Geometric and Topological Methods for Quantum Field Theory
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Proceedings of the Summer School
Geometric and Topological Methods for Quantum Field Theory Villa de Leyva, Colombia
9-27 July 2001
Editors
Alexander Cardona Universite Blaise Pascal, Clermont-Ferrand, France Universidad de Los Andes, Bogota, Colombia
Sylvie Paycha Universite Blaise Pascal, Clermont-Ferrand, France
Hernan Ocampo Universidad de Valle, Cali, Colombia
b
World Scientific New Jersey London Singapore Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd.
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British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.
GEOMETRIC AND TOPOLOGICAL METHODS FOR QUANTUM FIELD THEORY Proceedings of the Summer School Copyright 0 2003 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereoj 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 981-238-131-7
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V
INTRODUCTION This volume offers an introduction t o recent developments in several active topics of research at the interface between geometry, topology and quantum field theory. These include Hopf algebras underlying renormalization schemes in quantum field theory, noncommutative geometry with applications t o index theory on one hand and the study of aperiodic solids on the other hand, geometry and topology of low dimensional manifolds with applications t o topological field theory, Chern-Simons supergravity and the anti de Sitter/ conformal field theory correspondence. The volume is based on lectures delivered during a Summer School “Geometric and Topological Methods for Quantum Field Theory” held at Villa de Leyva, Colombia, in July 2001, complemented by some short communications by participants of the school. The lecture notes, which are aimed at students and researchers in mathematics and physics, are organized in seven contributions around three main topics, Noncommutative Geometry which comprises the first three, Topological Field Theory which includes two lectures, and Supergravity and String Theory with the last two contributions. Each lecture is self-contained and can be read independently. The volume starts with an introductory course by Joseph VBrilly on Hopf algebras in noncommutative geometry, in which, among other examples, the reader can find a description of the Hopf algebra underlying renormalization schemes in quantum field theory. It is followed by a lecture by Moulay-Tahar Benameur who, after recalling classical results in index theory, gives an introduction t o type I1 index theory in noncommutative geometry. The third lecture by Jean Bellissard implements methods of noncommutative geometry including index theorems over von Neumann algebras, to investigate the geometry of aperiodic solids. In the fourth lecture, Christian Blanchet shows how, from a modular category, one can derive a topological field theory, and in particular, invariants of links and 3-manifolds. It is followed by a survey by Marcos Mariiio, who gives a pedestrian introduction t o Donaldson-Witten theory based here again on topological field theory, thus entering the realm of invariants of 4-manifolds. The reader will be led to higher dimensions with Jorge Zanelli’s lectures on supergravity, where he shows how Chern-Simons gravity and supergravity can provide a firm ground for constructing a quantum theory of the gravitational field in more than 4 space-time dimensions. Finally, Stefan Theisen and Ari Pankiewicz present an introduction t o string theory, which claims to provide a quantum theory of gravity. In this lecture, the emphasis is, however, on the possibility t o solve QCD at low energies via string theory. This is within the context of the anti de Sitter/ conformal field
vi
theory correspondence, to which the reader is introduced. Three short communications close the volume; the first one by Mauricio Ayala Sbnchez and Richard Haase on group contractions and their applications t o the de Sitter and Poincar6 groups, a second one by Alexander Cardona relating some anomalies in quantum field theory t o anomalies arising from regularized traces, and the final one by Ernest0 Lupercio and Bernardo Uribe on Deligne cohomology on orbifolds with an outlook on string theory using the language of gerbes. We thank the referees most warmly for reporting on these short communications. We are indebted to various organizations for their financial support. Let us first thank the French organization C.I.M.P.A., without which this school would not have taken place. We also thank Ecos-Nord, this school being a long term scientific program between the Universit6 Blaise Pascal in ClermontFerrand and the Universidad de Los Andes in Bogotb in the areas of mathematics and physics. We are also grateful to the French embassy in Bogoti, and especially t o the cultural attach&,Jean-Yves Deler, for showing interest in this project and supporting us at difficult times. We are also indebted to the Universidad de Los Andes which was our main source of financial support in Colombia. We also received financial support from the I.C.T.P. in Trieste, Italy, from C.L.A.F. in Brasil and from different organizations in Colombia, namely Colciencias, I.C.E.T.E.X., I.C.F.E.S. which we would like t o thank here. Special thanks t o Sergio Adarve (Universidad de Los Andes) coorganizer of the school who dedicated much time and energy t o make this school possible in a country like Colombia, where many difficulties are bound to arise along the way, due t o social, political and economic problems. Many thanks to Jos6 Rafael Tor0 (Universidad de Los Andes), whose support was essential for the success of the school. We would furthermore like to express our gratitude t o Ronald0 Roldin (Universidad de Los Andes) , Carlos Montenegro (Universidad de Los Andes), and Bernardo G6mez (Universidad de 10s Andes). We are very grateful t o Juana Vall-Serra and Marta Kovacsics who did a wonderful job for the practical organization of the school. Let us also address our thanks to Germbn Barragin, Roger Jimknez and NataIia Albino who helped them in their task. Special thanks to Andrks Garcia and Juan Esteban Martin. We are also grateful t o Luis Fernbndez, Andrks Reyes, Andrhs Vargas and Bernardo Uribe for assistin nd lecturing younger participants during the school. Without all the p ople named here, all of whom helped with the organization in some way or other, before, during and after the school, this event would not have left such vivid memories in the lecturers’ and the participants’ minds. Last but not least, thanks t o all the participants who gave
fi
vii
us lectures, contributors of short communications and editors, the impulse t o prepare this volume through the enthusiasm they showed during the school. We hope that these lectures will give -as much as the school itself seems to have given- young students the desire t o pursue what might be their first acquaintance with some of the problems on the frontier of mathematics and physics presented in this volume. On the other hand, we hope that the more advanced reader will find some pleasure reading about different outlooks on related topics and seeing how well-known geometric and topological tools prove to be very useful in some areas of quantum field theory.
The Editors Alexander Cardona, HernAn Ocampo, Sylvie Paycha.
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IX
CONTENTS Lec tures
Noncommutative Geometry Hopf Algebras in Noncommutative Geometry Joseph C. VCirilly The Noncommutative Geometry of Aperiodic Solids Jean Bellissard Noncommutative Geometry and Abstract Integration Theory Moulay- Tahar Benameur
1 86
157
Topological Field Theory Introduction to Quantum Invariants of 3-Manifolds, Topological Quantum Field Theories and Modular Categories Christian Blanchet An Introduction to Donaldson-Witten Theory Marcos Marinlo
228
265
Supergravity and String Theory (Super)-Gravities Beyond 4 Dimensions Jorge Zanelli Introductory Lectures on String Theory and the AdS/CFT Correspondence Ari Pankiewica and Stefan Theisen
312
372
Short Communications Group Contractions and its Consequences upon Representations of Different Spatial Symmetry Groups
435
Mauricio A yala-Sdnchea and Richard W. Haase Phase Anomalies as Trace Anomalies in Chern-Simons Theory Alexander Cardona
450
Deligne Cohomology for Orbifolds, Discrete Torsion and B-Fields Ernest0 Lupercio and Bernard0 Uribe
468
Geometric and Topological Methods for Quantum Field Theory Eds. A. Cardona, H. Ocampo & S. Paycha @ 2003 World Scientific Publishing, pp. 1-85
HOPF ALGEBRAS IN NONCOMMUTATIVE GEOMETRY JOSEPH c. VARILLY' Depto. de Matemcitica, Universidad d e Costa Rica, 2060 Sun Jose', Costa Rica We give an introductory survey to the use of Hopf algebras in several problems of noncommutative geometry. The main example, the Hopf algebra of rooted trees, is a graded, connected Hopf algebra arising from a universal construction. We show its relation t o the algebra of transverse differential operators introduced by Connes and Moscovici in order to compute a local index formula in cyclic cohomology, and to the several Hopf algebras defined by Connes and Kreimer to simplify the combinatorics of perturbative renormalization. We explain how characteristic classes for a Hopf module algebra can be obtained from the cyclic cohomology of the Hopf algebra which acts on it. Finally, we discuss the theory of noncommutative spherical manifolds and show how they arise as homogeneous spaces of certain compact quantum groups.
Contents 2
Introduction 1 Noncommutative Geometry and Hopf Algebras
3
1.1 The algebraic tools of noncommutative geometry 1.2 Hopf algebras: introduction 1.3 Hopf actions of differential operators: an example
3
2 The Hopf Algebras of Connes and Kreimer
2.1 The Connes-Kreimer algebra of rooted trees 2.2 Hopf algebras of Feynman graphs and renormalization
3 Cyclic Cohomology
27 27 41 46
3.1 Hochschild and cyclic cohomology of algebras 3.2 Cyclic cohomology of Hopf algebras 4 Noncommutative Homogeneous Spaces
4.1 Chern characters and noncommutative spheres 4.2 How Moyal products yield compact quantum groups 4.3 Isospectral deformations of homogeneous spin geometries
46 54 61
61 65 72 78
References 'Regular Associate of the AS-ICTP, Trieste.
9
22
Email: v a r i l l y Q c a r i a r i.ucr. ac .c r
2
Introduction These are lecture notes for a course given at the Summer School on Geometric and Topological Methods for Quantum Field Theory, sponsored by the Centre International de Math6matiques Pures et Appliquhes (CIMPA) and the Universidad de Los Andes, at Villa de Leyva, Colombia, from the 9th to the 27th of July, 2001. These notes explore some recent developments which place Hopf algebras at the heart of the noncommutative approach to geometry and physics. Many examples of Hopf algebras are known from the literature on “quantum groups”, some of which provide algebraic deformations of the classical transfmmation groups. The main emphasis here, however, is on certain other Hopf algebras which have recently appeared in two seemingly unrelated contexts: in the combinatorics of perturbative renormalization in quantum field theories, and in connection with local index formulas in noncommutative geometry. These Hopf algebras act on “noncommutative spaces”, and certain characteristic classes for these spaces can be obtained, by a canonical procedure, from corresponding invariants of the Hopf algebras. This comes about by pulling back the cyclic cohomology of the algebra representing the noncommutative space, which is the receptacle of Chern characters, to another cohomology of the Hopf algebra. Recently, some interesting spaces have been discovered, the noncommutative spheres, which are completely specified by certain algebraic relations. They turn out to be homogeneous spaces under the action of certain Hopf algebras: in this way, these Hopf algebras appear as “quantum symmetry groups’’. We shall show how these symmetries arise from a class of quantum groups built from Moyal products on group manifolds. Section 1 is introductory: it offers a snapshot of noncommutative geometry and the basic theory of Hopf algebras; as an example of how both theories interact, we exhibit the Connes-Moscovici Hopf algebra of differential operators in the one-dimensional case. Section 2 concerns the Hopf algebras which have been found useful in the perturbative approach to renormalization. We develop at length a universal construction, the Connes-Kreimer algebra of rooted trees, which is a graded, commutative, but highly noncocommutative Hopf algebra. Particular quantum field theories give rise to related Hopf algebras of Feynman graphs; we discuss briefly how these give a conceptual approach to the renormalization problem. The third section gives an overview of cyclic cohomology for both associative and Hopf algebras, indicating how the latter provide characteristic classes for associative algebras on which they act. The final Section 4 explains
3
how cyclic-homology Chern characters lead t o new examples of noncommutative spin geometries, whose symmetry groups are compact quantum groups obtained from the Moyal approach t o prequantization.
I am grateful t o Josk M. Gracia-Bondia, Chryssomalis Chryssomalakos and Sylvie Paycha for several remarks on earlier versions, and t o Jean Bellissard for helpful comments at the time of the lectures. I wish t o thank Sergio Adarve, H e r n h Ocampo, Marta Kovacsics and especially Sylvie Paycha for affording me the opportunity t o talk about these matters in a beautiful setting in the Colombian highlands. 1
Noncommutative Geometry and Hopf Algebras
Noncommutative geometry, in the broadest sense, is the study of geometrical properties of singular spaces, by means of suitable LLcoordinate algebras” which need not be commutative. If the space in question is a differential manifold, its coordinates form a commutative algebra of smooth functions; but even in this case, adding a metric structure may involve operators which do not commute with the coordinates. One learns t o replace the usual calculus of points, paths, integration domains, etc. , by an alternative language involving the algebra of coordinates; by focusing only on those features which do not require that the coordinates commute, one arrives a t an algebraic (or operatorial) approach which is applicable t o many singular spaces also.
