable and, therefore, admits a countable orthonormal basis. Given such a basis {q^} for Q, the Heisenberg CCR (3.7.4) - (3.7.5) take the form [0(9j), 4>{
[n(qj),
-iSjkI.

The CCR group G(Q) contains two nuclear Abelian subgroups T(Q) and P(Q). Following the representation algorithm in [164], we first construct representations of the nuclear Abelian group T(Q). These representations under certain conditions can be extended to representations of the whole CCR group G(Q). One can think of the nuclear Abelian group T(Q) as being the group of translations in the nuclear space Q. Its cyclic strongly continuous unitary representation n in a Hilbert space (E, (.|.)s) with a (normed) cyclic vector 8 £ E defines the complex function Z(q) = (ir(T(q))e\6)E on Q. This function is proved to be continuous and positive-definite, i.e., Z(0) = 1 and

Y^Z{qi-qj)ciCj>Q for any finite set qi,... ,qmoielements of Q and arbitrary complex numbers Ci,.. . , c m .

In accordance with Bochner's theorem for nuclear spaces in Example 10.5.11, any continuous positive-definite function Z(q) on a nuclear space Q is the Fourier transform

Z{q) = J exp{i{q,u)}n(u)

(3.7.6)

of a positive measure \i of total mass 1 on the dual Q' of Q [164]. Then the above mentioned representation n of T(Q) can be given by the operators Tz(q)p(u) = ex.p[i(q, u)]p(u)

(3.7.7)

Chapter 3 Algebraic Quantization

241

in the Hilbert space L%.(Q',[i) of the equivalence classes of square p,integrable complex functions p(u) on Q'. The cyclic vector 9 of this representation is the ^-equivalence class 6 « M 1 of the constant function p{u) = 1. Then we have Z{q) = (Tz(q)e\6)fl = J exp[i(q, u)]fi.

(3.7.8)

Conversely, every positive measure /x of total mass 1 on the dual Q' of Q defines the cyclic strongly continuous unitary representation (3.7.7) of the group T(Q). By virtue of the above mentioned Bochner theorem, it follows that every continuous positive-definite function Z(q) on Q characterizes a cyclic strongly continuous unitary representation (3.7.7) of the nuclear Abelian group T(Q). We agree to call Z(q) a generating function of this representation. It should be emphasized that the representation (3.7.7) need not be (topologically) irreducible. For instance, let p(u) be a function on Q' such that the set where it vanishes is not a jii-null subset of Q'. Then the closure of the set Tz(Q)p is a T(Q)-invariant closed subspace of L^(Q', /i). One can show that distinct generating functions Z(q) and Z'(q) determine equivalent representations Tz and Tz> (3.7.7) of T(Q) in the Hilbert spaces LQ(Q', /x) and L^(Q', pi') if and only if they are the Fourier transform of equivalent measures on Q' [164]. Indeed, let / / = s2fi,

(3.7.9)

where a function s(u) is strictly positive almost everywhere on Q', and fi(s2) = 1. Then the map Ll(Q', fi') 9 p(u) -> s(u)p(u) e L2C(Q', IM)

(3.7.10)

provides an isomorphism between the representations Tz> and TzThe representation Tz (3.7.7) of the nuclear Abelian group T(Q) in the Hilbert space L}.(Q',n) determined by the generating function Z (3.7.6) can be extended to the CCR group G(Q) if the measure y, possesses the following property. Let uq, q £ Q, be an element of Q' given by the condition (ql,ug) = (q'\q),

q' € Q.

(3.7.11)

These elements form the range of the monomorphism Q —> Q' determined by the Hermitian form (.|.) on Q. Let the measure /x in (3.7.6) remains

242

Geometric and Algebraic Topological Methods in Quantum Mechanics

equivalent under translations Q' 3u^u

+ uqeQ',

UqtQcQ',

in Q', i.e., fi[u + ug) = a2(q, u)n{u),

uq £ Q C Q\

(3.7.12)

where a function a(q, u) is square /u-integrable and strictly positive almost everywhere on Q'. This function fulfils the relations a(0,u) = l,

a(q + q',u) = a(q,u)a(q',u + uq).

(3.7.13)

A measure on Q' obeying the condition (3.7.12) is called translationally quasi-invariant, but it does not remains equivalent under an arbitrary translation in Q', unless Q is finite-dimensional. Let a generating function Z of a cyclic strongly continuous unitary representation of the nuclear group T(Q) be the Fourier transform (3.7.6) of a translationally quasi-invariant measure /i on Q'. Then one can extend the representation (3.7.7) of this group to the unitary strongly continuous representation of the CCR group in the Hilbert space L^.(Q',/J,) by operators (3.2.7) in Example 3.2.2. These operators read Pz(q)p(u) = a(q, u)p(u + «,).

(3.7.14)

Remark 3.7.1. Let // (3.7.9) be a /u-equivalent positive measure of total mass 1 on Q'. The equality (i'(u + uq) = s~2(u)a2(q,u)s2(u + uq)n' (u) shows that it is also translationally quasi-invariant. Then the isomorphism (3.7.10) between representations Tz and Tz1 of the nuclear Abelian group T(Q) is extended to the isomorphism Pzr(q) = s~1Pz(q)s : p(u) i-> s~1(u)a(q,u)s(u + ug)p(u + uq) of the corresponding representations of the CCR group G(Q).

O

Similarly to the case of a finite-dimensional Lie group, any strongly continuous unitary representation Tz (3.7.7), Pz (3.7.14) of the nuclear CCR group G{Q) implies a representation of its Lie algebra Q{Q) by (unbounded)

243

Chapter 3 Algebraic Quantization

operators in the same Hilbert space L$.(Q',p,) [389]. This representation reads 1 = 1, (q)p(u) = (q,u)p(v),

ir{q)p(u) = -i(6q+T)(q,u))p(u),

5gp(u) = lim a~1[p(u + auq) — p(u)],

(3.7.15)

a € M,

r)(q,u) = lim a " 1 [a(a
(3.7.16)

a—»0

One derives at once from the relations (3.7.13) that SgSq' = Sq-6q,

5g(r){g',u)) = 6g,(r)(q,u)),

5q = -6-g, ij(0,u)=0,

6g({q',u)) = (q'\q), uGQ',

5qe = 0,

qeQ.

With the aid of these relations, it is easily justified that the operators (3.7.15) fulfil the Heisenberg CCR (3.7.4) - (3.7.5). The unitarity condition implies the conjugation rule (q,u)* = (q,u),

6* = -Sq - 2r)(q,u).

Hence, the operators (3.7.15) are Hermitian. Let us further restrict our consideration to representations with generating functions Z(q) such that I S i H Z(tq)

(3.7.17)

is an analytic function on R at t — 0 for all q £ Q. Then one can show that the function (q\u) on Q' is square /z-integrable for all q £ Q and that, consequently, the operators (q) (3.7.15) are bounded everywhere in the Hilbert space LQ(Q',(J,). Moreover, the mean values of operators 4>{q) can be computed by the formula W ? i ) • • • *(<&.)> = * - " ^ I • • • ^ ^ " V J O U ' - O =

(3.7.18)

(qi,u)---{qn,u)fi(u). The operators Tr(q) (3.7.15) act in the subspace E^ of all smooth complex functions in L^.(Q',fi) whose derivatives of any order also belongs to LciQ'iP)- However, Eoa need not be dense in the Hilbert space L^(Q',p,), unless Q is finite-dimensional. The space E^ is also the carrier space of a representation of the universal enveloping algebra Q{Q) of the CCR algebra

244

Geometric and Algebraic Topological Methods in Quantum Mechanics

Q{Q). The representations of G{Q) and Q{Q) in E^ need not be irreducible. Therefore, let us consider the subspace Eg = G(Q)0 of E^, where 6 is a cyclic vector for the representation of the CCR group in L^(Q',fi). Obviously, the representation of the CCR algebra Q(Q) in Eg is (algebraically) irreducible. If 6' is another cyclic vector in L^(Q',/x), the representations of G{Q) in Eg and Eg' are equivalent. One also introduces creation and annihilation operators ^(q)

= -T=0(g) T«r(g)] = ~7|[T^ T»?(«,«) + (<*•«)].

(3.7.19)

They obey the conjugation rule (a±(q))* = aT(q) and the commutation relations [o+ (q), a+ (q')} = [a~ (q),a~ (q1)} = 0.

[a-(q),a+(q')} = (q\q')l,

The particle number operator N in the carrier space Eg is defined by the conditions [N,a±(q)}=±a±(q) up to a summand Al. With respect to a countable orthonormal basis {%}, this operator TV is given by the sum N=^a+(qk)a-(qk),

(3.7.20)

k

but need not be denned everywhere in Eg, unless Q is finite-dimensional. Gaussian measures given by the Fourier transform 10.5.24 exemplify the physically relevant class of translationally quasi-invariant measures on the dual Q' of a nuclear space Q. Let fix denote a Gaussian measure on Q' whose Fourier transform is the generating function ZK=exp[-±BK(q)}

(3.7.21)

BK(q) = {K-xq\K-\),

(3.7.22)

with the covariance form

where K is a bounded invertible operator in the Hilbert completion Q of Q with respect to the Hermitian form (.|.). The Gaussian measure fix is translationally quasi-invariant, i.e., HK{U + uq) = a2K(q, u)fj.K(u).

Chapter 3 Algebraic Quantization

245

Using the formula (3.7.18), one can show that aK{q,u) = exp[-±BK(Sq) - ^(Sq,u)],

(3.7.23)

where 5 = KK* is a bounded Hermitian operator in Q. Let us construct the representation of the CCR algebra G(Q) determined by the generating function ZK (3.7.21). Substituting the function (3.7.23) into the formula (3.7.16), we find •n{
--{Sq,u).

Hence, the operators (q) = (q,u),

*(q) = -i{8q-^(Sq,u)).

(3.7.24)

Accordingly, the creation and annihilation operators (3.7.19) read a±(g) = -~=[TSq ± \(Sq,u) + (q,u)}.

(3.7.25)

They act on the subspace Eg, 9 «Mi(. 1, of the Hilbert space L^Q' ,IIK), and they are Hermitian with respect to the Hermitian form (.\.)^K on

Remark 3.7.2. If a representation of the CCR is characterized by the Gaussian generating function (3.7.21), it is convenient for a computation to express all operators into the operators 5q and {q), which obey the commutation relation

K4>tf)] = {Sq). The mean values {4>{qi) • • • (qn)dq) vanish, while the meanvalues {<j>(qi) • • • {qn)), defined by the formula (3.7.18), obey the Wick theorem relations ((qi) • • •
(3-7.26)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

where the sum runs through all partitions of the set 1,... ,n in ordered pairs (ii < 12), • • • (in-i < in), and where

imtw)) = (K-WK-W). D In particular, let us put K = \/2 • 1. Then the generating function (3.7.21) takes the form ZF(q)=exp[-^(q\q)},

(3.7.27)

and determines the Fock representation of the CCR algebra G(Q)- It is given by the operators (Q) = {q,u),

n(q) =

+

a {q) = -^[-5q+2(q,u)},

-i{6q-(q,u)), a,-{q) =-j=5q.

Its carrier space is the subspace Eg, 9 « w 1 of the Hilbert space L^(Q', /ip), where /XF denotes the Gaussian measure whose Fourier transform is (3.7.27). We agree to call it the Fock measure. The Fock representation up to an equivalence is characterized by the existence of a cyclic vector 9 such that a-(q)9 = 0,

qeQ.

(3.7.28)

For the representation in question, this is 9 « MF 1. An equivalent condition is that the particle number operator TV (3.7.20) exists and its spectrum is lower bounded. The corresponding eigenvector of N in E$ is 9 itself so that N9 = 0. Therefore, one often interprets this eigenvector as a vacuum state. A glance at the expression (3.7-25) shows that the condition (3.7.28) does not hold, unless ZK is Zp (3.7.27). For instance, the particle number operator in the representation (3.7.25) reads

TV" = XV( % >~(<7j) = Y}-6qiSqj + Si(qk,u)dqj + 3

3

(6km - ^S£,)< g f c l u>< 9 m ,u) - (Sjj - ^ ) ] , where {%} is the orthonormal basis for the pre-Hilbert space Q. One can show that this operator is denned everywhere on Ee and is lower bounded

247

Chapter 3 Algebraic Quantization

only if the operator S is a sum of the scalar operator 21 and a nuclear operator in Q, in particular, if

Tr(l - ±S) < oo. This condition is also sufficient for the measures ^K and fip (and, consequently, the corresponding representations) to be equivalent [164]. For instance, the generating function Zc(q)=exp[~(q\q)],

c2 / 1,

determines a non-Fock representation of the nuclear CCR. Since the Fock measure fip on Q' remains equivalent only under translations by vectors uq £ Q c Q', the measure Ha = /iF(u -
a£Q'\Q,

on Q' determines a non-Fock representation of the nuclear CCR. Indeed, this measure is translationally quasi-invariant: ^(u + Ug) =al(q,u)fia(u),

aa(q,u) =

aF(q,u-a),

and its Fourier transform Za(q)=exp[i(q,a)}ZF(q) is a positive-definite continuous function on Q. Then the corresponding representation of the CCR algebra is given by operators a+(q) = -j=(-6q + 2(q,u} - (q,*)),

a~(q) = - L ( J , + (q, a)). (3.7.29)

In comparison with all the above representations, these operators possess non-vanishing vacuum mean values (a±(q)e\e}^=^(q,a). If a G Q C Q', the representation (3.7.29) becomes equivalent to the Fock representation (3.7.25) due to the morphism p{u) H-> exp[-(q', u)]p(u + uq>).

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Geometric and Algebraic Topological Methods in Quantum Mechanics

Let us note that the non-Fock representation (3.7.24) of the CCR algebra (3.7.4) - (3.7.5) in the Hilbert space L%.(Q',fj,K) is the Fock representation K(q) = 4>{q) =

(q,u),

irK(q) = niS-'q) = -i(*f - l(q,u)),

6? = 6s-lq,

of the CCR algebra {4>K(q)^K(q),I}, where [K(q),*K(q)]=i(K-1q\K-1q')IThis fact motivates somebody to regard representations of the CCR group (3.7.3) as representation of two mutually commutative Abelian groups T(Q) and P(Q) up to phase multipliers [447]. For instance, one can consider a fc-deformation of the CCR algebra (see [229] for the finite-dimensional case). Let Q be the universal enveloping algebra of the Heisenberg CCR algebra Q{Q). It is the quotient of the tensor algebra ®W of the vector space W (3.7.2) by the relation (3.7.30)

I<8>I = I

and by the commutation relations (3.7.4) - (3.7.5), written with respect to the tensor product . In accordance with Example 10.2.1, the universal enveloping algebra Q is brought into the classical Hopf algebra UQ, characterized by the comultiplication A (to) = t o ® l + l ® t o ,

w G.W,

the counit e(w) = 0, the antipode S(w) = —to, and the universal matrix R=1®1. It is a quasi-triangular cocommutative Hopf algebra. Now let us consider the quotient Qk,c of the tensor algebra ®W by the relations (3.7.4), (3.7.30) and the commutation relations

HqUW)} = -W^k^k-iy

< 3 - 7 - 31 )

where k and c are strictly positive real numbers. Due to the relation (3.7.30), the right-hand side of the relations (3.7.31) is well defined on ®W, and we have

Hq),4>{q')} = - W ^ f c l f c - ! ) 7 '

(3-7'32)

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Chapter 3 Algebraic Quantization

Hence, Gk,c is the universal enveloping algebra of the Heisenberg CCR algebra Gk,c given by the commutation relations (3.7.4) and (3.7.32). This CCR algebra is modelled over the same nuclear space Q, but provided with the Hermitian form (qW)k,c = CktC(q\Q'),

( 3 - 7 - 33 )

Ck,c = ^ l - i y

The universal enveloping algebra Gk,c admits both the structure of the classical Hopf algebra UGk,c and the Hopf algebra UkGk,c, which differs from the classical one in the comultiplication law A (7) = 7 ® 1 + 1<8) 7, A(#?)) = 4>{q) ® kcI/2 + k~cI/2 ® 4>(q), A(n(q)) = *(q) ® kcI/2 + k~cI'2 ® 7r(g). One can think of UkGk,c as being a Hopf algebra of the /c-deformed CCR. It is readily observe that, if c = 1, the CCR algebras G{Q) and Gk,i coincide for any k, but the Hopf algebra UkGk,i differs from the classical one UGk,i = UG. If k = 1, then Gi,c = G{Q) and C/fc^i,c = W for any c. Since the Hopf algebra UkGk,c is the universal enveloping algebra of the CCR algebra Gk,c, its representations are determined in full by representations of Gk,c Comparing the commutation relations (3.7.5) and (3.7.32), one can show that, given a representation p of the CCR algebra G(Q), the CCR algebra Gk,c admits a representation pk,c by the operators pkA4>(q)) = p(0(«)).

pk,c(*(q)) = p(*{Ck,cg)),

PkAi) = PW =!.

where Ck,c is given by the expression (3.7.33). For instance, if p is the Fock representation of the CCR algebra G(Q), the representation pkiC is not equivalent to the Fock representation of the CCR algebra Gk,c, unless Q is finite-dimensional. 3.8

Automorphisms of quantum systems

Let us consider uniformly and strongly continuous one-parameter groups of automorphisms of C*-algebras. In particular, they characterize evolution of quantum systems. Forthcoming Remarks 3.8.1 and 3.8.2 explain why we restrict our consideration mainly to these automorphism groups.

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Geometric and Algebraic Topological Methods in Quantum Mechanics

Remark 3.8.1. Let V be a Banach space and B(V) the set of bounded endomorphisms of V. The normed, strong and weak operator topologies on B(V) are defined in the same manner as in Section 3.ID. Automorphisms of a C*-algebra are obviously its isometries as a Banach space. Any weakly continuous one-parameter group of endomorphism of a Banach space is also strongly continuous and their weak and strong generators coincide with each other [63]. • Remark 3.8.2. There is the following relation between morphisms of a C*-algebra A and the set E{A) of its states which is a convex subset of A'. A linear morphism 7 of a C*-algebra A as a vector space is called the Jordan morphism if the relations -y(ab + ba) =7(0)7(6) +7(6)7(0),

a,b e A.

hold. One can show the following [144]. Let 7 be a Jordan automorphism of a unital C*-algebra A. It yields the dual weakly* continuous affine bijection 7' of E(A) onto itself, i.e., 7 '(A/

+ (1 - A)/') = A 7 '(/) + (1 - A)7'( A f,f',£E(A), Ae[0,l]. Conversely, any such a map of E(A) is the dual to some Jordan automorphism of A. However, we are not concerned with groups of Jordan automorphisms because of the following fact. If G is a connected group of weakly continuous Jordan automorphisms of a unital C*-algebra A which is provided with a weak operator topology, then it is a weakly continuous group of automorphisms of A. D One says that a one-parameter group G(K) is a uniformly (resp. strongly) continuous group of automorphisms of a C*-algebra A if it is a range of a continuous map of M. to the group Aut (A) of automorphisms of A which is provided with the normed (resp. strong) operator topology and whose action on A is separately continuous. The problem is that, if a curve G(M.) in Aut (A) is continuous with respect to the normed operator topology, then the curve G(M)(a) for any a £ A is continuous in the C*algebra A, but the converse is not true. At the same time, a curve G(R) is continuous in Aut (A) with respect to the strong operator topology if and only if the curve G(M)(a) for any a £ A is continuous in A. By this reason, strongly continuous one-parameter groups of automorphisms of C*-algebras

Chapter 3 Algebraic Quantization

251

are most interesting. However, the infinitesimal generator of such a group fails to be bounded, unless this group is uniformly continuous. Remark 3.8.3. If G(R) is a strongly continuous one-parameter group of automorphisms of a C*-algebra A, there are the following continuous maps [63]: • R 3 t n (Gt(a), / ) G C is continuous for all a G A and / G A'; • A 3 a —» Gt(a) G A is continuous for all t G M; • R 3 t i - » Gt(a) G A is continuous for all o G A. • Let A be a C*-algebra. Without loss of generality, we will assume that A is a unital algebra. The space of derivations of A is provided with the involution u >—> u* defined by the equality 5*{a) := -6(a*)*,

a £ A.

(3.8.1)

Throughout this Section, by a derivation S of A is meant an (unbounded) symmetric derivation of A (i.e., S(a*) = S(a)*, a £ A) which is defined on a dense involutive subalgebra D(S) of A. If a derivation 6 on D(S) is bounded, it is extended to a bounded derivation everywhere on A. Conversely, every derivation defined everywhere on a C*-algebra is bounded [129]. For instance, any inner derivation S(a) = i[b,a], where b is a Hermitian element of A, is bounded. There is the following relation between bounded derivations of a C*-algebra A and one-parameter groups of automorphisms of A [63]. THEOREM 3.8.1. Let 5 be a derivation of a C*-algebra A. The following assertions are equivalent: • S is defined everywhere and, consequently, is bounded; • S is the infinitesimal generator of a uniformly continuous oneparameter group [Gt] of automorphisms of the C*-algebra A. Furthermore, for any representation n of A in a Hilbert space E, there exists a bounded self-adjoint operator H G it {A)" in E and the uniformly continuous representation n(Gt) = exp(-itH),

t £ R,

(3.8.2)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

of the group [Gt] in E such that 7r(<5(a)) = -i[H,7r(o)], itH

a £ A, itn

7r(Gt(a)) = e- n{a)e ,

t e R.

(3.8.3) (3.8.4)

a A C*-algebra need not admit non-zero bounded derivations. For instance, no commutative C*-algebra possesses bounded derivations. The following is the relation between (unbounded) derivations of a C*-algebra A and strongly continuous one-parameter groups of automorphisms of A [62; 361]. THEOREM 3.8.2. Let 5 be a closable derivation of a C*-algebra A. Its closure 5 is an infinitesimal generator of a strongly continuous one-parameter group of automorphisms of A if and only if (i) the set (1 + XS)(D(5) for any A G R \ {0} is dense in A, • (ii) ||(1 + A<5)(a)|| > ||a|| for any A 6 M and any a € A.

It should be noted that, if A is a unital algebra and 5 is its closable derivation, then 1 G D(5). Let us mention a more convenient sufficient condition of a derivation of a C*-algebra to be an infinitesimal generator of a strongly continuous one-parameter group of its automorphisms. A derivation S of a C*-algebra A is called well-behaved if, for each element a G D(5), there exists a state / of A such that /(o) = ||a||,

/(*(*)) = 0.

If 6 is a well-behaved derivation, it is closable [247], and obeys the condition (ii) in Theorem 3.8.2 [62; 361]. Then we come to the following. 3.8.3. If 5 is a well-behaved derivation of a C*-algebra A and obeys the condition (i) in Theorem 3.8.2, its closure 5 is an infinitesimal generator of a strongly continuous one-parameter group of automorphisms of A. D PROPOSITION

For instance, a derivation 5 is well-behaved if it is approximately inner, i.e., there exists a sequence of self-adjoint elements {bn} in A such that 6(a) = limi[6 n ,o], n

a & A.

Chapter 3 Algebraic Quantization

253

In contrast with the case of a uniformly continuous one-parameter group of automorphisms of a C*-algebra A, a representation of A does not imply necessarily a unitary representation (3.8.2) of a strongly continuous oneparameter group of automorphisms of A, unless the following. PROPOSITION 3.8.4. Let Gt be a strongly continuous one-parameter group of automorphisms of a C*-algebra A and S its infinitesimal generator. Let A admit a state / such that

|/(<5(a))| < A[/(a*a) + f{aa*)\1'2

(3.8.5)

for all a £ A and a positive number A, and let (it/,Of) be a cyclic representation of A in a Hilbert space Ef determined by / . Then there exist a self-adjoint operator H o n a domain D{H) C AOj in Ej and a strongly continuous unitary representation (3.8.2) of Gt in Ef which fulfils the relations (3.8.3) - (3.8.4) for 7r = vr/. • Let us note that the condition (3.8.5) in Theorem 3.8.4 is sufficient in order that the derivation 5 to be closable [247]. There is a general problem of a unitary representation of an automorphism group of a C*-algebra. For instance, let B{E) be the C*-algebra of bounded operators in a Hilbert space E. All its automorphisms are inner. Any (unitary) automorphism U of a Hilbert space E yields the inner automorphism a-+UaU-\

a£B(E),

(3.8.6)

of B{E). Herewith, the automorphism (3.8.6) is the identity if and only if U = Al, |A| = 1, is a scalar operator in E. It follows that the group of automorphisms of B(E) is the quotient PU(E) = U(E)/U(1),

(3.8.7)

called the projective unitary group of the unitary group U(E) with respect to the circle subgroup U(l). Therefore, given a group G of automorphisms of the C*-algebra B(E), the representatives Ug in U(E) of elements g £ G constitute a group up to phase multipliers, i.e., UgUg. = exp[ia(g,g')}Ugg/,

a(g,g') £ K.

Nevertheless, if G is a one-parameter weakly* continuous group of automorphisms of B(E) whose infinitesimal generator is a bounded derivation

254

Geometric and Algebraic Topological Methods in Quantum Mechanics

of B(E), one can choose the multipliers exp[ia(g,g')} = 1. Representations of groups by unitary operators up to phase multipliers are called projective representations [91; 429]. In a general setting, let G be a group and A a commutative algebra. An A-multiplier of G is a map £ : G x G —> A such that £(1G,3) = £(S,1G) = :U,

geG,

^(91,9293)^(92,93) = ^(91,92)^(9192,93),

9i G G.

For instance, £:GxG->lA£A is a multiplier. Two A-multipliers £ and £' are said to be equivalent if there exists a map f : G —> A such that

An ,4-multiplier is called exact if it is equivalent to the multiplier £ = 1A. The set of ^.-multipliers is an Abelian group with respect to the pointwise multiplication, and the set of exact multipliers is its subgroup. Let HM(G, A) be the corresponding factor group. If G is a locally compact topological group and A a Hausdorff topological algebra, one additionally requires that multipliers £ and equivalence maps / are measurable maps. In this case, there is a natural topology on HM(G, A) which is locally quasi-compact, but need not be Hausdorff [321]. PROPOSITION 3.8.5. [91]. Let G be a simply connected locally compact Lie group. Each f/(l)-multiplier £ of G is brought into the form £ = exp ia, where a is an R-multiplier. Moreover, £ is exact if and only if a is well. Any R-multiplier of G is equivalent to a smooth one. •

Let G be a locally compact group of strongly continuous automorphisms of a C*-algebra A. Let M(A) denote the multiplier algebra of A, i.e., the largest C*-algebra containing A as an essential ideal, i.e., if a € M(A) and ab = 0 for all b G A, then a — 0 [441]). For instance, M(A) = A if A is a unital algebra. Let £ be a multiplier of G with values in the center of M(A). A G-covariant representation n of A [132; 336] is a representation 7T of A (and, consequently, M(A)) in a Hilbert space E together with a

Chapter 3 Algebraic Quantization

255

projective representation of G by unitary operators U(g), g G G, in E such that <9{a)) - U(g)n(a)U*(g),

U(g)U(g') = n(t{g, g'))U(gg').

Chapter 4

Geometry of algebraic quantization

Algebraic quantum theory usually deals with Hilbert spaces. This Chapter addresses quantum models involving Hilbert manifolds and Hilbert bundles, but we restrict our consideration to particular Hilbert bundles over smooth finite-dimensional manifolds. For instance, this is the case of timedependent quantum systems and quantum models depending on classical parameters. Since a Hilbert space E is a real Banach space and the unitary group U(E) is a Banach-Lie group, we in fact deal with Banach manifolds and bundles. Their differential geometry is similar to differential geometry of finite-dimensional manifolds and bundles in main. In particular, the inverse mapping theorem and the Frobenius theorem hold. For instance, this is not the case of Frechet manifolds. We refer the reader to [262] for the wider class of infinite-dimensional smooth manifold and principal bundles modelled on so called convenient locally convex topological vector spaces. 4.1

Banach and Hilbert manifolds

We start with the notion of a real Banach manifold [273; 422]. Banach manifolds are defined similarly to finite-dimensional smooth manifolds, but they are modelled on Banach spaces, not necessarily finite-dimensional. Passing later to complex Hilbert manifolds, we also refer the reader to Section 2.6 for analogy tofinite-dimensionalHermitian and Kahler manifolds. Let us recall some particular properties of (infinite-dimensional) real Banach spaces (see Section 10.5A for a general case of topological vector spaces). Let us note that afinite-dimensionalBanach space is always provided with an Euclidean norm. • Given Banach spaces E and H, every continuous bijective linear map 257

258

Geometric and Algebraic Topological Methods in Quantum Mechanics

of E to H is an isomorphism of topological vector spaces. • Given a Banach space E, let F be its closed subspace. One says that F splits in E if there exists a closed complement F' of F such that E = F®F'. In particular, finite-dimensional andfinite-codimensionalsubspaces split in E. As a consequence, any subspace of a finite-dimensional space splits. • Let E and H be Banach spaces and / : E —> H a continuous injection. One says that / splits if there exists an isomorphism g:H-*H1xH2 such that g o f yields an isomorphism of E onto Hi x {0}. • Given Banach spaces (E, \\.\\E) and (H, ||.||ff), one can provide the set Hom°(£, H) of continuous linear morphisms of E to H with the norm ||/||:=

sup \\f(z)\\H,

f€Rom°(E,H).

(4.1.1)

||*|| E =i

In particular, the norm (10.5.1) on the topological dual E' of E is of this type. If E, H and F are Banach spaces, the bilinear map Hom°(£,F) x Horn °(F, H) -» Horn °(E,H), obtained by the composition / o g of morphisms 7 € Horn (E, F) and / £ Hom°(jF, H), is continuous. Let us note that this assertion is false for more general spaces, e.g., the Frechet ones. • Let (E, \\.\\E) and (H, \\.\\H) be real Banach spaces. One says that a continuous map / : E —> H (not necessarily linear and isometric) is a differentiable function between E and H if, given a point z G E, there exists an R-linear continuous map df(z)

:E^H

(not necessarily isometric) such that

IV) = /M + <M(')V -») + o(z' - z),

lim

y-«)a. = O i

for any 2' in some open neighborhood U of z. For instance, any continuous linear morphism / of E to H is differentiable and df(z)z = f{z). The linear map d/(z) is called a differential of / at a point z £ U. Given an element v £ E, we obtain the map E 3 z >-+dvf(z) = df(z)v € H,

(4.1.2)

Chapter 4 Geometry of Algebraic Quantization

259

called the derivative of a function / along a vector v G E. One says that / is two-times differentiable if the map (4.1.2) is differentiable for any v G E. Similarly, r-times differentiable and infinitely differentiable (smooth) functions on a Banach space are denned. The composition of smooth maps is a smooth map. The following inverse mapping theorem enables one to consider smooth Banach manifolds and bundles similarly to the finite-dimensional ones. 4.1.1. Let / : E —> H be a smooth map such that, given a point z € E, the differential df (z) : E —> H is an isomorphism of topological vector spaces. Then / is a local isomorphism at z. • THEOREM

Let us turn to the notion of a Banach manifold, without repeating the statements true both for finite-dimensional and Banach manifolds. 4.1.2. A Banach manifold B modelled on a Banach space B is defined as a topological space which admits an atlas of charts V&e = {(Ui,4>i)}, where the maps 4>L are homeomorphisms of UL onto open subsets of the Banach space B, while the transition functions (j>^(j)^1 from t(C/t n U{) C B to 4>^{Utr\U^) C B are smooth. Two atlases of a Banach manifold are said to be equivalent if their union is also an atlas. • DEFINITION

Unless otherwise stated, Banach manifolds are assumed to be connected paracompact Hausdorff topological spaces. A locally compact Banach manifold is necessarily finite-dimensional. Remark 4.1.1. Let us note that a paracompact Banach manifold admits a smooth partition of unity if and only if its model Banach space does. For instance, this is the case of (real) separable Hilbert spaces. Therefore, we restrict our consideration to Hilbert manifolds modelled on separable Hilbert spaces. • Any open subset U of a Banach manifold B is a Banach manifold whose atlas is the restriction of an atlas of B to U. Morphisms of Banach manifolds are defined similarly to those of smooth finite-dimensional manifolds. However, the notion of the immersion and submersion need a certain modification (see Definition 4.1.3 below). Tangent vectors to a smooth Banach manifold B are introduced by analogy with tangent vectors to a finite-dimensional one. Given a point Z G B , let us consider the pair (v; (Ut, ,,)) of a vector v £ B and a chart (£/t 9 z, 4>t)

260

Geometric and Algebraic Topological Methods in Quantum Mechanics

on a Banach manifold B. Two pairs (v; {Ut,4>L)) and {v'\ (f/ c ,0 c )) are said to be equivalent if

v'= dfc^XM*))*-

(4-1-3)

The equivalence classes of such pairs make up the tangent space TZB to a Banach manifold B at a point z e B. This tangent space is isomorphic to the topological vector space B. Tangent spaces to a Banach manifold B are assembled into the tangent bundle TB of B. It is a Banach manifold modelled over the Banach space B® B which possesses the transition functions

(^rSdo^:1))Any morphism / : B —> B' of Banach manifolds yields the corresponding tangent morphism of the tangent bundles Tf : TB —> TB'. 4.1.3. Let / : B —> B' be a morphism of Banach manifolds. (i) It is called an immersion at a point z £ B if the tangent morphism Tf at z is injective and splits. (ii) A morphism / is called a submersion at a point z £ B if Tf at z is surjective and its kernel splits. • DEFINITION

In the case of finite-dimensional smooth manifolds, the split conditions are superfluous, and Definition 4.1.3 recovers the notion of the immersion and submersion in Section 10.6A. The range of a surjective submersion / of a Banach manifold is a submanifold, though / need not be an isomorphism onto a submanifold, unless / is an imbedding. One can think of a surjective submersion n : B —» B' of Banach manifolds as a fibred Banach manifold. For instance, the product B x B' of Banach manifolds is a fibred Banach manifold with respect to prx and pr2. Let B be a Banach manifold and E a Banach space. The definition of a (locally trivial) vector bundle with the typical fibre E and the base B is a repetition of that of finite-dimensional smooth vector bundles. Such a vector bundle Y is a Banach manifold and Y —> B is a surjective submersion. The above mentioned tangent bundle TB of a Banach manifold exemplifies a vector bundle over B. Let Bnh be the category of Banach spaces and Vect(S) denotes the category of vector bundles over a Banach manifold B with respect to their

Chapter 4 Geometry of Algebraic Quantization

261

morphisms over idB. Let F : Bnh x Bnh -> Bnh

(4.1.4)

be a functor of two variables which is covariant in the first and contravariant in the second. Then there exists a functor VF : Vect(tf) x Vect(B) -> Vect(£) such that, if YE —> B and YR —» B are vector bundles with the typical fibres E and H, then VF(YE, YH) is a vector bundle with the typical fibre F(E,H). For instance, the Whitney sum, the tensor product, and the exterior product of vector bundles over a Banach manifold are defined in this way. In particular, since the topological dual E' of a Banach space E is a Banach space with respect to the norm (10.5.1), one can associate to each vector bundle YE —* B the dual Yg = YE> with the typical fibre E'. For instance, the dual of the tangent bundle TB of a Banach manifold B is the cotangent bundle T*B. Sections of the tangent bundle TB —> B of a Banach manifold are called vectorfieldson a Banach manifold B. They form a locally free module T\{B) over the ring C°° (B) of smooth real functions on B. Every vector field •# on a Banach manifold B determines a derivation of the R-ring C°°(B) by the formula f{z) -> dtf(z) = df(z)ti(z),

z G B.

Different vector fields yield different derivations. It follows that T\ (B) possesses a structure of a real Lie algebra, and there is its monomorphism T1(B)^-0Coo(B)

(4.1.5)

to the derivation module of the R-ring C°°(B). Let us consider the Chevalley-Eilenberg complex of the real Lie algebra 71 (S) with coefficients in C°°(B) and its subcomplex O*[TX{B)\ of C°°(S)multilinear skew-symmetric maps by analogy with the complex C*[0^4] in Section 1.6. This subcomplex is a differential calculus over a R-ring C°°(B) where the Chevalley-Eilenberg coboundary operator d (1.6.6) and the prod-

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Geometric and Algebraic Topological Methods in Quantum Mechanics

uct (1.6.9) read r

# ( 0 O , . . . , 0 r ) = ^ ( - 1 ) ^ Wtfo,.-., £ , . . . ,

tfr))+

(4.1.6)

i=0

X J ( - l ) i + J > ( [ t f i , ^ l . ^ o , . . . , * , . . . , 0 , , . . . ,
(4.1.7)

*l<-"
0 G Or[7i(B)],

0'GO'[TI(B)],

^GTi(B).

There are the familiar relations *\
= d*f,

feC°°(B),

0G7I(B),

d(4> A 4>') = d4>A<j>' + (-1)1*10 Ad<j>\

,4>' € 0 * [7I(B)].

The differential calculus O*[Ti(B)] contains the following subcomplex. Let O1(B) be the C°°(B)-module of global sections of the cotangent bundle T*B of B. Obviously, there is its monomorphism O1{B)-^-QC°°{B)*

(4.1.8)

to the dual of the derivation module T)C°°(B). Furthermore, let f\T*B be the r-degree exterior product of the cotangent bundle T*B and Or{B) the C°°(S)-module of its sections. Let O*(B) be the direct sum of C°°(B)modules Or{B), r £ N, where we put O°(B) := C°°(B). Elements of O*(B) are obviously C°°(S)-multilinear skew-symmetric maps of T\(B) to C°°(B). Therefore, the Chevalley-Eilenberg differential d (4.1.6) and the exterior product (4.1.7) of elements of O*(B) are well denned. Moreover, one can show that dcp and <j> h<j>', ,<j>' € O*(B), are also elements of O*(B). Thus, O*(B) is a differential graded commutative algebra, called the algebra of exterior forms on a Banach manifold B. At the same time, one can consider Chevalley-Eilenberg differential calculus O*[QC°°(B)] over the M-ring C°°(B). Because of the monomorphism (4.1.5), we have the homomorphism of C°°(5)-modules Ol$C°°{B)) = 3C°°(B)* -+ Ti(B)* = Ol[T1{B)]^Ol{B).

(4.1.9)

It follows that the differential calculi 0*[7i(B)], O*{B) and Ol[DC°°{B)\ over the M-ring C°°(B) are not mutually isomorphic in general. However, it is readily observed that the minimal differential calculi in O* [7i (B)] and

263

Chapter 4 Geometry of Algebraic Quantization

O*(B) coincide with the minimal Chevalley-Eilenberg differential calculus O*CC°(B) over the R-ring C°°(B) because they are generated by the elements df, f G C°°{B), where d is the restriction (4.1.6) to 71 (B) of the Chevalley-Eilenberg coboundary operator (1.6.6). A connection on a Banach manifold B is defined as a connection on the C°°(B)-module 71 (B) [422]. In accordance with Definition 1.3.2, it is an R-module morphism V : 7i(B) -• O1COO(B) ®Ti(B), which obeys the Leibniz rule V ( / 0 ) = # ® 0 + /V(tf),

f£C°°(B),

0€7i(B).

(4.1.10)

In view of the inclusions, OlC°°{B) c O\B) c TX{B)*,

Ti(B) C 71(B)** c

O\B)*,

it is however convenient to define a connection on a Banach manifold as an R-module morphism V:r 1 (B)-»O 1 (B)®Ti(B),

(4.1.11)

which obeys the Leibniz rule (4.1.10). Let us turn now to Hilbert manifolds. These are particular Banach manifolds modelled on complex Hilbert spaces, which are assumed to be separable (see Remark 4.1.1). An important case of Hilbert manifolds are infinite-dimensional Kahler manifolds [105; 327] (see Section 2.6 for geometry of finite-dimensional Kahler manifolds). Remark 4.1.2. We refer the reader to [273] for the theory of real Hilbert and (infinite-dimensional) Riemannian manifolds. A real Hilbert manifold is a Banach manifold B modelled on a real Hilbert space V. It is assumed to be connected Hausdorff and paracompact space admitting the partition of unity by smooth functions (this is the case of a separable V). A Riemannian metric on B is defined as a smooth section g of the tensor bundle V2T*B such that g(z) is a positive non-degenerate continuous bilinear form on the tangent space TZB. This form yields two maps TZB —» T*B. It is said to be non-degenerate if these maps are continuous isomorphisms. In infinite-dimensional geometry, the most of local results follow from general arguments analogous to those in the finite-dimensional case. In particular, a Riemannian metric makes B into a metric space. As

264

Geometric and Algebraic Topological Methods in Quantum Mechanics

in the finite-dimensional case, B admits a unique Levi-Civita connection. The global theory of real Hilbert manifolds is more intricate. For instance, an infinite-dimensional real (and, consequently, complex) Hilbert space V is diffeomorphic to V \ {0}, and the unit sphere in V is a deformation retract of V [36]. • A complex Hilbert space (E, {•].)) can be seen as a real Hilbert space E 3 v i-> vR e

CUR,«R)

ER,

=Re(v\v'),

in Remark 3.1.7 equipped with the complex structure JUR = (W)R. We have {Jvu, Jv^) = (vu,vR),

(JVR,I4)

= Im(vR,vR).

Let E JER denote the complexification of EM provided with the Hermitian form (.|.)c (3.1.4). The complex structure J on ER is naturally extended to Ec by letting J o % = i o J. Then Ec is split into the two complex subspaces Ec = E1*0 © E 0 ' 1 ,

(4.1.12)

10

E ' = {vu- Uvm : vu G Eu}, which are mutually orthogonal with respect to the Hermitian form (.|.)cSince {vR - iJvK)\v'R - Uv^) = 2(v\v'),

{vR + iJMvii + iJvn) = 2 ( w » ,

there are the following linear and antilinear isometric bijections EBv^vu-*

-/=(VR

-

IJVR)

e E1'0,

They make ^ l l 0 and E0'1 isomorphic to the Hilbert space E and the dual Hilbert space E", respectively. Hence, the decomposition (4.1.12) takes the form (4.1.13)

EC = E®~E. The complex structure J on the direct sum (4.1.13) reads J : E © E 3 i ) + u i - > iv-iu€

E®E~,

(4.1.14)

Chapter 4 Geometry of Algebraic Quantization

265

where E and E are the (holomorphic and antiholomorphic) eigenspaces of J characterized by the eigenvalues i and —i, respectively. Let / be a function (not necessarily linear) from a Hilbert space E to a Hilbert space H. It is said to be differentiable if the corresponding function /K between the real Banach spaces J5R and H^ is differentiable. Let dfR(z), z G £7R, be the differential (4.1.2) of /R on £ R which is a continuous linear morphism EM 3 vu H-> dfR(z)vu

E Hu

between real topological vector spaces _ER and H$L- This morphism is naturally extended to the C-linear morphism £ C 3 B C H dfa(z)vc G Hc

(4.1.15)

between the complexifications of ER and HR- In view of the decomposition (4.1.13), one can introduce the C-linear maps

dfo(z)(v + u) := dfa{z)v,

df(z){v + u) := dfR{z)u

from E © E to .He such that

<#kO)«c = dfu(z)(v + u) = dfR(z)v + dfR(z)u. Let us split

fu(z) = f(z)+J(Z) in accordance with the decomposition He = H ® H. Then the morphism (4.1.15) takes the form dfa{z)(v + u) = df(z)v + df{z)u + d]{z)v + dj{z)u,

(4.1.16)

where df = df, d f = df. A function / : E —> H is said to be holomorphic (resp. antiholomorphic) if it is differentiable and df(z) = 0 (resp. df(z) = 0) for all z & E. A holomorphic function is smooth, and is given by the Taylor series. If / is a holomorphic function, then the morphism (4.1.16) is split into the sum

dfa(z)(v + u)= df(z)v + dj(z)u of morphisms E —> H and E~ —>H~. Example 4.1.3. Let / be a complex function on a Hilbert space E. Then /R = (Re/,Im/)

266

Geometric and Algebraic Topological Methods in Quantum Mechanics

is a map of E to M2. The differential dfa{z), z € E, of / R yields the complex linear morphism E®T5 3vc^

(dRef(z)vc,dLmf(z)vc)

>-> d(Re/ + ilm/)(z)v c e C ,

which is regarded as a differential df(z) of a complex function / on a Hilbert space E. O A Hilbert manifold V modelled on a Hilbert space E is denned as a real Banach manifold modelled on the Banach space Em. which admits an atlas {(Ui, (pi)} with holomorphic transition functions (j>$<j>~1. Let CTV denote the complexified tangent bundle of a Hilbert manifold V. In view of the decomposition (4.1.13), each fibre CTZV, z e V, of CTV is split into the direct sum CTZV = TZV®TZV of subspaces TZV and TZV, which are topological complex vector spaces isomorphic to the Hilbert space E and the dual Hilbert space E, respectively. The spaces CTZV, TZV and TZV are respectively called the complex, holomorphic and antiholomorphic tangent spaces to a Hilbert manifold V at a point z £ V. Since transition functions of a Hilbert manifold are holomorphic, the complex tangent bundle CTV is split into a sum CTV = TV®TV of holomorphic and antiholomorphic subbundles, together with the antilinear bundle automorphism TV®TV3v + u^>v + ueTV®TV and the complex structure J : TV ®TV 3 v + u H^ iv - iu € TV © TV.

(4.1.17)

Sections of the complex tangent bundle CTV —» V are called complex vector fields on a Hilbert manifold V. They constitute the locally free module CT\{V) over the ring C°°(V) of smooth complex functions on V. Every complex vector field d + v on V yields a derivation

f(z)^df(z)(0 + v) = df(z)4(z) + df(z)v(z),

feC°°(V), z&V,

oftheC-ring
267

Chapter 4 Geometry of Algebraic Quantization

complex vector spaces isomorphic to E © E. Since Hilbert spaces are reflexive, the complex tangent bundle CTV is the dual of CT*V. The complex cotangent bundle CT*V is split into the sum CT*V = T*V © T*V

(4.1.18)

of holomorphic and antiholomorphic subbundles, which are the annihilators of antiholomorphic and holomorphic tangent bundles TV and TV, respectively. Accordingly, CT*V is provided with the complex structure J via the relation (v, Jw) = (Jv,w),

v G CTZV,

w G CT^V,

zeV.

Sections of the complex cotangent bundle CT*V —> V constitute a locally free C°°(-p)-module Ol{V). It is the C°°(P)-dual O1(V) = CTi(Vy

(4.1.19)

of the module CTi(V) of complex vector fields on V, and vice versa. Similarly to the case of a Banach manifold, let us consider the differential calculi O*[TiCP)], 0*{V) (further denoted by C*{V)) and O1[5COC(P)] over the C-ring C 0 0 ^ ) . Due to the isomorphism (4.1.19), 0*{Ti(V)] is isomorphic to C*(V), whose elements are called complex exterior forms on a Hilbert manifold V. The exterior differential d on these forms is the Chevalley-Eilenberg coboundary operator k

d(-d0,..., 0 fc ) = ^ ( - i r # ( t f 0 , . . . , £ , . . . , 0fc)0i +

(4.1.20)

t=0

x ; ( - i ) i + W i . ^ - ] . 0 o , . . . , & , . . . , £ , • , . . . ,,>fc),

^ G cTi(p).

In view of the splitting (4.1.18), the differential graded algebra C*(V) admits the decomposition C*{V) = © Cp'"(V) p,q=o

pq

into subspaces C ' {V) of p-holomorphic and q-antiholomorphic forms. Accordingly, the exterior differential d on C* (T7) is split into a sum d = d + 9 of holomorphic and antiholomorphic differentials d : Cp'q(V) -> Cp+ll9(73),

9 : Cp'q{V) -> C P ' 9+1 (P),

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Geometric and Algebraic Topological Methods in Quantum Mechanics

A Hermitian metric on a Hilbert manifold V is denned as a complex bilinear form g on fibres of the complex tangent bundle CTV which obeys the following conditions: • g is a smooth section of the tensor bundle CT*V CT*V —> V\ • gi'dzi'&'z) = 0 if complex tangent vectors i?z,i9z € CTZV are simultaneously holomorphic or antiholomorphic; • gi'dz^z) > 0 for any non-vanishing complex tangent vector -dz € CTZV; • the bilinear form 0(tfz,i9z), i9j,,i9z e CTZV, defines a norm topology on the complex tangent space CTZV which is equivalent to its Hilbert space topology. As an immediate consequence of this definition, we obtain

WM

= g@zX),

g(J#z, J#'z) = 9(W*)-

A Hermitian metric exists, e.g., on paracompact Hilbert manifolds modelled on separable Hilbert spaces. The above mentioned properties of a Hermitian metric on a Hilbert manifold are similar to properties of a Hermitian metric on a finite-dimensional complex manifold in Section 2.6. Therefore, one can think of the pair (V, g) as being an infinite-dimensional Hermitian manifold. A Hermitian manifold (V, g) is endowed with a non-degenerate exterior two-form tyW,)

= 9(J#z,K),

tiz,ti'z£CTzV,

z€V,

(4.1.21)

called the fundamental form of the Hermitian metric g. This form satisfies the relations

WM

= to@*X), £l(Jz#z,Jz#'z)=n(#z^'z)-

If ft (4.1.21) is a closed (i.e., symplectic) form, the Hermitian metric g is called a Kdhler metric and 0. a Kdhler form. Accordingly, (V, g, Q) is said to be an infinite-dimensional Kdhler manifold. A Kahler metric g and its Kahler form ( l o a a Hilbert manifold V yield the bundle isomorphisms £ : CTV 9 ^ 1 / j f l e CT*V, nb : CTV 3v»-v\Q,£ CT*V. Let g" and fi" denote the inverse bundle isomorphisms CT*V —> CTV.

269

Chapter 4 Geometry of Algebraic Quantization

They possess the properties fi" = Jg*,

fi«(«;,).K = -n J K)jw z , wz,w'z G cr*p. In particular, every smooth complex function / € C 0 0 ^ ) on the Kahler manifold (V,g) determines: • the complex vector field 9Hdf) = g\df) + g*(df),

(4.1.22)

which is split into holomorphic and antiholomorphic parts g^(df) and

gHdf);

• the complex Hamiltonian vector field n«(#) = J(gHdf)) = -ig*(df) + igHdf);

(4.1.23)

• the Poisson bracket {/,/'} =

fi«(d/)J4f',

/ , / ' e C°°(P).

(4.1.24)

By analogy with the case of a Banach manifold, we modify Definition 1.3.2 and define a connection V on a Hilbert manifold V as a C-module morphism V : C7I0P) -» C\V) ® CTi(P), which obeys the Leibniz rule V(/0) = # ® 0 + /V(i9),

fsC°°(V),

tiGCT^P).

Similarly, a connection is introduced on any C°°(P)-module, e.g., on sections of tensor bundles over a Hilbert manifold V. Let D and D denote the holomorphic and antiholomorphic parts of V, and let V^ = #J V, D$ and D-g be the corresponding covariant derivatives along a complex vector field 1? on V. For any complex vector field i? = v + v on 7>, we have the relations A? = V,,,

A , = Vu,

£>j,9 = i£>0,

Dj^ = -iDtf.

There is the following generalization of Theorem 2.6.8 to infinitedimensional Kahler manifolds.

270

Geometric and Algebraic Topological Methods in Quantum Mechanics

PROPOSITION 4.1.4. Given a Kahler manifold (P,g), there always exists a metric connection on V such that Vg = 0,

Vfi = 0,

V J = 0,

where J is regarded as a section of the tensor bundle CT*V ® CTV-

•

Example 4.1.4. If V = E is a Hilbert space, then CTV = Ex ( £ © £ ) . A Hermitian form (.|.) on E defines the constant Hermitian metric g : (E © E) x (E © E) -> C, 5(0,0') = H u ' } +

0 = v + u, tf' = T / + <

(4.1.25)

onV = E. The associated fundamental form (4.1.21) reads fi(#,tf')= j( v | u ') _ i( w '|u).

(4.1.26)

It is constant on E. Therefore, dO = 0 and g (4.1.25) is a Kahler metric. • The metric connection on E is trivial, i.e., V = d, D — d, D = d. Example 4.1.5. Given a Hilbert space E, a projective Hilbert space PE is made up by the complex one-dimensional subspaces (i.e., complex rays) of E. This is a Hilbert manifold with the following standard atlas. For any non-zero element x £ E, let us denote by x a point of PE such that x G x. Then each normalized element h £ E, \\h\\ = 1, defines a chart (Uh,4>h) of the projective Hilbert space PE such that 0h(x) = T ^ T - h. (4.1.27) (x\h) The image of Uh in the Hilbert space E is the one-codimensional closed (Hilbert) subspace Uh = {x£PE

: (x\h)^0},

Eh = {z£E

: {z\h)=0},

(4.1.28)

where z(x) + /i G x. In particular, given a point x £ PE, one can choose the centered chart Eh, h £ x, such that 4>h(x) = 0. Hilbert spaces Eh and Eh> associated to different charts Uh and Uh' are isomorphic. The transition function between them is a holomorphic function (4.1.29)

271

Chapter 4 Geometry of Algebraic Quantization

from (f>h(Uh nUh>) C Eh to <j>h>{Uh r\Uh>) C Eh>. The set of the charts

{(Uh,h)} with the transition functions (4.1.29) provides a holomorphic atlas of the projective Hilbert space PE. The corresponding coordinate transformations for the tangent vectors to PE at x € PE reads (4.1.30)

v' = -^[(x\h)v-x{v\h)].

The projective Hilbert space PE is homeomorphic to the quotient of the unitary group U(E) equipped with the normed operator topology by the stabilizer of a ray of E. It is connected and simply connected [104]. The projective Hilbert space PE admits a unique Hermitian metric g such that the corresponding distance function on PE is p(x,x') = v/2arccos(|(a;|3;/)|),

(4.1.31)

where x,x' are normalized elements of E. It is a Kahler metric, called the Fubini-Studi metric. Given a coordinate chart (Uh,4>h), this metric reads _ (A A \ fol|"2) + fal"l) 9FS{VUM2) = 1 + ||Z||2

(Z\u2)(vi\z} + (z\Ul){V2\z) (1 + INI2)2 '

(4.1.32) for any complex tangent vectors $i = V\ + u\ and ^2 = V2+U2 in CTZPE. The corresponding Kahler form is given by the expression

nFS(#M

=r

fTpp

i

(1+ N | 2 ) 2

• (4-1-33)

It is readily justified that the expressions (4.1.32) - (4.1.33) are preserved under the transition functions (4.1.29) - (4.1.30). Written in the coordinate chart centered at a point z(x) = 0, these expressions come to the expressions (4.1.25) and (4.1.26), respectively. •

4.2

Dequantization

The standard approach to canonical quantization of classical Hamiltonian systems implies the assignment of Hermitian operators / to smooth functions / on a Poisson manifold (Z,{,}) such that the Poisson bracket of functions {/,/'} is replaced with the commutator of operators [/, /'] and Dirac's condition llf'} = -i(fJr},

fi=l,

(4.2.1)

272

Geometric and Algebraic Topological Methods in Quantum Mechanics

holds. Here, we show that, conversely, any C*-algebra can be reproduced as a Poisson algebra of smooth functions on a projective Hilbert space [104; 105]. One can think of this procedure as being dequantization [238]. As is well-known (see Theorem 4.2.1 below), any commutative C*-algebra is represented by continuous functions on its spectrum. The above mentioned dequantization in fact generalizes this theorem to non-commutative C*algebras. Let X be a locally compact space. Continuous complex functions vanishing at infinity on X are assembled into a C*-algebra A with respect to the norm | | / | | = sup 1/(^1, feA. (4.2.2) xex If X is compact, A = C°(X) is the unital algebra of continuous complex functions on X. Remark 4.2.1. If X is a compact smooth manifold, the algebra C°°(X) of smooth complex functions on X is a dense subalgebra of the C*-algebra C°(X) of continuous functions on X, but it is not a C*-algebra. D Let A be an arbitrary commutative C*-algebra. Any its irreducible representation is one-dimensional and, consequently, is a character of A, i.e., a homomorphism \ : A —> C of a C-algebra A to the field C. Therefore, the spectrum A of a commutative C*-algebra reduces to the set of characters of A provided with the weak topology as a subset of the dual A' of A. The spectrum A is Hausdorff and, consequently, locally compact. Then A — Prim(j4) in accordance with Proposition 3.1.4. Given an element a G A, the complex function Ta : X - X(a)

(4.2.3)

on A is the well-known Gelfand map. THEOREM 4.2.1. The Gelfand map (4.2.3) is an isomorphism of A to the C*-algebra of continuous complex functions vanishing at infinity on the • spectrum A [129].

In particular, the spectrum of the C*-algebra of continuous complex functions vanishing at infinity on a locally compact space X is homeomorphic to X. Let us turn now to the above-mentioned dequantization procedure. We start with the algebra B(E) of bounded operators in a Hilbert space

273

Chapter 4 Geometry of Algebraic Quantization

(E, (.|.)). Let PE be the projective Hilbert space in Example 4.1.5. Let us assign to each element a G B(E) the complex function / o (x) = (ax\x)

(4.2.4)

on PE, where x £ E is a normalized representative of x G PE. This assignment is a monomorphism of the complex vector space B(E) to the complex vector space C°°(PE) of smooth complex functions on PE such i > fa. In order to make the monomorphism (4.2.4) into a morphism that a* — of algebras, let us provide C°°(PE) with the multiplication

/ * / ' = / / ' + i4s(9/W.

(4-2-5)

where fips is the Kahler form (4.1.33). Then one can show that (4.2.6)

fab = fa*fb for all a, b G B(E) [104; 105]. Moreover, we have \\a\\2= s u p ( 7 a * / o ) ( x ) . x€P£

It follows from the expressions (4.1.24) and (4.2.5) that {/,/'}FS

= -t(/*/'-/'*/),

(4.2.7)

where {,}FS is the Poisson bracket with respect to the Kahler form f2pg (4.1.33) on the projective Hilbert space PE. Then combining the formulae (4.2.6) and (4.2.7) gives the relation {/a,/fc}FS = -»/[o,6],

a,b G B{E).

(4.2.8)

This relation is viewed as the converse Dirac 's condition, when one assigns to operators in a Hilbert space E the smooth functions on the projective Hilbert space PE such that a^

fa,

b>-+ fb,

[a, b] i-> i{fa,

fb}Fs-

Remark 4.2.2. The relation (4.2.8) is converse to Dirac's condition (4.2.1) up to complex conjugation, that can be corrected by passing to • complex conjugate functions a —> (x\ax). Let now A be an arbitrary unital C*-algebra, and let us consider the open surjection P{A) -> A, where P{A) is the space of pure states of A and A is the spectrum of A. In accordance with item (iii) of Theorem 3.1.10,

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Geometric and Algebraic Topological Methods in Quantum Mechanics

the inverse image PV{A) of P(A) —> A over a point TT £ A, (i.e., the set of pure states of A associated to the same irreducible representation n) is isomorphic to the projective Hilbert space PEn, where En the carrier space of IT. This isomorphism is a homeomorphism if P{A) is provided with the weak* topology and P^(A) C p(A) is endowed with the relative one [105; 398]. Let us assign to each element a £ A the complex function 4> - » fa(d>) =
(4.2.9)

on P(A) —» A. These functions can be subject to the fibrewise multiplication * (4.2.5) on each fibre P^(A) = PEn, where Qps is the Kahler form of the Fubini-Studi metric on PEn [105]. Then a H /„ is a monomorphism of A to the space of continuous complex functions C°(P(A)) on P(A) such that fab = fa*fb, 2

a,b€A,

sup (fa*fa){4>).

\\a\\ =

4>€P(A)

In particular, if A is a commutative C*-algebra, then P(A) = A and the function (4.2.9) restarts the Gelfand map (4.2.3). If a Hilbert space E admits a reproducing system, we have observed a different variant of dequantization of the C*-algebra B{E) by the product (3.3.11) of the mean values (3.3.9). Another dequantization is exemplified by Berezin's quantization in forthcoming Section.

4.3

Berezin's quantization

Berezin's quantization [4; 32; 145] in fact is a particular variant of dequantization. Given a complex Poisson algebra C°°(Z) on a symplectic manifold (Z,£l), one aims to construct an involutive algebra A whose dequantization by functions on Z belongs to C°°(Z). Herewith, A must satisfy the following conditions. (1) There is a family of involutive algebras A^ such that: (la) the index h runs through a subset R C M+ of strictly positive numbers which has 0 in its closure; (lb) A is a subalgebra of © AhheR

Chapter 4 Geometry of Algebraic Quantization

275

(2) There is a linear morphism $ : A —> C°°(Z) with the following properties: (2a) for any two distinct points z, z' € Z, there exists a function / e $(Jl) such that f{z) ^ f{z'); (2b) any two elements a, a1 £ A fulfil the condition ih $([a,a']) = - - { $ ( a ) , $ ( a ' ) } ; (2c) $(a*) = $(a) for all a € A. Prom the physical viewpoint, h plays the role of Plank's constant. The properties (2a) - (2b) are called the correspondence principle. Berezin's quantization is said to obey the weak correspondence principle if the morphism $ is defined on a linear subspace A' of A. Berezin's quantization is called special if it satisfies the following additional properties. (3) Each Ah is a subalgebra of the algebra of complex (not necessarily continuous) functions on Z. (4) The algebra A consists of complex functions a(h, z) on R x Z such that their restrictions to {h} x Z belong to A^ for any h € R. (5) The morphism *:J4-»C°°(Z)

is given by the formula $(a) = )im a(h,z),

a € A.

(4.3.1)

h—>0

It is readily observed that Berezin's quantization axioms do not direct to finding a map / —» / from a classical observable to its quantum counterpart, but to obtaining a quantum algebra A with the appropriate dequantization properties. These axioms are rather restrictive. Therefore, one mainly provides Berezin's quantization only of a particular class of Kahler manifolds (see Remark 4.3.2 below). Let Z be a non-compact Kahler manifold of complex dimension m. Let Z be coordinated by (za,~za) and equipped with a Riemannian metric 9 = 9apdza V dzp

and the Kahler form 9. = iga-pdza

hdz13.

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The corresponding Poisson bracket of smooth complex functions on Z reads {/, /'} = -ig^idafdpf - daf'd0f).

(4.3.2)

Let U C Z be an open dense subset, where the Kahler metric g is expressed

into a potential function F (see Remark 2.6.2). Let us consider the Hilbert space L^(U, fj,h) of the equivalence classes of complex functions on U which are square integrable with respect to the measure (4.3.3)

Let us provide L^U,^)

with the scalar product

(f\f')h = ch Jfj'vh, u which differs from the standard one (10.5.13) in a normalization constant Ch to be specified later. Let Eh c L^iUi^h) be the subspace of holomorphic functions (called the Bergman space). However, it may happen that Eh consists of only the zero constant. One can show that Eh is a closed Hilbert subspace of L^(U,/J,h), and that it admits the reproducing kernel Kh{z,z')=YJfk{z')h&),

(4.3.4)

where {fk}^L\ is a countable orthonormal basis for Eh such that

]T|/ fc (z)| 2 < oo,

zGZ,

k

[4; 145; 216]. It follows that the set Sh = {L=Kh(z,z'), ||k||£= £|/ fc (z)| 2 , zeZ}

(4.3.5)

k

is a reproducing system in the Hilbert space Eh with respect to the measure iih (4.3.3). Let us note that the reproducing kernel (4.3.4), called the Bergman kernel function, is independent of the choice of an orthonormal basis, and it is antiholomorphic in the first variable and holomorphic in the second

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Chapter 4 Geometry of Algebraic Quantization

one. The relations (3.3.5) - (3.3.6) hold. As a consequence, any function f E Eh obeys the equality (3.3.8) which reads

/(*) = /, = / f(z')K(z', z)nh(z').

(4.3.6)

Given the reproducing system (4.3.5), one can associate to each bounded operator a in the Hilbert space Eh the mean value function fa{z) (3.3.10) on Z. However, the product (3.3.11) of these functions need a minor modification in order to be utilized in Berezin's quantization procedure. Let us put

alz) = A£f) = (<**&)* _ A a{z) HU 2 «.IOh " El/*(*)l a

'

(4 3 7 )

(

}

fc

It is an element of Eh called the covariant symbol of an operator a. The product of covariant symbols (4.3.7) reads (a*b)(z) = ||6|| 2 (/a * fb)(z) = ab(z).

(4.3.8)

Example 4.3.1. Let Ph denote a projector onto the Hilbert subspace Eh- Then any bounded measurable function g on Z yields the bounded operator

Tg : f -» Ph(gf) = J(gmz)kZ*Hk{z)

(4.3.9)

in Eh- It is called the Toeplitz operator. The covariant symbol (4.3.7) of the Toeplitz operator Tg (4.3.9) coincides with the Berezin transform fg = Bhg(z) = fg(z')lK^Z'^\h(z') J Kh(z,z)

(4.3.10) •

of the function g.

Here, we restrict ourselves to the most interesting case when a potential function F is the logarithm of the reproducing kernel (4.3.4) and when, in accordance with the hypothesis (la), there exists a set R C K + such that Kh(z,z)=Xhex.p\—1F(Z)\,

L "

J

Afc = const.,

(4.3.11)

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for any h e R. With some additional technical assumptions, it can be shown that, choosing a normalization constant Ch in an appropriate way, one obtains A^ = 1. Let Ah denote the algebra (under the product (4.3.8)) of covariant symbols of all bounded operators in the Hilbert space Eh- One can think of elements of © Ah as being complex functions on R x Z. Let A be the linear h

subspace of all functions a in © A^ which can be represented in the form h

a(h,z) = a{0,z) + ^^(z)

+ —a2(z),

where a, ai, a-i are functions admitting analytic continuations to Z x Z. One can justify that (a*b)(h,z) =a{0,z)b(0,z) + o(h), ~ ~ ih ~ (a*b-b* a)(h, z) = - — {a, 6}(0, z) + o(h2), where {, } is the Poisson bracket (4.3.2). Consequently, A is desired special Berezin's quantization obeying the weak correspondence principle. Remark 4.3.2. It should be emphasized that the condition (4.3.11) is very restrictive. There are only a few instances where it is satisfied, e.g., Z = C m and Z is a bounded symmetric domain (i.e., it is a bounded open connected subset of Cn whose each point is the isolated fixed point of an involutive holomorphic automorphism of the domain; it is homogeneous, i.e., the group of its holomorphic automorphisms acts transitively) [51; 79; 248]. In order to make Berezin's quantization procedure applicable to a wide class of Kahler manifolds, one studies the case when the condition (4.3.11) is fulfilled only asymptotically as h -> 0 [145]. •

4.4

Hilbert and C*-algebra bundles

Due to the inverse mapping Theorem 4.1.1 for Banach manifolds, fibred Banach manifolds similarly to the finite-dimensional smooth ones can be defined as surjective submersions of Banach manifold onto Banach manifold. Locally trivial fibred Banach manifolds are called (smooth) Banach vector bundles. They are exemplified by vector (e.g., tangent and cotangent) bundles over Banach manifolds in Section 4.1. Here, we restrict our

Chapter 4 Geometry of Algebraic Quantization

279

consideration to particular Banach vector bundles whose fibres are C*algebras (seen as Banach spaces) and Hilbert spaces, but the base is a finite-dimensional smooth manifold. Note that sections of a Banach vector bundle B —> Q over a smooth finite-dimensional manifold Q constitute a locally free C°° (Q)-module B(Q). Following the proof of Proposition 1.8.9, one can show that it is a projective C°° (<2)-module. In a general setting, we therefore can consider projective locally free C°°(Q)-modules, locally generated by a Banach space. In contrast with the case of projective C°°(X) modules of finite rank, such a module need not be a module of sections of some Banach vector bundle. Let C —+ Q be a locally trivial topological fibre bundle over a finitedimensional smooth real manifold Q whose typical fibre is a C*-algebra A, regarded as a real Banach space, and whose transition functions are smooth. Namely, given two trivializations charts (Ui,ipi) and (U2,ip2) of C, we have the smooth morphism of Banach manifolds ipi o ip'1 : Ui n U2 x A -> Ui n U2 x A, where "01 °1p21\q€U1nU2

is an automorphism of A. We agree to call C ^ Q a bundle of C* -algebras. It is a Banach vector bundle. The C°°(Cj)-module C(Q) of smooth sections of this fibre bundle is a unital involutive algebra with respect to fibrewise operations. Let us consider a subalgebra A(Q) C C(Q) which consists of sections of C —> Q vanishing at infinity onQ. It is provided with the norm ||a|| = sup||a( 9 )||
a€A(Q),

(4.4.1)

but fails to be complete. Nevertheless, one extends A(Q) to a C*-algebra of continuous sections of C —» Q vanishing at infinity on a locally compact space Q. Remark 4.4.1. Let C —* Q be a topological bundle of C*-algebras over a locally compact topological space Q, and let C°(Q) denote the involutive algebra of its continuous sections. This algebra exemplifies a locally trivial continuous field of C* -algebras in [129]. Its subalgebra A°(Q) of sections vanishing at infinity on Q is a C*-algebra with respect to the norm (4.4.1). It is called a C* -algebra defined by a continuous field of C*-algebras. There

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are several important examples of C*-algebras of this type. For instance, every liminal C*-algebra A with a Hausdorff spectrum A is isomorphic to a C*-algebra defined by a continuous field (not necessarily locally trivial) of C*-algebras on A [129]. By virtue of Theorem 4.2.1, any commutative C*-algebra is isomorphic to the algebra of continuous complex functions • vanishing at infinity on its spectrum. Remark 4.4.2. One can consider a locally trivial bundle of C*-algebras C —> Q as a fibre bundle with the structure topological group Aut(A) of automorphisms of A. If a fibre bundle C is smooth, this group is necessarily • provided with a normed operator topology. Hilbert bundles over a smooth manifold are similarly defined. Let £ —> Q be a locally trivial topological fibre bundle over a finite-dimensional smooth real manifold Q whose typical fibre is a Hilbert space E, regarded as a real Banach space, and whose transition functions are smooth functions taking their values in the unitary group U(E) equipped with the normed operator topology. We agree to call £ —> Q a Hilbert bundle. It is a Banach vector bundle. Smooth sections of £ —> Q constitute a C°°(Q)-module £(Q), called a Hilbert module. Continuous sections of £ —> Q constitute a locally trivial continuous field of Hilbert spaces [129]. There are the following relations between bundles of <7*-algebras and Hilbert bundles. Let T(E) C B(E) be the C*-algebra of compact (completely continuous) operators in a Hilbert space E (see Remark 3.1.10). It is called an elementary C* -algebra. Every automorphism <j> of E yields the corresponding automorphism

T{E) -> Q with transition functions E - > Pli(q)E,

q£U,D

Uo

over a cover {Ut} of Q, we have the associated locally trivial bundle of elementary C*-algebras T(E) with transition functions T(E) - pa0(q)T(E)(paP(q))-\

qeUan

Up,

(4.4.2)

which are proved to be continuous with respect to the normed operator topology on T(E) [129]. The proof is based on the following facts.

Chapter 4 Geometry of Algebraic Quantization

281

• The set of degenerate operators (i.e., operators of finite rank) is dense in T(E). • Any operator of finite rank is a linear combination of operators P(,*ta • C - (C\v)S,

t,vX£E,

and even the projectors Pj onto £ 6 E. • Let £i,...,£2™ be variable vectors of E. If &, i — 1,..., 2n, converges to 77$ (or, more generally, (&|£j) converges to (r]i\r]j} for any i and j), then

uniformly converges to Pm,V2 + " ' " + -Pf)2n-l,f?2n-

Note that, given a Hilbert bundle E —> Q, the associated bundle of C*algebras B(E) of bounded operators in E fails to be constructed in general because the transition functions (4.4.2) need not be continuous. The opposite construction however meets a topological obstruction as follows [73; 86]. Let C —> Q be a bundle of C*-algebras whose typical fibre is an elementary C*-algebra T(E) of compact operators in a Hilbert space E. One can think of C —> Q as being a topological fibre bundle with the structure group of automorphisms of T(E). This is the projective unitary group PU(E) (3.8.7. With respect to the normed operator topology, the groups U(E) and PU(E) are the Banach Lie groups [213]. Moreover, U(E) is contractible if a Hilbert space E is infinite-dimensional [263]. Let {Ua,pap) be an atlas of the fibre bundle C —> Q with PU(E)-va\ued transition functions pa0- These transition functions give rise to the maps pap :UanUp^

U(E),

which however fail to be transition functions of a fibre bundle with the structure group U(E) because they need not satisfy the cocycle condition. Their failure to be so is measured by the [/(l)-valued cocycle ealpt =90-y9ay9a0-

This cocycle defines a class [e] in the cohomology group H2(Q;U(1)Q) of the manifold Q with coefficients in the sheaf U(1)Q of continuous maps of Q to U(l). This cohomology class vanishes if and only if there exists a

282

Geometric and Algebraic Topological Methods in Quantum Mechanics

Hilbert bundle associated to C. Let us consider the short exact sequence of sheaves 0 - + Z —> o, where CQ is the sheaf of continuous real functions on Q and the morphism 7 reads 7 : C°Q 3 f .-» exp(27ri/) e

U{\)Q.

This exact sequence yields the long exact sequence (1.7.16) of the sheaf cohomology groups O ^ Z —+ C Q —»tf(l)g —>ff 1 (Q;Z) —»••• tfp(Q;Z)

_ > f r P ( Q ; C § ) — > f P ( Q ; t f ( l ) g ) —>if p + 1 (Q;Z) — > . . . .

Since the sheaf C Q is fine and acyclic, we obtain at once from this exact sequence the isomorphism of cohomology groups

H2(Q,U(1)Q) = H3(Q,Z). The image of [e] in H3(Q,Z) is called the Dixmier-Douady class [129]. One can show that the negative — [e] of the Dixmier-Douady class is the obstruction class of the lift of Pf/(£;)-principal bundles to the C/(£')-principal ones [86]. Thus, studying Hilbert and C*-algebra bundles, we come to fibre bundles with unitary and projective unitary structure groups. 4.5

Connections on Hilbert and C*-algebra bundles

There are different notions of a connection on Hilbert and C* -algebra bundles which need not be obviously equivalent, unless bundles are finitedimensional. These are connections on structure modules of sections, connections as a horizontal splitting and principal connections. Given a bundle of C* -algebras C —> Q with a typical fibre A over a smooth real manifold Q, the involutive algebra C(Q) of its smooth sections is a C°° (Q)-algebra. Therefore, one can introduce a connection on the fibre bundle C —> Q as a connection on the C°°(Q)-algebra C(Q). In accordance with Definition 1.3.4, such a connection assigns to each vector field r on Q a symmetric derivation VT of the involutive algebra C(Q) which obeys the

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Chapter 4 Geometry of Algebraic Quantization

Leibniz rule VT(fa) = (T\df)a + fVTa,

/ € C°°(Q),

aeC(Q),

and the condition VTa* - (VTa)*. Let us recall that two such connections V r and V'r differ from each other in a derivation of the C°°(Q)-algebra C(Q). Then, given a trivialization chart C\u^UxA of C —> Q, a connection on C(Q) can be written in the form VT=Tm(q)(dm-6m(q)),

(4.5.1)

q£U,

where (qm) are local coordinates on Q and Sm(q) for all q € U are symmetric bounded derivations of the C*-algebra A. Remark 4.5.1. Bearing in mind the discussion in Section 4.2, one should assume that, in physical models, the derivations 8m(q) in the expression (4.5.1) are unbounded in general. This leads us to the notion of a generalized connection on bundles of C*-algebras [15]. • Let £ —> Q be a Hilbert bundle with a typical fibre E and £{Q) the C°°((3)-module of its smooth sections. Then a connection on a Hilbert bundle 8 —> Q is defined as a connection V on the module £{Q). In accordance with Definition 1.8.7, such a connection assigns to each vector field r on Q a first order differential operator V r on £{Q) which obeys both the Leibniz rule V r W ) = (T\df)1> + /V T V,

/ G C°°(Q),

V G S(Q),

and the additional condition ((VrVOfa) W«)> + W?)l(VTV)(g)> = T(q)}dU>(q)\1>(q)). Given a trivialization chart £\u =UxEoi£—>Q,a, reads Vr = rm(q)(dm + iHm(q)),

(4.5.2)

connection on £{Q) q € U,

(4.5.3)

where TCm(q) for all q £ J7 are bounded self-adjoint operators in the Hilbert space E.

284

Geometric and Algebraic Topological Methods in Quantum Mechanics

In a general setting, let B —> Q be a Banach vector bundle over a finitedimensional smooth manifold Q and B(X) the locally free C°°(Q)-module of its smooth sections s(q). By virtue of Definition 1.8.7, a connection on B(Q) assigns to each vector field r on Q a first order differential operator VT on B(Q) which obeys the Leibniz rule VT(fs) = (T\df)s + fVTs,

/£CM(0),

seB(Q).

(4.5.4)

In accordance with Proposition 1.8.11, such a connection exists. Connections (4.5.1) and (4.5.3) exemplify connections on Banach vector bundles C —> Q and S —> Q, but they obey additional conditions because these bundles possess additional structures of a C* -algebra bundle and a Hilbert bundle, respectively. In particular, the connection (4.5.3) is a principal connection whose second term is an element of the Lie algebra of the unitary group U(E). In a different way, connection on a Banach vector bundle B —> Q can be defined as a splitting of the exact sequence 0 -> VB -^ TB -> TQ®B -> 0, Q

where VB denotes the vertical tangent bundle of B —> Q. In the case of finite-dimensional vector bundles, both definitions are equivalent (see Section 1.8). This equivalence is extended to the case of Banach vector bundles over a finite-dimensional base. We leave the proof of this fact outside the scope of our exposition because it involves the notion of jets of Banach fibre bundles. Turn now to principal connections. Given a Banach-Lie group G, a principal bundle over a finite-dimensional smooth manifold Q, a principal connection, its curvature form and a holonomy group are defined similarly to those in the case offinite-dimensionalLie groups. The main difference lies in the facts that there are Banach-Lie algebras without Lie groups and the holonomy group of a principal connection need not be a Lie group. Referring the reader to [262] for theory of Lie groups and principal bundles modelled over so called convenient locally convex vector spaces (including Frechet spaces), we here formulate some statements adapted to the case of Banach-Lie groups and Banach principal bundles over a finite-dimensional manifold. • Any Banach-Lie group G admits an exponential mapping which is a diffeomorphism of a neighborhood of 0 in the Lie algebra g of G onto a neighborhood of the unit in G. In a general setting, one can always

Chapter 4 Geometry of Algebraic Quantization

285

associate to a Banach-Lie algebra a local Banach-Lie group which however fails to be extended to the global one in general [213]. • Let G be a Banach-Lie group and fl its Lie algebra. If f) is a closed Lie subalgebra offl,there exists a unique connected closed Banach-Lie subgroup H of G with the Lie algebra I) [373]. • Given a Banach-Lie group, the definition of a G-principal bundle P —> Q over a finite-dimensional smooth manifold Q, a principal connection and its curvature form in [262] follows those in the case of a locally compact Lie group [250]. A principal connection T on P defines the global parallel transport and a holonomy group. In particular, the following generalizations of the reduction theorem ([250], Theorem 7.1) and the AmbroseSinger theorem ([250], Theorem 8.1) to Banach principal bundles hold [291; 431]. THEOREM 4.5.1. Let P -* Q be a principal bundle with a Banach-Lie structure group G over a simply connected finite-dimensional manifold Q. Let H be & Banach-Lie subgroup of G. Let us assume that there exists a principal connection on P whose curvature form u possesses the following property. For any smooth one-parameter family of horizontal paths Hcs starting at a point p e P and arbitrary smooth vector fields u, u' on Q,

[0,l}23s,t^ujC3it)(u,u')

(4.5.5)

is a smooth (j-valued map. Then the structure group G of P is reduced to

•

H.

4.5.2. Let us consider closed Lie subalgebras of the Lie algebra Q which contain the range of the map (4.5.5). Their overlap is the minimal closed Lie subalgebra gred of Q possessing this property. The corresponding Banach-Lie group Gred is the minimal Banach-Lie group which contains the holonomy group of a connection T. By virtue of Theorem 4.5.1, the structure group F of P is reduced to GredD THEOREM

• Theorem 10.10.5 holds when G and H are Banach-Lie groups [86]. • Given a trivialization chart of a Banach principal bundle P —> Q with a structure Banach-Lie group G, a principal connection on P is represented by a g-valued local connection one-form Tmdqm with the corresponding transition functions. Let B = {Px V)/G

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Geometric and Algebraic Topological Methods in Quantum Mechanics

be a Banach vector bundle associated to P whose typical fibre V is a Banach space provided with a continuous effective left action of the structure group G. Then a principal connection T on P yields a connection on B given by the first order differential operators Vr = r m ( 9 m - r m )

(4.5.6)

on the C°°(Q) module B(Q) of sections of B —> Q which obey the Leibniz rule (4.5.4). For instance, let G = U(E) be the unitary group of a Hilbert space E. Its Lie algebra consists of the operators i7i, where Ji are bounded selfadjoint operators in the Hilbert space E. It follows that a t/(£')-principal connection takes the form (4.5.3). In conclusion, let us mention the straightforward definition of a connection on a Hilbert bundle as a parallel displacement along paths lifted from a base [228]. Roughly speaking, such a connection corresponds to parallel displacement operators whose infinitesimal generators are (4.5.3). Due to the condition (4.5.2), these operators are unitary. If a path is closed, we come to the notion of a holonomy group of a connection on a Hilbert bundle. 4.6

Example. Instantwise quantization

As will be shown in Section 5.3, geometric quantization of time-dependent Hamiltonian mechanics takes the form of the instantwise quantization, and results in a quantum system described by a Hilbert bundle over the time axis R. This Section addresses the evolution of such quantum systems which can be viewed as a parallel displacement along time. It should be emphasized that, in quantum mechanics based on the Schrodinger and Heisenberg equations, the physical time plays the role of a classical parameter. Indeed, all relations between operators in quantum mechanics are simultaneous, while computation of mean values of operators in a quantum state does not imply integration over time. It follows that, at each instant ( € l , there is an instantaneous quantum system characterized by some C*-algebra At. Thus, we come to instantwise quantization. Let us suppose that all instantaneous C*-algebras At are isomorphic to some unital C*-algebra A. Furthermore, let they constitute a locally trivial smooth bundle C of C*-algebras over the time axis M. Its typical fibre is A. This bundle of C*-algebras is trivial, but need not admit a canonical trivial-

287

Chapter 4 Geometry of Algebraic Quantization

ization in general. One can think of its different trivializations as being associated to different reference frames (see Section 2.4). Let us describe evolution of quantum systems in the framework of instantwise quantization. Given a bundle of C*-algebras C —» R, this evolution can be regarded as a parallel displacement with respect to some connection on C —> R [15; 228; 385]. Following previous Section, we define V as a connection on the involutive C°° (R)-algebra C(R) of smooth sections of C —> M. It assigns to the standard vector field dt on R a symmetric derivation V t of C(R) which obeys the Leibniz rule V t (/a) = dtfa + / V t a ,

a e C(R),

/ € C°°(R),

and the condition V t a* = (V t a)'. Given a trivialization C — R x A, a connection V t reads Vt = $ - * ( « ) ,

(4.6.1)

where 6(t), t G R, are symmetric derivations of the C*-algebra A, i.e., <5t(ab) = 6t(a)b + aSt(b),

5t{a*) = 6t(a)*,

a,b £ A.

We say that a section a of the bundle of C*-algebras C —> R is an integral section of the connection Vt if V t o(t) = [dt - 6(t)]a(t) = 0.

(4.6.2)

One can think of the equation (4.6.2) as being the Heisenberg equation describing quantum evolution. In particular, let the derivations S(t) = 5 in the Heisenberg equation (4.6.2) be the same for all £ £ M, and let 5 be an infinitesimal generator of a strongly continuous one-parameter group [Gt] of automorphisms of the C*algebra A (see Theorem 3.8.2). The pair (A, [Gt]) is called the C*-dynamic system. It describes evolution of a conservative quantum system. Namely, for any a £ A, the curve a(t) — Gt(a), t £ R, in A is a unique solution with the initial value a(0) — a of the Heisenberg equation (4.6.2). It should be emphasized that, if the derivation 5 is unbounded, the connection Vt (4.6.1) is not defined everywhere on the algebra C(R). In this case, we deal with a generalized connection. It is given by operators of a parallel displacement, whose generators however are ill defined [15]. Moreover, it may happen that a representation w of the C*-algebra A does not

288

Geometric and Algebraic Topological Methods in Quantum Mechanics

carry out a representation of the automorphism group [Gt] (see Proposition 3.8.4). Therefore, quantum evolution described by the conservative Heisenberg equation, whose solution is a strongly (but not uniformly) continuous dynamic system (A, Gt), need not be described by the Schrodinger equation (see Remark 4.6.1 below). If 6 is a bounded derivation of a C*-algebra A, the Heisenberg and Schrodinger pictures of evolution of a conservative quantum system are equivalent. Namely, by virtue of Theorem 3.8.1, S is an infinitesimal generator of a uniformly continuous one-parameter group [Gt] of automorphisms of A, and vice versa. For any representation TT of A in a Hilbert space E, there exists a bounded self-adjoint operator Tim E such that n(5{a)) = -i[H,ir(a)],

n(Gt) = exp(-iiW),

a e A,

The corresponding conservative Schrodinger equation reads (dt + iH)ip = 0,

tgt. (4.6.3) (4.6.4)

where ip is a section of the trivial Hilbert bundle R x E —» R. Its solution with an initial value tp(O) S E is ip(t) = exp[-itH]tp(0).

(4.6.5)

Remark 4.6.1. If the derivation S is unbounded, but obeys the assumptions of Proposition 3.8.4, we also obtain the unitary representation (4.6.3) of the group [Gt], but the curve ip(t) (4.6.5) need not be differentiate, and the Schrodinger equation (4.6.4) is ill denned. • Let us return to the general case of a non-conservative quantum system characterized by a bundle of C*-algebras C —> R with the typical fibre A. Let us suppose that a phase Hilbert space of a quantum system is preserved under evolution, i.e., instantaneous C*-algebras At are endowed with representations equivalent to some representation of the C*-algebra A in a Hilbert space E. Then quantum evolution can be described by means of the Schrodinger equation as follows. Let us consider a smooth Hilbert bundle £ —> R with the typical fibre E and a connection V on the C°°(R)-module £ (R) of smooth sections of £ —> R (see previous Section). This connection assigns to the standard vector field dt on R an M-module endomorphism Vt of £ (R) which obeys

Chapter 4 Geometry of Algebraic Quantization

289

the Leibniz rule

V t (/V) = dtfip + /V t V,

i> e £{R),

/ e C°°(R),

and the condition

((vtv)(i)|VW) + (t)l(vtV0(t)> = $Mt)Mt)>Given a trivialization £ = R x E, the connection Vt reads VtV = (dt+i«W)V>,

(4.6.6)

where W(t) are bounded self-adjoint operators in E for all t £ l . It is a C/(£)-principal connection. We say that a section ip of the Hilbert bundle II —» M is an integral section of the connection V t (4.6.6) if it fulfils the equation VtiKO = (dt + iH{t))ijt(t) = 0.

(4.6.7)

One can think of this equation as being the Schrodinger equation for the Hamiltonian H. Its solution with an initial value ^(0) G E exists and reads (4.6.8)

1>{t) = U(t)1>{0),

where U(t) is an operator of a parallel displacement with respect to the connection (4.6.6). This operator is a differentiable section of the trivial bundle

R which obeys the equation

v

TJ(

T7l\

TIT)

X U \-tli) —> K

Uo = 1.

dtU{t) = -iH(t)U(t),

(4.6.9)

The operator U(t) plays the role of an evolution operator. It is given by the time-ordered exponential t

U{t)=Texp -ifn(t')dt' o

,

(4.6.10)

which uniformly converges in the operator norm [118]. Under certain conditions, U(t) can be written as a true exponential U(t) = exp S(t)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

of an anti-Hermitian operator S(t) which is expressed as the Magnus series oo

s(*)=5>fc(t) fc=l

of multiple integrals of nested commutators [266; 348]. It should be emphasized that the evolution operator [/(£) has been defined with respect to a given trivialization of a Hilbert bundle £ —> R. 4.7

Example. Berry connection

We refer the reader to [7; 43; 246; 320; 443] and references therein for the geometric and topological analysis of the Berry's phase phenomenon in quantum systems depending on classical time-dependent parameters. In Section 2.4F, classical mechanical systems with time-dependent parameters have been described in terms of composite fibre bundles and composite connections. Here, this description is extended to quantum systems. Let us consider a quantum system depending on a finite number of real classical parameters given by sections of a smooth parameter bundle £ —> R. For the sake of simplicity, we fix a trivialization S = R x Z, coordinated by (t, am). Although it may happen that the parameter bundle E —> R has no preferable trivialization. In previous Section, we have characterized the time as a classical parameter in quantum mechanics. This characteristic is extended to other classical parameters. In a general setting, one assigns a C*-algebra Aa to each point a £ £ of the parameter bundle E, and treat Aa as a quantum system under fixed values (£, am) of the parameters. However, we will simplify repeatedly our consideration in order to single out a desired Berry's phase phenomenon. Let us assume that all algebras Aa are isomorphic to the algebra B{E) of bounded operators in some Hilbert space E, and consider a smooth Hilbert bundle £ —> S with the typical fibre E. Smooth sections of £ —> S constitute a module £(E) over the ring C°°(E) of real functions on S. A connection V on £(E) assigns to each vector field r on E a first order differential operator VTGDiff !(£(£),£(£))

(4.7.1)

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Chapter 4 Geometry of Algebraic Quantization

which obeys the Leibniz rule V T (/s) = (rJd/) S + / V r s ,

SGf(E),

/€C°°(£).

Let r be a vector field on £ such that dt\ r = 1. Given a trivialization chart of the Hilbert bundle £ —> E, the operator V T (4.7.1) reads V r (s) = (ft + iW(t, ff'))s + r m ( 9 m - ^ ( t ,
(4.7.2)

where H(t,al), Am(t,al) for each cr G S are bounded self-adjoint operators in the Hilbert space E. Let us consider the composite Hilbert bundle £ —+ £ —> R. Similarly to the case of smooth composite fibre bundles (see Section 10.6A), every section h(t) of the fibre bundle £ —> R defines the subbundle £h — h*£ —> R of the Hilbert bundle £ —> R whose typical fibre is E. Accordingly, the connection V (4.7.2) on the C°°(S)-module £(£) yields the pull-back connection Vh(V) = [ft - <(An(t, V(t))ftftro - W(t, ^(i))]V

(4.7.3)

on the C°°(R)-module £h(M) of sections V of the fibre bundle £h -» R (cf. (2.4.64)). As in previous Section, we say that a section ip of the fibre bundle £h —> R is an integral section of the connection (4.7.3) if

Vh(V) = [ft - i(Am(t, h'itydth™ - H{t, h'it))}^ = 0.

(4.7.4)

One can think of the equation (4.7.4) as being the Shrodinger equation for a quantum system depending on the parameter function h(t). Its solutions take the form (4.6.8) where U(t) is the time-ordered exponent "

t

U{t)=Texp i f(Amdt'hm-H)dt' . o

.

(4.7.5)

The term iAm(t, hl(t))dthm in the Shrodinger equation (4.7.4) is responsible for the Berry's phase phenomenon, while "H. is treated as an ordinary Hamiltonian of a quantum system. To show the Berry's phase phenomenon clearly, we will continue to simplify the system under consideration. Given a trivialization of the fibre bundle £ —» R and the above mentioned trivialization £ = R x Z of the parameter bundle E, let us suppose that the components Am of the connection V (4.7.2) are independent of t and that

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the operators 7i(a) commute with the operators Am{a') at all points of the curve h(t) C S. Then the operator U(t) (4.7.5) takes the form

r * £/(*) = Texp i

.Texp -i jH{t')dt'

f Am{ais)dam

. h([o,t})

\

.

(4.7.6)

o

I

One can think of the first factor in the right-hand side of the expression (4.7.6) as being the operator of a parallel transport along the curve h([0, i\) C Z with respect to the pull-back connection V = i*V = dm-iAm(t,ai)

(4.7.7)

on the fibre bundle £ —» Z, defined by the imbedding i : Z <-> {0} x Z c E. Note that, since Am are independent of time, one can utilize any imbedding of Z to {*} x Z. Moreover, the connection V (4.7.7), called the Berry connection, can be seen as a connection on some principal fibre bundle P —» Z for the unitary group U(E). Let the curve /i([0,£]) be closed, while the holonomy group of the connection V at the point h(t) = h(0) is not trivial. Then the unitary operator Texp i

I Amio-^da™

(4.7.8)

. M[o,t]) is not the identity. For example, if (4.7.9)

iAmio-') = iA^a^lds

is a [/(l)-principal connection on Z, then the operator (4.7.8) is the wellknown Berry phase factor exp i

f Am{ai)dam

.