1.1
The algebraic tools of noncommutative geometry
The first step is t o replace a topological space X by its algebra of complexvalued continuous functions C ( X ) . If X is a compact (Hausdorff) space, then C ( X ) is a commutative C*-algebra with unit 1 and its norm llfll := The first GelfandsupzEx lf(z)I satisfies the C*-property llf1I2 = Ilf’fll. Naymark theorem [49] says that any commutative unital C*-algebra A is of this form: A = C ( X ) where X = M ( A ) is the space of characters (nonzero homomorphisms) p : A -+ @, which is compact in the weak* topology determined by the maps p H p ( a ) , for a E @. Indeed, the characters of C ( X ) are precisely the evaluation maps E, : f H f(z)a t each point z E X . We shall mainly deal with the compact case in what follows. A locally compact, but noncompact, space Y can be handled by passing to a compactification (that is, a compact space in which Y can be densely embedded). For instance, we can adjoin one “point a t infinity”: if X = Y H {m}, then { f E C ( X ) : f ( m ) = 0 ) is isomorphic t o Co(Y),the commutative C*-algebra of continuous functions on Y “vanishing at infinity”; thus, by dropping the
4
constant functions from C ( X ) ,we get the commutative nonunital C*-algebra Co(Y)as a stand-in for the locally compact space Y.There is also a maximal compactification PY := M ( C b ( Y ) ) ,called the Stone-Cech compactification, namely, the character space of the (unital) C*-algebra of bounded continuous functions on Y . This construction X +-+ C ( X ) yields a contravariant functor: to each continuous map h : X I + X2 between compact spaces there is a morphism’ vh : C ( X 2 ) + C ( X 1 ) given by v h ( f ) := f o h. By relaxing the commutativity requirement, we can regard noncommutative C*-algebras (unital or not) as proxies for “noncommutative locally compact spaces”. The characters, if any, of such an algebra may be said to label “classical points” of the corresponding noncommutative space. However, noncommutative C*-algebras generally have few characters, so these putative spaces will have correspondingly few points. The recommended course of action, then, is to leave these pointless spaces behind and to adopt the language and techniques of algebras instead. There is a second Gelfand-Naimark theorem [49], which states that any C*-algebra, commutative or not, can be faithfully represented as a (normclosed) algebra of bounded operators on a Hilbert space. The data for a “noncommutative topology” consist, then, of a pair ( A , % ) where ‘H is a Hilbert space and A is a closed subalgebra of L(’H). b
Vector bundles over a compact space also have algebraic counterparts. If
X is compact and E 1 , X is a complex vector bundle, the space r ( X ,E ) of continuous sections is naturally a module over C ( X ) ,which is necessarily of the form e C ( X ) m , where e = e2 E M,(C(X)) is an idempotent matrix of elements of C ( X ) . More generally, if A is any algebra over C, a right Amodule of the form eAm with e = e2 E M m ( A ) is called a finitely generated projective module over A. The Serre-Swan theorem [loll matches vector bundles over X with finitely generated projective modules over C ( X ) . The idempotent e may be constructed from the transition functions of the vector bundle by pulling back a standard idempotent from a Grassmannian bundle: see [46, 51.11 or [53, 52.11 for details. A mare concrete example is that of the tangent bundle over a compact Riemannian manifold M : by the Nash embedding theorem [103, Thm 14.5.11, one can embed M in some Rm so that the metric on T M is obtained from the ambient Euclidean metric; if e(x) is the orthogonal projector on Rn with range T,M, then e = e2 E M m ( C ( M ) )and the module r ( M , T M ) of vector fields on M may be identified with the range of e. ‘By a morphism of unital C*-algebras we mean a *-homomorphism preserving the units.
5
In the noncompact case, one can use Rennie's nonunital version of the Serre-Swan theorem [86]: Co(Y)-modules of the form eC(X)", where X is some compactification of Y and e = e2 E M,(C(X)), consist of sections vanishing at infinity (i.e., outside of Y) of vector bundles E -+ X. One can take X to be the one-point compactification of Y only if E is trivial at infinity; as a rule, the compactification to be used depends on the problem at hand. If A is a C*-algebra, we may replace e by an orthogonal projector (i.e., a selfadjoint idempotent) p = p* = p2 so that eA" N pA" as right A-modules. If A is faithfully represented by bounded operators on a Hilbert space 1-1, then M,(A) is an algebra of bounded operators on 1-1" = 1-1 @ . . . @ 1-1 (m times), so we can schematically write e =
then p :=
(h :)
(h :)
as an operator on ex" @ ( 1 - e ) x m ;
is the range projector on e1-1".
The correspondence E H r(X, E ) is a covariant functor which carries topological invariants of X to algebraic invariants of C(X). In particular, it identifies the K-themy group Ko(X), formed by stable equivalence classes of vector bundles where [El [ F ] := [E @ F ] -here @ denotes Whitney sum of vector bundles over X- with the group Ko(C(X)) formed by stable isomorphism classes of matrix projectors over C(X) where [p] [q] := [p @ q] and @ now denotes direct sum of projectors. The K-theory of C*-algebras may be developed in an operator-theoretic way, see [8,78,110] and [53, Chap. 31, for instance; or purely algebraically, and the group Ko(A) turns out to be the same in both approaches. (However, the group Kl(A), formed by classes of unitaries in Mm(A),does not coincide with the algebraic K1-group in general: see, for instance, [97]or [53, p. 1311.) The salient feature of both topological and C*-algebraic K-theories is Bott periodicity, which says that two K-groups are enough: although one can define Kj(A) is a systematic way for any j E N, it turns out that Kj+2(A) 2 Kj(A) by natural isomorphisms (in marked contrast to the case of purely algebraic K-theory).
+
+
b To deal with a (compact) differential manifold M (in these notes, we only treat differential manifolds without boundary), we replace the continuous functions in C(M) by the dense subalgebra of smooth functions d = Cm(M). This is no longer a C*-algebra, but it is complete in its natural topology (that of uniform convergence of functions, together with their derivatives of all orders), so it is a Frkchet algebra with a C*-completion. Likewise, given a vector bundle E M, we replace the continuous sections in r ( M , E ) by the d-module of smooth sections r m ( M ,E ) . The SerreSwan theorem continues to hold, mutatis mutandis, in the smooth category.
-
6
In the noncommutative case, with no differential structure a priori, we need to replace the C*-algebra A by a subalgebra A which should (a) be dense in A; (b) be a FrCchet algebra, that is, it should be complete under some countable family of seminorms including the original C*-norm of A; and (c) satisfy &(A) N &(A). This last condition is not automatic: it is necessary that A be a pre-C*-algebra, that is to say, it should be stable under the holomorphic functional calculus (which is defined in the larger algebra A ) . The proof of (c) for pre-C*-algebras is given in [lo]; see also [53, 53.81. b The next step is to find an algebraic description of a Riemannian metric on a smooth manifold. This can be done in a principled way through a theory of '(noncommutative metric spaces" at present under construction by Rieffel [93-961. But here we shall take a short cut, by defining metrics only over spin manifolds, using the Dirac operator as our instrument; this was, indeed, the original insight of Connes [24]. A metric g = [g2j] on the tangent bundle T M of a (compact) manifold M yields a contragredient metric 9-l = [ g r S ]on the cotangent bundle T * M ; so we can build a Clifford algebra bundle Cl(M) M , whose fibre at z is Cl((T,"M)@,gil), by imposing a suitable product structure on the complexified exterior bundle (A'T*M)@. We assume that M supports a spinor bundle S M, on which Cl(M) acts fibrewise and irreducibly; on passing to smooth sections, we may write .(a) for the Clifford action of a 1-form a on spinors. The spinor bundle comes equipped with a Hermitian metric, so the squared norm ll$1I2 := J, l$(z)I2m d z makes sense; the completion of I'"(M, S ) in this norm is the Hilbert space 3.1 = L 2 ( M ,S) of square-integrable spinors. Locally, we may write the Clifford action of 1-forms as c(dz') := h: yo, where the "gamma matrices'' y a satisfy y"yp yBya = 2 6"p and the coefficients h: are real and obey h:b@h; = gTs. The Dirac operator is locally defined as
-
-
+
@ := -i c(dz,)
-
d (a
XT - w,),
where w, = frfoyayp are components of the spin connection, obtained from the Christoffel symbols ?fa (in an orthogonal basis) of the Levi-Civita connection. The manifold M is spin whenever these local formulae patch together to give a well-defined spinor bundle. There is a well-known topological condition for this to happen (the second Stiefel-Whitney class w2(TM) E H2(M,Z2) must vanish [69]),and when it is fulfilled, @ extends to a selfadjoint operator on 'H with compact resolvent [53,69]. Apart from these local formulae, the Dirac operator has a fundamental algebraic property. If $ is a spinor and a E C"(M) is regarded as a multipli-
7
cation operator on spinors, it can be checked that
?(a$) = -2 c ( d a )
+ + a P$,
or, more simply,
[ P , a ]= -ic(da). (1-2) Following [6], we call a “generalized Dirac operator” any selfadjoint operator D on ‘H satisfying ID,.] = -ic(da) for a E C - ( M ) . Now c(da) is a bounded operator on L 2 ( M ,S) whenever a is smooth, and its norm is that of the gradient of a, i.e., the vector field determined by g(grad a , X ) := d a ( X ) = X ( a ) . A continuous function a E C ( M ) is called Lipschitz (with respect to the metric g ) if its gradient is defined, almost everywhere, as an essentially bounded measurable vector field, i.e., 11 gradall, is finite. Now the Riemannian distance d,(p, q ) between two points p , q E M is usually defined as the infimum of the lengths of (piecewise smooth) paths from p to q ; but it is not hard to show (see 153, 59.31, for instance) that the distance can also be defined as a supremum: d,(p,q) = SUP{ l4P) - 4411 : a E C ( M ) , II gradall, 5 1}. (1.3) The basic equation (1.2) allows to replace the gradient by a commutator with the Dirac operator: d,(PA) = SUP{ b ( P > - 4 q ) l : a E C ( M ) , II[P,alllI 11. (1.4) Thus, the Riemannian distance function d, is entirely determined by 49. Moreover, the metric g is in turn determined by d,, according to the MyersSteenrod theorem [79]. From the noncommutative point of view, then, the Dirac operator assumes the role of the metric. This leads to the following basic concept. Definition 1.1. A spectral triple is a triple (A,X,D), where A is a preC*-algebra, 7-i is a Hilbert space carrying a representation of A by bounded operators, and D is a selfadjoint operator on A, with compact resolvent, such that the commutator [D,a]is a bounded operator on ‘H, for each a E A. (The kernel of D is then a finite-dimensional subspace of ‘H. Since we often compute with D-l, we could either redefine D to be nonzero on this subspace, or subtract the kernel from 7-i: to avoid such fine points, we shall simply assume from now on that ker D = {0},so that D is invertible.) Spectral triples comes in two parities, odd and even. In the odd case, there is nothing new; in the even case, there is a grading operator x on 7-i (a bounded selfadjoint operator satisfying x2 = 1, making a splitting 7-i = ‘H+@ X )such , that the representation of A is even (XU= ax for all a E A) and the operator D is odd, i.e., X D = -Dx; thus each [D,a]is a bounded odd operator on ‘H.
8
A noncommutative spin geometry is a spectral triple satisfying several extra conditions, which were first laid out by Connes in the seminal paper [26].These conditions (or "axioms", as they are sometimes called) arise from a careful consideration of the algebraic properties of ordinary metric geometry. Seven such properties are put forward in 1261; here, we shall just outline the list. Some of the terminology will be clarified later on; a more complete account, with all prerequisites, is given in [53, 510.51. 1. Classical dimension: There is a unique nonnegative integer n, the Wassical dimension" of the geometry, for which the eigenvalue sums UN := COlk
-
---f
2. Regularity: Not only are the operators a and [D,a]bounded, but they lie in the smooth domain of the derivation S(T) := [IDI,T]. (When A is an algebra of functions and D is a Dirac operator, this smooth domain consists exactly of the C" functions.)
3. Finiteness: The algebra A is a pre-C*-algebra, and the space of smooth vectors 'H" := nkDom(Dk) is a finitely generated projective left Amodule. (In the commutative case, this yields the smooth spinors.)
4. Reality: There exists an antiunitary operator C on 'H, such that [a,Cb*C-l] = 0 for all a,b E A (thus b H Cb*C-l is a commuting representation on 'H of the "opposite algebra" A", with the product reversed); and moreover, C2 = f l , C D = f D C , and Cx = f x C in the even case, where the signs depend only on n mod 8. (In the commutative case, C is the charge conjugation operator on spinors.)
5. First order: The bounded operators [D,a] commute with the opposite algebra representation: [ [ Da, ] ,Cb*C-l] = 0 for all a, b E A. 6. Orientation: There is a Hochschild n-cycle c on A whose natural representative is TD(C) = x (even case) or TD(C)= 1 (odd case). More on this later: such an n-cycle is usually a finite sum of terms like a0 @a1@. . .@a, which map to operators r o ( a o @ a l@ . . . @ a , ) : = a o [ D , a l ] . . . [ D , a , ] , and c is the algebraic expression of the volume form for the metric determined by D .
9
7. Poincare‘ duality: The index map of D determines a nondegenerate pairing on the K-theory of the algebra A. (We shall not go into details, except to mention that in the commutative case, the Chern homomorphism matches this nondegeneracy with Poincar6 duality in de Rham co/homology.)
It is very important to know that when A = C - ( M ) the usual apparatus of geometry on spin manifolds (spin structure, metric, Dirac operator) can be fully recovered from these seven conditions: for the full proof of this theorem, see [53, chap. 111. Another proof, assuming only that A is commutative, is developed by Rennie in [85]. 1.2
Hopf algebras: introduction
The general scheme of replacing point spaces by function algebras and then moving on to noncommutative algebras also works for symmetry groups. Now, however, the interplay of algebra and topology is much more delicate. There are at least two ways of handling this issue. One is to leave topology aside and develop a purely algebraic theory of symmetry-bearing algebras: these are the Hopf algebras, sometimes called “quantum groups”, about which there is already a vast literature. At the other extreme, one may insist on using C*algebras with special properties; in the unital case, there has emerged a useful theory of “compact quantum groups” [115],which only very recently has been extended to the locally compact case also [68]. We begin with the more algebraic treatment, keeping t o the compact case, i.e., all algebras will be unital unless indicated otherwise. The field of scalars may be taken as C , JR or Q, according to convenience; to cover all cases, we shall denote it by F. In this section, @ always means the algebraic tensor product. Definition 1.2. A bialgebru is a vector space A over F which is both an algebra and a coalgebra in a compatible way. The algebra structure is given by IF-linear maps m : A @ A -+ A (the product) and 77: IF -+ A (the unit map) where sy := m ( z , y )and ~ ( 1=) 1 ~ The . coulgebra structure is likewise given by linear maps A: A A @ A (the coproduct) and E : A + IF (the counit map). We write L : A + A , or sometimes L A ,to denote the identity map on A . The required properties are: -+
1. Associativity: m(m @ L ) = m ( &~I m) : A @ A @ A 2. Unity: m(q
L)
=m (@ ~ 77) = L : A
-+
+ A;
A;
3. Coassociativity: (A @ L ) A= ( L @ A)A : A
-+
A @ A @ A;
10
4. Counity:
(E
'8 L)A= ( L @ &)A= L : A -+ A;
5. Compatibility: A and
E
are unital algebra homomorphisms.
The first two conditions, expressed in terms of elements z, y, z of A, say that (zy)z = z(yz) and 1 ~ =z z 1 = ~ z. The next two properties are obtained by "reversing the arrows''. Commutativity may be formulated by using the "flip map" a : A '8 A -+ A '8 A : x '8 y H y '8 x: the bialgebra is commutative if ma = m : A '8 A + A. Likewise, the bialgebra is called cocommutatiwe if
aA = A : A
-+
A @A.
The (co)associativity rules suggest the abbreviations
m2 := m(m'8 L ) = m ( '8 ~m ) ,
A' := (A '8 L)A= ( L '8 A)A,
with obvious iterations m3: AB4
-+ A, A 3 : A --+ A@4;mT:A@(Tf1)-+ A, AT : A -+ A@(T+1). Exercise 1.1. If (C,A, E ) and (C',A', E ' ) are coalgebras, a counital coalgebra morphism between them is an F-linear map e : C -+ C' such that A'! = (e '8 [)A and E'[ = E. Show that the compatibility condition is equivalent to the condition that m and u are counital coalgebra morphisms. 0 Definition 1.3. The vector space Hom(C, A) of F-linear maps from a coalgebra (C, A, E ) to an algebra ( A ,m, q) has an operation of convolution: given two elements f,g of this space, the map f * g E Hom(C, A) is defined as
f * g := m(f 8 g ) A : C
-+
A.
Convolution is associative because
(f * 9) * h = m((f * g) '8 h)A = m(m'8 ~ ) ('8f g '8 h)(A '8 L)A = m ( '8 ~ m)(f '8 g '8 h ) ( '8 ~ A)A = m(f '8 (9* h))A = f * (9 * h). This makes Hom(C, A) an algebra, whose unit is ~ A E C :
f * VAEC = m(f '8 VAEC)A= m(LA '8 VA)(f '8 LF)(LC '8 EC)A = LA fLC = f, VAEC * f = m(77AEC '8 f ) A = m(qA '8 LA)(LF '8 f ) ( E C '8 Lc)A = LA fLC = f. A bialgebra morphism is a linear map e: H --+ H' between two bialgebras, which is both a unital algebra homomorphism and a counital coalgebra homomorphism; that is, e satisfies the four identities em = TI'([ '8 a),
tq
= q',
A'[ = (e@ [)A,
where the primes indicate coalgebra operations for HI.
E'[
= E,
11
A bialgebra morphism respects convolution, in the following ways; iff, g E Hom(C,H) and h , k E Hom(H’,A) for some coalgebra C and some algebra A , then
e ( f * 9 ) = em(f 8 g ) A c ( h * k)!
e ) ( f EI g ) n c = m i ( e j @ e g p C = e f * eg, = m ~ (8 hIc)A’e = m ~ ( EIhk)(e 8 e)A = mA(hC 8 Ice)A = he* Ice. = ml(e8
Definition 1.4. A Hopf algebra is a bialgebra H together with a (necessarily unique) convolution inverse S for the identity map L = L H ; the map S is called the antipode of H . Thus, L
* S = m ( EI~ S ) A = VE,
S
* L = m(S @ L ) A= V E .
A bialgebra morphism between Hopf algebras is automatically a Hopf algebra morphism, i.e., it exchanges the antipodes: CS = SIC. For that, it suffices to prove that these maps provide a left inverse and a right inverse for t in Hom(H, H I ) . Indeed, since the identity in Hom(H, H‘) is V I E , it is enough to notice that
es * e = e(s* L ) = e V E = V I E = ViEie= * sl)e= e * sle, (L1
and associativity of convolution then yields
sle= V I E * s‘e = es * e * sle = es * V I E = es. The antipode has an important pair of algebraic properties: it is an antihomomorphism for both the algebra and the coalgebra structures. Formally, this means
Sm = ma(S 8 S ) and A S = ( S @ S ) a A .
(1.5)
The first relation, evaluated on a@b, becomes the familiar antihomomorphism property S(ab) = S( b )S (a ).We postpone its proof until a little later. Example 1.1. The simplest example of a Hopf algebra is the “group algebra” FG of a finite group G. This is just the vector space over IF with a basis labelled by the elements of G; the necessary linear maps are specified on this basis. The product is given by m(x 8 y) := zy, linearly extending the group multiplication, and ~ ( 1:= ) 1~ gives the unit map. The coproduct, counit and antipode satisfy A ( x ) := x 8 x , ~ ( x:= ) 1 and S ( x ) := x-’, for each x E G. Exercise 1.2. In a general Hopf algebra H , a nonzero element g is called grouplike if A ( g ) := g @ g . Show that this condition entails that g is invertible and that E ( g ) = 1 and S(g) = g-’. 0 There are two main “classical” examples of Hopf algebras: representative functions on a compact group and the enveloping algebra of a Lie algebra.
12
Example 1.2. Now let G be a compact topological group (most often, a Lie group), and let the scalar field IF be either R or C. The Peter-Weyl theorem [13, 111.31 shows that any unitary irreducible representation T of G is finitedimensional, any matrix element f ( x ) := (u 1 T ( X ) V ) is a continuous function on G, and the vector space R ( G ) generated by these matrix elements is a dense subalgebra (*-subalgebra in the complex case) of C(G). Elements of R(G) can be characterized as those continuous functions f : G 4 IF whose translates ft : x H f (t-’x) generate a finitedimensional subspace of C(G); they are called representative functions on G. The algebra R(G) is a G-bimodule in the sense of Wildberger [112]under left and right translation; indeed, it is the algebraic direct sum of the finitedimensional irreducible G-subbimodules of C(G). The group structure of G makes R(G) a coalgebra. Indeed, we can identify the algebraic tensor product R ( G ) 8 R ( G ) with R(G x G) in the obvious way -here is where the finite-dimensionality of the translates is used [53, Lemma 1.271- by (f @ g)(x,y ) := f ( x ) g ( y ) ,and then
A f h Y ) := f ( X Y ) (1.6) defines a coproduct on R(G).The counit is E ( f ) := f(l),and the antipode is given by S f ( x ) := f ( x - l ) . Example 1.3. The universal enveloping algebra U ( g ) of a Lie algebra g is the quotient of the tensor algebra 7 ( g ) by the two sided ideal I generated by the elements X Y - Y X - [ X ,Y ] ,for all X , Y E g. (Here we write X Y instead of X 8 Y , t o distinguish products within 7 ( g ) from elements of 7 ( g ) 8 7(g).) The coproduct and counit are defined on g by A ( X ) := X 8 1 + 1 B X ,
(1.7) and E ( X ) := 0. These linear maps on g extend to homomorphisms of the tensor algebra; for instance,
A ( X Y ) = A ( X ) A ( Y )= X Y 8 1 + X 8 Y +Y 8 X
+ 1@ X U ,
and thus
A ( X Y - Y X - [ X , Y ] )= ( X U - Y X - [ X ,Y ] 81 ) + 1 8( X U - Y X - [ X , Y ] ) ,
+
so A ( I ) 5 I @U ( g ) U ( g )8 I . Clearly, & ( I )= 0, too. Therefore, I is both an ideal and a “coideal” in the full tensor algebra, so the quotient U ( g ) is a bialgebra, in fact a Hopf algebra: the antipode is given by S ( X ) := - X . From (1.7), the Hopf algebra U ( g ) is clearly cocommutative. The word “universal” is appropriate because any Lie algebra homomorphism qb : g 4 A , where A is an unital associative algebra, extends uniquely (in the obvious way) to a unital algebra homomorphism \Tr : U ( g )4 A.
13
Example 1.4. Historically, an important example of a Hopf algebra is Woronowiczl q-deformation of SU(2). The compact group S U ( 2 ) consists of complex matrices g
=
+
(: ic) ,
subject t o the unimodularity condition
a*a c*c = 1. The matrix elements a and c, regarded as functions of g , generate the *-algebra R ( S U ( 2 ) ) :that is, any matrix element of a unitary irreducible (hence finite-dimensional) representation of SU (2) is a polynomial in a , a*,c, c*. Woronowicz found [113] a noncommutative *-algebra with two generators a and c, subject to the relations ac = qca,
ac* = qc*u,
cc* = c*c, a*a + c*c = 1, aa*
+ q2cc* = 1,
where q is a real number, which can be taken in the range 0 < q 5 1. For the coalgebra structure, take A and E be *-homomorphisms determined by
Aa:=aQa-qc*@c, and & ( a ):= 1, E ( C ) matrix g :=
(z)::-
:= 0.
Ac:=c@a+a*@c,
One can check that, by applying A elementwise, the
satisfies A ( g ) = g
@ 9.
The antipode S is the linear
antihomomorphism determined by
S ( a ) := a*,
S(a*) := a ,
S ( C ) := -qc,
S(c*) := -q-lc*,
++ S ( x * ) is an antilinear homomorphism, indeed an involution: S ( S ( x * ) * )= x for all x. This last relation is a general property of Hopf algebras with an involution. The initial interest of this example was that it could be represented by a *-algebra of bounded operators on a Hilbert space, whose closure was a C*-algebra which could legitimately be called a deformation of C ( S U ( 2 ) )it; has become known as C(SUq(2)).In this way, the “quantum group” SUq(2) was born. Nowadays, many q-deformations of the classical groups are known, although q may not always be real: for example, t o define SLq(2,R),one needs selfadjoint generators a and c satisfying ac = qca, which is only possible if q is a complex number of modulus 1.
so that x
b If u Z 3 ( x:= ) (e, 17r(x)e3),for i,j = 1,.. . ,n, are the matrix elements of an n-dimensional irreducible representation 7r of a compact group G with respect to an orthonormal basis { e l , .. . , e n } , then (1.6) and 7r(zy) = 7r(x)r(y) show that
A%3
=
uak @ uk3 1
(1.8a)
14
and the coassociativity of A is manifested as
A2uij = Ck,lUzk
@ u k l '8 W j ,
(1.8b)
reflecting the associativity of matrix multiplication. This may be generalized by a notational trick due to Sweedler [102]: if a is an element of any Hopf algebra, we write
A a =: C a:1 @ a:2 (finite sum). (The prevalent custom is to write A a = Ca(l) @ a(2),leading to a surfeit of parentheses.) The equality of ( A @ L ) ( A ~ = )Ca:1:1@ a:1:28 a:2 and ( L @ A ) ( A a )= C a:1 @ a:2:1@ a:2:2is expressed by rewriting both sums as
A2a = C a,l @ a:2 @ a,3. The matricial coproduct (1.8b) is a particular instance of this notation. The counit and antipode properties can now be rewritten as
C.(.:l)
c
c
a:2 = a:1.(a:2) = a , S(a:1)a:2 = C a:1S(a:2)= .(a) 1.
(1.9a) (1.9b)
The coalgebra antihomomorphism property of S is expressed as
A ( s ( a )= ) C S(a:2)8 S ( a : i ) .
(1.10)
We can now prove the antipode properties (1.5). We show that the maps S m : a @ b ++ S(ab) and ma(S @ S) : a @ b H S(b)S(a)are one-sided convolution inverses for m in Hom(H @ H , H ) , so they must coincide. The coproduct in H @ H is ( L @ a @ L ) ( A @ A ) : a @ b H a:l @ b:l @ a:2 @ b,2, and so
c
( S m* m ) ( a@ b) = m(Sm @ m ) ( C a , i @ b,l @ a:2 @ b:2) = CS(a:lb,1)a:2b:2 = ( S * L ) ( u ~=) Q E ( U ~ = ) Q E H ~ H@( b). ~ On the other hand, writing
7 := ma(S @
S),
( m * T)(U @ b) = m(m @ T ) ( C a : l@ b:1 @ a:2 @ b : 2 ) = ca:lb:iS(b:a)S(a:2) = ~ ( bC) a : i S ( a : 2= ) &(a)E(b) 1~ = ~ ~ ( a=bQ) E
H ~ H@( b). ~
Thus, S m * m = 7 7 . ~= ~m 8~ * T, as claimed. In like fashion, one can verify (1.10) by showing that A S * A = Q H ~ H E= A * ( ( S @ S ) o Ain ) Hom(H, H @ H ) ; we leave the details to the reader. Exercise 1.3. Carry out the verification of A S = ( S @ S ) a A . 0
15
Notice that in the examples H = R(G)and H = U ( g ) , the antipode satisfies S2 = L H , but this does not hold in the SUq(2) case. We owe the following remark to Matthias Mertens [74, Satz 2.4.21: S2 = LH if and only if
C S(a:2)a : =~ C a
2
S(a1) = .(a) 1 for all
a E H.