. h([o,t])

If (4.7.9) is a curvature-free connection, Berry's phase is exactly the Aharonov-Bohm effect on the parameter space Z. The following variant of the Berry's phase phenomenon leads us to a principal bundle for familiar finite-dimensional Lie groups. Let a Hilbert

Chapter 4 Geometry of Algebraic Quantization

293

space E be the Hilbert sum of n-dimensional eigenspaces of the Hamiltonian H(a), i.e., oo

£=0£fc,

Ek = Pk(E),

where Pk are the projection operators, i.e., H(a)oPk=\k(a)Pk

(in the spirit of the adiabatic hypothesis). Let the operators Am(z) be time-independent and preserve the eigenspaces Ek of the Hamiltonian H, i.e., Am(z)='£Akm(z)Pk,

(4.7.10)

k

where A!^(z), Z € Z, are self-adjoint operators in Ek- It follows that Am(a) commute with Ti.(<j) at all points of the parameter bundle S —> R. Then, restricted to each subspace .£&, the parallel transport operator (4.7.8) is a unitary operator in Ek- In this case, the Berry connection (4.7.7) on the [/(-E)-principal bundle P —> Z can be seen as a composite connection on the composite bundle P -* P/U(n) - Z, which is defined by some principal connection on the [/(n)-principal bundle P —» P/U(n) and the trivial connection on the fibre bundle P/U(n) —> Z. Note that, since U(E) is contractible, the [/(n)-principal bundle U(E) —» U(E)/U(n) is universal (see Section 10.10) and, consequently, the typical fibre U(E)/U(n) of P/U(n) —> Z is exactly the classifying space B(U(n) (10.10.32) of [/(n)-principal bundles. Moreover, one can consider the parallel transport along a curve in the bundle P/U(n). In this case, a state vector ip(t) acquires a geometric phase factor in addition to the dynamical phase factor. In particular, if S = R (i.e., classical parameters are absent and Berry's phase has only the geometric origin) we come to the case of a Berry connection on the [/(n)-principal bundle over the classifying space B(U(n)) (see [43]). If n = 1, this is the variant of Berry's geometric phase of [7].

Chapter 5

Geometric quantization

The geometric quantization procedure follows the principle of canonical quantization by replacing a Poisson bracket of smooth functions on a Poisson manifold with a commutator product of operators in a Hilbert space such that Dirac's condition (4.2.1) holds. In the framework of geometric quantization, these operators are constructed by means of a suitable U(l)connection. 5.1

Leafwize geometric quantization

Subsections: A. Prequantization, 296; B. Polarization, 302; C. Quantization, 303. We refer the reader to [141; 257; 401; 438] for the basics on geometric quantization of symplectic manifolds. This quantization technique has been generalized to Poisson manifolds in terms of contravariant connections [425; 426; 427] (see Section 5.3). Though there is one-to-one correspondence between the (regular) Poisson structures on a smooth manifold and its symplectic foliations, geometric quantization of a Poisson manifold need not imply quantization of its symplectic leaves [427]. • Firstly, contravariant connections fail to admit the pull-back operation. Therefore, prequantization of a Poisson manifold does not determine straightforwardly prequantization of its symplectic leaves. • Secondly, polarization of a Poisson manifold is defined in terms of sheaves of functions, and it need not be associated to any distribution. As a consequence, its pull-back onto a leaf is not polarization of a symplectic manifold in general. • Thirdly, a quantum algebra of a Poisson manifold contains the center 295

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Geometric and Algebraic Topological Methods in Quantum Mechanics

of a Poisson algebra. However, there are models where quantization of this center has no physical meaning. For instance, the center of the Poisson algebra of a mechanical system with classical parameters consists of functions of these parameters [175]. Geometric quantization of symplectic foliations disposes of these problems. The quantum algebra of a symplectic foliation is also the quantum algebra of the associated Poisson manifold such that its restriction to each symplectic leaf is denned and quantized. Thus, geometric quantization of a symplectic foliation provides the leafwise quantization of a Poisson manifold. This is the case of systems whose symplectic leaves are indexed by non-quantizable variables. In particular, if a Poisson manifold is symplectic, we come to the familiar geometric quantization of symplectic manifolds (see its examples in Sections 5.2, 5.3 and 5.6). Geometric quantization of a symplectic foliation is phrased in terms of leafwise connections on a foliated manifold [175]. Firstly, we have seen that homomorphisms of the de Rham cohomology of a Poisson manifold both to the de Rham cohomology of its symplectic leaf and the LP cohomology factorize through the leafwise de Rham cohomology (Propositions 2.2.1 and 2.2.2). Secondly, any leafwise connection on a complex line bundle over a Poisson manifold is proved to come from a connection (see Theorem 5.1.3 below). Using these facts, we state the equivalence of prequantization of a Poisson manifold to prequantization of its symplectic foliation, which also yields prequantization of each symplectic leaf. On the contrary, polarization of a symplectic foliation is associated to a particular polarization of a Poisson manifold, and its restriction to any symplectic leaf is polarization of this leaf. Therefore, we define metaplectic correction of a symplectic foliation so that its quantum algebra restricted to each leaf is quantized. It is represented by Hermitian operators in the pre-Hilbert space of leafwise half-forms, integrable over the leaves of this foliation. A. Prequantization Prequantization of a symplectic foliation {T, Cl?) of a manifold Z provides a representation

/ -»if,

[/, /'] = -iUJ^r,

(5-1.1)

of the Poisson algebra (C°°(Z), {/, f'}?) by first order differential operators on sections of a complex line bundle TT : C —> Z, called the prequantization bundle. These operators are given by the Kostant-Souriau prequantization

297

Chapter 5 Geometric Quantization

formula / = -tV£+e/,

0/=n$r(d/),

£>O,

(5.1-2)

where V^ is a leafwise connection on C —» Z (see Definition 5.1.1 below) such that its curvature form obeys the prequantization condition (5.1.3)

R = ieQjr.

Using the above mentioned fact that any leafwise connection comes from a connection, we provide the cohomology analysis of this condition, and show that prequantization of a symplectic foliation yields prequantization of its symplectic leaves. DEFINITION 5.1.1. In the framework of the leafwise differential calculus, a (linear) leafwise connection on the complex line bundle C —> Z is defined as a connection V^ on the C°°(Z)-module C(Z) of global sections of this bundle, where C°°(Z) is regarded as a Sf(Z)-ring. It associates to each element r G T\{F) an 5^(Z)-linear endomorphism Vf oiC(Z) which obeys the Leibniz rule

Vf(/«) = (Tjd7)s + /Vf00,

f€C°°{Z),

seC(Z).

(5.1.4)

• A linear connection on C —> Z can be equivalently defined as a connection on the module C{Z) which assigns to each vector field r & T\(Z) on Z an R-linear endomorphism of C(Z) obeying the Leibniz rule (5.1.4). Restricted to Tx(J-), it obviously yields a leafwise connection. In order to show that any leafwise connection is of this form, we will appeal to an alternative definition of a leafwise connection in terms of leafwise forms. The inverse images 7r -1 (F) of leaves F of the foliation T of Z provide a (regular) foliation C? of the line bundle C. Given the (holomorphic) tangent bundle TCjr of this foliation, we have the exact sequence of vector bundles 0->VC —>TCr - ^ C x T f ^ O , c c z where VC is the (holomorphic) vertical tangent bundle of C —> Z.

(5.1.5)

DEFINITION 5.1.2. A (linear) leafwise connection on the complex line bundle C —> Z is a splitting of the exact sequence (5.1.5) which is linear over C. O

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Geometric and Algebraic Topological Methods in Quantum Mechanics

One can choose an adapted coordinate atlas {(U^;zx;zi)} (10.6.27) of a foliated manifold (Z, !F) such that U^ are trivialization domains of the complex line bundle C —> Z. Let (zx\zt;c), c € C, be the corresponding bundle coordinates on C —» Z. They are also adapted coordinates on the foliated manifold (C, C?). With respect to these coordinates, a (linear) leafwise connection is represented by a TCV-valued leafwise one-form Ayr = dz* (di + AiCdc),

(5.1.6)

where Ai are local complex functions on C. The exact sequence (5.1.5) is obviously a subsequence of the exact sequence 0 -» VC —^ TC —»C x TZ -» 0, C

C

Z

where TC is the holomorphic tangent bundle of C. Consequently, any connection r = dzx ® {dx + Txcdc) + dzi ® {di + Ticdc)

(5.1.7)

on the complex line bundle C —> Z yields a leafwise connection IV = dz* ® (di + Yicdc).

(5.1.8)

5.1.3. Any leafwise connection on the complex line bundle C —> Z comes from a connection on it. •

THEOREM

Outline of proof. Let A? (5.1.6) be a leafwise connection on C —> Z and IV (5.1.8) a leafwise connection which comes from some connection V (5.1.7) on C —> Z. Their affine difference over C is a section Q = Ayr - IV = dzi ® {Ai - Ti)cdc of the vector bundle TT* ®VC^C. c Given some splitting B:dzi<-> dzi - B{dzx

(5.1.9)

of the exact sequence (10.6.29), the composition (B ®Idvc)oQ

= (dzi - B\dzx) ® (Ai - Ti)cdc :C-^T*Z®VC

299

Chapter 5 Geometric Quantization

is a soldering form on the complex line bundle C —> Z. Then T + (B®Idvc)oQ

=

x

dz ® {dx + [rA - B\{Ai - Ti)]cdc) + dz* ® (ft + AiC9c) is a desired connection on C —> Z which yields the leafwise connection Ajr (5.1.6). QED In particular, it follows that Definition 5.1.1 and Definition 5.1.2 of a leafwise connection are equivalent, namely, V:Fs = ds-Aisdzi,

seC(Z).

The curvature of a leafwise connection V^ is defined as a C°°(Z)-linear endomorphism R(T, T') = V£ T ,, - [Vf, V?] = T ' 7 % - ,

Rij = ftyl,- - a ^ i , (5.1.10)

of C(Z) for any vector fields r, r ' € 7i (^"). It is represented by the complex leafwise two-form R=-Rijdzi

NdzK

(5.1.11)

If a leafwise connection V ^ comes from a connection V, its curvature leafwise form R (5.1.11) is the image R = i*-pR

of the curvature form R of the connection V with respect to the morphism %*T (2.2.2). Now let us turn to the prequantization condition (5.1.3). LEMMA 5.1.4. Let us assume that there exists a leafwise connection F> on

the complex line bundle C —> Z which fulfils the prequantization condition (5.1.3). Then, for any Hermitian form g on C —> Z, there exists a leafwise connection A9-p on C —> Z which: (i) satisfies the condition (5.1.3), (ii) preserves g, (iii) comes from a L/(l)-principal connection on C —> Z. • Outline of proof. For any Hermitian form g on C —> Z, there exists an associated bundle atlas * 9 = {(zx; z%, c)} of C with [/(l)-valued transition

300

Geometric and Algebraic Topological Methods in Quantum Mechanics

functions such that (5.1.12)

g(c,c')=cc'.

Let the above mentioned leafwise connection Tj? come from a linear connection r (5.1.7) on C —> Z written with respect to the atlas $9. The connection T is split into the sum A9 + 7 where A9 = dzx (8A + Im(r A )cd c ) + dz{ ® (<% + lm(Ti)cdc)

(5.1.13)

is a [/(l)-principal connection, preserving the Hermitian form g. The curvature forms R of T and R9 of A9 obey the relation R9 = lm(R). The connection A9 (5.1.13) defines the leafwise connection A% = i*TA = dzi ® (di + iA?cdc),

iA? = ha(Ti),

(5.1.14)

preserving the Hermitian form g. Its curvature fulfils a desired relation R? = ifR9

= Im(i^iZ) = iefijr.

(5.1.15) QED

Since A s (5.1.13) is a [/(l)-principal connection, its curvature form R9 is related to the first Chern form of integer de Rham cohomology class by the formula ci = i{2-K)-lR9 (10.10.34). If the prequantization condition (5.1.3) holds, the relation (5.1.15) shows that the leafwise cohomology class of the leafwise form (27r)^1e0jr is the image of an integer de Rham cohomology class with respect to the cohomology morphism [ij-] (2.2.3). Conversely, if a leafwise symplectic form 0.? on a foliated manifold (Z,!F) is of this type, there exist a complex line bundle C —> Z and a (/(l)-principal connection A on C —* Z such that the leafwise connection i*TA fulfils the relation (5.1.3). Thus, we have stated the following. 5.1.5. A symplectic foliation {T,Q.jr) of a manifold Z admits prequantization (5.1.2) if and only if the leafwise cohomology class of (2IT)~1EQ.F is the image of an integer de Rham cohomology class of Z. D PROPOSITION

In particular, let (Z,w) be a Poisson manifold and (!F, f2jr) its characteristic symplectic foliation. As is well-known, a Poisson manifold admits prequantization if and only if the LP cohomology class of the bivec-

Chapter 5 Geometric Quantization

301

tor field (2ir)~lew, e > 0, is the image of an integer de Rham cohomology class with respect to the cohomology morphism [iu"] (2.1.34) [425; 426]. By virtue of Proposition 2.2.2, this morphism factorizes through the cohomology morphism [ip] (2.2.3). Therefore, in accordance with Proposition 5.1.5, prequantization of a Poisson manifold takes place if and only if prequantization of its symplectic foliation does well, and both these prequantizations utilize the same prequantization bundle C —> Z. Herewith, each leafwise connection V^ obeying the prequantization condition (5.1.3) yields the admissible contravariant connection V£ := V £ , w >

<j> G O\Z),

on C —> Z whose curvature bivector equals iew. Clearly, V^ and V™ lead to the same prequantization formula (5.1.2). Let F be a leaf of a symplectic foliation {T, Cl?) provided with the symplectic form

In accordance with Proposition 2.2.1 and the commutative diagram of cohomology groups H*{Z;Z) —>tf-(Z)

I

H*(F;Z)

1 •

-^H*{F)

the symplectic form (2-K)~ SQ.F belongs to an integer de Rham cohomology class if a leafwise symplectic form Q.j: fulfils the condition of Proposition 5.1.5. This states the following. 1

PROPOSITION 5.1.6. If a symplectic foliation admits prequantization, each its symplectic leaf does well. •

The corresponding prequantization bundle for F is the pull-back complex line bundle i*FC, coordinated by (zi,c). Furthermore, let A9^ (5.1.14) be a leafwise connection on the prequantization bundle C —> Z which obeys Lemma 5.1.4, i.e., comes from a {/(l)-principal connection A9 on C —> Z. Then the pull-back AF = i*FA9 = dz* ® (dt + iiF{A9)cdc)

(5.1.16)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

of the connection A9 onto i*FC —> F satisfies the prequantization condition RF = i*FR = ieflF, and preserves the pull-back Hermitian form i*Fg on i*rC —» F.

B. Polarization Let us define polarization of a symplectic foliation (F, £V) of a manifold Z as a maximal (regular) involutive distribution T c TF on Z such that n?(u,v) = 0,

U,VGTZ,

zeZ.

(5.1.17)

Given the Lie algebra T(Z) of T-subordinate vector fields on Z, let Ar C C°°(Z) be the complexified subalgebra of functions / whose leafwise Hamiltonian vector fields •#/ (2.2.10) fulfil the condition

l#f,T(Z)]cT(Z). It is called the quantum algebra of a symplectic foliation (F, Qj?) with respect to the polarization T. This algebra obviously contains the center Sj?(Z) of the Poisson algebra (C°°(Z), {,}?), and is a Lie 5^(Z)-algebra. PROPOSITION 5.1.7. Every polarization T of a symplectic foliation (F, Q?) • yields polarization of the associated Poisson manifold (Z, w^i).

Outline of proof. Let us consider the presheaf of local smooth functions / on Z whose leafwise Hamiltonian vector fields $f (2.2.10) are subordinate to T. The sheaf $ of germs of these functions is polarization of the Poisson manifold (Z, wn). Equivalently, $ is the sheaf of germs of functions on Z whose leafwise differentials are subordinate to the codistribution fi^T. QED

Let us note that the polarization $ need not be maximal, unless T is of maximal dimension dim^-"/2. It belongs to the following particular type of polarizations of a Poisson manifold. Since the cochain morphism i*^ (2.2.2) is an epimorphism, the leafwise differential calculus 5* is universal, i.e., the leafwise differentials df of functions / € C°°(Z) on Z make up a basis for the C°°(Z)-module &{Z). Let $(Z) denote the structure M-module of global sections of the sheaf $. Then the leafwise differentials of elements of $(Z) make up a basis for the C°°(Z)-module of global sections of the codistribution fi^-T. Equivalently, the leafwise Hamiltonian vector fields of elements of $(Z) constitute a basis for the C°°(Z)-module T(Z). Then

303

Chapter 5 Geometric Quantization

one can easily show that polarization T of a symplectic foliation (!F, Qyr) and the corresponding polarization $ of the Poisson manifold (Z, WQ) in Proposition 5.1.7 define the same quantum algebra ATLet (F, SIF) be a symplectic leaf of a symplectic foliation (T, CIT)- Given a polarization T —> Z of (F, CIT), its restriction T F = i*FT c i*FTT = TF to F is an involutive distribution on F. It obeys the condition i*FSlr(u,v)=0,

u,v€

TFz,

zeF,

i.e., it is polarization of the symplectic manifold (F, £lF). Thus, we have stated the following. PROPOSITION 5.1.8. Polarization of a symplectic foliation defines polar• ization of each symplectic leaf.

Clearly, the quantum algebra AF of a symplectic leaf F with respect to the polarization TF contains all elements iFf of the quantum algebra A? restricted to F.

C. Quantization Since AT is the quantum algebra both of a symplectic foliation {T, QT) and the associated Poisson manifold (Z, WQ), let us start with the standard metaplectic correction technique [141; 438]. Assuming that Z is oriented and that H2(Z;Z2) = 0, let us consider the metalinear complex line bundle T> —» Z characterized by an atlas #z = { ( % z V ; c ) } with the transition functions d = 5c such that 5 5 is the inverse Jacobian of coordinate transition functions on Z. Global sections of this bundle are half-forms on Z. The metalinear bundle V belongs to the category of natural bundles, and the Lie derivative L r = rxdx + Tldi + i(9 A r A + diT*)

(5.1.18)

of its sections along any vector field r on Z is defined. The quantization bundle is the tensor product Y = C <8> V. The space YK(Z) of its sections

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Geometric and Algebraic Topological Methods in Quantum Mechanics

of compact support is provided with the non-degenerate Hermitian form / . v dimZ/2

^

J PP1,

(5.1.19)

p,p'eYK(Z),

z written with respect to the atlases * s of C and * z of T>. Given the leafwise connection AgT (5.1.14) and the Lie derivative L (5.1.18), one can assign the first order differential operator / = -i[(V#f + ief) ® Id + Id Ltf/] =

(5.1.20)

on YK(Z) to each element of the quantum algebra Ap. These operators obey the Dirac condition (5.1.1), and provide a representation of the quantum algebra A? by (unbounded) Hermitian operators in the pre-Hilbert space YK{Z). Finally, this representation is restricted to the subspace E of sections p e YK{Z) which obey the condition (Vj ® Id + Id ® L#)p = ( V j + \di^)p

=0

for all T-subordinate leafwise Hamiltonian vector fields $. However, it may happen that the above quantization has no physical sense because the Hermitian form (5.1.19) on the carrier space E and, consequently, the mean values of operators (5.1.20) are defined by integration over the whole manifold Z. For instance, it implies integration over classical parameters. Therefore, we suggest a different scheme of quantization of symplectic foliations. Let us consider the exterior bundle A TT*, m = dim T. Its structure group GL(m,R) is reducible to the group GL+(m,R) since a symplectic foliation is oriented. One can regard this fibre bundle as being associated to a GL(m, C)-principal bundle P —> Z. As earlier, let us assume that H2(Z;Z2) = 0. Then the principal bundle P admits a two-fold covering principal bundle with the structure metalinear group ML(m,C) [141]. As a consequence, there exists a complex line bundle V? —> Z characterized by an atlas yr = {{Ui;zx;zi;c)}

305

Chapter 5 Geometric Quantization

with the transition functions c' = Sjrc such that

One can think of its sections as being leafwise half-forms on Z. The metalinear bundle Vj: —> Z admits the canonical lift of any T-subordinate vector field T on Z. The corresponding Lie derivative of its sections reads Lf = T% + \diT\

(5.1.21)

We define the quantization bundle as the tensor product Yjr = C® Vjr.

Given a leafwise connection A9T (5.1.14) and the Lie derivative ~LT (5.1.21), let us associate the first order differential operator / = - i [ ( v £ , + t e / ) ® I d + I d ®L£ / ] = -ilV^+ief+^drf)],

(5.1.22)

feAr,

on sections p? of Y? to each element of the quantum algebra A?. A direct computation with respect to the local Darboux coordinates on Z proves the following. LEMMA

5.1.9. The operators (5.1.22) obey the Dirac condition (5.1.1). •

LEMMA 5.1.10. If a section p? fulfils the condition ( V j ® H + Id ® l4)pF

= (Vj + i W A F = 0

(5.1.23)

for all T-subordinate leafwise Hamiltonian vector field •&, then fp? for any / G A?- possesses the same property. • Let us restrict the representation of the quantum algebra A? by the operators (5.1.22) to the subspace E? e YF(Z) of sections p? which obey the condition (5.1.23) and whose restriction to any leaf of T is of compact support. The last condition is motivated by the following. Since i*FTT* = T*F,

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Geometric and Algebraic Topological Methods in Quantum Mechanics

the pull-back iFV? of V? onto a leaf F is a metalinear bundle of halfforms on F. By virtue of Propositions 5.1.6 and 5.1.8, the pull-back ipY? of the quantization bundle Y? —> Z onto F is a quantization bundle for the symplectic manifold (F, ipd^). Given the pull-back connection AF (5.1.16) and the polarization T> = i*FT, this symplectic manifold is subject to the standard geometric quantization by the first order differential operators / = -i(iF^f

+ ief + i040>),

/ G A F,

(5.1.24)

on sections pF of iFYy? —» F of compact support which obey the condition (»fVj + ldi^)pF

=0

(5.1.25)

for all Tir-subordinate Hamiltonian vector fields $ on F. These sections constitute a pre-Hilbert space EF with respect to the Hermitian form

/ 1\m/2 f

(PF\P'F) = ( ^

)

J PFP'FF

The key point is the following. PROPOSITION

5.1.11. We have i*FEj: C EF, and the relation

iF(fpT) = WTWFP?) holds for all elements / 6 Ap and py? G Ejr.

(5-1-26) D

Outline of proof. One can use the fact that the expressions (5.1.24) and (5.1.25) have the same coordinate form as the expressions (5.1.22) and (5.1.23) where zx =const. QED The relation (5.1.26) enables one to think of the operators / (5.1.22) in Ef as being the leafwise quantization of the 5^(Z)-algebra Ayr in the pre-Hilbert Sy(Z)-module.

5.2

Example. Quantum completely integrable systems

We study geometric quantization of a CIS on a 2m-dimensional symplectic manifold (Z, fi) around a regular compact invariant manifold M. By virtue of the classical Arnold-Liouville theorem [12] (see Theorem 2.3.14 for the

Chapter 5 Geometric Quantization

307

general case of a PIS), a small neighborhood U of M in the ambient symplectic manifold Z is isomorphic to the symplectic annulus U = N x T m , where N c M™ is a non-empty domain and Tm is an m-dimensional torus. The product Nx Tm is equipped with the action-angle coordinates (Ik, (f>k). With respect to these coordinates, the symplectic form on TV x Tm reads Sl = dIkAd(j)k,

(5.2.1)

while a Hamiltonian of a CIS is a function H{Ik) of action variables only. Let us note that there are different approaches to quantization of autonomous CISs [185; 208]. The peculiarity of the geometric quantization procedure is that it remains equivalent under symplectic isomorphisms, but essentially depends on the choice of a polarization [38; 364]. Geometric quantization of CISs around an invariant tori has been studied with respect to polarization spanned by Hamiltonian vector fields of first integrals [329]. In fact, the well-known Simms quantization of the harmonic oscillator is also of this type [l4l]. The problem is that the associated quantum algebra includes functions which are not globally defined and that elements of its carrier space are not smooth sections of a quantization bundle. Indeed, written with respect to the action-angle variables, this quantum algebra consists of functions which are afnne in angle coordinates. Here, we use a different polarization spanned by locally Hamiltonian vector fields of angle variables [177]. It provides the Schrodinger representation of action variables by first order differential operators on functions of angle coordinates. Remark 5.2.1. Since the action-angle coordinates, by definition, are canonical for the symplectic form (5.2.1), geometric quantization of the symplectic annulus (N x T m ,fi) in fact is equivalent to geometric quantization of the cotangent bundle T*Tm = R m x Tm of the torus T m provided with the canonical symplectic form (2.1.3). In particular, the above mentioned angle polarization of N x Tm corresponds to the familiar vertical polarization VT*Tm of T*Tm -> Tm which leads to Schrodinger quantization of the cotangent bundle T*Tm. The associated quantum algebra A of N x Tm consists of functions which are affine in action variables /&. Therefore, applications of Schrodinger geometric quantization are limited by the fact that a Hamiltonian fails to belong to the quantum algebra A, unless it is afnne in momenta. In the case of a CIS, a

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Hamiltonian H depends only on action variables which mutually commute. Therefore, if Ti(h) is a polynomial function, it is uniquely represented by an element of the enveloping algebra A of the Lie algebra A, and is quantized as the operator H(Ik)- Moreover, this quantization is also extended to Hamiltonians which are analytic functions on R m because the action operators Ik are diagonal. • Remark 5.2.2. One usually mentions a harmonic oscillator as the simplest CIS whose quantization in the action-angle variables looks notoriously difficult because the eigenvalues of its action operator is expected to be lower bounded [209]. However, a harmonic oscillator quantized with respect to action-angle coordinates is defined on the momentum phase space R2 \ {0}, but it is not the standard oscillator on R2. There is a monomorphism, but not an isomorphism of the Poisson algebra of smooth complex functions on R2 to that on R2 \ {0}. Furthermore, the angle polarization on R2 \ {0} is not extended to R2. As a consequence, the quantum algebra associated to this polarization contains functions on R2 \ {0} which are not extended to R2, and the carrier space of this quantization consists of tempered distributions. • We follow the standard geometric quantization procedure in order to quantize the symplectic annulus (N x Tm, O) [401; 438]. Since the symplectic form fi (5.2.1) is exact, the prequantization bundle is a trivial complex line bundle C —> N x Tm. Let its trivialization C^(N

xTm)xC

(5.2.2)

hold fixed. Any other trivialization leads to equivalent quantization of N x Tm. Given the associated bundle coordinates (7fc,^fe,c), c G C, on C (5.2.2), one can treat its sections as smooth complex functions on N x Tm. The Kostant-Souriau prequantization formula associates to each smooth real function / on N x Tm the first order differential operator

f = -iV*,+f

(5.2.3)

on sections of C, where "df (2.1.4) is the Hamiltonian vector field of / and V is the covariant differential with respect to a [/(l)-principal connection A on C whose curvature form obeys the prequantization condition R = id. This connection reads A = A o + iclkd dc,

(5.2.4)

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Chapter 5 Geometric Quantization

where Ao is a flat [/(l)-principal connection on C —> V x Tm. The classes of gauge conjugate (i.e., linked by vertical automorphisms) flat principal connections on C are indexed by the set M m /Z m of homomorphisms of the de Rham cohomology group

F^JV xTm)

=Rm

of the annulus N x Tm (which equals its first homotopy group) to U(l) (see Example 10.6.4). We choose their representatives of the form A0{(Xk)\

= dlk ®dk + d<j>k g> (dfc + i\kcdc),

Afc e [0,1).

Then the connection A (5.2.4) on C up to gauge conjugation reads A[{Xk)} = dlk ®dk + d<j>k ® (dk + i(h + Xk)cdc).

(5.2.5)

For the sake of simplicity, let Afc in the expression(5.2.5) be arbitrary real numbers, but we will bear in mind that connections >l[(Afc)] and ^4[(A'fc)] with Afc — Afe £ Z are gauge conjugate. Given a connection (5.2.5), the prequantization operators (5.2.3) read f = -i#f

+

(f-(Ik

+

Xk)dkf).

(5.2.6)

It is readily observed that the prequantization operators / also keep their form (5.2.6) on sections of the quantum bundle C X'1/2, where

V1/2

-+NxTm

is a metalinear bundle whose sections are half-forms on N x Tm. Let us choose the above mentioned angle polarization. It is the vertical tangent bundle Vix of the fibration •K :

N x Tm -> Tm

spanned by the vectors dk. It is readily observed that the corresponding quantum algebra

A C C°°(N x Tm) consists of afnne functions

f = ak(4>T)h + b{4>T)

(5.2.7)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

of action coordinates Ik. The carrier space £ of its representation (5.2.6) consists of sections p of the quantization bundle C ® D1/2

-^NxTm

of compact support which obey the condition V^/9 = 0 for any Hamiltonian vector field -d subordinate to the distribution Vn. This condition reads dkfdkp

= 0,

/ e C°°(r m ).

It follows that elements of £ are independent of action variables and, consequently, fail to be of compact support, unless p = 0. This is the well-known problem of Schrodinger geometric quantization. It is solved as follows [39; 174]. Given an imbedding

iT:Tm -> Nx Tm, let CT = ij-C be the pull-back of the prequantization bundle C (5.2.2) onto the torus T m . It is a trivial complex line bundle CT = Tm x C whose sections are smooth complex functions on Tm. Let A[Xk] = iTA[Xk} = d<j>k ® {dk + i(h + \k)cdc) be the pull-back of the connection A[Xk] (5.2.5) onto CT, and let V denote the corresponding covariant differential. Let T>T be a metalinear bundle of complex half-forms on the torus Tm. It admits the canonical lift of any vector field r on T m , and the corresponding Lie derivative of its sections reads

L T = r kdk +

\dkrk.

Let us consider the tensor product Y = CT®/DT->Tm.

(5.2.8)

Since the Hamiltonian vector fields df = akdk - (Irdkar + dkb)dk of functions / (5.2.7) are projectable onto Tm, one can associate to each

Chapter 5 Geometric Quantization

311

element / of the quantum algebra A the first order differential operator / = (-iV w t f / + / ) ® Id + Id ® L Ttf/ =

(5.2.9)

-iafe9fc - -dkak - ak\k + b on sections of Y. A direct computation shows that the operators (5.2.9) obey Dirac's condition (4.2.1). Sections s of the quantization bundle Y _> I'm (5.2.8) constitute a pre-Hilbert space ET with respect to the non-degenerate Hermitian form r

( 1\

(s\s ) = ( — J

m

f

-

I ss,

S,SG£T.

Then / (5.2.9) are Hermitian operators in ET- They provide the desired geometric quantization of a CIS on the annulus N x Tm. Of course, this quantization depends on the choice of a connection ^[(^fc)] (5.2.5) and a metalinear bundle VT [312]. The latter need not be trivial. If T>T is trivial, sections of the quantization bundle Y —> Tm (5.2.8) obey the transformation rule s(0fc + 2TT) = s(4>k) for all indices k. They are naturally complex smooth functions on Tm. By virtue of the multidimensional Fourier theorem, the functions Wr)=exp[2(nr0r)],

(n r ) = ( n i , . . . , n m ) € Z " \

(5.2.10)

constitute an orthonormal basis for the pre-Hilbert space ET = C°°(T m ). The action operators h = -idk - Afc

(5.2.11)

4VVr) = K - Afc)VVr)

(5.2.12)

(5.2.9) are diagonal

with respect to this basis. Other elements of the algebra A are decomposed into the pull-back functions 7r*V'(rir) on N x Tm which act on C°°(T m ) by multiplications T ^ V W ^ K . ) = V>(nr)V>K.) = Anr+K)-

(5.2.13)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

If T>T is a non-trivial metalinear bundle, sections of the quantization bundle Y -> Tm (5.2.8) obey the transformation rule p r ( ^ + 27r) = -pr(0»)

(5.2.14)

for some indices j . In this case, the orthonormal basis of the pre-Hilbert space £T can be represented by double-valued complex functions iP(nunj)

= e x p [ i ( n ^ + (rij + ^)<jP)]

(5.2.15)

on Tm. They are the eigenvectors Iii>(ni,nj) = (ni ~ Xi)ip(nunj),

^{nunj)

= («j ~ Aj +

^(ni,nj)

of the operators Ik (5.2.11), while the pull-back functions Tr*^^,.) act on the basis (5.2.15) by the above mentioned law (5.2.13). It follows that the representation of the quantum algebra A determined by the connection ^4[(Afc)] (5.2.5) in the space of sections (5.2.14) of a nontrivial quantization bundle Y (5.2.8) is equivalent to its representation determined by the connection A[(Xi,Xj - |)] in the space C°°(T m ) of smooth complex functions on Tm. Therefore, one can restrict the study of representations of the quantum algebra A to its representations in C°°(T m ) associated to different connections (5.2.5). These representations are nonequivalent, unless Xk ~ X'k eZ for all indices k. Given the representation (5.2.9) of the quantum algebra A in C°°(r m ), a polynomial Hamiltonian Ti(Ik) and polynomial first integrals fi(h) of a CIS are uniquely quantized as Hermitian elements •H(h) = W(Jfc),

fi(h) = fi{h)

(5.2.16)

of the enveloping algebra A of A. They have the countable spectra ii{h)il)(nT) = ft(nfc ~ Afc)VVr)>

fi(h)^(nr)

= fi(nk - Afc)VW)-

Let us note that, since the action operators / are diagonal, the quantization formula (5.2.16) can be extended to analytic functions of Ik on R m . 5.3

Quantization of time-dependent mechanics

We study the covariant (frame-independent) geometric quantization of nonrelativistic mechanics, subject to time-dependent transformations (see Sec-

Chapter 5 Geometric Quantization

313

tion 2.4). In contrast with the existent geometric quantizations of nonrelativistic mechanics [401; 438], we do not fix a trivialization Q = RxM,

V*Q = R x T*M.

(5.3.1)

The key point is that, in this case, the evolution equation is not reduced to the Poisson bracket on V*Q, but can be expressed into the Poisson bracket on the cotangent bundle T*Q of Q. Therefore, covariant geometric quantization of time-dependent mechanics on a configuration bundle Q —> R requires compatible geometric quantization both of the cotangent bundle T*Q and the vertical cotangent bundle V*Q of Q. The relation (2.4.28) defines the monomorphism of Poisson algebras C : (C°°(V*Q), {, }v) - (C°°(T*Q), {, } T ).

(5.3.2)

Therefore, a compatibility of geometric quantizations of T*Q and V*Q implies that this monomorphism is prolonged to a monomorphism of quantum algebras of V*Q and T*Q. Of course, it seems natural to quantize C°°(V*Q) as a subalgebra (5.3.2) of the Poisson algebra C°°(T*Q). However, geometric quantization of the Poisson algebra (C°°(T*Q), {, }T) need not imply that of its Poisson sub-

algebra CC°°(V*Q). We show that the standard prequantization of the cotangent bundle T*Q yields the compatible prequantization of the Poisson manifold V*Q such that the monomorphism £* (5.3.2) is prolonged to a monomorphism of prequantum algebras. However, polarization of T*Q need not induce any polarization of V*Q, unless it contains the vertical cotangent bundle V^T*Q of the fibre bundle C (2.4.26) spanned by vectors d°. A unique canonical real polarization of T*Q, satisfying the above condition V(T*Q c T,

(5.3.3)

is the vertical tangent bundle VT*Q of T*Q —> Q. The associated quantum algebra AT consists of functions on T*Q which are affine in momenta p,\We show that this vertical polarization of T*Q yields the polarization of the Poisson manifold V*Q such that the corresponding quantum algebra Ay consists of functions on V*Q which are affine in momenta pk- It follows that Av is a subalgebra of AT under the monomorphism (5.3.2). After metaplectic correction, the compatible Schrodinger representations of AT and Ay by operators on complex half-forms on Q is obtained.

314

Geometric and Algebraic Topological Methods in Quantum Mechanics

The physical relevance of the Schrodinger quantization of T*Q however is open to question. The scalar product of half-forms on Q implies integration over time, though the time plays the role of a classical evolution parameter in quantum mechanics, based on Schrodinger and Heisenberg equations. At the same time, the Schrodinger quantization of V*Q provides instantwise quantization of non-relativistic mechanics. Indeed, a glance at the Poisson bracket (2.4.27) shows that the Poisson algebra C°°(V*Q) is a Lie algebra over the ring C°° (R) of functions of time alone, where algebraic operations in fact are instantwise operations depending on time as a parameter. We show that the Schrodinger quantization of the Poisson manifold V*Q induces geometric quantization of its symplectic fibres Vt*Q, ( £ K , such that the quantum algebra At of V*Q consists of elements / £ Ay restricted to Vt*Q. This agrees with the instantwise quantization of symplectic fibres {t} x T*M of the direct product (5.3.1) in [401]. Moreover, the induced geometric quantization of fibres V*Q, by construction, is determined by their injection to V*Q, but not projection of V*Q. Therefore, it is independent of a trivialization (5.3.1). Let us turn now to quantization of the evolution equation in timedependent mechanics. The problem is that the Poisson structure (2.4.27) fails to provide any dynamic equation on the momentum phase space V*Q because all Hamiltonian vector fields $/ (2.4.29) for functions / on V*Q are vertical. Hamiltonian dynamics of time-dependent mechanics is described as a particular Hamiltonian dynamics on fibre bundles where the evolution equation is brought into the form (2.4.41). The problem is that the covariant Hamiltonian H* (2.4.39) does not belong to the algebra AT, unless it is affine in momenta. Let us assume that H* is a polynomial of momenta. This is the case of all physical models. Then we show below that H* can be represented by a finite sum of products of elements of AT, though this representation by no means is unique. Thereby, it can be quantized as an element of the enveloping algebra AT of the Lie algebra ATRemark 5.3.1. An ambiguity of the operator representation of a classical Hamiltonian is a well-known technical problem of Schrodinger quantization as like as any geometric quantization scheme, where a Hamiltonian does not preserve a polarization (see [40l] for a general, but rather sophisticated analysis of such Hamiltonians). One can include the covariant Hamiltonian Ji* (2.4.39) in a quantum algebra by choosing polarization of T*Q which contains the Hamiltonian vector field of 7i*. This polarization always exists,

315

Chapter 5 Geometric Quantization

but does not satisfy the condition (5.3.3) and, therefore, does not define any polarization of the Poisson manifold V*Q. Let us note that, given a trivialization (5.3.1), symplectic fibres Vt*Q, t e R, of the Poisson bundle V*Q —> R can be provided with the instantwise polarization spanned by vectors {diHd1

- d ^ d i , ••• ,dmndm

-

dmHdm).

However, this polarization need not be regular and, by construction, is frame-dependent. It is a standard polarization in conservative Hamiltonian mechanics of one-dimensional systems, but it requires an exclusive analysis of each physical model. • Given a covariant Hamiltonian H* (2.4.39) and its representative H in AT, the map V:/^{W*,/}T is a derivation of the enveloping algebra Av C AT of the Lie algebra Av • Moreover, this derivation obeys the Leibniz rule V(r/) = dtrf + rV/,

r e C°°(R),

and, consequently, is a connection on the instantwise algebra Ay- Since this property is preserved under quantization, geometric quantization of nonrelativistic time-dependent mechanics leads to its instantwise quantization (see Section 4.6). We start with the standard prequantization of the cotangent bundle T*Q [141; 401; 438]. Since the symplectic form fi on T*Q is exact and, consequently, belongs to the zero de Rham cohomology class, a prequantization bundle is the trivial complex line bundle C = T*Q x C -> T*Q

(5.3.4)

of zero Chern class. Coordinated by (qx,p\, c), this bundle is provided with the admissible linear connection A = dpx ® dx + dqx ® (d\ + ipxcdc)

(5.3.5)

whose strength form equals — ifl. The ^-invariant Hermitian fibre metric on C —> Q is given by the expression (5.1.12). The covariant derivative of sections s of the prequantization bundle C (5.3.4) relative to the connection

316

Geometric and Algebraic Topological Methods in Quantum Mechanics

A (5.3.5) along the vector field u on T*Q reads V u s = ux{dx - ip\)s + uxdxs.

(5.3.6)

Given a function / € C°°(T*Q) and its Hamiltonian vector field

uf = dxfdx - dxf3x,

uf\n = -df,

the covariant derivative (5.3.6) along Uf is VUfs =

dxf(dx-ipx)s-dxfdxs.

Then, in order to satisfy the Dirac condition (4.2.1), one assigns to each function / e C°°(T*Q) the first order differential operator f(s) = - i ( v u / + if)s = [-i{dxfdx - dxfdx) + (/ - pxdxf)]s

(5.3.7)

on sections s of the prequantization bundle C (5.3.4). The prequantum operators (5.3.7) for elements / of the Poisson subalgebra CC°°(V*Q) C C°°(T*Q) read f(s) = [-i(dkfdk - dxfdx) + (/ -

Pkd

k

f)]s.

(5.3.8)

Let us turn now to prequantization of the Poisson manifold (V*Q, {, }v)The Poisson bivector w of the Poisson structure (2.4.27) on V*Q is w = dk Adk = -[w,-d]SN,

(5.3.9)

where [, ]SN is the Schouten-Nijenhuis bracket and "9 = pkdk is the Liouville vector field on the vertical cotangent bundle V*Q —> Q. The relation (5.3.9) shows that the Poisson bivector w is exact and, consequently, has the zero Lichnerowicz-Poisson cohomology class. Therefore, let us consider the trivial complex line bundle Cv = V*Q x C -> V*Q

(5.3.10)

of zero Chern class. Since the line bundles C (5.3.4) and CV (5.3.10) are trivial, C can be seen as the pull-back C*CV of CV, while Cy is isomorphic to the pull-back h*C of C with respect to a section h (2.4.31) of the affine bundle (2.4.26). Since Cy = h*C and since the covariant derivative of the

Chapter 5 Geometric Quantization

317

connection A (5.3.5) along the fibres of £ (2.4.26) is trivial, let us consider the pull-back h*A = dpk dk + dqk (9fc + ipkcdc) + dt® (dt - iHcdc)

(5.3.11)

of the connection A (5.3.5) onto Cy —> V*Q. This connection defines the contravariant derivative Vsv = V ^ s y

(5.3.12)

of sections sy of Cy —> V*Q along one-forms <j> on V*Q. This contravariant derivative corresponds to a contravariant connection Ay on the line bundle Cy —> y*Q [426]. Since the vector fields w^cfi = kdk — <j)kdk are vertical on V*Q —> R, this contravariant connection does not depend on the choice of a section h. By virtue of the relation (5.3.12), the curvature bivector of Ay is equal to —iw [427], i.e., Ay is an admissible connection for the Poisson structure on V*Q. Then the Kostant-Souriau formula f(sv) = (-iVet+f)sv

= [-i(dkfdk-dkfdk)

+ (f-pkdkf)}sy, (5.3.13)

defines prequantization of the Poisson manifold V*Q. In particular, the prequantum operators of functions r e C°°(R) of time alone are reduced to the multiplication fsy = rsy. Consequently, the prequantum algebra of V*Q inherits the structure of a C°°(R)-algebra. It is immediately observed that the prequantum operator / (5.3.13) coincides with the prequantum operator £*/ (5.3.8) restricted to the pullback sections s = C,*sv- Thus, the above mentioned prequantization of the Poisson algebra C°°(V*Q) is equivalent to its prequantization as a subalgebra of the Poisson algebra C°°(T*Q). Let us note that, since the complex line bundles C (5.3.4) and Cy (5.3.10) are trivial, their sections are simply smooth complex functions on T*Q and V*Q, respectively. Then the prequantum operators (5.3.7) and (5.3.13) can be written in the form / = -*L u / + ( / - L t f / ) ,

(5.3.14)

where i? denotes the Liouville vector field •& = p\dx on T*Q —+ Q or •d=pkdk on V*Q^Q. Given compatible prequantizations of the cotangent bundle T*Q and the vertical cotangent bundle V*Q, let us now construct their compatible polarizations and quantizations. We assume that Q is an oriented manifold and that the cohomology group H2(Q,Z2) is trivial.

318

Geometric and Algebraic Topological Methods in Quantum Mechanics

Let us recall that polarization of a Poisson manifold (Z, {, }) is defined as a sheaf T* of germs of complex functions on Z whose stalks T*, z £ 2 , are Abelian algebras with respect to the Poisson bracket {,} [427]. Let T*(Z) be the structure algebra of global sections of the sheaf T*; it is also called a Poisson polarization [425; 426]. A quantum algebra A associated to the Poisson polarization T* is defined as a subalgebra of the Poisson algebra C°°{Z) which consists of functions / such that {f,T*(Z)}cT*(Z). Polarization of a symplectic manifold yields its Poisson one. Let T* be a polarization of the Poisson manifold (T*Q, {, }T)- Its direct image in V*Q with respect to the fibration £ (2.4.26) is polarization of the Poisson manifold (V*Q,{,}v) if the germs of T* are constant along the fibres of C [427], i.e., are germs of functions independent of the momentum coordinate p. It follows that the corresponding symplectic polarization T of T*Q is vertical with respect to the fibration T*Q -> V*Q. The vertical polarization T = VT*Q of T*Q obeys this condition. The associated quantum algebra AT C C°°(T*Q) consists of functions which are affine in momenta p\. The algebra AT acts by operators (5.3.14) on the space of smooth complex functions s on T*Q which fulfill the relation V u s = 0 for any T-valued (i.e, vertical) vector field u = u\dx on the cotangent bundle T*Q —•» Q. Clearly, these functions are the pull-back of complex functions on Q with respect to the fibration T*Q —» Q. Following the general metaplectic technique, we come to complex half-forms on Q which are sections of the complex line bundle P1/2 —> Q with the transition functions c' = Sc such that SS is the Jacobian of coordinate transition functions on Q. Then the formula (5.3.14), where L^ / is the Lie derivative of half-forms, defines the Schrodinger representation of the Lie algebra AT by operators fp = {-iLa,dx f = a\q»)px +

+ b)p = (-iaxdx - l-dxax + b)p,

(5.3.15)

b{q»)£AT,

in the space £>i/2(Q) of complex half-forms p on Q. From now on, we assume that a coordinate atlas of Q and a bundle atlas of 2?i/2 —> Q are defined on the same covering of Q, e.g., by contractible open sets. Let E C T>i/2(Q) consists of half-forms of compact support, and let E be its

Chapter 5 Geometric Quantization

319

completion with respect to the non-degenerate Hermitian form (P|p')=(^)m/w'-

(5-3.16)

The (unbounded) operators (5.3.15) on the domain E in the Hilbert space E are Hermitian. The vertical polarization of T*Q defines the polarization Tv of the Poisson manifold V*Q which contains the germs of functions, constant on the fibres of V*Q —> Q. The associated quantum algebra Av consists of functions on V*Q which are affine in momenta. It is a C°°(R)-algebra. This algebra acts by operators (5.3.14) on the space of smooth complex functions Sy on V*Q which fulfill the relation V u sy = 0 for any vertical vector field u = ukdk on V*Q —» Q. These functions are also the pull-back of complex functions on Q with respect to the fibration V*Q —> Q. Similarly to the case of AT, we obtain the Schrodinger representation of the Lie algebra Av by the operators fp = {-iLakdk

+ b)p = {-iakdk - l-dkak + b)p,

(5.3.17)

on half-forms on Q and in the above Hilbert space E. Moreover, a glance at the expressions (5.3.15) and (5.3.17) shows that (5.3.17) is the representation of Av as a subalgebra of the Lie algebra ATAs was mentioned above, the physical relevance of the space of halfforms on Q with the scalar product (5.3.16) is open to question. At the same time, the representation (5.3.17) preserves the structure of Av as a C°°(K)-algebra. Therefore, let us show that this representation defines the instantwise quantization of AvFirstly, the prequantization (5.3.13) of the Poisson manifold V*Q yields prequantization of its symplectic leaves Vt*Q, t G R, as follows. The symplectic structure on V*Q is nt = (hoityn = dPkAdqk,

(5.3.18)

where h is an arbitrary section of the fibre bundle f (2.4.26) and it : Vt*Q —> V*Q is the natural imbedding. Since w^ is a vertical vector field on V*Q —* R for any one-form (p on V*Q, the contravariant derivative (5.3.12) defines a connection along each fibre Vt*Q, t £ M, of the Poisson bundle V*Q —* R.

320

Geometric and Algebraic Topological Methods in Quantum Mechanics

It is the pull-back At = i*th*A = dpk ®dk + dqk ® (dk + ipkcdc) of the connection h* A (5.3.11) onto the trivial pull-back line bundle i*Cv

= Vt*Q X C H

Vt*Q.

It is readily observed that this connection is admissible for the symplectic structure (5.3.18) on Vt*Q, and provides prequantization of the symplectic manifold (V*Q,Q.t) by the formula j t = -»L* /t + (ft - L*t) = -i(dkft3k

- dkftdk) + (ft~Pkdkft),

(5.3.19)

where 0 / t = dkftdk - dkftdk is the Hamiltonian vector field of a function ft on Vt*Q with respect to the symplectic form Clt (5.3.18). The operators (5.3.19) act on smooth complex functions st on V*Q. In particular, let ft, st and (fs)t be the restriction to V*Q of a real function / and complex functions s and f(s) on V*Q, respectively. We obtain from the formulae (5.3.13) and (5.3.19) that (fs)t = ftstThis equality shows that the prequantization (5.3.13) of the Poisson manifold V*Q is a leafwise prequantization. Let Ty be the above mentioned polarization of the Poisson manifold V*Q. It yields the pull-back polarization TJ1 = i*tT*v of a fibre Vt*Q with respect to the Poisson morphism it : Vt*Q -> VQ. The corresponding distribution T ( coincides with the vertical tangent bundle of the fibre bundle Vt*Q —> Qt. The associated quantum algebra At consists of functions on V*Qt which are affine in momenta. In particular, the restriction to V*Q of any element of the quantum algebra Ay of V*Q obeys this condition and, consequently, belongs At- Conversely, any element of At is of this type. For instance, using a trivialization (5.3.1) and the corresponding surjection nt : V*Q —> V*Q, one can define the pull-back 7Tt*/t of a function ft G At which belongs to the quantum algebra Ay and ft = i*(7Tj/t). Thus, At = i*Av and, therefore, the polarization Tv of the Poisson bundle V*Q —> M. is a fibrewise polarization.

321

Chapter 5 Geometric Quantization

In order to provide metaplectic correction and to complete geometric quantization of symplectic fibres of the Poisson bundle V*Q —> R, one can use the following fact. Any atlas {(£/; t, qk)} of bundle coordinates on the fibre bundle Q —> R induces a coordinate atlas {(Qt fl U; qk)} of its fibre Qt, t G R. Since

the Jacobian J of the transition function between coordinate charts (U;t,qk) and (U';t,q'k) on Q coincides with the Jacobian Jt of the transition function between coordinate charts (Qt fl U;qk) and (Qt H U';q'k) on Qt at points oi QtC\U n U'. It follows that, for any fibre Qt of Q, the pull-back ijP —> Qt of the complex line bundle V —> Q of complex densities on Q with transition functions c' = Jc is the complex line bundle of complex densities on Qt with transition functions Jt = J\Qt- Accordingly, any density L on Q yields the pull-back section Lt = L o it of the line bundle i*X> —> Q t , i.e., Lt is a density on Q t . The pull-back L —> Lt takes the coordinate form L = C(t, qk)dmq A dt -> Lt = £(*, 9fe)dm9|t=const = £(t, ^Jd""?, where {d9fc} are holonomic fibre bases for V*Q. It is maintained under transformations of bundle coordinates on Q. Let now T>\ji —> Q be a complex line bundle of complex half-forms on Q with transition functions S such that 55 = J on J7 D [/'. Its pull-back i*X>i/2 is a complex line bundle over a fibre Qt, t G R, with transition functions St = 5|Q 4 . These transition functions obey the relation StSt = J|Qt = Jt, i.e., i*T>\/2 —* Qt is the fibre bundle of half-forms on Qt. Then the formula (5.3.19) defines the Schrodinger representation of the quantum algebra At of the symplectic fibre Qt by (unbounded) Hermitian operators ft/H = (-iLafd,. + b)Pt = {-iakdk - %^dkak + b)pu

(5.3.20)

ft = ak(j)pk + btf) G Au

(5.3.21)

in the Hilbert space Et which is the completion of the pre-Hilbert space Et

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Geometric and Algebraic Topological Methods in Quantum Mechanics

of half-forms on Qt of compact support with respect to the scalar product

MM=(-^y j ptptQt

If Qt is compact, the operators (5.3.20) in Et are self-adjoint. Pre-Hilbert spaces Et constitute a trivial bundle ER over R. As in the above case of densities, any half-form p on Q yields the section p o it of the pull-back bundle i*Pi/ 2 —» Qt, i-e, a half-form on Qt. Given an element / £ Av and its pull-back ft = i*f £ At, we obtain from the formulae (5.3.17) and (5.3.20) that fp°H = ft(p°it)This equality shows that the Schrodinger quantization of the Poisson manifold V*Q can be seen as the instantwise quantization. Following this interpretation, let us choose the representation space ER(B.) for Av which consists of complex half-forms p on Q such that, for any ( e l , the halfform poit on Qt is of compact support. It is a C°°(R)-module. The ER(R) is also the carrier space for the Lie algebra AT, but its action on ER(R) is not instantwise. Let us turn now to quantization of the evolution equation (2.4.41). As was mentioned above, the problem is that, in the framework of the Schrodinger quantization, the covariant Hamiltonian 7i* (2.4.39) does not belong to the quantum algebra AT, unless it is affine in momenta. Let us restrict our consideration to the physically relevant case of 7i*, polynomial in momenta. We aim to show that such TC* is decomposed in a finite sum of products of elements of the algebra AT • Let / be a smooth function on T*Q which is a polynomial of momenta p\. A glance at the transformation laws (2.4.22) shows that it is a sum of homogeneous polynomials of fixed degree in momenta. Therefore, it suffices to justify a desired decomposition of an arbitrary homogeneous polynomial F of degree k > 1 on T*Q. We use the fact that the cotangent bundle T*Q admits a finite bundle atlas (see Theorem 10.6.2). Let {C/j}, £ = 1,... ,r, be the corresponding open cover of Q and {/^} a smooth partition of unity subordinate to this cover. Let us put

It is readily observed that {l^} is also a partition of unity subordinate to

Chapter 5 Geometric Quantization

323

{{/,}. Let us consider the local polynomials

{ai-ctfc)

Then we obtain a desired decomposition F

= E ^ = E E ^ ^ r ^ ^ ^ H ^ J • • • ikPak}, (5-3.22)

where all terms ^a? 1 '" afc p ai and /^pa are smooth functions on T*Q. Clearly, the decomposition (5.3.22) by no means is unique. The decomposition (5.3.22) shows that one can associate to a polynomial covariant Hamiltonian 7i* an element H of the enveloping algebra AT of the Lie algebra AT- Let us recall that A consists of finite sums of tensor products of elements of AT modulo the relations /®/'-/'®/-{/,/'}r = 0.

To be more precise, a representative TC belongs to AT + Av, where Av is the enveloping algebra of the Lie algebra Av C AT (see the decomposition (10.8.16) below). The enveloping algebra Av is provided with the antiautomorphism * : /i ® • • • ® fk -> (-l)fc/fc ® • • • ® / l . and one can always make a representative H Hermitian. Since the Dirac condition (4.2.1) holds, the Schrodinger representation of the Lie algebras AT and Av is naturally extended to their enveloping algebras AT and Av, and provides the quantization H* of a covariant Hamiltonian H* • Given an operator Ji*, the bracket V / = t[W*,/]

(5.3.23)

defines a derivation of the quantum algebra Av- Moreover, since p = —idt, the derivation (5.3.23) obeys the Leibniz rule V(r/j = dtrf + r V / ,

r 6 C°°(R).

Therefore, it is a connection on the instantwise algebra Av- In particular, / is parallel with respect to the connection (5.3.23) if [H*,/3 = 0.

(5.3.24)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

One can think of this equality as being the Heisenberg equation in timedependent mechanics, while the quantum constraint H*p = 0,

p e ER(R),

(5.3.25)

plays a role of the Schrodinger equation. It is readily observed that an operator / is a solution of the Heisenberg equation (5.3.24) if and only if it preserves the subspaces of solutions of the Schrodinger equation (5.3.25).

5.4

Example. Non-adiabatic holonomy operators

We address the Berry phase phenomena (Section 4.7) in a completely integrable system (CIS) of m degrees of freedom around its invariant tori Tm. The reason is that, being constant under an internal evolution, its action variables are driven only by a perturbation holonomy operator A. We construct such an operator for an arbitrary connection on a fibre bundle ExT^E,

(5.4.1)

without any adiabatic approximation [l8l]. In order that a holonomy and a dynamic Hamiltonian mutually commute, we first define a holonomy operator with respect to the initial data action-angle coordinates and, afterwards, return to the original ones. The key point is that both classical evolution of action variables and mean values of quantum action operators relative to original action-angle coordinates are determined in full by the dynamics of initial data action and angle variables. A generic phase space of a Hamiltonian system with time-dependent parameters is a composite fibre bundle

n

v"1 \^ TCP

i TCP

where FI -» S x R is a symplectic bundle and £ x R —> R. is a parameter bundle whose sections are parameter functions [175; 294; 385; 442]. Here, we assume that all the bundles are trivial and, moreover, their trivializations hold fixed. In the case of a CIS with time-dependent parameters, we have the product n = E x £ / = £ x l / x ( R x Tm)

-^ExR^l,

equipped with the coordinates (aa, Ik,t, ipk). Let us suppose for a time that parameters are also dynamic variables. The phase space of such a system

Chapter 5 Geometric Quantization

325

is the product

n' = T*S x u coordinated by (aa, aa, Ik,t, yfc), and the dynamics is characterized by the Hamiltonian form Hx = aadaa + Ikd
-Hs(al3,a0,Ij,t,
Hx=vaZ? + h(A$ + AkaZ?) + H,

(5.4.2)

where H is a function, dt + £f da is a connection on the parameter bundle S x R -> R, and A = dt ® {dt + Afdfc) + d
(5.4.3)

is a connection on S xRxr

-> S x I

[175; 181]. Bearing in mind that aa are parameters, one should choose the Hamiltonian 7is (5.4.2) to be affine in their momenta aa- Then, a Hamiltonian system with a fixed parameter function aa = £ a (t) is characterized by the connection E° = dt£a, and is described by the pull-back Hamiltonian form ff£ = ?HJ: = Ikdvk - (Jfc[A*(t,ipi) +

hka{e,t^)dtia] +

(5.4.4)

n{ejht^))dt

on the Poisson manifold U (2.4.55). Let H = 7i(Ii) be a Hamiltonian of an original autonomous CIS on the toroidal domain U (2.4.55) equipped with the action-angle coordinates (Ik,t,(pk). We introduce a desired holonomy operator by the appropriate choice of the connection A (5.4.3). For this purpose, let us choose the initial data action-angle coordinates (Ik,t,(f>k) by the converse to the canonical transformation (2.4.52). With respect to these coordinates, the Hamiltonian of an original CIS vanishes and the Hamiltonian form (5.4.4) reads Hz = Ikd4>k - Ik[Ak(t,
(5.4.5)

Let us put A* = 0 by the choice of a reference frame associated to the initial data coordinates (j>k, and let us assume that coefficients A£ are independent

326

Geometric and Algebraic Topological Methods in Quantum Mechanics

of time, i.e., the part A E = daa ® (da + Akadk)

(5.4.6)

of the connection A (5.4.3) is a connection on the fibre bundle (5.4.1). Then, the Hamiltonian form (5.4.5) reads Hz = Ikdk - IkAka(^,

4P)dt(,adt.

(5.4.7)

Its Hamilton vector field (2.4.34) is lH

= dt + Aidt^di - Ikdikkadted\

(5.4.8)

and it leads to the Hamilton equations

dt<^ = A ^ ( 0 , < m r , k

l

(5.4.9)

dth = -IkdiA a^(t), )dtC-

(5.4.10)

V*AE = daa ® (da + A^di - hdiKa&)

(5.4.11)

Let us note that

is the lift of the connection A s (5.4.6) onto the fibre bundle S x (V x Tm) -> S, seen as a subbundle of the vertical cotangent bundle V*{E x Tm) = E x T*Tm of the fibre bundle (5.4.1). It follows that any solution /»(£), 4>l(t) of the Hamilton equations (5.4.9) - (5.4.10) (i.e., an integral curve of the Hamilton vector field (5.4.8)) is a horizontal lift of the curve £(i) C £ with respect to the connection V*A-£ (5.4.11), i.e.,

Thus, the right-hand side of the Hamilton equations (5.4.9) - (5.4.10) is the holonomy operator A = (AadtSa, -IkdiAadtta)-

(5.4.12)

It is not a linear operator, but the substitution of a solution (£(£)) of the equation (5.4.9) into the Hamilton equation (5.4.10) results in a linear holonomy operator on the action variables Ii. Let us show that the holonomy operator (5.4.12) is well defined. Since any vector field $ on R x Tm such that $J dt = 1 is complete, the Hamilton

Chapter 5 Geometric Quantization

327

equation (5.4.9) has solutions for any parameter function £(£). It follows that any connection As (5.4.6) on the fibre bundle (5.4.1) is an Ehresmann connection, and so is its lift (5.4.11). Let us recall that a connection on a fibre bundle Y —> X is said to be an Ehresmann connection if, given an arbitrary smooth curve £([0,1]) C X, there exists its horizontal lift through any point of Y over £(0). Because V*A^ (5.4.11) is an Ehresmann connection, any curve £([0, l ] ) c S can play a role of the parameter function in the holonomy operator A (5.4.12). Now, let us return to the original action-angle coordinates (Ik,t,ipk) by means of the canonical transformation (2.4.52). The perturbed Hamiltonian reads

H' = /*A* (£(*), ¥>* -

t&nitfdteit)+n{ii),

while the Hamilton equations (5.4.9) - (5.4.10) take the form dtp* = d'Wj) + A\,m,
him),

¥>*(*) =
l

where /*(£(*)), (p (^(t)) is a solution of the Hamilton equations (5.4.9) (5.4.10). We observe that the action variables Ik are driven only by the holonomy operator, while the angle variables ipl have a non-geometric summand. Let us emphasize that, in the construction of the holonomy operator (5.4.12), we did not impose any restriction on the connection AE (5.4.6). Therefore, any connection on the fibre bundle (5.4.1) generates a holonomy operator in a CIS. However, a glance at the expression (5.4.12) shows that this operator becomes zero on action variables if all coefficients A^ of the connection A s (5.4.6) are constant, i.e., A s is a principal connection on the fibre bundle (5.4.1) seen as a principal bundle with the structure group i-pm

In order to quantize a time-dependent CIS on the Poisson toroidal domain (U,{,}v) (2.4.55) equipped with action-angle coordinates (/*,*,
328

Geometric and Algebraic Topological Methods in Quantum Mechanics

on Tm with those on R x Tm [155]. Namely, the corresponding quantum algebra A C C°°(U) consists of affine functions f = ak{t,)

(5.4.13)

of action coordinates Ik represented by the operators (5.2.9) in the space E = C°°(R x Tm)

(5.4.14)

of smooth complex functions ip(t,ip) on R x Tm. This space is provided with the structure of the C°°(R)-module endowed with the non-degenerate C°°(R)-bilinear form

W > = (JA I W'dm
rp,i>' G C°°(R x Tm).

We call it a pre-Hilbertian module. Its basis consists of the pull-back onto R x Tm of the functions V(«r) (5.2.10). Furthermore, this quantization of a time-dependent CIS on the Poisson manifold (U, {, }y) is extended to the associated homogeneous CIS on the symplectic annulus (U',0.) (2.4.53) by means of the operator Io — —idt in the pre-Hilbertian module E (5.4.14). Accordingly, the homogeneous Hamiltonian H is quantized as H = -idt + U. It is a Hamiltonian of a quantum time-dependent CIS. The corresponding Schrodinger equation is H-0 = -idtip + Hip = 0,

(5.4.15)

ip£E.

For instance, the quantum Hamiltonian of the original autonomous CIS seen as the time-dependent one is H = -idt + H(Tj). Its spectrum Hip{nr)

=

E(nr)il>lnr)

on the basis {^(nr)} for E (5.4.14) coincides with that of the autonomous Hamiltonian (5.2.16). The Schrodinger equation (5.4.15) reads H^ = -idtip + H(-idk - Xk)tp = 0,

ipGE.

329

Chapter 5 Geometric Quantization

Its solutions are the Fourier series V" = J2BM (nr)

s

exp[-t*S(nr)]V(nr),

(rv) € C.

Now, let us quantize this CIS with respect to the initial data actionangle coordinates (7j, <j>%). As was mentioned above, it is given on a toroidal domain U (2.4.55) provided with another fibration over R. Its quantum algebra A$ C C°°(U) consists of afnne functions f = ak(t,<j>j)Ik + b(t,j).

(5.4.16)

The canonical transformation (2.4.52) ensures an isomorphism of Poisson algebras A and AQ. Functions / (5.4.16) are represented by the operators / (5.2.9) in the pre-Hilbertian module EQ of smooth complex functions #(£, ^ ) o n R x r . Given its basis

the operators h and tp(nr) take the form (5.2.12) and (5.2.13), respectively. The Hamiltonian of a quantum CIS with respect to the initial data variables is H o = —idt. Then one easily obtains the isometric isomorphism R(1>M) = exp[tt£(nr)]tf ( n r ) ,

(fl(V) W ) > = W > ,

(5-4.17)

of the pre-Hilbertian modules E and -Bo which provides the equivalence

Ti = R'1TiR,

£ (nr) = j r

1

*^,

U = R-1HOR

(5.4.18)

of the quantizations of a CIS with respect to the original and initial data action-angle variables. In view of the isomorphism (5.4.18), let us first construct a holonomy operator in a quantum CIS (Ao, H o ) with respect to the initial data actionangle coordinates. Let us consider the perturbed homogeneous Hamiltonian H S = H 0 + H! = /„ + dtC(t)Aka (£(*), 4?)Ik of the classical perturbed CIS (5.4.7). Its perturbation term Hi is of the form (5.4.13) and, therefore, is quantized by the operator Hi = -idtZa&a = -idtCl^adk

+ \dk(Aka) - iAfeA*].

330

Geometric and Algebraic Topological Methods in Quantum Mechanics

The quantum Hamiltonian Hf = H 0 + Hi defines the Schrodinger equation 3ttf + dt£a[Akadk + \dk{Kka) - iAfeA*]tf = 0.

(5.4.19)

If its solution exists, it can be written by means of the evolution operator U{t) which reduces to the geometric factor •

t

Ui(t) = Texp i Idt'Za{t')Ka{t')dt' . o

.

(5.4.20)

The latter can be viewed as a displacement operator along the curve £[0, l ] c S with respect to the connection Ax=daa(da + Aa)

(5.4.21)

m

on the C°°(S)-module C°°(ExT ) of smooth complex functions o n S x T " 1 (see Section 4.6). Let us study weather this displacement operator exists. Given a connection As (5.4.6), let $%(t,) denote the flow of the complete vector field drt + Aa(t,<j>)dtta on R x Tm. It is a solution of the Hamilton equation (5.4.9) with the initial data <j>. We need the inverse flow (<3>~1)*(£,) which obeys the equation

dtp-yit,*) = - a t f AiCU*-1)*^)) = -dtFAfaftdkp-1)^,*). Let ^o be an arbitrary complex half-form \Po on Tm possessing identical transition functions, and let the same symbol stand for its pull-back onto R x Tm. Given its pull-back ($-!)•*o = det (^Qjr-)

MQ'Ht,4)),

(5-4-22)

it is readily observed that $ = (S-Y^oexptiAfc^]

(5.4.23)

obeys the Schrodinger equation (5.4.19) with the initial data ^o- Because of the multiplier exp[iAfc<^fc], the function ^ (5.4.23) however is ill defined, unless all numbers Afe equal 0 or ±1/2. Let us note that, if some numbers Afc are equal to ±1/2, then ^o exp[*Ajtfc] is a half-density on T m whose

331

Chapter 5 Geometric Quantization

transition functions equal ±1, i.e., it is a section of a non-trivial metalinear bundle over Tm. Thus, we observe that, if A^ equal 0 or ±1/2, then the displacement operator always exists and A = iHi is a holonomy operator. Because of the action law (5.2.13), it is essentially infinite-dimensional. For instance, let As (5.4.6) be the above mentioned principal connection, i.e., A^ =const. Then the Schrodinger equation (5.4.19) where A^ = 0 takes the form dt*(t,
*(t,^) = w - (rw-no))Ai). The corresponding evolution operator U(t) reduces to Berry's phase multiplier Ui*{nr) = e x p [ - m , ( r (t) - C(0))A£)]tf(np)>

nd G (nr).

It keeps the eigenvectors of the action operators Jj. In order to return to the original action-angle variables, one can employ the morphism R (5.4.17). The corresponding Hamiltonian reads H=

R-^R.

The key point is that, due to the relation (5.4.18), the action operators Ii have the same mean values

(7fcV#> = (4*1*),

* =%),

with respect both to the original and the initial data action-angle variables. Therefore, these mean values are denned only by the holonomy operator. In conclusion, let us note that, since action variables are driven only by a holonomy operator, one can use this operator in order to perform a dynamic transition between classical solutions or quantum states of an unperturbed CIS by an appropriate choice of a parameter function £. The key point is that this transition can take an arbitrary short time because we are entirely free with time parametrization of £ and can choose it quickly changing, in contrast with slowly varying parameter functions in adiabatic models. This fact makes non-adiabatic holonomy operators in CISs promising for several applications, e.g., quantum control and quantum computation. It also looks

332

Geometric and Algebraic Topological Methods in Quantum Mechanics

attractive that quantum holonomy operators in CISs are essentially infinitedimensional, whereas both the existent quantum control theory [119; 393] and the theory of quantum information and computation [240] involve only finite-dimensional operators. 5.5

Geometric quantization of constrained systems

We start with autonomous constrained systems. Let (Z, Q) be a symplectic manifold and ijv : N —» Z its closed imbedded submanifold such that the presymplectic form i^fi on TV is non-zero. We assume that N is a final constraint space and H. an admissible Hamiltonian on Z. In this case, the constrained Hamiltonian system (Z, Cl, H, N) is equivalent to the Dirac constrained system (N,i*NQ,i*NH). Therefore, it seems natural to quantize a symplectic manifold (Z, Cl) and, afterwards, replace classical constraints with the quantum ones. In algebraic quantum theory, quantum constraints are described as follows [203; 204]. Let £ be a Hilbert space and H € B(E) a Hermitian operator in E. By a quantum constraint is meant the condition He = 0,

eeE.

(5.5.1)

A Hermitian operator H defines the unitary operator exp(iW). Then the quantum constraint (5.5.1) is equivalent to the condition exp(iH)e = e. In a general setting, let A be a unital C*-algebra and 1 some subset of its unitary elements called state conditions. Let Sj denote a set of states / of A such that f(a) = 1 for all a eX. They are called Dirac states. One has proved that / G Sx if and only if f(ba) = f(ab) = f(b) for any a £ 1 and b £ A [203]. In particular, if / £ Sj, it follows at once from the relation (3.1.18) that |/(6(a - 1))|2 < f(bb*)f((a - l)(a* - 1)) = 0. One can similarly show that, if a, a' £ 2 and / £ Sj, then / ( ( O - l ) ( a ' - l ) ) = 0.

Chapter 5 Geometric Quantization

333

Thereby, elements a—I, a Gl, generate an algebra which belongs to Ker / for any / G Sj. The completion of this algebra in A is a C*-algebra A% such that f(a) = 0 for all a e Ax and / g l The following theorem provides the important criterion of the existence of Dirac states [203]. THEOREM 5.5.1. The set of Dirac states S% is not empty if and only if D 1 g AT. Let us return to quantization of constrained systems. In a general setting, one studies geometric quantization of a presymplectic manifold via its symplectic realization. There are the following two variants of this quantization [14; 40; 187]. (i) Let (N, u>) be a presymplectic manifold. There exists its imbedding iN:N^Z

(5.5.2)

into a symplectic manifold (Z, Q) such that u> = iNQ. This imbedding is not unique and different symplectic realizations (Z,Q.) of (N,w) fail to be isomorphic. They lead to non-equivalent quantizations of a presymplectic manifold (N,u>). For instance, if a presymplectic form ui is of constant rank, one can quantize a presymplectic manifold (N,u) via its canonical coisotropic imbedding in Proposition 2.1.4 [187]. Geometric quantization of (N,u>) via its imbedding into T*N has been studied in [40]. Given an imbedding i^ (5.5.2), we have a constrained system where classical constraints are smooth functions on Z vanishing on TV. They constitute an ideal IN of the associative ring C°°(Z). Then one usually attempts to provide geometric quantization of a symplectic manifold (Z, Q) in the presence of quantum constraints, but meets the problem how to associate quantum constraints to the classical ones. • Firstly, prequantization procedure / H-> / does not preserve the associative multiplication of functions. Consequently, prequantization IN of the ideal IN of classical constraints fails to be an ideal in a prequantum algebra, i.e., if / e IN then / ' / e IN for any / ' e C°°(Z), but / ' / £ ?N in general. Therefore, one has to choose some set of constraints <j>i,... i}, the quantum constraints {4>i} and {^} define different subspaces of a prequantum space in general. By the same reason, the direct adaptation of the notion of first and second class constraints to the quantum framework fails [203].

334

Geometric and Algebraic Topological Methods in Quantum Mechanics

• Secondly, given a set {fa} of classical constraints, one should choose a compatible polarization of the symplectic manifold (Z, Q) such that prequantum operators fa belong to the quantum algebra. Different sets of constraints imply different compatible polarizations in general. Moreover, a compatible polarization need not exist. (ii) If a presymplectic form u is of constant rank and its characteristic foliation is simple, there is a different symplectic realization (P, Q) of (N, u>) via a fibration N —> P (see Proposition 2.1.5 and Example 2.5.2). Then the reduced symplectic manifold (P, fi) is quantized [14]. Let us apply the above mentioned quantization procedures (i) - (ii) to the Poisson manifold (V*Q, {, }y) in Section 5.3. A glance at the equation (2.4.33) shows that one can think of the vector field 7# as being the Hamiltonian vector field of a zero Hamiltonian with respect to the presymplectic form dH on V*Q. Therefore, one can examine quantization of the presymplectic manifold (V*Q,dH). Given a trivialization (5.3.1), this quantization has been studied in [438]. (i) We use the fact that the range Nh = h(V*Q) of any section h (2.4.31) is a one-codimensional imbedded submanifold and, consequently, is coisotropic. It is given by the constraint H*=p + H(t,qk,Pk)=0. Then the geometric quantization of the presymplectic manifold (V*Q,dH) consists in geometric quantization of the cotangent bundle T*Q and setting the quantum constraint condition n*ip = 0

(5.5.3)

on physically admissible quantum states. It serves as the Shrodinger equation. The condition (5.5.3) implies that, in contrast with geometric quantization in Section 5.3, the Hamiltonian H* always belongs to the quantum algebra of T*Q. This takes place if one use polarization of T*Q which contains the Hamiltonian vector field uH. =dt + dkHdk - dk7idk.

(5.5.4)

Such a polarization of T*Q always exists. Indeed, any section h (2.4.31) of the afHne bundle £ (2.4.26) defines the splitting axdx = ak{dk - dknd°) + (o0 + akdkH)d°

335

Chapter 5 Geometric Quantization

of the vertical tangent bundle VT*Q of T*Q -> Q. Then elements (dk — dkHd°) together with the Hamiltonian vector field (5.5.4) span a polarization of T*Q. Clearly, this polarization does not satisfy the condition (5.3.3), and does not define any polarization of the Poisson manifold V*Q. (ii) In application to (V*Q, dH), the reduction procedure leads to quantization along classical solutions as follows. The kernel of dH is spanned by the vector field 7# and, consequently, the presymplectic form dH is of constant rank. Its characteristic foliation is made up by integral curves of this vector field, i.e., solutions of Hamilton equations. If the vector field 7/f is complete, this foliation is simple, i.e., is a fibration of V*Q over a symplectic manifold N of initial values. In this case, we come to the instantwise quantization when functions on V*Q at a given instant t G R are quantized as functions on N.

5.6

Example. Quantum relativistic mechanics

Quantum relativistic Hamiltonian mechanics exemplifies geometric quantization of a constrained system [391]. A configuration space of relativistic mechanics is an oriented pseudoRiemannian manifold (Q,g), coordinated by (qx). The space of relativistic velocities of relativistic mechanics on Q is the tangent bundle TQ of Q equipped with the induced coordinates (qx,qx) with respect to the holonomic frames {d\}. Relativistic motion is located in the subbundle Wg of hyperboloids 9»u{qWqv

- 1 = 0

(5.6.1)

of TQ. It is described by a second order dynamic equation gA = 3 V , 9 " )

(5-6.2)

on Q which preserves the subbundle (5.6.1), i.e.,

(qxdx + E ^ X s ^ T - 1) = 0,

dx=

d/dq\

This condition holds if the right-hand side of the equation (5.6.2) takes the form

336

Geometric and Algebraic Topological Methods in Quantum Mechanics

where {,/„} axe Christoffel symbols of a metric g, while Fx obey the relation g^vF^q" — 0. In particular, if the dynamic equation (5.6.2) is a geodesic equation qx = Kxq» with respect to a (non-linear) connection K = dqx® (dx +

K^)

on the tangent bundle TQ —> Q, this connections splits into the sum ^

= {A}«"+fl%

(5-6-3)

of the Levi-Civita connection of g and a soldering form F = ^"F^dq"

® dx,

F^ = -FVil.

The momentum phase space of relativistic mechanics on Q is the cotangent bundle T*Q provided with the canonical symplectic form n = dpxAdqx.

(5.6.4)

Let us note that one also considers another symplectic form Q + F where F is the strength of an electromagnetic field [401]. A relativistic Hamiltonian is defined as follows [294; 380; 384]. Let H be a smooth real function on T*Q such that the morphism H : T*Q -> TQ,

(f o H = d»H,

(5.6.5)

is a bundle isomorphism. Then the inverse image

of the subbundle of hyperboloids Wg (5.6.1) is a one-codimensional (consequently, coisotropic) closed imbedded subbundle N oiT*Q given by the constraint HT = g^d^Hff'H - 1 = 0.

(5.6.6)

We say that H is a relativistic Hamiltonian if the Poisson bracket {H, HT} vanishes on N. This means that the Hamiltonian vector field 7 = dxHdx - dxHdx

(5.6.7)

337

Chapter 5 Geometric Quantization

of H preserves the constraint N and, restricted to N, it obeys the Hamilton equation (5.6.8)

•y\QN+i*NdH = 0

of a Dirac constrained system on N with a Hamiltonian H. The morphism (5.6.5) sends the vector field 7 (5.6.7) onto the vector field 7T

= qxdx + {dfiH8xdliH - d^Hdxd^H)dx

on TQ. This vector field defines the second order dynamic equation qx = dfiHdxdtlH - d^Hdxd"H

(5.6.9)

on Q which preserves the subbundle of hyperboloids (5.6.1). Example 5.6.1. The following is a basic example of relativistic Hamiltonian systems. Let us put

# = ^
Tit

-9au{p»-K),

Pa = -^-dag^(Pll

- bjfrv - K) + - < T ( P M -

bJdcK.

The corresponding second order dynamic equation (5.6.9) on Q is 9A = { / A M V - ^9XuF^, { M M = —j9Xfi{dn9pv + 9^90^ - dpg^), Ffj,u = dpbi, — dvb^.

It is a geodesic equation with respect to the affine connection K

of type (5.6.3).

n - U u)q - —9 tp.v flC

(5.6.10)

338

Geometric and Algebraic Topological Methods in Quantum Mechanics

For instance, let g be a metric gravitational field and let b^ = eA^, where A^ is an electromagnetic potential whose gauge holds fixed. Then the equation (5.6.10) is the well-known equation of motion of a relativistic massive charge in the presence of these fields. D Let us turn now to quantization of relativistic mechanics [391]. We follow the standard geometric quantization of a cotangent bundle [39; 401; 438]. Because the canonical symplectic form fi (5.6.4) on T*Q is exact, the prequantum bundle is denned as a trivial complex line bundle C over T*Q. Let us note that this bundle need no metaplectic correction since T*X is endowed with canonical coordinates for the symplectic form Q. Thus, C is a quantum bundle. Let its trivialization C^T*QxC

(5.6.11)

x

hold fixed, and let (q ,p\,c), c £ C, be the associated bundle coordinates. Then one can treat sections of C (5.6.11) as smooth complex functions on T*Q. Let us note that another trivialization of C leads to an equivalent quantization of T*Q. The Kostant-Souriau prequantization formula associates to each smooth real function / G C°°(T*Q) on T*Q the first order differential operator f = -iVtf+f

(5.6.12)

on sections of C, where

0, = dxfdx - dxfdx is the Hamiltonian vector field of / and V is the covariant differential with respect to a suitable [/(l)-principal connection A on C. This connection preserves the Hermitian metric g(c, c') (5.1.12) on C, and its curvature form obeys the prequantization condition R = i£l. For the sake of simplicity, let us assume that Q and, consequently, T*Q is simply connected. Then the connection A up to gauge transformations is A = dpx ® dx + dqx ® (dx + icpxdc),

(5.6.13)

and the prequantization operators (5.6.12) read f = -Wf + {f-pxdxf).

(5.6.14)

Let us choose the vertical polarization on T*Q. It is the vertical tangent bundle VT*Q of the fibration n : T*Q —> Q. The corresponding quantum

Chapter 5 Geometric Quantization

339

algebra .4 C C°°(T*Q) consists of affine functions of momenta (5-6.15)

f = a\q*)px + b(q*) on T*Q. They are represented by the Schrodinger operators / = -iax6x - %-dxax - -^axdxH-g)

+ b,

g = det(ga0),

(5.6.16)

in the space E of sections p of the quantum bundle C of compact support which obey the condition V^p = 0 for any vertical Hamiltonian vector field •d on T*Q. This condition takes the form dxfdxp = Q,

f£C°°{Q).

It follows that elements of E are independent of momenta and, consequently, fail to be compactly supported, unless p = 0. This is the above mentioned problem of Schrodinger quantization which is solved as follows. Let IQ : Q —> T*Q be the canonical zero section of the cotangent bundle T*Q. Let CQ = i*QC be the pull-back of the bundle C (5.6.11) over Q. It is a trivial complex line bundle CQ = Q x C provided with the pull-back Hermitian metric g(c, c') = cc' and the pull-back AQ = i*QA = dqx ® (dx + icpxdc) of the connection A (5.6.13) on C. Sections of CQ are smooth complex functions on Q, but this bundle need metaplectic correction. Let the cohomology group H2(Q; Z2) of Q be trivial. Then a metalinear bundle V of complex half-forms on Q is defined. It admits the canonical lift of any vector field r on Q such that the corresponding Lie derivative of its sections reads LT = TX8x + \dXTX. Let us consider the tensor product Y — CQ T> over Q. Since the Hamiltonian vector fields 1?/ = axdx - (P)1dxa» + dxb)dx

340

Geometric and Algebraic Topological Methods in Quantum Mechanics

of functions / (5.6.15) project onto Q, one can assign to each element / of the quantum algebra A the first order differential operator / = (-iV**, + /) ® Id + Id L7rtf/ == -iaxdx - l-d\ax + b on sections PQ of Y. For the sake of simplicity, let us choose a trivial metalinear bundle V —> Q associated to the orientation of Q. Its sections can be written in the form

PQ = (-s) v V, where ip are smooth complex functions on Q. Then the quantum algebra A can be represented by the operators / (5.6.16) in the space C°°(Q) of these functions. It is easily justified that these operators obey the Dirac condition

lf,f'} = -i{fJ7}. Remark 5.6.2. One usually considers the subspace EQ C C°°(Q) of functions of compact support. It is a pre-Hilbert space with respect to the non-degenerate Hermitian form

(V|V') = /#'(-
It is readily observed that / (5.6.16) are symmetric operators / = /* in EQ, i.e.,

(/V#'> = WV/>. In relativistic mechanics, the space EQ however gets no physical meaning.

•

Let us note that the function HT (5.6.6) need not belong to the quantum algebra A. Nevertheless, one can show that, if HT is a polynomial of momenta of degree k, it can be represented as a finite composition HT=Y^fii---fki i

(5-6.17)

341

Chapter 5 Geometric Quantization

of products of affine functions (5.6.15), i.e., as an element of the enveloping algebra A of the Lie algebra A [174]. Then it is quantized 5Z/ii---/fci

HT^HT=

(5.6.18)

i

as an element of A. However, the representation (5.6.17) and, consequently, the quantization (5.6.18) fail to be unique. The quantum constraint HTip = 0.

(5.6.19)

serves as a relativistic quantum equation. Example 5.6.3. Let us consider a massive relativistic charge in Example 5.6.1 whose relativistic Hamiltonian is H=

±g»'(jpli-eAll)(pv-eAv).

It defines the constraint HT = - ^ " " ( P M - eAM)(jpv

- eAv) - 1 = 0 .

(5.6.20)

Let us represent the function H? (5.6.20) as the symmetric product HT

=^

^

• (pM - eA.) • H O V4 • « T . {-g) V4 .

{pv - eAv) • ( ~ g ) ~ 1 / 4 - 1 m of affine functions of momenta. It is quantized by the rule (5.6.18), where {-9)1/4odao(-g)-V* = -ida. Then the well-known relativistic quantum equation (-
- ieAu) + m2)^ = 0. •

342

5.7

Geometric and Algebraic Topological Methods in Quantum Mechanics

Geometric quantization of holomorphic manifolds

This Section addresses geometric quantization of holomorphic manifolds. The key point is that holomorphic manifolds admit the canonical complex polarization. Let Z be an m-dimensional holomorphic manifold and C - > Z a complex line bundle equipped with complex coordinates (c, z*). It means that the coordinate transition functions on Z obey the condition dzH _ dzH

_

Let us assume that the line bundle C admits a linear connection r = dzi ® (di + Ticdc) + dzi ® (di - Ticdc) with the curvature form R = iQ, where f2 is a non-degenerate real 2-form on Z. It implies that the manifold Z is provided with a symplectic structure, given by the symplectic form Cl, and with a F-invariant Hermitian fibre metric g. With respect to the local complex canonical coordinates, this symplectic form reads n = 2i ^dzj i

Adzj,

while the Hermitian metric g is given by the expression (5.1.12). Therefore, geometric quantization of a holomorphic manifold is similar to that of a real symplectic manifold if complex and real canonical coordinates are connected with each other by the relations ^ = l^Vi+W)The holomorphic manifold Z has the canonical polarization T = {v e TZ : Jv = -iv}, whose sections i? e TQ are complex vector fields d = i? di on Z (see Section 2.6 for the notation). As in the case of a real symplectic manifold, let us consider the subalgebra AT of the Poisson algebra A{Z) which consists of smooth complex functions / on Z such that [i?/,Tf,]cr fl ,

Chapter 5 Geometric Quantization

343

where i

is the Hamiltonian vector field for a function / . It is readily observed that holomorphic functions on Z always belong to Aria order to construct the carrier space of the algebra AT, we take the sections s of the complex line bundle C —» Z such that V,?s = 0,

i? G Tn,

i.e., {di+Ti)3 = 0, and the holomorphic sections P = Pi.^mdz1 A • • • A dzm

of the exterior product m

A TZi )0 -> z.

Let g

-

s

(g, p — spx...mdzx A • • • A dzm

(5.7.1)

be sections of the tensor product C®ATZioz Then the operator f(s ®p) = -i[(V tf/ + z/)(s) ® /o + s ® Ltf//o]

(5.7.2)

can be assigned to any function / G AT- With respect to local complex canonical coordinates, this operator reads

?{s®p) = \( Y^dkfidk -Tk + if)s) ® p + \s ® ( J2 dk(pdkf)\ . The operators (5.7.2) obey Dirac's condition (4.2.1).

344

Geometric and Algebraic Topological Methods in Quantum Mechanics

Sections (5.7.1) of compact support form a pre-Hilbert space Sz with respect to the Hermitian form i™2 f g(si,s2)pip2 {si ® pi\s2 ® p2) = — z

(5.7.3)

The operators / (5.7.2) fulfill the relation fSz C Sz, and they are Hermitian in SzThere is the standard procedure of geometric quantization of a holomorphic manifold Z which admits an imbedding to a projective Hilbert space. Let Z be an m-dimensional holomorphic manifold provided with a symplectic form fi and C —> Z a, complex line bundle over Z with a Hilbert space E realized as a vector subspace of the space C of smooth sections of the dual C —> Z to the complex conjugate C of C —> Z. Given a trivialization chart Ua of C —> Z, let us assume that, for each section ip £ E, the evaluation functionals ea,z{ija) = ipa{z) : E ^ C are non-zero and continuous, i.e.,

hM*)l < Ak.zllV'll,

Ma,zeR.

In accordance with the Riesz theorem, there exists non-zero Ka{z) e E such that

[345]. Then we have the smooth map K : Z 9 Z H

[Ka(z)\ € PE,

where [Ka(;z)] denotes the one-dimensional subspace spanned by Ka(z) ^ 0 and PE is the projective Hilbert space in Example 4.1.5. It is readily observed that the definition of K. is independent of a trivialization chart. Since a projective Hilbert space admits the canonical complex line bundle (see (5.7.4) below), we have the following standard procedure of its geometric quantization [4]. Let be E a Hilbert space and PE the corresponding projective Hilbert space. For the sake of simplicity, E is assumed to be separable. Let us

345

Chapter 5 Geometric Quantization

consider the fibre bundle C = {(z, z) € E x PE : z G z}

(5.7.4)

together with the corresponding projections C : C -» PE,

K:C-+E.

It is called the universal line bundle. There is its subbundle C o = E \ { 0 } - > PE,

n : z ^ z ,

with the typical fibre C \ {0}. Let {Uh,iph) be a chart (4.1.27) of the projective Hilbert space PE. Recall that its image in E is the subspace Eh (4.1.28). We provide this chart with the following coordinate system. Let us take the orthonormal basis {h, efc}, k = 1,..., dimE — 1, for the Hilbert space E. Then {e^} is an orthonormal basis for its subspace Eh, while the coordinates zfc with respect to this basis are the coordinates on the above mentioned chart of the projective Hilbert space PE. The projective Hilbert space PE is provided with the metric Fubini-Studi (4.1.32) and with the corresponding Kahler form u (4.1.33). The universal line bundle (5.7.4) has the connection T = dz>® (dj + j ^ j p ^ c ) <& ® (dj - YVWCd)

'

which preserves the Hermitian fibre metric (5.1.12) in C and satisfies the condition R = iui. Now let Z be a holomorphic manifold and K. its imbedding into the projective Hilbert space PE. The above manifested standard procedure of geometric quantization of Z consists in the following. Let us propose that the pull-back form K.*u> on Z is non-degenerate, i.e., this is a symplectic form. Then one can consider the pull-back line bundle fC*C over Z. This line bundle admits the pull-back connection K.*T with the curvature form iK,*ui, i.e., K,*C is a prequantization bundle.

Chapter 6

Supergeometry

Supergeometry is an ingredient in many quantum models, e.g., SUSY mechanics and BRST formalism. Supergeometry is phrased in terms of Z2graded modules and sheaves over Z2-graded commutative algebras. Their algebraic properties naturally generalize those of modules and sheaves over commutative algebras, but this is not a particular case of non-commutative geometry because of the peculiar definition of graded derivations. Here, we restrict our consideration to geometry of graded manifolds. They are not supermanifolds, though every graded manifold determines a DeWitt JY^-supermanifold, and vice versa [21] (see Theorem 6.9.5 below). 6.1

Graded tensor calculus

Unless otherwise stated, by a graded structure throughout this Chapter is meant a Z2-graded structure, and the symbol [.] stands for the Z2-graded parity. Let us recall some basic notions of the graded tensor calculus [21; 102]. An algebra A is called graded if it is endowed with a grading automorphism 7 such that 7 2 = Id. A graded algebra seen as a Z-module falls into the direct sum A = Ao © Ai of two Z-modules AQ and A\ of even and odd elements such that 7 (a)

= (-l) i a,

aeAu

t = 0,l.

One calls Ao and A\ the even and odd parts of A, respectively. In particular, if 7 = Id, then A = Ao- Since -y(aa') = 7(0)7(0'), 347

348

Geometric and Algebraic Topological Methods in Quantum Mechanics

we have [aa'j = ([o] + [o'])mod2 where a £ A[a], a' € A[aiy It follows that A§ is a subalgebra of A and A\ is an .Ao-module. If A is a graded ring, then [1] = 0. A graded algebra A is said to be graded commutative if aa' = (-i)MM o 'a, where a and a' are arbitrary homogeneous elements of A, i.e., they are either even or odd. Given a graded algebra A, a left graded A-module Q is a left .4-module provided with the grading automorphism 7 such that -y(aq) = 7(0)7(9).

a€A,

q&Q,

i.e., H = ([a] + [<7])mod2. A graded module Q is split into the direct sum Q = Qo ® Qi of two AQmodules Qo and Qi of even and odd elements. Similarly, right graded modules are denned. If /C is a graded commutative ring, a graded /C-module can be provided with a graded K-bimodule structure by letting qa := {-l)^^aq,

a G AC,

q£Q.

A graded /C-module is called free if it has a basis generated by homogeneous elements. This basis is said to be of type (n, m) if it contains n even and m odd elements. In particular, by a (real) graded vector space B — BQ ® B\ is meant a graded K-module. A graded vector space is said to be (n, m)-dimensional if Bo = R" and Bx = Rm. The following are standard constructions of new graded modules from old ones. • The direct sum of graded modules over the same graded commutative ring and a graded factor module are defined just as those of modules over a commutative ring.

349

Chapter 6 Supergeometry

• The tensor product P®Q of graded /C-modules P and Q is an additive group generated by elements p®q, p G P, q G Q, obeying the relations {p + p') ® q = p ® q + p' ® q, p®{q + q') = p®q+p®q', ap®q = (-l)[p][a]pa ®q = (-l) [pI[al p ® aq = aeK.. (-l)(W+[9])Wp®9o, In particular, the tensor algebra <3 of a graded /C-module Q is denned as that of a module over a commutative ring in Example 1.1.3. Its quotient /\Q with respect to the ideal generated by elements q®q'

+ {-l)[q][q']q'®q,

9,«'eQ,

is the bigraded exterior algebra of a graded module Q with respect to the graded exterior product qAq'

= -(-1)WI»VA?.

• A morphism $ : P —> Q of graded /C-modules is said to be even (resp. odd) if $ preserves (resp. change) the graded parity of all elements P. It obeys the relations $(ap) = (-l)l*" a l$(p),

p£P,

OG/C.

The set Hom^(P, Q) of graded morphisms of a graded /C-module P to a graded /C-module Q is naturally a graded /C-module. The graded /C-module P* = Horn £(P, /C) is called the duaJ of a graded /C-module P. A graded commutative JC-ring A is a graded commutative ring which is also a graded /C-module. A graded commutative R-ring is said to be of rank N if it is a free algebra generated by the unit 1 and iV odd elements. A graded commutative Banach ring A is a graded commutative R-ring which is a real Banach algebra whose norm obeys the additional condition ||ao + ai|| = ||ao|| + ||ai||,

a0 G Ao,

ai G Ai-

Let V be a real vector space. Let A = AV be its (N-graded) exterior algebra provided with the Z2-graded structure

A = A o eAi,

Ao = R 0 A V , fc=i fc=i

Ai=®2fc/\V.

(6.1.1)

350

Geometric and Algebraic Topological Methods in Quantum Mechanics

It is a graded commutative K-ring, called the Grassmann algebra. A Grassmann algebra, seen as an additive group, admits the decomposition A = R®R = R®R0®R1=R®

(A^2 © Ai,

(6.1.2)

where R is the ideal of nilpotents of A. The corresponding projections a : A —» R and s : A —> R are called the body and soul maps, respectively. Remark 6.1.1. Let us note that there is a different definition of a Grassmann algebra [231] which is equivalent to the above one only in the case of an infinite-dimensional vector space V [102]. Let us mention the Arens-Michael algebras of Grassmann origin [69] which are most general graded commutative algebras, suitable for superanalysis (see Remark 6.9.3 below). • Hereafter, we restrict our consideration to Grassmann algebras of finite rank. Given a basis {c1} for the vector space V, the elements of the Grassmann algebra A (6.1.1) take the form

« = E E

a il ... it c il ---cS

(6.1.3)

fc=o (n-ifc)

where the second sum runs through all the tuples (ii • • • ik) such that no two of them are permutations of each other. The Grassmann algebra A becomes a graded commutative Banach ring if its elements (6.1.3) are endowed with the norm fc=0 (u-i f c )

Let B b e a graded vector space. Given a Grassmann algebra A of rank N, it can be brought into a graded A-module AB = (AB)0 © (AS)! = (Ao ® Bo 8 Ax ® B{) ® {Al ®BQ®AQ®

B{),

called a superspace. The superspace Bn\m =

[ (( | ^

e (™ A i ) ] e

[(£ Ai) © (© A0)]

is said to be (n, m)-dimensional. The graded A0-module 5"-" l = (ffiAo)©(©A1) is called an (n, m)-dimensional supervector space.

(6.1.4)

Chapter 6 Supergeometry

351

Whenever referring to a topology on a supervector space Bn'm, we will mean the Euclidean topology on a 2N~1[n + m]-dimensional real vector space. Given a superspace _B"lm over a Grassmann algebra A, a A-module endomorphism of £?™lm j s represented by an (n + m) x (n + m) matrix (6.1.5) with entries in A. It is called a supermatrix. One says that a supermatrix L is • even if L\ and L4 have even entries, while L 2 and L3 have the odd ones; • odd if L\ and L4 have odd entries, while L2 and L3 have the even ones. Endowed with this gradation, the set of supermatrices (6.1.5) is a graded A-ring. Unless otherwise stated, by supermatrices are meant homogeneous ones. The familiar notion of a trace is extended to supermatrices (6.1.5) as the supertrace StrL = T r L i - ( - l ) [ L l T r £ , 4 . For instance, Str( 1) — n — m. A supertransposition Lst of a supermatrix L is the supermatrix t

_(

L\

(-l)MLl\

where L* denotes the ordinary matrix transposition. There are the relations Str(Ls*) = StrL, (6.1.6)

(LL')st = (-i)MlL']L'stLs\ Str(LL') = (-l) lL1[L/1 Str(L'L)

or Str([L,L']) = 0.

(6.1.7)

In order to extend the notion of a determinant to supermatrices, let us consider invertible supermatrices L (6.1.5). They are never odd. One can show that an even supermatrix L is invertible if and only if either the matrices Li and L4 are invertible or the real matrix a(L) is invertible, where a is the body morphism. Invertible supermatrices constitute a group GL(n\m; A), called the general linear graded group. Then a superdeterminant of L € GL(n\m; A) is

352

Geometric and Algebraic Topological Methods in Quantum Mechanics

defined as SdetL = det(L! -L 2 ^4 1 i3)(detL4 1 ). It satisfies the relations Sdet(LL') = (SdetL)(SdetZ/)> Sdet(L st )=SdetZ,, Sdet(exp(L)) = exp(Sdet (L)). Let K. be a graded commutative ring. A graded commutative (nonassociative) /C-algebra g is called a Lie IC-superalgebra if its product, called the superbracket and denoted by [.,.], obeys the relations ( _l)W[e"] [e7 [ £ V / ] ] + ( _l)[ £ 'l[e] [£ ' ) [e»>e]]

+ ( - 1 ) 1 ^ ' ] [e ", [£)£']] = 0 .

Obviously, the even partfloof a Lie /C-superalgebra g is a Lie /Co-algebra. A graded /C-module P is called a g-module if it is provided with a ^-bilinear map g x P B (e, p) t-> ep e P, [ep] = ([e] + [p])mod2) [e,e']p = ( e o e / - ( - l ) I e " e V o e ) p . 6.2

Graded diflferential calculus and connections

Linear differential operators and connections on graded modules over graded commutative rings are denned similarly to those in commutative geometry. Let K, be a graded commutative ring and A a graded commutative /C-ring. Let P and Q be graded ,4-modules. The graded /C-module Horn A; (P, Q) of graded /C-module homomorphisms $ : P —•» Q can be endowed with the two graded ,4-module structures (a$)(p) := a$(p),

($ • a)(p) := $(ap),

a € A,

p € P,

(6.2.1)

called A- and .4#-module structures, respectively. Let us put 5a$:=a$-(-l)[al[*I$«a,

a & A.

(6.2.2)

353

Chapter 6 Supergeometry

An element A e Hom;c(P, Q) is said to be a Q-valued graded differential operator of order s on P if <5a o °---°^ s A = 0 for any tuple of s +1 elements ao, • • •, as of A- The set Diff S(P, Q) of these operators inherits the graded module structures (6.2.1). In particular, zero order graded differential operators obey the condition SaA(p) = aA(p) - (-l)W( A ]A(ap) = 0,

a £ A,

p € P,

i.e., they coincide with graded ,4-module morphisms P —» Q. A first order graded differential operator A satisfies the condition 5a o Sb A(p) = abA(p) - (-l)(l6WA»Ia)&A(ap) - {-lf^aA(bp) peP. (_l)[b][A]+([A]+[6])H = 0> a,bzA,

+

For instance, let P = A. Any zero order Q-valued graded differential operator A on A is defined by its value A(l). Then there is a graded .4-module isomorphism DiSQ(A,Q) = Q via the association QBq^Aq

eDiff o (.4,Q),

where A, is given by the equality A g (l) = q. A first order Q-valued graded differential operator A on ^4 fulfils the condition A{ab) = A{a)b + (-l)t°][A]aA(6) - (-l)([ b ] + [ Q ]" A la6A(l),

a.fisA

It is called a Q-valued graded derivation of A if A(l) = 0, i.e., the graded Leibniz rule A{ab) = A(a)b + {-l)WWaA{b),

a,b e A,

(6.2.3)

holds (cf. (1.2.10)). One obtains at once that any first order graded differential operator on A falls into the sum A(o) = A(l)a + [A(a)-A(l)o] of a zero order graded differential operator A(l)a and a graded derivation A(a) — A(l)a. If d is a graded derivation of A, then ad is so for any a G A. Hence, graded derivations of A constitute a graded .4-module D(A,Q), called the graded derivation module.

354

Geometric and Algebraic Topological Methods in Quantum Mechanics

If Q = A, the graded derivation module T)A is also a Lie superalgebra over the graded commutative ring /C with respect to the superbracket [uItt/]=uou'-(-i)M[«Vou,

u,u'eA.

(6.2.4)

We have the graded .4-module decomposition (6.2.5)

DiS1(A)=A®DA.

Let us turn now to jets of graded modules. Given a graded ,4-module P, let us consider the tensor product A ®tc P of graded ^-modules A and P. We put 8b(a®p) :=(ba)®p-(-l)[a]lb]a®(bp),

p e P,

a,b e A.

(6.2.6)

The k-order graded jet module Jk{P) of the module P is defined as the quotient of the graded /C-module A ®K P by its submodule generated by elements of type 5bo

o---o5bk{a®p).

In particular, the first order graded jet module Jl{P) consists of elements a®ip modulo the relations ab®lP-

(—I)ta][6l6 ®t (ap) -a®x (bp) + 1 i (abp) = 0.

(6.2.7)

For any h G Horn A(A ® P,Q), the equality Sb(h(a®p)) = (-l)WWh(6b(a®p)) holds. By analogous with Theorem 1.2.3, one then can show that any Qvalued graded differential operator A of order A; on a graded ^-module P factorizes uniquely A : P ^Jk(P)

—*Q

through the morphism Jk

:p3p^l®kpeJk(P)

and some homomorphism fA : Jk(P) —> Q. Accordingly, the assignment A w f A defines an isomorphism Diff s (P,Q)=Hom^(J s (P),Q).

(6.2.8)

355

Chapter 6 Supergeometry

Let us focus on the first order graded jet module Jx of A consisting of the elements a ®\ b, a, b G A, subject to the relations ab i 1 - (-l)ta]M& ®! a - a ®i 6 + 1 ®i (ab) = 0.

(6.2.9)

It is endowed with the A- and .4*-module structures c(a ®i b) := (ca) i 6,

c • (a (g>i b) :— a ®i (c6).

There are canonical .4- and ^'-module monomorphisms

ii : A3 a>-* a®il € Jl, :A3a^>l®laeJ1, J1 such that Jl, seen as a graded «4-module, is generated by the elements Jla, a e A. With these monomorphisms, we have the canonical ^.-module splitting J1 = ii(A)®O\ 1

aJ ^)

(6.2.10)

= a 0i b — ab i 1),

where the graded ,4-module Ol is generated by the elements I®i6-6®il,

JeA

Let us consider the corresponding ,4-module epimorphism h1 : J1 3 1 <s>i b H^ 1 0 ! 6 - b ®j 1 e O1

(6.2.11)

and the composition d = h1oJ1:A3bt-^l®1b-b®1leO1.

(6.2.12)

The equality d(a6) = a ®i 6 + 6 <&! a - a6 ! 1 - 6a ®j 1 = ( - l ) ( ° 1 H M a + adb shows that d (6.2.12) is an even O 1 -valued derivation of A. Seen as a graded v4-module, O1 is generated by the elements da for all a £ A. In view of the splittings (6.2.5) and (6.2.10), the isomorphism (6.2.8) reduces to the isomorphism i}A = Ou = Rom A{0\

A)

(6.2.13)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

ofVA to the dual 01* of the graded .4-module O1. It is given by the duality relations X)A3u^(j)u£

Ou,

a € A.

(6.2.14)

Using this fact, let us construct a differential calculus over a graded commutative /C-ring A. Let us consider the bigraded exterior algebra O* of a graded module Ol. It consists of finite linear combinations of monomials of the form cj) = aodai A • • • A dak,

Oj G A,

(6.2.15)

whose product obeys the juxtaposition rule (agdai) A (podbi) = aodiaibo) A db\ — aoa\dbo A db\ and the bigraded commutative relations 4> A <£' = (_l)l*ll*'l+[*H*y A .

(6.2.16)

In order to make O* to a differential algebra, let us define the coboundary operator d : Ol -> O2 by the rule # ( « , « ' ) = -u'(u(4>)) + (-1)H[«'] U ( U '(0)) + [u',u](), where u, v! G T>A, <j> £ O1, and u, u' are both graded derivatives of A and A-valued forms on O1. It is readily observed that, by virtue of the relation (6.2.14), (d o d)(a) = 0 for all a e A. Then d is extended to the bigraded exterior algebra O* if its action on monomials (6.2.15) is defined as d(aodai A • • • A dak) = dao A da\ A • • • A dakThis operator is nilpotent and fulfils the familiar relations d{cf> A tf) =dcpA
(6.2.17)

It makes O* into a differential bigraded algebra, called a graded differential calculus over a graded commutative /C-ring A. Furthermore, one can extend the duality relation (6.2.14) to the graded interior product of u S VA with any monomial <j> (6.2.15) by the rules u\(bda) =

(-l)M[b]u(a),

u}(

),

* e C

0 € 7 £ , , (6.4.5)

where dc and d-p are exterior differentials on the differential algebras C^ and V^, respectively. We obtain at once from the relation (6.4.3) that d{(j>®i)) = {dv)®ip + {-i)w®{dc^),

V - e C

4>^V^.

The exterior differential d (6.4.5) is nilpotent. It obeys the equalities d{

if, iff G S£>,

and makes S^ into a differential bigraded algebra, which is locally generated by the elements (1,c%,y\,dx\6 a h = del ~ cl+Adx\6\

= dy\ - y\+Adxx),

|A| > 0.

Hereafter, let the collective symbols s^ and 9^ stand both for even and odd generating elements c^, y\, Q\, 6\ of the C°°(X)-algebra S^ which, thus, is locally generated by (l,s^,dxx,6£), |A| > 0. We agree to call elements of S^ the graded exterior forms on X. Similarly to O^, the differential bigraded algebra S^ is decomposed into S^ -modules S^r of fc-contact and r-horizontal graded forms together with the corresponding projections hk and hr. Accordingly, the exterior differential d (6.4.5) on 5 ^ is split into the sum d = dn + dy of the total and vertical differentials dH{(j>)=dxx

hdx{<j)),

dv(cj)) = e£Ad%ct>,

The projection endomorphism Q of S^ is given by the expression ?o/lfeo/l Q= X^T "' K

fc>o

m

= Y, ( - i ) | A | ^ A [dA(^j

°-",

|A|>0

similar to (10.7.18). The graded variational operator 6 = god is introduced. Then the differential bigraded algebra ££, is split into the Z2-graded vari-

370

Geometric and Algebraic Topological Methods in Quantum Mechanics

ational bicomplex (O(X),S^,Ek

= g(S^);d,dH,dv,Q,6),

(6.4.6)

analogous to the variational bicomplex (10.7.20). It should be emphasized that, in contrast with the Remark 6.4.1. differential graded algebra O^ of exterior forms on J°°Y, the differential bigraded algebra C^ consists of sections of sheaves on X. In order to regard these algebras on the same footing, let us consider the open surjection 7r°° : j°°y -» X and the direct image T T ^ Q ^ on X of the sheaf Q ^ °f exterior forms on J°°Y (see Theorem 10.7.3). Its stalk at a point x G X consists of the equivalence classes of sections of the sheaf £2^ which coincide on the inverse images {K°°)~1{UX) of open neighborhoods Ux of X. Since (n°a)~1(Ux) is the infinite order jet space of sections of the fibre bundle ir~1(Ux) —> X, every point x G X has a base of open neighborhoods {Ux} such that the sheaves £2^,* of Q^-modules are acyclic on the inverse images (Tr°°)~1(Ux) of these neighborhoods. Then, in accordance with the Leray theorem [184], cohomology of J°°Y with coefficients in the sheaves £2^* is isomorphic to that of X with coefficients in their direct images TT^Q^, i.e., the sheaves •^oo-doo on X are acyclic. If Y —> X is an affine bundle, then X is a strong deformation retract of J°°Y and the inverse images (7roo)~1(t/x) of contractible neighborhoods Ux are contractible and 7r£°R. = R. Then, by virtue of the algebraic Poincare lemma, the variational bicomplex H ^ of sheaves on (7roo)~1(C/x), except the terms R, is exact, and the variational bicomplex TT^O^, of sheaves on X is so. There is the R-algebra isomorphism of the differential graded algebra of sections of the sheaf n^Q^ on X to the differential graded algebra Q^ in the proof of Theorem 10.7.3. Thus, Q£o and its subalgebra O^ can be regarded as algebras of sections of sheaves on X, and they keep their d-, dn- and <5-cohomology expressed • into the de Rham cohomology H{X) = H(Y) of X [172]. Remark 6.4.2. If Y —> X is a vector bundle, one can obtain the differential bigraded algebra S^ = C^ A P^ in a different way. Let us consider the Whitney sum S = Q © Y of vector bundles Q —> X and Y —> X regarded as a bundle of graded vector spaces Qx © Yx, x € X. Let us define the

371

Chapter 6 Supergeometry k

quotient S of the tensor product sk =

R®S®®S®---®®S*

xx by the elements q®q'+q'®q,

xx

y* ®y'* - y" ® y',

q ®y* -y* ®q*

for all q,q' G Q, jy,y'* £ y^, and x E X. The C°° (X)-modules ^ | of sections of the vector bundles S —> X make up a direct system with respect to the natural monomorphisms As —> Ag1. Its direct limit A'g' is endowed with a structure of a graded commutative C°° (X)-ring generated by odd and even elements. Generalizing the above technique for a graded manifold (X, AQ) to (X, Ag3), one obtains the differential bigraded algebra • isomorphic to S ^ [296; 387]. We aim to study the cohomology of the short variational complex 0—•R_+S£ ^ S ^ 1 - - - ^ S £ , n --£!

(6.4.7)

and the complex of one-contact graded forms U —> 6 ^ — • " o o " "

— >< - ) oo

—

>

£'i^U

(b.4.8)

of the differential bigraded algebra S^ • For this purpose, one however must: (i) enlarge the differential bigraded algebra 5^, to the differential bigraded algebra r(6J o ) of graded exterior forms of locally finite jet order, (ii) compute the cohomology of the corresponding complexes of r(SJ 0 ), (iii) prove that this cohomology of r(6J o ) coincides with that of S^. Following this procedure, we show that cohomology of the complex (6.4.7) equals the de Rham cohomology of X, while the complex (6.4.8) is globally exact. Note that the exactness of the short variational complex (6.4.7) on X = Rn has been repeatedly proved [20; 60; 134]. One has also considered its subcomplex of graded exterior forms whose coefficients are constant on R n . Its dn-cohomology is not trivial [20]. One can think of the elements L = Cu e S^n,

5(L)=

^2(-l)^eAAdA(d^L)eE1

|A|>0

372

Geometric and Algebraic Topological Methods in Quantum Mechanics

of the complexes (6.4.7) - (6.4.8) as being a graded Lagrangian and its Euler-Lagrange operator, respectively. Therefore, the exactness of the complex (6.4.8) enables us to generalize the first variational formula and Lagrangian conservation laws in the calculus of variations on fibre bundles to graded Lagrangians and contact supersymmetries. THEOREM 6.4.1. The cohomology of the complex (6.4.7) equals the de Rham cohomology H(X) of X. The complex (6.4.8) is exact. • The proof of Theorem 6.4.1 falls into the three steps, (i) We start by showing that the complexes (6.4.7) - (6.4.8) are locally exact. LEMMA

6.4.2. The complex (6.4.7) on X = Rn is exact.

•

Referring to [20], Theorems 4.1 - 4.2, for the proof, we summarize a few / S S^* admits formulae quoted in the sequel. Any horizontal graded form <> the decomposition l

4> = d>0 + l

^ = / y E s A a ^> 0

(6.4.9)

|A|>0

where cf>o is an exterior form on R™. Let 4> £ <S^ m < n be dn-closed. Then its component — dx£, where £ is given by the following expressions. Let us introduce the operator l

Jo

fc>°

(6.4.10)

The relation

holds, and leads to the desired expression (6.4.11)

P 0 = l,

Pk=dVl.--dVkD+^---D+v.

Now let <j> 6 S^m = 0. Then its component o (6.4.9) is an exact form on R™ and 4> = dtft;, where £ is given

Chapter 6 Supergeometry

373

by the expression

£= E

E

|A|>O£+E=A

(-l) |E| 2d E 0£ +A &V

(6-4-12)

Remark 6.4.3. Since elements of S^ are polynomials in s^, the sum in the expression (6.4.11) is finite. However, the expression (6.4.11) contains a d/f-exact summand which prevents its extension to O^. In this respect, we also quote the homotopy operator (5.107) in [346] which leads to the expression l

£= Jl()(x»,\si,dx^,

(6.4.13)

o

i(4,) = y y — ^ t ± l —

(6.4.14)

d

At E (- 1 ) 5 ( r + A A vS ) ! s A ^^ + A + E ^^^i' |H|>0

VM

;

'~"

where A! = AMl! • • • A/in! and AM denotes the number of occurrences of the index /i in A. The graded forms (6.4.12) and (6.4.13) differ in a d^-exact graded form. D LEMMA

6.4.3. The complex (6.4.8) on X = W1 is exact.

D

Outline of proof. The fact that a dn-closed graded (1, m)-form (f> £ S^7lAAA.

(6.4.15)

where <$>\ 6 S^m are horizontal graded m-forms. Let us introduce additional variables s^ of the same Grassmann parity as s^. Then one can associate to each graded (l,m)-form <j> (6.4.15) a unique horizontal graded m-form

^=E^5A>

(6-4-16)

whose coefficients are linear in the variables s^, and vice versa. Let us consider the modified total differential

dH = dH+dxXA

E^A+A^l |A|>0

374

Geometric and Algebraic Topological Methods in Quantum Mechanics

acting on graded forms (6.4.16), where dA is the dual of ds^. Comparing the equality dHs£ = dxxs+A and the last equality (10.8.30), one can easily justify that dH =

^H4>-

Let a graded (l,m)-form (6.4.15) be enclosed. Then the associated horizontal graded m-form <j) (6.4.16) is d/y-closed and, by virtue of Lemma 6.4.2, it is d#-exact, i.e., <j> = djjf;, where £ is a horizontal graded (m — 1)form given by the expression (6.4.11) depending on additional variables s^. A glance at this expression shows that, since is linear in the variables s^, so is

It follows that <j> = dif£ where

It remains to prove the exactness of the complex (6.4.8) at the last term Ex. If

Q(a)= ^2(-l)^9AA[dA(di\a)}

=

|A|>0

53 (-l)lAl0A A [d^]u = 0,

ae S^n,

|A|>0

a direct computation gives

c =- E

E (-i) |E| ^Ad E ^ + V-

|A|>0S+E=A

(6-4-17) QED

(ii) Let us associate to each open subset U C X the differential bigraded algebra Sjj of elements of the C°° (X)-algebra S^ whose coefficients are restricted to U. These algebras make up a presheaf over X. Let & ^ be the sheaf of germs of this presheaf and F(©^o) its structure module of sections. One can show that 6 ^ inherits the variational bicomplex operations, and r(S^ o ) does so. For short, we say that F(SJO) consists of polynomials

375

Chapter 6 Supergeometry

in s\, ds\ of locally bounded jet order |A|. There is the monomorphism S^ —» r(6^ o ). Let us consider the complexes of sheaves

o — R _ ^ 6 ^ ^ e ^ 1 - - - £+&£ - ^ d ,

(6.4.18)

0 -> ©M> ^

(6 4 1 9 )

g u . .. ^

gi,n ^

£i

_

0;

over X and the complexes of their structure modules

O ^ R —>r(62j ^r(e°J)--- ^ r ( s ^ " ) -^r(€!),(6.4.20) o-+r(e^°) ^ r ( s ^ ) . . . ^ r ( s ^ " ) - ^ ( d j - o . (6.4.21) By virtue of Lemmas 6.4.2 - 6.4.3 and Theorem 1.4.2, the complexes of sheaves (6.4.18) - (6.4.19) are exact. The terms 6^, of the complexes (6.4.18) - (6.4.19) are sheaves of C°°(X)-modules. Therefore, they are fine and, consequently, acyclic. By virtue of Theorem 1.7.7, the cohomology of the complex (6.4.20) equals the cohomology of X with coefficients in the constant sheaf M, i.e., the de Rham cohomology H(X) of X, whereas the complex (6.4.21) is globally exact. (iii) It remains to prove the following. 6.4.4. Cohomology of the complexes (6.4.7) - (6.4.8) equals that of the complexes (6.4.20) - (6.4.21). • PROPOSITION

Let the common symbol D stand for the operators dn, 5 and g in the complexes (6.4.20) - (6.4.21), and let F ^ denote the terms of these complexes. Since cohomology groups of these complexes are either trivial or equal to the de Rham cohomology of X, one can say that any D-closed element cf> G F ^ takes the form 4> = i, + D^

CeC

(6.4.22)

where ip is a closed exterior form on X which is not necessarily exact. Since all D-closed elements of F ^ of finite jet order are also of form (6.4.22), it suffices to show that, if an element (p £ «S£> ^s -f-exact in the module F ^ (i.e., = D£,£e F^), then it is also in S£ (i.e., <j> = Dip, if e S^). Let X be a (contractible) domain, and let an element <j> £ S^ be Dexact in F^,. Then, being .D-closed, it is D-exact in S^ in accordance with Lemmas 6.4.2 and 6.4.3. Moreover, a glance at the expressions (6.4.11), (6.4.12) and (6.4.14) shows that the maximal jet order [ip] of
\ of <j>.

376

Geometric and Algebraic Topological Methods in Quantum Mechanics

It follows that, if
property. • Outline of proof. Let <j> e S^ be a D-exact graded form on X. The finite exactness on UUa holds since = D<pa on every Ua and all [ipa] < JV( [<£]). QED

6.4.6. Suppose that the finite exactness of an operator D takes place on open subsets U, V of X and their non-empty overlap U(~\V. Then it is also true on U U V. • LEMMA

Outline of proof. Let = Dtp 6 S^ be a £)-exact graded form on X. By assumption, it can be brought into the form Dipu on U and Dipv on V, where ipu and ipv are graded forms of bounded jet order. Due to the decomposition (6.4.22), one can choose the forms
where a is also of bounded jet order. Lemma 6.4.7 below shows that a = °~u + °~v where o\j and av are graded forms of bounded jet order on U and V, respectively. Then, putting y'u =

Dav,

we have the graded form <j>, equal to Dip'y on U and D
I
377

Chapter 6 Supergeometry

is of bounded jet order.

QED

6.4.7. Let U and V be open subsets of X and a a graded form of bounded jet order on U D V. Then a splits into the sum o\j + oy of graded exterior forms ou on U and ay on V of bounded jet order. • LEMMA

Outline of proof. By taking a smooth partition of unity on U U V subordinate to its cover {U, V} and passing to the function with support in V, we get a smooth real function / on U U V which is 0 on a neighborhood Uu-v of U — V and 1 on a neighborhood Uv-u of V — U in U U V. The graded form fa vanishes on Uu-v n{UDV) and, therefore, can be extended by 0 to U. Let us denote it a\j. Accordingly, the graded form (1 — f)a has an extension ay by 0 to V. Then a = au +ay is a desired decomposition because au and ay are of finite jet order which does not exceed that of a. QED Lemma 9.5 in [65], Chapter V, states that, if some property holds on a domain and obeys the conditions of Lemmas 6.4.5 and 6.4.6, it holds on any open subset of R n . Hence, the operator D has the jet exactness property on any open subset of R™ and, consequently, on any chart of the fibre bundle QxY-^X. x Since the latter admits a finite bundle atlas with the transition functions (10.7.11) and (6.4.1) preserving the jet order, the finite exactness of D takes place on the whole manifold X in accordance with Lemma 6.4.6. This proves Proposition 6.4.4 and, consequently, Theorem 6.4.1. Remark 6.4.4. Let us consider the complex O - R ^ S ^ J L ^ - . - J U ^

—>•••,

(6.4.23)

which we agree to call the de Rham complex because («!>£,, d) is the differential calculus over the R-ring ££,. If X = W1, it is exact [134]. Similarly to the proof of Theorem 6.4.1, one can show that the cohomology of the de Rham complex (6.4.23) equals the de Rham cohomology of X. O

378

Geometric and Algebraic Topological Methods in Quantum Mechanics

COROLLARY 6.4.8. Every ^ - c l o s e d graded form <$> G <S^m<™ falls into the sum 4> = il> + dHK,

(6.4.24)

^€S^m-\

where tp is a closed m-form on X. Every 5-closed graded Lagrangian L £ S^n is the sum £e<S£"-\

cf> = iP + dHt,,

(6.4.25)

where ip is a non-exact n-form on X.

D

The global exactness of the complex (6.4.8) at the term S^n results in the following. PROPOSITION

6.4.9. Given a graded Lagrangian L =

CUJ,

there is the

decomposition dL = 5L-dHZ,

(6.4.26)

EGS^1'1,

3= EC.^^ 8 "" 1 ^,

(6A27)

s=0

where local graded functions h obey the relations h"a = 0,

/ i l " ' " ' - 1 ' - " 1 = 0.

• Outline of proof. The decomposition (6.4.26) is a straightforward consequence of the exactness of the complex (6.4.8) at the term 5^," and the fact that Q is a projector. The coordinate expression (6.4.27) results from a direct computation

- d H E = - d H \ 6 A F \ + OfF?

+ ••• + tfM...UlF%'--v +

[9AdxF>l + Q${F\ + dxF]?) + ••• + 0 A t + l V . . . . V l ( K ' + 1 ' " " " 1 + d x F ? - + 1 ' " - V l ) + ---]Aw = [U d\tA +tiv{oAL) -\

h oVa+il/a

Ul

[OA

L)-\ ] A w -

6A(dxF% - dAC) Au> + dL = -6L + dL. QED

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Chapter 6 Supergeometry

Proposition 6.4.9 states the existence of a global finite order Lepagean equivalent Hi, = H + L of any graded Lagrangian L. Locally, one can always choose H (6.4.27) where all functions h vanish. By analogy with a contact symmetry (see Proposition 10.8.2), an infinitesimal contact supersymmetry or, simply, a contact supersymmetry is defines as a graded derivation v € 05^, of the K-ring 5^, such that the Lie derivative L v preserves the ideal of contact graded forms of the differential bigraded algebra S^ (i.e., the Lie derivative Lv of a graded contact form is a graded contact form). 6.4.10. With respect to the local basis (xx,sA,dxx,6A) for the differential bigraded algebra S^, any contact supersymmetry takes the form PROPOSITION

v = vH + vv = vxdx + (vAdA + J2 d^Ad^),

(6.4.28)

|A|>0

where vx, vA are local graded functions.

•

Outline of proof. The key point is that, since elements of C^, can be identified as sections of a finite-dimensional vector bundle over X, so can elements of the C°°(X)-algebra 5£,. Moreover, any graded form is a finite composition of df, f e S^. Therefore, the proof follows that of Proposition 10.8.2. QED The interior product v\ and the Lie derivative liv, 4> £ <S£, are defined by the same formulae v\4> = vxct>x + {-i)^vAA,

teSl,, +

v\(Ao-) = (v\(j>)Ao- + (-l)M MM
,o-&S^

Lvcf> = v\d(f) + d(v}(/>), Lv{
) A a + ( - l ) H I ^

A

Lv{a),

as those on a graded manifold. Following the proof of Lemma 10.8.3, one can justify that any vertical contact supersymmetry v (6.4.28) satisfies the relations (6.4.29)

v\dH(f> = -dH(v\(f>), Lv(dH) = dH(Lv),

4>eSZ,.

(6.4.30)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

6.4.11. Given a graded Lagrangian L 6 S^1, its Lie derivative LVL along a contact supersymmetry v (6.4.28) fulfills the first variational formula PROPOSITION

LVL = w\SL + dH{h0{v]EL)) + dv{vH\u)C,

(6.4.31)

where S L = 2 + L is a Lepagean equivalent of L given by the coordinate expression (6.4.27). D Outline of proof. The proof follows that of Proposition 10.8.4 and results QED from the decomposition (6.4.26) and the relation (6.4.29). In particular, let v be a variational symmetry of a graded Lagrangian L, i.e., LvL = dHa,

aeS0^-1.

Then the first variational formula (6.4.31) restricted to Ker<5L leads to the weak conservation law 0dH(h0(v\ZL)-a).

(6.4.32)

Remark 6.4.5. Let us consider the gauge theory of principal connections on a principal bundle P —> X with a structure Lie group G. These connections are represented by sections of the quotient C = JlP/G —> X (see Section 10.10). This is an afflne bundle coordinated by (xx,arx) such that, given a section A of C —» X, its components A\ — arx o A are coefficients of the familiar local connection form (i.e., gauge potentials). Let J°°C be the infinite order jet manifold of C —> X coordinated by {xx,ar A ) , 0 < |A|, and let P^,(C) be the polynomial subalgebra of the differential graded algebra 0^(0). Infinitesimal generators of one-parameter groups of vertical automorphisms (gauge transformations) of a principal bundle P are G-invariant vertical vector fields on P —» X. They are associated to sections of the vector bundle VGP = VP/G - X of right Lie algebras of the group G. Let us consider the simple graded manifold (X, 2lyGy) constructed from this vector bundle. Its local basis is (xx, Cr). Let CJTVGY be the differential bigraded algebra of graded exterior

Chapter 6 Supergeometry

381

forms on the graded manifold (X,2ljryoP), and C^VQP) the direct limit of the direct system (6.4.2) of these algebras. Then the graded product S^VcC)

= C(VCP)AP;(C)

(6.4.33)

describes gauge potentials, odd ghosts and their jets in the BRST theory. With respect to a local basis (ar\a^,C r ) for the differential bigraded algebra S^VcC) (6.4.33), the BRST symmetry is given by the contact supersymmetry

v = v\$ + vrdr + £ (dAVrxOy + dAvrd£),

(6.4.34)

|A|>0

where c£g are structure constants of the Lie algebra of G and d£, dr, <9^'A d£ are the duals of darx, dCr, darA x and dCrK, respectively. A remarkable peculiarity of this contact supersymmetry is that the Lie derivative Lv along v (6.4.34) is nilpotent on the module S^* of horizontal graded forms.

•

In a general setting, a vertical contact supersymmetry v (6.4.28) is said to be nilpotent if

ML^) =

£

(vid%(v£)d% + (-l)[sB][vA]viv£d%db = 0

|E|>0,|A|>0

(6.4.35) for any horizontal graded form (f> £ S^. 6.4.12. A contact supersymmetry v is nilpotent if and only if it is odd and the equality LEMMA

U{vA) - Y, vld%(vA) - 0 |S|>0

holds for all vA. Outline of proof. There is the relation dxoV\d$=v\d$odx,

D (6.4.36)

similar to (10.8.31). Then the lemma follows from the equality (6.4.35) where one puts <j> — sA and 0 = sAs^. QED

382

6.5

Geometric and Algebraic Topological Methods in Quantum Mechanics

Lagrangian supermechanics

Lagrangian supermechanics exemplifies Lagrangian systems on graded manifolds in previous Section. Moreover, following the BRS mechanics of [191; 192], one can construct the canonical SUSY extension of Lagrangian and Hamiltonian time-dependent mechanics [294; 386]. The first step toward this extension is the vertical extension of Lagrangian and Hamiltonian mechanics on a configuration space Q —> R onto the vertical tangent bundle VQ - • R in Section 2.4G. The SUSY-extended mechanics is formulated in terms of simple graded manifolds whose characteristic vector bundles are

VJXVQ - JlVQ in Lagrangian formalism (see the isomorphism (2.4.75)) and

VVVQ -> VVQ in Hamiltonian formalism (see the isomorphism (2.4.73)). The SUSY extension adds to (t,gl,ql) the odd variables (c\cl). The corresponding SUSY extensions of a Lagrangian and a Hamiltonian, by construction, are invariant under the one-parameter supergroup with the infinitesimal generator UQ=c%+iqi-^.

(6.5.1)

Let us consider the vertical tangent bundle VVQ —> VQ of VQ —> R and the simple graded manifold (VQ,QIVVQ) whose body manifold is VQ and the characteristic vector bundle is VVQ —> VQ. Its local basis is (t, g 4 ,^ 1 ,^,?) where {cl,c1} is the fibre basis for VVQ, dual of the holonomic fibre basis {di,di} for VVQ —> VQ. Graded vector fields and graded exterior one-forms are introduced on (VQ,$IVVQ) as sections of the vector bundles VVVQ and VyVQ, respectively. We complexify these bundles as C VWQ and C VVVQ. The velocity phase space of the SUSY-extended mechanics is the simple graded manifold (VJlQ, S V V J I Q ) whose characteristic vector bundle is the vertical tangent bundle

VVJXQ -> VJlQ of VJlQ -> R. Its local basis is

383

Chapter 6 Supergeometry

where {c1 ,c, c^cl} is the fibre basis for VVJ1Q dual of the holonomic fibre basis {di,di,dj,dl} for

VVJlQ -> VJlQ. The affine fibration TT* : VJXQ - • 1/g and the corresponding vertical tangent morphism V-KI : V V ^ Q -» VVQ

yields the associated morphism (6.3.6) of graded manifolds

(VJlQ,KVVJiQ)

-»

(VQ,VVQ).

In a similar way, the simple graded manifold with the characteristic vector bundle

VVJkQ -» VJkQ is defined. Its local basis is the collection {t,q\qiq\qlc\T?,^,c\),

A = (t...t),

0 < |A| < fc.

Let us introduce the operator of the total derivative dt = dt + qidi+ci—.+7?t-^r

+...,

With this operator, the coordinate transformation laws of c\ and cj read d? = dtc'\

c'j = dtcH.

(6.5.2)

Then one can treat c\ and c[ as the jets of the odd variables c% and c\ The transformation law (6.5.2) shows that the graded vector field UQ (6.5.1) on (VQ,QIVVQ) gives rise to the complex graded vector field

JluQ = uQ + cidt + iyi-jjg

(6.5.3)

on (VJ^Q, ^•VVJ1Q)- The Lie derivative along this graded vector field plays a role of the BRS operator in SUSY mechanics. In accordance with Lemma 6.4.12, it is nilpotent.

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Geometric and Algebraic Topological Methods in Quantum Mechanics

As in the BRS mechanics [191; 192], the main criterion of the SUSY extension of Lagrangian mechanics is its invariance under the BRS transformation (6.5.1). Let us introduce the operators

dc = cidi + etd\ + ettdf + ---, dc = cidi +
(6.5.4)

dtdc — dcdt.

Then the desired BRS-invariant extension of the vertical Lagrangian Ly (2.4.77) is the graded horizontal density (6.5.5)

Ls = Csdt = LV + idcdcCdt such that IJJUQLS

=

0.

The corresponding Euler-Lagrange equations are defined as the kernel of the Euler-Lagrange operator

They read

SLs = (dq% + dq% + dc'^7 + dc'^Cs ocl 6c

A dt.

5iCs = 5iC = 0, 5iCs = SiCv + idcdAC = 0,

(6.5.6a) (6.5.6b)

-^-Cs = -idcSiC = 0, oc1 ^-Cs = idcSiC = 0,

(6.5.6c) (6.5.6d)

where the relations (6.5.4) are used. The equations (6.5.6a) are the EulerLagrange equations for the original Lagrangian L, while (6.5.6b) - (6.5.6d) can be seen as the equations for a Jacobi field 5yl = ecl + c£ + ieey% modulo terms of order > 2 in the odd parameters e and e.

385

Chapter 6 Supergeometry

6.6

Graded Poisson manifolds

Let us start with the notion of a graded symplectic manifold. Let (Z,%IE) be a simple graded manifold, E its characteristic vector bundle and CE the differential bigraded algebra of graded exterior forms on (Z,21JS). A graded two-form
u\(t) e CE

(6.6.1)

has the zero kernel Ker<^>. In this case, the map (6.6.1) is an isomorphism of graded modules. Let <j>0 and 4>i be even and odd parts of (p. A graded two form cfi is said to be homogeneously non-degenerate if Kerfo n K e r ^ i = 0. Of course, any non-degenerate graded two-form is also homogeneously nondegenerate, while a homogeneous (even or odd) graded two-form is nondegenerate if and only if it is homogeneously non-degenerate. In a general setting, a closed graded two-form is called the graded symplectic form if it is homogeneously non-degenerate. However, by graded symplectic form is often meant a non-degenerate closed graded two-form. For the sake of simplicity, we here restrict our consideration to even symplectic forms (see, e.g., [420; 243] for the non-homogeneous and odd ones). Let Q, be an even symplectic form on a simple graded manifold (Z, 21E). Similarly to the case of symplectic manifolds in Section 2.1 A, one can associate to any graded function / GCE the graded vector field tf/jn = -df,

-Of = « » ( # ) ,

(6.6.2)

where fi" is the inverse map to the even map fib (6.6.1). It is called the Hamiltonian graded vector field of a graded function / . Given another graded function g and its Hamiltonian graded vector field dg, one can define the (even) graded Poisson bracket {f,g} = $f\dg =

tig\$f\n.

(6.6.3)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

It obeys the relations {/,
(6.6.4a) (6.6.4b)

= 0,

{/, 9h) = {/, g}h + (-1)1/1 W p {/, h}, [^/A] =#{f,g}-

(6.6.4c) (6-6.4d)

One can prove that a graded manifold (Z, 21E) can be endowed with a graded symplectic structure if its body manifold Z admits an exact symplectic form [182; 318]. The graded version of the Darboux theorem assures that one can find local canonical basis for a graded symplectic structure. One says that a graded symplectic form on a simple graded manifold (Z, 21JE) is projectable onto Z if the restriction of the graded Poisson bracket (6.6.3) to the R-subring C°°(Z) C C% of functions on Z is a Poisson bracket associated to some symplectic structure on Z. Theorem 6.6.1 below shows that a graded symplectic structure cannot be projectable, unless the characteristic bundle E —» Z admits a flat connection. Turn now to the notion of a graded Poisson manifold. As before, we restrict our consideration to the case of even Poisson brackets (see, e.g., [241; 242] for the odd ones). A glance at the relation (6.6.4c) shows that a graded Poisson bracket on a simple graded manifold (Z, 2lg) can be defined by associating to each graded function / £ CE a graded derivation

#f.CEBf~{f,f}eC°E

(6.6.5)

of the M-ring C\ which satisfies the graded skew-symmetry condition (6.6.4a) and the graded Jacobi identity (6.6.4b). Herewith, "df (6.6.5) is the Hamiltonian graded vector field of / . A graded Poisson bracket is said to be projectable if its restriction to the M-subring C°°(Z) C C°E of functions on Z is a Poisson bracket on Z. We quote the following theorem [225]. 6.6.1. Let (Z,QlE) be a simple graded manifold and {-,-}z a (regular) Poisson bracket on Z. Then a graded Poisson bracket on (Z, 21^) projected to {.,.}% on Z exists if and only if the characteristic vector bundle E —> Z of (Z, QIE) admits a connection whose curvature vanishes along the symplectic leaves of {., .}zd THEOREM

Chapter 6 Supergeometry

387

The proof of this theorem is based on the following facts. Bearing in mind that the ring C% is the ring /\E(Z) of sections of the exterior bundle AE, let us consider the following two types of derivations of the ring AE{Z). (i) Let 7 (6.3.5) be a linear connection on E and 75 (6.3.15) the corresponding graded connection on (Z, Slg). Given a section 5 = g r of the tensor product AE* ®TZ->Z, one can associate to s the endomorphism V, : AE{Z) 3 / ' -> gVTf

€ AE(Z),

(6.6.6)

where V r is given by the expression (6.3.12). Clearly, V s (6.6.6) is a graded derivation of the E-ring AE(Z). (ii) Let us consider the tensor product AE®E. Given a section r = g®e of AE* ®E -> Z, one can associate to r the endomorphism Tr : AE(Z) 9 / ' -» ge\' E AE(Z).

(6.6.7)

This is a graded derivation of the C°°(Z)-ring AE(Z). 6.6.2. Given a connection 7 on E —> Z, each graded derivation A of the K-ring AE(Z) is uniquely split into the sum A = V s + Tr of the • graded derivations (6.6.6) and (6.6.7) [317].

LEMMA

In particular, for each graded function / G C%, there exist the above mentioned sections Sf and rf such that the graded Hamiltonian vector field •df (6.6.5) of / is brought into the form tif = VSf+Trj. Then the graded skew-symmetry condition (6.6.4a) and the graded Jacobi identity (6.6.4b) lead to the statement of Theorem 6.6.1.

388

6.7

Geometric and Algebraic Topological Methods in Quantum Mechanics

Hamiltonian supermechanics

This Section addresses the SUSY extension of Hamiltonian time-dependent mechanics on a configuration space Q —> R. The momentum phase space of the SUSY-extended mechanics is the complexified simple graded manifold (VVQ,QIVVVQ) whose characteristic vector bundle is VVVQ^ VVQ. Its local basis is (t,ql,Pi,ql,Pi,cl,^,Ci,Ci),

where Cj and Cj have the same transformation laws as Pi and pi, respectively. The corresponding graded vector fields and graded one-forms are introduced as sections of the vector bundles on the graded manifold {VVQ,%IVVVQ) C®VVVVQ and C<8)VVV.VQ, respectively. X

X

In accordance with the above mentioned transformation laws of c^ and c, the graded vector field UQ (6.5.1) on (VQ,%LVVQ) can give rise to the complex graded vector field (6.7.1)

uQ=dc + iq'-^r + ipi—

on {VVQ,%VVVQ)The BRS-invariant extension of the canonical threereads form fly (2.4.80) onto the graded manifold (VVQ,QIVVVQ) Qs = [dpi A dqi + dpi A dql + i(ddi A dc* - d ? A dci)} A dt,

(6.7.2)

where (cl, —ici) and (c,icj) are the conjugate pairs. Let 7 H be the Hamilton vector field (2.4.34) for a Hamiltonian form H (2.4.32) on the original momentum phase space VQ. Its double vertical prolongation VV-y on VVVQ -> R (see the formula (2.4.76)) defines the graded connection 7S

= v,H + 8&H^ - a ^ A + de#H£ _

dAn^

on (VV<2,2lyvVQ)) where V^H is given by the expression (2.4.82) and dc = 0% + Cid\

dc = tfdi + Cid\

This graded connection satisfies the relation 75 Jfi s

= -dHs,

389

Chapter 6 Supergeometry

and can be regarded as the Hamilton graded vector field for the Hamiltonian graded form Hs = \pidqi + pidc? + iicidd + d?a)] - Hsdt, Hs = {dv + idcdc)H,

(6.7.3)

on (VVQ,%LVVVQ)- It is readily observed that this graded form is BRSinvariant, i.e., LizQHs = 0. Thus, it is the desired SUSY extension of the original Hamiltonian form H (2.4.32). The Hamiltonian graded form Hs (6.7.3) defines the corresponding SUSY extension of the Lagrangian L# (2.4.42) as follows. The fibration JlVVQ->VVQ yields the pull-back of the Hamiltonian graded form Hs (6.7.3) onto J1VVQ. Let us consider the graded generalization of the operator h0 (10.7.15) such that ho '• dcl H+ c\dt,

dci >—> cudt.

Then the graded horizontal density LHS = ho(Hs) = LHV + i[{ciCi + cla) - dedjQdt = LHV +

(6.7.4)

i[ci(4 ~ aca'w) + (cj - dc&tyci + 2ic^aiajw - cicjdidjn}dt on JXVVQ -» X is the SUSY extension (6.5.5) of the Lagrangian LH (2.4.42). The Euler-Lagrange equations for Lus coincide with the Hamilton equations for Hs, and read q\ = &HS = dirH, Pti = -diHs = -diH, Pu = -diHs = q\ = &Hs = (dv + idcdJ&H, ~{dv + idcdc)diH,

(6.7.5a) (6.7.5b)

4 = i ^ - = -dc&H,

(6.7.5c)

3 = - t ^ =-^W, dci

cu = i ^

= -dcdiH,

cti = -i^f = -ckdiH. (6.7.5d) d?

The equations (6.7.5a) are the Hamilton equations for the original Hamiltonian form H, while (6.7.5b) - (6.7.5d) describe the Jacobi fields 5ql = ecl + cle + ieeq1,

dpi = ecj + Cj£ + ieepi.

390

Geometric and Algebraic Topological Methods in Quantum Mechanics

Let now Q —> M. be an affine bundle. In this case, transition functions of the holonomic coordinates q1 on VQ are independent of q%, the transformation laws of the frames {di} and {di} coincide, and so do the transformations laws of the coframes {c1} and {
U«=~Cid

are globally denned. The graded vector fields (6.5.1) and (6.7.6) constitute the Lie superalgebra of the supergroup ISp(2): [UQ,UQ] = [UQ,UQ] = [UQ,UQ] = {UK,UQ} = [UW,UQ] [UK,UQ]

= uQ,

[uc, uK) = 2uK,

[u-j?, uQ] = U-Q,

[UK, ujr] = uc,

= 0, (6.7.7)

[uc, U-K\ = -2uj(.

Similarly to (6.5.3), let us consider the jet prolongation of the graded vector fields (6.7.6) onto VJlY. Using the compact notation u = uada, we have the formula

Jlu = u + dtuadl and, as a consequence, obtain

(6.7.8) Jiuc=uc

= cl—-c\

—.

It is readily observed that the SUSY-extended Lagrangian Ls (6.5.5) is invariant under the transformations (6.7.8). The graded vector fields (6.5.3) and (6.7.8) make up the Lie superalgebra (6.7.7). The graded vector fields (6.7.6) can give rise onto ( V * V Q , 2 1 V V V Q ) by the formula 5 = u _(_i)[«-](w+[« 6 ])a o U «'p 6 ^_.

OPa

391

Chapter 6 Supergeometry

We have

(6.7.9)

id ac*

3 9Q

,

8 . 3 9c* oci

A direct computation shows that the BRS-extended Hamiltonian form Hs (6.7.3) is invariant under the transformations (6.7.9). Accordingly, the Lagrangian LHS (6.7.4) is invariant under the jet prolongation Jlu of the graded vector fields (6.7.9). The graded vector fields (6.7.1) and (6.7.9) make up the Lie superalgebra (6.7.7). With the graded vector fields (6.7.1) and (6.7.9), one can construct the corresponding graded currents I{u) = u\Hs = u\Hs, which read I{uQ) = dpi - fa, I(UK)

= -icjC,

I(UQ) = tin /(JJJ)

- q%,

= iCiZ\

I{uc) = ifac1 - clCi).

These currents form the Lie superalgebra (6.7.7) with respect to the product [/(«),/(«')]=/([«,«'])•

The following construction is similar to that in the SUSY and BRS mechanics. Given a function F on the Legendre bundle VQ, let us consider the operators Fp = e0F OUQO e~0F = uQ- (3dcF, Fp = e-0FouQoe0F =v,Q+pdcF, called the SUSY (VVQ,$IVVVQ)-

/3 > 0,

charges, which act on graded These operators are nilpotent, i.e., Ff}oFi3 = 0,

functions on

TpoFp = 0.

(6.7.10)

By the BRS-invariant extension of a function F is meant the graded function fs = ~(f/joF/J+F/,oF/1). We have the relations F0oFs-FsoF(s

= 0,

T(3 o Fs - Fs o ~F0 = 0.

392

Geometric and Algebraic Topological Methods in Quantum Mechanics

These relations together with the relations (6.7.10) provide the operators Fp, Ffj, and Fs with the structure of the Lie superalgebra sl(l/l) [94]. In particular, let F be a local function % in the expression (10.9.4). Then W5 = -i(WioW 1 +W 1 oWi) is exactly the local function Hs in the expression (6.7.3). The similar splitting of a super-Hamiltonian is the corner stone of the SUSY mechanics [ill; 265]. 6.8

BRST complex of constrained systems

We start with conservative constrained systems on a Poisson manifold and, afterwards, consider Lagrangian constraints (2.5.39) in time-dependent mechanics in Section 2.5D. Given a constrained system on a Poisson manifold (Z, {,}), one can associate to this system a BRST charge 0 by introducing a set of ghosts and antighosts and extending the Poisson bracket {,} to a superbracket [, ] [156; 245; 404]. The BRST charge obeys the condition [0,0] = 0 and defines the nilpotent classical BRST operator D = [6,.], which can be quantized [153]. In order to obtain the BRST charge, one starts with constructing the Koszul-Tate complex of constraints. Let Z be a (finite-dimensional) smooth manifold and J an ideal of the associative ring CO°(Z) whose elements are regarded as constraints. Let J be finitely generated by elements {/i}, i = 1,... ,n. One associates to J the Grassmann C°°(Z)-algebra B° freely generated by odd elements {&?}. These elements are treated as antighosts of level 0, and are endowed with the antighost number 1. The algebra B° is provided with the bigraded structure B° = C°°(Z)®B%®---B°

(6.8.1)

with respect to the antighost number agh= l , . . . , n and the Grassmann parity [&] = (agh(6) + l)mod2,

b G B°.

The algebra B° is brought into the chain complex

0 ^C°°(Z) J-B° J-

B° — 0

(6.8.2)

393

Chapter 6 Supergeometry

with respect to the C°° (Z)-linear boundary operator 6(b°) = fh

(6.8.3)

S(b A b') = 5{b) A b' + (-lfh

A 6{b'),

b, b' e B°.

This is exactly the Koszul boundary operator (1.5.19) - (1.5.20), and the chain complex (6.8.1) exemplifies the Koszul complex (1.5.21) in Section 1.5B. One obtains at once its homology group (6.8.4)

H0(Bi) = C°°(Z)/J.

For instance, if J = /JV is the ideal of functions vanishing on a closed imbedded submanifold N of Z, then (6.8.5)

H0(B°)=C°°(N).

One says that the constraints {/j} are irreducible if the equality g'fi = 0,

g{£ C°°(Z),

(6.8.6)

implies the equalities 7 ij = - 7 j i ,

9 =-?!,,

lijeC°°(Z).

In this case, the tuple {ft} is regular in the ring C°°(Z) and, in accordance with Theorem 1.5.6, the homology groups Hp>o(B®) of the Koszul complex (6.8.2) are trivial. It follows that the Koszul complex (6.8.2) is acyclic and, thus, it provides a finite resolution of the ring CCO(Z)/J. In the case of reducible constraints {/j}, one can also obtain the resolution of the ring C°°(Z)/J by extending the chain complex (6.8.2) to the Koszul-Tate one as follows [156; 245]. If constraints {/j} are reducible, there exists a collection of functions x \ G C°°(Z), i\ = 1,... , m , which do not belong J \ 0, but XUiJi=0,

ii = l , . . . , n i ,

(6.8.7)

and the equality (6.8.6) implies the equalities Si = X 1 ! 1 P i l +7 i j 7j,

7

ij

= -7ji,

ph,Jij

eC°°(Z).

(6.8.8)

In this case, the term B® of the Koszul complex (6.8.2) contains the homologically non-trivial cycles generated by the elements X1^,

i1 = l,...,n1.

(6.8.9)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

In order to remove these non-trivial cycles, let us extend the algebra B° to the C°°(Z)-algebra B1 of polynomials in the odd variables &9, i = 1,... ,n, of antighost number 1 and the even ones b}^, i\ = 1,... , m, of antighost number 2. Then B1 is a bigraded algebra B1 = C°°(Z) © Bj © B e • • •

(6.8.10)

with respect to the antighost number and the Grassmann parity [b] = (agh(b) + l)mod 2,

beB1.

For instance, B\ = Bj. The Koszul boundary operator 5 (6.8.3) is extended to B1 as a C°°(Z)-linear Z2-graded derivation S1 mapping b\t into the nontrivial cycles (6.8.9), i.e., ii = 1 ni, S\bil)=xUilbl S\b A b') = 8(b) A 6' + (-l)M& A 5(b'),

(6.8.11) b, b' e B 1 .

This boundary operator makes the bigraded algebra B1 (6.8.10) into the chain complex 0 ^— C°°(Z) £- B\ £- B\ A

.

(6.8.12)

The homology group HQ{B\) of this chain complex equals the homology group H0(B°) (6.8.4) of the Koszul complex, while the first homology group Hi(Bl) is trivial. However, there may be nontrivial cycles in B\. This procedure can be iterated to higher antighost numbers. Let us assume that we have constructed the collection of chain complexes Bm, m = 0,..., L, graded in the antighost number k = 0 , 1 , . . . such that: • the complex Bm consists of polynomials with coefficients in C°°(Z) over the antighosts bik,

H = l,...,nfc,

k =

0,...,m,

of antighost number k + 1 and Grassmann parity (fc + I)mod2; • B^ = B-1 for all k = 1,... m; • Hk{Bm) = 0 for 0 < k < m, while B%+1, m = 1,... ,L, contains non-trivial cycles generated by the elements of the form X

m+lim tm _i_ im + l »m ~l~

pm+1 tm+l'

where functions X m+1 i" +1 € C°°(Z) do not belong to J \ 0, but

xm+i£+1xrir1=0modJ

Chapter 6 Supergeometry

395

and the equality

implies the equalities

• the coboundary operator 6m of the complex Bm is the extension of the coboundary operator 5m~1 of the complex B"1'1 by the rule

Given the chain complex (BL,8L), one can prove the existence of the complex (BL+1, SL+1) which obeys the above mentioned conditions for m = L + 1 [156]. Let us emphasize the key points of the proof. (i) One assumes that the constraints {fi} can be locally split into the independent and dependent ones. For instance, this is the case of the ideal J — IN of functions vanishing on a closed imbedded submanifold N of Z. (ii) One shows that any <5fe-closed element of Bk is a (5fc+1-exact element of£ f c + 1 . (iii) If all functions XL+17i+1

= 0,

(6.8.13)

then the complex BL is acyclic. If the equalities (6.8.13) hold, the constraints are called finitely reducible. In this case, the iteration procedure is finite, and the complex (BL,SL) forms an acyclic resolution of C°°(Z)/J. It is called the Koszul-Tate complex with respect to the Koszul-Tate differential SL. If the constrained system in question if infinitely reducible, the above mentioned procedure is iterated an infinite number of times by introducing an infinite number of antighosts. It also gives the Koszul-Tate resolution B°° of C°°(Z)/J. For instance, this is the case of second-class constraints [245]- We here restrict our consideration to finitely reducible constraints. Given the Koszul-Tate complex (BL,SL), one then aims to extend the algebra BL in order to replace the graded derivation SL with the inner one D = [9,.]. For this purpose, given the C°°(Z)-module Vfc of antighosts of level k, let us consider its dual Vfe whose elements are called the ghosts of level k. It is a free module generated by the set of elements {c^}. The ghosts c^° possess the same Grassmann parity as the antighosts b\k and the antighost number 0, but they are characterized by the ghost number

396

Geometric and Algebraic Topological Methods in Quantum Mechanics

gh(c^) = k + 1. Accordingly, the antighosts bkk are endowed with the negative ghost number — (k + 1). Let us consider the C°°(Z)-aigebra CL of polynomials in ghosts Ck and antighosts bk, k = 0,... ,L. It is called the BRST algebra. This algebra is provided with the C°°(Z)-bilinear superbracket [, ]gh denned by the relations

M V = [c,c']gh = 0,

[ # , & £ > = £•<&£,#) = SI-.

(6.8.14)

Given a Poisson bracket {,} on a manifold Z, the superbracket (6.8.14) is extended to the M-bilinear superbracket [»] = {.} +Ugh

(6-8.15)

The Koszul-Tate differential 8L is prolonged onto CL by letting 5Lc = 0 for all ghosts c. Let C denote the C°°(Z)-linear restriction morphism CL —> BL by One can show the following [156; putting ck = 0, k = Q,...,L. 245]. 6.8.1. Let J be a coisotropic ideal of the Poisson algebra (C°°(Z),{,}). There exists an element 6 e CL of ghost number 1 called the BRST charge such that: • [0,0] = 0 where [,] is the superbracket (6.8.15); THEOREM

• the operator, called the BRST differential, aeCL, £>(a) = [0,a],

(6.8.16)

is a nilpotent graded derivation of the R-algebra CL; • the operator BL9&^C([e,&]gh)e£L coincides the Koszul-Tate differential 8L on BL; • the BRST charge takes the coordinate form

e = 4/i+Ectx fc S- 16 t- 1l + efcc fc=i

where etc. consists of terms with at least two ghosts and one antighost or terms with at least two antighosts and one ghost; • the BRST charge is defined up to the Koszul-Tate boundary 5La, D where a E CL is of ghost number 2.

397

Chapter 6 Supergeometry

The BRST differential makes the BRST algebra CL into a cochain complex graded by the ghost number. It is called the BRST complex. It is important that, at the zero ghost number, the cohomology group H°(CL) of the BRST complex equals J"/J, where J" is the normalize (2.1.24) of the ideal J, i.e., H°(CL) coincides with the reduction of the Poisson algebra C°°(Z) via the ideal J (see Definition 2.1.16). Example 6.8.1. Let {/,} be irreducible constraints on a manifold Z provided with the zero Poisson structure, i.e., {/,g} = 0 for all f,g £ C°°{Z). In this case, the BRST algebra C° is generated by odd ghosts CQ of ghost number 1 and odd antighosts 6° of antighost number 1 and ghost number -1. The Koszul-Tate differential S° is the Koszul boundary operator 5°(b°i) = fi,

S°(4) = 0-

(6-8.17)

In this case, the BRST charge and the BRST differential (6.8.16) read

0 =/4,

D = 6°. a

Example 6.8.2. Let Z be a symplectic manifold which admits a Hamiltonian action ££ of a finite-dimensional real Lie algebra $j. Let {e,} be a basis for Q and fi the functions on Z possessing £Ei as their Hamiltonian vector fields (see Proposition 2.3.3). The functions fi constitute the Lie algebra g with respect to the Poisson bracket {fiJj} = <$jfk, where c^ are constant on Z. Let the representation e i-> ££ be a Lie algebra isomorphism. Then {/»} is an irreducible system. As in Example 6.8.1, the BRST algebra C° is generated by ghosts cj, and antighosts b°, and the Koszul-Tate differential 5° on CL takes the form (6.8.17). The BRST charge is

© = fA - \ckiA4A-

(6-8.18)

The corresponding BRST differential reads

D(f) = {/„ f}cl

D($) =U- 4^6°,

D(4) =

-\AA^

398

Geometric and Algebraic Topological Methods in Quantum Mechanics

It can be written in the form D = 5 + d + etc., where 5 is the Koszul boundary operator (6.8.17) and d is the Chevalley-Eilenberg differential • (1.5.55). Turn now to Lagrangian constraints Ki=Pi-Oij<4kPk=0

(6.8.19)

(2.5.39) in Hamiltonian time-dependent mechanics on a configuration bundle Q —> R (see Section 2.5D). We aim to construct their Koszul-Tate and BRST complexes. In accordance with the splitting Pi = Hi + Vi = fa - Oij4kPk] + [oijOo'Pk]-

(6-8.20)

(2.5.37b) and the relation aij
(6.8.21)

(2.5.32), let us introduce the projection operators

R = Si- aijalk,

P =

k aij4

such that PfUk = 0,

RkUk = TZi.

(6.8.22)

A glance at the relations (6.8.22) shows that the constraints TZi (6.8.20) are reducible. Moreover, since PR) = 0,

(6.8.23)

these constraints are infinitely reducible in general. The corresponding algebra B of antighosts is generated by the elements &£, r = 0 , 1 , . . . of antighost number agh(&£) = r + 1 and Grassmann parity (r + I)mod2. They are introduced as follows [295]. Given the vertical tangent bundle VQVQ of VQ —> Q, let us consider the infinite Whitney sum E = E0®E1

= VQVQ © VQVQ®--VQ

(6.8.24)

over VQ of the copies of VQVQ. This vector bundle is provided with the holonomic coordinates (t,qi,pi,p'i), r = 0 , 1 , . . . , where (t,q\pi,pfr+1), are

399

Chapter 6 Supergeometry

coordinates on Eo, while {t,ql,Pi,pfr) are those on E\. Following Remark 6.4.2, let us define the quotient E of the tensor product Er

= R © E* VQ

© ®E

© ••• © <£>E*

V'Q

V'Q

V'Q

by the elements ei e[ + e[ ex,

e0 ® e'o - e'o ® e0

e0 ® ei - ei ® e0

for all eo,e(, G ^op, ei,ei G Ei p , and p G VQ. The Croo(VQ)-modules Br of sections of the vector bundles E —> VQ make up a direct system with respect to the natural monomorphisms Br —> Br+1. Its direct limit B is endowed with a structure of a graded commutative C°°(VQ)-ring generated by odd and even elements b\, r = 0,1,..., which have the same linear coordinate transformation law as the coordinates pi. Furthermore, one defines the graded differential calculus over the graded commutative C°°(yQ)-ring B. The C°°(^(5)-module B is graded by the antighost number as B = C°°{VQ) ®B° ®Bl © • • • , and is brought into a chain complex ^Af1

0 ^C°°(VQ)

<

(6.8.25)

with respect to the Koszul-Tate differential 5 : C°°(VQ) 6(b2ir+1)=Plkb2kr,

-

0,

S(b°)=R^pk, r+2

5{bl

)

= RkiblT+\

r>0.

(6.8.26)

The nilpotency property 8 o S = 0 of this differential is the corollary of the relations (6.8.23). 6.8.2. The chain complex (6.8.25) with respect to the differential (6.8.26) is the Koszul-Tate resolution, i.e., its homology groups are

PROPOSITION

#fc>o = O,

HQ = C°°(VQ)/IN

= COO(NL).

a Let us note that, in particular cases of the degenerate quadratic Lagrangian (2.5.24), the complex (6.8.25) may have a subcomplex, which is also the Koszul-Tate resolution. For instance, if the fibre metric a in

400

Geometric and Algebraic Topological Methods in Quantum Mechanics

VQ —> Q is diagonal with respect to a holonomic atlas of VQ, the constraints (6.8.19) are irreducible and the complex (6.8.25) contains a subcomplex which consists only of the antighosts 6°. Now, we aim to construct the BRST charge G such that {e,9}

s

= 0,

8(b) = { Q , b } s ,

b € B

with respect to some Poisson superbracket {, }s- The problem is to find the Poisson superbracket such that {f,g}s = 0 for all f,g e C°°(VQ). In order to solve this problem, we follow the SUSY extension of Hamiltonian formalism in Section 6.7. We assume that Q —> R is a vector bundle, and further denote II = VQ. Let us consider the vertical tangent bundles VII and W I I . The latter admits the canonical decomposition

vvn = vn © vn ^ vn.

(6.8.27)

Let choose the bundle E = Eo © E\ over Vn which is the infinite Whitney sum over VII of the copies of VVII. In view of the decomposition (6.8.27), we have

E = vn©vn©---p-^vn. R

This vector bundle is provided with the holonomic coordinates {t,q\Pi,qr,Vri), r = 0' 1 )---! where {qlr+nPT^) a r e bundle coordinates on EQ —> VII and {q\r,p\T) are those on Ex -* VII. As a repetition of the construction of B, we define the graded commutative C°°(VII)-ring C generated by odd and even elements (6£, bt, clr, clr), r = 0,1,..., where: • c, clr are ghosts of ghost number r + 1 and Grassmann parity (r + I)mod2, • b\, VI are antighosts of antighost number r +1, ghost number — (r+1) and Grassmann parity (r + I)mod2. One can think of C as being the BRST algebra. Furthermore, we define the graded differential calculus over the graded commutative C°°(VQ)ring C. It contains the graded three-form tts = [dpi A dqi + dpi A dql + oo

i Yffi r=0

A ~H

- dbl A d°r)\ A dt>

401

Chapter 6 Supergeometry

generalizing the graded three-form (6.7.2). The corresponding Poisson superbracket on C reads

{c,c'}s = {c,c'}v + i J2(-l)[f]{r+1)[^~ r=0 (_1)r+1dc_d£

_ dc_ dd_ _

(6-8.28)

+

Ot)i

r

+1

dc_dd_

where {, }y is the Poisson bracket (2.4.81). Then the desired BRST charge takes the form

e = i[cjiipfc + £ ( 4 r + i W + 4- + 2W +1 )]r=0

The corresponding BRST differential D = {6, .}s reads D ( ^ ) = 0,

D(bl) = 6(bl), i?(4-) = ^ r + l .

6.9

r = 0,l,...,

^>(4r+l) = - 4 4 + 2 ,

£>(<£) = 0.

Appendix. Supermanifolds

Subsections: A. Superfunctions, 401; B. Supermanifolds, 405; C. DeWitt supermanifolds, 409; D. Supervector bundies, 411; E. Superconnections, 415; F. Principal superconnections, 417. There are different types of supermanifolds. These are H°°-, G°°-, GH°°-, G-, and DeWitt supermanifolds [21]. By analogy with smooth manifolds, supermanifolds are constructed by gluing together of open subsets of supervector spaces Bn'm with the aid of transition superfunctions. Therefore, let us start with the notion of a superfunction. A. Superfunctions Though there are different classes of superfunctions, they can be introduced in the same manner as follows. Let Bn<m = A£©A?\

n,m>0,

402

Geometric and Algebraic Topological Methods in Quantum Mechanics

be a supervector space, where A is a Grassmann algebra of rank 0 < N > m. Let an,m

_>

. Bn,m

Rn^

g

. Q^m

_>

Rn,m

=

Rn

Q

Rrn

be the corresponding body and soul maps (see the decomposition (6.1.2)). Then any element q £ Bn>m is uniquely split as q = (x,y) = (a(i 4 ) + s(xi))e? +yje),

(6.9.1)

where {e?, e]} is a basis for Bn'm and a(xi) € R, six) £ Ro, yj £ Ri. Let A' be another Grassmann algebra of rank 0 < N' < N which is treated as a subalgebra of A, i.e., the basis {ca}, a — 1,..., N', for A' is a subset of the basis {c}, i = 1,..., N, for A. Given an open subset U C M", let us consider a A'-valued function N' fc=0

(6.9.2) '

on U with smooth coefficients fai-.akiz), z £U. It is a graded function on U. Its prolongation to (cr"' 0 )-" 1 ^) C Bn'° is defined as the formal Taylor series i r

N>

fc=o

N

' LP=O

i

F>vf

y'

(6.9.3) Then a superfunction F(q) on (cr"' 7 ")-^^) C Bn'm is given by a sum m

F{q) = F(a;,i/) = ]T - / ^ . . j , ^ • • V ' , r=0

(6.9.4)

'

where fj1...jrix) are functions (6.9.3). However, the representation of a superfunction Fix, y) by the sum (6.9.4) need not be unique. The germs of superfunctions (6.9.4) constitute the sheaf SJV of graded commutative A'-algebras on Bn<m, but it is not a sheaf of Cg^.m-modules since superfunctions are expressed in Taylor series. Using the representation (6.9.4), one can define derivatives of superfunctions as follows. Let fix) be a superfunction on Bn<°. Since / , by

403

Chapter 6 Supergeometry

definition, is the Taylor series (6.9.3), its partial derivative along an even coordinate xl is defined in a natural way as dif(x) = (dif)(a(x), s(x)) = N

'

i r

N

(6.9.5)

1

i

8P+ 1

L, k\ [L p\ dzidzii...dziP

MX)MX )

& Y

c .

This even derivative is extended to superfunctions F on Bn'm in spite of the fact that the representation (6.9.4) is not necessarily unique. However, the definition of odd derivatives of superfunctions is more intricate. Let &°N, C &N1 be the subsheaf of superfunctions F(x, y) = f(x) (6.9.3) independent of the odd arguments yi. Let AMm be a Grassmann algebra generated by ( a 1 , . . . , am). The expression (6.9.4) implies that, for any open subset U C Bn'm, there exists the sheaf morphism A : 6 ^ , AMm -> &N,, m

X X

( ^)

:

E 3 / - * • ( * ) ® K • --ajr) r=0

m

(6.9.6)

-

(6-9-7)

1

r=0

'

n m

over 5 - . Clearly, the morphism A (6.9.6) is an epimorphism. One can show that this epimorphism is injective and, consequently, is an isomorphism if and only if iV - TV' > m

(6.9.8)

[21]. Roughly speaking, in this case, there exists a tuple of elements y1,..., yir € A for each superfunction / such that \(f®(ajl at t h e point (x, yjl,...,

•••«>)) ^ 0

yjm) of Bn<m.

If the condition (6.9.8) holds, the representation of each superfunction F(x, y) by the sum (6.9.4) is unique, and it is an image of some section / ® a of the sheaf 6 ^ , ® ARm with respect to the morphism A (6.9.7). Then an odd derivative of F is defined as

^y(A(/®y)) = A(/®A(a)).

404

Geometric and Algebraic Topological Methods in Quantum Mechanics

This definition is consistent only if A is an isomorphism, i.e., the relation (6.9.8) holds. If otherwise, there exists a non-vanishing element / ® a such that A(/®a) = 0 , whereas A(/®a,(a))#0. For instance, if N-N'

=

m-\,

such an element is /<8>a = c 1 ---c w '®(a 1 ---a i n ). In order to classify superfunctions, we follow the terminology of [21; 377]. • If N' = N, one deals with G°° -superfunctions, introduced in [376], In this case, the inequality (6.9.8) is not satisfied, unless m = 0. • If the condition (6.9.8) holds, SJV = QHN' is the sheaf of GH°°superfunctions. • In particular, if N' = 0, the condition (6.9.8) is satisfied, and &N> — H°° is the sheaf of H°° -superfunctions

r=0

Lp=O

(6.9.9) where fj-i...jr are real functions [25; 126]. Superfunctions of the above three types are called smooth superfunctions. The fourth type of superfunctions is the following. Given the sheaf GTi-N' of Gfi"°°-superfunctions on a supervector space Bn'm, let us define the sheaf of graded commutative A-algebras QN, =gnN> ®A, A'

(6.9.10)

where A is regarded as a graded commutative A'-algebra. The sheaf Q^1 (6.9.10) possesses the following important properties [2l]. • There is the evaluation morphism 5:GN' 3 F®a>-> Fa£C£n,m,

(6.9.11)

Chapter 6 Supergeometry

405

where

is the sheaf of continuous A-valued functions on Bn>m. Its image is isoraorphic to the sheaf G°° of G°°-sup erfunctions on Bn'm. • For any two integers N' and N" satisfying the condition (6.9.8), there exists the canonical isomorphism between the sheaves GN' and GN" • Therefore, one can define the canonical sheaf Gn,m of graded commutative A-algebras on the supervector space Bn'm whose sections can be seen as tensor products F($>a of iJ°°-superfunctions F (6.9.9) and elements a £ A. They are called G-superfunctions. • The sheaf 5^ n , m of graded derivations of the sheaf Gn,m is a locally free sheaf of 5 nm -modules of rank (n, m). On any open set U C Bn'm, the Gn,m{Uymod\i\e dGn,m{U) is generated by the derivations d/dxl, d/dyj which act on Gn,m{U) by t h e rule 7— ( F <8> a) = 77-- a, — {F a) = - — <8> a. (6.9.12) ox1 oyi oyi dx% These properties of G-superfunctions make G-supermanifolds most suitable for differential geometric constructions. B. Supermanifolds A paracompact topological space M is said to be an (n, m)-dimensional smooth supermanifold if it admits an atlas = {C/C,0C},

^:Uc^Bn'm,

such that the transition functions (p^ o (p'T1 are supersmooth. Obviously, a smooth supermanifold of dimension (n, m) is also a real smooth manifold of dimension 2N~1(n + m). If transition superfunctions are H00-, G°°- or G#°°-superfunctions, one deals with H°°-, G°°- or Gff^-supermanifolds, respectively. By virtue of Theorem 1.8.2 extended to graded local-ringed spaces, this definition is equivalent to the following one. 6.9.1. A smooth supermanifold is a graded local-ringed space (M,&) which is locally isomorphic to (Bn'm,S), where S is one of the sheaves of smooth superfunctions on Bn'm. The sheaf S is called the structure sheaf of a smooth supermanifold. • DEFINITION

406

Geometric and Algebraic Topological Methods in Quantum Mechanics

In accordance with Definition 6.9.1, by a morphism of smooth supermanifolds is meant their morphism () as graded local-ringed spaces, where $ is an even graded morphism. In particular, every morphism ip : M —> M' yields the smooth supermanifold morphism ( Cfa, where U

M — ° M <& A

is sheaf of continuous A-valued functions on M, and S is locally isomorphic to the evaluation morphism (6.9.11). • Example 6.9.1. The triple (Bn'm,Gn,m,5), where S is the evaluation morphism (6.9.11), is called the standard G-supermanifold. For any open subset U c Bnm(U) =* H°°{U) ® A £ C°°(an'm(U)) nm

C°°(a ' {U))

A ® AR m ^

(6.9.13)

N+m

®AR

.

a Remark 6.9.2. Any Gif^-supermanifold (M, GH^) with the structure sheaf GH'M is naturally extended to the G-supermanifold (M, GH'jfi A). Every G-supermanifold defines an underlying G°°-supermanifold

Chapter 6 Supergeometry

407

(M,5(GM)), where S(GM) — G^ is the sheaf of G°°-superfunctions on M. 0 As in the case of smooth supermanifolds, the underlying space M of a G-supermanifold (M,GM) is provided with the structure of a real smooth manifold of dimension 2N~1(n + m), and morphisms of G-supermanifolds are smooth morphisms of the underlying smooth manifolds. However, it may happen that non-isomorphic G-supermanifold have isomorphic underlying smooth manifolds. Remark 6.9.3. Let us present briefly the axiomatic approach to supermanifolds which enables one to obtain all the previously known types of supermanifolds in terms of i?°°-supermanifolds [21; 22; 69]. This approach to supermanifolds refines that in [378]. The i?°°-supermanifolds are introduced over the above mentioned Arens-Michael algebras of Grassmann origin [69], but we here omit the topological side of their definition, though just the topological properties differ i?°°-supermanifolds from Rsupermanifolds in [378]. Let A be a graded commutative algebra of the above mentioned type (for the sake of simplicity, the reader can think of A as being a Grassmann algebra). A superspace over A is a triple (M,?R°°,6), where M is a paracompact topological space, 91°° is a sheaf of graded commutative Aalgebras, and S : $H°° —> C^ is an evaluation morphism to the sheaf C^ of continuous A-valued functions on M. Sections of EH°° are called R°°superfunctions. Let Mq denote the ideal of the stalk fH£°, q e M, formed by the germs of i?°°-superfunctions / vanishing at a point q, i.e., such that <5(/)(?) = 0. An i?°°-super-manifold of dimension (n, m) is a superspace (M, 91°°, S) satisfying the following four axioms [69]. Axiom 1. The graded O^-dual (59^°°)* of the sheaf of derivations is a locally free sheaf of graded ^""-modules of rank (n,m). Every point q 6 M has an open neighborhood U with sections x1, • • • ,xn G 9\°°(U)o, y1, • • • , ym € m°°(U)i such that {dx\ dyj} is a graded basis for (W\°°)(U) over m°°(U). Axiom 2. Given the above mentioned coordinate chart, the assignment determines a homeomorphism of U onto an open subset of Bn'm. Axiom 3. For every q G M, the ideal Mq is finitely generated. Axiom 4. For every open subset U
408

Geometric and Algebraic Topological Methods in Quantum Mechanics

An .R-supermanifold over a graded commutative Banach algebra satisfying Axiom 4 is an -R^-supermanifold. The standard G-supermanifold in Example 6.9.1 is an i?°°-supermanifold. Moreover, in the case of a finite Grassmann algebra A, the category of i?°°-supermanifolds and the category • of G-supermanifolds are equivalent. Let (M, GM) be a G-supermanifold. As was mentioned above, it satisfies the Axioms 1-4. Sections u of the sheaf C)GM of graded derivations are called super-vectorfieldson the G-supermanifold (M,GM), while sections <j> of the dual sheaf VGM are one-superforms on (M, GM). Given a coordinate chart () on U C M, supervector fields and one-superforms read u = uldi,

<j> =
where coefficients ul and fa are G-superfunctions on U. The graded differential calculus in supervector fields and superforms obeys the standard formulae (6.2.4), (6.2.16), (6.2.17) and (6.2.18). Given a GLet us consider cohomology of G-supermanifolds. supermanifold (M,GM),

let S A M = OJW® A

be the sheaves of smooth A-valued exterior forms on M. These sheaves are fine, and they constitute the fine resolution 0^A-Cfi®A->D1M®A---of the constant sheaf A on M. We have the corresponding de Rham complex

0-A->CnM)-»OyUM)-"of A-valued exterior forms on M. By virtue of Theorem 1.7.6, the cohomology H^ (M) of this complex is isomorphic to the sheaf cohomology H* (M; A) of M with coefficients in the constant sheaf A and, consequently, is related to the de Rham cohomology as follows: HX(M) = H{M;A) = H(M)®A.

(6.9.14)

Thus, the cohomology groups of A-valued exterior forms do not provide us with information on the G-supermanifold structure of M. Let us turn to cohomology of superforms on a G-supermanifold k

(M,GM)-

The sheaves ADG^f of superforms constitute the complex 0-+ A ^ G M - ^ * G M - * ••• •

(6.9.15)

Chapter 6 Supergeometry

409

The Poincare lemma for superforms is proved to hold [21; 68], and this complex is exact. However, the structure sheaf GM need not be acyclic, and the exact sequence (6.9.15) fails to be a resolution of the constant sheaf A on M in general. Therefore, the cohomology HS{M) of the de Rham complex of superforms are not equal to cohomology H(M; A) of M with coefficients in the constant sheaf A, and need not be related to the de Rham cohomology H{M) of the smooth manifold M. In particular, cohomology Hg(M) is not a topological invariant, but it is invariant under G-isomorphisms of G-supermanifolds. 6.9.3. The structure sheaf Qn,m of the standard Gsupermanifold (Bn>m, Gn,m) is acyclic, i.e.,

PROPOSITION

n

' , yn,m)

\B

— u.

D The proof is based on the isomorphism (6.9.13) and some cohomological constructions [21; 69]. C. DeWitt super manifolds There exists a particular class of supermanifolds, called DeWitt supermanifolds. Their notion implies that a supervector space Bn'm is provided with the De Witt topology, which is coarser than the Euclidean one. This is the coarsest topology such that the body map an,m

. Bn,m _^ R n

(6.9.16)

is continuous. The open sets in the DeWitt topology are of the form V x Tln'm, where V are open sets in Rn. Clearly, this topology is not Hausdorff. DEFINITION 6.9.4. A smooth supermanifold (resp. a G-supermanifold) is said to be a De Witt supermanifold if it admits an atlas such that the local morphisms <j>^ : U^ —> Bn'm in Definition 6.9.1 (resp. Definition 6.9.2) are continuous with respect to the DeWitt topology, i.e., ^(U^) C Bn'm are open in this topology. •

Given an atlas (U^,<j>^) of a DeWitt supermanifold in accordance with Definition 6.9.4, it is readily observed that its transition functions 4>^o(p~1 must preserve the fibration an'm (6.9.16) whose fibre (an'm)~1 (z) over z €

410

Geometric and Algebraic Topological Methods in Quantum Mechanics

Rn is equipped with the coarsest topology, where only 0 and ( ZM over an n-dimensional smooth manifold ZM with the typical fibre Rn<m = a(Bn<m). The base ZM of this fibre bundle is said to be a body manifold of a DeWitt supermanifold, while the surjection CJM is called a body map. There is the above mentioned correspondence between the graded manifolds and the DeWitt //°°-supermanifolds. It is based on the following facts. (i) The structure sheaf 21 of a graded manifold (Z, 21) is locally isomorphic to the sheaf Cg? ARm. (ii) Given a DeWitt i?°°-supermanifold (M,H^), the direct image <J{H'M) of its structure sheaf onto the body manifold ZM is locally isomorphic to the sheaf Cg? ARm. The expression (6.9.9) provides this isomorphism in an explicit form. (iii) The graded local-ringed spaces (M,H^) and (ZM,o-f(H^)) determine the same element of the cohomology set H1(ZM', Aut (AMm)°°). Thus, we come to the following statement [21; 25]. THEOREM 6.9.5. Given a DeWitt H°°-supermanifold (M, H^), the associated pair {ZM,O{HM)) is a graded manifold. Conversely, for any graded manifold (Z, 21), there exists a DeWitt if°°-supermanifold, whose body manifold is Z and whose structure sheaf 21 is isomorphic to cr(HM'). •

Then by virtue of Batchelor's Theorem 6.3.1 and Theorem 6.9.5, there is one-to-one correspondence between the isomorphism classes of DeWitt H°°supermanifolds of odd rank m with a body manifold Z and the equivalence classes of m-dimensional vector bundles over Z. This result is extended to DeWitt GH°°-, G°°- and G-supermanifolds because their isomorphism classes are in one-to-one correspondence with isomorphism classes of DeWitt i?°°-supermanifolds [2l]. Let us say something more on DeWitt G-supermanifolds. PROPOSITION 6.9.6. The structure sheaf GM of a DeWitt G-supermanifold is acyclic, and so is any locally free sheaf of graded GM-modules [21; 69].

•

PROPOSITION 6.9.7. There is an isomorphism of the cohomology Hg(M) of superforms on a DeWitt G-supermanifold to the cohomology (6.9.14) of A-

411

Chapter 6 Supergeometry

valued exterior forms on its body manifold [363].

ZM,

i-e., Hg(M) =

H{ZM)®^-

•

These results are based on the fact that the structure sheaf GM on M, provided with the DeWitt topology, is fine. However, this does not imply automatically that GM is acyclic since the DeWitt topology is not paracompact. Nevertheless, it follows that the image ZM is contractible, then H(M) = H(ZM) such that the isomorphism in Proposition 6.9.7 takes place. D. Supervector bundles As was mentioned above, supervector bundles are considered in the category of G-supermanifolds. We start with the definition of the product of two G-supermanifolds seen as a trivial supervector bundle. Let (Bn'm,Qn,m) and (Br>s,QrtS) be two standard G-supermanifolds in Example 6.9.2. Given open sets U C Bn'm and V C Br>s, we consider the presheaf UxV-^

(6.9.17)

gn,m(U)®Gr,s(V),

where denotes the tensor product of modules completed in Grothendieck's topology (see Remark 1.8.4). Due to the isomorphism (6.9.13), it is readily observed that the structure sheaf Gn+r,m+s of the standard Gsupermanifold on Bn+r'm+s is isomorphic to that, defined by the presheaf (6.9.17). This construction is generalized to arbitrary G-supermanifolds as follows. Let (M,GM) and (M',GM') be two G-supermanifolds of dimensions (n,m) and (r,s), respectively. Their product {M,GM)x(M',GM>) is defined as the graded local-ringed space (M x

M',GM§>GM'),

where

412 GM®GM'

Geometric and Algebraic Topological Methods in Quantum Mechanics

is the sheaf determined by the presheaf

UxU'

^GM(U)®GM>(U'),

5 : GM{U)®GMI{U') - CSfogC^,) =

C^myiauVJI),

for any open subsets U C M and U' C M ' . This product is a Gsupermanifold of dimension (n + r,m + s) [21]. Furthermore, there is the epimorphism pri

: {M,GM) x (M',GM>) - (M,GM).

One may define its section over an open subset U C M as the Gsupermanifold morphism su • {U,GM\u)

-> ( M , G M ) x

(M'.GM')

such that pr1 o sy is the identity morphism of (C/, G M It/)- Sections su over all open subsets U C M determine a sheaf on M . This sheaf should be provided with a suitable graded commutative GM-structure. For this purpose, let us consider the product (M,GM)x(Br^,gris),

(6.9.18)

where Br^s is the superspace (6.1.4). It is called a product G-supermanifold. are isomorphic, Br^s has a natural Since the Ao-modules Br\" and Br+3'r+s structure of an (r + s,r + s)-dimensional G-supermanifold. Because Br\8 is a free graded A-module of the type (r, s), the sheaf S^ of sections of the fibration (M, GM) x (Br^, gr]s) -> (M, G M )

(6.9.19)

has the structure of the sheaf of free graded GM-modules of rank (r,s). Conversely, given a G-supermanifold (M, GM) a n d a sheaf S of free graded Gjw-modules of rank (r, s) on M, there exists a product G-supermanifold (6.9.18) such that S is isomorphic to the sheaf of sections of the fibration (6.9.19). Let us turn now to the notion of a supervector bundle over Gsupermanifolds. Similarly to smooth vector bundles (see Theorem 1.8.5), one can require of the category of supervector bundles over Gsupermanifolds to be equivalent to the category of locally free sheaves of graded modules on G-supermanifolds. Therefore, we can restrict ourselves to locally trivial supervector bundles with the standard fibre Br^s.

413

Chapter 6 Supergeometry

DEFINITION 6.9.8. A supervector bundle over a G-supermanifold (M,GM) with the standard fibre (Br^,Gr\s) 1S denned as a pair ((Y,Gy),7r) of a G-supermanifold (Y,Gy) and a G-epimorphism

n : (Y,GY) -+ (M,GM)

(6.9.20)

such that M admits an atlas {(U^, V'c) °f local G-isomorphisms A • (^" 1 (^c)^y| 7 r -i(c/ c )) -> {U(,GM\U()

x

(Br^,gr]s).

a It is clear that sections of the supervector bundle (6.9.20) constitute a locally free sheaf of graded Gjvf-modules. The converse of this fact is the following [21]. 6.9.9. For any locally free sheaf S of graded GM-modules of rank (r, s) on a G-supermanifold (M, GM), there exists a supervector bundle over (M, GM) such that 5 is isomorphic to the structure sheaf of its sections. •

THEOREM

The fibre Yq, q G M, of the supervector bundle in Theorem 6.9.9 is the quotient SJMq

<* S%/{Mq • SMsq) * Br^

of the stalk Sg by the submodule Aiq of the germs s € Sq whose evaluation "K/X?) vanishes. This fibre is a graded A-module isomorphic to Br^s, and is provided with the structure of the standard G-supermanifold. Remark 6.9.4. The proof of Theorem 6.9.9 is based on the fact that, given the transition functions p^£ of the sheaf S, their evaluations 9a = HPtt)

(6-9.21)

define the morphisms U(nUt-+GL{r\s;A), and they are assembled into a cocycle of G°°-morphisms from M to the general linear graded group GL(r\s;A). Thus, we come to the notion of a G°°-vector bundle. Its definition is a repetition of Definition 6.9.8 if one replaces G-supermanifolds and G-morphisms with the G°°- ones. Moreover, the G°°-supermanifold underlying a supervector bundle (see Remark 6.9.2) is a G°°-supervector bundle, whose transition functions g^ are related to

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Geometric and Algebraic Topological Methods in Quantum Mechanics

those of the supervector bundle by the evaluation morphisms (6.9.21), and are GL(r\s; A)-valued transition functions. • Since the category of supervector bundles over a G-supermanifold (M, GM) is equivalent to the category of locally free sheaves of graded C?M-modules, one can define the usual operations of direct sum, tensor product, etc. of supervector bundles. Let us note that any supervector bundle admits the canonical global zero section. Any section of the supervector bundle TT (6.9.20), restricted to its trivialization chart (U,GM\u)x(Br^,grls),

(6.9.22)

is represented by a sum s = sa(q)ea, where {ea} is the basis for the graded A-module Br\s, while sa(q) are G-superfunctions on U. Given another trivialization chart U' of n, a transition function s'b(q)e'b = sa(q)hba(q)eb,

(6.9.23)

q€UnU', b

is given by the (r + s) x (r + s) matrix h whose entries h a(q) are Gsuperfunctions on U D U'. One can think of this matrix as being a section of the supervector bundle over U D U with the above mentioned group GL(r\s; A) as a typical fibre. Example 6.9.5. Given a G-supermanifold (M,GM), let us consider the locally free sheaf QGM of graded derivations of GM- In accordance with called supertanTheorem 6.9.9, there is a supervector bundle T(M,GM), gent bundle, whose structure sheaf is isomorphic to QGM- If (q1, • • •, qm+n) and (q11,..., q'm+n) are two coordinate charts on M, the Jacobian matrix , dqH /i. = — ,

i,j = l , . . . , n + m,

(see the prescription (6.9.12)) provides the transition morphisms for T(M,GM). It should be emphasized that the underlying G°°-vector bundle of the called G°°-supertangent bundle, has the supertangent bundle T(M,GM), transition functions <$(/iJ) which cannot be written as the Jacobian matrices since the derivatives of G°°-superfunctions with respect to odd arguments • are ill denned and the sheaf OG^ is not locally free.