(1.11)
Indeed, if S2 = L H , then
C S ( Q ) ail = &(a)1 implies that ( S * S 2 ) ( a )= c S(a:1)S2(a:2)= S(C S(a:n).:I)
while the relation
= S(.(a) 1) = .(a) 1,
so that (1.11) entails S * S2 = S 2 * S = V E , hence S 2 = LH is the (unique) convolution inverse for S . Now, the relations (1.11) clearly follow from (1.9b) if H is either commutative or cocommutative (in the latter case, Aa = C aZl@ a:2 = C a : 2 @ .:I). It follows that S2 = LH if H is either commutative or cocommutat ive. b Just as locally compact but noncompact spaces are described by nonunital function algebras, one may expect that locally compact but noncompact groups correspond to some sort of “nonunital Hopf algebras”. The lack of a unit requires substantial changes in the formalism. At the purely algebraic level, an attractive alternative is the concept of “multiplier Hopf algebra” due to van Daele [105,106]. If A is an algebra whose product is nondegenerate, that is, ab = 0 for all b only if a = 0, and ab = 0 for all a only if b = 0, then there is a unital algebra M ( A ) such that A M ( A ) , called the multiplier algebra of A , characterized by the property that xu E A and ax E A whenever x E M ( A ) and a E A. Here, M ( A ) = A if and only if A is unital. A coproduct on A is defined as a homomorphism A : A 4 M ( A @ A ) such that, for all a, b, c E A ,
(Aa)(l @ b) E A @ A , and
( a @ l)(Ab) E A @ A ,
and the following coassociativity property holds:
( a @ 1 8 1) (A @ L ) ((Ab)(1 @ c))
= (L @
A) ( ( a@ l)(Ab))(1 @ 1 @ c).
There are then two well-defined linear maps from A @ A into itself
Tl(a@ b)
:= ( A a ) ( l @b ) ,
and T2(a @ b) := ( a @ l)(Ab).
We say that A is a multiplier Hopf algebra [lo51 if TIand
T2
are bijective.
16
When A is a (unital) Hopf algebra, one finds that T c l ( a @ b ) = S ) A a ) ( 1@ b) and T . l ( a @ b) = ( a @ 1 ) ( ( S@ L ) A ~In) .fact,
((L
@
T~(((L @ S ) A a ) ( 1 @b ) ) = CTl(a:l@ S(a:z)b)= Ca:1@ a:2S(a:3)b =
C a:1 @ &(a:p)b= a @ b,
and T2((a @ 1)((S@ L ) A ~=) )a @ b by a similar argument. The bijectivity of Tl and T2 is thus a proxy for the existence of an antipode. It is shown in [lo51 that from the stated properties of A, TI and T2, one can construct both a counit E : A -+ IF and an antipode S, though the latter need only be an antihomomorphism from A to M ( A ) . The motivating example is the case where A is an algebra of functions on a locally compact group G (with finite support, say, to keep the context algebraic), and A f ( z ,y) := f ( z y ) as before. Then Tl(f@ g ) : (2,y) ++ f(zy)g(y) also has finite support and the formula ( T y l F ) ( zy) , := F ( z y - l , y) shows that TI is bijective; similarly for T2. A fully topological theory, generalizing Hopf algebras to include Co(G) for any locally compact group G and satisfying Pontryagin duality, is now available: the basic paper on that is [68]. b Duality is an important aspect of Hopf algebras. If (C,A, E ) is a coalgebra, the linear dual space C* := Hom(C, IF) is an algebra, as we have already seen, where the product f @ g c) (f @ g ) A is just the restriction of At to C* @ C*; the unit is E ~ where , denotes transpose. (By convention, we do not write the multiplication in IF, implicit in the identification IF @ F N F.) However, if ( A ,m, u)is an algebra, then ( A * ,mt,u t ) need not be a coalgebra because mt takes A* to ( A @ A)* which is generally much larger than A* @ A*. Given a Hopf algebra ( H ,m, u , A, E , S ) , we can replace H* by the subspace H" := { f E H* : m t ( f )E H* @ H * } ; one can check that ( H o , A t l r t , m t , u t , S t )is again a Hopf algebra, called the finite dual (or "Sweedler dual") of H . To see why H" is a coalgebra, we must check that m t ( H " ) C H" @ H". So suppose that f E H * satisfies m t ( f ) = CEl g j @ hj, a finite sum with g j , hj E H * . We may suppose that the gj are linearly independent, so we can find elements a l l . . . ,am E H such that g j ( a k ) = 6 j k . Now m
m
j=1
j=l
so mt(hk)= CY==, f j k @ h j , where f j k ( a ) := gj( aka);thus h k E H " . A similar argument shows that each g j E H " , too. However, H" is often too small to be useful: in practice, one works with two Hopf algebras H and HI, where each may be regarded as included in the dual of the other. That is to say, we can write down a bilinear form
17
( a , f ) := f ( a ) for a E H and f E H’ with an implicit inclusion H’ L) H*. The transposing of operations between the two Hopf algebras boils down t o the following five relations, for a, b E H and f,g E H‘:
(ab,f) = ( a @ b, A’f), f @ g ) , ( S ( a ) ,f) = (a,W ) ) , (a, fg) = € ( a ) = ( a ,l w ) , and ~ ’ ( f=) ( I H , f). The nondegeneracy conditions which allow us t o assume that H’ C H* and H G H’* are: (i) (a,f ) = 0 for all f E H’ implies a = 0, and (ii) (a,f ) = 0 for all a E H implies f = 0. Let G be a compact connected Lie group whose Lie algebra is g. The function algebra R ( G ) is a commutative Hopf algebra, whereas U ( g ) is a cocommutative Hopf algebra. On identifying g with the space of left-invariant vector fields on the group manifold G, we can realize U ( g ) as the algebra of left-invariant differential operators on G. If X E g, and f E R ( G ) ,we define
( X , f ) := Xf(1) = and more generally, (XI . . . X,, f ) := X l ( . . . ( X n f ) ) ( l )we ; also set (1,f) := f(1). This yields a duality between R(G) and U ( g ) . Indeed, the Leibniz rule for vector fields, namely X(fh) = (Xf)h f(Xh), gives
+
+
+
(X,fh) = X f ( l ) h ( l ) f ( l ) X h ( l ) = (X @ 1 163 X)(f @ h ) ( l Cz3 1) (1.12) = A x ( f @ h ) ( l @ 1) = (AX, f Cz3 h ) , while
= X(Yf)(1) = (XY7.f).
If (D,f) = 0 for all D E U ( g ) , then f has a vanishing Taylor series at the identity of G. Since representative functions are real-analytic [64], this forces f = 0. On the other hand, if ( D ,f ) = 0 for all f ,the left-invariant differential operator determined by D is null, so D = 0 in U ( g ) . The remaining properties are easily checked. Definition 1.5. The relation (1.12) shows that AX = X @ 1 1@ Xencodes the Leibniz rule for vector fields. In any Hopf algebra H , an element h E H for which Ah = h @ 1 1@ h is called primitive. It follows that ~ ( h=) 0 and that S ( h ) = -h. In the enveloping algebra U ( g ) ,elements of g are obviously
+
+
18
primitive. If a and b are primitive, then so is ab - ba, so the vector space Prim(H) of primitive elements of H is actually a Lie algebra. Indeed, since the field of scalars F has characteristic zero, the only primitive elements of U ( g ) are those in g, i.e., Prim(U(g)) = g: see [ll],153, Lemma 1.211 or [76, Prop. 5.5.31. (Over fields of prime characteristic, there are other primitive elements in U ( g ) [76].) b If H is a bialgebra and A is an algebra, and if q5, 11,: H --+ A are algebra homomorphisms, their convolution q5 * 11, E Hom(H,A) is a linear map, and will be also a homomorphism provided that A is commutative. Indeed, q5 * 11, = m(q5 @ $)A is a composition of three homomorphisms in this case; the commutativity of A is needed to ensure that m : A @ A -+ A is multiplicative. A particularly important case arises when A = IF. Definition 1.6. A character of an algebra is a nonzero linear functional which is also multiplicative, that is,
p(ab) = p(a) p ( b ) for all a , b; notice that p ( 1 ) = 1. The counit E of a bialgebra is a character. Characters of a bialgebra can be convolved, since p * v = ( p @ v ) A is a composition of homomorphisms. The characters of a Hopf algebra H form a group G ( H ) under convolution, whose neutral element is E ; the inverse of p is pS. A derivation or “infinitesimal character” of a Hopf algebra H is a linear map 6 : H IF satisfying -+
6(ab) = 6 ( a ) ~ ( b+) ~ ( a ) b ( b )for all a,b E H . This entails 6 ( 1 ~ = ) 0. The previous reiation can also be written as rnt(6) = 6 @ E E @ 6, which shows that 6 belongs to H” and is primitive there; in particular, the bracket [6, a] := 6 * d - d * 6 of two derivations is again a derivation. Thus the vector space Der,(H) of derivations is actually a Lie algebra. In the commutative case, there is another kind of duality to consider: one that matches a Hopf algebra with its character group. A compact topological group G admits a normalized left-invariant integral (the Haar integral): this can be thought of as a functional J : R ( G ) R, where the left-invariance translates as ( L @ J ) A = q J . (We leave it as an exercise to show that this corresponds to the usual definition of an invariant integral.) The evaluations at points of G supply all the characters of this Hopf algebra: G(R(G))N_ G. Conversely, if H is a commutative Hopf algebra possessing such a left-invariant functional J , then its character group is compact, and H N_ R ( G ( H ) ) .These results make up the Tannaka-Kreh duality theorem -for the proofs, see [53] or [56]- and it is important either to use real scalars, or to consider
+
--+
19
only hermitian characters if complex scalars are used. The totality of all @valued characters of R ( G ) , hermitian or not, is a complex group G@called the complexification of G [13, 111.81; for instance, if G = S U ( n ) ,then G' 21 sqn,@). b The action of vector fields in g and differential operators in U ( g ) on the space of smooth functions on G, and more generally on any manifold carrying a transitive action of the group G, leads t o the notion of a Hopf action of a Hopf algebra H on an algebra A. Definition 1.7. Let H be a Hopf algebra. A (left) Hopf H-module algebra A is an algebra which is a (left) module for the algebra H such that h . 1 =~ ~ ( h ) and l ~
h . (ab) = C(h:l. a)(h:2 b)
(1.13)
'
whenever a, b E A and h E H . Grouplike elements act by endomorphisms of A , since g.(ab) = (g.a)(g.b) and g . 1 = 1 if g is grouplike. On the other hand, primitive elements of H act by the usual Leibniz rule: h . (ab) = ( h . a)b a ( h . b) and h . 1 = 0 if Ah = h @ 1 1@ h. Thus (1.13) is a sort of generalized Leibniz rule.
+
+
b Duality suggests that an action of U ( g ) should manifest itself as a coaction of R ( G ) . Definition 1.8. A vector space V is called a right comodule for a Hopf algebra H if there is a linear map @ : V -+V @ H (the right coaction) satisfying ( @ @ L ) @ = (L@A)@ :V +V
@H @ H,
(L@E)@
=L
:
V
-+
V. (1.14)
In Sweedler notation, we may write the coaction as @(v) =: CV:O @ v : ~so , C V : OE ( V : I ) = v and C Y:O:O @ v:0:1@ v:1 = C V : O@ v:1:1@ v,1,2; we can rewrite both sides of the last equality as CV:O @ v:1 @ v:2, where, by convention, v:, E H for r # 0 while v,oE V . Left H-comodules are similarly defined; a linear map : V -+ H @ V is a left coaction if (L
@
@)@
=
(A @ L ) @
and
(E
@ L ) @ = L;
it is convenient t o write @(v) =: C v : - 1 @ V:O in this case. If a H-comodule A is also an algebra and if the coaction @ : A -+ A @ H is an algebra homomorphism, we say that A is a (right) H-comodule algebra. In this case, C(ab):o@ ( a b ) , = ~ C a , o b , o@ a,lb,l. If H and U are two Hopf algebras in duality, then any right H-comodule algebra A becomes a left U-module algebra, under
x . a := C a:o ( X ,a : 1 ) ,
20
for X E U and a E A . In symbols: X acts as the operator Indeed, it is enough to note that
(L
@
( X I ) @on A.
X . (ab) = C a:ob:o( X ,a:ib:i)= C a:ob:o( A x ,a:i @ b:i) = C a:ob:o(X:i @ x:2,a:i @ hi) = C G O (X:i,.:I) b:o (X:27b:i) = C(X:,. a) (X:2. b). The language of coactions is used to formulate what one obtains by applying the Gelfand cofunctor (loosely speaking) to the concept of a homogeneous space under a group action. If a compact group G acts transitively on a space M , one can write M M G I K , where K is the closed subgroup fixing a basepoint zo E M (i.e., K is the “isotropy subgroup” of zo). Then any function on M is obtained from a function on G which is constant on right cosets of K. If F ( G ) and T ( M ) denote suitable algebras of functions on G and M (we shall be more precise about these algebras in a moment), then there is a corresponding algebra of right K-invariant functions
F ( G ) K := { f
E F ( G ) : f ( z w ) = f(z)whenever w E
If 3 E M corresponds to the right coset
K, z E G } .
XK in G I K , then
Cf@) := f(z) defines an algebra isomorphism C: F ( G ) K F ( M ) .