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Chapter 6 Supergeometry

E. Superconnections Given a supervector bundle n (6.9.20) with the structure sheaf S, one can follow suit of Definition 1.8.10 and introduce a connection on this supervector bundle as a splitting of the the exact sequence of sheaves 0 -> X>GM ® 5 -^{GM®X>GM) ®S-^S^0.

(6.9.24)

Its splitting is an even sheaf morphism V :S->-0GM®S

(6.9.25)

satisfying the Leibniz rule V ( / s ) = d / ® a + /V(s),

feGM(U),

s€S(U),

(6.9.26)

for any open subset U € M. The sheaf morphism (6.9.25) is called a superconnection on the supervector bundle n (6.9.20). Its curvature is given by the expression R = V 2 : S -> A dGM ® S,

(6.9.27)

similar to the expression (1.8.26). The exact sequence (6.9.24) need not be split. One can apply the criterion in Section 1.8 in order to study the existence of a superconnection on supervector bundles. Namely, the exact sequence (6.9.24) leads to the exact sequence of sheaves 0 -» Horn (5, T)GM ® S) -> Horn (5, (G M © QGM) ® S) -» Horn (5,5) -> 0 and to the corresponding exact sequence of the cohomology groups 0 -> # ° ( M ; Horn (5, clC^ ® S)) -> ff°(M; Horn (5, (G M © 5G^) <8> S)) -> if o (M; Horn (5, 5)) -> F : ( M ; Horn (51, QGM ® S1)) -» • • • . The exact sequence (6.9.24) defines the Atiyah class At(7r) Gi? 1 (M;Hom(S,DGM®S)) of the supervector bundle TT (6.9.20). If the Atiyah class vanishes, a superconnection on this supervector bundle exists. In particular, a superconnection exists if the cohomology set H1(M;Hom(S,QGM ® S)) is trivial. In contrast with the sheaf of smooth functions, the structure sheaf GM of a G-supermanifold is not acyclic in general, cohomology

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H(M; Horn (S, VGM ® S)) is not trivial, and a supervector bundle need not admit a superconnection. Example 6.9.6. In accordance with Proposition 6.9.3, the structure sheaf of the standard G-supermanifold (Bn'm, Gn,m) is acyclic, and the trivial supervector bundle {Bn'm, Qn,m) X (B r|S , Qr\s) - (Bn'm, Gn,m)

(6.9.28)

has obviously a superconnection, e.g., the trivial superconnection.

•

Example 6.9.7. By virtue of Proposition 6.9.6, the structure sheaf of a DeWitt G-supermanifold (M,GM) is acyclic, and so is the sheaf Horn (S, f)GM (g> S). It follows that any supervector bundle over a DeWitt G-supermanifold admits a superconnection. • Example 6.9.6 enables one to obtain a local coordinate expression for a superconnection on a supervector bundle n (6.9.20), whose typical fibre is Br\s and whose base is a G-supermanifold locally isomorphic to the standard G-supermanifold (B n ' m , Gn,m)- Let 17 C M (6.9.22) be a trivialization chart of this supervector bundle such that every section s of n\u is represented by a sum sa(q)ea, while the sheaf of one-superforms X)GM\U has a local basis {dq1}. Then a superconnection V (6.9.25) restricted to this trivialization chart can be given by a collection of coefficients Via&: V(ea) = dq1 ® (Vi6oe6),

(6.9.29)

which are G-superfunctions on U. Bearing in mind the Leibniz rule (6.9.26), one can compute the coefficients of the curvature form (6.9.27) of the superconnection (6.9.29). We have R(ta) Rij\

= ^dq{ A dqj

Rijbaeb,

= (-l)Mb-]0. V / 6 - djViab + (-l)Wb]+M+W) V / f c VA (_1)W(H+W)Via)cVjfeb.

In a similar way, one can obtain the transformation law of the superconnection coefficients (6.9.29) under the transition morphisms (6.9.23). In particular, any trivial supervector bundle admits the trivial superconnection Viba = 0.

Chapter 6 Supergeometry

417

F. Principal superconnections In contrast with a supervector bundle, the structure sheaf Gp of a principal superbundle (P, Gp) —» (M, GM) is not a sheaf of locally free G^-modules in general. Therefore, the above technique of connections on modules and sheaves is not applied to principal superconnections in a straightforward way. Principal superconnections are introduced on principal superbundles by analogy with principal connections on smooth principal bundles [2l]. For the sake of simplicity, let us denote G-supermanifolds (M, GM) and their morphisms (ip:M->N,

$ : GN-+
by M and H,

DEFINITION

together with the natural identifications e x H = H x e = H, which satisfy the associativity m o (Id x in) = m o (m x Id) : H x H x H —> H x H —> H, the unit property (mo (ex Id))(e x H) = (mo (Id x e))(H x e) = Id if, and the inverse property (m o (S,ld))(H) = (fho (Id, S))(H) = e(e).

• Given a point g e H, let us denote by g : e —> H the G-supermanifold morphism whose range in H is g. Then one can introduce the notions of

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Geometric and Algebraic Topological Methods in Quantum Mechanics

the left translation Lg and the right translation Rg as the G-supermanifold morphisms Lg:H = exHd^HxH I

Rg:H = Hxe ^HxH

^H, ^H.

Remark 6.9.8. Given a G-Lie supergroup H, the underlying smooth manifold H is provided with the structure of a real Lie group of dimension 2 (n + m), called the underlying Lit group. In particular, the actions on the underlying Lie group H, corresponding to the left and right translations • by g, are ordinary left and right translations by g. Let us reformulate the group axioms in Definition 6.9.10 in terms of the structure sheaf Ti of the G-Lie supergroup (H, "H). We observe that H has properties of a sheaf of graded Hopf algebras as follows. If {H,H) is a G-Lie supergroup, the structure sheaf % is provided with the sheaf morphisms: • a comultiplication in :H —> m(W'H), • a counit e : H —» e* (A), • a coinverse S : H. —> s+Ti. Let us denote k = m o (Id x m) = m o (m x Id) : H x H x H —> H. Then the group axioms in Definition 6.9.10 are equivalent to the relations ((Id ® in) o fh)(H) = {(in* ® Id) o fh){H) = K(H®H%H), (rh o (Id (8> r))(W®e.(A)) = (m* o (e* ® Id))(e,(A)§W) = Id W, (Id • S) o ih = {S* • Id) o in* = e. Comparing these relations with the axioms of a Hopf algebra in Section 10.2, one can think of the structure sheaf of a G-Lie group as being a sheaf of graded topological Hopf algebras. Example 6.9.9. The general linear graded group GL(n\m; A) is endowed with the natural structure of an i7°°-supermanifold of dimension (n2 + m 2 ,2nm). The matrix multiplication gives the i?°°-morphism m : GL(n\m;A) x GL(n\m; A) —> GL(n\m\K)

Chapter 6 Supergeometry

419

such that F(g,g') >-> F(gg'). It follows that GL(n\m;A) is an H°°-Lie supergroup. It is trivially extended to the G-Lie supergroup GL(n | m; A), O called the general linear supergroup. A Lie superalgebra h of a G-Lie supergroup H is defined as an algebra of left-invariant supervector fields on H. Let us recall that a supervector field u on a G-supermanifold H is a derivation of its structure sheaf H. It is called left-invariant if (Id u) o m = m* o u. If u and u' are left-invariant supervector fields, so are [u, u'} and au + a'u', a, a' £ A. Hence, left-invariant supervector fields constitute a Lie superalgebra. The Lie superalgebra f) can be identified with the supertangent space Te{H). Moreover, there is the sheaf isomorphism H<8>t)=QH,

(6.9.30)

i.e., the sheaf of supervector fields on a G-Lie supergroup H is the globally free sheaf of graded W-modules of rank (n,m), generated by left-invariant supervector fields. The Lie superalgebra of right-invariant supervector fields on H is introduced in a similar way. Let us consider the right action of a G-Lie supergroup H on a Gsupermanifold P. This is a G-morphism p: P x H -> P such that po (p x Id) = p o (Id x m) : P x H x H -> P, po(ldxe)(Pxe)=ldP. The left action of H on P is defined similarly. Example 6.9.10. Obviously, a G-Lie supergroup acts on itself both on the left and on the right by the multiplication morphism m. The general linear supergroup GL(n\m; A) acts linearly on the standard supermanifold Bn\n on the left by the matrix multiplication which is a G• morphism. Let P and P' be G-supermanifolds that are acted on by the same G-Lie supergroup H. A G-supermanifold morphism (p : P —> P' is said to be

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Geometric and Algebraic Topological Methods in Quantum Mechanics

if-invariant if
P'. DEFINITION 6.9.11. A quotient of an action of a G-Lie supergroup on a G-submanifold P is a pair (M,TT) of a G-supermanifold M and a Gsupermanifold morphism 7?: P —» M such that: (i) there is the equality

7rop = 7ropr1 :PxH^>M,

(6.9.31)

(ii) for every morphism (p : P —> • M' such that (p o p = <J? o pi-j, there is a unique morphism 7j : M —> M' with (p = 7j OK. D The quotient (M, 7?) does not necessarily exists. If it exists, there is a monomorphism of the structure sheaf GM of M into the direct image 7rGp. Since the G-Lie group H acts trivially on M, the range of this monomorphism is a subsheaf of ir^Gp, invariant under the action of H. Moreover, there is an isomorphism GM = {^GP)H

(6.9.32)

between GM and the subsheaf of Gp of if-invariant sections. The latter is generated by sections of Gp on •K~1{U), U C M, which are if-invariant as G-morphisms U —> A, where one takes the trivial action of H on A. Let us denote the morphism in the equality (6.9.31) by -d. It is readily observed that the invariant sections of Gp{-K~l(U)) are exactly the elements which have the same image under the morphisms p : GP{K-X{U))

-+

pr? : GP(n-\U)) -

{H%GP)W-\U)), (WSGPXIT1^)).

Then the isomorphism (6.9.32) leads to the exact sequence of sheaves of A-modules on M 0 -^GM ^nGPP-^l-d(GM®H).

(6.9.33)

DEFINITION 6.9.12. A principal superbundle of a G-Lie supergroup H is defined as a locally trivial quotient n : P —> M, i.e., there exists an open covering {U^} of M together with if-invariant isomorphisms

$(:P\0(-+U(xH,

Chapter 6 Supergeometry

421

where H acts on C/c x # -> [/c by the right multiplication.

(6.9.34) •

Remark 6.9.11. In fact, we need only the condition (i) in Definition 6.9.11 of the action of H on P and the condition of local triviality of P. • In an equivalent way, one can think of a principal superbundle as being glued together of trivial principal superbundles (6.9.34) by ^-invariant transition functions

which fulfill the cocycle condition. As in the case of smooth principal bundles, the following two types of supervector fields on a principal superbundle are introduced. DEFINITION 6.9.13. A supervector field u on a principal superbundle P is said to be invariant if

p o u = (u Id) o u : GP -» p»(G P §H).

• One can associate to every open subset V C M the GM (V)-module of all iJ-invariant supervector fields on n~1(V), thus defining the sheaf DH(7rGp) of Gjw-niodules. DEFINITION 6.9.14. A fundamental supervector field v associated to an element v S f) is defined by the condition

v = (Id ® v) o p : GP -• GP®e(A) = GP.

n Fundamental supervector fields generate the sheaf VGp of Gp-modules of vertical supervector field on the principal superbundle P, i.e., UOTT = 0 . Moreover, there is an isomorphism of sheaves of Gp-modules G p ® h 9 F ® w i - > F u G VGP, which is similar to the isomorphism (6.9.30).

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Geometric and Algebraic Topological Methods in Quantum Mechanics

Let us consider the sheaf (nVGP)H

= 7T(VGP) n VH(nGP)

on M whose sections are vertical if-invariant supervector fields. PROPOSITION 6.9.15. [21]. There is the exact sequence of sheaves of GMmodules 0 -> (nVGp)H

-* UH{nGP)

-» DGM -» 0.

(6.9.35)

D The exact sequence (6.9.35) is similar to the exact sequence (10.10.23) and the corresponding exact sequence of sheaves of C°°-modules 0 -> (VGP)x

- (TGP)x

- VC% - 0

in the case of smooth principal bundles. Accordingly, we come to the following definition of a superconnection on a principal superbundle. DEFINITION 6.9.16. A superconnection on a principal superbundle IT : P —> M (or simply a principal superconnection) is defined as a splitting V : DGM - QH{KGP)

(6.9.36)

of the exact sequence (6.9.35).

D

In contrast with principal connections on smooth principal bundles, principal superconnections on a //-principal superbundle need not exist. A principal superconnection can be described in terms of a h-valued one-superform w : DGp ^GP®\]

on P called a superconnection form. defines the morphism of Gp-modules

=

VGP,

Indeed, every splitting V (6.9.36) H

TT{DGM) -> 7f(5 (7rGF)) ^ X)GP

which splits the exact sequence 0 -> VGP -> OGp -> Tf(T)GM) -> 0. Therefore, there exists the exact sequence 0 -> ? ( D G M ) V G P -» GP

^ VGP

-» 0.

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Chapter 6 Supergeometry

Let us note that, by analogy with associated smooth bundles, one can introduce associated superbundles and superconnections on these superbundles. In particular, every supervector bundle of fibre dimension (r, s) is a superbundle associated with GL(r\s; A)-principal superbundle [2l].

6.10

Appendix. Graded principal bundles

Graded principal bundles and connections on these bundles can be studied similarly to principal superbundles and principal superconnections, though the theory of graded principal bundles preceded that of principal superbundles [6; 258]. Therefore, we will touch on only a few elements of the graded bundle technique (see, e.g. [405] for a detailed exposition). Let (Z, 21) be a graded manifold of dimension (n, m). A useful object in the graded manifold theory, not mentioned above, is the finite dual 2l(Z)° of the algebra %l(Z) which consists of elements a of the dual 2l(Z) vanishing on an ideal of 2l(Z) of finite codimension. This is a graded commutative coalgebra with the comultiplication (A°(o))(/®/'):=a(//'),

/,/'€»(£),

and the counit c°(a):=a(l a ). In particular, 2l(Z)° includes the evaluation elements 5Z such that

Given an evaluation element Sz, elements u € 2l(Z)° are called primitive elements with respect to 5Z if they obey the relation (6.10.1)

A°(v)=u®8z+Sz®u. These elements are derivations of 2l(Z) at z, i.e., « ( / / ' ) = ( « / ) & / ' ) + (-i)M[/]

(Szf)(uf).

DEFINITION 6.10.1. A graded Lie group (G,Q) is defined as a graded manifold such that G is an ordinary Lie group, the algebra Q(G) is a graded Hopf algebra (A, e, S), and the algebra epimorphism a : Q(G) -> C°°(G) is a morphism of graded Hopf algebras. •

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Geometric and Algebraic Topological Methods in Quantum Mechanics

One can show that Q{G)° is also equipped with the structure of a Hopf algebra with the multiplication law od:=(o«i)oA,

a,beQ(G)°.

(6.10.2)

With respect to this multiplication, the evaluation elements Sg, Q € G, constitute a group 6g5g> = 5ggi isomorphic to G. Therefore, they are also called group-like elements. It is readily observed that the set of primitive elements of Q(G)° with respect to 5e, i.e., the tangent space Te(G,Q) is a Lie superalgebra with respect to the multiplication (6.10.2). It is called the Lie superalgebra g of the graded Lie group (G, Q).

One says that a graded Lie group (G, Q) acts on a graded manifold (Z, 21) on the right if there exists a morphism (¥>,$): (Z, 21) x ( G , S ) ^ ( Z , 21) such that the corresponding algebra morphism $ : 2l(Z) -+ 2l(Z) ® Q(G) defines a structure of a right C/(G)-comodule on 2l(Z), i.e., (Id A) o $ = ($ ® Id) o $,

(Id ® e) o $ = Id.

For a right action (tp, <E>) and for each element a G G(G)°, one can introduce the linear map $ a = (Id®a)o$:2l(Z)-^2l(Z).

(6.10.3)

In particular, if a is a primitive element with respect to 5e, then $ a G t)2l(Z). Let us consider a right action of (G, Q) on itself. If $ = A and a = 5g is a group-like element, then $ o (6.10.3) is a homogeneous graded algebra isomorphism of degree zero which corresponds to the right translation G —> Gg. If a Gfl,then $ a is a derivation of Q{G). Given a basis {UJ} for JJ, the derivations $ Ui constitute the global basis for dG(G), i.e., T)G{G) is a free left £/(G)-module. In particular, there is the decomposition g{G) = Q'{G) ®R G"(G), G'(G) - {/ G G(G) : $„(/) = 0, u G fl0}, G"(G) = {feG(G) : $ « ( / ) = 0, uGfli}. Since G'{G) = C°°{G), one finds that every graded Lie group (G,G) is the sheaf of sections of some trivial exterior bundle G x Q\ -* G [6; 59; 258].

Chapter 6 Supergeometry

425

Let us turn now to the notion of a graded principal bundle. A right action ((p, $) of (G, G) on (Z, 21) is called free if, for each z £ Z, the morphism $ z : 2l(Z) -> G(G) is such that the dual morphism $** : G(G)° -> 2l(Z)° is injective. A right action ( (Z, 21) such that the corresponding morphism 2l'(Z')° —> 2l(Z)° is an inclusion. A graded • submanifold is called closed if dim (Z1,21') < dim (Z, 21). Then we come to the following variant of the well-known theorem on the quotient of a graded manifold [6; 405]. THEOREM 6.10.2. A right action ( (Z/G, Ql/G) compatible with • the surjection Z —» Z/G.

In view of this Theorem, a graded principal bundle (P, 21) can be defined as a locally trivial submersion (P,B)->(P/G,a/0) with respect to the right regular free action of (G,Q) on (P, 21). In an equivalent way, one can say that a graded principal bundle is a graded manifold (P, 21) together with a free right action of a graded Lie group (G, Q) on (P, 21) such that the quotient (P/G, 2l/£) is a graded manifold and the natural surjection

(P,a)->(P/G,a/e) is a submersion. Obviously, P —> P/G is an ordinary G-principal bundle. A graded principal connection on a graded (G, S)-principal bundle (P, 21) —> (X, 8) can be introduced similarly to a superconnection on a

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Geometric and Algebraic Topological Methods in Quantum Mechanics

principal superbundle. This is defined as a (G, £)-invariant splitting of the sheaf 921, and is represented by a g-valued graded connection form on (P, 21) [405]. Remark 6.10.2. In an alternative way, one can define graded connections on a graded bundle (Z, 21) —> (X, 03) as sections T of the jet graded bundle J\Z/X)^{Z,%) of sections of (Z, 21) -> {X, 03) [6], which is also a graded manifold [381]. In the case of a (G, £)-principal graded bundle, these sections T are required D to be (G, C/)-equivariant.

6.11

Appendix. The Ne'eman-Quillen superconnection

In this Section, we consider the class of superconnections introduced by Y.Ne'eman in the physical literature [337; 338] and by D.Quillen in the mathematical literature [306; 362]. The fibre bundles that they consider belong neither to above studied graded manifolds and bundles nor superbundles. Ne'eman-Quillen superconnections have been applied to computing the Chern character in K-theory (see below), to non-commutative geometry [342], BRST formalism [275] and some particle unification models [337], Let X be an iV-dimensional smooth manifold and /\TX the exterior bundle. Let EQ, and E\ be two vector bundles over X of dimensions n and m, respectively, One constructs the vector bundle Q = /\TX®E = ATX®(E0®E1), (6.11.1) xx x called hereafter the body of an NQ-superbundle (see Definition 6.11.1 below) or simply an NQ-superbundle. The typical fibre of the NQ-superbundle Q is V = ARN®(B0®Bi),

(6.11.2)

where Bo and Bx are the typical fibres of the vector bundles Eo and E± respectively. This typical fibre can be provided with the structure of the superspace B"l m over the Grassmann algebra A — /\RN. This is the graded envelope of the graded vector space B = Bo © Bi, where Bo and Bi are regarded as its even and odd subspaces, respectively. The NQ-superbundle

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Chapter 6 Supergeometry

Q inherits this gradation since transition functions of EQ and E\ are mutually independent, while the transition functions of TX preserve the Zgradation of the Grassmann algebra AR^. Nevertheless, the NQ-superbundle (6.11.1) is not a supervector bundle over X since its transition functions are not A-module morphisms. Obviously, one can think of the NQ-superbundle Q as the tensor product of the vector bundle E and the characteristic vector bundle ATX of the simple graded manifold (X,OX), where Ox is the sheaf of exterior forms on X. The vector bundle Vg of graded vector fields and the vector bundle V% of graded one-forms (see Section 6.3) have a local structure of an NQ-superbundle (see local isomorphisms (6.3.9) and (6.3.18), respectively). Let us denote by Q(X) and Qx the space of global sections of the NQ-superbundle Q and the sheaf of its sections, respectively. Of course, Q(X) = Qx(X). The space Q(X) has the natural structure of a locally free C°°(X)-module, while Qx is the locally free sheaf of C^-modules of rank 2N(n + m). At the same time, bearing in mind that (X, Ox) is a graded local-ringed space, one can provide Q(X) = O{X)®E(X) with the structure of the graded locally free 0(X)-module, while Qx = CX°<E) ARN ®B = OX®B can be seen as the graded locally free sheaf of 0^-modules of rank (n + m). We will denote Q(X) and Qx endowed with the above mentioned structures by Q{X) and Qx, respectively. DEFINITION 6.11.1. The pair Q = (Q,Q(X)) (or the pair (Q,QX)) is called an NQ-superbundle. •

Given a trivialization domain U C X of the vector bundle E —» X, let {CA} and {CJ} be fibre bases for the vector bundles Eo and E\ over U, respectively. Then every element q of Q{X) reads q = qAcA + qlCi,

where qA, ql are local exterior forms on U. An element q is homogeneous, if its Grassmann degree is [?] = [iA\ = fe'] + 1.

[A = |
[
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Geometric and Algebraic Topological Methods in Quantum Mechanics

Given another trivialization domain U' C X of E, the corresponding transition functions read

Q'A = PUB,

qH = pW>

(6-H-3)

where p^, p- are local smooth functions on U D U'. We call the triple (U; qA, q1) together with the transition functions (6.11.3) a splitting domain of the NQ-superbundle Q. A connection on the the NQ-superbundle (Q,Q(X)) can be defined in accordance with Definition (1.3.2). It is easily seen that, in the case of the ring O(X), the derivation d1 (1.3.4), is exactly the exterior differential d. DEFINITION 6.11.2. A connection on the NQ-superbundle {Q,Q(X)), called an NQ-superconnection, is denned as a morphism V : Q —> Q which obeys the Leibniz rule

V(to) = (d0)g + (-l)M/V(g),

qeQ(X),

^O'(I).

(6.11.4)

• It should be emphasized that an NQ-superconnection is defined as a connection on C(X)-module Q(X), but not on the C°°(X)-module Q(X). Therefore, it is not an ordinary connection on the smooth vector bundle Q->X. The Leibniz rule (6.11.4) implies that an NQ-superconnection is an odd morphism. For instance, if E —> X is a trivial bundle, we have the trivial NQ-superconnection V = d. Let To and Fi be linear connections on the vector bundles Eo and Ei, respectively, and F o © Fi a linear connection on E —> X. Then the covariant differential V r (10.6.54) relative to the connection F is an NQ-superconnection Vr(q) = dxxA(dx-T)(q).

(6.11.5)

Given a splitting domain U, this superconnection reads

Vr(
(6.11.6)

where FOA^B, ^ixzj a r e local functions on U with the familiar transformation laws under the transition morphisms p^ and /?* of the vector bundles Eo and Ei, respectively. As it follows directly from the Leibniz rule (6.11.4), the NQsuperconnections on an NQ-superbundle constitute an affme space modelled over the C(X)0-module ~End(Q(X))i of odd degree endomorphisms

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Chapter 6 Supergeometry

ofQ(X),i.e., V' = V + L,

(6.11.7)

where L is an odd element of End(Q(X)). It is easy to see that the O (X)-module End(Q(X)) is a C°°(X)-module of sections of the vector bundle ATX®E®E ^ X. X

X

Given a splitting domain U of the NQ-superbundle Q, every element of End(Q(X)) is represented by a supermatrix function (or simply a supermatrix) (6.11.8)

whose entries are local exterior forms on U. The transformation law of this supermatrix under the transition morphisms (6.11.3) is

L' = pLr\ where p is the (n + m) x (n + m) matrix

?-(f°)

(6.U,,

whose entries are the transition functions p^ and pj. Due to the relation (6.11.7), any NQ-superconnection on a splitting domain of the NQ-superbundle Q can be written in the form V = d+

tf,

(6.11.10)

where r? is a local odd supermatrix (6.11.9), i.e., entries of $i, $4 are exterior forms of odd degree, while those of $2, i?3 are exterior forms of even degree. Obviously, the splitting (6.11.10) is not maintained under the transition morphisms (6.11.3), and we have the transformation law ti^p&p-i-dpp-1, where dp is the supermatrix whose entries are one-forms dp^ and dp. Similarly to the expression (1.8.26), the curvature of the NQsuperconnection V is the morphism fl=V2 :Q-+Q.

(6.11.11)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

For instance, the curvature of the trivial NQ-superconnection V = d is equal to zero. The curvature of the NQ-superconnection V r (6.11.6) is f \R\»ABdxx V

A dx» A qBcA 0

0 \ ±RXlSjdxxAdxi1Aq1ciJ' (6.11.12) where R\^,AB, R\^lj are the curvatures (10.6.63) of the linear connections To and Fi, respectively. Given the local splitting (6.11.10) of an NQ-superconnection V, its curvature (6.11.11) takes the local form B(a) W

'

=

R = d(-d) +

ti2,

(6.11.13)

and has the transformation law R' =~pRp~lunder the transition morphisms (6.11.3). It follows that the curvature of an NQ-superconnection is an even endomorphism of the O(X)-module Q(X). Remark 6.11.1. The notions of an NQ-superbundle and an NQsuperconnection are extended in a straightforward manner to the case of a complex vector bundle E —> X and the exterior algebra C ® O(X) of complex exterior forms on X. D Let us discuss briefly the application of NQ-superconnections to the computation of the Chern character [285; 306; 362]. Given an NQsuperbundle Q (6.11.1), let us consider an NQ-superconnection V = Vr + iI,

(6.11.14)

where V r is the superconnection (6.11.6), L is an odd element of End(Q(X)), and t is a real parameter. The curvature (6.11.11) of the connection (6.11.14) reads R = t2L2 + t[Vr,L] + (Vr)2, where ( V r ) 2 is the curvature (6.11.12) of the NQ-superconnection V r (6.11.6). Then we have cht = Str(exp(i?)) = J^ Str((V r + L)2k).

(6.11.15)

fc=0

This series converges since R is a section of a bundle of finite-dimensional algebras over X. It is readily observed that ch t=0 = c h ( £ 0 ) - c h ( £ i ) ,

(6.11.16)

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Chapter 6 Supergeometry

where ch(I?o), ch(Ei) are the Chern characters (10.10.39) of the complex vector bundles Eo and E\ (with accuracy to the customary factor i/2n). This is the Chern character of the difference element EQ Q E\ in the Ktheory of complex vector bundles on a manifold X [16; 237] (see the relation (10.11.2)). The key point is that the De Rham cohomology classes of the exterior forms cht (6.11.15) and chj=o (6.11.16) are the same. This fact issues from the following two assertions. 6.11.3. Given an NQ-superconnection V and an odd endomorphism T G End(Q(X)), one has

PROPOSITION

d(Str(T)) = Str([V,T]).

(6.11.17)

• Outline of proof. Since the relation (6.11.17) is local, one can assume that V is split as in (6.11.10) and T is a sum of supermatrices of the form 4>kTk where 4>k is are exterior forms, while Tk are constant supermatrices. Bearing in mind the relation (6.1.7), we have Str([d + 0, kTk) =

d(Sti(kTk)). QED

Let T = Rk, where R is the curvature of the NQ-superconnection V. Since Str([V,-R/c]) = 0, we obtain from the relation (6.11.17) that the exterior form Str(Rk) is closed. [362]. Given the curvature R of an NQPROPOSITION 6.11.4. superconnection V, the De Rham cohomology class of the exterior form Str(Rk) is independent of the choice of the NQ-superconnection V. • The coincidence of the De Rham cohomology classes of the forms cht (6.11.15) and cht=o (6.11.16) enables one to analyze the Chern character under the different choice of the supermatrix L and the parameter t in the expression (6.11.14).

Chapter 7

Deformation quantization

In a general setting, a deformation of an algebra A over a commutative ring K. is its Gerstenhaber extension to an algebra Ah over the ring K.[[h\] of formal power series in a real variable h such that Ah/hAh — A [167]. By deformation quantization is meant a deformation of a Poisson algebra of functions on a Poisson manifold where h is treated as a Plank constant [30]. One also considers a generalized deformation where a deformation parameter no longer commutes with elements of the original algebra [330; 332; 358]. 7.1

Gerstenhaber's deformation of algebras

Subsections: A. Formal deformation, 433; B. Deformation of associative algebras, 436; C. Relative deformation, 439; D. Commutative deformation; 441; E. Deformation of Lie algebras. 442. This Section summarizes the relevant material on deformations of algebraic structures, especially, associative algebras and Lie algebras [152; 167; 168]. A deformation parameter h throughout is real. A. Formal deformation Let A be a (not necessarily associative) algebra. Let us consider the set of formal power series ah = a + hai + h2a2 H

,

ai & A,

in a real variable h whose coefficients are elements of A. This set is customarily denoted by A[[/i]]. It is naturally an algebra, called the power series 433

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Geometric and Algebraic Topological Methods in Quantum Mechanics

algebra, with respect to the formal sum and product of power series. For instance, let K, be a commutative ring. Then /C[[/i]] is also a commutative ring. Without a loss of generality, we further assume that any algebra A under consideration is an algebra over some commutative ring K.. Then the set A[[h]] is a /C[[/i]]-module and, in particular, the power series algebra A[[h\] is /C[[/i]]-algebra. It is also a /C-algebra, and there is a /C-algebra monomorphism A —> ^4[[/i]]. The power series algebra A[[/i]] may be provided with a different algebraic structure. The key point is that a multiplication ah{a,b) : A x A —> A in any /C[[/i]]-algebra A is automatically represented by a power series oo

ah = a+ J~^hrar, r=l

where a are /C[[/i]]-bilinear maps Ay- A—* A. Given a /C-algebra A over a commutative ring /C, let A[[h]] be the power series module over the power series ring /C[[/i]]. Let (7.1.1)

B:AxA^A

be a /C-bilinear map. It is naturally extended to a /C[[/i]]-bilinear map B : A[[h}} x A[[h]] -> A[[h}\.

(7.1.2)

By a formal deformation of B (7.1.1) is meant a K.[[h]]-bilinear map Bh : A[[h]] x A[[h]] -> A[[h}}, oo

hrBr(a, b),

Bh(a, b) = B(a, b) + £

(7.1.3)

r=l

where Bi are /C-bilinear maps A x A —• A. In particular, if B(a,b) = ab (7.1.1) is a multiplication in A, then its formal deformation oo

a * b = ah(a, b) = ab + ^

hrar(a, b)

(7-1-4)

r=l

(7.1.3) is a multiplication in the /C[[/i]]-module A[[h}] which makes it into a /C[[/i]]-algebra. This algebra is called a formal deformation of A. This term is also abbreviated with the power series (7.1.4).

Chapter 7 Deformation Quantization

435

For instance, the power series algebra J4[[/I]] with the multiplication law a * b = ab is a formal deformation of A called the null deformation. Let us note that a /C[[/i]]-module P is isomorphic to a power series module Q[[h]} for some /C-module Q if an only if Q = P/hP. Therefore, given a /C-algebra A, a /C[[/i]]-algebra A is a formal deformation of A if and

only if A/hA = A. Two formal deformations Ah and A'h of a /C-algebra A are called equivalent if they are isomorphic £[[/i]]-algebras, i.e., there exists an isomorphism of /C[[/i]]-modules fa : A[[h}\ -> A[[h]\

(7.1.5)

such that the relation

fa(a'h(a, b)) = ah(fa(a), h{b))

(7.1.6)

holds. For the sake of brevity, let us write a'h — ah o fa. Any /C[[/i]]-linear morphism (7.1.5) is necessarily a formal power series

fa = + h<j)1 + h2(l>2 + --- ,

(7.1.7)

whose coefficients are /C-linear maps j : A —> A. Substituting this power series into the relation (7.1.6), one easily obtains that (7.1.8)

4> = IdA, a b

0i(a6) + ai(a,&) = ai(a,b) + i( ) + a
(7.1.9)

A formal deformation Ah of A is said to be trivial if it is equivalent to the null deformation A[[h]] of A. Remark 7.1.1. A /C[[/i]]-algebra automorphism fa of the power series algebra A[[h}} obeys the relation

fa(ab) =

fa(a)fa(b),

a,beA.

(7.1.10)

In particular, the equality (7.1.9) takes the form 0 i {ab) = cj>i (a)b + a<j>i (6),

i.e., 0i is a derivation of the algebra A into itself. Furthermore, if K. is a Q-ring, any formal automorphism fa of A[[h}] is brought into the form

fa =

e^.e^;

where 5it i — 1 , . . . , are derivations of A.

•

436

Geometric and Algebraic Topological Methods in Quantum Mechanics

Let S —> A be an algebra morphism. A deformation Ah of A is said to be the S-relative deformation if as = as,

sa = sa

(7.1.11)

for all o G A and s G S (see Subsection C below). For instance, let A be a unital algebra and S = {1}. One can show that any formal deformation of A has a unit (see Theorem 7.1.6 below). Given an algebra A, one usually consider its formal deformations Ah possessing some properties of A. Namely, let 8. be a full subcategory of the category of (arbitrary) algebras, and let A £ Ob£. One considers the formal deformations of A in the category .8, but such a deformation need not exists. We further restrict our consideration to deformations of associative algebras and Lie algebra. B. Deformation of associative algebras Let A be a unital /C-algebra over a commutative ring fC. We aim to study its formal deformations which are associative algebras. They are called associative deformations or, simply, deformations. It is readily observed that the null deformation of an associative algebra is associative. An algebra which admits only trivial deformations is called rigid. One can show the following. • Any deformation of A is equivalent to that where the unit element coincides with the unit element of A. • Invertible elements of A remain invertible in any deformation. In order to be associative, a formal deformation Ah of A must satisfy the associativity condition oo

(ai * a2) * a3 - ai * (a2 * a3) = ^ hkDk(ai,a2, a3) = 0, (7.1.12) fc=i

(7.1.13)

Dr{ai,a2,a3)= ^2

[ar(aa(ai,a2),a3) - a r (ai,a 5 (a2,a 3 ))],

s+r=k,s,r>0

i.e., A ( a i I O 2 , a 3 ) = 0)

k = l,...,

a* £ A.

(7.1.14)

This condition is phrased in terms of the Hochschild cohomology as follows. Let Ek{ai, a 2 , a3) denote the sum of the terms with indices 1 < s,r < k in the right-hand side of the expression (7.1.13). It is easily observed that

437

Chapter 1 Deformation Quantization

Ei = 0, while each Ek, k = 2 , . . . , depends on the terms a,, i = 1 , . . . ,fc— 1. Then the condition (7.1.14) takes the form Dk(ai,a2,a3) = Ek(ai,a2,a3) - (dak)(ai,a2,a3)

= 0,

(7.1.15)

where (dak)(ai,a2,a3)

= aiak(a2,a3) - ak(aia2,a3) +

(7.1.16)

afc(oi,O2a3) - afc(ai, 02)03 is the Hochschild coboundary operator (1.5.40) of the Hochschild complex B(A, A) (see Section 1.5C). Thus, one can think of the terms ak of the power series (7.1.4) as being two-cochains of the above mentioned Since E\ = 0, a glance at the condition Hochschild complex B(A,A). (7.1.15) shows that a\ is a two-cocycle of the Hochschild complex. Then we can obtain an associative deformation in the framework of the following recurrence procedure. Let us assume that a,, i < k, are two-cochains such that Di = 0 for all i < k. One can show that dEk+i = 0, i.e., Ek+\ is a three-cocycle whose cohomology class depends only on that of Q!j, i = 1, ...,fc. If this cocycle is a coboundary, i.e., it belongs to the zero element of the Hochschild cohomology group H3(A, A), then a desired cochain ak+x can be found. For instance, let Ah be a deformation of A such that oti = 0,

i = 1 , . . . ,k - 1,

k > 1.

Then Ek = 0, and the condition (7.1.15) shows that ak is a two-cocycle. As was mentioned above, this is also the case of fc = 1. The Hochschild cohomology class [ak] of ak (or, if there is no danger of confusion, ak itself) is called the infinitesimal of a deformation. Let us assume that ak = d<j> is a coboundary. It is easily verified that a'h = aho (U A-hk4>)

(7.1.17)

is an equivalent deformation such that a< = 0, i = 1 , . . . , fc. If H2(A, A) = 0 and ah is a deformation, one can use the equivalences (7.1.17) in order to remove all the terms cti follows. THEOREM 7.1.1. Equivalent deformations of A possess the same infinitesimal [Qi] e H2{A, A). •

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Geometric and Algebraic Topological Methods in Quantum Mechanics

7.1.2. If A is a unital /C-algebra with H2{A, A) = 0, then any deformation of A is trivial, i.e., A is rigid. • THEOREM

An algebra A is called absolutely rigid if H2(A, A) = 0. Let H3(A, A) = 0. For any two-cocycle a, there exists a deformation of A such that ot\ = a. Indeed, ai = a defines E2 by the formulae (7.1.13) and (7.1.15). This three-cocycle is a coboundary. Therefore, there exists a twocochain a2 such that the term D2 (7.1.15) vanishes. The two-cochains «i and a2 define the three-cocycle E3 by the formulae (7.1.13) and (7.1.15). It is also a coboundary. Consequently, there exists a two-cochain a$ such that the term £>3 (7.1.15) vanishes, and so on. Thus, elements of the Hochschild cohomology group H3 (A, A) provide the obstruction to a Hochschild twococycle be the infinitesimal of a deformation. Example 7.1.2. Let K, be a Q-ring. Let u and v be derivations of a /C-ring A. They are Hochschild one-cocycles. Their cup-product u -^ v (1.5.41) is a two-cocycle. This two-cocycle need not be the infinitesimal of a deformation of A, unless u and v mutually commute. If the derivations u and v commute, they define a deformation of A given by the formula ab = exp{hu^-v)(a,b)

°° hr

:=ab+ ] T — ur{a)vr{b),

a,b e A. (7.1.18)

r=l

It should be emphasized that the notation exp(/m •—' v) is somewhat misleading. It may happen that u —- v = v! •—' v' for another pair u' and v' of derivations, while exp(hu ^ v) ^ exp(hu' --- v'). D Since deformations of a /C-ring A are characterized by its Hochschild cohomology H(A, A), there is the following relation between deformations of Morita equivalent rings [167]. PROPOSITION 7.1.3. If/C-rings A and A' are Morita equivalent, then there is an isomorphism of their Hochschild cohomology groups

H(A, A) -> H(A', A'), which preserves the cup-product. The isomorphism (7.1.19) leads to the following.

(7.1.19) •

439

Chapter 7 Deformation Quantization

PROPOSITION 7.1.4. If/C-rings A and A' are Morita equivalent, then there is a bijection between their sets of equivalence classes of deformations. •

C. Relative deformation In order to describe the above mentioned relative deformations, we start with relative Hochschild cohomology, which is a straightforward generalization of Hochschild cohomology in Section 1.5C. Let /C be a commutative ring, A a AC-ring and Q an .4-bimodule, which a commutative /C-bimodule. Let B(A, Q) be the Hochschild complex (1.5.36). Given a JC-subring S of A, the S-relative Hochschild complex is the subcomplex

(7.1.20)

B{A,S,Q)->B(A,Q)

of the Hochschild complex B(A, Q) where: • B°(A, S, Q) consists of elements q € Q = B°(A, Q) such that sq = qs for all s G <S, • Bk>0(A, 5 , Q) consists of cochains fk € Bk(A, Q) obeying the conditions fk{sau..., k

f (ai,...

ak) = sfk(au ,aiS,ai+i,...

k

.. .,ak), k

,ak) = f (a\,..

s€S, .,a,i,sai+i,...

,ak),

k

f (ai,...,aks)

=f (ai,...,ak)s.

Cohomology H(A,S,Q) Hochschild cohomology.

of B(A,S,Q)

Example 7.1.3. If S = /C, clearly Bk>0(A,

are called the

JC, Q) = Bk(A, Q).

S-relative

•

The inclusion (7.1.20) is a cochain morphism which yields a homomorphism of cohomology groups

H(A,S,Q)-^H(A,Q).

(7.1.21)

It however need not be a monomorphism, but we always have

H°(A,S,Q) = H°(A,Q). THEOREM 7.1.5. If the algebra <S is separable (see its definition below), the morphism (7.1.21) is an isomorphism [167]. •

440

Geometric and Algebraic Topological Methods in Quantum Mechanics

Generalizing the notion of an allowable morphism in Section 1.5C, we say that, given /C-ring A and its subring B, a morphism 3> : Q —> P between A-modules is B-allowable if there is a B-module morphism A : P —> Q such that $ o A o $ = $.

(7.1.22)

For an epimorphism $ , the condition (7.1.22) reduces to $ o A = $. Likewise, for a monomorphism $ , it becomes A o $ = $ . In general, $ is Ballowable if and only if, in some (and, hence, every) factorization $ through an epimorphism and a monomorphism, both these maps are B-allowable. It should be emphasized, that a composition of allowable maps need not be allowable. A /C-ring A is called separable (in the terminology of [167]) if any /C-allowable map of modules over A is A-allowable. A finite product of separable algebras is separable. Moreover, if /C is a field, a separable /C-ring is necessarily finite-dimensional. The 5-relative Hochschild cohomology coincides with cohomology of the normalized subcomplex C(A, S, Q) of the relative Hochschild complex , a;) vanish if any a* € <S. In particB(A,S, Q) whose elements fk(a,i,... ular, any normalized 5-relative one-cocycle f1 is a Q-valued derivation of A regarded as a <S-ring, i.e., fl(a) = 0 for all a E S. The relative Hochschild complex B(A,S,Q) is functorial in Q, i.e., every ,4-bimodule morphism Q —> P yields a cochain /C-module morphism B(A,S,Q)-B'(A,S,P) and, consequently, a homomorphism of 5-relative Hochschild cohomology groups H(A,S,Q)^H(A,S,P). Each short exact sequence 0 -> Q' —> Q —-> Q" -> 0 of <S-allowable morphisms yields the long exact sequence •••Hk(A,S,Q') k+1

H

(A,S,Q')

—+Hk(A,S,Q)

-^Hk{A,S,Q")

—>

—....

Turn now to the <S-relative associative deformations of a /C-ring A [167]. THEOREM

7.1.6. Let S be a /C-subring of A.

441

Chapter 7 Deformation Quantization

(i) If the map (7.1.21) is an isomorphism for k = 2 and k = 3, then any deformation of A is equivalent to an S-relative deformation. (ii) Let Ah and A'h be equivalent 5-relative deformations of A. If the map (7.1.21) is an isomorphism for k = 1 and k = 2, then there is an S-relative equivalence between Ah and A'h, i.e., 0(a) — a for all a G S. • COROLLARY 7.1.7. By virtue of Theorems 7.1.5 and 7.1.6, if a subring S of a /C-ring A is separable, any (associative) deformation of A is 5-relative, • and any two equivalent deformations of A are <S-relatively equivalent.

D. Commutative deformation Let A be a commutative /C-ring. Its deformations need not be commutative, but its null deformation is certainly commutative. Commutative deformations of a commutative ring are characterized by its Harrison cohomology. Let Q be an j4-bimodule. The Harrison cohomology Har(yl, Q) is defined as a cohomology of a certain subcomplex S (A, Q) of the Hochschild complex B(A, Q). Unless K, is a Q-algebra, there are problems with the Harrison cohomology theory in degrees greater than 3. Therefore, we here restrict our consideration to the terms S-3(A,Q) of the Harrison complex. They are defined as follows: . S i ( A Q ) = B ' ( i , Q ) , « = O,l; • S2(A, Q) consists of the symmetric Hochschild two-cochains f2 € 2 B (A,Q), i.e., / 2 (ai,a 2 ) - / 2 ( o 2 , a i ) = 0,

a\,a2 G A;

• S3(A, Q) consists of the Hochschild three-cochains / 3 G B2(A, Q) such that / 3 (a x , o 2) o3) + / 3 (a 3 , a l5 o2) - / 3 (a 2 , oi, o3) = 0, / 3 (ai,a 2 ,a 3 ) + / 3 (a2,a3,ai) - / 3 ( a i , a 3 , a 2 ) = 0

a i , a 2 , a 3 € A;

As a consequence, we have Har°(A Q) = H°(A, Q) 3 Q,

Bar1 (A, Q) = Hl{A, Q).

A glance at the formula (1.5.39) shows that, if A is commutative and Q is an A-bimodule, any Hochschild two-coboundary is symmetric. It follows

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Geometric and Algebraic Topological Methods in Quantum Mechanics

that the Harrison cohomology group Ha,i2(A,Q) is a submodule of the Hochschildonetf2(A,Q). With the Harrison cohomology groups Uai2(A, A) and Har3(A, A), the commutative deformations of a commutative /C-ring A are characterized as follows. • The infinitesimal of a commutative deformation of A is a Harrison two-co cycle. • Elements of the Harrison cohomology group Hai3(A, A) provide the obstruction to a Harrison two-cocycle be the infinitesimal of a deformation. • If Hai3(A,A) = 0, every Harrison two-cocycle is the infinitesimal of a deformation of A and, moreover, this deformation is equivalent to a commutative one. E. Deformation of Lie algebras Let A be a Lie algebra over a commutative ring /C. Let us study its formal deformations which are Lie algebras over the power series ring JC[[/i]]. They are called the Lie deformations. It is readily observed that the null deformation of a Lie algebra is a Lie deformation. In order to be a Lie algebra, a formal deformation Ah (7.1.4) of A must be skew-symmetric ab=—ba,

ar(a,b) = —ar(b,a),

a,b € A,

and must satisfy the Jacobi identity (ai * o2) * a3 + (a2 * a3) * a\ + (a3 * ai) * a2 = oo

J2hkTk(ai,a2,a3)

(7.1.23)

= 0,

fc=i

T fc (ai,a 2 ,a 3 ) =

^

[ar(ai,as(a2,a3)) +

(7.1.24)

r+s=fc, s,r>0

ar(a2, as(a3, ax)) + ar(a3, aa{a\, a2))], i.e., Tfc(a1)o2>a3) = 0> fc = l , . . . ,

a* e A.

(7.1.25)

This condition is phrased in terms of the Chevalley-Eilenberg cohomology just as the associativity condition (7.1.14) has been done in terms of the Hochschild cohomology.

443

Chapter 7 Deformation Quantization

Let J/c(ai, 0,2,0,3) denote the sum of the terms with indices 1 < s,r < k in the right-hand side of the expression (7.1.24). It is easily observed that Jj = 0, while each Jk, k = 2,..., depend on the terms on, % = 1,..., k — 1. Then the condition (7.1.25) takes the form Tfc(ai,a2,a3) = Jk(ai,a2,a3)

- (5ak)(ai,a2,a3)

= 0,

(7.1.26)

where (5ak)(ai, 0,2,0,3) = aia fc (a2,a 3 ) + a2ak(a3,ai) + a3ak(ai,a2) ak{a\, a2a3) + ak{a2, a 3 ai) + ak[a3, aia2)

+

is the Chevalley-Eilenberg coboundary operator (1.5.47) of the ChevalleyEilenberg complex C* [A] of A-valued cochains on A in item (i) of Section 1.5D. Thus, one can think of the terms ak of the power series (7.1.4) as being two-cochains of the Chevalley-Eilenberg complex C[A]. Since Tx = 0, the condition (7.1.26) shows that a\ is a two-cocycle of the above mentioned Chevalley-Eilenberg complex. Then one can obtain a Lie deformation of A in the framework of the following recurrence procedure. Let us assume that oci, i < k, are two-cochains such that % = 0 for all i < k. One can show that 5Jk+i = 0, i.e., Jk+i is a three-cocycle whose cohomology class depends only on that of ai} i = 1,..., k. If this cocycle is a coboundary, i.e., it belongs to the zero element of the Chevalley-Eilenberg cohomology group H3(A), then a desired cochain afc+i can be found. THEOREM 7.1.8. If H2(A) = 0, then any Lie deformation of A is trivial.

•

Example 7.1 A. Let K be a Q-ring and A a semisimple finite-dimensional Lie algebra. Then Hl{A) = H2{A) = 0. • Let H3(A) = 0. For any two-cocycle a, there exists a deformation of A such that ai = a. Indeed, ai = a defines J2 by the formulae (7.1.24) and (7.1.26). This three-cocycle is a coboundary. Therefore, there exists a twocochain a2 such that the term T2 (7.1.26) vanishes. The two-cochains a^ and 0.2 define the three-cocycle JS3 by the formulae (7.1.24) and (7.1.26). It is also a coboundary. Consequently, there exists a two-cochain a3 such that the term T3 (7.1.26) vanishes, and so on. Thus, elements of the ChevalleyEilenberg cohomology group H3(A) provide the obstruction to a ChevalleyEilenberg two-cocycle be the infinitesimal of a Lie deformation.

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Geometric and Algebraic Topological Methods in Quantum Mechanics

Let us note that, if A is an associative algebra provided with the Lie algebra structure with respect to the bracket [a,b] = ab — ba,

a,b € A,

any associative deformation a b of A yields the Lie deformation [a, 6]* =ab-ba

(7.1.27)

of the Lie algebra A. 7.2

Star-product

Let (Z, {,} be a Poisson manifold. By a star-product on Z is meant an associative deformation oo

/ * / ' = / / ' + $>rar(/,/')

(7.2.1)

r=l

of the R-ring C°°(Z) of smooth real functions on Z such that <i(/,/') -<(/',/) = 2{/,/'},

/,/'eC°°(Z).

(7.2.2)

One usually requires that a star-product (7.2.1) - (7.2.2) is a differential are bidifferential operators of finite deformation of C°°{Z), i.e., ar(f,f) order. Moreover, up to the equivalence, one can always choose * such that / * l = l * / = /,

feC°°(Z).

(7.2.3)

Given a star-product (7.2.1) - (7.2.3), the commutator

[/, f% = {2hT\f * / ' - / ' * / ) = {/, /'} + 1

(7.2.4)

oo

- J> r (a r+1 (/,/')-a r+ i(/',/)) r=l

provides a Lie deformation of the Poisson bracket {,} on Z. This deformation is treated as deformation quantization. Remark 7.2.1. Due to formulae like (7.2.1), it is sometimes convenient to take the Laurent series £[/i~\/i]] in h, i.e., a polynomial in h~l and a formal series in h. O In order to study star-products, we start with associative deformations of the R-ring C°°(Z) of smooth real functions on a smooth manifold Z.

Chapter 7 Deformation Quantization

445

Since one usually ignores the algebraic case which is too general to be handled, there are two cohomology and deformation theories of interest, namely, the continuous and differential ones. Because C°°(Z) is a Prechet ring (see Remark 1.8.4), by its continuous deformation is meant the deformation (7.2.1) where the bilinear maps ar are continuous in each argument. Continuous cochains constitute a subcomplex B(C°°{Z),COO{Z)) of the Hochschild complex B(C°°{Z), C°°(Z)) whose cohomology H{C°°{Z), C°°{Z)) is called the continuous Hochschild cohomology. Another subcomplex BA{CCO{Z),CCO{Z)) of the Hochschild complex B(C°°(Z), C°°(Z)) consists of cochains which are multidifferential operators (i.e., differential operators in each argument) of finite order. Since every differential operator is continuous with respect to the Prechet topology of C°°(Z), the complex B^C00 (Z), C°° (Z)) is also a subcomplex of the complex B(C°°(Z),CO°(Z)) of continuous Hochschild cochains. Cohomology HZ(C°°{Z),C°°{Z)) of the complex Bd(Cco{Z),C°°(Z)) is called the differential Hochschild cohomology. A deformation (7.2.1) of C°°(Z) is said to be differential if all cochains ar are bidifferential operators of finite order. LEMMA 7.2.1. Any multivector field i? e %{Z) (10.6.16) of degree r defines a differential Hochschild r-cocycle 0(/i, • • • Jr) := W i A • • • Ad/ r € Brd(C°°(Z),C°°(Z)).

(7.2.5)

• Thus, there is the inclusion %{Z) c Bd(C°°(Z), C°°(Z)).

(7.2.6)

Moreover, any differential Hochschild cocycle is cohomologous to some multivector field (7.2.5) as follows [221; 433]. THEOREM

7.2.2. The inclusion (7.2.6) induces C°°(Z)-module isomor-

phisms Hrd(C°°(Z), C°°(Z)) = %(Z).

(7.2.7)

• This result is also extended to continuous Hochschild cohomology [331; 352; 357].

446

Geometric and Algebraic Topological Methods in Quantum Mechanics

THEOREM 7.2.3. Continuous and differential Hochschild cohomology groups are isomorphic

Hrc(C°°(Z),C°°{Z)) = H^(COO(Z),COO{Z)) =%{Z).

(7.2.8)

• As a consequence, one gets the following theorem [331; 357]. THEOREM 7.2.4. Every continuous deformation of C°°(Z) is equivalent to a differential one. A continuous equivalence between two differential deformations of C°°(Z) is differential (i.e., the maps fa (7.1.7) are linear differential operators). • In view of Theorem 7.2.4, by deformations of the ring C°°(Z) of smooth functions are further meant its differential deformations. Let us turn to a star-product (7.2.1) - (7.2.3) on a Poisson manifold (Z, {, } defined by a differential associative deformation of the M-ring C°°(Z). It is a Lie deformation of the Poisson algebra C°°(Z). Of course, an arbitrary deformation (7.2.1) need not be a star-product. The key point is that, in contradistinction with the Hochschild cohomology, the needed Chevalley-Eilenberg cohomology is small [433]. Remark 7.2.2. By virtue of Theorem 7.2.2, any Hochschild two-cocycle is cohomologous to its skew-symmetric part. Therefore, we can restrict our consideration to deformations of C°°(Z) whose infinitesimal is the Poisson bracket «i(/, /') = {/, / ' } .

/, / ' e C°°(Z),

(7.2.9)

in order to obtain a star-product. The fact that a (regular) Poisson bracket is a Hochschild two-cocycle is easily justified when it is written with respect to local Darboux coordinates (pi,ql) and, thus, is a sum of cup-products dl •-' di of mutually commutative vector fields dl and di (see Example 7.1.2). • Let us consider star-products on a symplectic manifold (Z,Q). The following assertions provide their comprehensive description. THEOREM 7.2.5. Any symplectic manifold admits a star-product [125], e.g., Fedosov's one [149]. •

Chapter 7 Deformation Quantization

447

THEOREM 7.2.6. Any star-product on a symplectic manifold is equivalent to some Fedosov star-product [34; 446], described in next Section. • 7.2.7. The equivalence classes of star-products on a symplectic manifold constitute an affine space modelled on the linear space //2(Z)[[/i]] of power series in h whose coefficients are elements of the de Rham coho• mology group H2(Z) of Z [123; 206; 340]. THEOREM

Namely, given two different star-products * and ', one associate to them a unique Cech cohomology class (',) Gff2(Z;R)[[/i]],

(7.2.10)

called Deligne's relative class. It is defined as follows [123; 206]. 7.2.8. Star-products and ' are equivalent if they are equivalent as associative deformations, i.e., LEMMA

{fg) = H'9,

f,g&C°°(Z),

oo

(j) = Id + ] T hrT,

(7.2.11)

(7.2.12)

r=l

where <j>r are differential operators.

•

One calls 4> (7.2.12) the formal differential operator. PROPOSITION 7.2.9. Let (Z,Cl) be a symplectic manifold, and let us suppose that the Cech cohomology group H2(Z;1BL) is trivial. Then any two star-products on Z are equivalent. D PROPOSITION 7.2.10. Let be a star-product on (Z, Q), and let us suppose that HX(Z;R) = 0. Then any self-equivalence (7.2.12) of is inner, i.e.,

4> = exp{ad,7>, for some 7 G C°°{Z)[[h}}, where ad, 7 (/) •=\9,f\.=9f-f9, is the star adjoint representation.

/ G C°°(Z)[[h]], •

Let {Ui} be a locally finite open cover of Z by Darboux coordinate charts such that Ui and all their non-empty intersections are contractible.

448

Geometric and Algebraic Topological Methods in Quantum Mechanics

Let us denote d = C°°(Ui). A star-product * in C°°(Z)[[h]] certainly yields a star-product * in Ci on a symplectic manifold {Ui, £1). Let * and ' be two star-products on Z. By virtue of Proposition 7.2.9, their restrictions to Ui are equivalent star products, i.e., there exists the formal differential operator 4>i : Ci[[h]] -» d[[h]]

(7.2.12) such that & ( / s ) = &/&
f,geCi{[h}}.

On Ui fl Uj, we accordingly have a self-equivalence (pj1 o » of in Cjj[[/i]]. By virtue of Proposition 7.2.10, this self-equivalence is inner, i.e., there exists an element fji = — 7y € ^[[/i]] such that fij1 ° 4>i — exp{ad»7jj}. On Ui fl Uj n C/fc, the composition ad7fcji = ad7jfc o a d , 7 ^ o ad7ji is the identity morphism of Cijjt[[/i]] and, consequently, is represented by an element jkji in the center R[[h}] of Ctjfc[[/i]]. The standard arguments show that the set of the elements jkji define a Cech two-cocycle with values in K[[/i]]. Its cohomology class [jkji] G H2(Z;R) is desired Deligne's relative class (7.2.10). PROPOSITION 7.2.11. If , ' and " are three star-products on (Z,Q,), then

D The moduli space of equivalent star-products on a symplectic manifold (Z, fi) is usually identified with

±{a} + H\z)[[h}}. Let us point out to the isomorphism H2(C°°(Z)[[h]],C°°(Z)[[h\]) = Z\Z)

+ H\Z)[[h\],

where Z2(Z) is the space of closed two-forms on Z.

(7.2.13)

Chapter 7 Deformation Quantization

449

Remark 7.2.3. One also introduces the notion of Morita equivalent star-products [77]. Star-products * and ' are said to be Morita equivalent if the associative algebras (C°°{Z)[[h]],) and (C°°{Z)[[h]},') are Morita equivalent. Two star products and ' on a symplectic manifold (Z,Q) are proved to be Morita equivalent if and only if there exists a symplectomorphism ip such that Deligne's relative class t(,ip(')) is integral. It provides an isomorphism []~t(,r

(')) + []

of the Hochschild cohomology space (7.2.13).

D

Remark 7.2.4. Given a symplectic manifold (Z, fi), one also considers a star-product * of smooth complex functions on Z which obeys the additional condition 7^9 = 97,

/,S€C°°(Z),

where / —> / denotes the pointwise complex conjugate. Such a star-product exists. It is utilized in order to define the GNS representation of deformation quantization [48; 434]. • Remark 7.2.5. Let (Z,Q) be a 2m-dimensional symplectic manifold. A star-product on Z is said to be closed if / ( / * < 7 - S * / ) A O = 0. z Closed star-products exist on any symplectic manifold [347], and any starproduct is equivalent to the closed one. For instance, the Moyal product on R 2m is closed. Cyclic cohomology replaces the Hochschild one for closed star products. Obstructions to the existence and equivalence of closed star• products live in the third and second cyclic cohomology spaces [106].

Remark 7.2.6. Let * be a star-product on a 2m-dimensional symplectic manifold (Z,fi). Given the ideal Cf{Z) c C°°(Z) of compactly supported smooth real functions on Z, by a trace of * is meant an R[[/i]]-linear map

450

Geometric and Algebraic Topological Methods in Quantum Mechanics

satisfying the relation

T(fg) = T(gf),

f,geC™(Z)[(h}}.

Any star-product on Z has a trace which is unique up to multiplication by an element of E[/i - 1 , h]} [150; 207; 340]. It takes the form f

AO

r(/)= J fp-f^, z

PeC°°{Z)[h-\h)\.

a Let us turn to star-products on Poisson manifolds. A star-product on an arbitrary (smooth, not necessarily regular) Poisson manifold exists. Its general construction was first suggested in [252; 256]. If (Z, {,} is a regular Poisson manifold, there exists a tangential starproduct on Z [150; 302]. It is denned as follows. Let S?{Z) be the center of the Poisson algebra C°°(Z). Let us consider an <S^(Z)-relative associative deformation of the R-ring C°°{Z). A star-product on Z and a Lie deformation of C°°(Z)) which come from an <Sj-(2")-relative deformation are called tangential because Sp{Z) consists of functions constant on leaves of the characteristic foliation of (Z, {,})• A tangential star-product can be introduced either as the tangential version of Vey's work [281; 302] or the straightforward generalization of Fedosov's deformation quantization of symplectic manifolds to symplectic foliations.

7.3

Fedosov's deformation quantization

The Moyal product on Z = R 2 m was the first example of a star-product [211; 236]. Let R 2 m be provided with the coordinates (g\Pi) and the canonical symplectic form fi = dpi Adq%.

Let us consider a differential associative deformation of C°°(R 2 m ) whose infintesimal is the Poisson bracket (7.2.2). This infinitesimal is a sum of cup-products of mutually commutative vector fields d% and <9. Then, generalizing Example 7.1.2, one can show that such a deformation exists and

451

Chapter 7 Deformation Quantization

it is given by the expression

//'=exp[^{/,/'}]=/exp[^(?a i -a i ?)]/'= i r D - 1 ) 7 " ^ • • • d^dir+1

£

fc=0 ' r=0

• • • dik^dh

(7.3.1) • • • d i r d i ^ • • • diJ')-

This is a star-product, called the Moyal product. Since the de Rham cohomology group H2{R2m) of R 2m is trivial, all star-products on (K 2m ,fi) are equivalent to the Moyal one (7.3.1). This star-product defines the corresponding Lie deformation (7.2.4). Fedosov's deformation quantization [149] generalizes the construction of the Moyal product (7.3.1) to an arbitrary symplectic manifold as follows. Let Z be a 2m-dimensional symplectic manifold coordinated by (zA) and provided with the symplectic form n = -£lapdza

A dz13.

A formal Weyl algebra Az over the tangent space TZZ is the unital associative algebra whose elements are the formal sums a(y,h)=

J2 2fc+r>0

hkak,a1...lyai---yar,

(7-3.2)

where yM = iM are holonomic coordinates on the tangent space TZZ. The algebra Az is provided with the Weyl product aoa' = exp \^na" (z)-^-~^

/c=0

v

a(y, h)a'(y', h)\v=y. =

(7.3.3)

'

a

where U P is the inverse to the matrix Q,a0- Of course, this definition is independent of a coordinate system. The disjoint union of the Weyl algebras Az, z £ Z, is the Weyl algebra bundle A —> Z whose sections read a{z,y,h)=

Yl 2fc+r>0

hka

k,ai...ar(z)yai---yar,

(7.3.4)

where afclQl...Qr(^) are sections of the tensor bundles VTZ. The set A of sections (7.3.4) is also an associative algebra with respect to the fibrewise

452

Geometric and Algebraic Topological Methods in Quantum Mechanics

multiplication (7.3.3). Its unit element is a(z,y,h) = 1. The center Z of the algebra A consists of the elements oo

a=^2hkak(z),

(7.3.5)

fc=o

independent of coordinates yx. There is a filtration A D Ai D • • • in the algebra A with respect to the total degree 2k + r of the terms of the series (7.3.4). Let us consider the tensor product A® O(Z) whose elements are Avalued exterior forms on the manifold Z: a(z,y,dz,h) =Y^hkak,al...ar,fi1...i3s{z)yai

• • -y^dz01 A- • • Adz0-, (7.3.6)

called simply .4-forms. Their multiplication is defined as the exterior product A of exterior forms and the Weyl product o (7.3.3) of polynomials in ya. Let the symbol o also stand for this multiplication. With this multiplication law, the algebra A®O(Z) has the structure of a graded algebra over the graded commutative ring O(Z). The corresponding bracket of two «4-forms a, a' (7.3.6) is defined as [a,a'}=aoa'-

( - l ) l a " ° V oa,

(7.3.7)

where \a\ e A® OM(Z), \a'\ € A® O^a'\Z). One says that an element a belongs to the center of the algebra A ® O(Z) if its bracket (7.3.7) with any element of this algebra vanishes. This center is Z ® O(Z). There are two projections of an ,4-form a(z,y,dz, h) (7.3.6) to this center. These are a0 =o(0,O, dz,h),

aOo = a(z,0,0,h).

We also have the following two operators acting on A-forms: 6 : A ® OS(Z) -> Ar-i ® OS+1(Z), a, 5a — dzx A - — dya 5:AT® 03{Z) -» A-+1 ®

O'~\Z),

These operators possess the following properties: • the operator 5 is an antiderivation, i.e., 5(aoa') = (5a)oa'+ (-l)Mao6a',

(7.3.8)

Chapter 7 Deformation Quantization

453

and is represented by the bracket 6a =

-[^na0yadzl3,a};

• 62 = (5)2 = 0; • for any monomial a = y a i • • • ya-dz01

A • • • A dz0",

(7.3.9)

we have (65* +66)a = (r + s)a. One also introduces the operator (T 1 : Ar ® OS{Z) -» Ar+l ® Os~l{Z) which acts on monomials (7.3.9) by the law x-i

_ ({r + s)~15a ~ \8-1a = 0

r + s>0 r+s= 0'

Then any ^-form (7.3.6) has the decomposition a = (66-1 + d-15)a + a0o.

(7.3.10)

Now let us consider a symplectic connection (7.3.11) on a symplectic manifold (Z,Q). It defines a connection on the graded O(Z)-algebra A <8> O(Z) of .4-forms (7.3.6) by the rule

Vo = dzx A V\a. In local canonical coordinates (za), we have

Va = da+ \TG,a\ , [h J where

d = dzxA^,

G=-kXlxvy»y»dz\

454

Geometric and Algebraic Topological Methods in Quantum Mechanics

and kx^L, are the covariant coefficients (2.1.7). In accordance with Definition 1.3.4 extended to an algebra over a graded ring, this connection is a graded derivation V(a o a') = V a o a ' + ( - l ) | a | a o Va' which obeys the Leibniz rule V(0 Ao) = # A a + ( - l ) V A Va for any exterior form <j> G Ok(Z). Its curvature reads

R = \R\a^yvdzx

Adza,

RXa^ =

O^-RAA.

We have the relation (V<5 + <5V)a = 0. Given a symplectic connection K on the symplectic manifold (Z, Q), let us consider more general connections on the algebra A <8> O(Z), namely, connections of the form Va = Va + [JT, a}=da+ [i(G + r), a],

(7.3.12)

where T is an ^.-valued one-form on Z. The .A-form T in the expression (7.3.12) is not denned uniquely. For the uniqueness of r, we require that its projection To (7.3.8) to the center Z®O(Z) vanishes. Then the curvature

of the connection V (7.3.12) takes the form ^R=^(R

+ VT + T2).

(7.3.13)

The connection (7.3.12) is called an Abelian connection if V2a = 0 for all elements a of the algebra A ® O(Z), i.e., the curvature form R (7.3.13) belongs to the center of this algebra.

455

Chapter 7 Deformation Quantization

PROPOSITION 7.3.1. For any symplectic connection K o n a symplectic manifold (Z,Q), there exists an Abelian connection on the algebra A <S> O(Z) which takes the form

V = V - 5 + [ir,.] = V + [±(na/3yadzP + r),.],

(7.3.14)

where r is an .4-valued one-form such that r 0 = 0.

•

The curvature form of the connection V (7.3.14) reads

R = -lna0yadz13 2i

+ R-8r + Vr + \r2. ft

The connection V is Abelian if

Sr = R + Vr + ]-r2.

(7.3.15)

h Then R = —fl is a central form. LEMMA

7.3.2. The equation (7.3.15) has a unique solution r such that 5 - ^ = 0.

(7.3.16)

• Let V (7.3.14) be a desired Abelian connection where r obeys the relations (7.3.15) and (7.3.16). Let us consider the subalgebra Ay of the algebra A which consists of the elements a, called flat, such that Va = 0. 7.3.3. For any element b G Z (7.3.5), there exists a unique flat element a(z, y, 7) G A\r such that THEOREM

a (a) :=a o (z, 0,7) = b.

a Then the associative deformation for elements a, a' £ Z (7.3.5) on the symplectic manifold Z is defined as aa'=
(7.3.17)

In particular, let Z = K2m and K be the zero symplectic connection. The corresponding Abelian connection (7.3.14) takes the form V = -S + d.

456

Geometric and Algebraic Topological Methods in Quantum Mechanics

It is readily observed that, in this case, the associative deformation (7.3.17) restarts the Moyal product (7.3.1). As was mentioned above, Fedosov's deformation construction has been generalized to regular Poisson manifolds. In a general setting, one extends it to symplectic ringed spaces [428]. Let K. be a commutative ring and A a commutative /C-ring. Let g be a Lie /C-algebra which is also an ,4-module, and is endowed with a Lie algebra and ^.-module homomorphism (7.3.18)

Q:Q^-0A

to the Lie algebra of derivations of the /C-ring A such that [u,fv} = f[u,v] + (g(u)f)v,

u,v£$,

feA.

(7.3.19)

One says that g is a Herz-Reinhart Lie algebra (henceforth a HRL-algebra) over the pair (/C, A). Of course, g = DA is a HRL-algebra. Example 7.3.1. Let L - Z be a Lie algebroid with an anchor g on a smooth manifold Z (see Section 10.3). Then the Lie algebra g = L(Z) of sections of L is a HRL-algebra over the pair (/C = R, C°°(Z)). • Example 7.3.2. Let L = T be a regular involutive distribution on Z in Example 10.3.11. Section of T -> Z make up a HRL-algebra T(Z) over the pair (/C = R,A = C°°(Z)). In particular, the Lie algebra T\{Z) of vector fields on a manifold Z is such a HRL-algebra. D Example 7.3.3. Let Z be a smooth manifold endowed with a regular foliation F. One can associate to F the following two HRL-algebras. (i) Let L = TJ- be the tangent bundle to IF. It exemplifies a Lie algebroid in Example 7.3.2. The corresponding HRL-algebra TIF{Z) consists of sections of TIF —> Z, i.e., vector fields tangent to leaves of IF. (ii) Let VF —• Z be the normal bundle to F and VFN(Z) the space of its sections which are constant on leaves of F. It is both a real Lie algebra and a module over the R-ring Syr(Z) of smooth real functions which are constant on leaves of F. Moreover, there is the homomorphism g of VFN(Z) to the Lie algebra of derivations of S?(Z) given by the relation g(u)f := u\df,

u G VFN{Z),

(7.3.20)

457

Chapter 7 Deformation Quantization

where u is a vector field on Z which projects onto u in accordance with the exact sequence (10.6.28). Then we have a HRL-algebra g = VT^{Z) over the pair (K. = 9., A = Sp{Z)). In contrast with the previous examples of HRL-algebras, it does not come from a Lie algebroid. • Given a HRL-algebra g over (/C, .4), one can associate to g the following differential calculus over A which coincides with the Chevalley-Eilenberg one if g = liA. Due to the homomorphism g (7.3.18), the ring A is a gmodule. Let us consider a subcomplex O[g] of the Chevalley-Eilenberg complex C[JJ;.4.] (1.5.46) which consists of ^-multilinear cochains. Its coboundary operator reads k

dcj>(u0,...,

ujb) = ^ ( - l ^ u O M u o , • • • > « * > • • • . « * ) ) +

(7-3.21)

t=0

^(-l)t+^(/)([

U i

,U

: 7

- ] , u 0 , ...,Ui,...,Uj,..

.,Uk),

i<j

(cf. the coboundary operator (1.6.6)). One can think of elements of O[g] as being differential forms over a HRL-algebra g whose exterior product (1.6.9) obeys the relations (1.6.10) - (1.6.11). Let w € H2[fl] be a two-form. It yields an A-module morphism tob :g-^n1[g],

J(v)(u) =u(u,v),

u,v e g.

(7.3.22)

A two-form u> is called non-degenerate if J" is an isomorphism. Let w" denote the inverse morphism to LJ° . If a non-degenerate two form u> is closed (i.e., dui = 0), it is called a symplectic form. Clearly, a symplectic form need not exists. Example 7.3.4. Let L —» Z be a Lie algebroid and g = L(z) the HRLalgebra in Example 7.3.1. If the differential calculus Q[L(Z)] over C°°(Z) contains a symplectic form, one says that L is a symplectic Lie algebroid.

•

Example 7.3.5. Let (Z,w) be a symplectic manifold and g = T\{Z) the HRL-algebra in Example 7.3.2. Then the differential calculus ft[7I(Z)] over C°°(Z) is exactly the algebra O(Z) of exterior forms on Z, and w is a symplectic form over the HRL-algebra g = T\(Z). • Example 7.3.6. Let (Z, w) be a regular Poisson manifold, J- its characteristic foliation and wy? the corresponding symplectic leafwise form (2.2.13).

458

Geometric and Algebraic Topological Methods in Quantum Mechanics

Let TT(Z) be the HRL-algebra in Example 7.3.3, item (i). Then the differential calculus Q[TF(Z)] over C°°(Z) is the leafwise differential calculus y(Z) (2.2.1), and wjr is a symplectic form over the HRL-algebra TT(Z). This is also the case of Example 7.3.5. • Example 7.3.7. Let w be a presymplectic form of constant rank on a manifold Z and T its characteristic foliation. Let V!FN{Z) be the HRLalgebra in Example 7.3.3, item (ii). The differential calculus £l[VFN(Z)} over Sf(Z) is the subcomplex of the de Rham complex O(Z) which consists of exterior forms <j> such that: (i) u\4> = 0 whenever u is a vector field tangent to the foliation T (see the exact sequence (10.6.29)), and (ii) are constant on leaves of T. Then u) is a symplectic form over the HRL-algebra VTN{Z). • Given a symplectic form u> over a HRL-algebra g, one can associate to each element f & A the element (7.3.23)

df=J{df)£2 (cf. (2.1.2)), and can define the Poisson bracket {/,/'} : = ^ / ) ( / ' ) >

(7.3.24)

fJ'eA,

on the ring A (cf. (2.1.14)). This bracket obeys the Jacobi identity, but the relation (2.1.15) need not hold. We have

P/Al-'WeKere. For instance, if (Z, {, }) is a Poisson manifold and Q is the HRL-algebra TJ-(Z) provided with the symplectic tojr in Example 7.3.6, then the Poisson bracket (7.3.24) coincides with the original one. Given the differential calculus n [9] over A, one can introduce a connection on an ,4-module g as an ^-module morphism 0 B u h-» Vu € Diff ! (fl) g)

(7.3.25)

such that the first order differential operators V u obey the Leibniz rule V u (/i;) = u(/)v + /V u «,

f£A,

veg,

(cf. Definition 1.3.3). In this case, one can also define the torsion T(u, v) — Vuv - Vvu - [u, v],

u, v G 0,

(7.3.26)

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Chapter 7 Deformation Quantization

of a connection on g and the torsionless connection V£v = -(V u i> + V,,u+ [«,«]). If a HRL-algebra g = L(Z) comes from a Lie algebroid, a connection on g is called a connection on a Lie algebroid. Any connection V on g yields a connection on the ,4-module £1[Q\ of differential forms over the HRL-algebra g by the law (V u )(ui,...,u r ) :=u(4>(vi,...,vr))

-

r

^^(ui,...,VuVi,...,ur),

u,ViCg,

(jxEQ.r[g}.

i=l

In particular, given a symplectic form u> G fi2[fl], a torsionless connection V on g is called a symplectic connection if Vuw = 0 for all u £ g. Let g be a HRL-algebra over the pair (fC,A) such that the morphism Q (7.3.18) is a monomorphism. Let w be a symplectic form over g and {,} (7.3.24) the corresponding Poisson bracket on A- For instance, this is the case of regular Poisson manifolds and presymplectic manifolds. With a symplectic connection V, one can straightforwardly generalize Fedosov's deformation procedure on a symplectic manifold in order to obtain a starproduct on the ring A. We refer the reader to [428] for the detailed exposition in terms of sheaves of iJi?L-algebras called HRL-ringed spaces (or sheaves of twisted Lie algebras in the terminology of [235]). 7.4

Kontsevich's deformation quantization

Subsections: A. Differential graded Lie algebras, 460; B. L^-algebras, 461; C. Formality theorem, 464; D. Kontsevich's formula, 468; E, Globalization of Kontsevich's deformation, 470; F. Deformation of algebraic varieties, 471. Kontsevich's deformation [252; 256] generalizes the Moyal star-product on R2m to a generic Poisson structure on M2m which fails to be regular and does not admit the Darboux coordinates. Recently, Kontsevich's deformation quantization has been extended to an arbitrary Poisson manifold [92; 130; 255]. The key point of Kontsevich's deformation is the formality theorem (see Theorem 7.4.24 below). This theorem and Theorem 7.4.1 establish

460

Geometric and Algebraic Topological Methods in Quantum Mechanics

the relations between algebras of multivector fields and multidifferential operators on a smooth manifold. A. Differential graded Lie algebras We start with the relevant algebraic constructions (see, e.g., [93; 127]). Unless otherwise stated, all algebras are over a field K of characteristic zero. By a graded structure is meant a Z-graded structure, and the symbol |. stands for the Z-graded parity. A differential graded Lie algebra (henceforth a DGLA) is a differential graded algebra 3 = © fl\ iez whose multiplication operation is a graded Lie bracket [,]:fl®fl->fl,

[a,&]€fll°l+'6l,

satisfying the relations [o,6] = -(-l)l-ll 6 l[6,o], a,b,ceg, (-1)I°HCI [a, [b, c\] + (-l) |6| l a l [b, [c, a}} + (-l)lcHfcl [c, [a, b}} = 0, (7.4.1) and the differential d of degree one obeys the graded Leibniz rule d[a,b] = [da,&] + (-l) |a| [a,d6].

(7.4.2)

The degree zero part g° and the even part of a DGLA g are Lie algebras. Any Lie algebra is a DGLA concentrated in degree zero with d = 0. A morphism of DGLAs is a graded linear map of degree zero which commutes with differentials and brackets. In particular, it is a cochain morphism. The cohomology H(g) of a DGLA g is a DGLA with respect to the zero differential d = 0 and the bracket [ a , b ] H ••= [a,b],

a,b£g,

where a denotes the cohomology class of a € g. It is evident that the cohomology of the DGLA H(g) coincides with H(g), i.e., H(H(2)) = H(2).

(7.4.3)

461

Chapter 7 Deformation Quantization

Every morphism $ : g —> g' of DGLAs yields a morphism of the cohomology DGLAs $:fr( f l )->JT(fl').

(7.4.4)

A DGLA morphism $ is called the quasi-isomorphism if the induced morphism $ (7.4.4) is an isomorphism. Let us emphasize that the existence of a quasi-isomorphism fl —>fl'does not imply the existence of the 'quasi-inverse' Q1 —> 0 which induces the inverse morphism $ . If the quasi-inverse exists, the DGLAs g and Q' are called quasi-isomorphic. A DGLA Q is called formal if it is quasi-isomorphic to the DGLA H(g). A graded coalgebra is a graded vector space f provided with the following graded co-operations: • a graded comultiplication A:f-»f®f such that

A(f)c

© f®f;

• a counit e : f -> K such that e(fi>0) = 0. These operations obey the relations (10.2.1). A graded coalgebra is called graded cocommutative if P o A = A, where P{a®b)

= (-l)^%®a,

a,bef,

is the graded transposition operator. A coderivation of degree r on a graded coalgebra f is a graded linear map d : f —> f+r which satisfies the co-Leibniz rule: (A o d){v) = {{d ® Id + (-l) r Hld ® d) o A)(v),

v G f.

By a codifferential on a graded coalgebra is meant a nilpotent coderivation of degree one. B. Loo-algebras The fact that the quasi-inverse of DGLAs need not exists has motivated one to call into play a wider class of algebras, which are Loo-algebras. Let V be a graded vector space and ®V its tensor algebra (1.1.7). It is a graded vector space such that v®v'\ = \v\ + \v'\,

v,v'GV,

462

Geometric and Algebraic Topological Methods in Quantum Mechanics

and |A| = 0, A £ K. We call ®+V = ®V \ K the reduced tensor algebra. The tensor algebra ®V is brought into a graded coalgebra with respect to the counit e : ®V —> K and the comultiplication A(t>i ® • • • vr) = 1 ® (DI ® • • • ® u r ) + r-l ] P ( ( u i (2) • • • u,) ((uj+i ® • • • ® vr) + (ui ® • • •
The graded symmetric algebra \/V and the graded exterior algebra f\V are defined as the the quotients of &V by the two-sided ideals generated by homogeneous elements of the form v <S>v' — P(v v') and

v®v' + P(v®v'), respectively. Accordingly, we have the reduced algebras \J+V and A+V^. The graded symmetric algebra VV is brought into a graded coalgebra provided with the comultiplication given on elements of V by A(u) = l ® u + t ; ® l , and extended as an algebra homomorphism with respect to the tensor product. Given a graded vector space V, one can obtain a new graded vector space V[k] by shifting the degree byfe,i.e.,

V[kY = Vi+k,

|i7|[fc] = M-fc,

v&V.

Then we have the decalage isomorphism between the graded symmetric and exterior algebras. It is given on the fc-symmetric power of V shifted by one by the expression (7.4.5)

Ck:\fV[l]^AV[k], Cfc : vi V • • • V vk ^ ( - l ) £ t i ( k - ) ( K I - i V l

A...AVk_

An Loo-algebra is a graded vector space g endowed with a codifferential D on the reduced symmetric space V+fl[l]. Let us abbreviate the symbol Dj with the projection of D to the com%

3

ponent V_|_g[l] of the target vector space and V + g[l] of the domain space.

463

Chapter 7 Deformation Quantization

A coderivation D is proved to be uniquely determined by its components D]., and it is a codifferential if and only if n

5 3 ^ 0 ^ = 0,

n>l.

(7.4.6)

i=i

In particular, D\ o D\ = 0, i.e., a codifferential D defines a complex on Q. One can show that any DGLA (g, d) can be brought into an Loo-algebra where D\a := (-l)[alda, Dl{a V b) := {-l)^^'^[a,b], Dln = 0, n > 3. Remark 7.4.1. If one had chosen £>! non-vanishing, the bracket [, ] would fulfill the graded Jacobi identity up to a term of the form dh(a, b, c) ± h(da, b, c) ± h(a, db, c) ± h(a, b, dc),

a,b,c G g,

where h: Afl->fl[-l]. In this case, g is said to become a homotopy Lie algebra.

•

An Loo-algebra morphism (a pre-Loo-algebra morphism in another terminology)

$:( fl) D)->( fl M>') is a morphism of graded coalgebras $ : V+fl[l] - V +0 '[l],

(7.4.7)

which intertwines the codifferentials, i.e., $ o £) = L>' o $. An L^-algebra morphism is uniquely determined by its components $] : V+fl[l] ->

fl'[l]

(7.4.8)

which obey the relations (7.4.9) i=l

t=l

464

Geometric and Algebraic Topological Methods in Quantum Mechanics

In particular, a morphism $ of DGLAs induces their morphism $ as Loo-algebras such that $J = $. Let us note that Loo-algebras are customarily introduced as strongly homotopy algebras [297]. Due to the decalage isomorphism (7.4.5), the codifferential D defines a sequence {Ik} of morphisms ** : Afl->fl[2-fc],

k>l.

The relation (7.4.6) puts an infinite family of conditions on the Ik's. These conditions imply that h :0 -

fl[l]

(7.4.10)

is a differential of degree one obeying the graded Leibniz rule. It follows that (g,h) is a complex. The morphism l^ defines a graded bracket [•,.]:Afl->fl

(7.4.11)

on g compatible with l\ (i.e., the relation (7.4.2) holds). This bracket obeys the graded Jacobi identity up to homotopy given by I3. Hence, any Looalgebra (g, D) such that Ik = 0 for k > 3 is a DGLA. Every Loo-algebra morphism $ provides a cochain morphism $1 between the complexes (g,h) and (g',l[). One says that an Loo-morphism $ (7.4.7) of Loo-algebras is a quasi-isomorphism if $1 is a quasi-isomorphism of complexes (g,h) and (g'Ji), i.e., it yields an isomorphism of their cohomology. In contrast with morphisms of DGLAs, any quasi-isomorphism of Loo-algebras possesses the quasi-inverse. Thus, quasi-isomorphisms of Loo-algebras define an equivalence relation, i.e., two Loo-algebras are quasiisomorphic if and only if there is an Loo-quasi-isomorphism between them. The notion of formality of Loo-algebras is formulated similarly to that of DGLAs. C. Formality theorem First, let us show that multivector fields on a smooth manifold Z make up a DGLA. The graded commutative algebra T(Z) of multivector fields is customarily provided with the Schouten-Nijenhuis bracket [., .]SN (10.6.17) which obey the relations (10.6.18) - (10.6.20). However, there is another sign convention used in the definition of the Schouten-Nijenhuis bracket [298]. This bracket, denoted by [., .]SN', is [i?,v]sN'=-(-l) |tf| [i?,u]sN-

(7-4.12)

465

Chapter 7 Deformation Quantization

The relation (10.6.18) for this bracket reads [MSN' = - ( - ^ " " " ^ ' - " M l s N ' -

(7-4.13)

The relation (10.6.19) keeps its form, while the relation (10.6.20) is replaced with the one (-l)(M-i)(M-i) [ l / ) [ t f ) t ; ] S N / ] S N , + ( _l)(M-i)(ll-D [ i ; i M ] S N

+

,

(_ 1 )(W-i)(M-i) [ t f ) [ t ; j I / ] S N / ] s N ,

] S N

, =

o.

(7.4.14)

The equalities (7.4.13) and (7.4.14) show that, with the modified SchoutenNijenhuis bracket (7.4.12), the graded vector space V=T,(Z)[1]

(7.4.15)

of multivector fields on a manifold Z is precisely a graded Lie algebra. This graded Lie algebra is brought into a DGLA by setting the differential d to be identically zero. Clearly, this DGLA is formal. In particular, let (Z, w) be a Poisson manifold. Then the Poisson bivector w obeys the Maurer-Cartan equation dw + ^[w,w]sw = 0

(7.4.16)

on the DGLA V* (7.4.15). Let us describe the DGLA of multidifferential operators on the ring C°°(Z) of smooth real functions on a manifold Z. In a general setting, let A be a K-ring and B(A, A) its Hochschild complex. Let us consider the complex (7.4.17)

C = B{A,A)[l}. It inherits the Hochschild coboundary operator (1.5.37): (6k){a0,..., o fc+1 ) = a o 0 f c (ai,..., ak+1) + ^ ( - l )

J +

V

f e

( a o , . . . , CLjdj+u

•••, ak+i)

(7.4.18) +

i

(-l) f e + V(a 0 ,...,afc)afc+i, the composition product (1.5.43): <^ m o (? !,"(ao,...,a m+n )=

(7.4.19)

ro

^(-l)

m

0

m

( a o , . . . , Oi-i,it>n(ai,...,

an+i),

an+i+i,...,

am+n),

466

Geometric and Algebraic Topological Methods in Quantum Mechanics

and the Gerstenhaber bracket (1.5.44):

[0,0'] G = W - ( - l ) l N * V o 0 ,

(t>,'eC.

(7.4.20)

One can show that this bracket obeys the graded Jacobi identity (7.4.1). Thus, (C, [., .]G) is a graded Lie algebra. Furthermore, let a one-cocycle / ^ a o , ai) = m(a0, a{) = a o ai be the multiplication in A. We consider the operator dm={m,4>}G,

cj>eC,

(7.4.21)

of degree one on C. A direct computation shows that

dm = (-l)^S,

eC,

and the relation (7.4.2) holds. Then the complex C (7.4.17) is a DGLA with respect to the bracket (7.4.20) and the differential (7.4.21). Let A = C°°(Z), and let us consider a subcomplex T>* of the complex B(C°°(Z), C°°(Z))[1] whose cochains are multidifferential operators on C°°(Z). This subcomplex is closed with respect both to the Gerstenhaber bracket (7.4.20) and the action of dm. Thus, it is a desired DGLA of multidifferential operators. Given a bidifferential operator a £ P 1 , one can think of

/ / ' = (m + a)(/,/') = / / ' + «(/./'),

f,f'eC°°(Z),

(7.4.22)

as being a deformation of the original product m in C°°(Z). One can show that the associativity constraint on this deformation is given by the equality [m + a,m + a\G = 0 , which takes the form of the Maurer-Cartan equation dma + ^{a,a}G=0.

(7.4.23)

Now, the goal is to construct a morphism of the DGLA V* (7.4.15) of multivector fields to the DGLA V* of multidifferential operators which intertwines their differential graded Lie algebra structures and solutions of the Maurer-Cartan equations (7.4.16) and (7.4.23). One has proved the following [221].

Chapter 7 Deformation Quantization

467

7.4.1. For any smooth manifold Z, there is an isomorphism between the cohomology H(D) of the algebra P and the algebra V. Since V coincides with its cohomology H(V), we have an isomorphism THEOREM

H(V) -» V* S #(V).

(7.4.24)

• The next step is the above mentioned formality theorem. 7.4.2. The DGLA P* of multidifferential operators on a smooth manifold Z is formal. •

THEOREM

It follows that there exists a quasi-isomorphism of the DGLA P* to H{V) = V* and, consequently, there is the inverse Loo-quasi-isomorphism Wof V* toP. Remark 7.4.2. There is a natural quasi-isomorphism U{ of complex V to the complex P. This morphism associates to a multivector field i?o A • • -Ai?n the multidifferential operator whose action on functions fo, • • • ,fn & C°°(Z) is given by the expression

where s runs through all permutations of the numbers (0,..., n) and sgn(s) is the sign of a permutation s. For instance, li[ assigns to a Poisson bivector w the Poisson bracket | { , } . However, the morphism li[0' fails to preserve the Lie structure. • An Loo-quasi-isomorphism U of V to P* intertwines their differential graded Lie algebra structures and solutions of the Maurer-Cartan equations (7.4.16) and (7.4.23). Let us note that this Loo-quasi-isomorphism fails to be canonical. It is represented by a power series whose first term of differential operators of minimal order coincides with the morphism U^ in Remark 7.4.2. This morphism associates to each Poisson bivector field ID 6 V1 on Z a certain bidifferential operator aw G P 1 which obeys the Maurer-Cartan equation (7.4.23) and, thus, defines a differential associative deformation m + aw of the ring C°°(Z).

468

Geometric and Algebraic Topological Methods in Quantum Mechanics

D. Kontsevich's formula An Loo-quasi-isomorphism U\ of V1 to V1 on Z = Rr in the explicit form has been obtained in [252; 256]. Given a Poisson bivector field w eV1, the corresponding deformation of C°°(Z) reads oo

« » = E E

«=i reG n , 2

^Sr,

(7.4.25)

where Bp, T £ Gni2, are bidifferential operators of order n in each argument and Wr are the weight coefficients. These operators and weight coefficients are indexed by the elements F of a suitable subset C?2,n of the set of graphs on n + 2 vertices. They are called admissible graphs. An oriented graph F belongs to the set G n , m , n, m > 0, In — m + 2 > 0, if the following holds. • The set of its vertices has n + m elements indexed, by {1,... ,n; 1,... , m}, where the vertices {1,... ,n} are said to be of first type, while {1,... ,m} are said to be of the second type. • The £Y is the set of 2n — m + 2 oriented edges of F. There is no edge starting at a vertex of the second type. The cardinality of the set of edges starting at a vertex of the first type k is denoted by (j(fc); hence

J2 lt(fc) = 2n + m - 2. l
The edges of F starting at vertex k are denoted by {e\,..., ejj. }. We also abbreviate (s(v),e(v)) with an edge possessing the starting vertex s(u) G {1,... ,n} and the ending one e(v) £ {1,... ,n : 1,... ,m}. • The F has no loop and no parallel multiple edges. In the case of a star-product, only the sets Gn,2, n>0, are of interest. Moreover, there are exactly 2 edges starting at each vertex of the first type, i.e., tt(fc) = 2, k = 1,... ,n. Hence, £p admits 2n edges for V € Gn,2- For the sake of convenience, we write the pair of edges {e£, e^} as {ik,jk}- The set Gn<2 has (n(n + 1))" elements, while Go,2 consists of only one element, which is the graph having as set of vertices {1,2} and no edges. Given a Poisson manifold (W, w) coordinated by (z 1 ,..., zr), one associates to each graph F € Gn,2, n > 1, the following bidifferential operator ^r(/iflO) f,9 £ C°°(Rr). The components w%k'ik of a Poisson bivector w are assigned to each vertex k of first type where the edges {ik,jk} start from. Functions / and g are associated to the vertices 1 and 2, respec-

469

Chapter 7 Deformation Quantization

tively. Every edge ik acts as dik on its ending vertex. Then Br is given by the following sum over all maps I : ET —> {1, • • •, r}:

\fc=ifc'=i

i

fft^vf) (ftdn^) •

/

The weight W(T) of a graph T in the formula (7.4.25) is determined as follows. Let H = {x G C : Im (a;) > 0} be the upper half-plane with its Lobachevsky hyperbolic metric. The configuration space of n distinct points in 7i is denoted by Hn. It is an open submanifold of C n with its natural orientation. Let „ = ip{s(v),e(v)). The weight W(T) is defined by integrating a 2n-form over Tin. We have

W9T) = ,/n .„

/ A nn

feAfo),

where {e\, e\}, 1 < k < n, is the set of edges of T. It should be emphasized that the weights depend neither on a Poisson structure nor the dimension r. With the deformation aw (7.4.25), we obtain the star-product fg

= fg + aw(f,9)

(7.4.26)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

on the Poisson manifold (W,w). It should be emphasized that, if a is a solution of the Maurer-Cartan equation (7.4.23), ha is not so, unless h = 0,1. Therefore, athw / haw, but takes the form oo

ahw= $ > n n=i

£

W B

^ ^

(7.4.27)

J2 WTBr{f,g).

(7.4.28)

reG n , 2

The corresponding star-product reads oo

f9 = f9+Y,hn n=i

reG n , 2

E. Globalization of Kontsevich's deformation As was mentioned above, the globalization of Kontsevich's deformation quantization to an arbitrary Poisson manifold has been performed both as a corollary of the formality theorem [255] and in an explicit form [92] similar Fedosov's one [150] (see also [130]). In the last case, local data given by the Poisson structure are described in terms of infinite order jets of functions, multivector fields and multidifferential operators. Let EQ be the fibre bundle of infinite jets of smooth functions on Z. It is endowed with the canonical flat connection Vo (see Remark 10.7.3) whose integral sections are the jets of globally defined functions. Moreover, the jets of a Poisson bracket w yield a Poisson bracket {, }oo on each fibre of Eo such that, whenever / , g are sections of Eo —> Z, the relations Vo(/s) = Vo(/)<7 + /Vo(<7), V 0 ({/, 9}oo) = {V 0 (/), g}oo + {/, V0( Z of associative R[[/i]]-algebras using Kontsevich's local star-product * (7.4.28) in each fibre. In order to avoid the dependence of coordinates, the fibre bundle E is defined by means of the quotient of jets of coordinate systems on Z by the action of the group GL(dim Z, R) of linear diffeomorphisms. Afterwards, the connection VQ is deformed to a flat connection Vo = VQ +

471

Chapter 7 Deformation Quantization

o(h) on E obeying the Leibniz rule V o (/ * 9) = V o (/) * 9 + f * Vo(s),

/, 9 G E{Z).

The star-product * induces a product on the space H°(E,V) of integral sections of the connection V. Finally, one obtains the R[[/i]]-module isomorphism Q:H°(Eo,Vo)[[h}}^H°(E,V){[h}},

which induces the star-product

in H°(Eo,Vo) and, consequently, C°°(Z). F. Deformation of algebraic varieties Kontsevich's deformation of a smooth Poisson manifold is generalized to deformation of algebraic varieties [255]. First, let us recall that, given a commutative ring .4, an algebroid over K is a small category £ such that: • £ is non-empty and all objects of C are isomorphic; • the homsets of C are endowed with the structure of ^4-modules: • composition of morphisms are ^.-bilinear. For instance, an algebroid over /C with one object is an associative /C-ring. Let A be a field of characteristic zero and V a smooth affine algebraic variety. Given its function field R\>, let us consider the module AdRy of multiderivations of Ry provided with the Schouten-Nijenhuis bracket [, ]SN2

Then a Poisson biderivation w £ A0i?v such that [W,U;]SN = 0 defines an algebraic Poisson structure on V. Let us consider a formal deformation of the Poisson algebra Ry, i.e., a non-commutative associative /C[[/i]]-linear product on i?y[[h]]. For this purpose, one is to choose a connection on f?y. Such a connection exists. Then one can associate to (V,w) an algebroid £-v,w o v e r ^[[^l] whose objects are connections V on i?y and the homsets Hom(V, V ) are canonically identified as ^t[[/i]]-modules with flv[[/i]]. Composition of morphisms is given by power series in h whose terms are bidifferential operators.' Furthermore, one considers the canonical Abelian category of -4[[/i]]linear functors from £yitu to the category of .4[[/i]]-modules, and treats this Abelian category as a deformation of the the category of quasi-coherent sheaves on V.

472

Geometric and Algebraic Topological Methods in Quantum Mechanics

7.5

Deformation quantization and operads

A different proof of the formality theorem on R r has been suggested in [41l]. One of its main idea is to consider the Lie algebras of multivector fields and multidifferential operators not as just DGLAs, but as homotopy Gerstenhaber algebras [183; 219; 411]. Another important perculiarity of Tamarkin's proof of the formality theorem is that it is phrased in terms of operads [253; 411]. Roughly speaking, one states that, for any associative algebra over a field of characteristic zero, its Hochschild complex and its Hochschild cohomology are algebras over the same operad. As was mentioned above, this observation has been the starting point of 'operad renaissance'. In a general setting, operads and algebras over operads are defined in terms of monoidal categories (see next Section), but we start with the following particular variant of (symmetric) operads. 7.5.1. An operadP of vector spaces consists of the following: • a collection of vector spaces P(n) indexed by integers n > 0, • an action of the symmetry group Sn (of permutations of n elements) on P(n) for every n, • an identity element 1 £ P(l), • compositions DEFINITION

m{ni...nk) • P(k) ® (P(ni) ® • • • ® P{nk)) -> P(m + • • • nk)

(7.5.1)

for every k > 0 and n 1 ; . . . ,nk > 0 obeying certain conditions [307].

D

There are the following basic examples of operads. Example 7.5.1. Given a vector space V, let us put P(n) = Horn (®nV,

V)

and m{ni...nk)(e

(ei <8> • • • efc))(vi <8> • • • u n i + . . . n J : = e(ei(«i • • • ® e n i ) ® • • • ek(vni+...nk_1+i

(7.5.2) (8) • • • (8)

vni+...nk)),

where e € P(k) and e, G P(rii), i — 1,..., k. The symmetry group Sn acts on ei ® • • • ® en € P(n) by permutations. D

Chapter 7 Deformation Quantization

473

Example 7.5.2. The nth component of the operad Associ is denned as the collection of functorial n-linear operations A®" —> A defined on all associative rings A, and it is spanned by the operations a i • • • an

t-> aa(i)

• • • aff(n),

where a £ Sn is a permutation.

•

Example 7.5.3. Let C be the symmetric monoidal category of Z-graded cochain complexes (cf. Example 7.6.5 below). Operads in this category are called differential graded operads or, simply, dg-operads. Passing from complexes to their cohomology, one obtains the cohomology operad H* (P) of complexes with zero differential. • An algebra over an operad P (or, simply a P-algebra) consists of a vector space A and a collection of polylinear maps /„ : P{n) ® A®n -» A,

n G N,

obeying the following axioms: • the map /„ is Sn-equi variant; • fi (Id a) = a for all a £ A; • all compositions in P map to compositions of polylinear operations on A. For example, an algebra over the operad Associ is a ring. A point of our interest here is the operads of chains and the little discs operad. Operads in the monoidal category (Top, x) (see Example 7.6.2 below) are called topological. One can associate to a topological operad P the following operad of chain. Given a topological space X, let Chains(X) be the chain complex concentrated in non-positive degrees whose (—fc)th component consists of the formal finite additive combinations N

Y^riifi,

N>0,

of continuous maps fi-.

modulo the following relations

[0,l}k^X

riiSZ,

474

Geometric and Algebraic Topological Methods in Quantum Mechanics

• / o a = sgn)o-)/ for any a G S^ acting on the standard cube [0, l]k by permutations of coordinates; • / ' o pr^fe,!) = 0, where pr^fc-DrlO.l]-^!]"1 is the projection onto first (k — 1) coordinates and f':[0,l]k-1-+X

is a continuous map. The boundary operator on cubical chains is introduced in the usual way. There is the map ® (Chains^)) -> Chains( TT X4)

i€l

tki

which is a natural homomorphisms of complexes for any finite collection (Xj)j e / of topological spaces. Let now P be a topological operad. Then the collection of complexes Chains(P(n)) possesses a natural operad structure in the category of complexes of Abelian groups. It is the operad of chains Chains(P). The compositions in this operad are defined using the external tensor product of cubical chains. Turn now to the definition of the little discs operad. Let d > 1 and Gd the (d + l)-dimensional Lie group acting on Rd by affine transformations u h-» \u + v, where A > 0 is a real number and v £ Rd is a vector. This group acts simply transitively on the space of closed discs with a center v and a radius A in M.d. The little discs operad C4 is a topological operad defined as follows: • Cd(0) = 0; • Cd(l) = {IdCd} is a singleton set; • Cd(n > 2) is the space of configurations of n disjoint discs inside the standard unit disc centered at the zero point of Rd; • the composition Cd(fc) x (Cd(rai) x • • • x Cd(rifc)) -> C d (ni + • • • + nk) is obtained applying elements of Gd associated with discs in the configuration in Cd(k) to configurations in all Cd(rij), i = 1,..., k, and putting the resulting configurations together;

475

Chapter 7 Deformation Quantization

• the symmetry group Sn acts on Cd(n) by renumerations of indices of discs in Cd(n). With these preliminaries, we can formulate the main steps of Tamarkin's proof of the formality theorem [253; 411]. (i) There exists a natural action of the operad of chains Chains(C2) of the little discs operad C2 on the Hochschild complex B(A,A) for an arbitrary associative algebra (Deligne's conjecture). (ii) The operad of chains of the little discs operad is formal, i.e., it is isomorphic to its cohomology. This is true only for rings over a field of characteristic zero. (iii) It follows from items (i) and (ii) that, for any ring A over a field of characteristic zero, its Hochschild complex B'(A,A) and its Hochschild cohomology H(A, A) are algebras over the same operad up to a homotopy. Herewith, if one chooses an explicit homotopy between these complexes, one obtains two different structure of a homotopy Gerstenhaber algebra on H(A, A) which are always equivalent for the case of A = M.[[h]]. These two structures give equivalent structures of homotopy Lie algebras. It follows that B(A,A) is equivalent to H(A,A) as a homotopy Lie algebra which is the statement of the formality Theorem 7.4.2. Let us note that Deligne's conjecture has several proofs [254; 309]. Tamarkin's proof is based on the Etingof-Kazhdan theorem of quantization of Lie bialgebras [146],

7.6

Appendix. Monoidal categories and operads

In a general setting, operads and algebras over operads are defined in terms of monoidal categories [276; 297; 307; 308].

DEFINITION 7.6.1. A class of objects of a monoidal category 27T is provided with a product K K', K, K' e Ob 971 and contains a unit object lm such that there are isomorphisms

(K®K')®K" ->K®{K'®K"),

lm®K-K,

K®\m-^

K (7.6.1)

476

Geometric and Algebraic Topological Methods in Quantum Mechanics

for all K, K', K" e ObSD? which obey the natural coherence relations (K®lm)®K' \

— K®(lm®K') / K®K' (lm®K)®K' —* lm ® (K ® K') \

JRT (8) i=f'

/

(if ® K') ® 1OT —> K ® (K1 ® lm) \ / K®K' (lm ® K) ® 1OT —> lm ® (K ® lm) K®lm

I

—>

I

-FST <—lart®-^ D

If the isomorphisms (7.6.1) are actually the identities, 9Jt is said to be a strict monoidal category. A monoidal category or a strict monoidal category 971 is said to be symmetric if there are isomorphisms K ® K' —» K' ® K for all K, K' € SDT. Let us note that the word "strict" qualifies "monoidal", but not "symmetric" in the term " symmetric monoidal category". The coherence theorem [234] states that any monoidal category (resp. symmetric monoidal category) is equivalent in a suitable sense to a strict monoidal category (resp. strict symmetric monoidal category). An object M in a monoidal category 9Jt is said to be a monoid if there are morphisms M®M -> M,

lgn -+ Af,

which obey the commutative diagrams (M®M)®M

I

—>M®(M®M)

I

M®M —> M <—M®M lm®M —>M®M <—M®lm

\

j /

M

Chapter 7 Deformation Quantization

477

Example 7.6.1. The category of sets Ens in Example 10.1.3 is a symmetric monoidal category where ® is the Cartesian product x of sets and the unit object is some singleton set. • Example 7.6.2. Similarly to Ens, the symmetric monoidal category Top of topological spaces is denned. Top, is the category of topological based spaces. Its object are topological spaces with a base point whose maps are continuous functions preserving base points. This category can be provided with the following two symmetric monoidal structures. Firstly, there is that given by the Cartesian product x and a singleton set. Secondly, there is the join V of two spaces by their base points whose unit is also a singleton set. • Example 7.6.3. The objects of the sceletal category N of finite sets are finite sets (n) = (0,..., n — 1) for each integer n € N and the empty set 0. Their morphisms are functions. The disjoint union (the addition) provides N with a monoidal product, whose unit object is 0. Then N becomes a symmetric monoidal category. Let N* be the category whose objects are also finite sets (n), but their morphisms are order-preserving functions. Then the disjoint union makes it into a monoidal category, which however is not symmetric. • Example 7.6.4. The category Mod of modules over a commutative ring K is a symmetric monoidal category where is the tensor product of modules and the unit object is K. This example is extended to the category GrMod of Z-graded ^-modules {Qn} where the unit object is the graded module {Qo = K,Qnito = 0}. There are at least two possible symmetries on GrMod. One can define either q®q'^>q'®qox q®q' ->(-l)Ml9V®«-

(7.6.2)

We will use the latter because this is the case of the category of chain complexes in Example 7.6.5 below. D Example 7.6.5. Let us denote ChC the category of Z-graded chain complexes of JC-modules. The tensor product and the unit object are the usual ones. The symmetry is given by the formula (7.6.2) because the other one is not a chain map. •

478

Geometric and Algebraic Topological Methods in Quantum Mechanics

DEFINITION 7.6.2. Let Wl and £ be monoidal categories. A monoidal functor consists of a functor F : 9Jt —» £ together with the isomorphisms Co : F(lm)

CKK>-F{K®K')->F{K®K'),

-> 1 £

(7.6.3)

such that the following diagrams commute for any objects K, K'', K" of 971: F(K ®K'® K")

I

—> F(K ® K') F(K")

I

F(K) ® F(if' O A"") — • F{K) ® f (JFT') <8> /^(/if")

F(/r) ® F(1 OT ) <—-F(tf ® IOR)

I ^

If 371 and £ are symmetric monoidal categories, the additional commutative diagram

a

F{K®K') —>F(K)®F{K') a monoidal transformation F —»orFnot) ' is amonoidal natural transformation such—» that must hold. Given two (symmetric functors F, F' : 271 £, a monoidal transformation F —>F' is a natural transformation such that F{K'®K) the following coherence diagrams —F{K')®F(K) commute:

I

I

F{K if')

—> F'(K ® K')

I

F{K)®F{K')

I

-^F'{K)®F'{K') -^F'(lm) F(lm) \ / Thus, if 97T and £ are monoidal categories l(resp. symmetric monoidal cat£ egories), there is a category Mon(27T, £) (resp. SMon(27l, £)) of monoidal functors from 971 —> £.

479

Chapter 7 Deformation Quantization

Replacing isomorphisms C,KK' (7.6.3) in Definition 7.6.2 with maps, we come to the notion of a colax monoidal functor. DEFINITION 7.6.3. A non-symmetric operad P consists of a sequence {P(n)}, n G N, of sets (objects in Ens) together with an element 1 G P(l) and a function P{n)

x • • • x P(kn)

x Pih)

- > P{h

+

---kn)

satisfying the natural unit and associativity axioms. A symmetric operad consists of a non-symmetric operad P together with the right action of the symmetry group Sn on P(n), n G N, satisfying sone compatibility laws [307]. • Let us note that, in contrast with Definition 7.5.1, the operad P in Definition 7.6.3 is an operad of sets. Though a symmetric operad is usually just called an operad, we further abbreviate this term with non-symmetric operads. There is no requirement that P(0) is a singleton set. Example 7.6.6. • There is a unique operad Obj with

• There is a unique operad Sem with fln>l a Sem = < rt n . \0 n = 0 • There is a unique operad Mon with Mon(n) = 1 for all n G N. • There is a unique operad P t with f l n = 0,l n

"|«n>2

'

• Let G be a monoid in Ens. There is an operad Algo with .,

, .

(G n = 1

•

480

Geometric and Algebraic Topological Methods in Quantum Mechanics

7.6.4. Let P be an operad and DJt a monoidal category. An algebra over P (or, simply, a P-algebra) in 971 consists of an object A in 9Xt with a function DEFINITION

P{n) -> Horn (A®n, A),

neN,

satisfying some axioms [307]. If P is a symmetric operad, a P-algebra in a symmetric monoidal category is considered. • There is an obvious notion of maps of P-algebras, and we come to the category of algebras over operads Alg(P,9Jl). Example 7.6.7. The following are some examples of algebras over the operads in Example 7.6.6. • Alg(Obj,9Jt) is isomorphic to 9Jt. • Alg(Mon, Ens) is the category of monoids and Alg(Mon, Top) is the category of topological monoids. • Alg(Mon, Cat) is the category of strict monoidal categories. • Alg(Sem, Ens) is the category of semigroups and Alg(Sem, Top) is the category of topological semigroups. • Alg(Pt,Ens) is the category of pointed sets (i.e., sets with a distinguished element). • Alg(Pt,Top) is isomorphic to Top,,. • Alg(Pt, ChC) is a chain complex together with a chosen zero-cycle. • Alg(Actc, Mod) is the category of /C-linear representations of G. D

DEFINITION 7.6.5. A monoidal category with equivalence is a monoidal category 9Tt equipped with a subclass £ of morphisms in 371, called equivalences or homotopy equivalences, such that the following properties hold: (El) any isomorphism is an equivalence; (E2) if 7 = a o (3 is a composition of morphisms in 9Jt and if any two of 7, a, (3 are equivalences, then so is the third; (E3) if a : Ki —> K[ and (3 : K K'2 are equivalences then so is

a ® (3 : Ki ® K2 -» K[ ® K'2.

D Clearly, the class of isomorphisms (resp. all morphisms) in 371 is the smallest (resp. largest) possible class of equivalences in 9Jt.

Chapter 7 Deformation Quantization

481

Example 7.6.8. • The 9JI is (Top, x, 1) and equivalences are homotopy equivalences. • The 9Jt is (ChC,®,/C) and equivalences are chain homotopy equivalences. • The SDT is (ChC,®,/C) and equivalences are quasi-isomorphisms of chain complexes. Axioms El and E2 are easily justified, but axiom E3 • holds only if K. is a field. One can associate to an operad P the following monoidal category P. Its objects are the natural numbers 0,1,... and the monoidal structure on these objects is addition, with the unit 0. The homsets are given by TT

Horn (m, n) = P(m, n) = w»H

P{mi) x • • • x P(mn).

\-mn=m

Thus, if 6i G P(i) and mi H h mn = m, there is an element (#i,..., 6n) of P(m, n). The identity element of P(m, m) is (1,... ,1), consisting of m copies of the unit 1 of P. The tensor product of morphisms is defined by (e1,...,en)®(6'1,...,e'nl)

=

(61,...,dn,6'1,...,eln,).

In order to define the composition rule, let us take fa G P{ki) with k\ + • • • + km — k so t h a t (i,..., m) S P(k, m ) , and let us take (0\,...,

9n) G

P(fc, m) as above. One can rewrite the collections (fci,..., km) as I / £ } , . . . , K^

, . . . , Kn, . . . , Kn

)

and, similarly, do (<j>i,..., <j>m) as

where J G P(kjf).

T h e n we have

6io{4>l...,(j)^)eP{kl

+ --- + kT%

and the composition rule reads

(«i,...,y»fc..,w = («iow!r.,r)r..,»i°(^..,ff")). 7.6.6. Let P be an operad and Tt a monoidal category with equivalences. A homotopy P -algebra in 9JI is a colax monoidal functor DEFINITION

where £o and (,m,n

are

equivalences.

•

482

Geometric and Algebraic Topological Methods in Quantum Mechanics

It is evident that, if the only equivalences in 371 are isomorphisms, the homotopy P-algebra in 971 is just a P-algebra in OJl. Conversely, any P-algebra is a homotopy P-algebra, and the categories Alg(P,9)I) and Mon(P,9Jt) are equivalent.

Chapter 8

Non-commutative geometry

Non-commutative geometry is developed in main as a generalization of the calculus in commutative rings of smooth functions [107; 194; 267; 290]. Accordingly, a non-commutative generalization of differential geometry is phrased in terms of the differential calculus over a noncommutative ring which replaces the exterior algebra of differential forms. The Chevalley-Eilenberg differential calculus over a commutative ring in Section 1.6 is straightforwardly generalized to a non-commutative /C-ring A- However, the extension of the notion of a differential operator in Section 1.2 to modules over a non-commutative ring meets difficulties (see Section 8.3). In a general setting, any non-commutative ring can be called into play, but one often follows the more deep analogy to the case of commutative smooth function rings. In Connes' commutative geometry, A is the algebra CCO(X) of smooth complex functions on a compact manifold X. It is a dense subalgebra of the C-algebra of continuous complex functions on X. Generalizing this case, Connes' non-commutative geometry [107; 109] addresses the differential calculus over an involutive algebra A of bounded operators in a Hilbert space E and, furthermore, studies a representation of this differential calculus by operators in E (see Section 8.5). Section 8.6 is devoted to another variant of non-commutative geometry, where one assigns to a Poisson manifold P = ^4(<$) coming from a Lie groupoid <5 the C-algebra of <J5 (see Section 3.4). The key point is that these assignments are functorial if one considers certain categories of Poisson manifolds, groupoids and C-algebras.

483

484

8.1

Geometric and Algebraic Topological Methods in Quantum Mechanics

Modules over C-algebras

Let us point out some features of modules over non-commutative algebras and, in particular, C-algebras. Let K. throughout be a commutative ring and A a /C-ring which need not be commutative. Let Z^ denote its center. An ,4-bimodule throughout is assumed to be a commutative ,2,4-bimodule. Sometimes, it is convenient to use the following compact abbreviation [137]. We say that right and left .A-modules, .4-bimodules and Z^-bimodules are A-modules of type (1,0), (0,1), (1,1) and (0,0), respectively (or (Ai — A,)-modules where AQ = Z^ and A\ = A). Of course, A-modules of type (1,1) are also of type (1,0) and (0,1), while ^-modules of type (1,0), (0,1) (1,1) are also of type (0,0). With this abbreviation, the basic constructions of new modules from old ones are phrased as follows. • If P and P' are ^-modules of the same type (i,j), so is its direct sum P®P'. • Let P and P' be ^-modules of type (i,k) and (k, j), respectively. Their tensor product P <8> P' is an 4j-module of type (i,j) (see Remark 10.4.1). • Given an «4j-module P of type (i,j), its .4-dual P* = Horn Ai-Aj (P, -A) is a module of type (i + 1, j + l)mod 2. Let A be a complex involutive algebra. Any module over A is also a complex vector space. An ylj-module of type (1,1) is called an involutive module if it is equipped with an antilinear involution p t-> p* such that (apb)* = bpa,

a,b £ A,

p £ P.

Due to this relation, an involutive module is reconstructed by its right or left module structure. In particular, an involutive module is said to be a projective module of finite rank if, seen as a right (or left) module, it is a finite projective module. Given a right module P over an involutive algebra A, a Hermitian form on P is defined as a sesquilinear A-valued form (8.1.1)

(.\.):PxP^A, (pa|pV> = a(p|pV.

{p\p') = (p'\p),

P,p'eP,

a,a'£A,

(see Remark 3.1.5). A Hermitian form (8.1.1) on P yields an antilinear morphism h of P to its A-dual P given by the formula (hp)(p') := (p\p'),

p,p' e P

(8.1.2)

Chapter 8 Non-Commutative Geometry

485

A Hermitian form (8.1.1) is called invertible if the morphism h is invertible. Let A be a C-algebra. A Hermitian form (8.1.1) on a right A-module P is called positive if {p\p} for all p G P is a positive element of a C-algebra A, i.e., (p\p) = aa,

a £ A.

Let A be a unital C-algebra. Any projective A-module P of finite rank admits an invertible positive Hermitian form. Moreover, all these forms on P are isomorphic [313]. A positive Hermitian form on a right ^4-module P endows P with the semi-norm

l|plh=llll1/2,

peP,

(8.1.3)

where ||(p|p)|| is the C-algebra norm of (p\p) G A. Equipped with this seminorm and the corresponding topology, P is called the (right) pre-Hilbert module. It is a Hilbert module (a C-module in the terminology of [107]) if the seminorm (8.1.3) is a complete norm. Example 8.1.1. A C-algebra A is provided with the structure of a Hilbert ^-module with respect to the action of A on itself by right multiplications and the positive Hermitian form (a\a') := aa',

a,a' G A.

(8.1.4)

• Example 8.1.2. Let A = C°(X) be the C-algebra of continuous complex functions on a compact space X, and let E —> X be a (topological) complex vector bundle endowed with a Hermitian fibre metric {.\.)x- Then the space E{X) of continuous sections of E —> X is a Hilbert C°(X)-module with respect to the C°(X)-valued Hermitian form (s\s')(x) := (s(x)\s'(x))x,

s,s'€E(X).

a Given a Hilbert A-module P, by its endomorphism T is meant a continuous ^-linear endomorphism of a right module P which admits the adjoint endomorphism T, which is uniquely given by the relation
p,p'&P

486

Geometric and Algebraic Topological Methods in Quantum Mechanics

The set BA(P) of yl-linear endomorphisms of a Hilbert A-module P is a C-algebra with respect to the operator norm (3.1.14). Compact endomorphisms of P are defined as the closer of its endomorphisms of finite rank (see Remark 3.1.10). Let us consider endomorphisms TP:Q € BA(P) of P of the form TP,gp'=p(p'\q),

(8.1.5)

p,p',q£P.

They obey the relations S

p,q

=

J

9,P>

J-p,q-l-p',q' — 1p(q\p'),q'

=

J

p,T,,p/
The linear span of endomorphisms (8.1.5) is a two-sided ideal of BA(P) [370]. Its closure is the set TA(P) of compact A-linear endomorphisms of P. In conclusion, let us turn to projective Hilbert modules of finite rank over a unital C-algebra A. One can show the following [313; 370]. • Let P be a right Hilbert A-module such that I d P G TA{P)- Then P is a projective module of finite rank. • Conversely, let P be a projective right ^4-module of finite rank. Then P admits a positive Hermitian form which makes it into a Hilbert module such that IdP &TA(P). • Given two positive Hermitian forms (.|.) and (.|.)' on a projective right yl-module P, there exists an invertible A-linear endomorphism of P such that (p\p'Y = (Tp\Tp'),

8.2

p,p' G P.

Non-commutative differential calculus

The notion of a differential calculus in Section 1.6 has been formulated for any /C-ring A. Let us generalize the Chevalley-Eilenberg differential calculus over a commutative ring in Section 1.6 to a non-commutative Airing .4. For this purpose, let us consider derivations u £ Z>A of A. They obey the Leibniz rule u(ab) =u(a)b + au(b),

a,b £ A,

(8.2.1)

(see Remark 1.2.1). By virtue of the relation (8.2.1), the set of derivations QA is both a Z^-bimodule and a Lie /C-algebra with respect to the Lie

487

Chapter 8 Non-Commutative Geometry

bracket [u, u'} = uu' - u'u.

(8.2.2)

It is readily observed that derivations preserve the center Z^ of A. Remark 8.2.1. If A is an involutive ring, the differential calculus over A fulfills the additional relations (a-P)=(3-a,

a,/3eft.

(6a)* = -5a,

(8.2.3)

In particular, the second relation (8.2.3) shows that, in contrast with derivations in Section 3.8, 5 is an antisymmetric derivation of an involutive ring

•

A.

Let us consider the extended Chevalley-Eilenberg complex (1.6.4) of the Lie algebra DA with coefficients in the ring .4, regarded as a T)Amodule. This complex contains a subcomplex O[5<4] of -Z^-multilinear skew-symmetric maps (1.6.5) with respect to the Chevalley-Eilenberg coboundary operator d (1.6.6). Its terms (^[O-A] are .4-bimodules. The graded module 0[i).4] is provided with the product (1.6.9) which obeys the relation (1.6.10) and makes 0[D.A] into a differential graded algebra. Let us note that, if A is not commutative, there is nothing like the graded commutativity of forms (1.6.11) in general. Since O^HA] = Horn ZA (It A, A),

(8.2.4)

we have the following non-commutative generalizations of the interior product

(u\)(ui,...,uk-i) = k(f>(u,ui,...,uk_i),

u£T)A,

and the Lie derivative

Lu()=d(WJ)+uJ/(<£). Then one can think of elements of O 1 ^^] as being the non-commutative generalization of exterior one-forms. The minimal Chevalley-Eilenberg differential calculus OA over A consists of the monomials aodai A • • • A dak,

&% € A,

whose product A (1.6.9) obeys the juxtaposition rule (aodai) A (bodbi) = a0d(a,ib0) A db\ — aoaxdbo A db\,

aiy 6j e A.

488

Geometric and Algebraic Topological Methods in Quantum Mechanics

For instance, it follows from the product (1.6.9) that, if a, a' £ ZA, then da A da' = -da' A da, PROPOSITION

ado! = (da')a.

(8.2.5)

8.2.1. There is the duality relation J>A = 1OomA-.A(O1A,A),

(8.2.6)

generalizing the relation (1.3.5) to non-commutative rings.

•

Outline of proof. It follows from the definition (1.6.6) of the ChevalleyEilenberg coboundary operator that (da)(u) = u(a),

a & A,

u£X)A.

(8.2.7)

This equality yields the morphism UA 3 u H-
A),

<j>u(da) := u(a),

a £ A.

This morphism is a monomorphism because the module O U is generated by elements da, a £ A. At the same time, any element £ HomA_A(O1A,A) induces the derivation Uj,(a) := <j>(da) of A. Thus, there is a morphism B.omA_A{OlA,A)^A, which is a monomorphism since OlA is generated by elements da, a £ A. QED

Example 8.2.2. Matrix geometry over the algebra A = Mn of complex nxn matrices provides an important example of a non-commutative system of finite degrees of freedom [135; 290]. Let {er}, 1 < r < n 2 — 1, be an anti-Hermitian basis for the (right) Lie algebra su(n). All derivations of the algebra Mn are inner, and ur = ade r constitute a basis for the complex Lie algebra X)Mn of derivations of Mn, together with the commutation relations [ur,uq]

=csrqus,

where csrq are structure constants of the Lie algebra su{n). Since the center ZMU of Mn consists of matrices cl, c £ C, the derivation module X)Mn is an (n2 — l)-dimensional complex vector space. Let us consider the minimal Chevalley-Eilenberg differential calculus (OMn,d) over the algebra Mn with respect to the Chevalley-Eilenberg coboundary operator d (1.6.6). In

489

Chapter 8 Non-Commutative Geometry

particular, 0°Mn = Mn, while OlMn is a free left Mn-module of rank n 2 — 1 whose basis {9r} is the dual of the basis {ur} for the complex Lie algebra DMn, i.e., er{uq) = srqi. It is readily observed that elements 8r of this basis belong to the center of the Mn-bimodule O1Mn, i.e., a6r = 6ra,

a e Mn.

(8.2.8)

It also follows that er A 9" = -6q A 9r.

(8.2.9)

The morphism

d:Mn^

OlMn,

given by the formula (1.6.7), reads der(uq) = &deq{er) = csqres,

that is, (8.2.10)

der = cqres6«. The formula (1.6.8) leads to the Maurer-Cartan equations d6r = -\crq3eq

K6S.

(8.2.11)

If we put 6 = er9r, the equality (8.2.10) can be brought into the form da = aB — 9a,

a £ Mn.

It follows that the Mn-bimodule O1Mn is generated by only one element

•

0.

Let us turn now to a different differential calculus over a noncommutative ring which is often used in non-commutative geometry [107; 267]. Let A be a. (non-commutative) /C-ring over a commutative ring /C. Let us consider the tensor product A® A of /C-modules. It is brought into K,

an ,4-bimodule with respect to the multiplication b(a a')c := (ba) ( a ' c ) ,

a, a', b,cG A .

490

Geometric and Algebraic Topological Methods in Quantum Mechanics

Let us consider its submodule ^(A) generated by the elements l®a-al,

a&A.

It is readily observed that d:A3a^l®a-a®l€fl1(A)

(8.2.12)

is a f21(^4)-valued derivation of A. Thus, Q1(A) is an .4-bimodule generated by the elements da, a £ A, such that the relation (da)b = d(ab) - adb,

a,b e A ,

(8.2.13)

holds. Let us consider the tensor algebra SI* (A) of the ,4-bimodule ft1 (A). It consists of the monomials aodai • • • dak,

o-i £ A ,

(8.2.14)

whose product obeys the juxtaposition rule (aodai)(bodbi) = aad{a-\bo)db\ — aaa\dbob\,

a,,6j £ A,

because of the relation (8.2.13). The operator d (8.2.12) is extended to SI* {A) by the law d(aoda\

• • • dak) '•— da^dai

• • • dak,

(8.2.15)

that makes Q* (A) into a differential graded algebra. Its de Rham cohomology groups are H°(n(A)) = /C,

Hr>0(Sl(A)) = 0.

If A is not a unital algebra, one can consider its unital extension A in Remark 1.1.2, and then construct the differential graded algebra Q(A). This algebra contains the differential graded subalgebra Cl(A) of monomials (8.2.14). The de Rham cohomology groups of fl(A) are trivial. Of course, Q.(A) is a minimal differential calculus. One calls it the universal differential calculus over A because of the following property [267]. Let P be an ,4-bimodule. Any P-valued derivation A of A factorizes as A = fA o d through some (A — .4)-module homomorphism fA : Q^A) -> P.

(8.2.16)

Moreover, let A' be another /C-algebra and (SI1, 6') its differential calculus over a /C-ring A'. Any homomorphism A —> A' is uniquely extended to a

Chapter 8 Non-Commutative Geometry

491

morphism of differential graded algebras

P

: n(A) -> n"

such that /9fe+1 od = 5' o pk. Indeed, this morphism factorizes through the morphism of $1 (A) to the minimal differential calculus in Q,'* which sends da —> 5'p(a).

Elements of the universal differential calculus Q* (A) are called universal forms. However, they can not be regarded as the non-commutative generalization of exterior forms because, in contrast with the Chevalley-Eilenberg differential calculus, the monomials da, a e Z^, of the universal differential calculus do not satisfy the relations (8.2.5). In particular, if A is a commutative ring, the module O1 (1.3.2) of exterior one-forms over A is the quotient of the module ^(A) (8.2.12) of universal forms by the relations (1.3.1). At the same time, if P — A, the morphism (8.2.16) takes the form fA{da) = A(a). This relation defines the monomorphism of fix(«4) to C 1 ^^.] (8.2.4) by the formula (8.2.7). Therefore, its range coincides with the term O1A of the minimal Chevalley-Eilenberg differential calculus, i.e., there is an isomorphism Q1(A)=O1A.

(8.2.17)

R e m a r k 8.2.3. There is the following relation between the universal differential calculus £l(A) over a /C-ring A and the Hochschild chain complex B{A,A) in Remark 1.5.3 [283; 368]. Let us assign to each Hochschild chain a 0 <8> • • • ® afc €

Bk(A,A)

the universal cochain UJ - a 0 ® • • • ® afc e £lh(A). This assignment provides an isomorphism of graded /C-modules £?* (A, A) and Q(A). Let us write u = u'dak. Then the Hochschild boundary operator (1.5.35) on SI(A) reads 3fc(w) = (-l) fc (o fc w' - w'ofc).

(8.2.18)

•

492

8.3

Geometric and Algebraic Topological Methods in Quantum Mechanics

Differential operators in non-commutative geometry

It seems natural to regard derivations of a non-commutative /C-ring A and the Chevalley-Eilenberg coboundary operator d (1.6.6) as particular differential operators in non-commutative geometry. Definition 1.2.1 provides a standard notion of differential operators on modules over a commutative ring. However, there exist its different generalizations to modules over a non-commutative ring [50; 136; 286]. Let P and Q be ,4-bimodules over a non-commutative /C-ring A. The K,-module Horn K. (P, Q) of /C-linear homomorphisms $ : P —> Q can be provided with the left A- and ^'-module structures (1.2.1) and the similar right module structures ($o)(p) := $(p)a,

(a«$)(p) :=$(pa),

a £ A,

p G P. (8.3.1)

For the sake of convenience, we will refer to the module structures (1.2.1) and (8.3.1) as the left and right .4 —.4* structures, respectively. Let us put 5o$:=$a-a»$,

a G A,

$eHomic(P,Q).

(8.3.2)

It is readily observed that 5aoSb

= Sb° Sa,

a,b e A.

The left ^.-module homomorphisms A : P —» Q obey the conditions 5aA = 0, for all a G A and, consequently, they can be regarded as left zero order Q-valued differential operators on P. Similarly, right zero order differential operators are defined. Utilizing the condition (1.2.3) as a definition of a first order differential operator in non-commutative geometry, one however meets difficulties. If P = A and A(l) = 0, the condition (1.2.3) does not lead to the Leibniz rule (1.2.8), i.e., derivations of the /C-ring A are not first order differential operators. In order to overcome these difficulties, one can replace the condition (1.2.3) with the following one [136]. 8.3.1. An element A e Homx;(-P, Q) is called a first order differential operator on a bimodule P over a non-commutative ring A if it obeys the condition DEFINITION

a,beA, <5ao5hA = <56o<5aA = 0, aA{p)b - aA(pb) - A{ap)b + A{apb) = 0,

p G P.

(8.3.3)

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Chapter 8 Non-Commutative Geometry

• First order Q-valued differential operators on P make up a Z^-module Diffi(P,Q). If P is a commutative bimodule over a commutative ring .4, then Sa = 5a and Definition 8.3.1 comes to Definition 1.2.1 for first order differential operators. In particular, let P = A. Any left or right zero order Q-valued differential operator A is uniquely defined by its value A(l). As a consequence, there are left and right .A-module isomorphisms Q 9 ^ A « S Diff ${A, Q), Q3q^A^£Di$^(A,Q),

Af(a) = qa,

a G A,

A^(a)=aq.

A first order Q-valued differential operator A o n ^ l fulfils the condition A(o6) = A{a)b + aA(b) - aA(l)6.

(8.3.4)

It is a derivation of A if A(l) = 0. One obtains at once that any first order differential operator on A is split into the sums A(a) = oA(l) + [A(a) - oA(l)], A(o) = A(l)a + [A(a) - A(l)a] of the derivations A(a) - oA(l) or A(a) — A(l)a and the left or right zero order differential operators aA(l) and A(l)a, respectively. If u is a Qvalued derivation of A, then au (1.2.1) and ua (8.3.1) are so for any a G Z^. Hence, Q-valued derivations of A constitute a Z^-module T)(A,Q). There are two Z^-module decompositions JM1(A,Q) = BiS^{A,Q)®J)(A,Q), Di$1(A,Q) = DiS(A,Q)®(A,Q). They differ from each other in the inner derivations a — i > aq — qa. Let Hom5(P, Q) and Hom^(P, Q) be the modules of right and left Amodule homomorphisms of P to Q, respectively. They are provided with the left and right ,4-.4#-module structures (1.2.1) and (8.3.1), respectively. 8.3.2. An element A G Hom c (F, Q) is a first order Q-valued differential operator on P in accordance with Definition 8.3.1 if and only if it obeys the condition

PROPOSITION

A(apb)

= (da)(p)b

+ aA(p)b + a(db)(p),

p G P,

a,b G A,

(8.3.5)

494

Geometric and Algebraic Topological Methods in Quantum Mechanics

where d and d are Kom^(P, Q)- and Hom^(P, Q)-valued derivations of A, respectively. Namely, (da){pb) = (da)(p)b,

(db)(ap)=a(db)(p). D

Outline of proof. It is easily verified that, if A obeys the equalities (8.3.5), it also satisfies the equalities (8.3.3). Conversely, let A be a first order Qvalued differential operator on P in accordance with Definition 8.3.1. One can bring the condition (8.3.3) into the form A(apb) = [A(ap) - aA(p)]b + aA(p)b + a[A(pb) - A(p)b], and introduce the derivations (9a)(p) := A(op) - oA(p),

(db)(p) := A(pb) - A(p)b. QED

For instance, let P be a differential calculus over a /C-ring A provided with an associative multiplication o and a coboundary operator d. Then d exemplifies a P-valued first order differential operator on P by Definition 8.3.1. It obeys the condition (8.3.5) which reads d(apb) = (da op)b + a{dp)b + a({-l)Mp

o db).

For instance, let P = OA be the Chevalley-Eilenberg differential calculus over A. In view of the relations (8.2.4) and (8.2.6), one can think of derivations u £ T)A as being vector fields in non-commutative geometry. A problem is that QA is not an «4-module. One can overcome this difficulty as follows [50]. Given a non-commutative /C-ring A and an .A-bimodule Q, let d be a Q-valued derivation of A. One can think of Q as being a first degree term of a differential calculus over A. Let Q^ be the right ^4-dual of Q. It is an .A-bimodule: (bu){q) :=bu(q),

(ub){q) := u{bq),

b £ A,

q£Q.

One can associate to each element u G Q^ the /C-module morphism u : i 6 a H u(da) € A.

(8.3.6)

Chapter 8 Non-Commutative Geometry

495

This morphism obeys the relations (bu)(a) = bu(da),

u(ba) = u{b)a + (ub)(a).

(8.3.7)

One calls (QR,U H-> U) the Aright Cartan pair, and regards u (8.3.6) as an A-valued first order differential operator on A [50]. Let us note that u (8.3.6) need not be a derivation of A and fails to satisfy Definition 8.3.1, unless u belongs to the two-sided .A-dual Q C QR of Q. Morphisms u (8.3.6) are called into play in order to describe (left) vector fields in noncommutative geometry [50; 232]. In particular, if Q = OlA, then au for any u € DA and a G A is a left non-commutative vector field in accordance with the relation (1.6.7). Similarly, the ,4-left Cartan pair is denned. For instance, ua for any u G T)A and a G A is a right non-commutative vector field. If .A-valued derivations u\,.. .ur of a non-commutative /C-ring A or the above mentioned non-commutative vector fields u\,... ur on A are regarded as first order differential operators on A, it seems natural to think of their compositions ui o • • • ur or u\ o • • • uT as being particular higher order differential operators on A. Let us turn to the general notion of a differential operator on ,4-bimodules. By analogy with Definition 1.2.1, one may try to generalize Definition 8.3.1 by means of the maps 5a (1.2.2) and Sa (8.3.2). A problem lies in the fact that, if P = Q = A, the compositions Sa o <5b and 5a o Sb do not imply the Leibniz rule and, as a consequence, compositions of derivations of A fail to be differential operators. This problem can be solved if P and Q are regarded as left ^-modules [286]. Let us consider the /C-module Horntc{P,Q) provided with the left A — A* module structure (1.2.1). We denote by ZQ its center, i.e., 5Q$ = 0 for all $ g Zo and a & A. Let J o = ^o be the A - A* submodule of Horn A; (P, Q) generated by Zo. Let us consider: (i) the quotient Hom^P, Q)/Io, (ii) its center Z\, (iii) the A-A* submodule ~Z\ of Hom K (P, Q)/Xo generated by Zx, (iv) the A — A' submodule 1\ of Hom^(P, Q) given by the relation I\/1Q = Z\. Then we define the A — A* submodules 2r, r — 2,..., of Horn K(P, Q) by induction as J r / I r _ i = ZT, where ZT is the A- A' module generated by the center Zr of the quotient HornK.{P,Q)/Z r -iDEFINITION

8.3.3. Elements of the submodule 1T of Rom ,c(P, Q) are said

496

Geometric and Algebraic Topological Methods in Quantum Mechanics

to be left r-order Q-valued differential operators on an 4-bimodule P [286].

•

PROPOSITION 8.3.4. An element A G Horn K(P, Q) is a differential operator of order r in accordance with Definition 8.3.3 if and only if it is a finite sum A(p) = bi&{p) + A P _i(p),

bi G A,

(8.3.8)

where A r _i and 8a$l for all a G A are (r - l)-order differential operators if r > 0, and they vanish if r = 0. • Outline of proof. If r = 0, the statement is a straightforward corollary of Definition 8.3.3. Let r > 0. The representatives $ r of elements of Zr obey the relation 5c$r = A'r_v

(8.3.9)

ceA,

where A'r_1 is an (r — l)-order differential operator. Then representatives $ r of elements of ZT take the form

$ r (p) = Y, ci&(ciP) + A;'_I(P),

i

where $ satisfy the relation (8.3.9) and A"_! is an (r —l)-order differential operator. Due to the relation (8.3.9), we obtain

^r(p) = ^ i ( p ) + A;/L1(p), ^ = c , 4

A - ^ - ^ C ^ ^ + A;'^.

(8.3.10) Hence, elements of Ir modulo elements of Ir~i take the form (8.3.10), i.e., they are given by the expression (8.3.8). The converse is obvious. QED If A is a commutative ring, Definition 8.3.3 comes to Definition 1.2.1. Indeed, the expression (8.3.8) shows that A G iiomic(P,Q) is an r-order differential operator if and only if 6a A for all a G A is a differential operator of order r — 1. PROPOSITION 8.3.5. If P and Q are ,4-bimodules, the set Ir of r-order Qvalued differential operators on P is provided with the left and right A —A' module structures. • Outline of proof. This statement is obviously true for zero order differential operators. Using the expression (8.3.8), one can prove it for higher order differential operators by induction. QED

497

Chapter 8 Non-Commutative Geometry

Let P = Q = A. Any zero order differential operator on A in accordance with Definition 8.3.3 takes the form a H-» cac' for some c, c' £ A. PROPOSITION 8.3.6. Let Ai and A2 be n- and m-order .4-valued differential operators on A, respectively. Then their composition Ai o A2 is an (n + m)-order differential operator. •

Outline of proof. The statement is proved by induction as follows. If n = 0 or m = 0, the statement issues from the fact that the set of differential operators possesses both left and right A — A' structures. Let us assume that A o A' is a differential operator for any fc-order differential operators A and s-order differential operators A' such that k + s < n + m. Let us show that Ai o A2 is a differential operator of order n + m. Due to the expression (8.3.8), it suffices to prove this fact when 6aAi and <5aA2 for any a £ A are differential operators of order n — 1 and m — 1, respectively. We have the equality 6a(A1 o A2)(6) = o(A! o A2)(6) - (A, o A2)(o6) = Ai(oA2(6)) + {8aA1 o A2)(6) - (Ax o A2)(o6) = (A1o«5aA2)(6) + (5 a A 1 oA 2 )(6), whose right-hand side, by assumption, is a differential operator of order QED n + m — 1. Any derivation u G QA of a /C-ring A is a first order differential operator in accordance with Definition 8.3.3. Indeed, it is readily observed that (Sau)(b) = au{b) — u(ab) = —u(a)b,

b € A,

is a zero order differential operator for all a G A. The compositions au, u» a (1.2.1), ua, a»u (8.3.1) for any u £ DA, a € A and the compositions of derivations uio- • -our are also differential operators on A in accordance with Definition 8.3.3. At the same time, non-commutative vector fields do not satisfy Definition 8.3.3 in general. First order differential operators by Definition 8.3.1 also need not obey Definition 8.3.3, unless P = Q = A. By analogy with Definition 8.3.3 and Proposition 8.3.4, one can define differential operators on right ,4-modules as follows. 8.3.7. Let P and Q be seen as right ^-modules over a noncommutative /C-ring A. An element A e Hom^(P, Q) is said to be a right

DEFINITION

498

Geometric and Algebraic Topological Methods in Quantum Mechanics

zero order Q-valued differential operator on P if it is a finite sum A = $'&», bi G A, where <Sa$* = 0 for all a £ A. An element A G Hom;c(P, Q) is called a right differential operator of order r > 0 on P if it is a finite sum &(p) = &(p)bi

+ Ar-1(p),

bte A ,

(8.3.11)

where A r _i and (5a$l for all a G A are right (r — l)-order differential operators. • Definition 8.3.3 and Definition 8.3.7 of left and right differential operators on .4-bimodules are not equivalent, but one can combine them as follows. 8.3.8. Let P and Q be bimodules over a non-commutative /C-ring A. An element A G Horn K.(P, Q) is a two-sided zero order Q-valued differential operator on P if it is either a left or right zero order differential operator. An element A G Horn jt(P, Q) is said to be a two-sided differential operator of order r > 0 on P if it is brought both into the form DEFINITION

A ^ b i ^ + Ar-i,

he

A,

A = $ % + A r _i,

he

A,

and

where A r _i, A r _j and 5a$l, <5a$ for all a G A are two-sided (r — l)-order differential operators. • One can think of this definition as a generalization of Definition 8.3.1 to higher order differential operators. It is readily observed that two-sided differential operators described by Definition 8.3.8 need not be left or right differential operators, and vice versa. At the same time, »4-valued derivations of a /C-ring A and their compositions obey Definition 8.3.8.

8.4

Connections in non-commutative geometry

This Section is devoted to the definitions of a connection in noncommutative geometry. We follow the notion of an algebraic connection in Section 1.3 generalized to modules over non-commutative rings [107; 137; 138].

499

Chapter 8 Non-Commutative Geometry

Let (fi, 5) be a differential calculus over a /C-ring A, and let P be a left .A-module. Following Definition 1.3.2, one can construct the tensor product of modules Q,1 P and then define a left connection on P as a /C-module morphism (8.4.1)

VL : P-> Q,1 <8> P, A

which obeys the Leibniz rule V L (ap) = 5a®p + dVL(p),

p £ P,

a £ A,

[267; 430]. If fi = Q* (.4) is the universal differential calculus over A, the connection (8.4.1) is called a universal connection. Similarly, a right non-commutative connection on a right .4-module P is defined as a /C-module morphism VR:P^P®n\ A

which obeys the Leibniz rule VR(pa) =p®5a + VR(p)a,

p £ P,

a £ A.

The forthcoming theorem shows that a connection on a left (or right) module over a non-commutative ring need not exist [115; 267]. 8.4.1. A left (resp. right) universal connection on a left (resp. right) module P of finite rank exists if and only if P is projective. • THEOREM

In contrast with connections on left and right modules, there is a problem how to define a connection on an .4-bimodule over a non-commutative ring. If A is a commutative ring, then the tensor products Q1 (g) P and P ® fi1 are naturally identified by the permutation morphism Q : a®p — i > p<S> a,

asfi1,

p £ P,

so that any left connection V^ is also the right one goVL, and vice versa. If A is a non-commutative ring, neither left nor right connections are connections on an .4-biinodule P since V L (P) £ fi^P, whereas V f i (P) £ P&H1. As a palliative, one may assume that there exists some ,4-bimodule isomorphism £>:ft 1 P->P®ft 1 .

(8.4.2)

500

Geometric and Algebraic Topological Methods in Quantum Mechanics

Then a couple (V L , V fl ) of right and left non-commutative connections on P is said to be a ^-compatible if go\7L =VR [138; 267; 324] (see also [117] for a weaker condition). However, this couple is not a true connection on an ,4-bimodule. The problem is not simplified even if P —ft1,together with the natural permutation <j> ® ' H-> 4>' ® <j>,

4>,<j>' e f i 1 .

A different construction of a connection on .A-bimodules has been suggested in [137]. Let (Q,A, 5) be the minimal Chevalley-Eilenberg differential calculus over a /C-ring A. Due to the isomorphism (8.2.6), any element u G DA determines the morphisms u:O1A®PBa®p^u(a)peP,

(8.4.3)

u: P® O1A 9 7r a H-> pu(a) <E P.

(8.4.4)

DEFINITION 8.4.2. A connection on an ,4-bimodule P with respect to the minimal Chevalley-Eilenberg differential calculus O A over A is defined as a ,2,4-bimodule morphism

V:M3MHVu6HomK(P,P),

(8.4.5)

which obeys the Leibniz rule Vu(aipbj) = u(ai)pbj + ciiVu(p)bj + aipu(bj),

p G P,

ak,bk G Ak. (8.4.6)

•

Comparing the formulae (8.3.5) and (8.4.6) shows that, by virtue of Proposition 8.3.2, V u (8.4.6) is a first order differential operator on P in accordance with Definitions 8.3.1 and 8.3.7. Let us recall that, if a G Zj., then 6a belongs to the center Zp of the module P and u(5a) G Zj^ for all u € QA. If A is a commutative ring, Definition 8.4.2 is identic to Definition 1.3.3. Remark 8.4.1. Due to the isomorphism (8.2.17), Definition 8.4.2 can be also extended to connections with respect to the universal differential • calculus. Remark 8.4.2. Definition 8.4.2 can be applied to the case of a nonminimal differential calculusfiover A by replacing the derivation module

501

Chapter 8 Non-Commutative Geometry

3,4 with the dual (ft1) of the ,4-bimodule ft1. Definition 8.4.2 is straightforwardly applied to the Chevalley-Eilenberg differential calculus O*[t)^4] over A due to the inclusion HA C VA** = (O[dA\).

D We agree to call a connection in Definition 8.4.2 the Dubois-Violette connection. It is readily observed that any left (resp. right) connection V on an .4-module of type (1,0) (resp. (0,1)) determines the Dubois-Violette connection V u = u o V, u G XiA, on P due to the morphism (8.4.3) (resp. (8.4.4)). Example 8.4.3. If P = A, the morphisms V u (a)=u(o),

uevA,

(8.4.7)

aeA,

define the canonical Dubois-Violette connection on the ring A. Then, due to the Leibniz rule (8.4.6), a Dubois-Violette connection on any .4-bimodule P is also a Dubois-Violette connection on P, seen as a 2U-bimodule. D Example 8.4.4. derivations

If P is an .A-bimodule and the ring A has only inner ad b(a) =ba — ab,

the morphisms Vad6(p) = bP-pb,

peP,

(8.4.8)

define the canonical Dubois-Violette connection on P.

•

be A,

The curvature R of a Dubois-Violette connection V (8.4.5) on an Amodule P is defined as the Z_4-module morphism R-.DAxdAB RU,U'{P)

(u,u') i-> Ruy <E Horn Ai_A.(P,P),

= V u (V u /(p)) -

V U <(V«(P))

- v [UiUi (p),

(8.4.9) peP,

[137]. We have the relations Rau,a'u' = aa'Ru,u',

a, a' G ZA,

Ru,u'(o.ipbj) = a,iRu,u'(p)bj,

a.k,bk G Ak-

For instance, the curvature of the connections (8.4.7) and (8.4.8) vanishes.

502

Geometric and Algebraic Topological Methods in Quantum Mechanics

There are the following standard constructions of new Dubois-Violette connections from old ones. (i) Given .4-bimodules P and P' and two Dubois-Violette connections V and V on them, there is an obvious Dubois-Violette connection V © V on the direct sum P © P'. (ii) Let P be an ,4-bimodule and P* its .4-dual. For any Dubois-Violette connection V on P, there is a unique dual Dubois-Violette connection V on P* such that U({P,P'))

= (VU(P),P')

+ (P,V'(P')),

p e P ,

P'GP,

U

G

U

(iii) Let P and P' be .4-bimodules, and let V and V' be Dubois-Violette connections on these modules. For any u G DA, let us consider the endomorphism (VV')u = V u ® I d P ' + I d P ® V ;

(8.4.10)

of the tensor product P P' of /C-modules P and P'. This endomorphism preserves the subset of P ® P' generated by elements pa®p'-p®ap',

p&P,

p' G P'

a £ A k.

Consequently, the endomorphisms (8.4.10) define a Dubois-Violette connection on the tensor product P P' of modules P and P'. (iv) Let A be a unital -algebra and P an involutive module over ALet us recall that, in this case, the derivation module DA consists of only symmetric derivations of A. For any Dubois-Violette connection V on P, the conjugate Dubois-Violette connection V* on P is denned by the relation Vi(p) := (V u (p)r.

(8.4.11)

A Dubois-Violette connection V on an involutive module P is said to be real if V = V.

Let now P = O1A- Any Dubois-Violette connection on an .A-bimodule O A is called a linear connection [137]. Let us note that this is not the term for a left or right connection on OXA [138]. If O1A is an involutive module, a linear connection on it is assumed to be real. Given a linear connection V on OlA, there is an ,4-module homomorphism X

T : OlA -* O2A, {T4>)(u,u') = {d<j>){u,u') - V u (^)(«') + VU'(0)(u),

(8.4.12)

Chapter 8 Non-Commutative Geometry

503

called the torsion of the linear connection V. Example 8.4.5. Let us consider linear connections on the Mn-bimodule O1Mn in matrix geometry in Example 8.2.2. Such a connection V is given by the relations V«=crUr = C r V r , Vr(0")=<0,

<eMn.

(8.4.13)

Bearing in mind the equalities (8.2.8) - (8.2.9), we obtain from the Leibniz rule (8.4.6) that

oV r (^) = Vr(6p)a,

a€Mn.

It follows that elements u)^g in the expression (8.4.13) are proportional to the identity 1 6 Mn, i.e., are complex numbers. Then the relations V r (0 p ) = w?,0«,

u>% e C,

(8.4.14)

determine any linear connection on O1Mn. Let us point out the following two particular linear connections on OlMn. (i) Since all derivations of the algebra Mn are inner, we have the curvature-free connection (8.4.8) given by the relations V r (0 p ) = 0. However, this connection is not torsion-free. The expressions (8.2.11) and (8.4.12) result in

(T0)( Ur , «,) = -<£,. (ii) In matrix geometry, there is a unique torsion-free linear connection V r (0 p ) = - c ^ ' .

• 8.5

Connes' non-commutative geometry

Connes' non-commutative geometry addresses the representation of the universal differential calculus (il(A),d) over an involutive algebra A in a Hilbert space E so that the coboundary operator da, a € A, is represented by the bracket [P,7r(a)j, a G A, where V is a certain operator in E. Thus, one comes to the notion of a spectral triple [107; 267; 430].

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Geometric and Algebraic Topological Methods in Quantum Mechanics

DEFINITION 8.5.1. A spectral triple (A, E, V) is given by a unital -algebra A C B(E) of bounded operators in a separable Hilbert space E and a (unbounded) self-adjoint operator V in E such that: (i) the brackets [T>,a], a G A, are bounded operators in E, (ii) the resolvent (T> — X)'1, A e C \ R, is a compact operator in E, •

The triple (A,E,T>) is also called the K-cycle over A [107]. Remark 8.5.1. In Connes' commutative geometry (see Example 8.5.3 below), A is the algebra C°°(X) of smooth complex functions on a compact manifold X. It is a dense subalgebra of the C-algebra of continuous complex functions on X. Generalizing this case, one usually assume that an algebra A in Connes' spectral triple is a dense involutive subalgebra of some C-algebra. • Remark 8.5.2. In many cases, £ is a Z2-graded Hilbert space whose grading automorphism 7 obeys the conditions jV + D7 = 0,

[a, 7] = 0,

aeA,

i.e., elements of A are even operators, while V is the odd one. The spectral • triple is called even if such a grading exists and odd otherwise. Given a spectral triple (A,E,V) in Definition 8.5.1, let (Q(A),d) be the universal differential calculus over the algebra A. Let us consider the representation of the graded algebra D, (A) by bounded operators n(aodai • • • dak) = ao[V, a : ] • • • [D, ak]

(8.5.1)

in the Hilbert space E. Since [D,a}* = -{V,a}, we have

However, the representation (8.5.1) fails to be a representation of the differential graded algebra Q(A) because n(w) = 0 need not imply that ir(dui) = 0. Therefore, one should construct the appropriate quotient of fl(A) in order to obtain a differential graded algebra of operators in E. Let Jg be the graded two-sided ideal of fi (A) where Jofc = {w G nk(A)

: TT(W) = 0}.

Chapter 8 Non-Commutative Geometry

505

Then it is readily observed that J* = J* + dJ£ is a differential graded two-sided ideal of Q(A). By Connes' differential calculus is meant the pair (ti^A, d) such that Q>A = ilA/J,

d[w]:=[dw],

where [u] denotes the class of u £ Q{A) in 0£>A It is a differential calculus overfi!p.A= A. Its representation ir(Q^A) is given by the classes ir[w] of operators

'Eab[V,ai}...[V,a{],

a} € A

modulo the operators 3

such that

536j[2>,&i] • - - [2>,6i_J = 0}. i

It should be emphasized that 7r([w]) are not operators in E. The coboundary operator in ir(£HpA) reads d(oo[D,oi] • • • [2>,ofc]) := [P,oo][P,oi] • • • [V,ak]. Example 8.5.3. Connes' commutative geometry is characterized by the spectral triple (A, E, V) where: • A = C°° (X) is the above mentioned algebra of smooth complex functions on a compact manifold X, • E = L2(X,S) is the Hilbert space of square integrable sections of a spinor bundle 5 —> - X, • P is the Dirac operator on this spinor bundle. In this case, the representation 7r (8.5.1) of the universal differential calculus Q(A) over A (up to a possible twisting by a complex line bundle) is the complex Clifford algebra Cliff (TX) C of the cotangent bundle TX and n(Q^A) = O(X) ® C is the algebra of complex exterior forms on X [108; 158; 368].

•

506

Geometric and Algebraic Topological Methods in Quantum, Mechanics

Spectral triples have been studied, e.g., for non-commutative tori, the Moyal deformations of Mn, non-commutative spheres 2-, 3- and 4-spheres [95; 110], and quantum Heisenberg manifolds [96]. Remark 8.5.4. Let us recall that, seen as a graded complex space, the universal differential calculusft(A) over an algebra A is isomorphic to the Hochschild chain complex B{A, A) in Remark 1.5.3. If A is a commutative involutive algebra in a spectral triple (A, E, T>), the relations

[[D,a],a']=0 for all o, a' G A hold. Then it follows from the relation (8.2.18) that the representation n (8.5.1) sends the Hochschild boundaries to zero. Hence, we have the injection of the Hochschild homology H(A, A) to 7r(ft(.A)) and, consequently, the representation of H(A, A) in a Hilbert space E [368]. • Given Connes' differential calculus (ft£,.4, d) over an algebra A, let P be a right projective ,4-module of finite rank. In the spirit of the Serre-Swan theorem 1.8.5, one can think of P as being a non-commutative vector bundle. By virtue of Theorem 8.4.1, it admits a connection. Let us construct this connection in an explicit form. Given a generic right finite projective module P over a complex ring A, let p:CN

iP:P-+CN®A c

®A^P, c

be the corresponding projection and injection, where c denotes the tensor product over C, while <8>^ stands for the tensor product over the ring A. There is the composition of morphisms P ^£N®AlA-^CN®Sll{A) C

^->P®&{A),

C

(8.5.2)

A

where the canonical module isomorphism

C

A

is used. It is readily observed that the composition (8.5.2) denoted briefly as p o d is a right universal connection on the module P.

Chapter 8 N'on- Commutative Geometry

507

Given the universal connection p o d on a right finite projective module P over a -algebra A, let us consider the morphism P

^p®V}(A)ld-^P®SlxvA. A

A

This is a right connection Vo on the module P with respect to Connes' differential calculus. Any other right connection V on P with respect to Connes' differential calculus takes the form V = V 0 +CT= (Id®7r)opo(i + (T

(8.5.3)

where a is an A module morphism a : P ->

P®VL1VA. A

The term a in the connection V (8.5.3) is called a non-commutative gauge field. 8.6

Landsman's quantization via groupoids

Landsman's quantization provides the functor from the category of integrable Poisson manifolds to some category of Lie groupoids and then to a certain category of C-algebras [270; 271]. One can think of this functor as being the functorial quantization. Let us start with the category Poisson of integrable Poisson manifolds whose Horn-sets Horn (P, P'), P, P' e ObPoisson, consist of the classes of isomorphic regular symplectic (P' — P)-bimodules (see Definition 2.7.4). We consider the subcategory LPoisson of this category defined as follows. Let us recall that a Lie groupoid <3 has an associated Lie algebroid A(<3) —* (25° which is the restriction V<3 of the vertical tangent bundle of the fibration r : <& -> 0° to 0°, together with the anchor V<& -> T0° (see Section 10.3). The key point is that the dual A{<&) -> 0° of A(<S) is equipped with the (linear) canonical Poisson structure [113], generalizing the Lie-Poisson structure on a Lie coalgebra. The above mentioned category LPoisson is a subcategory of the category Poisson whose objects are Poisson manifolds A(<3) associated to arbitrary Lie groupoids. Let us note that this subcategory fails to be full. Namely, the set Horn (A($j), A(<S)) consists of the classes of isomorphic regular symplectic bimodules of the type A(<S)^-TS-^A{F>),

508

Geometric and Algebraic Topological Methods in Quantum Mechanics

where S is some principal symplectic (<5 — J5)-bibundle [268]. THEOREM 8.6.1. The /-connected and /-simply connected symplectic groupoid over the Poisson manifold A(<&) (see Theorem 2.7.1) is the cotangent bundle T(3 (see^ Example 2.7.3) of the /-connected and /-simply connected Lie groupoid 0 with the Lie algebroid A(0) = A{<5) [270]. •

In view of Theorem 8.6.1, let us consider the full subcategory LG of the category LG of Lie groupoids whose objects are /-connected and /simply connected Lie groupoids and the Horn-sets consist of the classes of isomorphic principal bibundles (see Section 10.4). Example 2.7.3 and Theorem 8.6.1 state one-to-one correspondence A : <S H-» A'(G),

(A)-1

: A(<5) .-> 0

(8.6.1)

between the objects of the categories LG and LPoisson, while Theorem 2.7.3 shows that this correspondence is functorial. Thus, we have an isomorphism of the categories A* : LG -> LPoisson.

(8.6.2)

Now, let us assign to a Lie groupoid <5 the convolution C-algebra C{<5) of this groupoid (corresponding to the trivial two-cocycle a) (see Section 3.4). One can show that this assignment provides the functor C* : LG -> C*

(8.6.3)

from the category of Lie groupoids LG to the category C* of C-algebras in Section 8.8 whose morphisms are the classes of isomorphic Hilbert bimodules [270; 326]. Restricting this functor to the full subcategory LG of LG, we obtain the composite functor T : LPoisson (i-lC LG ^

C.

(8.6.4)

As was mentioned above, one can think of this functor as being the functorial quantization of Poisson manifolds in the category LPoisson. Let us note that, since the functors A (8.6.2) and C* (8.6.3) preserve the Morita equivalence in the categories LG, LPoisson and C, the functor T (8.6.4) does well, i.e., Morita equivalent Poisson manifolds in LPoisson are quantized by Morita equivalent C-algebras. However, this property fails if one tries to extend the functor T to the category Poisson. Basing on some particular results [107], one conjectures that there exists a functor from the category LPoisson=LG (and, more generally, Poisson)

509

Chapter 8 Non- Commutative Geometry

to the category KK of separable C-algebras whose Horn-sets are the KKgroups (see Section 8.10) [271]. Namely, this functor should map objects 0 into the convolution C-algebras C(<3), while principal <8 — 5>bibundles S are mapped into the elements S\ of the KK-groups KK(C{<8), C(£). Herewith, for the tensor product (5 S") of S and S' and the intersection product S\S'\ of 5! and S'\, the relation

(s * sy. = s\s'\ must hold. 8.7

Appendix. K"-Theory of Banach algebras

if-Theory of C-algebras as like as that of vector bundles in Section 10.11 is the particular if-theory of an additive category [16; 237]. Let 9Jt be a strict symmetric monoidal small category (see Section 7.4). It is called additive if, given any two objects E and E' of Tl, the set Horn (E, E') is an Abelian group and composition of morphisms provides a bilinear map Horn (E, E') x Horn (E1, E") -> Horn (E, E").

(8.7.1)

It is convenient to denote the product in SOT by © and its unit object by Oart- Let C(9Jt) be the set of classes of isomorphic objects of Wl. It is an additive monoid. Let K0(M) = K(C{m)) be its Grothendieck group (see Section 10.11). Then the assignment K : m -> K0(M) is a covariant functor. An additive category VJl is said to be a Banach category if, given any two objects E and E' of 9Jt, the set Horn (E, E') is a (complex or real) Banach group and the maps (8.7.1) are continuous. One can associate to a Banach category Wt the following Abelian group Ki(Wl). Let us consider the pairs (E, a) of objects of SOT and their automorphisms. Two pairs (E,a) and (E',a') are said to be isomorphic if there exists an isomorphism h : E —> E' such that a' o h = h o a. A pair (E, a) is called elementary if the morphisms a and Id E are homotopic in the set

510

Geometric and Algebraic Topological Methods in Quantum Mechanics

of automorphisms of E. Let us define the sum of pairs (E, a) and (E, a') by the formula (E, a) + (E, a') := (E ®E',a® a'). The set K\ (9Jt) consists of the equivalence classes [a] of pairs a with respect to the relation a ~ a' if and only if there exist elementary pairs r and r ' such that the pairs a + r and a' + T' are isomorphic. The set Ki (9Jt) is an Abelian group with respect to the operations l(E,a)} + l(E',a')):=[(E,a) + (E',a')}, -[(25, a)] :=[(£?, a" 1 )]. The assignment Ki :m-Kx{m) is a covariant functor. An additive category 2Jt is said to be pseudo-Abelian if, given its object E and a projector p : E —> E, there exists the object Kerp. One can associate to any additive category 9JI a certain pseudo-Abelian category Tt and an additive functor $ : 971 —> SO? which obey the following condition. Given a arbitrary pseudo-Abelian category 8. and an additive functor F : 9JT —> £, there exists a unique functor F :OT—* & such that F = F o $ . The pair (9JI, $) is uniquely defined up to an equivalence. The functor $ yields the group homomorphisms K0(M) -> K0(W) and Ki(Wt) -> i^i(OT). The latter is an isomorphism. The category of our interest is the following. Let A be a unital complex Banach algebra and 9Jt(A) the category of free A-modules offiniterank. The 971(^4) is a Banach additive category, but it need not be pseudo-Abelian. The associated pseudo-Abelian category ty(A) = 9Jt(j4) is the category of protective A-modules of finite rank. In particular, there is an isomorphism KtWiA)) = K1(Wt(A)). Let us denote KQ{A) = KQ{${A)),

K,{A) =

KiWW)-

They are called the K -theory groups of a unital complex Banach algebra A. Example 8.7.1. Let X be a compact space and
511

Chapter 8 Non-Commutative Geometry

to the norm (4.2.2). Accordingly, ty(C°(X)) is the category of projective C°(X)-modules of finite rank. By virtue of the Serre-Swan theorem [237; 409], this category is equivalent to the category of (topological) finitedimensional vector bundles over X (see Section 10.11 for the case of smooth vector bundles, where C°°(X) is a normed (but not Banach) algebra in accordance with Remark 4.2.1). Let us denote

K°{X) = K0{C°{X)),

K\X) = Ki(C°(X)).

Then the group K°(X) coincides with the group Kc{X) in Jsf-theory of vector bundles in Section 10.11. • Given the category VJl(A), the group Ki(V3l(A)) and, consequently, the group Ki(A) can be described as follows. Let Mn(A) denote the algebra of nx n matrices whose entries are elements of A. Let GL(n, A) be the group of invertible elements of Mn(A). It is an open subset of Mn(A) since A is a Banach algebra. There is the monomorphism

GL{n,A)3g^(9QOi)e

GL(n + l,A),

(8.7.2)

and we have the direct system of the groups GL(l,A)

—>•••.

-^•••GL(n,A)

Its direct limit GL(oo,A) as a set is the union of the groups GL(n,A), n € N+, modulo the injections (8.7.2). Then one can show that the set 7To(GL(oo, A)) of connected components of GL(oo,^4) is an Abelian group which is isomorphic to Kx{Tl{A)) [237]. Furthermore, Bott's periodicity theorem states the isomorphisms of homotopy groups nk(GL(oo, A)) = 7Tfc+2(GL(oo, A)),

keN,

[16; 237]. It follows from this theorem that the Grothendieck group K0(^i(A)) of the category ty(A) is isomorphic to the homotopy group

MGL{oo,A)). Another important corollary of Bott's periodicity [107] is that, given a closed two-sided ideal / of A, there is the commutative diagram

K0(I)

t

-^K0{A)

KM/1) ^K,{A)

—+K0(A/I) —

I

K^I)

512

Geometric and Algebraic Topological Methods in Quantum Mechanics

of K-theory groups. For instance, if A = C°(X) in Example 8.7.1 and N C X is a closed set, one obtains the commutative diagram

K°(X\N) K^N)

—>K°{X) —> K<-N) <—K^X)

<—K^XXN)

If A is a unital C-algebra, the A'-theory group K0(A) of the algebra A is isomorphic to the Witt group W(A) of classes of isomorphic invertible Hermitian forms (see Section 8.1) on objects ofty(A)[107].

8.8

Appendix. The Morita equivalence of C-algebras

Referring to Section 8.1 for the notation, we start with the notion of a Hubert bimodule. Let A and A' be C-algebras, and P be a right Cmodule over A together with a non-degenerate involutive algebra homomorphism (8.8.1)

p:A'^BA(P),

i.e., p(A)p = 0 implies p = 0. Then P is called a Hilbert (A' — A)-bimodule (C* -bimodule in the terminology of [107]). An isomorphism between Hilbert bimodules is a bimodule isomorphism preserving the inner product (8.1.1). For instance, a C-algebra A is provided with the structure of a Hilbert (A - A)-bimodule, denoted by 1A, with respect to the action of A on itself by left and right multiplications and the positive Hermitian form (8.1.4). Let P be a Hilbert (A' - ^l)-bimodule and P' be a right Hilbert A'module. Then the algebraic tensor product P' <S> P can be endowed with the positive Hermitian form (pi ®Pl\p'2 ®P2) •= (/0((P2|pi»Pl|P2>,

A'

Pl,P2 & P,

p\,p'2 & P',

which makes this tensor product into a right pre-Hilbert A-module [370]. Let the notation P' P also stand for the completion of this tensor product A'

to a Hilbert module. Furthermore, any endomorphism T of P' yields the endomorphism T®1 of the Hilbert module P' ® P denned by the relation A'

(T l)(p' p) : = Tp' <8> p,

p€ P,

p' e P'.

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Chapter 8 Non-Commutative Geometry

It follows that if P' is a Hilbert {A" - A')-bimodule, then P ' g P i s a A'

Hilbert (A" — J4)-bimodule. It is called the Rieffel tensor product of Hilbert bimodules. 8.8.1. Let A and A' be C-algebras. Their (strong) Morita equivalence is given by a pair (P, P') of Hilbert (A' — A)- and (A — A')bimodules P and P' such that DEFINITION

P'®P^1A, A'

P®P"^lA'. A

•

If the Morita equivalence takes place, we can choose the following particular pair of Hilbert bimodules in Definition 8.8.1 [107]. One can show that the linear span in A of the elements {pi\p2), Pi,P2 £ P, is a dense two-sided ideal, and that p (8.8.1) is an isomorphism of A' onto the algebra TA(P) of compact endomorphisms of P. Let P be the complex conjugate of the complex vector space P. It is provided with the Hilbert (A — A')-bimodule structure pa' := p{a')(p),

p(a)(p) ~pa,

a£ A,

a' e A',

and the positive Hermitian form

Then one can take P = P in Definition 8.8.1 [371]. In this case, P is called the equivalence Hilbert (A' — A)-bimodule.

Let us mention some properties preserved under the Morita equivalence [107]. • Let A an A' be the Morita equivalent C-algebras and P the equivalence Hilbert [A' — A)-bimodule. Any representation of A in a Hilbert space E yields the representation of A' in the Hilbert space P®E. Moreover, A

there is the categorial equivalence between classes of unitary equivalent representations of A and A'. • It follows that the Morita equivalent C-algebras A and A' have the same spectrum A = A'. In particular, if a C-algebra A is Morita equivalent to a commutative C-algebra A', then its spectrum A is locally compact and A' is isomorphic to the algebra of continuous complex functions vanishing at infinity on A.

514

Geometric and Algebraic Topological Methods in Quantum Mechanics

• The Morita equivalent C-algebras A and A' have the same space Prim A — Prim A' of primitive ideals. • The if-theory groups of the Morita equivalent unital C-algebras A and A' are isomorphic, i.e.,

K0(A) = Ko(A'),

K^A) = KX{A')-

Following the constructions of the categories Alg and LG in Section 10.4, one can think of the Rieffel tensor product as being the composition of Hilbert bimodules. This composition is associative on classes of isomorphic Hilbert bimodules. Therefore, one considers the category C* whose objects are C-algebras and whose morphisms A —» A', A, A' eObC are isomorphism classes of Hilbert (A' — j4)-bimodules. The unit element in Horn (A, A) is represented by the bimodule 1^. An invertible morphism in Horn (A, A') is represented by an equivalence Hilbert (A' — A)-bimodule. 8.8.2. Two C-algebras are Morita equivalent if and only if they • are isomorphic in the category C [268].

THEOREM

In particular, isomorphic C-algebras are necessarily Morita equivalent.

8.9

Appendix. Cyclic cohomology

In this Section, the relevant basics on cyclic cohomology is summarized [107]. Let Abe a complex algebra. Let us consider the Hochschild complex (1.5.36) where Q = A is the C-dual of A- Let us note that A* is an A-bimodule with respect to the multiplication (&/c)(a):=/(ca&),

feA,

a,b,ceA.

One can replace the fc-cochains fk of the complex Bk(A, A) with the (k + l)-linear functions (j)k(a0, ...,ak)=

fk(ai,...

,ak)(a0)

(8.9.1)

on A. The coboundary operator Sh (1.5.37) on these functions reads fc

(5k)(a0,...,

a k + 1 ) = J 2 ( - l ) j ( f > k ( a o , . . . , a j a j + 1 , . . . , a k + 1 ) (8.9.2) j=o k+l

k

+{-l) (f> (ak+ia0,...,ak).

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Chapter 8 Non-Commutative Geometry

Given a fc-cochain 4>k G Bk(A, A) and its cyclic permutation (f>(r)(ao,...,ak)

= <j>k(ar,... ,ak,ai,...

,ar-i),

(8.9.3)

sgn(r) = (-l)kr,

(8.9.4)

let us define the map k X(l> ) =

£sgn(r)tf>fr), r=0

where sgn(r) is the sign of the cyclic permutation (8.9.3). One can show that the range B£(A) of this map is a subcomplex of the Hochschild complex B(A, A) whose elements cj>k obey the relation

tfn) =sgn(n)^ fc . The cohomology HC(A) of this subcomplex are called the cyclic cohomology of the algebra A-

Example 8.9.1. Since B°X(A) =

B(A,A),

the cyclic cohomology group HC°(A) equals the Hochschild one H°(A, A). It consists of the elements 4> °f A which obey the relation (f>(aa') = <j>(a'a),

a, a' € A,

i.e., (f) are traces on A.

D

Example 8.9.2. If A = C, we have HCk(C) = 0 for odd k and HCk{C) = C for even k. This cohomology differs from the Hochschild cohomology of C which is trivial for all k > 0. • A homomorphism of algebras a : A —> A induces a homomorphism of the cyclic cohomology groups

a* : HC{A) - HC(A'). For instance, a is the identity if a is an inner automorphism of A. There is the following relation between the cyclic cohomology and the Hochschild one. Given a unital algebra A and the Hochschild complex Bk(A,A), let us consider the map B = x o Bo : Bk+1(A,

A) -> Bk(A, A),

(8.9.5)

516

where \

Geometric and Algebraic Topological Methods in Quantum Mechanics 1S

the cyclic skew symmetrization (8.9.4) and

(B0<j>k+1)(a0,..., ofc) := 0 f c + 1 (l, a 0 , . . . , ak) - (-l) f c +V(ao, •. •, afcl 1). One can show that the range of B (8.9.5) is B^.(A) and 6koB

-Bo5k+1.

=

As a consequence, the map B (8.9.5) yields a homomorphism of the Hochschild cohomology groups Hk+1(A, A) to the cyclic ones HCh(A). Let us introduce the following notion. A cycle of dimension n is defined as a triple (fi,rf, T), where

(n,d),

n = © nj',

is a complex differential graded algebra and T : Q n —> C is a dosed graded trace onfi,i.e., T-(dw) = 0 , r

w G fi,

r

r(w w"- ) = (-l)r("-r)r(w"-ra;r),

w G fi.

Let ^4 be a complex algebra. A cycie over A is given by a cycle (fi, d, T) of dimension n and a homomorphism p : A —> Cl°. One defines its character

h(a0, ...,an):=

T(p(ao)dp(ai)

• • • dp{an)).

PROPOSITION 8.9.1. If f is a closed graded trace of dimension n on the universal differential calculus £l(A) over A, then there exists a cycle of dimension n over A whose character is r(a0,...

,an) = ?(aodai • • • dan),

and vice versa.

D

Given two cycles (Q,d, r ) and (fl',d',Tr) of dimension n, their direct sum fl © Q,' is defined as the direct sum of two differential graded algebras with the closed graded trace TB(U + U'):=T{W) + T'(U'),

w£fi,

u'eSl'.

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Chapter 8 Non-Commutative Geometry

Given two cycles (£l,d,T) and (Q',d',T') of dimension n and n', their tensor product ft ®ft'of dimension (n + n') is the tensor product of differential graded algebras (ft,d) and (ft',oC) with the closed graded trace T®(W® J) = (-1)""'T(W)T(CJ'),

w' e ft'.

to en,

The direct sum and the tensor product of cycles over an algebra A are defined in a similar way. A cycle (ft,d, T) is said to be vanishing if j9"C(ft°) = 0. There is the following relation between cyclic cohomology of A and the cycles over A. 8.9.2. Given the complex 5(.4), its element 4>r is a cocycle (resp. a coboundary) if and only if it is a character of some cycle (resp. a vanishing cycle) of dimension r. D PROPOSITION

The following is a corollary of Propositions 8.9.1 and 8.9.2. PROPOSITION 8.9.3. Given the complex B(«4), its element (jf is a cocycle if and only if there exists a closed graded trace of dimension r on the universal differential calculusfl(.4) over A such that

r (a 0 ,..., ar) = (j)(aodax • • • dar).

(8.9.6)

D There is the following relation between the cyclic cohomology groups and the K-theory groups. Let us recall that, given differential graded algebras (Cl,d) and (Q,',d'), their tensor product ((fi ® n'),^®) is the particular tensor product of cochain complexes, i.e., it is the differential graded algebra where

(n®n') fc =

© ft'sfi', i+j-k

d®(uj®uj') = (doj)®uj'+ (-l)Mco®(d'Lo'),

weft,

w'efi'.

Given algebras A and B and their tensor product A<8> B, the tensor product Q(A) <8> Sl(B) of the universal differential calculus Cl(A) and fl(B) is a differential graded calculus over A<8>B. Then, from the universal property of Q* (A <8> B), one obtains a homomorphism (•M{A®B)^tt(A)®n{B).

(8.9.7)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

By virtue of Proposition 8.9.3, the cocycles <j>m and ipn of the complexes -B(«4) and B^{B) are represented by closed graded traces 4>m and tpn on ft* (A) and Q* (B), respectively. They yield the closed graded trace {fC~^)m+n

•- (p* ® ^ " ) o £

(8.9.8)

on fT(.A S) and, consequently, the cocycle (# ^ i/0m+™ of the complex B^(A ® B). One calls <j> •- tp the cup-product of cyclic cocycles <> / and ip. This cup-product indices a homomorphism ® HC n (B) -» H C n + m ( . 4 ® B)

HCm{A)

of cyclic cohomology groups. For instance, let A = C. Then the cup-product (8.9.8) brings the cyclic cohomology HC(C) in Example 8.9.2 into a polynomial ring with one generator a of degree 2. Furthermore, because of the canonical isomorphism C B, we obtain the cup-product a ^-- =
F C r + 2 ( S )

(8.9.9)

of cyclic cohomology groups. Let us turn to the above mentioned relation between the cyclic cohomology groups and the if-theory groups. Let A be a unital Banach algebra. One can show that there is the pairing (.,.):

HCeven(A)xK0(A)^C,

(.,.):

HCodd(A)xK1(A)^C,

which obeys the equality

(Sf,e) = (f,e),

feHC(A),

eeK(A),

where S is the map (8.9.9). 8.10

Appendix. KK-Theory

We do not aim to review Kasparov's bivariant theory, but focus only on the notion of Kasparov bimodules and their products [37; 107]. Let A be a Z2-graded C-algebra and P a ^-graded right Hilbert module (see Section 6.1) whose grading automorphism 7 preserves the Hermitian

519

Chapter 8 Non-Commutative Geometry

form on P, i.e., <7(p)l7(p')) = .

p,j/eP.

The C-algebra BA(P) of endomorphisms P is also a Z2-graded algebra such that 7CO7(P)

- 7(rp),

r G sA(P),

PGP.

Given a pair of Z2-graded C-algebras A' and A, a ^-graded Hilbert (A' — A)-bimodule is a Z2-graded right Hilbert yl-module together with a graded involutive algebra homomorphism p (8.8.1). Hereafter, all Hilbert modules are assumed to be countably generated. 8.10.1. Let A', A be Z2-graded C-algebras. A Kasparov {A' — A)-bimodule is a pair (P, R) of a Z2-graded Hilbert (A' — A)-bimodule P and its odd endomorphism R € BA{P) such that, for any a' £ A', the endomorphisms [R,p(a')], p(a')(R2 —Id) and p(a')(R — R) are compact.

DEFINITION

•

Sometimes, the Kasparov bimodules are called the Kasparov cycles [271]. A Kasparov bimodule is said to be degenerate if the above mentioned three endomorphisms are zero for any a € A'. The direct sum of two Kasparov (A' — A)-bimodules (P, R) and (P1, R') is denned as the Kaspatov (A' - A)-bimodule (P@P',R@R'). Let (P, R) be a Kasparov (A[ — v4i)-bimodule. Given a Z2-graded Calgebra A2 and a homomorphism p : A\ — A2, one obtains the induced Kasparov (A[ - A2)-bimodule (P ® A2, R Id). Similarly, given a Z2-graded M

C-algebra A'2 and a homomorphism A[ — A'2, we have the restriction of any Kasparov {A\ — j4i)-bimodule to the (A'2 — Ai)-one. For instance, let A = B[0,l]=£C°[0,l] be the C-algebra of continuous functions on [0,1] taking their values into a Z2-graded C-algebra B. Let (P, R) be a Kasparov (A'-B[0, l])-bimodule. The evaluation morphism pt:B[0,l]3b®f-b®f{t)eB,

/eC°[0,l],

te[0,l],

yields the induced Kasparov (A' - jB)-bimodule (Pt,Rt). Furthermore, one can think of (P,R) as being a continuous field (Pt,Rt), t 6 [0,1],

520

Geometric and Algebraic Topological Methods in Quantum Mechanics

of Kasparov (A' — £?)-bimodules in the spirit of continuous fields of Calgebras and Hilbert spaces in [129] (see Section 4.4). One says that the Kasparov (A' — B[0, l])-bimodule (P, R) is the homotopy between Kasparov (A' - B)-bimodules (P0,RQ) and (Pi,Ri). One can show that the set KK(A', A) of homotopy classes of Kasparov (A' — yl)-bimodules is an Abelian group, called the KK-group, with respect to the direct sum of Kasparov bimodules. Its zero element is the homotopy class of the P = {0}, while the inverse of the class of a Kasparov bimodule (P,R) is the class (Pop,-R), where P o p is P with the opposite grading. Let us note that the bifunctor KK(A', A) is homotopy invariant both in A' and A. Remark 8.10.1. Given a Z2-graded Hilbert (A' — A)-bimodule, a homotopy (P,Rte[o,i]) of Kasparov (A' — j4)-bimodules is called operatorial if the map [0,1] 9

t^RteBA(P)

is norm continuous. Let A be a separable C-algebra. Two Kasparov (A' — A)-bimodules are homotopic if and only if they, up to adding degenerate bimodules, are operatorial-homotopic. D Example 8.10.2. Let g : A' —* A be a morphism of C-algebras. One can associate to g an element of KK (A', A) represented by the Kasparov (A' — A)-bimodule (P — A, R = 0), where A' acts on A by left multiplications g(a')a, a £ A, a' £ A'. D Let us turn now to the definition of the so-called intersection product of KK-groups. Let Pi be a Hilbert Aj-module, let Pi be a Hilbert {A\ — A2)bimodule, and let P = Pi ® Pi- For any p G Pi, let us consider an y^-linear M morphism Tp : Pi —> P defined by the relation Tp{p'):=p'®p,

p'GP2.

This morphism yields the endomorphism

^=(TJ)£^(P20P)Given R2 € BA2(P2), an element R e BA2(P) is called the lift of R2 onto P if the endomorphism [fp,R2®R]

£BA2{P2®P)

521

Chapter 8 Non-Commutative Geometry

is compact. Let (Pi, i?i) and (P 2l R2) be Kasparov (A-B)- and (B-C)-bimodules, respectively. Let us consider the Z2-graded Rieffel tensor product P = Pi ®P2 of Hilbert bimodules (see Section 1.8). It is a Hilbert (A - C)B

bimodule. The pair (P,R), R £ BQ{P), is called the intersection product of {PuRi) and {P2,R2)ii: (i) it is a Kasparov (A — C)-bimodule, (ii) R is the lift of R2 onto P , (iii) for any a £ A, we have a[R\ Id, R]a > 0 modulo compact endomorphisms of P . If A is separable, the intersection product (P, R) exists, and it is unique up to operatorial homotopy. Moreover, it yields the associative intersection product KK(A, B) x KK(B, C) -* KK(A, C)

(8.10.1)

of KK-groups. For instance, if A is separable, the intersection product (8.10.1) makes KK(A, A) into a ring. Since the intersection product (8.10.1) is functorial in all conceivable ways, one can consider the category KK whose objects are separable C* algebras and morphisms are the KK-groups, namely, Horn(B,A) = KK{A,B),

A,Be ObKK.

In particular, the if-theory of unital separable C-algebras can be restart in the framework of the category KK by K0(A)^KK{C,A). Moreover, the intersection product (8.10.1) yields the homomorphism KK(C,A) x KK(A,B) -> KK(C,B) = K0(B) of K-theory groups.

Chapter 9

Geometry of quantum groups

Quantum groups are particular Hopf algebras (see Section 10.2) and, consequently, particular non-commutative algebras provided with co-operations. There are various definitions of quantum groups. The most of them treats a quantum group as a certain Hopf algebra generated by elements assembled into a matrix whose entries are not numbers. Hopf algebras and quantum groups make a contribution to many quantum models [97; 249; 292; 293]. However, the development of differential calculus and differential geometry over Hopf algebras has met problems. Here, we are concerned with the two of them. These are differential calculus over Hopf algebras (Section 9.2) and quantum principal bundles (Sections 9.3). In the case of Lie groups, there are two equivalent definitions of a smooth principal bundle, which is both a set of trivial bundles glued together by means of transition functions and a bundle provided with the canonical action of a structure group on the right. In the case of quantum groups, these two notions of a principal bundle are not matched in general. 9.1

Quantum groups

Let us compile different definitions of quantum groups [97; 98; 249; 293]. In a general setting, any non-cocommutative Hopf algebra can be treated as a quantum group. For the sake of convenience, we call them general quantum groups. There are several methods of constructing general quantum groups. For instance, the following one is utilized in order to obtain quantum deformations of simple complex Lie algebras [97]. Let (A, m, A, e, S) be a Hopf algebra finitely generated by {a1} whose antipode S is invertible. Let B be a Hopf algebra finitely generated by {bj} and dually paired with At (10.2.3) relative to some interior product (.,.). There 523

524

Geometric and Algebraic Topological Methods in Quantum Mechanics

exists a unique Hopf algebra V such that: • A and B are subalgebras of V, • V is generated by {albj}, • we have the product bjtf = ^(a i ( 1 ),5 B (6 j ( 1 ) )}(a i ( 3 ),6, ( 3 ) )a i ( 2 )6 j ( 2 ) (see the notation (10.2.2)). The Hopf algebra T> is a general quantum group called the quantum double of the Hopf algebra A. Example 9.1.1. Let us describe quantum deformation Uq(su(2)), q E C \ {0}, of the Lie algebra su(2) whose Weyl basis (X±,H) obeys the commutation relations [H,X±] = ±2X±,

[X+,X_} = H.

There are two subalgebras (X+, H) and (X+, H) of this Lie algebra. Their quantum deformations are the Hopf algebras A± such that A(H) = H®l + l®H, &{X±) = X±®qH'2+q-H'2®X±,

(9.1.1)

e(H) = e(X±) = 0, S(H) = -H,

S(X±) =

-q±lX±.