4 [For aesthetic reasons, one may prefer to work with left K-invariant functions; for that, one should instead identify M with the space K\G of left cosets of K.] Suppose now that the chosen spaces of functions satisfy
T ( G )@ F ( M ) N F ( G x M ) ,
(1.15)
where @ denotes, as before, the algebraic tensor product. Then we can define p : F ( M ) -+ F ( G ) @ F ( M ) by pf(x,jj) := f(-). It follows that [ p < f ( z , Y) = Cf(W)= f ( z Y ) = A f ( z 7Y) = [ ( L €3 C)Af](z,S ) ,
(1.16)
so that pC = ( L @ <)A : F ( G ) K -+ F ( G ) @ F ( M ) . Notice, in passing, that the coproduct A maps .F(G)K into F ( G ) @ F (G ) K,which consists of functions h on G x G such that h(x,yw) = h(x,y) when w E K. [Had we used left cosets and left-invariant functions, the corresponding relations would be A ( F ( G ) KC ) F ( G ) K @ F ( G )p, : F ( M ) -+ F ( M ) @ F ( G )and , pC = ( ( 8 ~ ) A . l In Hopf algebra language, p defines a left [or right] coaction of F ( G ) on the algebra F ( M ) ,implementing the left [or right] action of the group G on M , and C intertwines this with left [or right] regular coaction on K-invariant functions induced by the coproduct A. We get an instance of the following definition.
21
Definition 1.9. In the lore of quantum groups -see, for instance, [63, 511.61- a (left) embedded homogeneous space for a Hopf algebra H is a left H-comodule algebra A with coaction p : A --t H @ A, for which there exists a subalgebra B C H and an algebra isomorphism i :B 4 A such t h a t p i = ( L @ < ) A: B -+ H @ A. A right embedded homogeneous space is defined, mutatis mutandis, in the same way. There are two ways t o ensure that the relation (1.15) holds. One way is to choose 3 ( G ) := R ( G ) ,which is a bona-fide Hopf algebra, and then t o define R ( M ) as the image < ( R ( G ) Kof) the K-invariant representative functions. For instance, if G = SU(2) and K = U ( l ) , so that M x S2 is the usual 2-sphere of spin directions, then R ( G ) is spanned by the matrix elements DL,,of the ( 2 j 1)-dimensional unitary irreducible representations of SU(2): see [7], for example. Now DLn is right U(1)-invariant if and only if j is an integer (not a half-integer) and n = 0; moreover, the functions qm:= are the usual spherical harmonics on the 2-sphere. In other words: R ( S 2 ) is the algebra of spherical harmonics on S2.
+
d w D k o
b To move closer t o noncommutative geometry, it would be better t o use either continuous functions (at the C*-algebra level) or smooth functions on G and M ; that is, one should work with 3 = C or with 3 = C". Notice that formulas like (1.16) make perfect sense in those cases; but the tensor product relation (1.15) is false in the continuous or smooth categories, unless the algebraic @ is replaced by a more suitable completed tensor product. In the continuous case, for compact G and M , the relation
C ( G )@ C ( M )N C(G x M ) is valid, where @ denotes the "minimal" tensor product of C*-algebras. (There may be several compatible C*-norms on a tensor product of two C*-algebras; but they all coincide if the algebras are commutative.) In the smooth case, we may fall back on a theorem of Grothendieck [55], which says that
C"(G)
C"(M)
E
C"(G x M ) ,
where % denotes the projective tensor product of Frkchet spaces. But then, it is necessary t o go back and reexamine our definitions: for instance, the coproduct need only satisfy A ( A ) & A @ Afor a completed tensor product, which is a much weaker statement than the original one - the formula A a = C a:1@ a:2 need no longer be a finite sum, but only some kind of convergent series. The bad news is that, in the C*-algebra case, the product map m : A @ A + A is usually not continuous; the counit E and antipode S become unbounded
22
linear maps and one must worry about their domains; and so on. We shall meet examples of these generalized Hopf algebras in subsection 4.2.
1.9
Hopf actions of differential operators: an example
The Hopf algebras which are currently of interest are typically neither commutative, like R ( G ) ,nor cocommutative, like U ( g ) . The enormous profusion of “quantum groups” which have emerged in the last twenty years provide many examples of such noncommutative, noncocommutative Hopf algebras: see [18,61,63,72] for catalogues of these. A class of Hopf algebras which are commutative but are not cocommutative were introduced a few years ago, first by Kreimer in a quantum field theory context [65], and independently by Connes and Moscovici [36] in connection with a local index formula for foliations; in both cases, the Hopf algebra becomes a device to organize complicated calculations. We shall discuss the QFT version at length in the next section; here we look at the geometric example first. If one wishes to deal with gravity in a noncommutative geometric framework [27], one must be able to handle the geometrical invariants of spacetime under the action of local diffeomorphisms. We consider an oriented ndimensional manifold M , without boundary. By local diffeomorphisms on M we mean diffeomorphisms $ : Dom 11, -+ Ran $, where both the domain Dom $ and range Ran$ are open subsets of M ; and we shall always assume that 11, preserves the given orientation on M . Two such local diffeomorphisms can be composed if and only if the range of the first lies within the domain of the second, and any local diffeomorphism can be inverted: taken all together, they form what is called a pseudogroup. We let r be a subpseudogroup (with the discrete topology), and consider the pair ( M ,I?). The orbit space M / r has in most cases a very poor topology. The noncommutative geometry approach is to replace this singular space by an algebra which captures the action of r on M . The initial candidate, a ‘‘crossed product” algebra C ( M ) >a I?, still has a very complicated structure; but much progress can be made [23] by replacing M by the bundle F --+ M of oriented frames on M . This is a principal fibre bundle whose structure group is GL+(n,R), the n x n matrices with positive determinant. Any $ E r admits a prolongation to the frame bundle described as follows. Let x = ( X I , . . . , 2), be local coordinates on M and let y = (y;, yf, . . . , y,”) be local coordinates for the frame at x. To avoid a “debauch of indices”, we mainly consider the l-dimensional case, where M M S1 is a circle and its frame bundle F is a cylinder (but we use a matrix notation to indicate how to proceed for higher dimensions; the details for the general case are carefully
23
laid out in [116]). Then $ acts locally on F through
i%,
Y) := (+(.)I
&, given by
$’WY).
The point is that, while it4 need not carry any r-invariant measure, the topdegree differential form v = y-2 dy A d x on F is I?-invariant:
-
$*v = Y-~$’(x)-~$’(x) dy A $’(x)dx = V , so we can build a Hilbert space L 2 ( F , v ) and represent the action of each $ E by the unitary operator U, defined by V&(x,y) := t(&-’(x7y)). It is slightly more convenient to work with the adjoint unitary operators U&$(X,y) := y)). These unitaries intertwine multiplication operators coming from functions on F (specifically, smooth functions with compact support) as follows:
c($(~, U, f U$
where
= f,,
f+(x,y) := f(&-’(x7y)).
(1.17)
The local action of I? on F can be described in the language of smooth groupoids [39], or alternatively by introducing a “crossed product” algebra which incorporates the groupoid convolution. This is a pre-C*-algebra A obtained by suitably completing the algebra span{fU$ : $ E
I?, f
E CF(Dom&)}.
The relation (1.17) gives the multiplication rule
(fu;)(9u;)
=
r(u$gu+)u;q
= f(gO
m;+,
(1.18)
Any two such elements are composable, since the support of f ( g 0 &) is a compact subset of Dom n &-l(Dom C Dom($G). This construction is called the smash product in the Hopf algebra books: if H is a Hopf algebra and A is a left Hopf H-module algebra, the smash product is the algebra A # H which is defined as the vector space A @ H with the product rule
&
F)
( a @ h)(b@ k) := C a ( h : l. b) 8 h:2k. If h is a grouplike element of HIthis reduces to ( a 8 h)(b@ k) := a ( h . b) @ h k , of which (1.18) is an instance. A local basis { X , Y } of vector fields on the bundle F is defined by the “vertical” vector field Y := y d / d y , generating translations along the fibres, and the “horizontal” vector field X := y d/dx,generating displacements transverse to the fibres. In higher dimensions, the basis contains n2vertical vector
24
fields yj" and n horizontal vector fields x Y is invariant:
k
[116]. Under the lifted action of
a
.-..
r,
a aY
$*Y = $'(x)y 7 = y - = Y,
a$ (X)Y
but X is not. To see that, consider the 1-forms a := y-' dx and w := y-' dy. The form a is the so-called canonical 1-form on F , which is invariant since $*a = y-'$'(z)-' d$(x) = y-' dx = a , whereas w is not invariant:
This transformation rule shows that w is a connection 1-form on the principal bundle F + M ; and the horizontality of X means, precisely, that w ( X ) = 0. Notice also that a ( X ) = 1. Now the vector field 6;'X can be computed from the two equations a($;'X) = @ a ( q ; ' X ) = a ( X ) = 1 and @w($;lX) = w ( X ) = 0; we get
( 1.19a) where
( 1.19b) Any vector field Z on F determines a linear operator on A, also denoted by
z,by wu;):= (Zf)u;,
which makes sense since supp(Zf) products, this operator gives
Z(f [I; gu;,
= Z(f (9 O
supp f
(1.20)
c Dom?.
When applied t o
4w;, w;* + m;* = (Zf)(g O
fZ(g O
= (2f)U;gu;
+ f u ; ( Z ( g o i z ) oG-')lJ;
= ( Z f , U $ gu;
+ f u; ii*Z(g)Ui.
(1.21)
Since the vector field Y is invariant, &Y = Y, so the lifted operator Y is a derivation on the algebra A:
w u ;gu;,
= (Yf)U$ d$
+ f u; (Wq.
25
Proposition 1.1. The operator X on A is not a derivation; however, there is a derivation X1 on A such that X obeys the generalized Leibniz rule
X ( a b ) = X ( a ) b+ a X ( b ) + A l ( a ) Y ( b ) for all a , b E A.
(1.22)
Proof. Using the invariance of Y and (l.l9a), we get
-
+*X - X = $*(X - $ c l X )
= $*(h+Y) =
(h,
o
$-')Y,
and it follows that
fU; ($*X(g) - Xg)U;
= fU; (h,
0
$-')(Yg)Ui = f h@u; (Yg)U;.
If we define
Xdfu;) := h,f u;,
(1.23)
then (1.21) for 2 = X now reads
x w ; gu;, = X ( f u;)gu; + f u; X W ; ) + Xl(fu;)y w ; ) . Thus, (1.22) holds on generators. We leave the reader t o check that the formula extends to finite products of generators, provided that X 1 is indeed a derivation. Now (1.19b) implies
a
h@,(",Y) = Y ~ ( l o g 4 % w ) + l o g + W ) = h@($(X,Y))+ h,(zlY),
so that h@+= P h b
+ h,,
and the derivation property of X I follows:
Xl(fu;gu;)= (@h@+ h+)f ( s
0
= f ( ( h 4 g )O $)$,
w;,
+ h,f u; &
= (fu;)(h@gu;) + (h+fU;)w;).
0
Consider now the Lie algebra obtained from the operators X , Y and X1. The vector fields X , Y have the commutator [ y a / a y l y a / a x ] = y a / d x and the corresponding operators on A satisfy [Y,XI = X . Next, [Y,Xl](fU;) =
f ( Y h + ) U ; , and from Y h , = h, we get [Y,XI] = XI. Similarly, [ X ,X I ] ( f U;) = f(Xh,)U$, where X h , = y d / d z ( y + " ( z ) / + ' ( x ) ) = y 2 d2/as2(log+'(z)). Introduce
hj
= yn
d" dx"
-log +'(z),
for n = 1 , 2 , .. . , and define Xn(fU;) := f h j U ; , then A 2 = [ X ,X I ] and by induction we obtain X n + l = [X,X,] for all n. Clearly Yh; = nh;, which
26
implies [Y,A,] = nA,. The operators A, commute among themselves. We have constructed a Lie algebra, linearly generated by X , Y , and all the A., We can make the associative algebra with these same generators into a Hopf algebra [36] by defining their coproducts as follows. Since Y and A1 act as derivations, they must be primitive: (1.24a) (1.24b) The coproduct of X can be read off from (1.22):
A X := X 8 1 + 1 8 X
+ A1 B Y .
(1.24~)
Moreover, E ( Y )= &(A,) = 0 since Y and A1 are primitive, and & ( X )= 0 since X = [ Y , X ]is a commutator; moreover, &(A,) = 0 for all n 2 2 for the same reason. The commutation relations yield the remaining coproducts; for instance,
A A z : = [AX,AA,]=A281+18Ax,+xi8A,. The antipode is likewise determined: S ( Y ) = -Y and S(A1) = -A1 since Y and A1 are primitive, and ( L * S ) ( X ) = &(X)1= 0 gives X S ( X ) A1Y = 0, so S ( X ) = - X +X1Y. The relation S(A,+1) = [S(A,), S ( X ) ]yields all S(A,) by induction. Definition 1.10. The Hopf algebra HCM generated as an algebra by X , Y and XI, with the coproduct determined by (1.24) and the indicated counit and antipode, will be called the Connes-Moscovici Hopf algebra. Exercise 1.4. Show that the commutative subalgebra generated by {A, : n = 1 , 2 , 3 , .. . } is indeed a Hopf subalgebra which is not cocommutative. 0 The example HCM arose in connection with a local index formula computation, which is already very involved when the base space M has dimension 1 (the case treated above). In higher dimensions, one may start [116] with the vertical vector fields yj” = yj”a/ayf and a matrix-valued connection 1-form w j = (y-’)k(dyj” +I’&yT dxp),which may be chosen torsion-free, with Christoffel symbols I’& = With respect t o this connection form, there are horizontal vector fields X I , = y:(a/axp - I‘;,yj” !lay$). One obtains the Lie algebra relations [x’,YL]= Siyz” - SiYi and [ x J , X ~ = ]S i X i , involving “structure constants”; however, [ X I , X , l ] = RiklY’ where Rjkl are the components of the curvature of the connection w , and these coefficients are in general not constant, for n > 1. At first, Connes and Moscovici decided to use flat connections only [36], which entails [ X I , X , , ] = 0; then, on lifting the and the X I , using (1.20), a
+
+
27
higher-dimensional analogue of HCM is obtained. For instance, one gets [116]:
AXk
=Xk @
1
+ 18.k
+ X i j @y’,
where the X i j are derivations of the form (1.23). A better solution was later found [39]: one can allow commutation relations like [Xk,Xl] = R;klY,’ if one modifies the original setup t o allow for “transverse differential operators with nonconstant coefficients”. The algebra A remains the same as before, but the base field CC is replaced by the algebra R = C m ( F ) of smooth functions on F . Now A is an R-bimodule under the commuting left and right actions
a(b): fUL H b . (fUi) := ( b f ) U L ,
P(b) : f UL H (fU,) t .b
:= ( b o
$) . (fUL) = ( f ( b o 4))U;.