Let us introduce the new bases (e+=qHX+,t+=q2h), (e_=X^q-H,t_=q2h) for A+ and A-, respectively. The Hopf algebras A\ and A- are dually paired with respect to the interior product (t + ,e_) = <e + I t_)=0, (e + , e _) = - ( 9 - 9 - 1 ) - 1 , (t+,t_)=q-2. The quantum double of A+ is the Hopf algebra Uq(su(2)) generated by the elements X+, X_ and H modulo the commutation relations

[H,X±] = ±2X±,

[X+,X.] =q"~f",

525

Chapter 9 Geometry of Quantum Groups

and provided with the co-operations (9.1.1). This Hopf algebra is noncocommutative and, therefore, exemplifies a general quantum group. Running q —> 1, we come to the classical Hopf algebra of the Lie algebra su(2). The universal .R-matrix 1Z of Uq(su(2)) reads H

=

V^ C1 - 9

)" \(H®H+n(H®l-X®H)) j n

Q

^n

where we follow the standard notation of the quantum factorial

[0],! = l,

[n],!= f[[m}q. m=l

D A quantum group is often defined as a Hopf algebra which is dually paired with some quasi-triangular Hopf algebra. For instance, let H be a Hopf algebra and p(7i) its representation by complex N x N matrices so that any non-zero element of H is represented by a non-zero matrix and, for any pair of indices i,j = 1 , . . . , N, there exists an element h € H such that plj{h) / 0. Then one can construct another Hopf algebra A which is dually paired with H. This Hopf algebra is generated by N2 elements Alj such that (h,Aij):=pij(h),

h£7i.

Since this interior product is non-degenerate, the elements Alj € A are uniquely defined. It is convenient to think of these elements as being the entries oi a,n N x N matrix, but this is not a number matrix because Ai • A k

TA

4 f c Ai

in general. The duality condition (10.2.10a) defines the multiplication operation of the Hopf algebra A, while the conditions (10.2.10b) - (10.2.10c) induce the co-operations of A. We have A^jO^fc®^-,

e{Aij)=Sij,

S(Aij) = (A-1)ij,

(9.1.2)

(cf. Example 10.2.5). If H is a quasi-triangular Hopf algebra possessing a universal il-matrix 7£, let us consider the iV2 x TV2 matrix Rijki •= (11, A\ ® A\) = {p® p)ijki(1l).

(9.1.3)

526

Geometric and Algebraic Topological Methods in Quantum Mechanics

Let us introduce the N3 x iV3 matrices Rap = (p <8> P ®

p)(Kap)

with the entries (Ru)ijklmn = Rijlm5t (Rn)ijkimn = RikiJL ijk

(R23) lmn = B?

k

(9.1.4)

mj\.

Since the universal .R-matrix 11 obeys the quantum Yang-Baxter equation (10.2.5) and R12R13R23 = (p®p® p)(n12TZ13TZ23), the matrices (9.1.4) fulfil the Yang-Baxter equation (9.1.5)

R12R13R23 = R23RisRi2Furthermore, the property (10.2.4c) leads to the relation 0 = {RAn(h) -Po An^A^ (h, ICimnAmkAnl

® Ajt) =

- A\AimITnki),

(9.1.6) heH.

Let us introduce the iV2 x N2 matrices Ai = A® 1,

A2 = 1 ® A

whose entries are A^k^A'kSf,

A2jkl^6iA^.

Then the relation (9.1.6) takes the compact form RA1A2 = A2A1R.

(9.1.7)

In a more general setting, let us consider the algebra C[^4j] of noncommutative polynomials in TV2 variables Azj, and let R be a complex iV2 x iV2 matrix which is a solution of the Yang-Baxter equation (9.1.5). Let AR be the quotient of C[Aj] with respect to the ideal generated by the matrix elements of the matrix RAXA2 ~ A2AXR,

527

Chapter 9 Geometry of Quantum Groups

i.e., AR is generated by elements Alj obeying the equation (9.1.7). It is a Hopf algebra with the co-operations (9.1.2). For instance, if R = 1 ® 1, we come to the commutative polynomial algebra C[Aj]. Example 9.1.2. The quantum group SLq{2), q ^ 0, is the Hopf algebra generated by four elements a, b, c, d which are assembled into the matrix (9.1.8)

They obey both the commutation relations ab = qba, be = cb,

ac = qca,

ad — da = Xbc,

bd = qdb,

A = q — q~~l,

(9.1.9)

cd = qdc,

derived from the formula (9.1.7) where

(

g000\ 0100 0A10 '

OOOqJ

and the polynomial equation

A(detqA) = (detqA)A = A, where detgA = ad - qbc is the quantum determinant of the matrix A (9.1.8). This quantum group possesses the Hopf algebra structure (9.1.2), i.e., A(a) = a®a + b®c, A(c) = cig)a + rf(g)c,

A(6) = a (8 b + b® d, A(d) = c®b + d®d,

c(a) = e(d) = l, e(b)=c(c)=0, S{a) = d, S{b) = -b, S(c) = - c ,

S(d) = a.

In particular, if 5L q=1 (2) is exactly the Hopf algebra C(SL(2) in Example 10.2.5. D Let us mention an important class of so called compact quantum groups [439]. Let A be a unital C-algebra and b an TV x N matrix whose entries 6belong to A. Let A be an involutive subalgebra of A generated by entries of

528

Geometric and Algebraic Topological Methods in Quantum Mechanics

b. One says that the pair (A, b) is a compact quantum group (or a compact matrix pseudogroup in the original terminology of [439]) if the following holds. (i) The subalgebra A is dense in A. (ii) There exists a C-algebra homomorphism A:A-A®A

(9.1.10)

A(b)) = bi®b1].

(9.1.11)

such that

(iii) The A is an involutive Hopf algebra with respect to the comultiplication A (9.1.11), the counit e(6-) = 5j and the antipode 5 which obeys the relations

S(&£)6£ = &£S(&$) = ajl,

S(S(ay) = a.

Remark 9.1.3. Due to the condition (i), the homomorphism A (9.1.10) is uniquely defined. It is coassociative. However, the counit and the antipode of A fail to be extended to A in general. • Example 9.1.4. Let G be a compact subgroup of the group GL(N,C) and C°(G) the commutative C-algebra of continuous complex functions on G (see Section 4.2). For any g € G, let &•(
Chapter 9 Geometry of Quantum Groups

529

• A € Mor(A, A® A), where M.ox{A,B) is the set of homomorphisms TV of a C-algebra A to the multiplier algebra M(B) of a C-algebra B such that 7r(yl)B is norm dense in B\ • A(a)(l a') and (a' l)A(o) belong to A ® .A for any a, a' £ A and, moreover, the sets of these elements are dense in A® A; • S is a closed densely denned linear map; • A admits a locally finite strictly faithful lower semicontinuous right invariant weight h which obeys certain conditions (see [305], Definition 1.5), • i.e., A is a weighted Hopf C-algebra. In conclusion, let us mention a class of Hopf algebras called the quantum Lie algebras. One says that a Hopf algebra A is a quantum Lie algebra, if there exists a finite-dimensional subspace B B C B; • A(6) G A (1 © B) for all b £ B; • e(6) = 0 for all 6 € B. One can derive from these conditions that A(6i) = 6 i ® l + aifc(2)6fc, j

h>bk = fik bj, ai >bj=R

(9.1.12)

/ife'eC,

km

k

o ^ e A, fl^-eC,

ijbm, n

bibj - R^ijhbrn = fij bn.

(9.1.13)

The equalities (9.1.13) are said to be quantum commutation relations, where fijn are called quantum structure constants. The multipliers Rkmij in these commutation relations are assembled into a complex iV2 x N2 matrix which obeys • the Yang-Baxter equation R12R13R23 — R23R13R12 (see the notation (9.1.4)), • the quantum Jacobi identity f kf s Jis Jjn

r>sr f k f m f mf k - rt ijjsm Jrn — Jij Jmn 1

530

Geometric and Algebraic Topological Methods in Quantum Mechanics

• the equalities r>sk f m T>sr jjmk p I rtmk js jysl rtmr J: k ft injjs — ft jiU slim — ft snjji ~ ft in ft jsjrl ,

pmi f s par r>li r m "• jsjkn — ft jkft rnJsl •

It should be emphasized that the commutation relations (9.1.13) need not be independent and that the matrix R is not necessarily a universal Rmatrix of the Hopf algebra A, i.e., a quantum Lie algebra need not be quasi-triangular. Example 9.1.6. The quantum Lie algebra Uq(sl(2)) is denned as the universal enveloping algebra of the four elements

fhb+\ \b-hj modulo the commutation relations bib-i. — b_i_6i -|—fc+&2 = ~b+, 1 2

b2b+- q b+b2 =-qb+,

bib- — fo_foi b^b- = — 6 _ ,

Q

Q

Q

&2&--^&-&2 = ~b-,

b+b- - b-b+ + -b2(bi - b2) = -(h - h), q q

hb2 - 626i = 0,

where \ = q — q~1. If q —> 1, these commutation relations are exactly those O of the Lie algebra si(2).

9.2

Differential calculus over Hopf algebras

Let (A,m,A,e,S) be a Hopf algebra with an invertible antipode S (see Section 10.2 for the notation). A differential calculus over A is that over A as a non-commutative ring which obeys some additional conditions related to the co-operations in A [249; 440]. Here, we focus on a minimal first order differential calculus (henceforth FODC) over A (see Remark 9.2.1 below). It is a left ^-module fl1 generated by elements da, a € A, and provided with the .4-bimodule structure by the rule (da)b := d{ab) - adb,

a,b G A.

(9.2.1)

Chapter 9 Geometry of Quantum Groups

531

The universal FODC Q,1 (A) over A is isomorphic to the sub-bimodule Ker m of the .4-bimodule A® A. Then any FODC fi1 over A is uniquely characterized by a sub-bimodule M = {^2(ak ®bk- akbk 1) e Ker m : Q1 9 ] T ofcd6fe = 0}

(9.2.2)

of the .4-bimodule Kerm. Obiously, ft1 = 9}{A)/N. A FODC (fi\d) over a Hopf algebra A is called left-covariant if it possesses the structure of a left .4-comodule A, :fi J

^A^Q1

(see relations (10.2.17) and (10.2.19)) such that Ai(adb) = A(a)(Id d)A(6),

a, 6 e A.

(9.2.3)

Let us note that the original notion of a left-covariant FODC in [440] implies the above mentioned one. Similarly, a FODC over a Hopf algebra A is called right-covariant if it possesses the structure of a right ^4-comodule A r : fi1 -+ fi1 A such that Ar(adb) = A(a)(d ® Id) A(6),

a.deA

A FODC over A is called bicovariant if it is both a left- and right-covariant FODC satisfying the relation (Id ® A r ) o A; = (A, ® Id) o Ar. We here restrict our consideration to left-covariant FODCs over a Hopf algebra A. An .4-bimodule V which is also an .4-comodule is called a covariant bimodule. Covariant bimodules possess the following important properties. THEOREM 9.2.1. Let (V,Aj) be a left-covariant bimodule over a Hopf algebra A, and let Vjnv denote the complex vector space of left-invariant elements of V obeying the condition (10.2.20). There exists an epimorphism of complex spaces p : V —> Vinv such that p(av) = e(a)p(v),

v&V,

Aj(u)=]Tafc®Ufc,

akeA,

a £ A-

(9.2.4)

Moreover, if vkeV,

(9.2.5)

532

Geometric and Algebraic Topological Methods in Quantum Mechanics

then

(9.2.6)

p(v) = J2S(ak)vk-

• THEOREM 9.2.2. Let {wi}iei be a basis for the vector space Vinv of leftinvariant elements of V in Theorem 9.2.1. Then the following hold. • Any element of V is uniquely decomposed into the finite sums

v = ^2/aivji,

v = s^wibi,

a,i,bi € A.

(9.2.7)

• There exist complex linear forms /y- £ A* on the complex vector space A such that wib='Y^{fijb)wj,

(9.2.8)

be A,

j

awi = ^2wj(fa °S~l a),

aeA,

(9.2.9)

j

where * is the convolution product (10.2.7) - (10.2.8). • The forms fa are uniquely denned. They obey the relations fij(ab) = J2 Ma)fkj(b),

fij(l) = 6it,

a,beA.

(9.2.10)

k

a Theorem 9.2.2 describes all left-covariant bimodules. Namely, given a family of complex linear forms {fijjijei obeying the conditions (9.2.8) (9.2.9), one can consider a free left ^-module V generated by elements Vi indexed by the set / and provided it with the operations Wib

:= ] T ( / y * b)v)j,

&i{awi) := A(a)(Id ®Wi).

(9.2.11)

3

For instance, if V = Cl1 is a left-covariant FODC over A, the condition A,(d6) = ^ ( i ) ®d6 (2) ,

be A,

(9.2.3) is exactly the relation (9.2.5) in Theorem 9.2.1. Therefore, the projection p of an element db £ fi1 to the space VSnv is p(db) = J2S(bii))db{2)

(9.2.12)

533

Chapter 9 Geometry of Quantum Groups

in accordance with the formula (9.2.4). It is called the Maurer-Cartan form. By virtue of the equality (9.2.5), the projection p of an arbitrary element (j> 6 Q,1 reads

P{4>) = p ( J 2 a k d b ^

=

k

5 Z e(afc) XZ -S'(6fc(i})
It follows that, for any element <j> £ fi1, there exists an element b S Kere such that p{) = p{db). Namely, we have

p(^2akdbk)

= p(db),

k

b= 53c(ofc)(6fc-e(6fc)l). fc

Consequently, any invariant element of a left-covariant FODC fi1 is of the form (9.2.12) where b G Kere. The covariance conditions impose some restrictions on the sub-bimodule M (9.2.2). The well-known Woronowicz theorem states the following [440]. THEOREM

9.2.3. Let V, be a right ideal of A contained in Kere and N = { ^ aS(b(i))

® & (2) :beTZ,aeA}.

Then A^ is a sub-bimodule of the universal FODC ^(A), ^

= Q\A)/Af

(9.2.13) and (9.2.14)

is a left-covariant FODC. Moreover, any left-covariant FODC can be obtained in this way. • It follows that fijy- 9 da = 0 if and only if either a = 1 of a £ 1Z. In particular, the universal FODC over a Hopf algebra is always left-covariant, right-covariant and bicovariant. Given the right ideal 1Z in Theorem 9.2.3, let us consider the subspace T = {X&A*

: x(l) = 0; x(a) = 0 , a 6 R }

(9.2.15)

of the complex dual A* of A. Somebody calls T the quantum tangent space [214]. Moreover, T is a complex Lie algebra with respect to the bracket

[x, x'] = x * x' - x' * x-

534

Geometric and Algebraic Topological Methods in Quantum Mechanics

Therefore, it is also called the quantum Lie algebra [395]. One can show the following. • There is the interior product (9.2.16)

(.,.):fi^xr-»C, (
p(da) = p(),

oeKere,

which obeys the relations

(a,X) = e(a)(-x),

(da,x) = x(a)-

• The interior product (9.2.16) makes the space V{m of left-invariant elements of the FODC fijy- (9.2.14) and the quantum tangent space T (9.2.15) into a dual pair (see Section 10.5A). • In order to simplify the notation, let us assume that T is finitedimensional. Let {xijiei be a basis for T, {wi} the dual basis for V;nv and {a,j} the set of elements of Ker e denned by the condition Xi{aj) — $ijThen the relations wi = p{dai),

fij{b) = Xi(ajb),

4>= ^2(4>,Xi)wu

da=^Txi(a)wi,

i Xiab=

b € A, a E A,

i

J2(Xi

a){fij

b)+ a(Xi b)

(9.2.17)

j

hold (see Theorem 9.2.2 for the notation). In view of the relation (9.2.17), one can think of the elements UG Horn (A, A),

u{a) := x * a = X^(i)X(a(2))

(9.2.18)

as being vector fields for the left-covariant FODC fijy- which obey the deformed Leibniz rule (9.2.17). They are called invariant vector fields [13], and can be denned in an intrinsic way as follows. PROPOSITION 9.2.4. An element u G Hom(«4, A) is an invariant vector field if and only if it satisfies the equality u = (Id ® e) o (Id u) o A.

D However, the notion of invariant vector fields meets the standard problem of non-commutative geometry investigated in Section 8.3. Namely, the

Chapter 9 Geometry of Quantum Groups

535

space of invariant vector fields fails to be an ^-module. For instance, one has suggested to overcome this difficulty by appealing to Cartan pairs in Section 8.3. The general problem of differential operators on Hopf algebras has been studied in [286]. Remark 9.2.1. Let us say a few words on a higher order differential calculus over a Hopf algebra. A minimal differential calculus (Q, d) over a Hopf algebra .4 seen as an associative algebra is said to be left-covariant if Q. possesses the structure of a left ,4-comodule

A;: n* -> A ® n* such that A ; (a o dai • • • dak) = A(ao)[(Id d)A(ax)] • • • [(Id d)A(ak)\,

a» & A.

Let Q(A) be the universal differential calculus over A and Af its differential graded ideal such that Aj(7V)C-4®;V. Then, the differential calculus SI* (A)/AT* is left-covariant. Similarly, rightcovariant differential calculi over a Hopf algebra A are considered. Let us note that different ideals Af* may lead to isomorphic quotients Q(A)/Af. We always choose Af* the kernel of the morphism D.(A) —> £1. •

9.3

Quantum principal bundles

As was mentioned above, there two notions of a quantum principal bundle which are not matched, unless the base is a smooth manifold [139; 355]. The first one is the construction of a locally trivial quantum principal bundle glued together from the trivial pieces. This construction falls into the following steps [76; 81; 82]. • The notion of coverings and gluings of algebras is introduced. A cover of an algebra B is denned as a (finite) set of its non-intersecting ideals Bi. Provided with such a cover, an algebra is reconstructed from its restriction to the elements of the cover by a gluing procedure. Such a reconstruction need not exist. Therefore, one introduces the notion of a complete cover. • A locally trivial quantum principal bundle has an algebra B as a base and a Hopf algebra (in particular, a compact quantum group) Ji as

536

Geometric and Algebraic Topological Methods in Quantum Mechanics

a structure group and a typical fibre. It is defined as a right H-comodule algebra V whose cover is adapted to that of the base B. • The particular differential calculus over a locally trivial quantum principal bundle is introduced. For this purpose, the differential calculus over an algebra which possesses covering and gluing is constructed in a general setting. Then one describes a covariant differential calculus on a locally trivial quantum principal bundle V which is adapted to a differential calculus over the base B. • A covariant derivative on a locally trivial quantum principal bundle is denned as a family of local derivations of H with values in the differential calculus over the base B, which obey the Leibniz rule, the covariance and compatibility conditions. Particular covariant derivatives on a locally trivial quantum principal bundle are connections. The key point is that a connection on quantum principal bundle can be reconstructed from connections on its local pieces. We start with the general notion of covering and gluing of algebras. Let finite families {i?j}i e /, {Bij = Bji}^ of unital algebras together with homomorphisms TT] : Bi —> B^ be given. Then the algebra B = ®Bi-.= {(bi) e ®Bi : 7r!-(bi) = 4(bj)} rrj

(9.3.1)

i

is called the gluing of the algebras Bi along the algebras B^ by means of the homomorphisms 7rj. A (finite) cover of an algebra B is defined as a finite set {Jj}ig/ of non-intersecting two-sided ideals of B. Let us put

Bi = B/Ji, Bn = BftJi + Jj), Bijk = B/(Ji + Jj + Jk), where by (Jj + Jj) is meant the minimal two-sided ideal of B containing both Jj and Jj. Let 7T; : B —> Bi, •Kij : B -> B^, Kijk '• B —> B^k,

ny.Bi^b + Ji^b + Ji + Jje B^, it)k '• ^ n 3

l

—> B^k,

'• Bij —» Bijk

537

Chapter 9 Geometry of Quantum Groups

be the corresponding homomorphisms. One can define the gluing (9.3.2)

~B = ®Bi

of Bi along B^ which is called the covering completion of B with respect to the cover { Bi, then the cover {Jj} is complete. For instance, any cover of a C-algebra is complete [8l]. This fact is essential for the case of locally trivial quantum principal bundles with compact quantum structure groups. A locally trivial quantum principal bundle is denned to be a tuple (9.3.3)

(P,Av,n,B,t,{^,Ji}ieI),

where B is a unital algebra, Ti is a Hopf algebra whose antipode is invertible, V is a right H-comodule algebra, {J»} is a complete cover of B, and £, and i are homomorphisms possessing the following properties: • £» : 'P ~ Bi ® Ti are surjections, • {Ker£j} is a complete cover of V, • (Id A) o & = (& ® Id) o A P , • t: B —> V is an injection, . t(£) = { ^ e P : Av((j>) = 4> ® 1}, • £i o t(a) = 7Tt(o) <8> 1, a £ S. Accordingly, P, JB and Ti are called the fo£a£ space, the 6ase and the structure group of a locally trivial quantum principal bundle. A locally trivial quantum principal bundle V (9.3.3) is called trivial if there exists an isomorphism ^:V^B®H such that £oi = Id®l,

(Id A)o£ = (£®Id) o Av.

Two locally trivial quantum principal bundles V and V (9.3.3) with the same structure group Ti, the same base B, and the same cover {J{\ of B are said to be isomorphic if there exists an isomorphism ip : V —> V such that ipoi = i',

(ip<S>ld)oAr = Av>oij;,

V(Ker&) =Ker^-.

538

Geometric and Algebraic Topological Methods in Quantum Mechanics

A locally trivial quantum principal bundle implies the following "local" trivial quantum principal bundles. • Let us define Vi:=V/Ker^, li-.Vi^Bi® H, Li-.Bi^Vi, AVi :Vi->Vi®n,

U
+ Ker &) := Av((/>) + Ker & ® H.

Then the tuple

is a trivial quantum principal bundle. • Let us put Vij := V/(Ker& +Ker£j). Then there exists an isomorphism &i:Pij^(Bi®H)/ti(Ker{J),

4-(0 + Ker & + Kerfc) := ^(0) + &(Ker&)F\irthermore, there is an isomorphism (Bi ® W)/6(Ker^) ^ By ® W, yielding the trivialization map Q : V —> Vij. Then one can show that V%j is a trivial quantum principal bundle. The following Theorem [76; 8l] is the key point of the analysis of locally trivial quantum principal bundle. THEOREM 9.3.1. A locally trivial quantum principal bundle with a structure group 7i and a base B provided with a complete cover {Jj} defines a family of homomorphisms Tij-.H^h- e(h)l € Btj,

(9.3.4)

called the transition functions, which obey the conditions 7if o T y = mBijk

o ((nf o Tik) (rrf o Tjfc)) o A,

(9.3.5)

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Chapter 9 Geometry of Quantum Groups

where mBijk denotes the multiplication in By,. Conversely, every family of homomorphisms r^- (9.3.4) obeying the relations (9.3.5) determines a locally trivial quantum principal bundle up to isomorphisms. D Let us point out the relations Tij(h) = (Id e) o Vy(l ® h), ipij(a h) = ^ a T i j ( h { l ) ) ®

heH, ft(2),

ae Bi:j,

between the transition functions T^ and the corresponding morphisms ipij : Bij ft -> Bij ® H.

We refer the reader to [81; 82] for a detailed exposition of the differential calculi and connections on a locally trivial quantum principal bundle. Let us turn now to a different notion of a quantum principal bundle [74; 293]. Let H be a Hopf algebra and V a right H-comodule algebra with respect to the coaction p:V

->V®H.

Let : (3(p) = p 1}

M = {peP

be its subalgebra of invariant elements. The triple (V, H, (3) is called an algebraic quantum principal bundle if the map ver:V®VB{p®q)>->pl3(q)£V®H M

M

(9.3.6)

is a linear isomorphisms. This condition, called the Hopf-Galois condition, is a key point of the algebraic notion of a quantum principal bundle. One can think of this condition as being a sui generis local trivialization as follows. The map ver (9.3.6) comes from the map ver : V V 9 (p q) ^ p(3{q) e f ® H .

(9.3.7)

Let Q}(V) = K e r m be the universal differential calculus over the algebra V. The map (9.3.7) restricted to Kerm reads ver : Q}{V) -> P ® K e r e .

540

Geometric and Algebraic Topological Methods in Quantum Mechanics

Regarding elements of ill(V) as universal one-forms over V, one can think of VQ}{M)VdSl\V) as being horizontal one forms. Then the Hopf-Galois condition (9.3.6) can be treated as the exact sequence Q-+VQ}{M)V —£ll{V)

^7>®Kere->0

(cf. the exact sequence (10.6.13b) and the corresponding exact sequence of sections of these bundles). The following statement provides many examples of an algebraic quantum principal bundle [293]. PROPOSITION 9.3.2. Let TT : V —> H be a Hopf-algebra surjection and /3:=(Id ®TT)A. If the product map m : Keie\M ®P —> Kerrr is a surjection, then (V,H,(3) is an algebraic quantum principal bundle. •

Chapter 10

Appendixes

For the sake of convenience of the reader, several mathematical topics are compiled in this Chapter. 10.1

Categories

We refer the reader to [45; 289; 314] for the basics of category theory. A category 8. consists of a class Ob .ft of objects and a class Mor& of morphisms which obey the following conditions. (i) A class Horn (K, K') of morphisms is assigned to each ordered pair of objects (K,K'). (ii) Any morphism a e ^ belongs to only one class Horn (.,.). (iii) Given pairs of objects (K,K') and (K',K"), there is the composition (3 o a G Horn (K, K") of morphisms a 6 Horn (K, K') and P G Horn (if ,K"). This composition is associative. (iv) Any class Horn (K, K) contains an identity morphism Id K such that, whenever K' € Ob£, we have ldKo/3 = j3e Horn (K1, K).

a o IdK = a G Horn (K,K'),

Remark 10.1.1. It should be emphasized that this definition of a category is formulated in the framework of a system of axiomatic set theory (e.g., the Godel-Bernays one) where the notion of a class differs from that of a set. Namely, classes unlike sets can not be elements of classes and sets. One often requires that all classes Hom(.,.) of a category are sets, called homsets. A category is called small if its morphisms constitute a set. Small categories and their functors make up the category of categories Cat. One 541

542

Geometric and Algebraic Topological Methods in Quantum Mechanics

also means by the category of categories the category cat whose objects and homsets are subsets of a fixed universal set. • Remark 10.1.2. The condition (iv) above associates a class of objects Ob R of a category to a subclass of a class of morphisms Mor R via the assignment ObRB K^ldK eMorR. This fact enables one to define a category as a class of morphisms obeying certain conditions. Namely, let M. be a class whose elements a, j3,7,... are provided with a partial product. An element K £ M is said to be the identity if na = a and (3K, = (5 whenever the product net and (In are defined. Then M. is called the abstract category if the following conditions holds. • The product a((3~f) is defined if and only if the product (a/3)j is so. If these products exist, then they equal each other. • The product a(Pj) exists if the products a/3 and (3^j are defined. • For any element a £ Ai, there exists the identities K and K' such that the products na and an' are defined. Morphisms of any category make up both an abstract category and a category (see Example 10.1.6 below). Conversely, any abstract category is • represented by morphisms of some category [288]. One says that K G ObR is a terminal object (resp. an initial object) of a category R if Horn (K1, K) (resp. Horn (K, K')) is a singleton set for any K' &R. Two objects K and K' of a category R are called isomorphic if there exist morphisms a : K —> K1 and /3 : K' —> K such that (3 o a — Id K and aof3 = I&K'. A category £ is said to be a subcategory of a category R if: (i) Ob £ C Ob R and Mor € C Mor R, (ii) the identity morphisms and composition of morphisms in a category C are induced by those in a category R. A subcategory € C R is called full if, whenever K and K' are objects of £, there is a bijection Hom£(JT, K') = Hom^-ft", K'). One can associate to any category R the dual one R* such that Ob R* = Ob R and Horn R. (K, K') = Horn R(K', K) for any K, K' e Ob R. Example 10.1.3. Sets and functions (i.e., mappings of sets) make up the category of sets Ens. Any singleton set is its terminal object. •

Chapter 10 Appendixes

543

Example 10.1.4. Topological spaces and continuous maps constitute the category of topological spaces Top. It is a subcategory of the category Ens of sets. D Example 10.1.5. Let A be a (non-commutative) ring. Left (resp. right) ^-modules and their morphisms make up the category ^9Jl (resp. VJIA) of left (resp. right) A-modules. Accordingly, the category A^A of Abimodules is defined. •

Example 10.1.6. Given a category 8, by Ar8 is denoted the category of morphisms of 8 whose objects are morphisms a G Mor 8 and whose morphisms a —> a' are pairs (7,7') of morphisms 7,7' G Mor 8 such that 70 a = a' 07'. In particular, morphisms of a category 8 are called isomorphic if they are isomorphic as objects of the category of morphisms Ai\£. D A morphism F : 8 —> £ from a category 8 to a category £ is said to be a covariant functor (or, simply, a functor) if it obeys the following conditions: • F{K) G Ob£ for any K G Obi?, • F{a) G Eom (F{K),F(K')) C MorC for any a e Horn{K,K') C Mor£, • F(ldK) = Id F(K) for any K G Ob£, • any composition of morphisms a o @ G Mor 8. yields the composition of morphisms F(a) o F(/3) = F ( a o ^ ) . A contravariant functor & —> £ is defined as a covariant functor £* —> £. The composition of two covariant functors and the composition of two contravariant functors are covariant functors, while the composition of covariant and contravariant functors and that of contravariant and covariant functors are contravariant functors. Let F and F' be covariant functors from a category 8. to a category £. A functorial morphism (a natural transformation) h : F —> F', by definition, associates to each object K of 8. a morphism h(K) : F(AT) -> F'(A") of £ such that, for each morphism a : K ^> K', the equality F'{a) o h(K) = &(#') o F(a) holds. If all morphisms h(K), K G Ob 8, are isomorphisms, /i is called the functorial isomorphism (the natural isomorphism). If .F and F' are

544

Geometric and Algebraic Topological Methods in Quantum Mechanics

contravariant functors from R to £, their natural transformation h is defined in a similar way, but obeys the equality F\a)oh{K') = h{K)oF{a). In particular, let F : R —>fi.be a functor. There exists a functorial morphism Id 8. —> F if and only if F possesses the property

(F o a){K) = (F(a) o F)(K) for all K £ Ob£ and a £ Horn (AT, .). Example 10.1.7. Let & and £ be arbitrary categories. Then functors & —» £ and their morphisms make up the category Hom(^, £). • Categories ^ and £ are called equivalent if there exist functors F:j?->(Z:,

$:£-£

and functorial isomorphisms h: Id.ft->$oF,

h': I d £ ^ F o $

such that the functorial morphisms F-Foh(Id&),

F->/i'(Id£)oF

of F to F o $ o F coincide with each other. Then the functorial morphisms $->$o/i'(Id€),

$->/i(Id£)o$

of $ to $ o F o $ also coincide. These functors F and $ are called the quasi-inverse to each other. They are the inverse to each other if $ o F = Id£,

Fo$=Id£.

In this case, categories R and £ are called isomorphic. More generally, functors F : £ —> € and $ : £ -> ^ are said to be adjoint if there exist functorial morphisms /i:Id£->$oF,

/i":Fo$-+Id£

such that the compositions F->Fo /i(Id£) = F o $ o F -> /i"(F o $) o F = F, $ -* h(Id&) o $ = $ o F o $ ^ $ o /i"(F o $) = $

Chapter 10 Appendixes

545

are the identity morphisms of the functors F and $, respectively. The definition of an (oriented) graph repeats items (i) and (ii) of that of a category whose classes of objects and morphisms are sets. Accordingly, morphisms between graphs are defined. They are called diagrams. A graph .ft is called finite if Ob R and Mor .ft are finite sets. A graph R is called small if Ob .ft and Mor .ft are subsets of a universal set. Graphs and diagrams make up a category. Given a small category .ft, its objects Ob .ft and morphisms Mor .ft make up a graph called the graph of a category R. A functor between categories is also a diagram of their graphs. Thus, assigning to each small category its graph, we have the functor from the category Cat to the category of graphs. A diagram F : R —» £ is said to be a diagram of type R over a graph £. This diagram is called finite (resp. small) if a graph R is finite (resp. small). Let F and F' be diagrams of the same type over the same graph. The definition of a morphism from F to F' repeats that of a functorial morphism. Accordingly, all diagrams of type R over a graph £ and their morphisms make up a category Hom(.ft, £). Let ?r : R —> £ be a diagram over a small category £, and C £ Ob £ an object of £. By a protective cone of TT with the vertex C is meant a family {PK} of morphisms PK £ Mor £ such that: • this family is indexed by objects K e Ob R of the graph R,

• fee Horn (C,7r(iO),

• given a morphism F of the graph R, we have the relation f3p(K) = C such that f3'K = / o /3R- for all K (E Ob R. This limit is unique up to an isomorphism. If any diagram over a sma21 category £ of type R has a limit, one says that £ admits limits of type R. If £ has limits of any small (resp. finite) type R, one says that R admits limits (resp. finite limits). A family {Ci,i £ 1} of objects of a category £ can be regarded as a diagram over £ whose type is the graph R such that Ob .ft = /, while Mor .ft = 0 . A limit of this diagram is called the direct product of the objects Ci- For instance, the direct product in the category of sets is the familiar Cartesian product of sets. Let 7r : R —> £ be a diagram over a small category £, and C € Ob £ an object of £. By an inductive cone of -n with the vertex C is meant a family

546

Geometric and Algebraic Topological Methods in Quantum Mechanics

{CXK} of morphisms Q ^ 6 Mor £ such that: • this family is indexed by objects K e 0 b £ of the graph £, • ak aKGHom(n(K),C), • given a morphism F of the graph £, we have the relation an = aF(K)°K(F). All inductive cones of a diagram n make up a category. An inductive cone ({a/s-},C) of TT is said to be an inductive limit of a diagram •K if it is an initial object of this category, i.e., for any inductive cone {{ot'K},C'), there exists a unique morphism g : C —> C' such that a'K = g o an for all K £ Ob R. An inductive limit is unique up to an isomorphism. If any diagram over a small category £ of type 8. has an inductive limit, one says that £ admits inductive limits of type &. If £ has inductive limits of any small (resp. finite) type £, one says that 8. admits inductive limits (resp. finite inductive limits). A family {Ci,i £ 1} of objects of a category £ can be regarded as a diagram over £ whose type is the graph & such that Ob& = /, while Mor.ft = 0. An inductive limit of this diagram is called the direct sum of the objects d. For instance, the direct sum in the category of sets (resp. groups and modules) is the disjoint union of sets (resp. the direct product of groups and the direct sum of modules). 10.2

Hopf algebras

In this Section, the relevant basics of Hopf algebras is summarized [98; 293]. Let A be a complex unital algebra, i.e., a C-ring. The tensor product A <8> A of algebras A is defined as that of vector spaces A provided with the multiplication (a ® b) <S> (a' ® b') = (aa1) ® (bb1).

Let us write the multiplication operation of the algebra A as a C-linear morphism m : a ®b —> ab,

a,b G A.

A coalgebra A is denned as a vector space A provided with the following linear morphisms: • a coassociative comultiplication A : A —> A <8> A, • a counit e : A —• C,

547

Chapter 10 Appendixes

which obey the relations (A<8»Id)oA = (Id <8>A)oA,

(eId) o A = (Id e) o A = Id. (10.2.1)

As a shorthand, one writes A(a) = ^2 a (1) o (2 ),

(A Id) o A(a) = ^ a{1) ® a (2) <E> a (3) . (10.2.2)

A comultiplication and a counit extend to tensor products pairwise, i.e., A(a ® a') = A(a) ® A(a'),

e(a ® a') = e(a)e(a').

A bi-algebra (A, m, A, e) is defined as an associative algebra A which is also a coalgebra so that A(1) = 1 ® 1 ,

(a6)(2) = 5Z( a d) fo (i)) ® (a(2)^(2)),

^2(ab){1) e(l) = 1,

o, 6 G A

e(ab) = e(a)e(b).

A Hopf algebra (A, m, A, e, 5) is a bi-algebra A endowed with a linear morphism S : A —> A, called the antipode, such that m ^ S ® Id )A(o)) = m((Id ® 5)A(a)) = e(o)l. It obeys the relations 5(1) = 1,

S(ab) = S{b)S(a),

e o 5 = e,

A o 5 = Po(5(2)5)oA,

a,b £ A,

where P : a6 — i > 6(g>a is the transposition operator. Let us note that, given a bi-algebra A, there is a unique antipode, if any, such that A becomes a Hopf algebra. For the sake of brevity, we call A, e and S the co-operations of a Hopf algebra. A Hopf algebra is said to be cocommutative if P o A = A. If a Hopf algebra is commutative or cocommutative, then S2 = Id . If (A, m, A, e, S) is a Hopf algebra whose antipode 5 is invertible, then A1 = (A,m,PoA,e,S-1) is also a Hopf algebra.

(10.2.3)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

Let A be an involutive algebra. A Hopf algebra (A, m, A,e, S) is said to be involutive if

A(a) = A(a) = 53°(D ® a(2)> e(o)=i(Z).

S(5(a))=a-

A Hopf algebra is called quasi-triangular if there exists an invertible which obeys the relations element H = ri®rlcA®A (A ® Id )(H) = H131Z23,

(10.2.4a)

(Id <8> A)(H) = 11x21123,

(10.2.4b)

1

(PoA)(a) = R A ( a ) r ,

(10.2.4c)

a £ A,

where K12 ••= rk ® rk ® 1,

1llz:=rk®l®rk,

H23 := 1 ® rk ® rk.

The element 1Z is said to be the universal R-matrix of a Hopf algebra A. It satisfies the quantum Yang-Baxter equation ^12^13^23 =

ft237?43ftl2-

(10.2.5)

One can show that (5®Id)(7e)=7e- 1 ,

(Id®Sr)(^-1)=^,

(5
The antipode of a quasi-triangular Hopf algebra is always invertible. Example 10.2.1. Let g be a finite-dimensional Lie algebra whose basis is {efc}. Its universal enveloping algebra g is provided with the structure of a cocommutative Hopf algebra Ug, called the classical Hopf algebra, with respect to the co-operations A(efc) :=ek®l

+ l<S>ek,

e(ek) := 0,

S(ek) := -ek,

extended by linearity to jj. It is a quasi-triangular Hopf algebra where

•

n = i®i.

Example 10.2.2. Let G be a finite group and CG the complex group ring (see Section 1.5). It is brought into the cocommutative Hopf algebra, called the group Hopf algebra, characterized by the co-operations A(s):=s<8>0,

e(s):=l,

S{g):=g-\

g&G,

549

Chapter 10 Appendixes

extended by linearity to CG.

•

Example 10.2.3. Given a finite group G, let us consider the algebra C(G) = C G of complex functions on G. Let us identify C(G)®C(G)=C(GxG). Then the algebra C(G) is a commutative Hopf algebra with respect to the co-operations A(/)(S,S') :=/(

£(/):=/(!),

S(f)(g) := /(fT 1 )

(10.2.6)

for all / € C(G). It is called the group function Hopf algebra.

•

Example 10.2.4. The Hopf algebra Uq(b+), where q is a non-zero real number, is generated by the elements 1, a, g and g~x obeying the relations gg~x = g~lg = i, 30 = w , A{g)=g®g, A^'1) A(o) = a 1 + 5® a, e(a) = 0, e(5) = e^" 1 ) = 1, 5(a) = - 5 " 1 a , S{g) = g-1, 5( 5 - 1 )=ff.

= g'1 ® g-1,

a Example 10.2.5. Let g be a finite-dimensional semisimple Lie algebra. It yields an associated complex Lie group G of n x n complex matrices b which obey a family of polynomial equations p(b) = 0. Correspondingly, we have an algebraic variety with the coordinate algebra C[G] of polynomials C[6J] in n2 variables modulo the relations p(b) = 0. It is a commutative Hopf algebra with respect to the co-operations

A(6j) = 6j®6,

e(6}) = 4

Its antipode is given algebraically via a matrix of cofactors of the matrix b), i.e.,

s{b\)tf
-CD

550

Geometric and Algebraic Topological Methods in Quantum Mechanics

and obey the polynomial equation AdetA = (detA)A = A, i.e., det^l = ad — cb is the unit element. The co-operations of this Hopf algebra can be written in the compact form

e(A) = (JJ) ,

A(A) =A®A,

S(A) = ^ = (

It is quasi-triangular Hopf algebra where 71 = 1 ® 1.

^ . •

Given a Hopf algebra (A,m,A,e,S), let us consider the space A = Horn (A, C) of complex linear forms on A. It is a complex ring with respect to the convolution product //':= (/»/')«A, ( / / »

/,/'£/,

(10.2.7)

= £/(<(i))/'(a(2)). " £ A

The unit of A is the counit e of A. Sometimes, the notion of the convolution product (10.2.7) includes the linear morphisms fa:=

[(Id® / ) o A}(a) = ^ a ( 1 ) / ( a ( 2 ) ) ,

a / : = [ ( / ® Id) o A](a) = ^ /(a ( i))a( 2 ),

(10.2.8) « e A

of the complex space A [439]. There are the relations (//')(«) = / ' ( « / ) = / ( / ' « ) .

« 6 A fJ'eA"-

(10-2.9)

A Hopf algebra (.4', m', A', e', 5") is said to be dually paired with a Hopf algebra (^4, m,A,e,S) if there exists a non-degenerate interior product (.,.):A®A'^C which satisfies the relations (a,a'b') = (A(a),a'®b'),

a,b € A,

(a b, A'(a')) = (ab, a'), /

c'(o') = (l,a >,

c(o) = (o,l>,

(a,S'(a')) = (S(a),a'),

a',b' £ A', (10.2.10a) (10.2.10b) (10.2.10c) (10.2.10d)

where (,) extends to tensor products pairwise, i.e., (a ® b,a' ® 6') = {a,a')(b,b').

(10.2.11)

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Chapter 10 Appendixes

If A is an involutive Hopf algebra, the involutive Hopf algebra A' dually paired with A should satisfy the additional condition

(a,a') = (a,(S'(a')r). For instance, the Hopf algebra Uq(b+) in Example 10.2.4 is dually paired with itself by (a,a) = l,

(g,g) = q,

(a,g) = (g,a) = 0.

If g is a finite-dimensional complex semisimple Lie algebra represented by nxn matrices p(a), a e g, the Hopf algebras Ug in Example 10.2.1 and C[G] in Example 10.2.5 are dually paired with respect to the interior product (10.2.12)

(a,b)) := pia)).

Let a Hopf algebra (A, m, A, e, S) be a finite-dimensional vector space, and let A' be its algebraic dual. Then the equality (10.2.11) yields an isomorphism (A® A)' = A' ®A'. In this case, the relations (10.2.10a) - (10.2.10d) provide A' with a unique Hopf algebra structure (m', A', e', S'), called the dual Hopf algebra. One says that a Hopf algebra H. acts on a complex algebra A on the left (or A is a module over a Hopf algebra 7i) if 7i acts h > a on A as a vector space such that h>(ab) = 53(/i(i)>a)(/i ( 2 )>&), h>l = e{h)l,

a,b £ A,

heH,

i.e., the multiplication and the unit in A commute with the action of H. In particular, any Hopf algebra acts on itself as an algebra. Namely, the left and right adjoint action of a Hopf algebra TL on itself is given by the formulae h > ti : h ® ti H-> J2 hwh'S(h(2)),

(10.2.13)

h'^5(/i (1 ))/i'/i(2).

(10.2.14)

For instance, the left and right adjoint action of the classical Hopf algebra Ug in Example 10.2.1 reads a>b=[a,b],

b
a,b e g.

(10.2.15)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

The left adjoint action of the group Hopf algebra CG in Example 10.2.2 coincides with the adjoint action of the group G on itself. Since the Hopf algebras C(G) in Example 10.2.3 and C[G) in Example 10.2.5 are commutative, their adjoint action is trivial. If a Hopf algebra Ti' is dually paired with a Hopf algebra Ti, its action on H reads h'>h = ^2h{1)(h',h{2)),

ti<=H',

heH.

(10.2.16)

For instance, the Hopf algebra C[G] acts on the classical Hopf algebra Ug by the law lp{a)ij+a6ij.

b)>a =

A coalgebra (A, A, e) is said to be a module of a Hopf algebra H if it is an Ti-module as an algebra and, additionally, the relations A(/i > a) = ^2{h(i) > a (1) ) (/i(2) > o ( 2 ) ),

e{h > a) = e{h)e{a)

hold for all h G H and o 6 A A left coaction of a Hopf algebra Ti on a vector space V is given by a map (3 : V -> K ® V such that (Id ®/?)o/3 = (AId)o/?,

( e ® I d ) o / 3 = Id.

(10.2.17)

A right coaction V —> V <2>H of TC on V is similarly defined. We write P(v) = ^ i J ( 1 ) ® w(2),

U(1) e W,

v{2) 6 V.

(10.2.18)

One says that V is a (left) comodule over a Hopf algebra 7i. For instance the trivial comodule is C where /3(A) = lA,

AeC.

If V is a W-bimodule, the additional relations p(hvti) = A(h)P(v)A(ti),

h, ti € W,

(10.2.19)

hold. If an element v eV satisfies the equality P(v) = l®v, one says that v is left-invariant.

(10.2.20)

553

Chapter 10 Appendixes

A Hopf algebra Ji coacts on a unital algebra A if A is a W-comodule and P is an algebra homomorphism of A to the algebra H <8> A, i.e., (3(ab) = P(a)p(b),

P(1A) = 1 ® U ,

a, b G A

Then A is called a (left) H-comodule algebra. A glance at the formulae (10.2.1) and (10.2.17) shows that any Hopf algebra coacts

on itself as an algebra. A Hopf algebra coacts H on a Hopf algebra A if it coacts on A as an algebra and, additionally, this coaction commutates with co-operations of A. In particular, any Hopf algebra coacts on itself on the left as a coalgebra by the law adj(ft) = J2hWS(h(3))

® h(2)

(10.2.21)

(see the shorthand (10.2.2)). Accordingly, the right coaction reads adP(/i) = 5^/1(2) ®5(/i (1) )/i (3) . 10.3

(10.2.22)

Groupoids and Lie algebroids

Though a groupoid can be defined as a small category whose morphisms are invertible, we follow the terminology of [287; 367]. A groupoid is a set <5 endowed with the following two operations: (i) a partial multiplication <32 3

(x,y)^xy£<5,

where (J52 is a subset of the Cartesian product (3 x (25 called the set of composable pairs, and (ii) the inversion x —> x~l. The set of composable pairs consists of all the pairs (re, x~l), (a;"1, a;) and the pairs (x,y) such that x~lx = yy"1 • Then the above mentioned operations obey the conditions: • ( z - 1 ) - 1 = x; • if (x,y) € 0 2 , then ( y " 1 , ! " 1 ) € <52 and (zy)" 1 =y'1x~1; • if (x,y) G <$2, then (x~x,xy) G <S2 and x~1(xy) = y; • if (x,y) € <S2, then (xy,y~x) € <82 and (xy)y~x — x; • if (x,y),(y,z) € (S2, then (xy,z),{x,yz) € (S2 and (xy)z = x{yz).

554

Geometric and Algebraic Topological Methods in Quantum Mechanics

One calls r(x) = xx~l the range and l(x) = x~xx the domain of x G 0. Thus, a pair (x, y) is composable if and only if the range of y is a domain of x. It is readily observed that l(x) (resp. r(y)) is the left unit (resp. the right unit) on elements y G 0 (resp. i £ 0 ) such that (x,y) G 02. The set 0° = r(<3) = 1(0) is called the um£ space of a groupoid 0. A groupoid & is said to be transitive if the map ( r , ! ) : « 3 i H (xx~1,x~1x) G 0° x 6°

(10.3.1)

is a surjection. This condition holds if and only if, whenever elements x, y G 0, there exists an element z £ 0 such that the pairs (x, z) and (z, y) are composable. A groupoid 0 is called principal if the map (10.3.1) is an injection. This condition is satisfied if and only if, for any two elements x and y ^ x~l of 0, the pairs (x,y) and (j/,:r) can not be simultaneously composable. Let u,v be elements of the unit space 0°. Let 0U = r~ x (u) denote the r-fibre, 0V = l~l(v) the l-fibre, and 0% = 0U n <5V. Each element

of 0 belongs only to one set <5". Given a subset N C 0°, one similarly introduces 0 W = r'^N), 0N = ^ ( / V ) , and 0% = 0N D (Sw. The following is easily justified. • The set <5£ is a group, called the vertex group (or the isotropy group) of an element u. Any element z € <S" yields the isomorphism of the vertex groups 0VV 9 x H^ .zzz-1 G <5£.

(10.3.2)

• If TV c 0°, then 0$ is a subgroupoid of 0 called the reduction of 0 by TV. • The relation R<& c 0° x 0° such that (u, u) € i?© if and only if 0^ 7^ 0 is an equivalence relation, called the equivalence relation of units. Its equivalence classes are said to be orbits of units. • The range of the map (10.3.1) coincides with the graph of the equivalence relation of units. A groupoid 0 is principal if and only if it is identified with the graph of the equivalence relation of units by means of the map (10.3.1). A groupoid 0 is transitive if and only if its unit space 0° consists of one orbit of units. Therefore, an orbit of unit is also called a transitivity component. It follows that any groupoid 0 is a disjoint union of transitive groupoids whose unit spaces are transitivity components of 0°. Example 10.3.1. Let G be a group and S a set. The Cartesian product

555

Chapter 10 Appendixes

5 x G x S can be provided with the following groupoid structure: • (x, g, x')-1 := (x^g-1^); • a pair ((x,g,y), (x',g',y')) is composable if and only if y = x'; • (x,g,x')(x',g',y) = (x,gg',y); • r(x, g, x1) = (x, 1, x) and l(x, g, x') = (x1, l,x'). It is called the trivial groupoid on S with the group G. This groupoid is transitive. In particular, the Cartesian product S x S is brought into the trivial groupoid with the group G = {1}. A group G is a trivial groupoid on any singleton set. Any set 5 can be identified with the diagonal of S x S and regarded as a subgroupoid of the trivial groupoid S x S whose composable pairs are (x,x), i £ l , the partial multiplication is (x,x) — x, and x~l := x. • PROPOSITION 10.3.1. Any transitive groupoid is isomorphic to a trivial • groupoid, though this isomorphism need be not canonical.

Outline of proof. Let 0 be a transitive groupoid. Given a point of u € (25°, let a : (55° —» <$u be the right inverse to the surjection r : <5U —> <S°, i.e., r o<7 = Id 6 ° . Then <8° x <&l x 0 ° 9 (u,x,iu) H^ ^ ^ ( u ) is a desired isomorphism of groupoids over Id 0 ° .

1

€0

(10.3.3) iS-ED

Thus, any groupoid is a disjoint union of transitive groupoids isomorphic to trivial groupoids. Example 10.3.2. Let a group G act on a set 5 on the right. The Cartesian product S x G can be endowed with the following groupoid structure, called the action groupoid: • {s,g)~l

••= (s^.ff" 1 );

• a pair {{s,g), (s',g')) is composable if and only if s' = sg; • {s,g){sg,g') = {s,gg'). It follows that r(s, g) = (s, 1) and l(s, g) = (sg, 1), while the map (s, 1) — i>s identifies the unit space with 5. This groupoid is principal (resp. transitive) • if and only if G acts freely (resp. transitively) on 5. Example 10.3.3. A groupoid 0 is called the group bundle if r(x) = l(x) for all x £ 0 , i.e., its r- and /-fibres 0 " and 0 U coincide with the vertex

556

Geometric and Algebraic Topological Methods in Quantum Mechanics

group 0™ for all u G 0°. Elements of 0 make up a composable pair if and only if they belong to the same fibre 0 " = 0 U = 0 " of the surjection •K

: 0 3 x H-> r{x) = l(x) G 0 ° .

It follows that the graph of the equivalence relation of units is exactly the diagonal of the product 0° x 0°, i.e., u ~ v if and only if u = v. Thus, a group bundle 0 is a disjoint union of the vertex groups whose base is the unit space 0°. Each point of this space is a transitivity component. An injection / : 0° —> 0 such that 7r o / = Id 0° is called a section of a group bundle. By the definition of a groupoid, there is the canonical section 0° —» 0 which sends 0° onto the unit elements of vertex groups. For instance, any group G is a group bundle 0 = G over the singleton unit space 0° = {1}. Any topological bundle Y —> X in groups (a topological group bundle) is a groupoid whose elements make up composable pairs if and only if they belong to the same fibre. Its unit space is the set of unit elements of fibres of Y —> X. D Example 10.3.4. Let IT : 0 —> 0° be a group bundle whose fibres 1 0 U = TT~ (W), u G 0°, are mutually isomorphic. Let Iso0 denote the set of all isomorphisms 7" : 0 V —> 0 U , u,v G 0°. This set possesses the natural groupoid structure such that: • (7")" 1 i s the inverse of 7"; • a pair (7", 72,) is composable if and only if v = q\ . r( 7 £) = U<8U and J(7«) = ld<8v. It is called the frame groupoid. Its unit space is identified with the base 0° of the group bundle by the assignment u H-» Id 0 U . • A homomorphism of groupoids, by definition, preserves their algebraic operations, sends composable pairs onto composable pairs, and maps a unit space to a unit space. Example 10.3.5. The orbits of units of a groupoid 0 make up a trivial groupoid 0°/i?© such that the surjection ip:<& ^ 0 ° —>©%R©

(10.3.4)

is a homomorphism of groupoids. It is readily observed that elements of 0 make up a composable pair if and only if i[> (10.3.4) sends them to the same element [u] G 0°/i?©. Conversely, let us associate to each orbit of

557

Chapter 10 Appendixes

units [u] G <5°/i?0 its representative u G 0°. Then the injection <&°/Re —> (5° C 0 is a homomorphism of groupoids. • Two homomorphisms 7, £ : (5 —» 0 ' of a groupoid 0 to a groupoid 0 ' are said to be similar if there exists a function $ : G5° —> & such that (or)(i)7(i)=((i)($o!)(i),

i£®.

(10.3.5)

Groupoids 0 and 0 ' are called similar if there exist homomorphisms 7 : © — 0 ' and 7' : 0 ' —> 0 such that 7' o 7 and 7 0 7 ' are similar to Id 0 and Id 0', respectively. The following is an important example of similar groupoids. PROPOSITION 10.3.2. If a subset N c 0° meets each orbit of units of a groupoid 0, then 0jy and 0 are similar groupoids. •

Outline of proof. It suffices to consider the case when N meets each orbit of units only once. There is the natural monomorphism iff : 0 $ —> 0. Let us construct an epimorphism 0 —> 0 ^ , which however is not canonical. Given a point u € ATI0 0 , let [u] be the corresponding orbit of units. Then the groupoid 0 ^ is a group bundle over TV whose vertex group over a point u G N is 0JJ. Given the homomorphism tp (10.3.4), one can define the following map

7 : 1>-l(M) ^ KL e t x G V - 1 ( M ) b e l o n g t o a s e t 0 ^ . T h e n t h e r e e x i s t e l e m e n t s y,z G 0 such t h a t

v = yy~1,

w = z~lz,

u = y~1y = zz~1.

It is easily verified that the pairs (z, x) and (x, y) are composable, and the element zxy exists and belongs to the set GJJ. Then 7 : x >-> zxy is a desired morphism 0 —> 0 $ . One can show that 7 is a homomorphism of groupoids and that joiN=ld&%. In order to prove that ijv 07 is similar to Id 0, let us construct the function $ : 0° —> 0 which obeys the condition (10.3.5), i.e., ($or)(i)7(i)=i($o|)(i),

a; G 0.

We put $(u) = y for any element u G [u], where v = yy~x and u = y~1y. QED

558

Geometric and Algebraic Topological Methods in Quantum Mechanics

Let 6 be a groupoid, G a group, and p : © -» G a homomorphism. The skew-product ©G is a groupoid ( 5 x G such that: :={x-l,gp(x)); • (x,g)~l • a pair ((x, g), (x1, g')) is composable if and only if (x, x') is a composable pair and g' = gp(x); • (x,g)(x',gp(x)) = (xx',g); • r(x,g) = (r(x),g) and l{x,g) = (l(x),gp(x)). The unit space of the skew-product © G is © G = ©° x G. Let 0 be a groupoid, G a group, and a : G —> Aut © a homomorphism. Let us denote x-g = a(g-1)(x),

i£«,

g e G.

The semidirect product © x a G is the groupoid © x G such that: • {x,g)-x := (x~l -g,g-1); • a pair ((x, g), (x', g')) is composable if and only if x' = x" • g and the pair (x, x") is composable; • (x,g){x'-g,gr) = {xx',gg')\ • r(x, g) = (r(x), 1) and l(x, g) = (l(x) • g, 1). For instance, let © G be the above mentioned skew-product. There are the canonical action of G on ©Q by the law a(g)(x,g') := (x,gg') and the natural homomorphism p(x, g) = g of the semidirect product © xa G to G. One can show that the groupoids © and ©G X a G are similar. Let © be a groupoid and n : 21 -> 21° = ©° a group bundle. A homomorphism ^ : © —> Iso2t is called a representation of © in 21 such that \t(:r) is some element 7 ^ of the frame groupoid Iso2l in Example 10.3.4. The pair (©,21) together with is called the 0-bundle. If 21 —> ©° is a bundle of Abelian groups, the ©-bundle (©,21) is called the &-module bundle. Unless otherwise stated, operations in a ©-module bundle will be written in the additive form. In particular, the unit space ©° is the zero space 21°. Example 10.3.6. A ©-module bundle 21 —> ©° is called trivial if 21 = ©° x A, where A is an Abelian group, and ^>(x) =ldA for all x € ©. •

559

Chapter 10 Appendixes

One can associate to each 0-module bundle (0, 21) thefollowingcochain complex. Its terms 0°, 0 1 = 0 and 0 2 have been denned. Let 0 n > 2 be n

the subset of elements (x 0 ,..., xn-i) of the product x 0 such that any pair (xi-i,Xi), i = 1,... ,n — 1, is composable. An n-cochain is denned as a map / : 0™ —+ 21 obeying the conditions: • (TT o / ) ( z 0 , . . . , x n _i) = r(xQ); • if n > 0 and Xi € 0° for some i, then f(xo, • • • ,£n-i) e 0°. The set Cn(0,21) of n-cochains is an Abelian group with respect to the pointwise addition. Then we have the following cochain complex 0 -» C°(0,21) ^U • • • Cn{(5,21) ^

• • •,

(10.3.6)

(5°/)(x) = * ( ! ) ( / o O ( a O - ( / o r ) ( x ) , («"/)(a;o, • • • ,xn) = V{xo)f(xu.

..,xn)

+

n

53(-l)V(l O , • • • , li-lXi, . . , ! „ ) + (-l)" +1 /(^0, • • • , In-l).

i=l

Cohomology -ff"(0,2l) of the complex (10.3.6) is called the cohomology of a groupoid & with coefficients in the ©-module bundle 21. For instance, zero-cochains are sections of the group bundle 21 —> 0°. Such a section / is called invariant if the equality (z)(/oJ)(aO-(/or)(a;)=0 holds. Thus, invariant sections are zero-cocycles. There exists a unique zero-coboundary represented by the canonical section 0° = 21° C 21 of a group bundle 21. Therefore, the cohomology group H°(<&, 21) is represented by the above mentioned invariant sections. Remark 10.3.7. It is readily observed that, if 0 and 21 are groups, the complex (10.3.6) is exactly the complex (1.5.8) of the group 0 with • coefficients in the 0-module 21. PROPOSITION 10.3.3. Similar groupoids have the same cohomology with coefficients in a trivial module bundle. •

Let us turn now to topological groupoids. A topological groupoid 0 is a topological space provided with a groupoid structure such that the partial multiplication and surjections r and I are continuous if 0 2 and 0° are equipped with the relative topologies. Let us list the main properties of a topological groupoid.

560

Geometric and Algebraic Topological Methods in Quantum Mechanics

• The inversion 8 3 2; n a;"1 € S is a homeomorphism. • The relative topology of 0° is the finest topology such that r : 0 —> 0° is continuous. • If (5 is Hausdorff, then 0° is closed in 0. • If 6° is Hausdorff, then the r- and /-fibres are closed in 0. • If U is an open subset of (25°, then 0 ^ is open in 0. • For any z € 0 , the left and right translations Lz : &M 3 x ^ zx e <3r{z),

Rz : « r ( z ) 3 x >-> xz € 0 i ( z )

(10.3.7)

are homeomorphisms. It follows that ^-fibres (resp. r-fibres) over points in a common transitivity component of 0° are homeomorphic under right (resp. left) translations. • The isomorphisms (10.3.2) are isomorphisms of topological groups. A topological groupoid 0 is called locally trivial if there exist a point u € 0°, an open cover {Ui} of 0°, and continuous maps cr> : Ui —» 0 U such that r o CTJ = Id f/j for all i. If one can choose C/j = (5°, a topological groupoid is said to be globally trivial. PROPOSITION

10.3.4. A locally trivial groupoid is transitive.

D

Outline of proof. Let v and v' be arbitrary two points of 0°. There are open sets Ui and Uj of the cover of 0° such that v £ Ui and v' G t/,-. Then v ~ u and t1' ~ u. Hence, v ~ v'. QED Furthermore, the map similar to (10.3.3) states an isomorphism of the topological subgroupoid (S^j to the trivial topological groupoid Ui x (5^ x UiA trivial topological groupoid is globally trivial. At the same time, a transitive topological groupoid need not be locally trivial though it is algebraically isomorphic to a trivial groupoid. For instance, a transitive topological groupoid whose unit space is discrete is globally trivial. 10.3.5. Let ® be a topological groupoid. The following conditions are equivalent: • 0 is locally trivial; • r : <5U —> <&° is a continuous surjective submersion (see Remark 10.6.1 below) for any u £ 0°; D • (r, I) : 0 —> <&° x 0° is a surjective submersion. PROPOSITION

Example 10.3.8. Let S x G be a topological action groupoid where G acts

Chapter 10 Appendixes

561

on S transitively. It is locally trivial if and only if the maps G 9 g >—> sg G S, • s G S, are smooth submersions. A Lie groupoid 0 is a smooth manifold provided with a groupoid structure such that the partial multiplication is smooth and the surjections r and / are submersions. Let us note that, in [287], this groupoid is called a differentiable groupoid, while by a Lie groupoid is meant a locally trivial differentiable groupoid. Let us list the main properties of a Lie groupoid. • The inversion <5 3 x >—> x~l G <5 is smooth. • ®2 C 6 x 6 is an imbedded submanifold. • 0° C 0 is a closed imbedded submanifold. • The r- and Z-fibres 0 U and X is a Lie groupoid with respect to the fibrewise addition. • Example 10.3.10. Let 5 x G —> S be a smooth action of a Lie group G on a manifold S on the right. Then the action groupoid 5 x G is a Lie groupoid. Its orbits of units are submanifolds of S. If G acts transitively, • then 51 is equivariently diffeomorphic to a homogeneous space. PROPOSITION 10.3.6. Let 0 be a locally trivial Lie groupoid and TV a transitivity component of 0°. Then there exists a manifold structure on 0 $ such that 0 $ is a submanifold of 0 and a locally trivial Lie groupoid.

•

It follows that a transitive Lie groupoid is locally trivial. Let 0 be a Lie groupoid. Then the tangent bundle T 0 of 0 is also a Lie groupoid, called the tangent groupoid of 0, where the partial multiplication is the tangent morphism T<&2 —> T 0 to the multiplication 0 2 —> 0, and so are the surjections Tr and Tl. The unit space of the tangent groupoid T 0 is the tangent bundle T0° of 0°. Let us turn now to Lie algebroids. Let Z be a smooth manifold. A Lie algebroid on Z is a real smooth vector bundle n : L —» Z such that: • the space L(Z) of sections of L —> Z is a real Lie algebra;

562

Geometric and Algebraic Topological Methods in Quantum Mechanics

• there is a vector bundle morphism g : L —> TZ over Z (called the anchor of L) which obeys the conditions

e(K,€']) = [e(0.e(O], ^ei(z), K,/e'] = /(K,f]) + (e(OJ4f)r, / e c~(Z). A Lie algebroid L is regular if £> is of locally constant rank, transitive if £> is a submersion, and totally transitive if £> = 0. Example 10.3.11. The simplest examples of Lie algebroids are: (i) a regular involutive distribution T C TZ on Z where g is the natural injection • T — TZ, and (ii) a Lie algebra bundle over Z with g — 0. If L is a Lie algebroid on Z and 17 is an open subset of Z, the restriction L\j of the vector bundle L to J7 is also a Lie algebroid, called the reduction of L to U. Example 10.3.12. Let g be a Lie algebra and Z x Q the trivial Lie algebra bundle over Z. Let us consider the pull-back vector bundle L = TZ®(Z xg)

z

together with the anchor

(10.3.8)

prx : L -> TZ. Given sections r © e of L —> Z, the bracket Q=

[r © e, r ' © e'] = [r, r'] © (TJ de' - r'Jrfe+ [e, e']) makes L (10.3.8) to a Lie algebroid, called the trivial Lie algebroid. Let $ : TZ®{Z xg)-* TZ®{Z x g')

z

z

be a morphism of trivial Lie algebroids. Then the condition g' o 3> = g implies that $ takes the form

$( T + e) = T + (0(T) + tf(e)), where ^ is a g'-valued one-form on TZ and (j>: Z x Q —> Z x

Q'

is a morphism of Lie algebra bundles over Z. Writing out the equation $([r © e, T' © e']) = [$(r © e), $ ( r ' © e')]

563

Chapter 10 Appendixes

and setting firstly T = T' = 0, then e = e' = 0, and lastly e = 0, we obtain the following relations [0(c),0(e')]=0([e,e / ]) ) d0(r, r') + [0(T), 0(T')] = 0, drj (e') - ^(rj de') + [rj 0,0(e')] = 0.

(10.3.9a) (10.3.9b) (10.3.9c)

Clearly, the relation (10.3.9a) is the condition that 0 is a morphism of Lie algebra bundles and the relation (10.3.9b) is the Maurer-Cartan equation. The relation (10.3.9c) is called the compatibility condition. • Example 10.3.13. Let E -* Z be a vector bundle and JlE the first order jet manifold of sections s £ E(Z) of E —> Z (see Section 10.6F below). The jet bundle J1E —> Z is also a vector bundle. Linear first order differential operators o n £ - * Z are defined as morphisms JlE —» E of vector bundles over Z. Hence, they are represented by sections of the vector bundle JXE* ®ZE->Z, where JlE* denotes the dual of the vector bundle JlE —» Z. There is the exact sequence of vector bundles Q^TZ®E —>^E —>E->0 z (1.8.18). Its dual is the exact sequence 0->£ —>JlE* —>TZ®E^0, z

which leads to the exact sequence 0^E®E —>J1E®E

z

z

-^TZ®E®E^0.

z

z

1

Let us denote DE — g~ (TZ) the inverse image of the diagonal TZ CTZ®E* ®ZE. z It is readily observed that it is a subbundle of the vector bundle JlE®E^> Z

z

whose sections are first order differential operators D on E which obey the Leibniz rule D{fs) = fD(s) + {e(D)\df)s,

s€E(Z),

feC°°(Z).

(10.3.10)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

They are called covariant differential operators on E —> Z. For instance, any covariant derivative on a vector bundle E —» Z is of this type. It is easy to check that covariant differential operators D £ DE{Z) obey the relations

[D,D'} e DE(Z), Q([D, D')) = [Q(D),Q(D% [DJD'} = f{D,D'} + (e(D)\df)D'. Consequently, DE is a transitive Lie algebroid, called the Lie algebroid of covariant differential operators on E. D Let L be a Lie algebroid on Z and £ - » 2 a vector bundle over Z. A representation of a Lie algebroid L on E is a morphism of L to the Lie algebroid DE in Example 10.3.13. For instance, if E —> Z admits a trivial connection T, there is a representation of the Lie algebroid TZ on E by the covariant derivatives T H-> V£. One can associate a Lie algebroid to any Lie groupoid as follows. Let © be a Lie groupoid and Tr<& the vertical tangent bundle of the fibration r : (5 —> <5°. Let us consider the restriction V<5 of T""<9 t o 0 o c 0 . There is the vector bundle morphism TLz-i

: Tz<Sr{z) -> Tl(z)<5l{z\

ze<&,

of Tr<3 to V<& over I : <S —> <S°, where TL is the tangent morphism to the left translation L (10.3.7). A vector field on a Lie groupoid is called vertical if it lives in T r 0 . A vertical vector field £ is said to be left-invariant if

S(zx) = TLz£(x) for all (z,x) € (3 2 . It is readily observed that a vertical vector field £ is left-invariant if

S(z) = TLzZ(l(z)) for all z G 0 . Thus, a left-invariant vector field on a Lie groupoid <J5 is determined by its values on the unit space (5°. Let us denote by Q the set of left-invariant vector fields on <S. It is a C°°(<5°) module which is isomorphic to the module V<25(6°) of section of the vector bundle V<5 -> (5°. The Lie bracket of left-invariant vector fields on (5 is also a left-invariant vector field. This bracket provides V<25(<5°) with the structure of a real Lie algebra such that

565

Chapter 10 Appendixes

for all f, £' G ^ 0 ( 0 ° ) and / G C°°(0 0 ). Then it is easily justified that the morphism Tl : TT<& DV<8^

T<5°

is an anchor of the Lie algebroid V& —> (25°. This is called the left Lie algebroid of a Lie groupoid <&. Similarly, the right Lie algebroid of (5 is defined. These Lie algebroids are locally trivial independently of local triviality of <S. 10.4

Algebraic Morita equivalence

This Section is devoted to the notion of the Morita equivalence of rings and Lie groupoids. Throughout this Section, /C is a commutative ring. Let A and B be /C-rings, and let ^9Jl and #501 denote the categories of left A- and S-modules, respectively. These rings are called Morita equivalent if the categories ^OT and sSDT are equivalent [52; 78], i.e., there exist quasiinverse functors F : e 9H -> .ASDT,

F : AWl ^

FoF^ldA9Jl,

FoF^UB9Jl.

(10.4.1)

BDJt,

Theorems 10.4.1 and 10.4.2 below state that one can construct the functor F (10.4.1) by means of tensor products of (A - Z?)-bimodules. Remark 10.4.1. Given rings A, B and C, let APB and BQC be (A — B)and (B — C)-bimodules, respectively. Their tensor product APB I BQC

(10.4.2)

is defined a s a n additive g r o u p g e n e r a t e d b y elements m o d u l o t h e relations (p + p') ® q = p <8> q + p' <E> q, pb ® q = p ® bq,

p®q,p£P,q£Q,

p® {q + q') = p® q + p® q',

p,p' € P,

q,q'&Q,

b G B,

and it is provided with the (A — C)-bimodule structure a(p ®q) = (ap) ®q, (p®q)c = p®(qc),

p G P, cGC.

q G Q,

a G A,

566

Geometric and Algebraic Topological Methods in Quantum Mechanics

Similarly, the tensor product of an (A— S)-bimodule P and a left B-module Q is a left .4-module, and the tensor product of a right .4-module Q and an (.4 — S)-bimodule P is a right B-module. • Any A — S)-bimodule APB yields the functor FP : Bm -» ADJl,

(10.4.3)

FP{BM) =A PB ® BM, FP{a) :APB® p®a(m)

BM

G ObBm,

BM 3p®mt-^> EAPB®

BM',

a G Horn

(BM,BM').

Furthermore, one can show that, given (A — Z?)-bimodules P and P', the corresponding functors Fp and Fp> are equivalent if and only if the bimodules P and P are isomorphic. It follows that, given an (A — 8)-bimodule P and a (B — ,4)-bimodule Q, the functors

FP

: Bm -> Am,

FQ: Am-^ Bm

are quasi-inverse if and only if there exist bimodule isomorphisms APB ® BQA = A B

BQA ® APB = B,

(10.4.4)

A.

where A and B are regarded as A- and S-bimodules, respectively. THEOREM 10.4.1. Two /C-rings A and B are Morita equivalent if and only if there exist (A — B)- and B — .A)-bimodules possessing the property (10.4.4). •

In order to say something more, let us consider the category Alg of /Crings where the set of morphisms Horn (S, A) is the set Bim(.4, B) of classes of isomorphic (A — $)-bimodules. The tensor product (10.4.2) of bimodules yields the composition Bim(.4, B) x Bim(B, C) -> Bim(A C),

(10.4.5)

which is associative on classes of isomorphic bimodules. The identity element of Horn (A, A) = Bim(.4, A) is represented by the ring A itself seen as an ^-bimodule. Accordingly, an invertible element of Horn (B, A) = Bim(.4., B) is represented by an (A - S)-bimodule (called invertible in Alg) which provides the isomorphisms (10.4.4). Thus, the notion of the isomorphism in the category Alg coincides with the Morita equivalence. Let us refer to the following theorems [52].

Chapter 10 Appendixes

567

10.4.2. A functor F : g9Jt —> ^371 is an equivalence of categories if and only if there exists an invertible (^4 — S)-bimodule P such that there is the natural isomorphism F = Fp. D THEOREM

10.4.3. An (.4 - #)-bimodule APB is invertible if and only if it is finitely generated and projective as a left ,4-module ^P and as a right fi-module PB, and A —> End(Pg) and B -> E n d ^ P ) are algebra isomorphisms. • THEOREM

In particular, isomorphic rings are necessarily Morita equivalent. Similarly, the Morita equivalence of Lie groupoids <& and f) is defined in terms of ( 0 - £)-bibundles [78; 315; 325]. We will use the standard notation XxY z

:= {(x,y) G X x Y : n(x) = r(y)}

(10.4.6)

for the fibre product of sets X and Y with respect to maps n : X —> Z and r : Y -> Z. Let (5 and S) be Lie groupoids (see the notation in Section 10.3) and X a smooth manifold. Let us require the following. (i) There are smooth maps r : X —> (5° and a : X —> Sj°. (ii) There is the left ©-action 0 lxXB (g,x)^gx£X

(10.4.7)

along the map a, i.e., a{gx) = a(x). (iii) There is the right i>action xTfi3(i,li)Hi/ieX

(10.4.8)

along the map T, i.e., r(xh) — T(X). (iv) The actions (10.4.7) and (10.4.8) mutually commute, i.e., (gx)h = g(xh) for all composable pairs (g, x) and (x,h). If the conditions (i) - (iv) hold, one says that X is a (<5 — Sj)-bibundle. The morphisms r and a in item (i) are sometimes called the moments. A (<3 — .ft)-bibundle is said to be (left) principal if: (i) a is a surjecive submersion (i.e., a fibred manifold), (ii) the (5-action (10.4.7) is free (i.e., gx = x, x E X, if and only if g e 0°), and (iii) this action is transitive

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along the fibres of a. Equivalently, the map &lX X 9 (g,x) H-> (gx,x) GX X X is a diffeomorphism. A principal {<& — i^)-bibundle X is called regular if the right action (10.4.8) is proper. Let us note that the left action (10.4.7) in a principal bibundle is automatically proper. Two (<25 — ,fj)-bibundles are isomorphic if there is a diffeomorphism between them which intertwines their moments and the groupoid actions. Similarly to the composition (the tensor product (10.4.2)) of bimodules, let us define the composition of principal bibundles. Let X be a principal (<25 — J5)-bibundle with the moments <8°^-X

^f)°,

and let Y be a principal (Sj — A)-bibundle with the moments

a,a

The fibre product X x Y carries the right action h : (x,y) H-»

(xh,h~1y)

of the groupoid Sj. Then the tensor product of the principal bibundles X and Y is denned as the orbit space XY

= XOxY/9),

(10.4.9)

which is provided with the principal ((5 — ^)-bibundle structure. One can think of the tensor product (10.4.9) as being the composition of principal bibundles. This composition is associative up to isomorphisms. Therefore, let us consider the category LG which has Lie groupoids as object and the classes of isomorphic principal ((25 — .£j)-bibundles as morphisms f) —> (25, (25, S) € ObLG. In particular, the unit element of Horn ( 0 , 0 ) is represent by the principal ((25 — <25)-bibundle X = <& with the moments r = r and a = I. Sometimes, one additionally requires that principal bibundles in the category L G are regular [270]. Just as in the case of rings, we say that two Lie groupoids are Morita equivalent if they are isomorphic in the category LG, i.e., if they are linked by means of an invertible principal bibundle, whose class is an invertible morphism in LG. One can show that a (left) principal bibundle is invertible

569

Chapter 10 Appendixes

if and only if it is also right principal. In particular, an invertible principal bibundle is necessarily regular. 10.5

Measures on non-compact spaces

Subsections: A. Infinite-dimensional topological vector spaces, 569; B. Measures on locally compact spaces, 572; C. Haar measures, 578; D. Measures on Hausdorff spaces, 581; E. Measures on infinite-dimensional vector spaces, 583. In this Section, we summarize some relevant material on measures on locally compact spaces, locally compact groups, Hausdorff spaces, and infinite-dimensional topological vector spaces. A. Infinite-dimensional topological vector spaces There are several standard topologies introduced on an infinitedimensional (complex or real) vector space and its dual [374]. Topological vector spaces throughout the book are assumed to be locally convex. Unless otherwise stated, by the dual V of a topological vector space V is meant its topological dual, i.e., the space of continuous linear maps of V —» R. Let us note that a topology on a vector space V is often determined by a set of seminorms. A non-negative real function p on V is called a seminorm if it satisfies the conditions p(Xx) = \X\p(x),

p(x + y)
x,yeV,

A € R.

A seminorm p for which p(x) = 0 implies x = 0 is called a norm. Given any set {pi}i£i of seminorms on a vector space V, there is the coarsest topology on V compatible with the algebraic structure such that all seminorms Pi are continuous. It is a locally convex topology whose base of closed neighborhoods consists of the set {a; : sup Pi(x) < e},

e > 0.

l
Let V and W be two vector spaces whose Cartesian product V x W is provided with a bilinear form (v,w), called the interior product, which obeys the following conditions: • for any element v ^ 0 of V, there exists an element w e W such that <«,tu)^0;

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Geometric and Algebraic Topological Methods in Quantum Mechanics

• for any element w ^ 0 of W, there exists an element v e V such that Then one says that (V, W) is a dual pair. If (V, W) is a dual pair, so is (W, V). Clearly, W is isomorphic to a vector subbundle of the algebraic dual V* of V, and V is a subbundle of the algebraic dual of W. Given a dual pair (V, W), every vector w 6 W defines the seminorm pw = \{v,w)\ on V. The coarsest topology a(V, W) on V making all these seminorms continuous is called the weak topology determined by W on V. It is also the coarsest topology on V such that all linear forms in W c V* are continuous. Moreover, W coincides with the (topological) dual V of V provided with the weak topology a(V, W), and a(V, W) is the coarsest topology on V such that V — W. Of course, the weak topology is Hausdorff. For instance, if V is a Hausdorff topological vector space with the (topological) dual V, then (V, V) is a dual pair. The weak topology a(V, V) on V is coarser than the original topology on V. Since (V, V) is also a dual pair, the dual V of V can be provided with the weak* topology a(V',V). Then V is the dual of V, equipped with the weak* topology. The weak* topology is the coarsest case of a topology of uniform convergence on V. A subset M of a vector space V is said to absorb a subset N C V if there is a number e > 0 such that N C AM for all A with |A| > e. An absorbent set is one which absorbs all points. A subset N of a topological vector space V is called bounded if it is absorbed by any neighborhood of the origin of V. Let (V, V") be a dual pair and N some family of weakly bounded subsets of V. Every N C TV yields the seminorm pN(v') = sup|(u,u')l on the dual V of V. The topology on V defined by the set of seminorms PN, N € A/", is called the topology of uniform convergence on the sets of J\f. When A/" is a set of all finite subsets of V, we have the coarsest topology of uniform convergence which is the above mentioned weak* topology cr(V', V). The finest topology of uniform convergence is obtained by taking Af to be the set of all weakly bounded subsets of V. It is called the strong topology. The dual V" of V, provided with the strong topology, is called the bidual. One says that V is reflexive if V = V". Since (V, V) is a dual pair, the vector space V can be also provided with the topology of uniform convergence on the subsets of V, e.g., the weak* and strong topologies. Moreover, any Hausdorff locally convex topology on

571

Chapter 10 Appendixes

V is a topology of uniform convergence. The coarsest and finest topologies of them are the weak* and strong topologies, respectively. There is the following chain weak* < weak < original < strong of topologies on V, where < means "to be finer". For instance, let V be a normed space. The dual V of V is also equipped with the norm \\v'\\':= sup \{v,v')\, Nl=i

VGV,

v'GV'.

(10.5.1)

Let us consider the set of all balls {v : \\v\\ < e, e > 0} in V. The topology of uniform convergence on this set coincides with strong and normed topologies on V because weakly bounded subsets of V are also bounded by the norm. Normed and strong topologies on V are also equivalent. Let V denote the completion of a normed space V. Then V is canonically identified to (V)' as a normed space, though weak* topologies on V and (V)' are different. Let us note that both V and V" are Banach spaces. If V is a Banach space, it is closed in V" with respect to the strong topology on V" and dense in V" equipped with the weak* topology. One usually considers the weak, weak and normed (equivalently, strong) topologies on a Banach space. It should be recalled that topology on a finite-dimensional vector space is locally convex and Hausdorff if and only if it is determined by the Euclidean norm. In conclusion, let us say a few words about morphisms of topological vector spaces. A linear morphism between two topological vector spaces is called weakly continuous if it is continuous with respect to the weak topologies on these vector spaces. In particular, any continuous morphism between topological vector spaces is weakly continuous [374]. A linear morphism between two topological vector spaces is called bounded if the image of a bounded set is bounded. Any continuous morphism is bounded. A topological vector space is called the Mackey space if any bounded endomorphism of this space is continuous (we follow the terminology of [374]). Metrizable and, consequently, normed spaces are of this type. Any linear morphism 7 : V —• W of topological vector spaces yields the

572

Geometric and Algebraic Topological Methods in Quantum Mechanics

dual morphism 7' : W —> V of the their topological duals such that (v,i'{w)) = (j(v),w),

v£V,

w € W.

If 7 is weakly continuous, then 7' is weakly continuous. If V and W are normed spaces, then any weakly continuous morphism 7 : V —• W is continuous and strongly continuous. Given normed topologies on V and W, the dual morphism 7' : W —> V is continuous if and only if 7 is continuous. B. Measures on locally compact spaces Measures on a (Hausdorff) locally compact space X are defined as continuous forms on spaces of continuous real (or complex) functions of compact support on X, and they are extended to a wider class of functions on X [55]. Let C°(X) be the ring of continuous complex functions on X. For each compact subset if of X, we have a seminorm PA-(/)=sup|/(x)|

(10.5.2)

xeK

on C°(X). These seminorms provide C°(X) with the Hausdorff topology of compact convergence. We abbreviate with /C(X, C) the dense subspace of C°(X) which consists of continuous complex functions of compact support on X. It is a Banach space with respect to the norm ||/|| = sup \f(x)\.

(10.5.3)

Its normed topology, called the topology of uniform convergence, is finer than the topology of compact convergence, and these topologies coincide if X is a compact space. The space K{X, C) can also be equipped with another topology, which is especially relevant to integration theory. For each compact subset K C X, let ICK(X, C) be the vector subspace of fC(X, C) consisting of functions of support in K. Let U be the set of all absolutely convex absorbent subsets U of K.(X, C) such that, for every compact K, the set U D K-K(X,C) is a neighborhood of the origin in fCx(X, C) under the topology of uniform convergence on K. Then U is a base of neighborhoods for a (locally convex) topology, called the inductive limit topology, on K{X, C) [374]. This is the finest topology such that the injection KK{X,C) —» K.(X,C) is continuous. The inductive limit topology is finer than the topology of uniform

Chapter 10 Appendixes

573

convergence, and these topologies coincide if X is a compact space. Unless otherwise stated, referring to a topology on fC(X, C), we will mean the inductive limit topology. A complex measure on a locally compact space X is defined as a continuous form \i on the space fC(X, C) of continuous complex functions of compact support on X. The value j(x(/), / € K.(X,C), is called the integral J J/J, of / with respect to the measure /it. The space M(X, C) of complex measures on X is the (topological) dual of K,(X, C). It is provided with the weak* topology. Given a complex measure /i, any continuous complex function h € C°(X) on X defines the continuous endomorphism / —> hf of the space tC(X,C) and yields the new complex measure h^{f) = fi(hf). Hence, the space M(X, C) of complex measures on X is a module over the ring C°(X). Let K(X) C K{X, C) denote the vector space of continuous real functions of compact support on X. The restriction of a complex measure fi on X to JC(X) is a continuous complex form on K.(X), equipped with the inductive limit topology. Any complex measure \i is uniquely determined as / i ( / ) = M R e / ) + z/z(Im/),

/e/C(X,C),

(10.5.4)

by its restriction to K{X). A complex measure \i on X is called a real measure if its restriction to /C(X) is a real form. A complex measure fi is real if and only if fi = JL, where /Z is the conjugate measure given by the condition M(/) = M7),

/G/C(X,C).

A measure on a locally compact space X is defined as a continuous real form on the space K{X) of continuous real functions of compact support on X provided with the inductive limit topology. Any real measure on K.(X, C) restricted to K.(X) is a measure. Conversely, each measure \i on K.{X) is extended to the real measure (10.5.4) on K(X, C). Thus, measures on K.{X) and real measures on K{X, C) can be identified. A measure /lona locally compact space X is called positive if /x(/) > 0 for all positive functions / G K(X). Any measure /z defines the positive measure

M(/) := K/)|,

574

Geometric and Algebraic Topological Methods in Quantum Mechanics

and can be represented by the combination

v = \{\n\ + v-)-\{\n\-v) of two positive measures. Any complex measure /J, on a locally compact space X defines the positive measure | M |(/) := sup{|/i(/i)| : h G /C(X,C), \h\ < / }

/ G K{X),

on. X. It is called the absolute value measure. A complex measure /J on a locally compact space X is called bounded if there is a positive number A such that |/u(/)| < A||/|| for all / G JC(X,C). A complex measure /i is bounded if and only if it is continuous with respect to the topology of uniform convergence on K,{X, C). Hence, the space M1(X,C) C M(X,C) of bounded measures is the dual of IC(X,C), provided with this topology. It is a Banach space with respect to the norm | H | = sup{| M (/)| : Il/H = 1}, / e K(X,C)}.

(10.5.5)

Of course, any measure on a compact space is bounded. If /U is a bounded complex measure and h is a bounded continuous function on X, the complex measure h\i is bounded. Similarly, the Banach space Ml(X) of bounded measures on X is defined. In particular, a complex measure n is bounded if and only if the absolute value measure \/J,\ is bounded. Example 10.5.1. while the measure

The Lebesgue measure dx on X = R is not bounded, + OO

2

1

H=(l+x )- dx,

/ '

u = 7r

'

— oo

is bounded.

D

Example 10.5.2. Given a point x £ X, the assignment £:/->/()>

/££().

(10-5-6)

defines the Dirac measure on X. Any finite linear combination of Dirac measures is a measure, called a point measure. The Dirac measure ex is bounded, and \\ex\\ — I. •

575

Chapter 10 Appendixes

Now we extend a class of integrable functions as follows. Let K±oo denote the extended real line, obtained from R by the adjunction of points {+00} and {—00}. It is a ring such that 0 • 00 = 0 and 00 — 00 = 0. Let J+ be the space of positive lower semicontinuous functions on X which take their values in the extended real line R±oo- These functions possess the following important properties: • the upper bound of any set of elements of J+ and the lower bound of a finite set of elements of J+ are also elements of J+; • any function / G J+ is an upper bound of a family of positive functions h £ K(X) such that h
= [ ffj, = s u p { / i ( / i ) : h e K ( X ) , 0 < h < f }

(10.5.7)

of R ± o c . Of course, /•(/) = /i(/) if / £ K{X). Example 10.5.3. Let U be an open subset of X and ipu its characteristic function, i.e., (10.5.8) It is readily observed that tpu £ J+. Given a positive measure / i o n l , the upper integral fi(U) — n{yu) is called the outer measure of U. For instance, the outer measure of a relatively compact set U (i.e., U is a subset of a compact set) is finite. The (finite or infinite) number fx(X) = /u(l) is called the total mass of the measure fx. In particular, a measure on a locally compact space is bounded if and only if it has a finite total mass.

•

Let / be an arbitrary positive R-too-valued function on a locally compact space X (not necessarily lower semicontinuous). There exist functions g £ J+ such that g > f (e.g., g — +00). Then the upper integral of / with respect to a positive measure /J, on X is defined as / / ( / ) = inf{/•(&) : h£j+,h>

/}.

(10.5.9)

Example 10.5.4. The outer measure /J,(V) = lJ.{fv) of an arbitrary subset V of X exemplifies the upper integral (10.5.9). In particular, one says that V C X is a fi-null set if ^{V) = 0. Two R±oo-valued functions

576

Geometric and Algebraic Topological Methods in Quantum, Mechanics

f and / ' on a locally compact space are called ^-equivalent if they differ from each other only on a ;u-null set; then fi(f) = M(/')- Two positive measures /i and / / are said to be equivalent if any compact /x-null set is also a /[/-null set, and vice versa. They coincide if fJ.(K) = fJ,'(K) for any compact set K C X. • Example 10.5.5. A real function / on a subset V C X is said to be defined almost everywhere with respect to a positive measure fi on X if the complement X \ V of V is a /i-null set. For instance, an R-too-valued function / which is finite almost everywhere on X exemplifies a real function defined almost everywhere on X. Conversely, one can think of a positive function defined almost everywhere on X as being ^-equivalent to some positive IR-too-valued function on X. Thereby, its upper integral is defined. However, it should be emphasized that functions defined almost everywhere • on X fail to constitute an additive group. The following classes of integrable functions (and maps) are usually considered. Let / be a map of a locally compact space X to a Banach space F, provided with a norm |.| (e.g., F is M. or C). Given a positive measure \x on X, let us define the positive (finite or infinite) number

NP(f) = \J I/IV] \

l
(10.5.10)

Clearly, Np(f) = Np(f) if / and / ' are /i-equivalent maps on X, i.e., if they differ on a /x-null subset of X. There is the Minkowski inequality NP(f + /') < Np(f) + Np(f). RPF{X,

(10.5.11)

• Let n) be the space of maps X —> F such that Np(f) < +00. In accordance with the Minkowski inequality (10.5.11), it is a vector space and Np (10.5.10) is a seminorm on R^(X,[i). Provided with the corresponding topology, RPF(X, n) is a complete space, but not necessarily Hausdorff. The space IC(X, F) of continuous maps X —> F of compact support belongs to RpF(X,fi). • The space £F(X, /i) is defined as the closure of IC(X, F) c RF(X, /i). Elements of £F(X,n) are called integrable F-valued functions of degree p. In particular, elements of £F(X, fi) are called integrable F-valued functions, while those of £F(X, fi) are square integrable F-valued functions. Any element of RF(X, n) which is /^-equivalent to an element of £F(X, fx) belongs

Chapter 10 Appendixes

577

to CF(X, fi). An F-valued map defined almost everywhere on X is also called integrable if it is /x-equivalent to an element of £PF(X, fi). • The space LF(X,n) consists of classes of //-equivalent integrable Fvalued maps of degree p. One usually treat elements of this space as Fvalued functions without fear of confusion, and call them integrable Fvalued functions of degree p, too. The LF(X,n) is a Banach space with respect to the norm (10.5.10). There are the following important relations between the spaces LF(X,fi), 1 \f\(1/q}~1f provides a homeomorphism between topological spaces CF(X, /u) and CqF(X, fx). Let the numbers 1 < p < +oo and 1 < q < +oo obey the condition p - 1 + g - 1 = l.

(10.5.12)

If / € ^(X, n) is an integrable complex function on X of degree p and / ' G Lqc{X,ii) is that of degree q, then / / is integrable, i.e., belongs to LQ(X, fi). In particular, the space LQ(X,/J,) of square integrable complex functions on a locally compact space X is a separable Hilbert space with respect to the scalar product

(/!/') =///V-

(10.5.13)

One can say something more in the case of real functions. Let numbers p and q obey the condition (10.5.12). Any integrable real function / £ C^X, fj) on X of degree q defines a continuous real form

tf.f'^Jff'n

(10.5.14)

on the space L^(X,/J,) such that Ng(f) = \\f\\- Conversely, each continuous real form on C^(X,fi) is of type (10.5.14) where / is an element of Cft(X,(j.) whose equivalence class in Ljpf./z) is uniquely defined. As a consequence, there is an isomorphism between the Banach spaces L^(X, /z) and (L^X, fj,))', and the Banach space L^(X,n) is reflexive. Remark 10.5.6. One can define the space L^(X, /x) of complex integrable functions on X of infinite degree as the (topological) dual of the Banach space L^(X,fi). In particular, any bounded continuous function belongs to L^(X,fj,). Let us note that the space L^(X,fx) is not reflexive, i.e.,

578

Geometric and Algebraic Topological Methods in Quantum Mechanics

the dual of L^(X, /x), provided with the strong topology, does not coincide with L^(X,fi). a We now turn to the relation between equivalent measures. Let / be a positive R±oo-valued function on a locally compact space X. Given a positive measure fi on X, the quantity £ ( / ) = supn(
(10.5.15)

KCX

where K runs through the set of all compact subsets of X, is called the essential upper integral of / . Since PK/ < / f° r any compact subset K, the inequality /!(/) < p,{f) holds. In particular, if V is a subset of X and V-ifv) = 0) one says that V is a locally pi-null set, i.e., any point x € X has a neighborhood U such that UC\ V is a /x-null set. Essential upper integrals coincide with the upper ones if X is a locally compact space countable at infinity. One says that a function on a subset V of X is defined locally almost everywhere if the complement of V is a locally null set. A real function / defined locally almost everywhere on X is called locally p-integrable if any point x £ X has a neighborhood U such that puf is a ^x-integrable function or, equivalently, if hf is a /x-integrable function for any positive function h G K,{X) of compact support. Let h be a locally /i-integrable function which is defined and nonnegative almost everywhere on X. There is a positive measure hfj, on X which obeys the relation hfi(f) = fi(hf) for any / e JC(X). One says that hfi is a measure with the basis pi and the density h. For instance, if h is a /i-integrable function, then hpt is a bounded measure on X. This construction is also extended to complex functions and complex measures, seen as compositions of the positive real ones. 10.5.1. Positive measures pt and p! on a locally compact space X are equivalent if and only if p! — fpL, where / is a locally ^x-integrable function such that / > 0 locally almost everywhere on X. •

THEOREM

The function / in Theorem 10.5.1 is called the Radon-Nikodym derivative. Of course, Jl'(f') = A(//') f°r a n v positive integrable function / ' on X. C. Haar measures Let us note the peculiarities of measures on locally compact groups [55].

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Chapter 10 Appendixes

Let G be a topological group acting continuously on a locally compact space X on the left, i.e., 7(3) : X 3 x >-> 5 z G X,

g&G,

(10.5.16)

so that this map j(g) is continuous for any g G G, and so is the map (10.5.17)

G3g^gx£X for any x £ X. It should be emphasized that the map

Gx X B (g,x)^

gx GX

need not be continuous. Let / be a real function on X and /i a measure on X. The group G acts on / and /u by the laws

(y(g)f)(x) = fig-'x), (7(fl)M)(/) = /x(7(ff-1)/)The measure 7(5)^ is the image of the measure /i with respect to the map (10.5.16). A measure /i on X, subject to the action of a group G, is said to be: • invariant if j(g)iJ, = fi for all g G G; • relative invariant, if there is a strictly positive number x{9) such that 7(S)M

= x{g)~ V

for each 5 G G; • quasi-invariant, if measures /i and 7(5)^ are equivalent for all g £ G. The strictly positive function 5 —• x(
: X 3 I H

xg'1

G X.

The corresponding transformations of functions and measures on X read

(r(g)f)(x) = f(xg), (T(g)n)(f) = n{T{g-l)f). Then invariant, relative invariant, and quasi-invariant measures on X are defined similarly to the case of G acting on X on the left.

580

Geometric and Algebraic Topological Methods in Quantum Mechanics

Now let G be a locally compact group acting on itself by left and right multiplications r{g) : q i-> qg"1, 7(3) = Q !-> 91, 7(<7I)T($2) = r(g2h{gi).

Accordingly, left- and right-invariant measures, relative left- and rightinvariant measures, left- and right-quasi-invariant measures on the group G are denned. Each measure /J, on G also yields the inverse measure n~l given by the relation

/ f(g)^\g) = I fig-'Ug),

/ e K.{G).

A positive (non-vanishing) left-invariant measure on a locally compact group G is called the left Haar measure (or, simply, the Haar measure). Similarly, the right Haar measure is denned. 10.5.2. A locally compact group G admits a unique Haar measure with accuracy to a number multiplier. The total mass n{G) of a Haar measure \x on G is finite if and only if G is a compact group. • THEOREM

Let us choose, once and for all, a left Haar measure dg on a locally compact group G. If G is a compact group, dg is customarily the Haar measure of total mass 1. Example 10.5.7. The Lebesgue measure dx is the Haar measure on the • additive group G = M. Its inverse is — dx. The equality (10.5.18) shows that, if dg is a Haar measure on G, the measure r(g')dg for any g' G G is also left-invariant. Therefore, there exists a unique continuous strictly positive function A(g') on G such that T(g')dg = A(g')dg,

g' G G.

It is called the modular function of G. If dg is a left Haar measure, its inverse (dg)-1 is a right Haar measure. There is the relation (dg)-1 = &{g)-ldg. If A (g) = 1, a group G is called unimodular. Left and right Haar measures on a unimodular group differ from each other in a number multiplier. For instance, compact, commutative, and semisimple groups are unimodular. There is the following criterion of a unimodular group. If the unit element

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Chapter 10 Appendixes

of a locally compact group has a compact neighborhood invariant under inner automorphisms, this group is unimodular. Measures / i i , . . . ,/xn on a locally compact group G are called mutually contractible if there exists the measure * fii on G given by the relation i

f G K(G). (10.5.19)

ff(9)li(9) = ff(9i---9n)Hi(9i)---Hn(gn), n

It is an image of the product measure /^i • • • \xn on x G with respect to the map x G 9 ( 5 1 , . . . , gn) i-> 5 l • • • gn e G. Let eg, g £ G,be the Dirac measure on G. The following relations hold for all x,y,z £ G: • e x * fi = j(x)fi

and fi* ex = T(X~1)[JL;

• if measures A, fjb, v are contractible, the pairs of measures A and /x, \x and ^, A * /i and u, A and /i * v are also contractible, and we have A * /i * v = (A * fi) * v — A * (fi * i/). For instance, any two bounded measures on G are contractible, and the space M1(G) of these measures is a unital Banach algebra with respect to the contraction * (10.5.19), where the Dirac measure E\ is the unit element. One also defines: • the contraction of a measure v and a dg-integrable function f on G as the density of the contraction v * (fdg) with respect to a Haar measure dg on G\ • the contraction of dg-integrable functions f\ and fi on G as the density of the contraction (fidg) * (fidg) with respect to a Haar measure dg on G, i.e., (/i/2)(
q,g£G.

(10.5.20)

D. Measures on Hausdorff spaces Let X be a Hausdorff topological space (not necessarily locally compact). A premeasure fi on X associates a measure [LK to each compact subset K of X such that if Q is another compact set and K C Q, the

582

Geometric and Algebraic Topological Methods in Quantum Mechanics

restriction [IQ\K of fiQ to K.(K) C K,(Q) coincides with fiK. A premeasure ji is called positive if all associative measures HK are positive. Given a positive premeasure /x, the essential upper integral £ ( / ) = sup fiK(
(10.5.21)

is denned for any positive M-too-valued function / on a Hausdorff space X. A premeasure /j on I is called a measure if any point x € X has a neighborhood U such that |/x|({7) = |/x|(y>y) is finite. In particular, if X is a locally compact space, each premeasure on X is a measure and, conversely, any measure /x on X defines a premeasure K —> /J,\K such that the essential upper integrals (10.5.15) and (10.5.21) coincide.

Example 10.5.8. The premeasure

where ex is the Dirac measure on K 9 x, is the Dirac measure on X such • that £x(/) = / ( z ) for any positive R-too-valued function / on X. A map 0 of a Hausdorff topological space X to a Hausdorff topological space Y is called measurable with respect to a positive measure /x on X if, for any compact subset K c X, there exists a partition of -ft' by a /^-null (locally /i-null) set and a (finite or infinite) sequence of compact sets Kn such that the restriction of <j> to every Kn is continuous. In particular, any continuous map is measurable. One says that a subset V of X is measurable if its characteristic function /u#- to the set of compact subsets of V. If X is a locally compact space and Y is its locally compact subspace, the induced measure [ly on Y is given by the relation M / ) = M(/),

/eAC(y),

where fx is a function on X which equals f onY and vanishes on X \ Y. Let 0 : X —> y be a /i-measurable map, and let every point 3/ G Y" have a neighborhood U such that ti(4>~l{U)) < oo. Then there exists a unique

583

Chapter 10 Appendixes

measure (/x) on Y which obeys the relation

M 5 ( / ) = M(/°(//) is called the image of the measure \i with respect to the map . Given Hausdorff spaces X and X' provided respectively with measures jit and n', there exists a unique product measure \i\i! on X x X' such that

W?(/®/') = £(/)£'(/') for all positive K-too-valued functions / on X and / ' on X'. E. Measures on infinite-dimensional vector spaces Throughout this Section, E denotes a real (locally convex) Hausdorff topological vector space. Infinite-dimensional topological vector spaces need not be locally compact, and measures on them are defined as follows [55; 164]. All measures are assumed to be positive. Let N(E) denote the set of closed vector subspaces of E of finite codimension, i.e., a vector subspace V of E belongs to N(E) if and only if there exists a finite set y\,..., yn of elements of the (topological) dual E' of E such that V consists of x € E which obey the equalities {x,yi) = 0, i — 1,... , n. A quasi-measure (or a cylinder set measure in the terminology of [164]) on E is defined as family /J, = {^v-, V £ N(E)} of bounded measures /zy on finite-dimensional vector spaces E/V such that if W C V, the measure /zy is the image of the measure fiw with respect to the canonical morphism E/W -> E/V. For instance, each bounded measure on E yields a quasi-measure {nv,V € N(E)}, where fiy is the image of the measure [i with respect to the canonical morphism ry : E —> E/V. There is one-to-one correspondence between the bounded measures on E and the quasi-measures on E which obey the following condition. For any e > 0, there exists a compact subset K C E such that Hv(E/V-rv(K))<e,

VGN(E).

Clearly, any quasi-measure on a finite-dimensional vector space is a measure.

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Let 7 : E —> F be a continuous morphism of topological vector spaces. For any W G N(F), the subspace V = 7~1(W) of E belongs to iV(£), and 7 yields a morphism 7 W : E/V -> F/W. Let ^ = {^y, V S N(E)} be a quasi-measure on E. Then one can assign the measure vw = 7fr(/V-i(W)) to each W G N(F). It is readily observed that the family ^ = {i>w,W G N(F)} is a quasi-measure on F. It is called the image of the quasi-measure [i with respect to the continuous morphism 7. In particular, let F = R, and let y £ E' be a. continuous form on E. The image /iy of a quasi-measure /i on E with respect to the form y is a measure on M. The Fourier transform of a quasi-measure // on E is defined as the complex function

Z(y) = J euny{t)

(10.5.22)

on the dual .E' of E. If /i is a bounded measure on E, its Fourier transform reads

Z(y) = J exp[i(x,y)]n(x).

(10.5.23)

Let us note the following variant of Bochner's theorem. A complex function Z on a topological vector space F is called positive-definite if

for any finite set J/J , . . . , ym of elements of F and any complex numbers Ci,...,Cm.

10.5.3. The Fourier transform provides a bijection of the set of quasi-measures on a Hausdorff topological vector space E to the set of positive-definite functions on the dual E' of E whose restriction to any finite-dimensional subspace of E' is continuous. • THEOREM

For instance, let B(y) be a seminorm on £". Then the function

Z(y) = exp[-iB(y)]

(10.5.24)

585

Chapter 10 Appendixes

on E' is positive-definite. By virtue of Theorem 10.5.3, there is a unique quasi-measure \IQ on E whose Fourier transform is Z(y) (10.5.24). It is called the Gaussian quasi-measure with the covariance form B. Example 10.5.9. Let E = M.n be a finite-dimensional vector space, coordinated by (xl), and let B be a norm on the dual of E. The Gaussian measure on E with the covariance form B is equivalent to the Lebesgue measure on E, and reads

/iB = d

(2^exp

[4(irl)yxV]dnx-

(10 5 25)

--

•

Example 10.5.10. Let E be a Banach space and E' its dual, provided with the norm ||.||' (10.5.1). The Gaussian quasi-measure on E with the covariance form ||.||' is called canonical. One can show that this quasimeasure fails to be a measure, unless E is finite-dimensional. Let T be a continuous operator in E'. Then V -> \\Ty\\

(10.5.26)

is a seminorm on E'. The Gaussian quasi-measure on E with the covariance form (10.5.26) is proved to be a measure if and only if T is a Hilbert-Schmidt operator. • Example 10.5.11. Let E be a real nuclear space and E' its dual, equipped with the topology of uniform convergence. Let us recall that all topologies of uniform convergence (including weak and strong topologies) on E' coincide, and E is reflexive. A quasi-measure on E' is a measure if and only if its Fourier transform on E (which is the dual of E') is continuous. A variant of Bochner's theorem for nuclear spaces states that the Fourier transform provides a bijection of the set of measures on the dual E' of a real nuclear space E to the set of continuous positive-definite functions on E • Example 10.5.12. Let E C E C E' be a real rigged Hilbert space, defined by a norm ||.|| on E. Let T be a nuclear operator in E and ||.|| T the restriction to E of the seminorm y —> ||Ty||, y 6 E, on E. Then the Gaussian measures /i and [IT on E' with the covariance forms ||.|| and \\.\\T are not equivalent. The Gaussian measures fi and JJLT are inequivalent if T

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Geometric and Algebraic Topological Methods in Quantum Mechanics

is a sum of the identity and a nuclear operator. In particular, all Gaussian • measures on a finite-dimensional vector space are equivalent.

10.6

Fibre bundles I. Geometry and connections

Subsections: A. Fibre bundles, 586; B. Vector bundles, 589; C. Differential forms and multivector fields, 593; D. Regular distributions, 597; E. Differential geometry of Lie groups, 599; F. First order jet manifolds, 602; G. Connections, 604; H. Composite connections, 609. This Section follows the exposition in [169; 296]. Smooth manifolds throughout obey the conditions in Remark 1.8.3.

A. Fibre bundles Let M and N be smooth manifolds of dimensions m and n, respectively. Let / : M —» N be a manifold morphism. Its rank at a point p e M is defined as the rank of the linear map of the tangent spaces Tpf : TPM -> Tf{p)N,

peM.

Since the function p —> rank p / is lower semicontinuous, a manifold morphism / of maximal rank at a point p is also of maximal rank on some open neighborhood of p. It is said to be an immersion if Tpf is injective and a submersion if Tpf is surjective. Let us note that a submersion is necessarily an open map. If / : M —> N is an injective immersion, its range is called a submanifold of N. A submanifold is said to be imbedded if it also a topological subspace. In this case, / is called an imbedding. If 7T : Y —-> X is a surjective submersion, its domain Y is said to be a fibred manifold over a base X. A surjective submersion is often called a fibration. By virtue of the well-known inverse function theorem [435], it admits an atlas of fibred coordinates (x^,yl) such that (x^) are coordinates on X. A fibred manifold Y —> X is said to be a fibre bundle if there exist a manifold V, called a typical fibre, and an open cover ii = {£/{} of X such that Y is glued from the splittings il>(:ir-1{Ut)-+UtxV

(10.6.1)

587

Chapter 10 Appendixes

by means of transition functions p^ = ^ o ip^1 on overlaps U^ (1 U^. Hence, all fibres Yx = TT~1(X), X € X, of a fibre bundle are diffeomorphic to a typical fibre V. Transition functions p^ fulfil the cocycle condition PK ° Pic = Pfr

(10.6.2)

on all overlaps U$ f~l U$ n Ut. Trivialization charts (U^,ip^) together with transition functions p^ form a bundle atlas * = {(^,V«).P«}

(10-6-3)

of a fibre bundle V —* X. Two bundle atlases ^ and \P' are said to be equivalent if their union is also a bundle atlas, i.e., there exist transition functions between trivialization charts of atlases \& and Vt'. A fibre bundle Y —> X is uniquely determined by a bundle atlas, and all its atlases are equivalent. The following two theorems describe the particular covers which one can choose for a bundle atlas [197]. 10.6.1. Every smooth fibre bundle Y —> X admits a bundle atlas over a countable cover il of X such that each member Ug of il is a domain (i.e., an open contractible set) whose closure Ug is compact. •

THEOREM

If X is compact, there is a bundle atlas of Y in Theorem 10.6.1 over a finite cover of X. THEOREM 10.6.2. Every smooth fibre bundle Y —> X admits a bundle atlas over a finite cover il of X, though its members need not be contractible and connected. •

Given a bundle atlas * (10.6.3), a fibre bundle Y is provided with bundle

coordinates xx(y) = (xX0Tr)(y),

yi(y) = (yioi>i)(y),

j/GTr"1^),

where xx are coordinates on [/ X to a fibre bundle n' : Y' —> X', by definition, preserves their fibrations. It is denned as a pair ( $ , / ) of manifold morphisms $ : Y —> Y' and / : X —> X' such that n' o $ = / o w, i.e., a jxfcre Yx = TT~1(X) is sent to a fibre Yi,y A bundle diffeomorphism is called an isomorphism. An automorphism is an isomorphism of a fibre bundle to itself. A bundle morphism over / = Id X is said to be a bundle

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Geometric and Algebraic Topological Methods in Quantum Mechanics

morphism over X. Two fibre bundles over the same base X are called equivalent if there exists their isomorphism over X. An automorphism of Y over X is said to be vertical. A bundle monomorphism $ : Y —» Y' over X is called a subbundle of the fibre bundle Y' —> X if $(Y) is a submanifold of y . In particular, a fibre bundle Y —> X is said to be trivial if it is equivalent to the manifold product X xV. THEOREM 10.6.3. A fibre bundle over a contractible base is always trivial [406]. • A section (or a global section) of a fibre bundle Y —» X is an injection s : X —> Y such that TT O S = idX, i.e., a section sends any point x e X into the fibre Yx over this point. A section is an imbedding and a closed map. Similarly, a section of a fibre bundle Y —> X over a submanifold of X is defined. By a local section is customarily meant a section over an open subset of X. A fibre bundle admits a local section around each point of its base, but need not have a global section. THEOREM 10.6.4. If a typical fibre of a fibre bundle Y —> X is diffeomorphic to R m , its section over a closed imbedded submanifold of X is extended to a global section. In particular, this fibre bundle always has a • global section [406].

Remark 10.6.1. A topological fibre bundle is defined as a continuous surjective submersion of topological spaces n : Y —> X. A continuous map n : Y —> X is called a submersion if, for any point y E Y, there is an open neighborhood U of the point ir(y) and a right inverse a : U —> Y of TT such that a o 7r(y) = y, i.e., there exists a local section of TT. The notion of a locally trivial topological fibre bundle is a repetition of that of a smooth fibre bundle, where trivialization morphisms Vf and transition functions Ptt are continuous. • The following are the standard constructions of new fibre bundles from old ones. • Given a fibre bundle TT : Y —> X and a manifold morphism / : X' —> X, the pull-back of Y by / is defined as a fibre bundle fY

= {(x1,y)eX'xY

: n(y) = f(x')}

(10.6.4)

Chapter 10 Appendixes

589

over X'. Its fibre over a point x' € X' is that of Y over the point f(x') G X. For instance, if / : X' —» X is a submanifold, then / y is called the restriction of Y to X'. • Let Y and Y' be fibre bundles over the same base X. Their fibred product Y x Y' is a fibre bundle over X whose fibres are products Yx x Y'x of fibres of Y and Y'. • Let us consider the compositefibrebundle Y -» E -> X.

(10.6.5)

It is provided with bundle coordinates (a;, X, i.e., transition functions of coordinates am are independent of coordinates yl. Let h be a global section of the fibre bundle E —> X. Then the restriction Yh = hY of the fibre bundle Y -> S to /i(X) C E is a subbundle of the fibre bundle Y -> X. B. Vector bundles A fibre bundle n : Y —•> X is called a vector bundle if both its typical fibre and fibres are finite-dimensional real vector spaces, and it admits a bundle atlas whose trivialization morphisms and transition functions are linear isomorphisms. Then the corresponding bundle coordinates (yl) on Y possess linear transition functions, and are coordinates with respect to the fibre bases {ei{x)} = ^\x){vi},

x = n(y)eUz,

where {vi} is a fixed basis for the typical fibre V of Y. By virtue of Theorem 10.6.4, a vector bundle has a global section, e.g., the canonical zero section 0(X) which sends every point x g X t o the origin of the fibre Yx over x. The following are the standard constructions of new vector bundles from old ones. • Let Y —> X be a vector bundle. By Y* —> X is denoted the dual vector bundle whose fibres are the duals of fibres of Y. • Given vector bundles Y and Y' over the same base X, their Whitney sum Y © Y' is a vector bundle over X whose fibres are the direct sums of fibres of Y and Y'. • Given vector bundles Y and Y' over the same base X, their tensor product Y<&Y' is a vector bundle over X whose fibres are the tensor products

590

Geometric and Algebraic Topological Methods in Quantum Mechanics

of fibres of Y and Y'. Similarly, the exterior product Y A Y is defined. Accordingly, AY = (X xR)®Y®AY®---®1AY, m = dimV. (10.6.6) x x > \ / x x is called the exterior bundle of Y. • If y is a subbundle of the vector bundle Y —> X, the factor bundle Y/Y' over X is defined as a vector bundle whose fibres are the quotients Yx/Y^ xeX. Vector bundles are subject to linear bundle morphisms, which are linear fibrewise maps. They possess the following properties. (i) Given vector bundles Y' and Y over the same base X, every linear bundle morphism $ : yx' 9 {e'iix)} h-> {$(x)ek(x)}

G Yx

over X defines a global section $ : x H-> $i(x)ek(x)

eft(z)

of the tensor product y K', and vice versa. (ii) Given a linear bundle morphism $ : Y' —> Y of vector bundles over X, its kernel Ker$ is denned as the inverse image $~ 1 (0(X)) of the canonical zero section 0(X) of Y. If $ is of constant rank, its kernel and its range are subbundles of the vector bundles Y' and Y, respectively. For instance, monomorphisms and epimorphisms of vector bundles fulfil this condition. A sequence Y'

i

>Y

J

> Y"

of vector bundles over the same base X is called exact at Y if Ker j = Imi. A sequence of vector bundles 0 -» y ' - U y -^-> Y" -^ 0

(10.6.7)

over X is said to be a short exact sequence if it is exact at all terms Y', Y, and Y". This means that i is a bundle monomorphism, j is a bundle epimorphism, and Ker j — Imi. Then Y" is the factor bundle Y/Y'. Given an exact sequence of vector bundles (10.6.7), there is the exact sequence of their duals r,

y//» _J

y* J

y/*

>

Q

591

Chapter 10 Appendixes

One says that an exact sequence (10.6.7) admits a splitting if there exists a bundle monomorphism s : Y" —> Y such that jos — Id Y" or, equivalently, Y

= i(y') © s(y") s y ' e y " .

THEOREM 10.6.5. Every exact sequence of vector bundles (10.6.7) is split [220; 224]. • Given a vector bundle Y —> X, an affine bundle modelled over Y —> X is a fibre bundle y —> X whose fibres Yx, x e X, are affine spaces modelled over the corresponding fibres Yx of the vector bundle Y, and Y admits a bundle atlas whose trivialization morphisms and transition functions are affine maps. The corresponding bundle coordinates possess affine transition functions. For instance, every vector bundle has a natural structure of an affine bundle. By virtue of Theorem 10.6.4, any affine bundle has a global section. Tangent and cotangent bundles exemplify vector bundles. The fibres of the tangent bundle -KZ '• TZ —> Z of a manifold Z are tangent spaces to Z. Given coordinates (zx) on a manifold Z, the tangent bundle TZ is equipped with holonomic coordinates (zx) with respect to the holonomic frames {d\} for tangent spaces TZZ, z G Z. Every manifold morphism / : Z —> Z' yields the linear bundle morphism over / of the tangent bundles Tf:TZ—^TZ',

i' A or/ = ^

.

(10.6.8)

It is called the tangent map to / . The cotangent bundle of a manifold Z is the dual TZ —> Z of the tangent bundle TZ —> Z. It is equipped with the holonomic coordinates (zx,z\) with respect to the coframes {dzx} for TZ which are the duals of {d\}. A tensor product m

k

T={® TZ) (® TZ),

m,ke N,

(10.6.9)

of tangent and cotangent bundles is called a tensor bundle. Tensor bundles belong to the category of natural bundles which admit the canonical lift of any diffeomorphism / of a base to a bundle automorphism, called a natural automorphism. For instance, a natural automorphism of the tangent bundle TZ over a diffeomorphism / of Z is the tangent map Tf (10.6.8). In view of the expressions (10.6.8), natural automorphisms are also called holonomic transformations.

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Geometric and Algebraic Topological Methods in Quantum Mechanics

Let Try : TY —> Y be the tangent bundle of a fibre bundle n : Y —> X. Given bundle coordinates (xx,yi) on Y, it is equipped with the holonomic coordinates (xx,y\xx,yl). The tangent bundle TY — Y has the subbundle VY = Ker TIT, which consists of the vectors tangent to fibres of Y. It is called the vertical tangent bundle of Y, and is provided with the holonomic coordinates (xx, y%, yl) with respect to the vertical frames {di}. Every bundle morphism $ : Y —> Y' yields the linear bundle morphism over $ of the vertical tangent bundles y $ : VY -» VY',

8& • y'1 o y $ = —

(10.6.10)

tf.

It is called the vertical tangent map. In many important cases, the vertical tangent bundle VY —> V of a fibre bundle Y —> X is trivial, and is equivalent to the bundle product (10.6.11)

VY^YxY

where Y —> X is some vector bundle. It follows that VY can be provided with bundle coordinates (xx ,y\yl) such that transition functions of coordinates y1 are independent of coordinates y\ One calls (10.6.11) the vertical splitting. For instance, every affine bundle Y —» X modelled over a vector - X admits the canonical vertical splitting (10.6.11). If Y is a bundle Y —> vector bundle, the vertical splitting (10.6.11) reads VY = Y®Y. Ji.

The vertical cotangent bundle VY —> Y of a fibre bundle Y — X is denned as the dual of the vertical tangent bundle VY —» Y. It is not a subbundle of the cotangent bundle TY, but there is the canonical surjection C : TY 9 xxdxx + yidy* .-> fcdj/* G Vy,

(10.6.12)

where {dy1} are the bases for the fibres of VY which are the duals of the holonomic frames {di} for the vertical tangent bundle VY. For any fibre bundle Y, there exist the exact sequences of vector bundles 0 -> VY -> TY -» Y x TX -> 0,

(10.6.13a)

o-»yxrx-^ry^ry^o.

(10.6.13b)

X

Their splitting, by definition, is a connection on V —» X.

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Chapter 10 Appendixes

C. Differential forms and multivector fields Vectorfieldson a manifold Z are global sections of the tangent bundle TZ —> Z. They are assembled into a real Lie algebra T\(Z) with respect to the Lie bracket [,]. Every vector field on a manifold Z can be seen as an infinitesimal generator of a local one-parameter group (a flow) of diffeomorphisms of Z along its integral curves [250]. A vector field is called complete if its flow is a one-parameter group of diffeomorphisms of Z. For instance, every vector field on a compact manifold is complete. A vector field u on a fibre bundle Y —> X is called projectable if it projects onto a vector field on X, i.e., there exists a vector field r on X such that T o 7r = Tn o u. Projectable vector fields take the coordinate form u = ux(x»)dx + u'Cz", yj)du

T = uxdx.

Their flows are local one-parameter groups of automorphisms of Y —> X. A projectable vector field is called vertical if its projection onto X vanishes, i.e., it lives in the vertical tangent bundle VY. A vector field r = TXd\ on a base X of a fibre bundle Y —> X gives rise to a vector field on Y by means of a connection (see the formula (10.6.48) below). Nevertheless, every natural fibre bundle Y —> X admits the canonical lift T onto Y of any vector field T on X. The flow of r is the canonical lift onto Y of theflowof the vector field r on X. For instance, the canonical lift onto the tensor bundle (10.6.9) reads T = T"3 M + [dvTi%;;£™ + ... - d^xX^l

- • • ]%\"£hm,

(10-6.14)

where we employ the compact notation d\ = d/dxx. An exterior r-form on a manifold Z is a section 0=-f0Ai...ArdzAl r!

A---AdzK

of the exterior product ATZ->Z. Let Or{Z) denote the vector space of exterior r-forms on a manifold Z. By definition, O°(Z) — C°°(Z) is the ring of smooth real functions on Z. All exterior forms on Z constitute the Ngraded commutative algebra O(Z) of global sections of the exterior bundle I\TZ (10.6.6). This algebra is provided with the exterior differential d<$> = dz» A d^

= ^dIM(j>Xl..,Xr.dz>i

A dzXi

A •••

dzK.

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Geometric and Algebraic Topological Methods in Quantum Mechanics

This obeys the relations

d(<j> A a) = d{) A a + {-l)W<j) A d(a),

where the symbol \ Z', any exterior A:-form <j) on Z' yields the pull-back exterior form f4> on Z given by the condition

r
Tf{vk)){f{z))

for an arbitrary collection of tangent vectors v1, • • • ,vk tions /(0ACT) =

G TZZ.

The rela-

/ > A / V ,

d/V = /•(#) hold. In particular, given a fibre bundle TT : Y —> X, the pull-back onto Y of exterior forms on X by n provides the monomorphism of graded commutative algebras O{X) —> O(Y). Elements of its range TTO*(X) are called basic forms. Exterior forms on Y such that u\4> = 0 for an arbitrary vertical vector field u on Y are said to be horizontal forms. Horizontal forms of degree n = dim X are called densities. The interior product (or contraction) of a vector field u and an exterior r-form i ^ o n a manifold Z is given by the coordinate expression u\4>=Y^ ~—1 r ' fc=i

uXk(PXl...\k-\rdzXl

A---Adz

A • • • A dzXr,

(10.6.15)

where the caret ~ denotes omission. It obeys the relations (ui,...,ur)

= ur\

•••ui\<j>,

u\{Aa +

{-l)W(j)Au\a.

The Lie derivative of an exterior form (f> along a vector field u is Lu = u\d(f> + d(u\cj)), Lu((/> A a) — Lu(f> A a + A Lu<7.

In particular, if / is a function, then L u / = u(f) = u\df. An exterior form 4> is invariant under a local one-parameter group of diffeomorphisms G(t) of Z (i.e., G(t)(p = (f>) if and only if its Lie derivative along the infinitesimal generator of this group vanishes.

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Chapter 10 Appendixes

Similarly to the graded commutative algebra of differential forms, the graded commutative algebra of multivector fields on a manifold Z is introduced. A multivectorfield$ of degree \d\ = r (or, simply, an r-vector field) on a manifold Z is a section d = ^-dM-KdXlA---AdXr

(10.6.16)

of the exterior product I\TZ — Z. Let Tr(Z) denote the vector space of r-vector fields on Z. By definition, %(Z) = C°°(Z). All multivector fields on a manifold Z make up the N-graded commutative algebra %(Z) of global sections of the exterior bundle f\TZ (10.6.6) with respect to the exterior product A. The graded commutative algebra %,(Z) is endowed with the SchoutenNijenhuis bracket [., .] S N : %{M) x TS{M) -> T r + s _i(M),

**v=

^(^X2-Xrd^vai-^dx2

(10.6.17)

A • • • A dXr A dai A • • • A da.).

This generalizes the Lie bracket of vector fields. The relations [IMSN = ( - 1 ) | I 9 | H M ] S N ,

[v,d A W]SN = h < % N A « + (_i)(M-DIl0 A \v,u]SN, (_l)H(M-i)[V> [^ u ] s N ] S N + (.ijltfld-l-D [^; [Vi v]SJi]SIi

(10.6.18) (10.6.19) (10.6.20)

+(-l)N«l- 1 )[t; l [i/,iJ] SN ] SN = 0> hold. In particular, the Lie derivative of a multivector field v along a vector field u reads Luv= [U,W]SN, LU(T? AV) = Lutf AU + IJA LUV. A generalization of the interior product (10.6.15) to multivector fields is the left interior product 4\4> = 4>(4),

|0|<M,

£O(Z),

0€%(Z),

of multivector fields and exterior forms. It is defined by the equalities ^(«iA---A« T .) = ^ ( u 1 , . . . l « r ) ,

eO(Z),

meT^Z),

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Geometric and Algebraic Topologicat Methods in Quantum Mechanics

and obeys the relation 0\v\4>=(vM>)\ = (-l)MWv\0\,

4>eO(Z),

0,veT.(Z).

If |0| < |T?|, the right interior product

0L0 = <W).

£O(Z),

i)er,(Z),

of exterior forms and multivector fields is given by the equalities

tf(01,...,0r) = tf^r---Li>
(

~ir

k=l

tieTr(Z),

#ai-c"'"ariakdai A • • • A 3ak A • • • A dar.

r

It satisfies the relations {•d A v)[4> = d A (v[) + (-l)M(d[A<j)=ti\al,

4> e

O\Z),

A tangent-valued r-form on a manifold Z is a section 0 = -^...x/z1

A • • • A d z ^ <8> 9M

of the tensor bundle }\TZ®TZ

(10.6.21)

^> Z.

Remark 10.6.2. Point out one-to-one correspondence between the tangent-valued one-forms 0 on a manifold Z and the linear bundle endomorphisms
$:TzZ3v^

0* : TZ -> TZ,

v\<j>{z) € TZZ,

(10.6.22)

0* : TZZ 3 v >-+ {z)\v* € T2Z, (10.6.23)

over Z. For instance, the canonical tangent-valued one-form 6Z = dzx ® dx

(10.6.24)

on Z corresponds to the identity morphisms (10.6.22) and (10.6.23).

•

Chapter 10 Appendixes

597

The space 0{Z) ® T\(Z) of tangent-valued forms is provided with the Prolicher-Nijenhuis bracket [, ] P N : Or{Z) ® 7I(Z) x OS{Z) ® Ti(Z) - 0 r + a (Z) ® Ti(Z), (10.6.25)

- r C...A r _ 1 ^V^ r + 1 ...A r + a which fulfils the relation [^]FN = (-1)WW+1[IMFN.

Given a fibre bundle V —> X, soldering forms on Y are vertical-valued horizontal one-forms a = a\{y)dxx ® di.

(10.6.26)

D. Regular distributions A (regular) distribution on a manifold Z is a subbundle T of the tangent bundle TZ of Z. A vector field u on Z is said to be subordinate to a distribution T if it lives in T. A distribution T is called involutive if the Lie bracket of T-subordinate vector fields is also subordinate to T. A codistribution T* on a manifold Z is a subbundle of the cotangent bundle TZ of Z. For instance, the annihilator AnnT of a distribution T is a codistribution whose fibre over z e Z consists of covectors w £ T such that v\w = 0 for all v e T 2 . There is the following criterion of an involutive distribution [435]. THEOREM 10.6.6. Given a distribution T and its annihilator AnnT, let AAnnT(Z) be the ideal of the graded commutative algebra 0{Z) which is generated by sections of Ann T —» Z. A distribution T is involutive if and only if the graded commutative algebra AAnnT(Z) is a differential ideal, i.e., d(AAnnT(Z)) C AAnnT(Z). •

The following local coordinates can be associated to an involutive distribution [435]. THEOREM 10.6.7. Let T be an involutive r-dimensional distribution on manifold Z, Am\Z = k. Every point z G Z has an open neighborhood U

598

Geometric and Algebraic Topological Methods in Quantum Mechanics

which is a domain of an adapted coordinate chart (z1,... ,zk) such that, restricted to U, the distribution T and its annihilator Ann T are spanned by the local vector fields d/dz1, ••• , d/dzr and the one-forms dzr+l,..., dzk, respectively. • A connected submanifold N of a manifold Z is said to be an integral manifold of a distribution T on Z if TN C T. Unless otherwise stated, by an integral manifold is meant an integral manifold of dimension of T. An integral manifold is called maximal if no other integral manifold contains it. The following is the classical theorem of Probenius [250; 435]. THEOREM 10.6.8. Let T be an involutive distribution on a manifold Z. Through any point z G Z, there passes a unique maximal integral manifold of T, and any integral manifold through z is its open subset. •

Maximal integral manifolds of an involutive distribution on a manifold Z are assembled into a (regular) foliation T of Z. Let us recall that a regular r-dimensional foliation T of a fc-dimensional manifold Z is defined as a partition of Z into connected r-dimensional submanifolds (leaves of a foliation) Ft, i S / , which possesses the following properties [366; 412]. A foliated manifold (Z,T) admits an adapted coordinate atlas {(EfcsV)},

A = l,...,n-r,

t = l,...,r,

(10.6.27)

such that transition functions of coordinates zx are independent of the remaining coordinates zl and, for each leaf F of a foliation T, the connected components of F n U^ are given by the equations zx =const. These connected components and coordinates (zl) on them make up a coordinate atlas of a leaf F. It follows that tangent spaces to leaves of a foliation T constitute an involutive distribution TT on Z, called the tangent bundle

to the foliation T. The factor bundle VT = TZ/TT, called the normal bundle to T, has transition functions independent of coordinates zl. Let TT —> Z denote the dual of TT —* Z. There are the exact sequences 0 -» TT ^-> TX ^VT-0,

(10.6.28)

0 -> A n n T ^ — TX -^> TT -* 0

(10.6.29)

of vector bundles over Z. It should be emphasized that leaves of a foliation need not be neither

599

Chapter 10 Appendixes

closed nor imbedded submanifolds. Every leaf has an open tubular neighborhood U, i.e., if z G U, then a leaf through z also belongs to U. For instance, any submersion / : Z —> M yields a foliation F={FP = r1(p)}Pef(z) of Z indexed by elements of f(Z), which is an open submanifold of M, i.e., Z —> f(Z) is a fibred manifold. Leaves of this foliation are closed imbedded submanifolds. Such a foliation is called simple. Each (regular) foliation is locally simple.

E. Differential geometry of Lie groups Let G be a real Lie group of dim G > 0, and let Lg : G —> gG and Rg : G —» Gg denote the action of G on itself by left and right multiplications, respectively. Clearly, Lg and Rgi for all g,g' EG mutually commute, and so do the tangent maps TLg and TRg>. A vector field £; (resp. £r) on a group G is said to be left-invariant (resp. right-invariant) if£/o£,fl = T L g o £ ; (resp. t,roRg =TRgo£r). Leftinvariant (resp. right-invariant) vector fields make up the left Lie algebra Qi (resp. the right Lie algebra gr) of G. There is one-to-one correspondence between the left-invariant vector field £; (resp. right-invariant vector fields £ r ) on G and the vectors £j(e) = TLg-i£i(g) (resp. f r (e) = TRg-i£i(g)) of the tangent space TeG to G at the unit element e of G. This correspondence provides TeG with the left and the right Lie algebra structures. Accordingly, the left action Lg of a Lie group G on itself defines its adjoint representation (10.6.30)

$r>-A.dg(tr)=TLgo£roLg-i

in the right Lie algebra g r . Let {e m } (resp. {em}) denote the basis for the left (resp. right) Lie algebra, and let c^ n be the right structure constants r

]

k

There is the morphism P-2i^em^em

=-em €

between left and right Lie algebras such that lemi enj = ~cmnek-

flr

(10.6.31)

600

Geometric and Algebraic Topological Methods in Quantum Mechanics

The tangent bundle TTG : TG —> G of a Lie group G is trivial. There are the following two canonical isomorphisms gi:TG3q^(g

= nG(q), TL~l{q)) £ G x Sh

gr:TG3q^(g

1

= nG(q),TR; (q)) £ G x

(10.6.32) flr.

(10.6.33)

Therefore, any action Gx

Z B(g,z)^

gz £ Z

of a Lie group G on a manifold Z on the left yields the homomorphism 0r9ei->£eeTi(Z)

(10.6.34)

of the right Lie algebra gr of G into the Lie algebra of vector fields on Z such that ^Adg(e)=Tgo^og-1.

(10.6.35)

Vector fields ££ are said to be the infinitesimal generators of a representation of the Lie group G in Z. In particular, the adjoint representation (10.6.30) of a Lie group in its right Lie algebra yields the adjoint representation E ' : E H ade'(e) = [e'.e],

ad £ m (e n ) = ckmnek,

(10.6.36)

of the right Lie algebra Qr in itself. An exterior form i^ona Lie group G is said to be left-invariant (resp. right-invariant)

if <j> = L<j> (resp. (f> = R4>) for all g £ G. T h e exterior

differential of a left-invariant (resp. right-invariant) form is left-invariant (resp. right-invariant). In particular, left-invariant one-forms satisfy the Maurer-Cartan equation #(e,e') = -
e,e'£

0l.

There is the canonical g; -valued left-invariant one-form (10.6.37)

9r.TeGBe^e£3i

on a Lie group G. The components 0[™ of its decomposition 8i = 0["em with respect to the basis for the left Lie algebra Qi make up the basis for the space of left-invariant exterior one-forms on G such that e m j#" = 5^. The Maurer-Cartan equation, written with respect to this basis, reads jam M

l

s,m an

— 2CnkUl

A

A

ak

fy

601

Chapter 10 Appendixes

(cf. the equation (1.5.55)). Accordingly, the canonical gr-valued rightinvariant one-form 6r:TeG3e^eeQr on the group G is defined. There are the relations vg e TgG,

e^vg) = e^TL-^Vg)) = TLj^Vg),

er(Vg) = er{TR-\Vg)) = TR;\Vg), p{9i{vg)) = -TLgoTR-%{vg)

=

-Adg(9r(vg)),

where p is the isomorphism (10.6.31). The dual g = TG of the tangent space TeG is called the dual Lie algebra (or the Lie coalgebra). It is provided with the basis {em} which is the dual of the basis {em} for TeG. The group G and the right Lie algebra 2r act on g by the coadjoint representation {Adg(e),e):=(e,Adg~1(e)), (adV(e),e> =-(e,[e',e]), ad £m (e") = - C f c e f c .

eG9, e' G flp,

e G gr, (10.6.38)

Remark 10.6.3. In the literature (e.g., [l]), one can meet another definition of the coadjoint representation in accordance with the relation (Adg(e),e) = (e,Adg(e)).

• The Lie coalgebra g of a Lie group G is provided with the canonical Poisson structure, called the Lie-Poisson structure [l; 279; 426]. It is given by the bracket {/,
:= <£, [df(e),dg(e)}),

f,g G C°°(g),

(10.6.39)

where df(e),dg(e) £ g,- are seen as linear mappings from TE.Q* = g* to R. Given coordinates zk on g* with respect to the basis {ek}, the Lie-Poisson bracket (10.6.39) and the corresponding Poisson bivector field w read {/,
wmn = ckmnzk.

602

Geometric and Algebraic Topological Methods in Quantum Mechanics

One can show that symplectic leaves of the Lie-Poisson structure on the coalgebra g* of a connected Lie group G are orbits of the coadjoint representation (10.6.38) of G on g* [436]. These leaves provide rather general examples of homogeneous symplectic manifolds as follows. Let us recall that a symplectic manifold (Z,fi)where a Lie group G acts transitively by symplectomorphisms is called a homogeneous G-symplectic manifold. A classification of homogeneous symplectic manifolds (Z, Q) subject to a Hamiltonian action of a simple connected real Lie group G is established by the following classical Kostant-Souriau-Kirillov theorem [257]. 10.6.9. Let the above mentioned group G have trivial cohomology groups i/^GjR) and H2(G;R) with coefficients in R (e.g., G is semisimple). Then the most general homogeneous Hamiltonian Gsymplectic manifold is locally isomorphic to an orbit Gq = G/Gq of G in g* where q € Q* and Gq is the stabilizer of q. • THEOREM

This classification has been refined and extended to homogeneous symplectic (not necessarily Hamiltonian) manifolds [198; 205]. F. First order jet manifolds Given a fibre bundle Y —> X with bundle coordinates (xx,yl), let us consider the equivalence classes j\s of its sections s identified by their values sl{x) and values of their derivatives dllsl{x) at points x € X. They are called the first order jets of sections. The particular choice of coordinates does not matter for this definition. The key point is that the set JlY of first order jets j\s, x £ X, is a smooth manifold with respect to the adapted coordinates (xx,yl,y\) such that y\{jls)

= dxs\x),

y'\ = -^(d,

+ yid^y'K

(10.6.40)

It is called the first order jet manifold of a fibre bundle Y —> X (see Section 10.7 for higher order jet manifolds) [169; 296; 392]. The jet manifold JXY admits the natural fibrations TT1 : JXY 3 jlxs ^xeX,

-KlQ : JXY 3 jlxs ^ s(x) £ Y,

(10.6.41)

where the second one is an afnne bundle modelled over the vector bundle

TX®VY -> Y. Y

If Y —> X is a vector bundle, the jet bundle u 1 (10.6.41) is so.

Chapter 10 Appendixes

603

There are the canonical imbeddings Ai : JlY -^TX®TY,

(10.6.42)

Ax = dxx ® (dx + y\di) = dxx ® dx, 6i : JY-^ TY®VY,

(10.6.43)

0i = (dy* - ^ d z A ) ® 3< = 9i <8> 9,, where d^ are total derivatives and #* are called contact forms. Identifying the jet manifold JlY to its images under the canonical morphisms (10.6.42) and (10.6.43), one can represent jets j^s = (xx,yl,ylll) by tangent-valued forms dxx®(dx+yidi),

{dyi-y\dxx)®di.

(10.6.44)

Sections and morphisms of fibre bundles admit prolongations to jet manifolds as follows. • Every section s of a fibre bundle Y —> X has the jet prolongation to the section {jtsM-jls,

y{oJh = dxsi(x),

of the jet bundle J1Y —> X. A section s of the jet bundle JlY —> X is called holonomic or integrable if it is the jet prolongation of some section of the fibre bundle Y —> X . • Every bundle morphism $ : Y —> Y' over a diffeomorphism / admits a jet prolongation to the bundle morphism over $ of the affine jet bundles

• Every projectable vector field u on a fibre bundle Y —> X has a jet prolongation to the projectable vector field Jlu = uxdx + rfdi + {dxu1 - yl,dxu^)dx X

on the affine jet bundle J Y —> Y.

(10.6.45)

604

Geometric and Algebraic Topological Methods in Quantum Mechanics

G. Connections A connection on a fibre bundle Y —> X is defined as a splitting T-.YxTX

->TY,

F : xxdx ~ xx(dx +Tix(y)di), (10.6.46)

±xdx + tfdi = ±\dx + r\di) + {f -

xxr\)di.

of the exact sequence (10.6.13a). Its range is a subbundle of TY —> Y called the horizontal distribution. By virtue of Theorem 10.6.5, a connection on a fibre bundle always exists. A connection F (10.6.46) is represented by the horizontal tangent-valued one-form r = dxx ® (dx + T\di)

(10.6.47)

on Y which projects onto the canonical tangent-valued form 6x (10.6.24) onl. Given a connection F on a fibre bundle Y —» X, any vector field r on a base X gives rise to the projectable vector field TT = r j r = Tx(dx + T\di)

(10.6.48)

on Y which lives in the horizontal distribution determined by T. It is called the horizontal lift of r by means of a connection F. The splitting (10.6.46) is also given by the vertical-valued form T = {dyi-r\dxx)®di,

(10.6.49)

which yields an epimorphism TY —> VY. It provides the corresponding splitting T : VY 3 dy >-> dy* - r\dxx

G TY,

(10.6.50)

xxdxx + mdtf = (xx + ViT\)dxx + yi{dy - T\dxx), of the dual exact sequence (10.6.13b). In an equivalent way, connections on a fibre bundle Y —> X are introduced as global sections of the affine jet bundle JlY —» Y. Indeed, any global section T of JXY —> Y defines the tangent-valued form Ai o F (10.6.47). It follows from this definition that connections on a fibre bundle Y —* X constitute an affine space modelled over the vector space of soldering forms a (10.6.26). One also deduces from (10.6.40) the coordinate transformation law of connections

Chapter 10 Appendixes

605

The following are two standard constructions of new connections from old ones. • Let Y and Y' be fibre bundles over the same base X. Given connections F on Y and V on Y', the bundle product Y x Y' is provided with the product connection

rxr' = ^ ® (dx + v\ A + r ^ ) .

(10.6.51)

• Given a fibre bundle y —> X, let / : X' —> X be a manifold morphism and fY the pull-back of Y over X'. Any connection F (10.6.49) onY ^ X yields the pull-back connection

f'T = (dy - r l A (nx'"),^) J^dz'") ® ^

(10.6.52)

on the pull-back bundle /Y -> X1. Every connection F on a fibre bundle Y —> X defines the first order differential operator Dr : JlY ^TX®VY,

(10.6.53)

Dr = A! - F o ^ = (y{ - T\)dxx ® di, on Y called the covariant differential. If s : X —> Y is a section, its covariant differential V r s := DT o Jls = {dxsi - T\ o s)dxx ® di

(10.6.54)

and its covariant derivative V^s = r J V r s along a vector field r on X are introduced. In particular, a (local) section s of Y —> X is called an integral section of a connection F (or parallel with respect to F) if s obeys the equivalent conditions V r s = 0 or J 1 s = F o S .

(10.6.55)

Let F be a connection on a fibre bundle Y —> X. Given vector fields r, r ' on X and their horizontal lifts F r and F T ' (10.6.48) on Y, let us consider the vertical vector field R(T,T')

= F[r, T'] - [FT, FT'] = T V " ^ ^ ,

«U = d^i ~ «

+ ri^rj, - Tfarl

(10.6.56)

(10.6.57)

0

Geometric and Algebraic Topological Methods in Quantum Mechanics

It can be seen as the contraction of vector fields r and r' with the verticalvalued horizontal two-form R

= g P ' r l™

=

2R^dxX

A

^

® 5i

(10.6.58)

on Y called the curvature of a connection F. Given a connection F and a soldering form a, the torsion of F with respect to a is defined as the vertical-valued horizontal two-form T = [F,CT]FN= (dxo-i + Tidjai - djTixal)dxx A d i " ® $ .

(10.6.59)

A ,/Zaf (or curvature-free) connection is a connection F on a fibre bundle Y —> X which satisfies the following equivalent conditions: • its curvature vanishes everywhere on Y; • its horizontal distribution is involutive; • there exists a local integral section for the connection F through any point y £ Y. By virtue of Theorem 10.6.8, a flat connection F yields a foliation of Y which is transversal to the fibration Y —» X. It called a horizontal foliation. Its leaf through a point y £ Y is locally defined by an integral section sy for the connection F through y. Conversely, let a fibre bundle Y —> X admit a horizontal foliation such that, for each point y &Y, the leaf of this foliation through y is locally defined by a section sy of Y —> X through y. Then the map T-.YBy^jl^SyGJ'Y sets a flat connection on Y —> X. Hence, there is one-to-one correspondence between the flat connections and the horizontal foliations of a fibre bundle Y->X. Given a horizontal foliation of a fibre bundle Y —> X, there exists the associated atlas of bundle coordinates (xx, y1) onY such that every leaf of this foliation is locally given by the equations yl =const., and the transition functions yl —> y'l{y^) are independent of the base coordinates xx [83; 169]. It is called the atlas of constant local trivializations. Two such atlases are said to be equivalent if their union is also an atlas of the same type. They are associated to the same horizontal foliation. Thus, the following is proved. PROPOSITION 10.6.10. There is one-to-one correspondence between the flat connections F on a fibre bundle Y —> X and the equivalence classes of

Chapter 10 Appendixes

607

atlases of constant local trivializations of Y such that T = dxx ® d\ relative to the corresponding atlas. D Example 10.6.4. Any trivial bundle has flat connections corresponding to its trivializations. Fibre bundles over a one-dimensional base have only flat connections. If a principal bundle over a simply connected base admits a flat (strength-free) principal connection, it is necessarily trivial [250]. In general, a fibre bundle admitting a flat connection need not be trivial. Namely, there is one-to-one correspondence between the set of gauge conjugate flat principal connections on a principal G-bundle P —» X and the set of conjugate homomorphisms of the first homotopy group iri(X) of X to the group G [2; 296]. • Example 10.6.5. Let {Z,T) be a foliated manifold endowed with the adapted coordinate atlas ^r = {(U;zx,zi)} (10.6.27). With respect to this atlas, the normal bundle VT —> Z to T is provided with coordinates ( r \ z\ zx) whose fibre coordinates zx have transition functions independent of coordinates zl on leaves of the foliation. Therefore, restricted to a leaf F, the normal bundle VF\F — F has transition functions independent of coordinates on its base F, i.e., it is equipped with a bundle atlas of local constant trivializations. In accordance with Proposition 10.6.10, this atlas provides the fibre bundle VT\p —* F with the corresponding flat connection, called Bott's connection. This connection is canonical in the sense that any two different adapted coordinate atlases tyj? andty'-pon Z also form an atlas of this type and, therefore, induce equivalent bundle atlases of constant local trivializations on V!F\F• Let Y —> X be a vector bundle equipped with linear bundle coordinates (xx,yz). It admits a linear connection r = dxx ® (dx + Txij(x)yjdi).

(10.6.60)

The associated covariant differential V r (10.6.53) is a linear first order differential operator on sections of a vector bundle Y —> X which obeys the Leibniz rule (1.8.21). There are the following standard constructions of new linear connections from old ones. • Any linear connection T (10.6.60) on a vector bundle Y —> X defines the dual linear connection V* = dxx ® (dx - TxiiWVjd)

(10.6.61)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

on the dual bundle Y —+ X. • Let F and V be linear connections on vector bundles Y —> X and Y' —> X, respectively. The direct sum connection F © F' on the Whitney sum Y © Y' of these vector bundles is defined as the product connection (10.6.51). • Similarly, the tensor product Y <8> Y' of vector bundles possesses the tensor product connection T®T' = dxx® [dx + (r A V ° + T'x\yib)~-}.

(10.6.62)

The curvature of a linear connection F (10.6.60) on a vector bundle Y —> X is usually written as a 7-valued two-form R = \Rx^j{x)yjdxx

A dx" ® e<)

(10.6.63)

due to the canonical vertical splitting VY = 7 x 7 , where {<%} = {ej}. For any two vector fields r and T' OH I , this curvature yields the zero order differential operator R(T, T')S = ([Vj:, V^,] - VfTir,j)S

(10.6.64)

on section s of a vector bundle Y —> X. An important example of linear connections is a connection K = dxx®{d\

+ KxilpXudtl)

(10.6.65)

on the tangent bundle TX of a manifold X. It is called a world connection or, simply, a connection on a manifold X. The dual connection (10.6.61) on the cotangent bundle TX is K = dxx ® (5A - K\vxp8r).

(10.6.66)

Remark 10.6.6. It should be emphasized that the expressions (10.6.65) - (10.6.66) for a world connection differ in a minus sign from those customarily used in the physics literature. D The curvature of a world connection K (10.6.65) reads R = ^RXlMa0x/3dxx Rx^p

= dxKfp

A dx» ® da, - d^Kxafi + Kx->pKp% -

(10.6.67) K^0Kxa7.

Chapter 10 Appendixes

609

Its Ricci tensor R\p = Rx^p is introduced. A torsion of a world connection is defined as the torsion (10.6.59) of the connection T (10.6.65) on the tangent bundle TX with respect to the canonical vertical-valued form dxx ® d\. Due to the vertical splitting of VTX, it is also written as a tangent-valued two-form

T = faxdx A dx» ® dv,

T/ A - KSx - Kx\,

(10.6.68)

on X. A world connection (10.6.65) is called symmetric if its torsion (10.6.68) vanishes. For instance, let a manifold X be provided with a non-degenerate fibre metric jeVO1^),

g=

gXfidxx®dxfi,

in the tangent bundle TX, and with the dual metric g&VT\X),

g = gx>idx®dn,

in the cotangent bundle TX. Then there exists a unique symmetric world connection K x\

= {x%} = -\gvp{dxgpll

+ d^gpx ~ 9 P 5 A M )

(10.6.69)

such that g is an integral section of K, i.e. Vxgaf3 = dxga0 - g^ix0-,}

- 907{x%} - 0.

This is the Levi-Civita connection, whose components (10.6.69) are called Christoffel symbols. H. Composite connections Let us consider a composite fibre bundle Y —> E —» X (10.6.5), coordinated by (xx,am,y%). Connections on fibre bundles Y —» X, Y — E and E -» X read l = dxx®{dx+ixndm + l\di), A s = dxx ® (dx + A\di) + dam ® (9 m + < 5 0 , r = dxA ® (0A + r ^ a m ) .

(10.6.70) (10.6.71) (10.6.72)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

A connection A s (10.6.71) on the fibre bundle Y —> £ and a connection F (10.6.72) on the fibre bundle £ —> X define a unique connection 7 = dxx ® (dx + Tfdm + (A\ + AiJDdi)

(10.6.73)

on the composite fibre bundle Y —> X such that the horizontal lift "yr on Y of a vector field r on X by means of the connection 7 (10.6.73) coincides with the composition A^(TT) of horizontal lifts of r onto £ by means of the connection F and then onto Y by means of the connection A^. For the sake of brevity, let us write 7 = A-£ o F. This is called the composite connection. Given a composite fibre bundle Y (10.6.5), there are the exact sequences 0 -> VT,Y - » y y -> y x v s -+ o,

(10.6.74)

0 - . 7 x y * S -4 VY -» V£y - • 0,

(10.6.75)

E

where V^y denotes the vertical tangent bundle of the fibre bundle Y —> S coordinated by (a^jO-771,!/,yl), and V^y is the dual of VsY —> y . Let us consider a splitting = yidi+ &mdm " v\B =

B:VY3v

B = (dj/* - B\J,am)

(10.6.76)

®di£ VY ® y E y,

of the exact sequence (10.6.74). Then a connection 7 (10.6.70) o n 7 - > X and a splitting B (10.6.76) define the connection AS

= B o 7 : TY -> y y -> v s y ,

^ E = da:A ® (9A + ( 7 j - B ^ T J 1 ) ^ ) +

(10.6.77)

m

Ar ®(^,+B^ai), on the fibre bundle Y —> S. Conversely, every connection A s (10.6.71) on the fibre bundle y —> S provides the splitting yy = vEy©4E(yxVE), y^i + &mdm = (y - A^&m)8i + &m(dm + A^di),

(10.6.78)

611

Chapter 10 Appendixes

(10.6.79)

VY = (YXVE)®AJ:(V£Y), I

ZJ

Vidy' + amdam

= y^dy' - A^da"1) + (&m +

Aimyi)dam,

of the exact sequences (10.6.74) - (10.6.75). Using this splitting, one can construct the first order differential operator D: JlY

(10.6.80)

^T*X®VY,Y, Y

D = dxx® (y\ - A\ -

A^di,

called the vertical covariant differential, on the composite fibre bundle Y —> X. The vertical covariant differential (10.6.80) possesses the following important property. Let h be a section of the fibre bundle S —» X and Yh —> X the restriction of the fibre bundle Y —» E to h(X) c E. This is a subbundle ih : Yh -> Y of the fibre bundle Y -> X. Every connection A E (10.6.71) induces the pull-back connection AA = ?hAs = dxx ® [9A + ( ( / 4 o ft)5Aftm + (A o h)\)di\

(10.6.81)

on yj, —> X . Then the restriction of the vertical covariant differential C JlY coincides with the familiar covariant D (10.6.80) to Jlih(JxYh) differential DAh (10.6.53) on Yh relative to the pull-back connection Ah (10.6.81). 10.7

Fibre bundles II. Higher and infinite order jets

The notion of a first order jet manifolds in Section 10.6F is naturally extended to higher order jets [169; 251; 392], Let Y —> X be a fibre bundle. Given its bundle coordinates (xx,yl), a multi-index A of the length |A| = r throughout denotes a collection of numbers (Ar...Ai) modulo permutations. By A + E is meant a multi-index (Ar • • • \\<Jk • • • O'I)- We will use the notation 9A = d\T o • • • o d\1. The r-order jet manifold JrY of sections of a fibre bundle Y —» X (or, simply, the jet manifold of Y —> X) is defined as the disjoint union of the equivalence classes j^s of sections s of Y such that different sections s and s' belong to the same equivalence class j^s if and only if **() = s'\x),

d^ix)

= d^x),

0 < |A| < r.

612

Geometric and Algebraic Topological Methods in Quantum Mechanics

In brief, one can say that sections of Y —» X are identified by the r + 1 terms of their Taylor series at points of X. The particular choice of coordinates does not matter for this definition. Given bundle coordinates {xx,yl) on a fibre bundle Y — X, the set JrY is endowed with an atlas of the adapted coordinates (s\tfX),

1/1+A =

0 < |A| < r,

y\os = dAsi(x),

(10.7.1)

(10-7.2)

|J<WA,

where the symbol d\ stands for the total derivative r-l

(10.7.3)

|A|=0

We will use the notation dA = d\r o • • • o dXl. The coordinates (10.7.1) bring the set JrY into a smooth manifold. They are compatible with the natural surjections Tif : JrY -* JlY,

r>l,

which form the composite bundle 7 r r. t 7 r y "bi t r -l y <Ei...

±>Y -=->X,

A glance at the transition functions (10.7.2) when |A| — r shows that the fibration JrY —> Jr~lY is an afflne bundle modelled over the vector bundle VTX ®

Jr~1Y

VY-Jr~lY.

Remark 10.7.1. In order to introduce higher order jet manifolds, one can use the construction of the repeated jet manifolds. Let us consider the r-order jet manifold JrJkY of the jet bundle JhY —> X. It is coordinated by {x^,y^A), |A| < k, |S| < r. There is the canonical monomorphism ark : J r + f c y - JrJkY,

y^A o <7rfc = y,+K.

a In the calculus in higher order jets, we have the r-order jet prolongation functor such that, given fibre bundles Y and Y' over X, every bundle

613

Chapter 10 Appendixes

morphism $ :Y —> Y' over a diffeomorphism / of X admits the r-order jet prolongation to the morphism of the r-order jet manifolds J r $ : JrY 3 fxs .-> ff{x){<S> o s o / - 1 ) G JrY'.

(10.7.4)

The jet prolongation functor is exact. If $ is an injection (resp. a surjection), so is J r $ . It also preserves an algebraic structure. In particular, if Y —> X is a vector bundle, so is JrY —> X. Every section s of a fibre bundle Y —> X admits the r-order jet prolongation to the section (Jrs){x) = j£s, called an integrable section, of the jet bundle JrY -> X. Every exterior form (j> on the jet manifold JkY gives rise to the pullback form ir£+i on the jet manifold Jk+iY. Let Ok = O{JkY) denote the algebra of exterior forms on the jet manifold JkY. We have the direct system of differential graded algebras O{X) ^O(Y)

^ O J ^...Ll>o;_>... .

(10.7.5)

Any vector field u = u^d^ + uldi on a fibre bundle Y —* X has the fc-order jet prolongation onto JkY as follows: Jku = uxdx + v.% + u\df, " U A = d\u\ - yl+^xu,

0 < |A| < k,

(10.7.6)

0 < |A| < k.

Remark 10.7.2. Higher order jet formalism provides the adequate formulation of theory of (non-linear) differential equations and operators [70; 26l]. A fc-order differential equation on a fibre bundle Y —> X is denned as a closed subbundle £ of the jet bundle JkY — X. Its classical solution is a (local) section s of Y —* X whose fc-order jet prolongation Jks lives in <£. One usually considers differential equations associated to differential operators. Let E —> X be a vector bundle coordinated by (xx,vA), A = 1 , . . . , m. A bundle morphism £:JkY

j+E,

vAoS = £A(x\y\yi,...,yik...Xi),

(10.7.7)

is called a A;-order differential operator on a fibre bundle Y —» X. It sends each section s of Y —> X onto the section (S o Jks)A(x)

= £A(x\si(x),

dxs'ix),

...,dXk---

Ox^ix))

of the vector bundle E —» X. Let us suppose that the canonical zero section 0{X) of the vector bundle E -> X belongs to the range £( JkY). Then the

614

Geometric and Algebraic Topological Methods in Quantum Mechanics

kernel of a differential operator 8 is defined as Kerf = 8-\Q{X)) C JkY, £A(x\y\yi,...,y\k..,Xi)

(10.7.8) = 0,

A =

l,...,m.

If Kerf (10.7.8) is a closed subbundle of the jet bundle JkY -> X, it is a fc-order differential equation, associated to the differential operator £. • Finite order jet manifolds make up the inverse system X ^ Y

?!-...<—

.

Jr-lY^JrY<

(10.7.9)

Its projective limit is denned similarly to the limit of an inverse system of modules. It is a set J°°Y together with surjections 7r°° : J°°Y -> X,

TT£°

: J°°Y -» Y,

7r£° : J°°Y - • JkY,

(10.7.10)

such that TT£° = irk o TT^° for any admissible k and r < k. Thus, elements of J°°Y represent oo-jets j%°s of local sections of Y —> X. These sections belong to the same jet j£°s if and only if their Taylor series at a point x € X coincide with each other. Therefore, J°°Y is called the infinite order jet space. The set J°°Y is provided with the coarsest topology such that the surjections (10.7.10) are continuous. The base of open sets of this topology in J°°Y consists of the inverse images of open subsets of JkY, fc = 0,..., under the mappings (10.7.10). This topology makes J°°Y into a paracompact Frechet manifold [410]. A bundle coordinate atlas {U, (xx,y1)} of Y —> X yields the manifold coordinate atlas 0<|A|,

{«r1(UY),(x\yiA)},

(10.7.11)

of J°°Y, together with the transition functions QXP-

Vx+A = ^ 7 I « ,

(10-7.12)

where d\ denotes the total derivative dx = dx + Y, Vx+Adt

(10.7.13)

|A|>0

Though J°°Y fails to be a smooth manifold, one can introduce the differential calculus on J°°Y as follows. Let us consider the direct system (10.7.5) of R-modules Oh of exterior forms on finite order jet manifolds JkY. The direct limit ( O ^ , TT^°) of this

615

Chapter 10 Appendixes

direct system is an R-module which consists of all the exterior forms on finite order jet manifolds module pull-back identification. The operations of the exterior product A and the exterior differential d also have the direct limits on 0 ^ . They provide 0 ^ with the structure of the differential graded algebra 0-,R—^O°oo^O1oo-^---,

(10.7.14)

where 0™ are the direct limits of the direct systems U

[A)

>U0

>UX

>---Cr

>•••

> Ur+1

of R-modules 0™ of exterior m-forms on r-order jet manifolds JrY. We agree to call elements of 0 ^ the exterior forms on the infinite order jet space J°°Y. Let us consider cohomology iJ(0J o ) of the de Rham complex (10.7.14) of these forms. By virtue of Theorem 1.4.4, it is isomorphic to the direct limit of the direct system Hm(O;)^Hm(Or+1)^--of the de Rham cohomology groups Hm(O) = Hm(JrY) JrY.

of jet manifolds

10.7.1. The cohomology groups ffm(0^,) of the complex (10.7.14) coincide with the de Rham cohomology groups Hm(Y) ofY—>X [9; 29]. •

PROPOSITION

Outline of proof. The proof is based on the fact that the fibre bundle JrY —> Jr~lY is affine, and it has the same de Rham cohomology as its base. It follows that the de Rham cohomology H(JrY) of jet manifolds JrY coincides with the de Rham cohomology H* (Y) of the fibre bundle Y —> X, and the cohomology of the complex (10.7.14) does so. QED Given a manifold coordinate atlas (10.7.11) of J°°Y, elements of the direct limit 0 ^ can be written in the coordinate form as exterior forms on finite order jet manifolds. The basic one-forms dxx and the contact forms 9\ = dy\ — y\+Kdxx make up the set of local generators of the 0^-algebra 0 ^ . There is the canonical decomposition Ooo = ® £&, k,s

0 < k,

0 < s < n,

616

Geometric and Algebraic Topological Methods in Quantum Mechanics

of 0^ into O^-modules O^s offc-contactand s-basic forms, together with the corresponding projections hk-.O^-tO1^,

0
h'-.O^^Og,

0<s
For instance, the horizontal projection h0 : O^ —» O^ reads hoidyi) = y\+Adxx.

ho(dx») = dx",

(10.7.15)

Accordingly, the exterior differential on O^ is split into the sum d = du + dy of the horizontal {total) and vertical differentials such that dt{ ° hk — hk odohk,

h0od = dH°h0,

dvohs

dv{(j)) = 6\ A aA,

= hs odo hs,

d}t ° df{ = 0,

dy o dy = 0,

dH(<j>) — dxx /\d\(), (j> £ Olc,

dy o du + d# o dy = 0.

Remark 10.7.3. Given a vector field r on X, let us consider the map

Vr:e£o3/^Tj(dff/)ee>2o.

(10.7.16)

This is a derivation of the ring 0%^. Moreover, if O^ is regarded as a C°°(X)-ring, the map (10.7.16) satisfies the Leibniz rule. Hence, the assignment V : r t->rjdtf = rxd\

(10.7.17)

is the canonical flat connection on the C°°(X)-ring 0 ^ .

•

Being nilpotent, the differentials dy and djj split the graded algebra O^ into the bicomplex { 0 ^ m } . One also introduces the K-module projector

r = J2lTohkohn' fc>o

(10.7.18)

k

T(O(-l)l A I^A[d A (a i A j^)], r o d n — 0,

Todor

— T o d = 0,

of O^Q and the nilpotent variational operator 5 = rod,

6oT-rod

= 0,

(10.7.19)

617

Chapter 10 Appendixes

on O^,". Let us put Ek = T(O^1). AS a consequence, the differential graded algebra 0 ^ is split into the variational bicomplex

<%°

^

O^ 1

^..-

O^n

-^

£?!-0

0^M->

0^

^

C&1

^•-.

O°in

=

0°j n

0-»R-»

o°(X) - ^

O^X) -^-»---

t

t

0-

0

O"(X) -2-> 0

t

0

0

(10.7.20) There are the well-known statements usually summarized as the algebraic Poincare lemma [346; 418]. LEMMA 10.7.2. If Y is a contactable bundle M.n+m -> R n , the variational bicomplex (10.7.20) is exact. D In the case of a general Y, one can show the following [173; 388]. THEOREM 10.7.3. (i) The rows of the variational bicomplex (10.7.20) starting from the third one from the bottom are exact sequences, (ii) The second row from the bottom and the last column of the complex (10.7.20) are assembled into the variational complex

U —> K —> U^ —> U^ • • • —> U^

—> hi —> t,2 —> • • • , (10.7.21)

whose cohomology equals the de Rham cohomology of Y

•

Outline of proof. One has to enlarge the differential graded algebra O^ in order to find its dH- and 5-cohomology. Let £) be the sheaf of germs of exterior forms on the r-order jet manifold JrY, and let D* be its canonical presheaf. There is the direct system of presheaves u

x

—^o

—> Ui • • • —> Vr

—y

Its direct limit 0^ is a presheaf of differential graded algebras on the infinite order jet space J°°Y. Let 0 ^ be a sheaf constructed from '3oo. The module Q^ = F(0^ o ) of sections of O ^ 1S a differential graded algebra

618

Geometric and Algebraic Topological Methods in Quantum Mechanics

such that, given an element e £j£o and a point z € J°°Y, there exist an open neighborhood U of z and an exterior form cf)^ on some finite order In particular, there is the jet manifold JkY so that (j>\v = (fc) °^\umonomorphism O^ —> Q ^ . The key point is that the paracompact space J°°Y admits a partition of unity by elements of the ring Q^, [410], and that Y is a strong deformation retract of J°°Y [9; 173]. These facts have enabled one to obtain dn- and <5-cohomology of Q ^ [8; 9; 410]. Finally, the duand 5-cohomology of Q ^ is proved to equal that of its subalgebra O^ [172; 173; 388]. QED

10.8

Fibre bundles III. Lagrangian formalism

We start with first order Lagrangian formalism since almost all Lagrangian systems in mechanics and field theory are of this type. The velocity phase space of a first order Lagrangian system on a fibre bundle Y —> X, coordinated by {xx,y'l,y\), is the first order jet manifold JXY of Y —> X, coordinated by (x^,y%,y\). A first order Lagrangian on JlY is defined to be a density L = CLJ:J1Y^ATX,

w = dx1 A-• • Adxn,

n = dimX.

Given a projectable vector field u on Y —* X and its jet prolongation Jxu (10.6.45) onto JlY, let LjiuL = [dxuxC + (uxdx + u'di + (dxu* - i/£0Au")^)£]w

(10-8.1)

be the Lie derivative of a Lagrangian L along Jlu. The first variational formula (see Proposition 10.8.4 below) provides its canonical decomposition LjiuL = uv\£L + dHh0(u\HL) = (u* - i/jyXfc - dxdx)CLj - dx[dxC(u%

(10.8.2) - u') - uxC]w,

where £L:J2Y-->TYA{ATX),

SL = (diC-dxd?C)0i/\w,

(10.8.3)

619

Chapter 10 Appendixes

is the Euler-Lagrange operator associated to the Lagrangian L, and HL : JlY-+ZY

= TYA(nA1TX),

(10.8.4)

i

HL = L + d^C6 Awx= d^Cdtf A ux + (C - d$Cy\)uj,

(10.8.5) LJX = dx\w,

is the Poincare-Cartan form. The kernel of the Euler-Lagrange operator £L (10.8.3) defines the system of second order Euler-Lagrange equations, given by the coordinate equalities (di - dxdl)C = 0.

(10.8.6)

A solution of these equations is a section s of the fibre bundle X —» Y, whose second order jet prolongation J2s lives in (10.8.6), i.e., diCos- (dx + dxsjdj + dxd^d^C

o s = 0.

Remark 10.8.1. The kernel (10.8.6) of the Euler-Lagrange operator £i need not be a closed subbundle of the second order jet bundle J2Y —> X. Therefore, it may happen that Euler-Lagrange equations are not differential • equations in a strict sense (see Remark 10.7.2). The Poincare-Cartan form Hi (10.8.5) is a particular Lepagean equivalent of a Lagrangian L = IIO(HL) [169; 190] which is a horizontal form on the affine jet bundle JlY —> Y. It is also a Lepagean equivalent of the Lagrangian I = ho(HL) = (£ + (y{ - yx)d^C)LO,

(10.8.7)

hoidy') = yxdx\ on the repeated jet manifold JlJxY, coordinated by (xx,yl,yi/1,yi x). The Euler-Lagrange operator for L (called the Euler-Lagrange-Cartan operator) reads £z : JlJlY

-> TJlY A (A TX),

£T = [(diC - dxd?C + did$£(yi - yifidy* +

$d?C(yi-yl)dyi]Au, dx^dx+yidi + yi^.

(10.8.8)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

Its kernel Ker£^ c J1J1Y is the Carian equations d?dfC(%-yil) = 0,

(10.8.9a)

diC - dxd?jC + (y{ ~ yi)did$C = 0,

(10.8.9b)

on JlY. Since £T \j2Y= £L, the Cartan equations (10.8.9a) - (10.8.9b) are equivalent to the Euler-Lagrange equations (10.8.6) on integrable sections of JYY —» X. These equations are equivalent if a Lagrangian is regular. The Cartan equations (10.8.9a) - (10.8.9b) on sections s : X —> JlY are equivalent to the relation s{u\dHL)=0,

(10.8.10)

which is assumed to hold for all vertical vector fields u on JXY —> X. The Poincare-Cartan form HL, (10.8.5) yields the Legendre morphism HL:JlY

^ZY,

The homogeneous Legendre bundle Zy admits the canonical multisymplectic Liouville form Sy =pto + pfdyi AUJ\. Let ZL = HL{J1Y)

(10.8.12)

be an imbedded subbundle IL • ZL —> Zy of Zy —»

Y. It is provided with the pull-back De Bonder form HL = i L Hy. There is the equality HL = HLZL = HL(ilZY).

(10.8.13)

By analogy with the Cartan equations (10.8.10), the Hamilton-De Bonder equations for sections r of ZL —> X are written as f(uJdSL)=0,

(10.8.14)

621

Chapter 10 Appendixes

where u is an arbitrary vertical vector field on Zj, —» X. 10.8.1. Let the Legendre morphism Hi be a submersion. Then s of JXY —» X is a solution of the Cartan equations (10.8.10) if and a section only if HL °S is a solution of the Hamilton-De Donder equations (10.8.14), i.e., the Cartan and Hamilton-De Donder equations are quasi-equivalent [190]. • THEOREM

A glance at the transition functions (10.8.11) shows that Zy (10.8.4) is a one-dimensional affine bundle (10.8.15)

TTZU :ZY^U

over the Legendre bundle II = ATX ® VY ®TX = VY A^A* TX), Y

Y

Y Y

endowed with holonomic coordinates (xx,yl,px). L = TTZnoHL:J1Y

(10.8.16)

The composition

-^U,

(10.8.17)

(x\yi,p$)oL = (x\yi,d?£), is called the Legendre map. In particular, a Lagrangian L is said to be: • hyperregular if the Legendre map L is a diffeomorphism; • regular if L is of maximal rank, i.e., det( Y and the Legendre map L : JlY —> NL is a fibred manifold with connected fibres. The first variational formula (10.8.2) provides the following standard procedure for the study of differential conservation laws in Lagrangian formalism. Let u be a projectable vector field on a fibre bundle Y —» X treated as the infinitesimal generator of a local one-parameter group of bundle automorphisms of Y -> X. On the kernel (10.8.6) of the EulerLagrange operator £L, the first variational formula (10.8.2) leads to the weak identity LjiuL w -dx1xw,

(10.8.18)

where X = 1xu>x,

1X = dxC{u%

- ul) - uxC,

(10.8.19)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

is the symmetry current along the vector field u. Let u be a variational symmetry of L, i.e., the Lie derivative LjiuL (10.8.1) vanishes. Then we obtain the weak conservation law d\1 w 0 of the symmetry current X (10.8.19). Let us turn now to higher order Lagrangian systems. Given the variational complex (10.7.21), one can think of its elements

|A|>0

as being a higher order Lagrangian and its Euler-Lagrange operator, respectively. A Lagrangian L is called variationally trivial if its Remark 10.8.2. Euler-Lagrange operator SL vanishes. By virtue of item (ii) of Theorem 10.7.3, every variationally trivial Lagrangian L e O°'n is the sum ZeO0^-1,

L = hov + dHt;,

(10.8.20)

where

•

The exactness of the row 0 ^OJ i »^oU..Ao^l,£ 1 ^0

(10.8.21)

of the variational bicomplex (10.7.20) at the term Q^g1 in accordance with item (i) of Theorem 10.8.2 provides the R-module decomposition 01oin =

E1®dH(01o?-1).

This decomposition yields the splitting (10.8.22)

dL = 5L-dH{Z)

of dL e Olg1. The form E in this splitting is not uniquely denned. It reads

S=£i?''-"'X...I,1Au;A

(10.8.23)

s=0 j?vk---v\ _ ay\f-v\

f

J , pAi/fc...i/i

,

where local functions h G 0 ^ obey the relations h" = 0,

/l(""-O->'i = o.

IMI,...V\

623

Chapter 10 Appendixes

It follows that S i = S+L is a Lepagean equivalent, e.g., a Poincare-Cartan form of a finite order Lagrangian L [190]. The decomposition (10.8.22) leads to the desired first variational formula as follows. A derivation v G J)O^, of the R-ring 0 ^ such that the Lie derivative hv preserves the contact ideal of the differential graded algebra 0 ^ (i.e., the Lie derivative Lv of a contact form is a contact form) is called an infinitesimal contact transformation or, simply, a contact symmetry (by analogy with C-transformations in [261] though v need not come from a morphism of J°°Y). Forthcoming Proposition 10.8.2 shows that, restricted to a coordinate chart (10.7.11) and a differential graded algebra 0* of finite jet order, any contact symmetry v is the jet prolongation of a generalized vector field in [346]. 10.8.2. (i) The derivation module c)0^ is isomorphic to the 0^-dual (Ojo)* of the module of one-forms O^. (ii) Relative to an atlas (10.7.11), a derivation v G 3 0 ^ is given by the expression

PROPOSITION

v = vxdx + v{di + ]T v\d£,

(10.8.24)

|A|>0

where t r \ vl, v\ are local smooth functions of finite jet order obeying the transformation law

^ = ^-
I^TM

dyJ

^

dx

dx»

(10.8.25)

v

'

*

(iii) A derivation v (10.8.24) is a contact symmetry if and only if W A = d A ( t / - i / X ) + i/* +A 'A

0<|A|.

(10.8.26)

• Outline of proof. (i) At first, let us show that O^ is generated by elements df,f£ 0%^. It suffices to justify that any element of O^ is a finite O^-linear combination of elements df, f e 0%^. Indeed, every <j> G O^ is an exterior form on some finite order jet manifold JrY. By virtue of the Theorem 1.8.5, the C°°{JTY)-module O of one-forms on JrY is a projective module of finite rank, i.e., cf> is represented by a finite C°°(JrY)linear combination of elements df, f G C°°(JrY) C 0 ^ . Any element $ G (O^)* yields a derivation u$(/) = $(d/) of the M-ring 0 ^ . Since

624

Geometric and Algebraic Topological Methods in Quantum Mechanics

the module 0^, is generated by elements df,fe 0 ^ , different elements of (C^,)* provide different derivations of 0^,, i.e., there is a monomorphism (Coo)* ~* W%o- By the same formula, any derivation v G 5 0 ^ sends df i-» v(f) and, since 0 ^ is generated by elements df, it defines a morphism $v : 0^-, —» 0 ^ . Moreover, different derivations w provide different morphisms $>v. Thus, we have a monomorphism and, consequently, an isomorphism (ii) Restricted to a coordinate chart (10.7.11), O^ is a free 0^-module generated by the exterior forms dxx, 6\. Then 00^, = ( 0 j j * restricted to this chart consists of elements (10.8.24), where dx, df are the duals of dxx, 9\. The transformation rule (10.8.25) results from the transition functions (10.7.11). The interior product v\ and the Lie derivative Lv(j), <j> G O^o, obey the standard formulae. Restricted to a coordinate chart, the Lie derivative Lv sends each finite jet order differential graded algebra O* to another finite jet order differential graded algebra 0. Since the atlas (10.7.11) is finite, hv4> preserves O^. (iii) The expression (10.8.26) results from a direct computation similar to that of the first part of Backlund's theorem [226]. One can then justify that local functions (10.8.26) satisfy the transformation law (10.8.25). QED

Remark 10.8.3. As follows from Proposition 10.8.2, the de Rham complex (10.7.14) of the differential graded algebra O^ is both the ChevalleyEilenberg complex of the Lie algebra 50^, with coefficients in 0 ^ and the universal differential calculus over the R-ring 0 ^ . • For instance, any vector field u on Y and its jet prolongations Jku (10.7.6) onto jet manifolds JkY of any finite order k is a particular contact symmetry, called a classical symmetry. In comparison with a classical symmetry, components vx and vl of a generic contact symmetry depend on jet coordinates y\. Let us note that symmetries of differential equations under transformations of dynamic variables depending on their derivatives have been intensively investigated [10; 226; 261; 346]. Contact symmetries of Lagrangian systems on a local coordinate domain have been also described in detail [346]. Any contact symmetry admits the horizontal splitting V

= VH

+ Vv

= vxdx + (tf'fli + Y, d^di |A|>0

)' & = yi ~ 2/X< (10-8-27)

625

Chapter 10 Appendixes

relative to the canonical connection V = dxx d\ on the C°°(X)-ring O^ [296]. LEMMA 10.8.3. Any vertical contact symmetry v = vy obeys the relations (10.8.28)

v\dH =-dH{v\), Lv(dH
= dH{Lv),

(10.8.29)

4> e 0^.

D Outline of proof. It is easily justified that, if
' satisfy the relation (10.8.28), then (p A <j>' does well. Then it suffices to prove the relation (10.8.28) when 4> is a function and <j> = 6\. The result follows from the equalities dH{v\) = v\+Kdxx,

dHe\ = dxx A 6\+A,

(10.8.30) (10.8.31)

dxov\dt =v\d^odx. The relation (10.8.29) is a corollary of the equality (10.8.28).

QED

PROPOSITION 10.8.4. Given a Lagrangian L = COJ e
LVL = w\SL + dH(ho(v\SL))

+ £dv(vH\uj),

(10.8.32)

where H/, is a Lepagean equivalent, e.g., a Poincare-Cartan form of L. • Outline of proof. The formula (10.8.32) comes from the splitting (10.8.22) and the relation (10.8.28) as follows:

LvL = v\dL + d(v\L) = [vv\dL - dyC A VH}UJ] + [dH(vH\L) + dv(CvH\u)\ = vv\dL + dH{vH\L) + Cdv(vH\w) = vv\SL - -uyJdnH + dH{vH\L) + Cdv(vH\u) = vv\SL + dH(vv\Z + vH\L) + £dv(vH\u>), where vv\E = /io(wJH) since E is a one-contact form, W//JL = ho(v\L), and EL = S + L. QED

626

Geometric and Algebraic Topological Methods in Quantum Mechanics

If v is a classical symmetry given by a projectable vector field u on Y —> X, the first variational formula (10.8.32) for an r-order Lagrangian takes the form LjruL = uv\6L + dH{h0{J2r-lu\EL)).

(10.8.33)

Let v be a variational symmetry of L (in the terminology of [346]), i.e., LVL = dHa,

a G C&"- 1 .

By virtue of the expression (10.8.32), this condition implies that a contact symmetry v is projected onto X. Then the first variational formula (10.8.32) reads dHa = vv\5L + dH(hQ(v\EL)).

(10.8.34)

Restricted to Ker 5L, it leads to the weak conservation law 0fsd fl (/ifl(i;JSi)-
(10.8.35)

Let us note that any classical variational symmetry of a Lagrangian L is also a symmetry of its Euler-Lagrange operator by virtue of the master identity Lj2ruSL = 5{LjruL). However, this equality is not true for contact symmetries [346]. 10.9

Fibre bundles IV. Hamiltonian formalism

Given a fibre bundle TT : Y —> X, the Legendre bundle Trny : II —» Y (10.8.16) plays a role of the momentum phase space of polysymplectic Hamiltonian systems onY—>X [169; 171]. It is provided with the canonical polysymplectic form, Hamiltonian connections and Hamiltonian forms as follows. Let us consider the canonical bundle monomorphism 6= -Pidy* Aw®a A :II - • " A V ' Y ® T X .

(10.9.1)

The polysymplectic form on II is defined as a unique TX-valued (n+2)-form Q = dp$ A dyi A u dx

(10.9.2)

627

Chapter 10 Appendixes

such that the relation

n\

(10.9.3)

on the fibre bundle II —• X is called a Hamiltonian connection if the exterior form 7JQ is closed. Let nzn • Zy —> II be the one-dimensional affine bundle (10.8.15). It admits a global section. Given such a section h, the bundle coordinate p' = p—h provides a trivialization of the bundle (10.8.15) if X is an oriented manifold. Given a section h of Zy —> II, the pull-back H = hEY=p$dyi Aux-Hw

(10.9.4)

of the canonical form Sy (10.8.12) is called a Hamiltonian form on the Legendre bundle II. Each Hamiltonian form H (10.9.4) yields the associated Hamiltonian map H : n -> JlY,

y{oH = d\H.

(10.9.5)

10.9.1. For every Hamiltonian form H (10.9.4), there exists an associated Hamiltonian connection such that THEOREM

1\Sl

= dH,

j{=d{n,

7k = -0iW.

(10.9.6)

Conversely, for any Hamiltonian connection 7, there exists a local Hamiltonian form H on a neighborhood of any point q G n such that the relation • (10.9.6) holds. Hamiltonian forms on n constitute an afnne space modelled over the linear space of horizontal densities H = HUJ on n —» X. This is an immediate consequence of the fact that Zy —> n is an affine bundle modelled over the pull-back vector bundle

nxArx^n. X

For instance, any connection V (10.6.47) on a fibre bundle Y —> X determines the section hr : 2yi H-> dyi - T\dxx

628

Geometric and Algebraic Topological Methods in Quantum Mechanics

of the affine bundle Zy —> II and the corresponding Hamiltonian form Hr = ftfSy = pxdyl Aux-PiT\oj.

(10.9.7)

As a consequence, given a connection T on Y —> X, every Hamiltonian form H admits the decomposition H = HT-Hr=

pxdyi A u A - pxT{uj - Hrw.

(10.9.8)

A Hamiltonian form H (10.9.4) on the Legendre bundle II can be seen as the Poincare-Cartan form of the Lagrangian LH = hQ{H) = {pxy\-'H)u

(10.9.9)

on the jet manifold J1!!. The corresponding Euler-Lagrange operator (10.8.3) reads £H:J1H^T'UA(ATX),

£H = \{y\ ~ d\H)dp* - (pxXi + diHW] A w.

(10.9.10)

It is called the Hamilton operator for a Hamiltonian form H. Its kernel defines the covariant Hamilton equations y\ = SiH,

(10.9.11a)

pxM = -diH.

(10.9.11b)

It is readily observed that all Hamiltonian connections (10.9.6) associated to a Hamiltonian form H live in the kernel of the Hamilton operator EHConsequently, every integral section Jxr = jor of a Hamiltonian connection 7 associated to a Hamiltonian form H is a solution of the covariant Hamilton equations (10.9.11a) - (10.9.11b). Let us note that, similarly to the Cartan equations (10.8.10), the covariant Hamilton equations (10.9.11a) - (10.9.11b) are equivalent to the condition r(u\dH)=0

(10.9.12)

for any vertical vector field u on n —» X. The covariant Hamilton equation (10.9.11a) can be also written as the equality J^TTny or) = Hor.

629

Chapter 10 Appendixes

Remark 10.9.1. In fact, the Lagrangian (10.9.9) is the pull-back onto J 1 !! of the form LH on the product II x JlY. • Y

Let us associate a Hamiltonian system on the Legendre bundle II (10.8.16) to a first order Lagrangian system on the jet manifold JlY. Given a Lagrangian L, let L (10.8.17) be the corresponding Legendre map. A Hamiltonian form H is said to be associated to a Lagrangian L if it satisfies the relations LoHoL = L,

(10.9.13a)

HLH = HL.

(10.9.13b)

It is immediately observed that, if a Lagrangian L is hyperregular, it admits a unique associated Hamiltonian form H such that H = L~x. Moreover, any solution s : X —> Y of the Euler-Lagrange equations (10.8.6) for L yields a solution r = L o Jls of the Hamilton equations (10.9.11a) (10.9.11b) and, conversely, any solution r : X —> II of these Hamilton equations defines a solution s = TTUY ° f of the Euler-Lagrange equations for L. Let further a Lagrangian need not be hyperregular. A glance at the relation (10.9.13a) shows that L o H is the projector rf{q) = d?C(x'l,yi,%H(q)),

q £ NL,

(10.9.14)

from n onto the Lagrangian constraint space NL = L(JlY). Accordingly, HoL is the projector from JY onto H(NL)- A- Hamiltonian form is called weakly associated to a Lagrangian L if the condition (10.9.13b) holds on the Lagrangian constraint space NL. The following assertion holds [169; 171]. PROPOSITION 10.9.2. Any Hamiltonian form H weakly associated to a Lagrangian L obeys the relation H\NL

=

HHL\NL,

where HL is the Poincare-Cartan form (10.8.5).

•

Example 10.9.2. Any Hamiltonian form is weakly associated to the Lagrangian L = 0, while the associated Hamiltonian forms are only Hr (10.9.7). It follows that the associated Hamiltonian form is not regular outside the Lagrangian constraint space NL. •

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Geometric and Algebraic Topological Methods in Quantum Mechanics

One can say something more in the case of semiregular Lagrangians. A Lagrangian L is called semiregular if the inverse image L~l(q) of any point q S NL is a connected submanifold of JlY. LEMMA 10.9.3. The Poincare-Cartan form Hi for a semiregular Lagrangian L is constant on the connected inverse image L~1(q) of any point <7 € NL. U COROLLARY 10.9.4. All Hamiltonian forms weakly associated to a semiregular Lagrangian L coincide with each other on the Lagrangian constraint space NL, while the Poincare-Cartan form HL (10.8.5) for L is the pullback

HL = LH,

(d^Cyi - £)CJ = H{x»,yi,d<C)u,

of any such a Hamiltonian form H.

(10.9.15) •

Corollary 10.9.4 enables us to connect the Euler-Lagrange and Cartan equations for a semiregular Lagrangian L with the covariant Hamilton equations for Hamiltonian forms weakly associated to L as follows. THEOREM 10.9.5. Let a section r of the Legendre bundle II —> X be a solution of the Hamilton equations (10.9.11a) - (10.9.11b) for a Hamiltonian form H weakly associated to a semiregular Lagrangian L. If r lives in the Lagrangian constraint space N^, the section s = TTUY °T of Y —> X satisfies the Euler-Lagrange equations (10.8.6), while s = H or obeys the Cartan equations (10.8.9a) - (10.8.9b). •

This results from the relation L = {J1L)LH, where L is the Lagrangian (10.8.7) on J1^ and LH is the Lagrangian (10.9.9) on J J n . THEOREM 10.9.6. Given a semiregular Lagrangian L, let a section s of the jet bundle JlY - » I b e a solution of the Cartan equations (10.8.9a) (10.8.9b). Let if be a Hamiltonian form weakly associated to L, and let H satisfy the relation HoLos

= JV^os).

(10.9.16)

Then the section r = L o s of the Legendre bundle n —> X is a solution of the Hamilton equations (10.9.11a) - (10.9.11b) for H. D

Chapter 10 Appendixes

631

Remark 10.9.3. Since H o L in Theorem (10.9.6) is a projector, the condition (10.9.16) implies that the solution I of the Cartan equations is actually an integrable section s = J ^ , where s is a solution of the EulerLagrange equations. • We will say that a set of Hamiltonian forms H weakly associated to a semiregular Lagrangian L is complete if, for each solution s of the EulerLagrange equations, there exists a solution r of the Hamilton equations for a Hamiltonian form H from this set such that s — TTUY ° r. By virtue of Theorem 10.9.6 and Remark 10.9.3, a set of weakly associated Hamiltonian forms is complete if, for every solution s on X of the Euler-Lagrange equations for L, there is a Hamiltonian form H from this set which fulfils the relation HoLoJls = Jls. As for the existence of complete sets of weakly associated Hamiltonian forms, we refer to the following theorem [45l]. 10.9.7. Let L be an almost regular Lagrangian. On an open neighborhood in n of each point q G NL, there exist local Hamiltonian forms associated to L which constitute a complete set. • THEOREM

In the case of an almost regular Lagrangian L, we can say something more on the relations between Lagrangian and Hamiltonian systems. Let us assume that the fibred manifold L : JlY -> NL

(10.9.17)

admits a global section <£. Let us consider the pull-back HN = VHL,

(10.9.18)

called the constrained Hamiltonian form. By virtue of Lemma 10.9.3, it does not depend on the choice of a section of the fibred manifold JXY —> NL and, consequently, HL = LHN. For sections r of the fibre bundle NL —> X, one can write the constrained Hamilton equations r{uN]dHN)=0,

(10.9.19)

where u^ is an arbitrary vertical vector field on NL —» X. These equations possess the following important properties.