(1.25a) (1.25b)
Letting H now denote the algebra of operators on A generated by these operators (1.25) and the previous ones (1.20), then we no longer have a Hopf algebra over CC, but ( H ,R, a, p) gives an instance of a more general structure called a Hopf algebroid over R [71]. For instance, the coproduct is an R-bimodule map from H into H @R H , where elements of this range space satisfy ( h . b) @R k = h @R ( b . k) by construction, for any b E R. Just as Hopf algebras are the noncommutative counterparts of groups, Hopf algebroids are the noncommutative counterparts of groupoids: see [71,117] for instance. For the details of these recent developments, we refer to [39]. 2
2.1
The Hopf Algebras of Connes and Kreimer The Gonnes-Kreimer algebra of rooted trees
A very important Hopf algebra structure is the one found by Kreimer [65] to underlie the combinatorics of subdivergences in the computation of perturbative expansions in quantum field theory. Such calculations involve several layers of complication, and it is no small feat to remove one such layer by organizing them in terms of a certain coproduct: indeed, the corresponding antipode provides a method t o obtain suitable counterterms. Instead of addressing this matter from the physical side, the approach taken here is algebraic, in order first to understand why the Hopf algebras which emerge are in the nature of things. A given Feynman graph represents a multiple integral (say, over momentum space) where the integrand is assembled from a definite collection of Rules, and before renormalization will often be superficially divergent, as determined by power counting. Even if not itself divergent, it may well contain
28
one or several subgraphs which yield divergent partial integrations: the first order of business is to catalogue and organize the various graphs according to this nesting of subdivergences. Kreimer’s coproduct separates out the divergences of subgraphs from those of the overall graph. In consequence, when expressed in terms of suitable generators of a Hopf algebra, the coproduct turns out to be polynomial in its first tensor factor, but merely linear in the second factor, and is therefore highly noncocommutative. Our starting point is to find a source of Hopf algebras with this kind of noncocommutativity. F We start with an apparently unrelated digression into the homological classification of (associative) algebras. There is a natural homology theory for associative algebras, linked with the name of Hochschild. Given an algebra A over any field IF of scalars, one forms a complex by setting C,(d) := and defining the boundary operator b: C,(A) 4 Cn-l(A)by n- 1
b(a0 @ a1 @ . . . @a,) := C ( - l ) j a o
@ I f .@ .
. a,
ajaj+l@.. @
j=O
+(-l)na,ao~al@...@aa,-l, where the last term “turns the corner”. By convention, b = 0 on Co(A)= A. One checks that b2 = 0 by cancellation. For instance, b(a0 @ a1) := [ao,al], while
b(ao @ a1 @ a2) := aOal@ a2 - uo
0102
+ a2ao
al.
There are two important variants of this definition. One comes from the presence of a “degenerate subcomplex” D.(A) where, for each n = 0,1,2,. . . , the elements of &(A) are finite sums of terms of the form a0 @ . . . @ a j @ . . . @ an, with aj = 1 for some j = 1 , 2 , .. . , n; elements of the quotient -@n R”A := C,(A)/D,(A) = A @ A , where 3 = A/F, are sums of expressions a0 dal . . . da, where d(ab) = dab+ adb. The direct sum R’A = P A is the universal graded differential algebra generated by A in degree zero; using it, b can be rewritten as n-1
b(ao da1 . . . da,) := aoa1 duz . . . da,
+ C(-l)jao da1.. . d ( a j a j + l ) . . . da, j=1
+ (-l),a,ao
da1.. . da,-1.
(2.1)
The second variant involves replacing the algebra A in degree 0 by any Abimodule €, and taking Cn(A,€) := E @ d@,; in the formulas, the products
29
anaO and aOa1 make sense even when a0 E E . We write its homology as H.(A, E ) and abbreviate HHn(A) := &(A, A). Hochschild cohomology, with values in an A-bimodule E l is defined using cochains in C" = Cn(A,E), the vector space of n-linear maps $: d" 4 E ; this itself becomes an A-bimodule by writing (a' . $ . a")(al, . . . ,a,) := a' . $ ( a l l . . . ,a,) . a". The coboundary map b : C" -+ Cn+' is given by b$(al,.
. . an+^)
:= a1 $ ( a 2 , . . . ,an+l) n
+ Z(-l)j$(a1,. .
' I
ajaj+1,.. . I %+l)
j=1
+ (-l)n+l$(all... , a n ) .
%+l.
(2.2)
The standard case is E = A* as an A-bimodule, where for $ E A* we put (a' a")(c) := $(a"ca'). Here, we identify $ E Cn(A,E ) with the (n 1)linear map cp: A"+' 4 C given by cp(uo,a~,...,u~) := $(a1 ,..., un)(uo); then, from the first summand in ( 2 . 2 ) we get a1 . $ ( a 2 , . . . ,a,+l)(ao) = $ ( a 2 , . . . , a,+l)(aoal) = cp(aoal,. . . ,a,+l), while the last summand gives $ ( a l l . . . , an) . an+l(ao) = $ ( a l l . . . ,an)(an+iao) = cp(a,+lao, . . . , a n ) . In this case, ( 2 . 2 ) reduces to
+
n
bv(ao, . . . ,an+i) :=
C(- I ) ~ P ( ~ O. ,ajaj+l,. .. . . an+1) I
j=O
+ (-l)n+lP(an+lao,.. . , a n ) .
(2.3)
The n-th Hochschild cohomology group is denoted H " ( d , E ) in the general case, and we also write H H n ( A ) := H"(d, A*). Suppose that /I: A F is a character of A. We denot,e by A,, the bimodule obtained by letting A act on itself on the left by the usual multiplication, but on the right through p: --f
a' . c . a''
:= a'cp(a")
for all
a', a", c E A.
In the formula (2.2), the last term on the right hand side must be replaced by (-l)n+lP(all ,an)/I(%+l). ' ' '
F We return now to the Hopf algebra setting, by considering a dual kind of Hochschild cohomology for coalgebras. Actually, we now consider a bialgebra B ; the dual of the coalgebra ( B ,A , E ) is an algebra B*, and the unit map 77 for B transposes t o a character vt of B*. Thus we may define the Hochschild cohomology groups H n ( B * ,B7Tt). An "n-cochain" now means a linear map
30
B -+ B@nwhich transposes to an n-linear map cp = (B*)"+ B* by writing cp(a1,.. . , a n ) := tt(al @ ' . . @ an). Its coboundary is defined by
!:
We compute be using (2.2). First,
Next, if A j : B@'" 4 B@(,+l) is the homomorphism which applies the coproduct on the j t h factor only, then (cp(a1,.. . , a j a j + l , .. . ,a,+l),z) = (a1 @ . . . @ a,+l,Aj(!(z))). Finally, notice that (cp(a1,.. . ,an)qt(an+l),z) = (al@---@a,+l, !(z)@l). Thus the Hoehschild coboundary operator simplifies to n
b!(z) := ( L @ !)A(z)
+ E(-l)'Aj(!(z)) + (-l)"+'!(z)
@ 1.
(2.4)
j=l
In particular, a linear form A : B 4 lF is a 0-cochain, and bX = ( L @ X)A - X @ 1 is its coboundary; and a 1-cocycle is a linear map !: B 4 B satisfying
+
A! = !@ 1 (L @ !)A.
(2.5)
The simplest example of a nontrivial 1-cocycle obeying (2.5) comes from integration of polynomials in the algebra B = F[X]; we make F[X] a cocommutative coalgebra by declaring the indeterminate X to be primitive, so that A(X) = X @I 1 1 @ X and E(X) = 0. We immediately get the binomial expansion A ( x ~ >= AX)^ = (:) x k - j @ xj. If x is any linear form on F[X],then
+
c,"=,
bX(Xk)= ( L @ X)A(Xk)- X(Xk)@ 1 =
' j=1
(k .) x(X"j) 3
Xj,
so bX is a linear transformation of polynomials which does not raise the degree. Therefore, the integration map !(Xk) := Xkf1/(k+ 1) is not a 1-coboundary,
31
but it is a 1-cocycle:
')
k+l
A(C(Xk))= k + 1 j=o (k
C
Xk+l --
Xk+l-j 8 xj
k+l
k+l
j=t k
= - t ( X k )8 1
+ ( L 8 C)(A(Xk)).
This simple example already shows what the "Hochschild equation" (2.5) is good for: it allows a recursive definition of the coproduct A, with the assistance of a degreeraising operation C. Indeed, F[X] is a simple example of a connected, graded bialgebra. Definition 2.1. A bialgebra H = @,"==, H ( " ) is a graded bialgebra if it is graded both as an algebra and as a coalgebra:
H ( m ) H ( n )C - H(mfn)
and A(H(n))C_ @ H(P) 8 H ( Q ) ,
(2.6)
p+q=n
It is called connected if the degree-zero piece consists of scalars only: H(O) = IF1 = imv. In a connected graded bialgebra, we can write the coproduct with a modified Sweedler notation: if a E H ( n ) ,then
Aa =a 8 1
+ 18 a + C aI1 8 a12,
(2.7)
where the terms dl and a12 all have degrees between 1 and n - 1. Indeed, for the counit equations (1.9a) to be satisfied, A a must contain the terms 8 IT(");the remaining terms have a 8 1 in 8 H(O) and 1 8 a in intermediate bidegrees. On applying E 8 L , we get a = ( E 8 L ) ( A = ~ )~ ( a ) l + a + C &(all)a$, so that & ( a )= 0 when n 2 1: in a connected graded bialgebra, so that H = IF 1 @ ker E . the "augmentation ideal" ker E is @,"=l In fact, a connected graded bialgebra is a Hopf algebra, since the grading allows us t o define the antipode recursively [75, '$81. Indeed, the equation m(S 8 L)A= QE may be solved thus: if a E Idn), we can obtain 0 = &(a)1 = S ( a ) a C S ( a l l )alz, where each term all has degree less than n, just by setting
+ +
S ( a ) := -a - C s ( a l l ) d2.
(2.8)
32
Likewise, m ( 8 ~ T ) A = V E is solved by setting T(l) := 1 and recursively defining T ( a ) := -a - C all. It follows that T = S * L * T = S, so we have indeed constructed a convolution inverse for L . In the same way, if there is a 1-cocycle C which raises the degree, then (2.5) gives a recursive recipe for the coproduct: start with A(1) := 1@ 1 in degree zero (since H is connected, that will suffice), and use
A(C(a)):= [ ( a ) @ 1
+
(L
@ C)A(a)
as often as necessary. The point is that, at each level, coassociativity is maintained: (L
+
8 A ) A ( [ ( a )= ) ( L 8 A ) ( [ ( a8 ) 1 ( L E I C)(Aa)) = [ ( a )@ 1 8 1 ( L @ AC)(Aa)
+ = [ ( a ) 8 18 1 +
(L
@ C)(Aa)8 1
+
+
(L
8 L 8 C ) ( L 8 A)(Aa);
( A @ ~ ) a ( C ( a= ) )( A 8 L ) ( C ( ~ 8 ) 1 ( L @ C)(Aa)) = [ ( a ) 8 181 ( L EI [)(nu)8 1 ( L 8 L
+
+
[ ) ( A@ L ) ( A ~ ) ,
where we have used the trivial relation ( A @ L ) ( L @ a) = ( L @ L 8 [ ) ( A@ L ) . The only remaining issues are (i) whether such a 1-cocycle [ exists; and (ii) whether any c E is a sum of products of elements of the form [(a) with a of degree at most n. Both questions are answered by producing a universal example of a pair ( H ,C) consisting of a connected graded Hopf algebra and a 1-cocycle .! It was pointed out by Connes and Kreimer [31] that their Hopf algebra of rooted trees gives precisely this universal example. (Kreimer had first introduced a Hopf algebra of “parenthesized words” [65],where the nesting of subdivergences was indicated by parentheses, but rooted trees are nicer, and both Hopf algebras are isomorphic by the same universality.) Definition 2.2. A rooted tree is a tree (a finite, connected graph without loops) with oriented edges, in which all the vertices but one have exactly one incoming edge, and the remaining vertex, the root, has only outgoing edges. Here are the rooted trees with a t most four vertices (up to isomorphism). To draw them, we place the root a t the top with a o symbol, and denote the other vertices with 0 symbols: b
tl
t2
t31
t32
t4l
t42
t43
t44.
33
The algebra of rooted trees HR is the commutative algebra generated by symbols T , one for each isomorphism class of rooted trees, plus a unit 1 corresponding to the empty tree. We shall write the product of trees as the juxtaposition of their symbols. There is an obvious grading making H R a graded algebra, by assigning t o each tree T the number of its vertices #T. The counit E : H R 4 F is the linear map defined by ~ ( 1:= ) 1 and 4TlT2 . . . T,) = 0 if TI,.. . , T, are trees; this ensures that H R = F l e k e r E. To get a coproduct satisfying ( 2 . 7 ) , we must give a rule which shows how a tree may be cut into subtrees with complementary sets of vertices. A simple cut c of a tree T is the removal of some of its edges, in such a way that along the path from the root t o any vertex, at most one edge is removed. Here, for instance, are the possible simple cuts of t44:
Among the subtrees of T produced by a simple cut, exactly one, the "trunk" R,(T), contains the root of T. The remaining "pruned" branches also form one or more rooted trees, whose product is denoted by Pc(T).The formula for the coproduct can now be given, on the algebra generators, as
+
CP,(T) BRJT),
AT := T '8 1 1 @ T +
(2.9)
C
where the sum extends over all simple cuts of the tree T ; as well as A1 := 1'81, of course. Here are the coproducts of the trees listed above:
+
At1 = t i '8 1 1 '8 t i , At2=t2'81++1tt,+tt,~~1t1, At31 = t31 '8 1 1 '8 t31 t 2 '8 t l t l '8 t2,
+ + + At32 t32 '8 1 + 1'8 t32 + 2tl '8 t 2 f '8 t l , At41 = t 4 l '8 1 + 1 8 t41 + t31 '8 t ] f- t2 '8 t2 + tl '8 t 31 At42 = t 4 2 '8 1 + 1 '8 t42 + t l '8 t32 + t 2 '8 t 2 + tl '8 t 3 1 f t 2 t l '8 t l + t2 1 '862. At43 = t 4 3 '8 1 + 1 '8 t43 + 3tl '8 t32 + 3t21 '8 t 2 + t; '8 t l , At44 t44 '8 1 + 1 '8 t44 + t32 '8 t l + 2tl '8 t31 + t: '8 t 2 . (2.10) >
In this way, H R becomes a connected graded commutative Hopf algebra; clearly, it is not cocommutative. In order to prove that this A is coassociative, we need only produce the appropriate 1-cocycle L which raises the degree by 1.
34
The linear operator L -also known as B+ [31]- is defined, on each product of trees, by sprouting a new common root. Definition 2.3. Let L : H R -+ H R be the linear map given by L ( l ) := tl and
L(T1 . . .Tk) := TI
(2.11)
where T is the rooted tree obtained by conjuring up a new vertex as its root and extending edges from this vertex t o each root of T I , . . , Tk. Notice, in passing, that any tree T with n 1 vertices equals L(T1 . . .Tk), where T I , . . ,Tk are the rooted trees, with n vertices in all, formed by removing every edge outgoing from the root of T . For instance,
+
Checking the Hochschild equation (2.5) is a matter of bookkeeping: see 131, p. 2291 or [53, p. 6031, for instance. Here, an illustration will suffice:
= ~ € 9 1 + 1 € 9 ~ + ~+ 2€0 9 € 4 10 + o o €9f
=L(A)@l+(L@L)
(
1€9A+a\.€91+20
By + o o €90
)
=L ( ~ , , ) @ ~ + ( L B L ) A ( , ~ J .
Finally, suppose that a pair (H,C) is given; we want t o define a Hopf algebra morphism p : H R .+ H such that
p ( L ( a ) )= W a ) ) ,
(2.12)
+
where a is a product of trees. Since L ( a ) may be any tree of degree #a 1, we may regard this as a recursive definition (on generators) of an algebra homomorphism, starting from p(1) := 1 ~The . only thing to check is that it also yields a coalgebra homomorphism, which again reduces to an induction on the degree of a:
A ( p ( L ( a ) ) )= A ( W a ) ) )= % ( a ) ) €9 1 + ( L €9 " ( d a ) ) = % ( a ) ) €9 1 + ( L €9 e) ( P €9 P ) (Aa) = d L ( a ) )€4 1 + ( P €9 P ) ( L €9 L ) ( A a ) = (P c 3 p ) ( L ( a ) €9 1
+ ( L €9 L ) ( W )= ( P €9 p ) A ( L ( a ) ) ,
35
where in the third line, by using t(p(a;,)) = p(L(a;,)), we have implicitly relied on the property (2.7) that the nontrivial components of the coproduct Aa have lower degree than a. F Since the Hopf algebra H R is commutative, we may look for a cocommutative Hopf algebra in duality with it. Now, there is a structure theorem for connected graded cocommutative Hopf algebras, arising from contributions of Hopf, Samelson, Leray, Borel, Cartier, Milnor, Moore and Quillen,' commonly known as the Milnor-Moore theorem, which states that such a Hopf algebra H is necessarily isomorphic to U ( g ) ,with g being the Lie algebra of primitive elements of H . (Notice that U ( g ) inherits an algebra grading from the obvious grading of the tensor algebra 7 ( g ) ; (1.7) entails that it is also a coalgebra grading; and moreover, U ( g ) is connected and cocommutative.) This dual Hopf algebra is constructed as follows. Each rooted tree T gives not only an algebra generator for HR, but also a derivation 2,: HR + F defined by
(ZT, TI . . . T , ) := 0 unless Ic (Z,,T) := 1.
=
1 and T I = T ;
Also, ( Z T ,1) = 0 since 2, E DerE(H) (Definition 1.6). Notice that the ideal generated by products of two or more trees is (ker&)2,and any derivation 6 vanishes there, since b(ab) = b(a)&(b) &(a)b(b)= 0 whenever a , b E ker &. Therefore, derivations are determined by their values on the subspace H R (1)
+
spanned by single trees -which equals L(HR), by the way- and reduce t o linear forms on this subspace; thus Der,(H) can be identified with the (algebraic) dual space Ha''*. We denote by 9 the linear subspace spanned by all the 2,. Let us compute the Lie bracket [ Z R ,Zs] := ( Z R@ 2 s - 2s @ ZR)A of two such derivations. Using (2.9) and ( Z R ,1) = (Zs,1) = 0, we get
(ZR @ z S , A T ) = ~
( Z RPc(T)) I (ZS,Rc(T)), C
where ( Z R ,P c ( T ) )= 0 unless Pc(T)= R and (Zs,R c ( T ) )= 0 unless R J T ) = S; in particular, the sum ranges only over simple cuts which remove just one edge of T . Let n ( R , S ; T )be the number of one-edge cuts c of T such that Pc(T)= R and R c ( T )= S ; then
( [ Z R ,Z s ] ,T ) = (ZR@ 2s - 2s @ ZR,A T ) = n ( R ,S; T ) - n ( S ,R;T ) , 2The historical record is murky; this list of contributors is due to P. Cartier.
36
and this expression vanishes altogether except for the finite number of trees T which can be produced either by grafting R on S or by grafting S on R. Evaluation of the derivation [ZR,Zs] on a product T I . .. T k of two or more trees gives zero, since each Tj E ker E. Therefore,
which is a finite sum. In particular, [ZR,Zs] E 4, and so b is a Lie subalgebra of Der,(H). The linear duality of Hg) with b then extends to a duality between the graded Hopf algebras HR and U ( b ) . It is possible to give a more concrete description of the Hopf algebra U ( 9 ) in terms of another Hopf algebra of rooted trees HGL, which is cocommutative rather than commutative. This structure was introduced by Grossman and Larson [54] and is described in [53, 514.21; here we mention only that the multiplicative identity is the tree tl and that the primitive elements are spanned by those trees which have only one edge outgoing from the root. Hoffman [59], correcting an earlier observation by Panaite [81], has shown that 9 is isomorphic to the Lie algebra of these primitive trees -by matching a certain multiple of each ZT to the tree L(T)- so that U ( b ) 11 HGL. In [31],another binary operation among the 2, was introduced by setting Z R * 2 s := n(R,S; T )2,. This is not the convolution ( Z R@ Zs)A, nor is it even associative, although it is obviously true that Z R * Zs - 2 s * ZR = [ZR,Zs]. This nonassociative bilinear operation satisfies the defining property of a pre-Lie algebra [16]:
CT
(ZR
* 2 s ) * ZT
- ZR
* ( 2 s * ZT) = (ZR * ZT)* zs - Z R * (ZT * 2 s ) .
Indeed, both sides of this equation express the formation of new trees by grafting both S and T onto R. The combinatorics of this operation are discussed in [17], and several computations with it are developed in [19] and [62]. b The characters of HR form a group G(HR)(under convolution): see Definition 1.6. This group is infinite-dimensional, and can be thought of as the set of grouplike elements in a suitable completion of the Hopf algebra U = U ( b ) . To see that, recall that U is a graded connected Hopf algebra; denote by e its counit. Then the sets (ker e ) m = bk, for m = 1,2,. . . , form a basis of neighbourhoods of 0 for a vector space topology on U , and the grading properties (2.6) entail that all the Hopf operations are continuous for this topology. (The basic neighbourhoods of 0 in U @ U are the powers of the ideal 1@ ker e ker e @ 1.) We can form the completion of this topological vector space, which is again a Hopf algebra since all the Hopf operations extend by
Ek,,,,
+
6
37
u A
continuity; an element of is a series x k > O Zk with Zk E bk for each k E N, Gnce the partial sums form a Cauchy sequence in U . The closure of b within U is Der,(H). For example, consider the exponential given by ' p ~:= expZT = Cn20(l/n!) 2;; in any evaluation
t_he series has only finitely many nonzero terms. More generally, 'p := exp b E U makes sense for each b E Der,(H); and 'p E G'(HR) since A'p = exp(Ab) = exp(E IZJ b b IZJ E ) = 'p IZJ 'p by continuity of A. In fact, the exponential map is a bijection between Der,(H) and G(HR ),whose inverse is provided by the logarithmic series log(1 - x) := - Ck,l xk/k;for if p is a character, the -
+
A
equation p = exp(10gp) holds in U , and A(l0gp) = A(log(E - ( E - p ) ) = log(&@ E - A(E - p ) ) = log(p €9 p ) = log(&€9 p )
+ log(p @
E)
= E €4 log p
+ log p €9
El
so that logp E Der,(H). See [56, Chap. X] or [57, Chap. XVI] for a careful discussion of the exponential map. In view of this bijection, we can regard the commutative Hopf algebra H R as an algebra of affine coordinates on the group G(HR),in the spirit of Tannaka-Kreh duality [56]. F In any Hopf algebra, whether cocommutative or not, the determination of the primitive elements plays an important part. If in any tree T, the longest path from the root to a leaf contains k edges, then the coproduct AT is a sum of at least k 1 terms. In the applications to renormalization, T represents a possibly divergent integration with k nested subdivergences, while the primitive tree t l corresponds to an integration without subdivergences. A primitive algebraic combination of trees represents a collection of integrations where some of these divergences may cancel. For that reason alone, it would be desirable to describe all the primitive elements of H R and then, as far as possible, to rebuild H R from its primitives. This is a work in progress [12,19,47], which deserves a few comments here. To begin with, since t l is primitive and At2 = t2 @I 1 1 IZt2 J t l €9 t l , the combination p2 := t2 - $tf is also primitive. One can check that p3 := t31 - tlt2 it; is primitive, too. For each k = 1 , 2 , . . . , let tl, denote the "stick" tree with k - 1 edges and k vertices in a vertical progression. (In particular, t3 and t4 are the trees previously referred to as t31 and t41, respectively: see the diagram after
+
+
+
+
38
Definition 2.2.) A simple cut severs t k into two shorter sticks, and so (2.13) O<_r
with t o := 1 by convention. Thus the sticks generate a cocommutative graded Hopf subalgebra Hl of HR. To find the primitives in Hl, we follow the approach of [19]. Consider the formal power series g ( z ) := Ck,,,t k X k whose coefficients are sticks. Then the equation (2.13) can be read assaying that g ( z ) is grouplike in H&i, that is, A g ( z ) = g(z) @ g ( z ) . If we can find a power series p ( z ) = CT,lpTzT, where each p , is homogeneous of degree r in the grading of Hi, such that exp(p(z)) = g(z), the corresponding equation will be Ap(z) = p(z)@ 1 1@ p ( z ) ; on comparing coefficients of each z T ,we see that each p , is primitive. The equation exp(p(z)) = g(z) is solved as
+
by developing the Taylor series of log(1 +z). Since a monomial ty1ty2. . .ty has degree ml 2m2 . . . rm,, the general formula [47, Prop. 9.31 is
+
PT =
c
+ +
m1+2m2+...+rmr=r
ml+...+m,+l
(ml
+ . . . + m,
ml! . . .m,!