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Geometric and Algebraic Topological Methods in Quantum Mechanics

10.9.8. For any Hamiltonian form H weakly associated to an almost regular Lagrangian L, every solution r of the Hamilton equations which lives in the Lagrangian constraint space NL is a solution of the constrained Hamilton equations (10.9.19). • THEOREM

Outline of proof. Such a Hamiltonian form H defines the global section = H o iN of the fibred manifold (10.9.17). Due to the relation (10.9.15), HN =i^H and the constrained Hamilton equations can be written as r(uN\diNH) = r(uN\dH

\NL) = 0.

(10.9.20)

They differ from the Hamilton equations (10.9.12) restricted to NL. These read r(u\dH\NL) = 0,

(10.9.21)

where r is a section of NL —> X and u is an arbitrary vertical vector field on II —> X. A solution r of the equations (10.9.21) obviously satisfies the weaker condition (10.9.20). QED

10.9.9. The constrained Hamilton equations (10.9.19) are equivalent to the Hamilton-De Donder equations (10.8.14). • THEOREM

Outline of proof. It is readily observed that L = nzn ° HL. Hence, the projection TTZU (10.8.15) yields a surjection of ZL onto NL. Given a section ^ of the fibred manifold (10.9.17), we have the morphism Hi ° \P : NL, —> ZL,. By virtue of Lemma (10.9.3), this is a surjection such that •zn ° HL o $ = Id NL . Hence, HL o * is a bundle isomorphism over Y which is independent of the choice of a global section $ . Combination of (10.8.13) and (10.9.18) gives HN = (HL O^))EL that leads to a desired equivalence. QED The above proof gives something more. Namely, since ZL and NL are isomorphic, the Legendre morphism HL fulfils the conditions of Theorem 10.8.1. Then combining Theorem 10.8.1 and Theorem 10.9.9, we obtain the following. THEOREM 10.9.10. Let L be an almost regular Lagrangian such that the fibred manifold (10.9.17) has a global section. A section s of the jet bundle

633

Chapter 10 Appendixes

JlY —» X is a solution of the Cartan equations (10.8.10) if and only if Los is a solution of the constrained Hamilton equations (10.9.19). • Theorem 10.9.10 is also a corollary of Lemma 10.9.11 below. The constrained Hamiltonian form HN (10.9.18) defines the constrained Lagrangian LN = ho(HN) = {JliN)LH on the jet manifold ^NL

(10.9.22)

of the fibre bundle NL —> X.

LEMMA 10.9.11. There are the relations I = {JlLYLN,

LN = (J 1 * )•!,

where L is the Lagrangian (10.8.7).

(10.9.23) •

The Euler-Lagrange equation for the constrained Lagrangian LN (10.9.22) are equivalent to the constrained Hamilton equations and, by virtue of Lemma 10.9.11, are quasi-equivalent to the Cartan equations.

10.10

Fibre bundles V. Characteristic classes

Due to the de Rham Theorem 1.8.4, the characteristic classes of principal bundles are represented by the De Rham cohomology classes of certain exterior forms expressed in the strength of principal connections [142; 220; 333]. One meets these characteristic forms in many quantum field models. Let G be a topological group. A locally trivial topological fibre bundle Y —> X with a typical fibre V is called a fibre bundle with a structure group G (or, simply, a G-bundle) if G acts effectively on V on the left and Y admits an atlas \? = {{U^,ip{),p^} whose transition functions p^{x) take their values in the group G. In view of the cocycle condition (10.6.2), the transition functions p^ of a G-bundle constitute a one-cocycle (1.7.12) (written in the multiplicative form) of the presheaf G({U}) of G-valued continuous functions on X. Let \t' be another atlas over the same cover. Atlases Vt and \&' are equivalent and characterize the same G-bundle if there is a 0-cochain {g^} of the presheaf G({£/}) such that

This implies

p'idx) = se(z)Ptt(a:)sr10i:)>

XGU^H

uc.

(10.10.1)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

Accordingly, the cocycles obeying the relation (10.10.1) axe said to be equivalent. The equivalence classes of cocycles associated to a fixed cover il of X constitute the cohomology set H 1 (il;G({t/})). Passing to the direct limit of these sets through all covers of X, we obtain the cohomology set Hl(X;Gx) of the sheaf Gx of G-valued continuous functions on X. It should be emphasized that, since a direct limit is not defined in the category of groups, H1(X;GX) fails to be a group, unless G is an Abelian group. THEOREM 10.10.1. [220]. There is one-to-one correspondence between the elements of the cohomology set H1(X; Gx) and the equivalence classes of G-bundles over X with the same typical fibre V. • It is readily observed that the construction of the cohomology set i/ 1 (X; Gx) is independent of a typical fibre. Two G-bundles over X which belong to the same element of the cohomology set H1(X;GX) are called associated. By virtue of Theorem 10.10.1, two associated G-bundles with the same typical fibre are isomorphic, but their isomorphism need not be canonical. Let / : X' —> X be a continuous map. Every topological G-bundle Y -> X yields the pull-back fibre bundle fY -> X' (10.6.4) with the same structure group G. Therefore, / induces the map [f}:H\X;Gx)-^H\X';Gx,). 10.10.2. [220; 406]. Given a G-bundle Y over a paracompact base X, let f\ and /s be two continuous maps of X' to X. If these maps are homotopic, the pull-back G-bundles f{Y and f%Y over X' are equivalent. THEOREM

•

This fact leads to the following classification theorem. THEOREM 10.10.3. For every topological group G, there exists a topological space BG, called the classifying space, and a G-bundle PG —> BG, called the universal bundle, which possess the following properties. • For any G-bundle Y over a paracompact base X, there exists a continuous map / : X —> BG such that Y is associated to the pull-back bundle fPG. • If two maps / i and fi °f X to BG are homotopic, then the pull-back bundles J{PG and /I-PG are equivalent, and vice versa. •

635

Chapter 10 Appendixes

A topological G-bundle np : P —> X is said to be a principal bundle if it is provided with the continuous free right action RG : PxG^

P,

Rg:p^>pg,

p e P,

g£G,

(10.10.2)

of its structure group G on P which is transitive on each fibre of P —> X. It follows that P/G = X and that the typical fibre of a principal G-bundle is isomorphic to the group space of G, but it is not a group since the structure group acts on the typical fibre by left multiplications which do not preserve the group structure. A principal G-bundle P is equipped with a bundle atlas ^p = {{Ua,^),pap},

(10.10.3)

whose trivialization morphisms ip^ obey the equivariance condition €(P9)=1P(P)9,

9£G,

peiTpHUa).

(10.10.4)

Due to this property, every trivialization morphism ipf^ determines a unique local section za of P over Ua such that pr 2 o-0^oz a = 1. The transformation rules for za read zp{x) = za(x)pa0{x),

xeUaC\ U0.

(10.10.5)

Conversely, the family {(Ua, za)} of local sections of P with the transition functions (10.10.5) determines a unique bundle atlas of P. In particular, it follows that only trivial principal bundles have global sections. An automorphism $ p of a principal G-bundle P, by definition, is equivariant under the canonical action (10.10.2) of the group G on P, i.e., Rg o $ P = $ P o Rg,

g e G.

Every vertical automorphism of a principal bundle P (called a gauge transformation) is represented as $p(p)=pf{p),

P&P,

(10.10.6)

where / is a G-valued equivariant function on P, i.e., f(pg) = g~1f(p)g,

geG.

(10.10.7)

There is one-to-one correspondence between these functions and the global sections s of the group bundle PG = {Px G)/G,

(10.10.8)

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Geometric and Algebraic Topological Methods in Quantum Mechanics

whose typical fibre is the group G, subject to the adjoint representation of the structure group G. It is given by the relation PGxP3(s(KP(p)),p)^pf(P)€P, where P is considered as a G-bundle associated to itself. Hence, the group of gauge transformations of a principal G-bundle P —> X is isomorphic to the group of global sections of the P-associated group bundle (10.10.8). Given a principal G-bundle P — X, let V be a topological space provided with a continuous effective left action of the structure group G. Let us consider the quotient Y = (P x V)/G

(10.10.9)

of the product P x V by identification of elements (p, v) and (pg, g~lv) for all g € G. Let [p] denote the restriction of the canonical surjection P x V -> (P x V)/G

(10.10.10)

to the subset {p} x V so that \p\(v) = \pg](g~lv). Then the map Y 3 [p}{V) .-> ITP(P) e X

makes the quotient Y (10.10.9) into a fibre bundle over X. This is a topological G-bundle with the typical fibre V which is associated to the principal G-bundle P. Let us note that, for any G-bundle, there exists an associated principal G-bundle [406]. The peculiarity of the G-bundle Y (10.10.9) is that it is canonically associated to a principal bundle P. Indeed, every bundle atlas "Jp = {(Ua,za)} of P determines a unique associated bundle atlas V = {{Ua,ipa(x) = \za(x)}-1)} of the quotient Y (10.10.9), and any automorphism of P yields the corresponding automorphism $y : (pG, G^v) ->

{$P(P)G,

G~lv)

(10.10.11)

of the P-associated bundle Y (10.10.9). Since the pull-back of a principal bundle is also a principal bundle with the same structure group, it follows from Theorem 10.10.3 that, for any

Chapter 10 Appendixes

637

topological group G, there exists a universal principal bundle PG —> BG such that: • any principal G-bundle P over a paracompact base X is isomorphic to the pull-back bundle fPG with respect to some continuous map f : X —> BG; • if two maps / i and /2 of X to J3G are homotopic then the principal bundles ffPG and f%PG are isomorphic, and vice versa. Let $ : H —> G be a topological group homomorphism. If PH —> X is a //-principal bundle, there is always exists a G-principal bundle PG —» X together with the principal bundle morphism $ : PH — PG o v e r X. This is the P#-associated fibre bundle PG = (PH X G)/H with the typical fibre G on which / / acts on the left by the rule h(g) = (j>(h)g, while G acts on PG as G3

5

':(

M

)/HH(P,

5 5

')/H.

Conversely, if PQ —» X is a G-principal bundle, a problem is to find a / / principal bundle PH —> -X' together with the principal bundle morphism PH —> PG- If H —* G is a subgroup, we have the structure group reduction. If if —> G is a group epimorphism (extension), one says that PG lifts to PH-

Here, we restrict our consideration to the reduction problem in the case of finite-dimensional Lie groups. Let irpx '• P —> X be a G-principal bundle and H a closed subgroup of G. There is the composite fibre bundle P -> P/H -> X, where PE = Pn~^P/H

(10.10.12)

Y, = P/H'K^>X

(10.10.13)

is a -ff-prmcipal bundle and

is a P-associated fibre bundle with the typical fibre G/H. One says that the structure group G of a principal bundle P is reducible to a Lie subgroup H if there exists a //-principal subbundle Ph of P. The following two theorems [250] characterize the structure group reduction. 10.10.4. A structure group G of a principal bundle P is reducible to its closed subgroup H if and only if P has an atlas ^ p with //-valued transition functions. Given a reduced subbundle Ph of P, such an atlas <J>p is defined by a family of local sections {za} which take their values into Ph. • THEOREM

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Geometric and Algebraic Topological Methods in Quantum Mechanics

10.10.5. There is one-to-one correspondence Ph = ^p%{h{X)) between the reduced .ff-prmcipal subbundles Ph of P and the global sections h of the quotient fibre bundle P/H -» X (10.10.13). • THEOREM

Remark 10.10.1. Theorem 10.10.5 also holds when G and H are Banach Lie algebras [86]. • In general, there are topological obstructions to the reduction of a structure group of a principal bundle to its subgroup. The following is a corollary of Theorem 10.6.4 and Theorem 10.10.5. THEOREM 10.10.6. The structure group G of a principal bundle P is always reducible to its closed subgroup H, if the quotient G/H is homeomorphic to an Euclidean space Rfc. In particular, a structure group G of a principal bundle is always reducible to its maximal compact subgroup H since the • quotient space G/H is contractible [406].

It should be emphasized that two ^-principal subbundles Ph and Ph> of a G-principal bundle P need not be isomorphic, unless the quotient G/H is homeomorphic to an Euclidean space M.k [406]. If H-principal bundles Ph and Ph are isomorphic, there exists a vertical automorphisms of P which sends Ph onto Ph' [296]. All that we have said so far holds in the category of smooth manifolds (see Remark 1.8.3), and we come to smooth principal bundles. In this case, since any smooth manifold has the homotopy type of a CW-complex [402], it sufficient to deal with the category CW of CW-complexes. The key point is that any weak homotopy equivalence between connected CW-complexes is a homotopy equivalence, and that a principal bundle PQ in this category is universal if and only if the topological space PQ is weakly contractible, i.e., all homotopy groups irq(PG) of PQ are trivial. For instance, the unitary group U{E) of any separable Hilbert space is contractible in the uniform topology [263]. This makes U(E) a candidate for a universal principal bundle in the category CW. Let us recall that, if H is a closed subgroup of a topological group G, the surjection G —» G/H is a principal i/-bundle if it admits a local section [406]. This is the case of Banach-Lie groups G and H. In particular, a closed subgroup of a real Lie group G is a Lie group, and G —> G/H is a smooth principal bundle. From now on, we consider smooth principal bundles n : P —> X whose structure group is a finite-dimensional (dim G > 0) Lie group G. Theorem

Chapter 10 Appendixes

639

10.10.3 also provides classification of these principal bundles due to the following fact [406]. 10.10.7. Let X be a smooth manifold and G a Lie group. There is bijection between the cohomology set H1(X; Gx) of the sheaf G°x of continuous G-valued functions on X and the cohomology set Hl(X; G~) of the sheaf G~ of smooth G-valued functions on X. • THEOREM

The key point is that, as was mentioned above, the characteristic classes of smooth principal bundles are characterized by the De Rham cohomology classes of certain characteristic exterior forms expressed in the strength of principal connections on these bundles. Given a principal bundle P —> X, the quotient of the tangent bundle TP —> P and that of the vertical tangent bundle VP of P by the tangent prolongation TRG of the canonical action RG (10.10.2) are vector bundles TGP = TP/G,

VGP = VP/G

(10.10.14)

over X. Sections of TQP -> X are naturally identified with G-invariant vector fields on P, while those of VGP —> X are G-invariant vertical vector fields on P. Owing to the equivariance condition (10.10.4), any bundle atlas (10.10.3) of P yields the associated bundle atlases {Ua,Tip£/G)} of TGP and {Ua, Vtp^/G)} of VGP. Given a basis {ep} for the right Lie algebra gr, let {d\, ep} and {ep}, where ep = {ipa/G)~l(ep), be the corresponding local fibre bases for the vector bundles TGP and VGP, respectively. Relative to these bases, the Lie bracket of sections

of the vector bundle TGP —> X reads

K, V) = (nr?

- rfd^dx

+ (edxvr ~ VXdxC + c ^ V K - (10.10.15)

Putting £A = 0 and r)1* = 0, we obtain the Lie bracket [C^c^Ver

(10.10.16)

of sections of the vector bundle VG —» P. A glance at the expression (10.10.16) shows that VQP —> X is a finite-dimensional Lie algebra bundle, whose typical fibre is the right Lie algebra gr of the group G. The structure group G acts on this typical fibre by the adjoint representation (10.6.30). Therefore, VGP is called the Lie algebra bundle.

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Geometric and Algebraic Topologicat Methods in Quantum Mechanics

Let JlP be the first order jet manifold of a principal G-bundle P —> X. Its quotient C = JlP/G

(10.10.17)

by the jet prolongation of the canonical action RQ (10.10.2) is a fibre bundle over X. Bearing in mind the canonical imbedding JlP —> TX TP (10.6.42) and passing to the quotient by G, we obtain the corresponding canonical imbedding A c : C -> TX ® TGP

(10.10.18)

of the fibre bundle C (10.10.17). It follows that C is an affine bundle modelled over the vector bundle C = TX VGP -» X.

(10.10.19)

Given a bundle atlas of P and the associated bundle atlas of VGP, the affine bundle C is provided with affine bundle coordinates (a;,a'), and its elements are represented by TQP-valued one-forms a = dxx® {dx + aqxeq)

(10.10.20)

on X. One calls C (10.10.17) the connection bundle because its sections are naturally identified with principal connections on the principal bundle P —» X as follows. A principal connection A on a principal bundle P —» X is a connection (i.e., a global section A of the affine jet bundle J 1 P —> P) which is equivariant JlRg o A = A o Rg,

g£G,

(10.10.21)

under the right action (10.10.2) of the group G on P. Therefore, there is one-to-one correspondence between the principal connections on a principal bundle P -+ X and the global sections of the bundle C - » I (10.10.17). Since C —> X is an affine bundle, a principal connection on a principal bundle always exists. It is represented by a T^P-valued form A = dxx (dx + Aqx(x)eq)

(10.10.22)

on X due to the imbedding (10.10.18), and it splits the exact sequence 0 -> VGP -> TGP -» TX -> 0

(10.10.23)

641

Chapter 10 Appendixes

(cf. the exact sequence (10.6.13a)). Given a principal G-bundle P —> X, the Frolicher-Nijenhuis bracket (10.6.25) on the space O{P) 71 (P) of tangent-valued forms on P is compatible with the canonical action RQ (10.10.2) of G on P , and yields the induced Prolicher-Nijenhuis bracket on the space O(X) ®TQP(X) of TQP-valued forms on X. Its coordinate form issues from the Lie bracket (10.10.15). Similarly to the curvature R (10.6.58), let us put FA = \dAA = \{A,A]m FA = ^

G 02{X)®VGP{X),

(10.10.24)

^F^dxxAdx»®er,

= 9xAr^ - d»Arx + t^A'A}.

(10.10.25)

It is called the strength of a principal connection A. Every principal connection A on a principal bundle P —* X induces a unique connection on the associated fibre bundle Y (10.10.9) [250]. If Y is a vector bundle, this connection takes the form A = dxx 0 (dx - ApxIp}^di),

(10.10.26)

where Ip are generators of the linear representation of the Lie algebra fjr in the vector space V. The curvature R (10.6.58) of this connection equals R= —FA, where FA = \FpxI^dxx

A dx» ® di

(10.10.27)

is also called the strength of the connection A (10.10.26). Let us consider the quotient JXP -+ JlP/G

= C.

One can show that it is a principal G-bundle which is canonically isomorphic to the trivial pull-back bundle Pc = C x P - • C.

x

The latter admits the canonical principal connection A = dxx® (dx + apxep) + da\ ® dx € O\C) ® TG(PC)(C),

642

Geometric and Algebraic Topological Methods in Quantum Mechanics

whose strength $ is the V^P-valued horizontal two-form (10.10.28)

S=^[A,A]FS,

S = ( d ^ A dx» + l-crpqalaldxx A d i " ) ® e r , on C [162; 296]. It is readily observed that, given a principal connection A on P, the pull-back A$ = FA is the strength (10.10.24) of A. We use tf (10.10.28) in order to construct the above-mentioned characteristic forms. Let ffc(e) = &n...rfc£ri •••e r be a G-invariant polynomial of degree fc > 1 on the Lie algebra gr written with respect to its basis {er}, i-e., -Pk(e) = £ b n . . . r , . e r i " ' ' < & ? ••-er"=

kbri...rk<%,eer

• • • sr« = 0.

3

Let us associate to I{e) the closed gauge-invariant 2/c-form P2fc(^) = br^.r.r1

A • • • A rk

(10.10.29)

on C. Let A be a section of C —> X . Then the pull-back (10.10.30)

Pn(FA) = A'PM($)

of P2fc(T?) is a closed form on X, called the characteristic form. Since C —+ X is an affine bundle, the de Rham cohomology of C equals that of X. It follows that V2k(-S) a n d ^fcC-fx) possess the same cohomology class [P2fc(ff)] = [P2fc(i^)]

(10.10.31)

for any principal connection A Thus, •Pk(e) H- [P 2 fc(^)] € H'(A-) is the familiar W^eii homomorphism. Henceforth, we will restrict our consideration to principal bundles with the structure Lie groups GL(k,C) and GL(k,M). By virtue of Theorem 10.10.6, these structure groups are reducible to the maximal compact subgroups U(k) and O(k), respectively. The classifying spaces for the above mentioned groups are B(U(k))=

lim <5(k,N;C), N—>oo

B(O(k)) = Urn <5(k,N;R),

(10.10.32)

N-+oo

where <3(k,N;C) and <S(k,N;R) are the Grassmann manifolds of kdimensional subspaces of CN and IR^, respectively. As a consequence, the equivalence classes of U(k)- and O(fc)-bundles over a manifold X are

643

Chapter 10 Appendixes

represented by elements of the cohomology algebra H* (X; Z) of X with coefficients in the constant sheaf Z [220]. They are called the characteristic classes. To be more precise, characteristic classes are elements of singular cohomology of X with coefficients in the ring Z, but one refers to Theorem 1.7.5 since X is paracompact. Due to the homomorphism (1.8.13), these characteristic classes are represented by elements of the de Rham cohomology groups of a manifold X. We start with the Chern classes a(P) G H2i(X; Z) of a principal U{k)bundle P. The corresponding characteristic form (called the total Chern form) is c ( F ) = d e t (l + ± F \

= 1 + a(F) + c2{F) + ••• ,

(10.10.33)

where F is the strength (10.10.27) of a principal connection A (10.10.26) on a P-associated vector bundle E -* X with the typical fibre Cfc, which provides the natural representation of the Lie group. The summands Ci(F) of c(F) are called Chern 1%-forms. For instance, Cl (F)

= ^TrF,

c2{F) = -^[Tr(FAF)-TrFATrF}.

(10.10.34)

All Chern forms Ci(F) are closed, and their cohomology classes are identified with the Chern classes a(E) € H2i(X,Z) of the {/(fc)-bundle P under the homomorphism (1.8.13). The total Chern form (10.10.33) is associated to the total Chern class c{E) = co(E) + Cl{E) + • • • . Example 10.10.2. Let us consider a complex line bundle L —> X. It is a fibre bundle with the structure group U(l) which acts on its typical fibre C by means of the infinitesimal generator I = il. The strength form (10.10.27) on this fibre bundle reads F = 1-FXildxx A da;".

(10.10.35)

Then the total Chern form (10.10.33) of a [/(l)-bundle c(F) = 1 + c i ( F ) ,

C l (F) =

L is

^ - T r F = -^-FXlldxx 2ir

47T

A dxP.

•

644

Geometric and Algebraic Topological Methods in Quantum Mechanics

The Chern classes possess the following properties (which easily issue from their representation by the Chern forms): (i) a{E) = 0 if 2z > n = dimX; (ii) Ci{E) = 0 if i > jfe; (Hi) c(E ® E1) = c{E) ^ c{E'); (iv) c\(L © L') = c\(L) + ci(L'), where L and U are complex line bundles; (v) if fE —> X' is the pull-back bundle generated by the morphism / : X' —» X, then c(fE) = fc(E), where / is the induced morphism of the cohomology groups /* : H(X; 1) -» H(X'; Z).

(10.10.36)

Properties (iii) and (v) are basic for the following theorem [220]. 10.10.8. For any U(k)-bundle E —> X, there exists a topological space X' and a continuous morphism / : X' —> X such that the pull-back fE — X' is the Whitney sum of complex line bundles THEOREM

fE

=

Ll®---®Lk,

and the induced morphism / (10.10.36) is an injection.

•

It follows that the total Chern class of any [/(fc)-bundle E can be represented by the product c(E) = fc(E) = c(Lx © • • • © Lfc) = c(L1)^...^c(Lk) = ( l + o i ) - - - ( l + afc),

(10.10.37)

where a» = ci(Lj) denotes the Chern class of the complex line bundle Li (see Example 10.10.2). The formula (10.10.37) is called the splitting principle. For instance, ci(£) = ^ a i , i

c2(E) = ^2 ahai2, »i<2

Cj(E) =

^

aix---air

ii<---
Example 10.10.3. Let E be a f/(A;)-bundle, the dual of E. In accordance with the splitting principle, we have c(E)

= c(L\) ~ • • • ~ c{Lk) = (I+ a{) •••(! + o j ) .

645

Chapter 10 Appendixes

Since the infinitesimal generator of the dual representation of U(l) is / = -il, then F* = -F and a* = -a,. It follows that Ci(E)

= (-iyCi(E).

(10.10.38)

• There are some other characteristic classes of [/(fc)-bundles which are expressed in the Chern classes. Let us mention the following two ones, (i) The Chern character ch(J5) is given by the characteristic form

^) = E^U^) '

(iai0'39)

possessing the properties ch(£ © E1) = ch(E) + ch(E'),

ch(E ® E') = ch(E) - ch(E')-

Using the splitting principle, one can express the Chern character in the Chern classes as follows: ch(E) = ch(Li © • • • © Lk) = ch(Li) + • • • + ch{Lk) = e x p a ! + • • • + e x p a f c = k + a ( E ) + -=[c\(E)

(10.10.40)

- 2 c 2 { E ) \ + ••• .

(ii) The Todd class is given by the characteristic form k

td(E) = Y ^

l + lc1(E)

—. T = 1 - exp(-Oi)

(10.10.41)

+ ^ ( a ( E ) 2 + c2(E)) + .-.,

possessessing the property td(E © E') = \d(E) - td(S'). Let us note that, because of rational coefficients in the series (10.10.40) and (10.10.41), the Chern character and the Todd class belong to cohomology H(X;Q>) with rational coefficients. Let us turn now to characteristic classes of a fc-dimensional real vector bundle E —> X with the structure group O(k).

646

Geometric and Algebraic Topological Methods in Quantum Mechanics

The Pontryagin classes of an O(fc)-bundle E in the de Rham cohomology algebra H(X) are associated to the characteristic polynomial p(F)

= D e t ( l - ± - F \ = 1 + P l ( F ) + P 2 ( F ) + •••

(10.10.42)

of the strength F (10.10.27) on E which takes its values in the Lie algebra of the group O(k). Its infinitesimal generators in Rfc obey the condition (1)1 = —(/)„. Therefore, only the summands of F of even degree in the decomposition (10.10.42) are different from zero, i.e., Pi(E) 6 Hil(X). Pontryagin classes possess the following properties: (i) pi(E) = 0 if 4z > n = dimX; (ii) Pi(E) = 0 if 2z > fc;

(Hi) p(E®E')=p(E)+p(E'). Let us note that, if Pontryagin classes are given by elements of cohomology H(X; Z) with integer coefficients, property (ii) is true only modulo cyclic elements of second order. Let us consider the relations between the Pontryagin classes of O(k)bundles and the Chern classes of [/(fc)-bundles. There are the following commutative diagrams of group monomorphisms

O(k) I GL(k,R) U(k)

—*

U(k) I

—>GL(k,C) —^

O(2k)

[

I

GL(k,C)

(10.10.43)

(10.10.44)

—>GL(2fc,R)

The diagram (10.10.43) implies the injection V

: H\X;O(k)^) ^ H\X;U(k)^),

and one can show that Pi(E)

= (-lYcnME)).

The diagram (10.10.44) yields the injection p:H1(X;U(k)^)-^H1(X;O(2k)^).

(10.10.45)

647

Chapter 10 Appendixes

Then we have
H\X; U(2k)%). This means that if E is a {/(fc)-bundle, then p(E) is an 0(2fc)-bundle, while
O(2k) —> C/(2A:) have the matrix form /ReM -IraM\ ^

M r

»,r

r> lit)

(M 0 \ >

n »r»

/ m m ^ •

(10.10.46)

\ I m M Re My \ 0 My A glance at this expression shows that the vector bundle tp o p(E) is the Whitney sum of vector bundles E and E. Therefore, we have c(vop(E))=c(E)^c(E). Then a combination of (10.10.38) and (10.10.45) gives the relation = c(V o p(E)) = c(E) - c(E) - (10.10.47)

Y^(-iyPi(p(E)) i

. i

J

[

j

between the Chern classes of a C/(/c)-bundle E and the Pontryagin classes of the O(2/c)-bundle p(E). Example 10.10.4. By Pontryagin classes Pt(X) of a manifold X are meant those of the tangent bundle TX. If X is a 2m-dimensional almost complex manifold (i.e., the structure group GL(2m, R) of its tangent bundle TX is reducible to its subgroup GL(m,C) (2.6.2)), one can associate to X the Chern classes Ci(X). They are the Ghern classes of the tangent bundle TX of X regarded as a GL(m,C)-bundle, i.e., a(X) = d(p(TX)). Then the formula (10.10.47) provides the relation

£(-iyPi(x)= $>(x) - £(-i)'Ci(x) i

Li

J

[ j

between Pontryagin and Chern classes of an almost complex manifold X.

•

In conclusion, let us also mention the Stiefel-Whitney classes Wi S Hl(X,Z2) of the tangent bundle TX of a manifold X. In particular, a

648

Geometric and Algebraic Topological Methods in Quantum Mechanics

manifold X is orientable if and only if w± = 0 . If X admits an almost complex structure, then W21+1 — 0, W2i = Ci mod 2. 10.11

X-Theory of vector bundles

ff-Theory of vector bundles is a particular /£T-theory of additive categories (see Section 8.7). Let C(X) denote the set of all classes of equivalent (finite-dimensional) vector bundles over a smooth manifold X. It is a commutative monoid with respect to the Whitney sum ©, where the 0-dimensional vector bundle plays the role of a zero element. The goal is to define the operation ©. Let us recall the following algebraic construction. Given an additive monoid A (i.e., without a subtraction operation '—'), let us consider the quotient K{A) of A x A with respect to the equivalence relation: (a, b) w (a', b') if there exists an element p £ A such that a + b' + p = b + a' + p. Then K(A) is an Abelian group, called the Grothendieck group of the monoid A. There is the homomorphism k : A —* K(A) such that k(a), a G A, is the equivalence class of the pair (a, 0) G A x A. The inverse element —A;(a) is defined as the equivalence class of the pair (0, a). Then any element (a, b) of the group K(A) is the difference k(a) — k(b), a,b £ A. It is easily seen that k(a) = k(b) if and only if there exists an element p £ A such that a + p = b + p. Let us construct the Grothendieck group K(X) = K(C(X)) of the monoid C(X) of classes of equivalent vector bundles over a manifold X [16; 237]. Let us denote [E] = k(E), E £ C{X). Then [E] = [E'\ if and only if there exists a vector bundle F —> X such that E®FK,E'

®F.

Let us recall the well-known structural theorem [197], which can be derived from Theorem 1.8.5. THEOREM 10.11.1. Given a vector bundle E —> X (whose base X need not be compact), there exists a vector bundle E' —> X such that the Whitney sum E © E' is a trivial vector bundle. • COROLLARY

10.11.2. The equality [E] — [E1] in the Grothendieck group

Chapter 10 Appendixes

649

K(X) holds if and only if E©i

m

« E' © r

for some trivial vector bundle Im of fibre dimension m over X.

•

It follows that two vector bundles E and E' belong to the same element of the Grothendieck group K(X) (they are called stably isomorphic) only if they are of the same dimension, but are not necessarily isomorphic. For example, let X = S2 be a two-dimensional sphere and TS2 the tangent bundle of S2. Then we have {TS2} = [I2] = 0 e K{S2), though the tangent bundle TS2 —» S2 is not trivial. This example shows that the morphism k : C(X) —> K(X) is not an injection. It becomes an injection on classes of equivalent real vector bundles of dimension m > dimX and on the classes of equivalent complex vector bundles of dimension m > dimX/2. There is another equivalence relation {E} on the monoid C(X) of classes of equivalent vector bundles over X. Let us put {E} = {E1} if and only if there exist trivial vector bundles Ik and Ip such that E © Ik w E' © P. These equivalence classes constitute the group K(X) whose zero element includes all trivial vector bundles over X. There is an isomorphism K(X)=Z®K{X). For instance, if X contractible, then K{X) = Z in accordance with Theorem 10.6.3, while K(X) = 0. THEOREM

morphisms

10.11.3. Let X and X' be compact manifolds, and let manifold

h •• x -> x',

h-.x^x'

be homotopic. Then / i and / 2 yield the same morphisms of groups K{X) —> K(X') and K(X) -> K(X'). • Let us now specify KC(X) (resp. KR(X)) as the Grothendieck group of complex (resp. real) vector bundles over a compact manifold X. The

650

Geometric and Algebraic Topological Methods in Quantum Mechanics

Chern character (10.10.39) of these vector bundles defines the map ch : KC(X) -» 0 F 2 i ( X , Q )

(10.11.1)

i>0

such that ch([E] - [£']) = ch(E) - ch(E').

(10.11.2)

The morphism (10.11.1) leads to the isomorphism Kc(X)®Q=($H2i(X,Q) i>0

for complex vector bundles and to the isomorphism

KR(X)®Q= 0# 4 (X,Q) i>0

for the real ones. 10.12

Elliptic complexes and the index theorem

The index theorem establishes the relationship between the analytic properties of differential operators on sections of fibre bundles and the topological properties of fibre bundles [53; 142; 333; 335]. Here, we restrict our exposition to the classical index theorem for linear elliptic operators on compact manifolds. These operators have a finite-dimensional kernel and, consequently, they are Fredholm operators. In order to generalize the index theorem to non-compact manifolds, one either imposes conditions sufficient to force the operators to be Fredholm ones or consider the operators which are no longer Predholm, but their index can be interpreted as a real number by some kind of averaging procedure [18; 375]. The reader is addressed to [350] for the index problem for non-linear elliptic operators. We start with the notion of a symbol of a linear differential operator. Let E = RnxVE,

F = RnxVF

651

Chapter 10 Appendixes

be trivial smooth complex vector bundles over W1 equipped with the Cartesian coordinates (xa). We follow the standard notation Dt =

11 = ! + ••• + «.

W—^'

t10-12-1)

for any n-tuple of positive numbers t = (ti,... ,tn). Let E(M.n) and F(R n ) be the structure C(R")-modules of global sections of the vector bundles E and F, respectively. Any r-order linear differential operator V : t,[K ) —> t [K ) takes the form V= 5^(-i)llo t i?,

(10.12.2)

|t|
where CLl . O/yK.

j —> r [K.

j

are C°°(M")-linear morphisms, i.e., 0-order linear differential operators. The linear r-order differential operator |t|=r

is called the leading term of the linear differential operator V (10.12.2). For instance, a first order linear differential operator (10.12.2) reads n

V = Y2 ~iaada + a,

(10.12.3)

a=l

while its leading term is n

T>=

^-iaada. J=I

Let us consider the Cartesian product E n x R", where Rn denotes the dual of R" coordinated by (pa)- Let E = Rn x E,

F = Rn x F

652

Geometric and Algebraic Topological Methods in Quantum Mechanics

be the pull-back fibre bundles over R n x W1. Any linear differential operator V (10.12.2) yields the linear bundle morphism a(V):E->F,

o{V)(x,p) = ^

at{x)ft • • -p^,

(10.12.4)

\t\ = T

over Rn x R". The fibrewise morphism a(T>) (10.12.4) is called the symbol of the differential operator T> (10.12.2) (or the leading symbol in the terminology of [142]). This is the Fourier transform of the leading term of the differential operator T>, namely, Vs(x) = (27r)-n

[a(V)(x,p)sF(P)e-ip°xadnp,

where sF is the Fourier transform of a section s of the vector bundle E. Now, let us generalize this notion to smooth complex vector bundles E and F over a compact oriented n-dimensional smooth manifold X, provided with a Riemannian metric g. Let (10.12.5)

V : E{X) -» F(X)

be an r-order linear differential operator. It takes the local form (10.12.2) with respect to atlases \p£ and typ of the fibre bundles E and F over the same cover of the base X. Let 7r: DX —> X and SX —> X be fibre bundles of n-dimensional unit discs and (n — l)-dimensional unit spheres, respectively, which are associated to the tangent bundle TX. Let us consider the pull-back bundles ftE and 7rF over DX. The symbol of the differential operator V (10.12.5) is denned as the bundle morphism a(V):irE

^nF,

a{V) : (y, VE) -> (y,a{V){ix{y),y)VE),

y £ DX,

(10.12.6)

where VE is the typical fibre of E and the fibrewise morphism a(D)(x — ir(y),y) takes the local form (10.12.4). One can justify the fact that the morphism (10.12.6) is globally defined. Symbols of differential operators obey the composition rule a{V2oV1) = a(V2)oa{V1).

(10.12.7)

DEFINITION 10.12.1. The linear differential operator V (10.12.5) is called elliptic if the morphism a(T>)\sx is an isomorphism of the fibre bundles nE\sx &ndTrF\SX•

653

Chapter 10 Appendixes

We point out the following two properties of elliptic operators. • An immediate consequence of Definition 10.12.1 and the composition rule (10.12.7) is that a composition of elliptic operators is an elliptic operator. • The kernel of an elliptic operator is finite-dimensional, i.e., any elliptic operator on a compact manifold is a Fredholm operator. Let us assume that complex vector bundles E —> X and F —> X are provided with non-degenerate positive Hermitian forms (., .)E and (-,-)F (or, simply, Hermitian metrics). Accordingly, complex vector spaces E(X) and F(X) are provided with Hermitian forms

(•\.)E

= / ( - , - )EV,

(.\.)F=

X

J(.,.)FV, X

where V is a volume form on X. Then, given a linear differential operator V (10.12.5), the adjoint differential operator V : F(X) -> E(X)

(10.12.8)

is uniquely denned by the equality (S\WS')E

= (Ds\s')F

for arbitrary sections s £ E(X), s' G F(X). Moreover, its symbol a{V) : irF -> nE is the adjoint morphism to the symbol cr(V) of the differential operator T> with respect to the pull-back Hermitian metrics on the fibre bundles nE and TTF. If V is an elliptic operator, V is well. DEFINITION 10.12.2. The index of an elliptic operator V (10.12.5) is defined as the difference T(£>) = d i m K e r P - d i m K e r D * .

(10.12.9)

D Let us turn now to the notion of an elliptic complex. Let {Ek\T>k, £>£}, k G N, be a finite sequence of complex smooth vector bundles E^ over a compact manifold X together with linear differential operators

Vk : Ek(X) -

Ek+l(X)

654

Geometric and Algebraic Topological Methods in Quantum Mechanics

and the adjoint ones Vk : Ek+1(X)

-> Ek{X).

Let us suppose that Vk+\ o Vk = 0 for all k. It is easily justified that 2?£ o Z?j£+1 = 0, too. Then there is the cochain complex ..

V

-^i Ek(X)

^Ek+1(X)V-^

••• ,

denoted simply by (E,V), together with linear differential operators Ak = VkoVk+Vk^1oVk_1 : Ek(X) - Ek(X),

(10.12.10)

called Laplacians. The complex (E, V) is said to be elliptic if all Laplacians Afc (10.12.10) are elliptic operators. Example 10.12.1. The Hodge star operator : Ok{X) - • On~k(X) on exterior forms on a Riemannian manifold (X, g) reads (dxXlA- • -AdxXk) = lgj_ k)ltXl"XkjUfc+1...Mn^^+1

A- • -Adx^,

(10.12.11)

where \g\ = det(g^) and e is the totally skew-symmetric Levi-Civita tensor with the component ei...n = 1. Let us recall the relations 1=V,

V = 1,

* 4> = ( - l ) " ( n - r V ,

^eOr(I).

With , we have the Hermitian form, called the Hodge inner product {<j>\<j>')= [<(> A <}>'= J X

f4>A(j>'

J X

(10.12.12)

on exterior forms of the same degree. The exterior differential d admits the adjoint operator with respect to this form. It is the nilpotent codifferential 6(t> = {-l)nk+n+1d(t>,

(f>GOk{X),

which obeys the relation d(f>A4>'-4>A

S<j>' = d{cj> A «/>')•

Chapter 10 Appendixes

655

Then the de Rham complex (O(X),T>k = d) of exterior forms on X, together with the codifferential T>k = 5 and the Laplace-Beltrami operator (or, simply, a Laplacian) Ak = A = do5 + 5od,

(10.12.13)

is elliptic. Elements of the kernel Bk of the Laplacian A^ are called harmonic k-forms. • The well-known Hodge theorem is extended to an arbitrary elliptic complex as follows. THEOREM 10.12.3. Given an elliptic complex (E,T>), every section sk G Ek{X) is split into the sum

sfc = Vk-iSk-i + Vtsk+i + hk, where hk is a harmonic section, i.e., Akhk

(10.12.14)

=0.

•

In particular, it follows that sk € Ek{X) is a harmonic section if and only if Vksk = 0,

Vk_lSk = 0.

Let us consider the cohomology H(E, V) of the elliptic complex (E, T>). One can derive from the Hodge theorem that the cohomology group Hk(E,T>) is isomorphic to the kernel Bk of the Laplacian Afe, i.e., any cohomology class of sections sk e Ek{X) contains only one harmonic section. It follows that the cohomology groups of an elliptic complex are finite-dimensional. DEFINITION 10.12.4. The index of an elliptic complex (E,V) is denned as the sum T(E, V) = ^ ( - l ) f c d i m Hk{E, V) = ^ ( - l ) f c d i m Ker Afc. k

(10.12.15)

k

• Example 10.12.2. If {E,T>) = (Cl,d) is the elliptic de Rham complex on a Riemannian manifold X in Example 10.12.1, its cohomology is the de

656

Geometric and Algebraic Topological Methods in Quantum Mechanics

Rham cohomology of X and its index is precisely the Euler characteristic T(Q,d) = J](-l) f c dimtf f e pO = ^ ( - l ) f c d i m 5 f e = k

k

fc

£(-l) 6 f c = XPO, k

where bk are Betti numbers. Let us note that bk = bn-k and X(X)=

Je(TX)

where e(TX) is the Euler characteristic form on X, corresponding to the Euler characteristic class of the tangent bundle TX of X. • Example 10.12.3. The index of an elliptic operator V (10.12.5) equals the index of the two-term complex 0 -> E{X) -^-> F(X) -> 0.

(10.12.16)

Indeed, Laplacians of this complex are elliptic operators A0 = X>o2?,

A1=DoD,

and its cohomology groups are H1=KeiV.

H° =Ker£>,

A manifested equality follows at once from the formulae (10.12.9) and (10.12.15). • Given an elliptic complex (E,T>), one can construct a suitable two-term elliptic complex with the same index as (E,T>). Letting E = ®E2k, k

F = ®E2k+i, k

V = ®{V2k+V2k_l), k

V

=®{V2k+V2k_l),

k

we easily observe that V : E{X) -+ F(X),

V : F) -^ E(X)

and that the sequence 0 —>E ^->F —>0

(10.12.17)

657

Chapter 10 Appendixes

is a complex. Its Laplacians k

k

are elliptic operators, and so is the operator V. (10.12.17) is elliptic. Its index

Hence, the complex

r = dimKerDo - dimKerDi = ^ ( - l ) * d i m K e r A* =

T{E,V)

k

coincides with the index of the original complex (E, V) and that of the operator V. Let us turn now to the index theorem for an elliptic differential operator V (10.12.5). We take two copies DX\ and DX2 of the unit disc bundle and glue them together along the common boundary SX in order to define a new fibre bundle p : C(X) —* X with the typical fibre 5". It is called the compactified tangent bundle. It inherits the orientation of DX\. Then we put P12 = p\oxi 2> a n d consider the pull-back bundles p\E and p^F- Let us glue these fibre bundles together over the overlap SX = DX\ n DX2 by means of the morphism a(D)\S(x)- We obtain the vector bundle £ —> CX, called the symbol bundle. 10.12.5. The topological index of an elliptic differential operator V is defined as

DEFINITION

7(P)=

j ch(E)A/3td(X), C(X)

where ch(S) is the characteristic form of the Chern character of the symbol bundle S and td(A") is the characteristic form of the Todd class of the tangent bundle TX of an even-dimensional manifold X. The topological index is zero if X is an odd-dimensional manifold. D The well-known Atiyah-Singer index theorem states the following. THEOREM 10.12.6. The index and the topological index of an elliptic operator are equal to each other, i.e.,

r(P)=7(P).

(10.12.18)

•

658

Geometric and Algebraic Topological Methods in Quantum Mechanics

In particular, it follows that the topological index of an elliptic operator is an integer number. Another corollary of the equality (10.12.18) is that the index of an elliptic operator is zero if X is of odd dimension. For a 2m-dimensional oriented compact manifold X in question, the equality (10.12.18) can be rewritten in the form

r(V) = (-i)=5±a JoHEGF)^, x

(10.12.19)

where e(X) is the Euler characteristic form on X. If V is the differential operator in the two-term complex (10.12.17) associated to an elliptic complex (E, V), we obtain the index of the complex (E, V) equal to that of the operator V. It reads

r(E,V) = (-1)=3> J E ( - D f c c h ( ^ ) ^ . x

k

In conclusion, let us consider the signature complex whose index coincides with the signature of a manifold X. Let X be a 4m-dimensional oriented compact manifold. The bilinear form

nx(4>,') = y^A^.', x

c/>,(t>'£O2m(x),

yields a non-degenerate real symmetric bilinear form Qx •• H2m(X)

x H2m(X)

-> H2m(X)

- H2m{X)

-> R

(10.12.20)

on the cohomology group H2m(X). It is called the intersection form. The signature or the index of a manifold X is denned as the signature T(X) = r(flx) of the intersection form Qx (10.12.20), i.e., the number of positive minus the number of negative eigenvalues of fix. It should be noted that QX{
(10.12.21)

659

Chapter 10 Appendixes

acting on exterior forms on X. A direct calculation shows that Cl2 = Id and (10.12.22)

n{d + 5) = -{d + S)n.

Let O±(X) denote the subspaces of exterior forms on X possessing the eigenvalues ±1 with respect to the operator f2 (10.12.21). In view of the relation (10.12.22), we have the two-term elliptic complex 0 -» Sl+(X) ^>fi_(X) -» 0,

(10.12.23)

called the signature complex. Its index is T(O ± (X),d + (J) = dimKerA|n + (x)-dimKerA| n _(x) = dimHlm(X) - dimH2_m{X) = T(SIX). Applying the index theorem to the index complex (10.12.23), one restarts the Hirzebruch signature theorem r(fi x )=

/L im {X), X

where L(X) is the Hirzebruch polynomial

3

J

Example 10.12.4. If dimX = 4, the signature of X is expressed in the first Pontryagin class T(X)=T(QX)

= i Pl (X) = - ^

JTr(RAR), x

where R is the curvature form of the Levi-Civita connection of a Rieman• nian metric on X.

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Index

/x-equivalent functions, 576 cr-representation of a groupoid, 228 6-tensor product, 230 6-algebra, 237 6-algebra, barreled, 237 dg-operad, 473 Z-fibre, 554 p-coboundary, 35 p-cocycle, 35 r-fibre, 554

C-algebra, 196 C -algebra defined by a continuous field of C-algebras, 279 C-algebra, elementary, 206, 280 C-algebra of a groupoid, 229 C-algebra of a locally compact group, 210 C-bimodule, 512 C"-dynamic system, 287 C-module, 485 G-bundle, 633 G-module, 40 G-superfunction, 405 G-supermanifold, 406 G-supermanifold, standard, 406 G#°°-superfunctions, 404 G°°-superfunctions, 404 G°°-tangent bundle, 414 G°°-vector bundle, 413 /f°°-superfunctions, 404 K-cycle, 504 -f^-theory group of an algebra, 510 Loo-algebra, 462 O -algebra, 236 Op* -algebra, 236 .ft-supermanifold, 407 fl°°-superfunction, 407 •R°°-supermanifold, 407 /C-algebra, 19 ©-bundle, 558 ©-module bundle, 558 ©-module bundle, trivial, 558

Atiyah class, 84 Darboux theorem for a symplectic structure, 93 Abelian connection, 454 absorbent set, 570 action-angle coordinates, 129 action-angle coordinates, time-dependent, 148 adjoint action of a Hopf algebra, 551 adjoint differential operator, 653 adjoint representation, 236 adjoint representation of a Lie algebra, 600 adjoint representation of a Lie group, 599

admissible graphs, 468 admissible Hamiltonian, 159 admissible vector, 215 affine bundle, 591 affine map, 250 683

684

Geometric and Algebraic Topological Methods in Quantum Mechanics

affine scheme, 71 affine variety, irreducible, 85 algebra, 17 algebra, N-graded, 56 algebra, N-graded, commutative, 56 algebra, Z2-graded, 347 algebra, Z2-graded, commutative, 348 algebra, absolutely rigid, 438 algebra over an operad, 480 algebra, rigid, 436 algebra, unital, 17 algebraic Poincare lemma, 617 algebraic scheme, 73 algebroid, 471 almost complex connection, 176 almost complex manifold, 175 almost complex structure, 174 almost complex structure, integrable, 175 almost Hermitian manifold, 179 almost hypercomplex structure, 187 almost Kahler manifold, 181 almost symplectic form, 91 almost symplectic manifold, 91 anchor, 562 angle polarization, 309 annihilation operator, 244 annihilator of a distribution, 597 anti-isomorphic Hilbert spaces, 201 antighost, 392 antighost number, 392 antiholomorphic cotangent bundle, 177 antiholomorphic differential, 178 antiholomorphic exterior form, 178 antiholomorphic exterior form on a Hilbert manifold, 267 antiholomorphic function, 178 antiholomorphic function on a Hilbert space, 265 antiholomorphic space, 173 antiholomorphic tangent bundle, 176 antiholomorphic tangent vector, 176 antiholomorphic vector field, 178 antilinear map, 200

approximate identity, 197 associated atlas, 636 Atiyah-Singer index theorem, 657 atlas of constant local trivializations, 606 atlases equivalent, 587 automorphism of a fibre bundle, 587 automorphism, vertical, 588 Banach algebra, 196 Banach category, 509 Banach manifold, 259 Banach vector bundle, 278 base, 586 base of a quantum principal bundle, 537 basic forms, 594 basis, adapted to a complex structure, 172 basis for a graded manifold, 360 basis for a module, 20 basis for a pre-Hilbert space, 198 Batchelor's theorem, 359 Berezin transform, 277 Berezin's quantization special, 275 Bergman kernel function, 276 Bergman space, 276 Berry connection, 292 Betti number, 656 bi-algebra, 547 bi-Hamiltonian PIS, 132 bibundle, 567 bibundle, invertible, 568 bibundle, principal, 567 bibundle, symplectic, 192 bibundles, isomorphic, 568 bicommutant, 203 bidual, 570 bimodule, 19 bimodule, commutative, 19 bimodule, covariant, 531 bimodule, graded, 348 bimodule, invertible, 566 bimodule, symplectic, regular, 192 biregular morphism, 88 Bismut connection, 180

Index Bochner's theorem, 584 Bochner's theorem for a nuclear space, 585 body manifold, 410 body map, 350 body map of a DeWitt supermanifold, 410 body of a graded manifold, 359 body of a ringed space, 70 Bott's periodicity theorem, 511 boundary, 31 boundary operator, 31 bounded morphism, 571 bounded subset, 570 bra-ket notation, 217 BRST algebra, 396 BRST charge, 396 BRST complex, 397 BRST differential, 396 bundle atlas, 587 bundle coordinates, 587 bundle morphism, 587 bundle of subgroups, 226 bundle universal, 634 Calabi-Yau manifold, 187 canonical complex, 108 canonical coordinates for a Poisson structure, 100 canonical coordinates for a symplectic structure, 94 canonical lift of a vector field, 593 canonical Liouville form, 93 canonical presheaf, 60 canonical quantization, 271 canonical symplectic form, 93 canonical tangent-valued form, 596 canonical vector field for a symplectic structure, 92 carrier space, 204 Cartan equations, 620 Cartan equations in mechanics, 140 Cartan pair, 495 categories, equivalent, 544 categories, isomorphic, 544 category, 541

685 category Vect(S), 260 category, abstract, 542 category, additive, 509 category, dual, 542 category, monoidal, 475 category, monoidal, strict, 476 category, monoidal, symmetric, 476 category of algebras over operads, 480 category of categories, 541 category of free modules, 510 category of modules, 543 category of morphisms, 543 category of projective modules, 510 category of sets, 542 category of topological spaces, 543 category, pseudo-Abelian, 510 category, sceletal, 477 category, small, 541 category Alg, 566 category Alg(P,3Jt), 480 category Ar, 543 category Bnh, 260 category Cat, 541 category cat, 542 category ChC, 477 category C, 514 category Ens, 542 category GrMod, 477 category Horn, 544 category KK, 521 category LG, 508 category LG, 568 category LPoisson, 507 category Mod, 477 category N , 477 category N», 477 category Poisson, 193 category SMon, 478 category Top, 543 category Top,, 477 Cauchy-Riemann conditions, 175 center of a module, 52 center of an algebra, 484 chain, 31 chain complex, 31 chain complex, acyclic, 31

686

Geometric and Algebraic Topological Methods in Quantum Mechanics

chain complex, standard, 39 chain homotopy, 33 chain splitting, 34 character, 272 character of a cycle, 516 character of a group, 216 characteristic class, 643 characteristic distribution, 100 characteristic distribution of a presymplectic form, 96 characteristic foliation of a Poisson manifold, 101 characteristic foliation of a presymplectic form, 96 characteristic form, 642 characteristic function of a set, 575 characteristic ideal of an affine variety, 85 characteristic vector bundle, 359 Chern character, 645 Chern class, 643 Chern class of a manifold, 647 Chern form, 643 Chern form, total, 643 Chevalley-Eilenberg coboundary operator, 54 Chevalley-Eilenberg cohomology, 54 Chevalley-Eilenberg cohomology with coefficients in a module, 54 Chevalley-Eilenberg cohomology with coefficients in the trivial representation, 55 Chevalley-Eilenberg complex, 54 Chevalley-Eilenberg differential calculus, 58 Chevalley-Eilenberg differential calculus, minimal, 58 Christoffel symbols, 609 CIS, time-dependent, 146 classical symmetry, 624 classification theorem, 634 classifying space, 634 closure, 235 closure of a representation, 236 co-Leibniz rule, 461 co-operation, 547

coaction, 552 coadjoint representation, 601 coalgebra, 546 coalgebra, graded, 461 coboundary operator, 35 cochain, 35 cochain complex, 35 cochain homotopy, 37 cochain morphism, 32, 37 cocycle condition, 64, 587 cocycles, equivalent, 634 coderivation, 461 codifferential, 461, 654 codistribution, 597 coefficient of a representation, 210 coframe, 591 coherent state, 220 coherent state, covariant, 222 coherent states, generalized, 223 cohomology group, 35 cohomology of a group, 41 cohomology of a Lie algebra, 54 cohomology with coefficients in a presheaf, 64 coisotropic, 96 coisotropic ideal, 108 coisotropic imbedding, 97 coisotropic submanifold of a Poisson manifold, 101 colax monoidal functor, 479 commutant, 203 comodule algebra, 553 comodule over a Hopf algebra, 552 comodule, trivial, 552 compact matrix pseudogroup, 528 compactified tangent bundle, 657 compatibility condition, 563 complete set of Hamiltonian forms, 631 completely integrable system, 125 complex, 35 complex, acyclic, 36 complex algebraic variety, 88 complex canonical coordinates, 342 complex conjugate morphism, 173 complex cotangent bundle, 177

Index

complex, dual, 36 complex, elliptic, 654 complex, functorial, 51 complex Hamiltonian vector field, 269 complex harmonic form, 184 complex manifold, 175 complex structure, 172 complex structure on a manifold, 175 complex tangent bundle, 176 complex tangent vector, 176 complex vector field on a Hilbert manifold, 266 complexification, 173 composable pair, 553 composite connection, 610 composite fibre bundle, 589 composition product, 53 comultiplication, 546 cone of a diagram inductive, 545 cone of a diagram projective, 545 connection, 604 connection, Bott's, 607 connection bundle, 640 connection, complete, 138 connection, curvature-free, 606 connection, flat, 606 connection, left, 499 connection, non-commutative, universal, 499 connection on a Banach manifold, 263 connection on a bundle of C-algebras, 282 connection on a graded commutative ring, 358 connection on a graded module, 358 connection on a Hilbert bundle, 283 connection on a Hilbert manifold, 269 connection on a Lie algebroid, 459 connection on a manifold, 608 connection on a module, 29, 30 connection on a ring, 30 connection on a sheaf, 83 connection, right, 499 connection, vertical, 155 Connes' differential calculus, 505 constant sheaf, 59

687

constrained Hamilton equations, 631 constrained Hamiltonian system, 157 constraint algorithm, 118 constraint, quantum, 332 constraint space, 157 constraint space of a constrained Hamiltonian system final, 159 constraint space, primary, 157 constraints, 159 constraints, complete, 159 constraints, finitely reducible, 395 constraints, irreducible, 393 constraints, primary, 158 constraints, reducible, 393 constraints, secondary, 159 constraints, tertiary, 159 contact form, 603, 615 contact symmetry, 623 continuous action of a group, 579 contraction, 594 contraction of a measure and a function, 581 contraction of functions, 581 contraction of measures, 581 contravariant connection, 301, 317 contravariant derivative, 317 convolution algebra, 227 convolution product, 550 coordinate ring of a variety, 86 correspondence principle, 275 correspondence principle, weak, 275 cotangent bundle, 591 cotangent bundle, antiholomorphic, 267 cotangent bundle, complex, 266 cotangent bundle, holomorphic, 267 cotangent bundle of a Banach manifold, 261 counit, 546 countably Hilbert space, 201 covariance form, 585 covariant curvature tensor, 95 covariant derivative, 605 covariant differential, 605 covariant differential of a section, 605 covariant differential on a module, 29

688

Geometric and Algebraic Topological Methods in Quantum Mechanics

covariant differential, vertical, 611 covariant symbol of an operator, 277 cover of an algebra, 536 cover of an algebra, complete, 537 covering completion of an algebra, 537 creation operator, 244 cup-product, 52, 57 cup-product of cyclic cocycles, 518 cup-product of the Chevalley-Eilenberg cohomology, 55 curvature, 606 curvature leafwise form, 299 curvature of a connection on a module, 30 curvature of a connection on sheaves, 83 curvature of a graded connection, 364 curvature of a leafwise connection, 299 curvature of a superconnection, 415 curvature of an NQ-superconnection, 429 cycle, 31 cycle of dimension n, 516 cycle over an algebra, 516 cycle, vanishing, 517 cyclic cohomology, 515 cyclic representation, 205 cyclic vector, 205 cyclic vector, strongly, 237 cylinder set measure, 583 decalage isomorphism, 462 Darboux coordinates for a Poisson structure, 100 Darboux coordinates for a symplectic structure, 94 De Donder form, 620 de Rham cohomology, 57 de Rham cohomology, leafwise, 111 de Rham cohomology of a manifold, 77 de Rham cohomology of a ring, 58 de Rham complex, 56

de Rham complex, leafwise, 111 de Rham complex of a ring, 58 de Rham complex of exterior forms, 77 de Rham complex of sheaves, 77 de Rham complex, tangential, 111 de Rham theorem, 77 de Rham theorem, abstract, 67 deformation, 436 deformation, associative, 436 deformation, continuous, 445 deformation, differential, 445 deformation, formal, 434 deformation, null, 435 deformation of an algebra, 434 deformation quantization, 444 deformation, relative, 436 deformation, tangential, 450 deformation, trivial, 435 deformations, equivalent, 435 degenerate operator, 204 Deligne class, 447 Deligne's conjecture, 475 dequantization, 272 derivation approximately inner, 252 derivation, graded, 361 derivation, inner, 25 derivation module, 25 derivation of a non-commutative algebra, 25 derivation, outer, 52 derivation, symmetric, 251 derivation, well-behaved, 252 derivation with values in a module, 25 derivative on a Banach space, 259 DeWitt supermanifold, 409 DeWitt topology, 409 dGLA, 460 DGLA, formal, 461 diagrams of graphs, 545 differentiable function between Banach spaces, 258 differentiable function on a Hilbert space, 265 differential bigraded algebra, 356 differential calculus, 56

Index

differential calculus, minimal, 57 differential calculus, universal, 490 differential equation, 613 differential graded algebra, 56 differential graded Lie algebra, 460 differential horizontal, 616 differential ideal, 597 differential on a Banach space, 258 differential operator, 613 differential operator, covariant, 564 differential operator, elliptic, 652 differential operator in non-commutative geometry, 492, 496 differential operator on a module, 24 differential operator on an affine variety, 89 differential, total, 616 differential, vertical, 616 dimension of a variety, 86 Dirac constrained system, 158 Dirac measure on a Hausdorff space, 582 Dirac state, 332 Dirac's condition, 271 Dirac's condition, converse, 273 direct image of a sheaf, 72 direct limit, 21 direct product of object of a category, 545 direct product of Poisson structures, 99 direct sequence, 21 direct sum connection, 608 direct sum of chain complexes, 32 direct sum of complexes, 36 direct sum of Kasparov bimodules, 519 direct sum of modules, 19 direct sum of object of a category, 546 direct system of modules, 21 directed set, 21 distribution, 597 distribution, involutive, 597 division algebra, 17 Dixmier-Douady class, 282

689

Dolbeault cohomology, 184 Dolbeault complex, 183 domain, 587 domain of an element of a groupoid, 554 domain of an operator, 235 dual connection, 607 dual Hopf algebra, 551 dual Lie algebra, 601 dual morphism, 572 dual pair, 190, 570 dual vector bundle, 589 Dubois-Violette connection, 501 Dubois-Violette connection, dual, 502 Dubois-Violette connection, linear, 502 Dubois-Violette connection, real, 502 dynamic equation, 138 dynamic equation, autonomous, 115 dynamical algebra, 124 Ehresmann connection, 327 endomorphism of a Hilbert module, 485 endomorphism of a Hilbert module, adjoint, 485 endomorphism of a Hilbert module, compact, 486 enveloping C* -algebra, 211 equivalence Hilbert bimodule, 513 equivalence relation of units of a groupoid, 554 equivalent fibre bundles, 588 equivalent measures, 576 equivalent representations of involutive algebras, 205 essential ideal, 254 Euler characteristic, 78, 656 Euler characteristic form, 656 Euler-Lagrange equations, 619 Euler-Lagrange operator, 619, 622 Euler-Lagrange-Cartan operator, 619 evaluation morphism, 404 even element, 347 even morphism, 349 evolution equation, 117

690

Geometric and Algebraic Topological Methods in Quantum Mechanics

evolution equation of time-dependent mechanics, 144 evolution operator, 289 exact sequence of modules, 20 exact sequence of vector bundles, 590 exact sequence, short, 21 extended real line, 575 extension factor, 43 extension of a group, 42 extension of an operator, 235 exterior (p,q)-form, 177 exterior algebra, 23 exterior algebra, bigraded, 349 exterior bundle, 590 exterior differential, 593 exterior differential, contravariant, 109 exterior form, 593 exterior form, complex, 177 exterior form on a Banach manifold, 262 exterior form on a Hilbert manifold, 267 exterior product of vector bundles, 590 factor algebra, 18 factor bundle, 590 factor chain complex, 32 factor complex, 36 factor module, 20 factor sheaf, 62 Fell's topology, 225 fibration, 586 fibre, 587 fibre bundle, 586 fibre bundle of C"-algebras, 279 fibre bundle, topological, 588 fibre bundle, topological, locally trivial, 588 fibre bundle with a structure group, 633 fibre bundles, associated, 634 fibred coordinates, 586 fibred manifold, 586 fibred manifold, Banach, 260

fibred product, 589 field, 17 field, algebraically closed, 85 field of C-algebras, 279 field of quotients, 18 final constraint space of a Dirac constrained system, 161 finite diagram, 545 finite dual, 423 finite exactness, 376 first integral of motion, 117 first variational formula, 618, 625 first-class constraints, 160 flow, 593 Fock measure, 246 Fock representation, 246 FODC, 530 FODC, bicovariant, 531 FODC, left-covariant, 531 FODC, right-covariant, 531 foliation, 598 foliation, simple, 599 form, dominated, 208 formal differential operator, 447 formal dimension, 215 formality theorem, 467 Fourier transform of a quasi-measure, 584 Frolicher-Nijenhuis bracket, 597 Frechet axiom, 206 Frechet ring, 76 fraction field, 18 frame, coherent, 223 frame, coherent, tight, 223 Fredholm operator, 653 Fubini-Studi metric, 271 function, defined almost everywhere, 576 function, defined locally almost everywhere, 578 function field, 86 functor, 543 functor, contravariant, 543 functor, covariant, 543 functorial isomorphism, 543 functorial morphism, 543

Index functors, adjoint, 544 fundamental form of a Hermitian metric, 180 fundamental form of a Hermitian metric on a Hilbert manifold, 268 Garding space, 217 gauge field, non-commutative, 507 gauge fields, 118 gauge transformation, 635 Gaussian quasi-measure, 585 Gaussian quasi-measure, canonical, 585 Gel'fand topology, 76 Gelfand map, 272 general linear graded group, 351 general linear supergroup, 419 generalized connection, 283 generalized Hamiltonian system, 161 generating function, 241 generator of a representation, 600 geometric space, 70 Gerstenhaber algebra, 53 Gerstenhaber bracket, 53 ghost, 395 ghost number, 396 gluing of algebras, 536 GNS representation, 207 graded cocommutative coalgebra, 461 graded commutative /C-ring, 349 graded commutative ring, Banach, 349

graded connection, 363 graded connection on a graded bundle, 426 graded de Rham cohomology, 366 graded derivation, 353 graded derivation module, 353 graded differential calculus, 356 graded differential operator, 353 graded envelope, 350 graded exterior algebra, 462 graded exterior differential, 365 graded exterior form, 365 graded exterior product, 349 graded function, 359

691

graded Hilbert bimodule, 519 graded Hilbert module, 518 graded Jacobi identity, 386 graded jet module, 354 graded Lie algebra, 460 graded Lie bracket, 460 graded manifold, 358 graded manifold, simple, 359 graded module, 348 graded module, dual, 349 graded module, free, 348 graded Poisson bracket, 385, 386 graded Poisson bracket, projectable, 386 graded principal bundle, 425 graded principal connection, 425 graded submanifold, 425 graded symmetric algebra, 462 graded symplectic form, 385 graded symplectic form, projectable, 386 graded vector field, 361 graded vector space, 348 grading automorphism, 347 graph, finite, 545 graph of a category, 545 graph, oriented, 545 graph, small, 545 graph-topology, 236 Grassmann algebra, 350 Grothendieck group, 648 Grothendieck's topology, 76 group algebra, 210 group bundle, 555, 635 group bundle, topological, 556 group function Hopf algebra, 549 group Hopf algebra, 548 group ring, 40 group-like elements, 424 groupoid, 553 groupoid action, 555 groupoid, differentiable, 561 groupoid, discrete, 226 groupoid, frame, 556 groupoid, globally trivial, 560 groupoid, locally trivial, 560

692

Geometric and Algebraic Topological Methods in Quantum Mechanics

groupoid, principal, 554 groupoid, symplectic, 190 groupoid, tangent, 561 groupoid, topological, 559 groupoid, transitive, 554 groupoid, trivial, 555 groupoids, similar, 557 Haar measure, 580 Haar measure, left, 580 Haar measure, right, 580 Haar system, left, 225 Haar system, right, 226 half-form, 303 half-form, leafwise, 305 Hamilton equations, constrained, 165 Hamilton equations, covariant, 628 Hamilton operator, 628 Hamilton vector field, 143 Hamilton-De Donder equations, 620 Hamiltonian, 115 Hamiltonian action, 119 Hamiltonian connection, 627 Hamiltonian form, 143, 627 Hamiltonian form, associated to a Lagrangian, 629 Hamiltonian form, associated to a Lagrangian in mechanics, 150 Hamiltonian form, associated weakly to a Lagrangian, 629 Hamiltonian form, associated weakly to a Lagrangian in mechanics, 150 Hamiltonian form, constrained, 165, 631 Hamiltonian form, graded, 389 Hamiltonian graded vector field, 385 Hamiltonian manifold, 120 Hamiltonian map in mechanics, 149 Hamiltonian of time-dependent mechanics, 143 Hamiltonian system, gauge-invariant, 119 Hamiltonian system, Poisson, 115 Hamiltonian system, presymplectic, 117 Hamiltonian system, reduced, 119

Hamiltonian system, symplectic, 116 Hamiltonian system, time-dependent, 144 Hamiltonian vector field for a Poisson structure, 100 Hamiltonian vector field for a symplectic structure, 92 harmonic form, 655 harmonic section, 655 Harrison cohomology, 441 Harrison complex, 441 Heisenberg CCR algebra, 239 Heisenberg equation, 287 Hermitian connection, 180 Hermitian element, 195 Hermitian form, 198 Hermitian form, non-degenerate, 198 Hermitian form on a module, 484 Hermitian form on a module, invertible, 485 Hermitian form on a module, positive, 485 Hermitian form, positive, 198 Hermitian manifold, 179 Hermitian manifold, infinite-dimensional, 268 Hermitian metric, 179, 653 Hermitian metric on a Hilbert manifold, 268 Herz-Reinhart Lie algebra, 456 Hilbert bimodule, 512 Hilbert bundle, 280 Hilbert dimension, 199 Hilbert manifold, 266 Hilbert module, 280, 485 Hilbert space, 199 Hilbert space, dual, 200 Hilbert space, real, 199 Hilbert space, separable, 199 Hilbert sum, 200 Hilbert sum of representations, 205 Hilbert-Schmidt norm, 204 Hilbert-Schmidt operator, 204 Hirzebruch polynomial, 659 Hirzebruch signature theorem, 659 HKT-connection, 188

693

Index HKT-manifold, 188 Hochschild chain complex, 50 Hochschild cohomology, 51 Hochschild cohomology, continuous, 445 Hochschild cohomology, differential, 445 Hochschild cohomology, relative, 439 Hochschild complex, 50 Hochschild complex, relative, 439 Hochschild homology, 50 Hodge inner product, 654 Hodge number, 184 Hodge star operator, 654 Hodge theorem, 655 holomorphic cotangent bundle, 177 holomorphic differential, 178 holomorphic exterior form, 178 holomorphic exterior form on a Hilbert manifold, 267 holomorphic function, 178 holomorphic function on a Hilbert space, 265 holomorphic manifold, 175 holomorphic space, 173 holomorphic tangent bundle, 176 holomorphic tangent vector, 176 holomorphic transition functions, 175 holomorphic vector field, 178 holonomic coordinates, 591 holonomic frame, 591 holonomic section, 603 holonomic transformation, 591 holonomy operator, 326 homogeneous element, 348 homogeneously non-degenerate graded two-form, 385 homology, canonical, 108 homology group, 31 homology, Koszul-Brylinski-Poisson, 108 homomorphism of groupoids, 556 homomorphism of groupoids, similar, 557 homotopic chain complexes, 33 homotopic chain morphisms, 33

homotopic cochain morphisms, 37 homotopic complexes, 37 homotopy algebra over an operad, 481 homotopy between Kasparov bimodules, 520 homotopy between Kasparov bimodules, operatorial, 520 homotopy equivalence of chain complexes, 33 homotopy equivalence of cochain complexes, 37 homotopy Lie algebra, 463 homotopy operator, 31, 36 homset, 541 Hopf algebra, 547 Hopf algebra, classical, 548 Hopf algebra, cocommutative, 547 Hopf algebra, dually paired, 550 Hopf algebra, involutive, 548 Hopf-Galois condition, 539 horizontal distribution, 604 horizontal foliation, 606 horizontal forms, 594 horizontal lift of a vector field, 604 horizontal projection, 616 horizontal vector field, 138 HRL-algebra, 456 HRL-ringed space, 459 hyper-Hermitian connection, 188 hyper-Hermitian manifold, 188 hyper-Kahler connection, 188 hyper-Kahler manifold, 188 hypercomplex structure, 187 ideal, 17 ideal, maximal, 18 ideal of nilpotents, 350 ideal, prime, 18 ideal, proper, 18 ideal, self-adjoint, 196 image of a measure, 583 image of a quasi-measure, 584 imbedded submanifold, 586 imbedding, 586 immersion, 586

694

Geometric and Algebraic Topological Methods in Quantum Mechanics

index of a differential operator, topological, 657 index of a manifold, 658 index of an elliptic complex, 655 index of an elliptic operator, 653 inductive limit, 23 inductive limit of a diagram, 546 inductive limit topology, 238, 572 infinitesimal generator, 593 infinitesimal of a deformation, 437 initial data coordinates, 144 input algebra, 229 instantwise quantization, 286 integrable function, 576 integrable function, locally, 578 integrable function of degree p, 576, 577 integrable function of infinite degree, 577 integrable section, 603, 613 integral, 573 integral manifold, 598 integral manifold, maximal, 598 integral of motion, 140 integral scheme, 71 integral section of a connection, 605 integral, upper, 575 integral, upper, essential, 578 integration of a Poisson manifold, 191 interior product, 569, 594 interior product, graded, 356 interior product, left, 595 interior product, right, 596 intersection form, 658 intersection product of KET-groups, 521 intersection product of Kasparov bimodules, 521 invariant of Poincare-Cartan, 143 invariant vector field, 534 inverse image of a sheaf, 72 inverse limit, 23 inverse mapping theorem, 259 inverse sequence, 23 inverse to a functor, 544 involution, 195

involutive algebra, 195 involutive algebra, normed, 196 irreducible component of an affine variety, 85 isomorphism of fibre bundles, 587 isotropic submanifold, 96 isotropy group, 554 Jacobi field, 156 Jacobi identity, 97 Jacobson topology, 206 jet manifold, 602 jet manifold, higher order, 611 jet manifold, repeated, 612 jet module, 26 jet, odd, 368 jet of sections, 602 jet prolongation, 612 jet prolongation of a vector field, 603 jet prolongation of morphisms, 603 jet prolongation of sections, 603 jet sheaf, 82 jet space, infinite order, 614 joint homomorphism, 35 Jordan algebra, 196 Jordan morphism, 250 juxtaposition rule, 57 Kahler form, 181 Kahler form on a Hilbert manifold, 268 Kahler manifold, infinite-dimensional, 268 Kahler metric, 180 Kahler metric on a Hilbert manifold, 268 Kasparov bimodule, 519 Kasparov bimodule, degenerate, 519 Kasparov bimodule, induced, 519 Kasparov cycle, 519 kernel of a morphism of vector bundles, 590 KK-group, 520 Kostant-Souriau-Kirillov theorem, 602 Koszul complex, 44

695

Index Koszul complex of a module, 45 Koszul-Tate complex, 395 Koszul-Tate differential, 395 KT-connection, 180 Lagrange equations, 139 Lagrangian, 618, 622 Lagrangian, almost regular, 621 Lagrangian constraint space, 621 Lagrangian constraint space in mechanics, 149 Lagrangian, graded, 372 Lagrangian, hyperregular, 621 Lagrangian, regular, 621 Lagrangian, semiregular, 630 Lagrangian submanifold, 96 Lagrangian submanifold of a Poisson manifold, 101 Lagrangian, variationally trivial, 622 Laplace-Beltrami operator, 655 Laplacian, 654, 655 Laplacian complex, 183 Laurent series, 444 leading term of a linear differential operator, 651 leafwise connection, 297 leafwise differential calculus, 111 leafwise exterior differential, 111 leafwise form, 111 leafwise reduction, 106 left-invariant form, 600 left-invariant form, canonical, 600 left-invariant vector field, 599 left-invariant vector field on a Lie groupoid, 564 left-regular representation, 214 Legendre bundle, 621 Legendre bundle, homogeneous, 620 Legendre map, 621 Legendre map in mechanics, 149 Legendre morphism, 620 Leibniz rule, 25, 29, 30, 97, 358 Leibniz rule, graded, 56 Leibniz rule, non-commutative, 25 Levi-Civita connection, 609 Levi-Civita tensor, 654

LP-algebra, 238 Lichnerowicz-Poisson (LP) cohomology, 109 Lichnerowicz-Poisson complex, 109 Lie algebra bundle, 639 Lie algebra, left, 599 Lie algebra, right, 599 Lie algebroid, 561 Lie algebroid, left, 565 Lie algebroid of covariant differential operators, 564 Lie algebroid, regular, 562 Lie algebroid, symplectic, 457 Lie algebroid, transitive, 562 Lie algebroid, transitive, totally, 562 Lie algebroid, trivial, 562 Lie coalgebra, 601 Lie deformation, 442 Lie derivative, 594 Lie derivative of a bigraded algebra, 357 Lie groupoid, 561 Lie superalgebra, 352, 419 Lie superalgebra of a graded Lie group, 424 Lie supergroup, 417 Lie-Poisson structure, 601 limit, 545 linear bundle morphism, 590 linear connection, 607 local basis for an ideal, 158 local ring, 18 local ring of a subvariety, 87 local-ringed space, 70 locally finite covering, 65 locally free module, 73 Mackey space, 571 Magnus series, 290 matrix geometry, 488 Maurer-Cartan equation, 600 Maurer-Cartan form on a Hopf algebra, 533 measurable map, 582 measurable set, 582 measure, 573

696

Geometric and Algebraic Topological Methods in Quantum Mechanics

measure, absolute value, 574 measure, bounded, 574 measure, conjugate, 573 measure, Dirac, 574 measure, induced, 582 measure, invariant, 226, 579 measure, inverse, 580 measure, left-invariant, 580 measure on a Hausdorff space, 582 measure, outer, 575 measure, positive, 573 measure, quasi-invariant, 226, 579 measure, real, 573 measure, relative invariant, 579 measure, right-invariant, 580 measure, translationally quasi-invariant, 242 measure with a basis and a density, 578 measures, contractible, 581 metalinear bundle, 303 metalinear group, 304 metric connection on a Hilbert manifold, 270 Minkowski inequality, 576 modular function, 580 module, 18 module, dual, 20 module, finitely generated, 20 module, free, 20 module, involutive, 484 module of finite rank, 20 module over a Hopf algebra, 551 module over a Lie algebra, 54 module, projective, 20 moment of a bibundle, 567 momentum mapping, 120 momentum mapping, equivariant, 121 momentum phase space, 626 momentum phase space, homogeneous, 141 momentum phase space of mechanics, 141 monoid in a category, 476 monoidal category with equivalence, 480

monoidal functor, 478 Morita equivalence of star-products, 449 Morita equivalent C-algebras, 513 Morita equivalent groupoids, 568 Morita equivalent Poisson manifolds, 193 Morita equivalent rings, 565 morphism, allowable, 51, 440 morphism of a category, 541 morphism of presheaves, 61 morphism of ringed spaces, 72 morphism of schemes, 72 morphism of schemes locally of finite type, 73 morphism of schemes of finite type, 73 morphism of sheaves, 61 morphisms isomorphic, 543 Moyal product, 451 multiplicative subset, 18 multiplier algebra, 254 multiplier of measure, 579 multiplier phase, 253 multipliers, equivalent, 254 multisymplectic Liouville form, 620 multivector field, 595 natural bundle, 591 natural isomorphism, 543 natural transformation, 543 Nijenhuis torsion, 176 Noether current, 140, 146 Noetherian ring, 71 non-commutative vector field, 495 non-degenerate graded two-form, 385 non-degenerate two-form, 91 norm, 569 normal bundle to a foliation, 598 normal element, 195 normal point of a variety, 88 normalizer, 107 normed operator topology, 203 NQ-superbundle, 427 NQ-superconnection, 428 nuclear algebra, 237

697

Index nuclear map, 202 nuclear operator, 204 nuclear space, 201 nuclear Weyl CCR group, 239 null set, 575 null set, locally, 578 Obata connection, 188 object, initial, 542 object of a category, 541 object, terminal, 542 objects, isomorphic, 542 odd element, 347 odd morphism, 349 operad, 472 operad i T ( P ) , 473 operad cohomology, 473 operad, differential graded, 473 operad, little discs, 474 operad, non-symmetric, 479 operad of chains, 474 operad, symmetric, 479 operad, topological, 473 operad AIgG, 479 operad Associ, 473 operad Chains(P), 474 operad Cd, 474 operad Mon, 479 operad Obj, 479 operad Sem, 479 operator, 235 operator, adjoint, maximal, 235 operator, bounded, 235 operator, closable, 235 operator, closed, 235 operator, compact, 203 operator, completely continuous, 203 operator, essentially self-adjoint, 235 operator, intertwining, 221 operator norm, 202 operator of a parallel displacement, 289 operator, positive, 204 operator, self-adjoint, 235 operator, symmetric, 235 operator, unbounded, 235

operator-valued integral, 211 operators adjoint, 235 orbit of units of a groupoid, 554 orthogonal relative to a symplectic form, 95 orthonormal family, 198 output algebra, 229 paracompact space, 65 parallel with respect to a connection, 605 parameter bundle, 151 partial multiplication, 553 partially integrable system, 124 partially integrable system on a symplectic manifold, 124 particle number operator, 244 partition of unity, 65 perfect space, 202 physical phase space, 119 PIS, 124 Poincare-Cartan form, 619 Poincare-Cartan form in mechanics, 139 point measure, 574 Poisson action, 122 Poisson algebra, 97 Poisson bivector field, 98 Poisson bracket, 97 Poisson codifferential, 108 Poisson connection, 102 Poisson manifold, 97 Poisson manifold, exact, 109 Poisson manifold, homogeneous, 109 Poisson manifold, integrable, 191 Poisson manifold, reduced, 104 Poisson morphism, 99 Poisson morphism, complete, 189 Poisson reduction, 104 Poisson structure, 97 Poisson structure, algebraic, 471 Poisson structure, coinduced, 99 Poisson structure, non-degenerate, 98 Poisson structure, regular, 98 polarization of a Poisson manifold, 302, 318

698

Geometric and Algebraic Topological Methods in Quantum Mechanics

polarization of a symplectic foliation, 302 polysymplectic form, 626 Pontryagin class, 646 Pontryagin class of a manifold, 647 positive element, 485 positive form, 206 positive-definite function, 584 positive-definite function on a group, 213 potential function, 181 power series algebra, 434 pre-Hilbert module, 485 pre-Hilbertian module, 328 premeasure, 581 premeasure, positive, 582 prequantization bundle, 296 prequantization condition, 297 prequantization formula, 297 presheaf, 59 presymplectic form, 96 presymplectic manifold, 96 primitive ideal, 205 principal bundle, 635 principal bundle, universal, 637 principal connection, 640 principal connection, canonical, 641 principal superbundle, 420 product G-supermanifold, 412 product connection, 605 product of G-supermanifolds, 411 product of measures, 583 projective Hilbert space, 270 projective representation, 254 projective unitary group, 253 proper cover, 63 pull-back connection, 605 pull-back fibre bundle, 588 pull-back form, 594 pure form, 208 quantization bundle, 303, 305 quantum algebra of a symplectic foliation, 302 quantum commutation relations, 529 quantum determinant, 527

quantum double, 524 quantum factorial, 525 quantum group, 525 quantum group compact, 528 quantum group general, 523 quantum group locally, compact, 528 quantum Jacobi identity, 529 quantum Lie algebra, 529 quantum Lie algebra of a Hopf algebra, 534 quantum principal bundle, algebraic, 539 quantum principal bundle, locally trivial, 537 quantum principal bundle, trivial, 537 quantum structure constants, 529 quantum tangent space, 533 quasi-compact space, 206 quasi-equivalent equations, 621 quasi-inverse to a functor, 544 quasi-isomorphic DGLA, 461 quasi-isomorphism, 461 quasi-measure, 583 qubit, 229 Radon-Nikodym derivative, 578 range of an element of a groupoid, 554 rank of a bivector field, 98 rank of a two-form, 91 rational function on an affine variety, 86 rational morphism of a variety, 88 real spectrum of a ring, 75 real tangent vector, 176 recursion operator, 133 reduction of a groupoid, 554 reduction of a Lie algebroid, 562 reduction of a Poisson algebra, 106 reduction of a Poisson map, 105 reduction of a structure group, 637 reductive structure, 104 reductive structure on a Poisson manifold, 104 refinement, 63 reflexive space, 570

Index regular function on a variety, 87 regular invariant manifold, 125 regular morphism of varieties, 88 regular point of a distribution, 125 regular tuple, 48 relative equivalence of deformations, 441 relatively compact set, 575 relativistic Hamiltonian, 336 relativistic quantum equation, 341 representation, G-covariant, 254 representation, closed, 236 representation, determined by a form, 208 representation, Hermitian, 236 representation of a groupoid, 558 representation of an involutive algebra, 204 representation of an involutive algebra, algebraically irreducible, 205 representation of an involutive algebra, non-degenerate, 204 representation of an involutive algebra, topologically irreducible, 205 representation, self-adjoint, 236 representation, square integrable, 215 representation, universal, 208 reproducing kernel, 218 reproducing system, 217 resolution, 36 resolution, chain, 32 resolution, fine, 69 resolution, flabby, 68 resolution of a sheaf, 67 resolution of the identity, 217 restriction morphism, 59 restriction of a fibre bundle, 589 restriction of a Kasparov bimodule, 519 Ricci form, 187 Ricci tensor, 609 Rieffel tensor product of Hilbert bimodules, 513 rigged Hilbert space, 202

699 right structure constants, 599 right-invariant form, 600 right-invariant form, canonical, 601 right-invariant vector field, 599 ring, 17 ringed space, 70 scalar product, 198 scalar product, Hermitian, 173 scheme, 71 scheme, irreducible, 71 scheme, locally Noetherian, 71 scheme, Noetherian, 71 scheme over, 72 scheme, reduced, 71 scheme, separated, 72 scheme, separated over, 72 Schouten-Nijenhuis bracket, 595 Schrodinger equation, 289 second adjoint representation, 236 second class constraints, 160 second-countable topological space, 75 section, 588 section, global, 588 section, local, 588 semidirect product of a groupoid and a group, 558 seminorm, 569 separable ring, 440 separable topological space, 75 separated morphism of schemes, 72 Serre-Swan theorem, 78 set of power series, 433 sheaf, 59 sheaf, acyclic, 67 sheaf, coherent, 74 sheaf, coherent of rings, 74 sheaf, cohomology, 64 sheaf, fine, 69 sheaf, flabby, 68 sheaf, Basque, 68 sheaf, locally free, 73 sheaf, locally free of constant rank, 73 sheaf, locally free of finite type, 73 sheaf of continuous functions, 59

700

Geometric and Algebraic Topological Methods in Quantum Mechanics

sheaf of derivations, 73 sheaf of finite presentation, 74 sheaf of jets, 82 sheaf of modules, 73 sheaf of smooth functions, 59 sheaf, quasi-coherent, 74 short exact sequence of vector bundles, 590 signature complex, 659 signature of a manifold, 658 simple point of a variety, 87 singular locus of a variety, 88 singular point of a variety, 87 skew-product of a groupoid and a group, 558 small diagram, 545 smooth manifold, 74 smooth supermanifold, 405 smooth vector, 216 soldering form, 597 solution of a Poisson Hamiltonian system, 115 soul map, 350 spectral triple, 504 spectral triple, even, 504 spectral triple, odd, 504 spectrum, 206 spectrum of a ring, 70 spectrum of a ring, maximal, 71 split (injection), 258 split (subspace), 258 splitting domain, 359 splitting domain for an NQ-superbundle, 428 splitting of an exact sequence, 21 splitting of an exact sequence of vector bundles, 591 splitting, principle, 644 square integrable function, 576 stably isomorphic vector bundles, 649 star adjoint representation, 447 star-product, 444 star-product, closed, 449 star-product, tangential, 450 state, 207 state condition, 332

Stiefel-Whitney class, 647 strength, 641 strong operator topology, 203 strong topology, 570 strongly continuous group, 250 strongly continuous unitary representation, 209 strongly homotopy algebra, 464 structural theorem, 648 structure group of a quantum principal bundle, 537 structure module of a sheaf, 59, 60 structure module of a simple graded manifold, 360 structure module of a vector bundle, 78 structure sheaf of a graded manifold, 359 structure sheaf of a ringed space, 70 structure sheaf of a supermanifold, 405 structure sheaf of a vector bundle, 77 structure sheaf of an affine variety, 87 subbundle, 588 subcategory, 542 subcategory, full, 542 submanifold, 586 submersion, 586 submersion, continuous, 588 subvariety, 85 superbracket, 352 superconnection, 415, 422 superconnection form, 422 superconnection, principal, 422 superdeterminant, 351 superform, 408 superfunction, 402 superfunctions, smooth, 404 supermatrix, 351 supermatrix, even, 351 supermatrix, odd, 351 supertangent bundle, 414 supertrace, 351 supertransposition, 351 supervector bundle, 413 supervector field, 408

Index supervector field, fundamental, 421 supervector field, invariant, 421 supervector field, left-invariant, 419 supervector space, 350 SUSY charge, 391 symbol bundle, 657 symbol of a differential operator, 652 symmetry current, 146, 622 symmetry current in mechanics, 140 symmetry group, 472 symplectic action, 119 symplectic action of a symplectic groupoid, 192 symplectic bimodule, 190 symplectic connection, 94 symplectic form, 92 symplectic Hodge operator, 108 symplectic leafwise form, 113 symplectic manifold, 92 symplectic manifold, Einstein, 95 symplectic manifold, homogeneous, 602 symplectic manifold, reduced, 103 symplectic manifold, simple, 95 symplectic module, 189 symplectic morphism, 92 symplectic realization of a Poisson structure, 102 symplectic realization of a presymplectic structure, 96 symplectic reduction, 103 symplectic submanifold, 96 symplectomorphism, 92 tangent tangent tangent tangent tangent 260 tangent tangent tangent tangent tangent

bundle, 591 bundle, antiholomorphic, 266 bundle, complexified , 266 bundle, holomorphic, 266 bundle of a Banach manifold, bundle to a foliation, 598 map, 591 space, antiholomorphic, 266 space, complex, 266 space, holomorphic, 266

701 tangent space to a Banach manifold, 260 tangent-valued form, 596 tensor algebra, 23 tensor algebra, reduced, 462 tensor bundle, 591 tensor product connection, 608 tensor product, infinite, 230 tensor product of C-algebras, 197 tensor product of algebras, 546 tensor product of bibundles, 568 tensor product of bimodules, 565 tensor product of chain complexes, 32 tensor product of complexes, 36 tensor product of differential graded algebras, 517 tensor product of graded modules, 349 tensor product of Hilbert spaces, 200 tensor product of modules, 19 tensor product of symplectic modules, 190 tensor product of vector bundles, 589 tensor product, topological, 76 time-ordered exponential, 289 Toeplitz operator, 277 topological dual, 569 topological invariant, 66 topology of compact convergence, 572 topology of uniform convergence, 572 topology of uniform convergence on sets, 570 torsion, 606 torsion of a non-commutative connection, 503 torsion of a world connection, 609 torsion of an almost complex structure, 176 total derivative, 603, 612 total family, 198 total mass of a measure, 575 total space of a quantum principal bundle, 537 trace, closed, 516 trace, graded, 516 trace norm, 204

702

Geometric and Algebraic Topological Methods in Quantum Mechanics

trace of a star-product, 449 trace on an algebra, 515 transition functions, 587 transition functions of a quantum principal bundle, 538 transitivity component, 554 trivial extension of a sheaf, 72 trivialization chart, 587 tubular neighborhood, 599 twisted Lie algebra, 459 typical fibre, 586 underlying G°°-supermanifold, 407 underlying Lie group, 418 uniformly continuous group, 250 uniformly continuous unitary representation, 209 unimodular group, 580 unit space of a groupoid, 554 unital extension, 19 unitary element, 196 unitary group, 210 universal R matrix, 548 universal form, 491 universal line bundle, 345 universal set, 542 upper integral, 575 vacuum state, 246 variation equation, 156 variational bicomplex, 617 variational bicomplex, graded, 370 variational complex, 617 variational operator, 616 variational symmetry, 622, 626 variety, 87 variety, affine, 85 variety, algebraic, 87 variety, non-singular, 87 variety, normal, 88 variety, prealgebraic, 87 variety, smooth, 87, 88 vector bundle, 589 vector field, 593 vector field, canonical for a Poisson structure, 100

vector field, complete, 593 vector field on a Banach manifold, 261 vector field, projectable, 593 vector field, subordinate, 597 vector field, vertical, 593 vector form, 207 vector space, 19 velocity phase space, 618 velocity phase space of mechanics, 139 vertex group, 554 vertical cotangent bundle, 592 vertical extension of a Hamiltonian form, 157 vertical extension of a Lagrangian, 155 vertical extension of an exterior form, 155 vertical polarization, 307 vertical splitting, 592 vertical tangent bundle, 592 vertical tangent map, 592 von Neumann algebra, 203 weak conservation law, 622 weak operator topology, 203 weak topology, 570 weak topology a(V,W), 570 weak* topology, 570 weakly continuous morphism, 571 weight coefficients, 468 weighted Hopf C*-algebra, 529 Weil homomorphism, 642 Weyl algebra, 451 Weyl algebra bundle, 451 Weyl product, 451 Whitney sum, 589 Witt group, 512 world connection, 608 Yang-Baxter equation, 526 Yang-Baxter equation, quantum, 548 Zariski topology on a spectrum, 71

Index

Zariski topology on an affine variety, 86 zero Poisson structure, 98 zero section, 589

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