- l)!
ty1
.. .t y ,
where the sum ranges over the partitions of the positive integer r. b Nonstick primitives are more difficult to come by, but an algorithm which provides many of them is found in \19], based on formal differential calculus. Indeed, this “differential” approach can be extended, in principle, to deal efficiently with the more elaborate Hopf algebras of Feynman diagrams discussed in the next subsection. For each a E H R , the expression
(2.14) IIa := C S(a:1)da:2 where d denotes an ordinary exterior derivative, may be regarded as a 1-form on G; it is a straightforward generalization of the familiar (matrix-valued) 1form 9-l dg on a group manifold, whose matrix elements are C j ( g - l ) i j dgjk. We can treat such expressions algebraically, as a “first-order differential calculus” on a Hopf algebra, in the sense of Woronowicz [114]. The commutativity of H R shows that these 1-forms have the following derivation property:
C S(a:lb:l)d(~:2b:2)= C S ( ~ : ~ > S ( U db:2 : I+ )U S(b:l)b:zS(a:l) :~ da:2 = &(a) + na&(b).
nab =
39
In particular, II, = 0 for u E (ker E ) ~ so , we need only consider II, for a E Hg’. Each II, can be thought of as a “left-invariant” 1-form, as follows. Exercise 2.1. Let G be a compact Lie group and let R ( G ) be its Hopf algebra of representative functions. If Lt denotes left translation by t E G, then L t f ( z ) = f(t-’z) = Af(t-l,z) = Cf:l(t-l)f:2(2), so that L t f = Ef:l(t-’)f:2 for f E R ( G ) . Let II, be the smooth 1-form on G defined by (2.14); prove that LFII, = II, for all t E G. 0 Each left-invariant 1-form (2.14) satisfies a “Maurer-Cartan equation” :
dn, = -
c
IIar1A & , z .
Indeed, since 0 = d(&(a)1) = C d ( S ( a : l )a:2) = Cd(S(a:1)) a:2 we find that d(S(a))=
+ S ( a : l )da,2,
c d(S(U:l) = c d(S(U:l))4a:2) c d(S(U:l)) c S(a:1) da:2 S ( a : 3 ) ,
a:2 S(a:3)
=
.(.:2))
=-
in analogy with d(g-l)
=
-g-l dg 9-l. Therefore,
dn, = c d ( S ( a : l ) )A da,2 = C S(a:l)da:2 A S(a:3)da:4 = - C IIaZlA IIaZz. -
Suppose now that we are given some element a E Hg’ for which dII, = 0. The bijectivity of the exponential map for G ( H R )suggests that this closed 1form should be exact: II, = db for some b E HR. It is clear from (2.14) that the equation II, = db can hold only if b is primitive. Theorem 2 of [19] uses the Poincark lemma technique t o provide a formula for b, namely,
b := -@-‘(S(a)), where Q is the operator which grades H R by the number of trees in a product: @(TI. . . T k ) := k TI . . . T k . Notice that b = a c, where c E (ker&)2is a sum of higher-degree terms.
+
Exercise 2.2. Show that a = compute that
A+
b=,+),-20-0B+p
A
-2
satisfies dII, = 0, and
f .
Verify directly that b is indeed primitive. 0 It is still not a trivial matter to find linear combinations of trees satisfying dII, = 0, but it clearly is much easier t o verify this property than t o check primitivity directly on a case-by-case basis. b Finally, we comment on the link between H R and the Hopf algebra HCM of differential operators, developed in [31]. This is found by extending H R to
40
a larger (but no longer commutative) Hopf algebra HR. Since H R is graded by the number of vertices per tree, we regard the subspace Hg’ of single trees as an abelian Lie algebra, and introduce an extra generator Y with the commutation rule
[Y,TI := ( # T )T .
+
For each simple cut c of T , it is clear that #Pc(T) #Rc(T)= #T; a glance at ( 2 . 9 ) then shows that A[Y,T ] = ( # T ) AT = [Y @ 1 1 @ Y ,AT]. This forces Y to be primitive:
+
AY:=Y@l++@Y,
(2.15)
in order to get A[Y,TI = [AY,AT] for consistency. Another important operator on 7 - l ~is the so-called natural growth of trees. We define N ( T ) , for each tree T with vertices v 1 , . . . ,v,, by setting N ( T ) := 7’1 T2 . . . T,, where each Tj is obtained from T by adding a leaf to vj. For example,
+ + +
In symbols, we write these relations as N(t1) =z
N
3
(tl)
N2(tl) =
t2,
= N(t31
+
t32)
N(t2) = t31
= t41
+ t32,
+ 3t42 + t43 + t44.
We rename these 61 := t l , 6 2 := N(&),63 := N2(61),6, := N3(h1),and in general 6,+1 := N”(61) for any n. Notice that S,,, is a sum of n! trees. N , defined on the algebra generators, extends uniquely to a derivation N : H R 4 H R . Now, we can add one more generator X with the commutation rule
[ X ,TI := N ( T ) . The Jacobi identity forces [Y,XI = X , as follows:
“Yl XI, TI = “Y,TI,XI + [Y,[ X ,TI1 = (#TI [T,XI + [Y,NT)I = - ( # T ) N ( T ) (#T 1 ) N ( T ) = N ( T )= [X,T].
+
+
What must the coproduct AX be? Proposition 3.6 of [31] -see Proposition 14.6 of [53]- proves that
+
A N ( T )= ( N @ L)AT
(L
@ N)AT
+ [61 @ Y,AT]
also (2.16)
41
for each rooted tree T . The argument is as follows: t o get A N ( T ) ,we grow an extra leaf on T and then cut the resulting trees in every allowable way. If the new edge is not cut, then it belongs either to a pruned branch or t o the trunk which remains after a cut has been made on the original tree T ; this amounts t o ( N @ L)AT ( L @ N ) A T . On the other hand, if the new edge is cut, the new leaf contributes a solitary vertex 61 t o Pc; the new leaf must have been attached t o the trunk Rc(T) a t any one of the latter’s vertices. Since (#Rc)Rc = [Y,Rc],the terms wherein the new leaf is cut amount t o [61 @ Y,A T ] . The equation (2.16) accounts for both possibilities. Then, since A [ X ,T ]= [ A X ,AT] must hold, we get
+
AX = X @ 1
+ 1 @ X + 61 @ Y.
(2.17)
Let f i be ~ the algebra generated by X , Y and HR. We can extend the counit and antipode t o it as follows. Since Y is primitive, we must take E ( Y ):= 0 and S ( Y ) := -Y. Then, on applying ( L @ E ) t o (2.17), E ( X ):= 0 follows; and by applying m ( @ ~ S ) or m(S @ L ) t o it, we also get 0 = X S ( X ) - S l y , which forces S ( X ) := - X 6lY. Now (2.15) and (2.17) reproduce exactly the coproducts (1.24) for the differential operators Y and X of the Hopf algebra H C M . Indeed, since 61, like A 1 E H C M ,is primitive and since 6,+1 = N(6,) = [ X ,S,], the - correspondence X HX , Y Y , A, H 6, maps HCM isomorphically into HR.
+
2.2
+
Hopf algebras of Feynman graphs and renormalization
In this subsection, we shall describe briefly some other Hopf algebras which underlie the structure of a renormalizable quantum field theory. Rather than going into the details of perturbative renormalization, we shall merely indicate how such Hopf algebras are involved. In a given QFT, one is faced with the problem of computing correlations (Green functions) from a perturbative expansion whose terms are labelled by Feynman graphs r, and consist of multiple integrals where the integrand is completely specified by the combinatorial structure of r (its vertices, external and internal lines, and loops) according to a small number of Feynman rules. Typically, one works in momentum space of D dimensions, and a preliminary count of the powers of the momenta in the integrand indicates, in many cases, a superficially divergent integral; even if the graph r itself passes this test, it may contain subgraphs corresponding t o superficially divergent integrals. The main idea of renormalization theory is t o associate a “counterterm” to each superficially divergent subgraph, in order to obtain a finite result by subtraction.
42
The first step in approaching such calculations is t o realize that all superficially divergent subgraphs must be dealt with, in a recursive fashion, before finally assigning a finite value t o the full graph F. Thus, each graph r determines a nesting of divergent subgraphs: this nesting is codified by a rooted tree, where the root represents the full graph, provided that the r does not contain overlapping divergences. (Even if overlapping divergences do occur, one can replace the single rooted tree by a sum over rooted trees after disentangling the overlaps: see [66] for a detailed analysis.) A “leaf” is a divergent subgraph which itself contains no further subdivergences. The combinatorial algebra is worked out in considerable detail in a recent article of Connes and Kreimer [32]: the following remarks can be taken as an incentive for a closer look a t that paper. See also the survey of Kreimer [67] for a detailed discussion of the conceptual framework. The authors of [32] consider 43 theory in D = 6 dimensions; but one could equally well start with 44 theory for D = 4 [50], or QED [14], or any other well-known theory. Definition 2.4. Let stand for any particular QFT. The Hopf algebra Ha is a commutative algebra generated by one-particle irreducible (1PI) graphs: that is, connected graphs with a t least two vertices which cannot be disconnected by removing a single line. The product is given by disjoint union of graphs: r1r2 means I?l & r2. The counit is given by &(I?) := 0 on any generator, with &(@) := l (we assign the empty graph to the identity element). The coproduct A is given, on any 1PI graph F, by
a r : = m i + m r +C 0
y8r/y, ~
7
(2.18) ~
where the sum ranges over all subgraphs which are divergent and proper (in the sense that removing one internal line cannot increase the number of its connected components); y may be either connected or a disjoint union of several connected pieces. The notation Fly denotes the (connected, 1PI) graph obtained from I’ by replacing each component of y by a single vertex. To see that A is coassociative, we may reason as follows. We may replace the right hand side of (2.18) by a single sum over @ C y 5 r, allowing y = @ or y = and setting F/F := 1. We observe that if y 5 yf C I?, then y‘/y can be regarded as a subgraph of Fly; moreover, it is obvious that
(W/(-Yf/4 = F/Y.
(2.19)
The desired relation (A 8 L)(Ar) = ( L 8 A)(Ar> can now be expressed as
43
so coassociativity reduces to proving, for each subgraph y of I?, that rCr'a-
PCY"C~IY
Choose yl so that y C 7' 5 I?; then 0 5 yl/y C Fly. Reciprocally, t o every y" C r / y there corresponds a unique yl such that y C yl C r and y'/y = 7''; the previous equality now follows from the identification (2.19). We have now defined Ha as a bialgebra. To make sure that it is a Hopf algebra, it suffices t o show that it is graded and connected, whereby the antipode comes for free. Several grading operators Y are available, which satisfy the two conditions (2.6):
y(rlr2)= T ( r l )+ Y(r2)
and
T(y)
+ T(r/y) = Y(r)
whenever y is a divergent proper subgraph of r. One such grading is the loop number !(I?) := I ( r )- V ( r ) 1, if I? has I ( r ) internal lines and V ( r ) vertices. If !(I?) = 0, then r would be a tree graph, which is never 1PI; thus ker! consists of scalars only, so Ha is connected. The antipode is now given recursively by (2.8):
+
s(r)= -r +
(2.20)
0575r
As it stands, the Hopf algebra Ha corresponds t o a formal manipulation of graphs. It remains to understand how to match these formulas to expressions for numerical values, whereby the antipode S delivers the counterterms. This is done in two steps. First of all, the Feynman rules for the unrenormalized theory can be thought of as prescribing a linear map
f:Ha+A, into some commutative algebra A, which is multiplicative on disjoint unions: f = f (rl)f (r2).In other words, f is actually a homomorphism of algebras. For instance, d is often an algebra of Laurent series in some (complex) regularization parameter E : in dimensional regularization, after adjustment by a mass unit p so that each f (r)is dimensionless, one computes the corresponding integral in dimension d = D E , for E # 0. We shall also suppose that A is the direct sum of two subalgebras:
+
A=d+@A-. Let T : A 4 A- be the projection on the second subalgebra, with ker T = A+. When A is a Laurent-series algebra, one takes A+ to be the holomorphic subalgebra of Taylor series and A- t o be the subalgebra of polynomials in 1 / ~
44
without constant term; the projection T picks out the pole part, as in a minimal subtraction scheme. Now T is not a homomorphism, but the property that both its kernel and image are subalgebras is reflected in a “multiplicativity constraint” :
+
T ( a b ) T ( a )T(b)= T(T(a)b)
+T(aT(b))
for all a, b E A.
(2.21)
Exercise 2.3. Check (2.21) by examining the four cases a E d+,b E d* separately. 0 The second step is t o invoke the renormalization scheme. It can now be summarized as follows. If I? is 1PI and is primitive (i.e., it has no subdivergences), we set
C(r) := -T(f(I’)), and then R ( r ):= f(r)+ C(r), where C(r) is the counterterm and R ( r )is the desired finite value: in other words, for primitive graphs one simply removes the pole part. Next, we may recursively define Bogoliubov’s ?Z-operation by setting
with the proviso that
C(Y1 . . .YT) := C(Y1).. . C(Y,),
(2.22)
whenever y = 71 . . . 7T is a disjoint union of several components. The final result is obtained by removing the pole part of the previous expression: C(r) := -T(R(F))and R ( r ):= R(r) C(r). In summary,
+
c(r):=
+
c
C(Y>f(r/r)]7
(2.23a)
05rV
R ( r ) := f(r)+
+
C
C(Y>f(F/Y).
(2.23b)
0575r The equation (2.23a) is what is meant by saying that “the antipode delivers the counterterm”: one replaces S in the calculation (2.20) by C t o obtain the right hand side, before projection with T . From the definition of the coproduct in Ha, (2.2313) is a convolution in Hom(H+,d),namely, R = C * f . To show that R is multiplicative, it is enough to verify that the counterterm map C is multiplicative, since the convolution of homomorphisms is a homomorphism because d is commutative. In other words, we must check that (2.22) and (2.23a) are compatible.
45
This is easy t o do by induction on the degree of the grading of HQ.We shall use the modified Sweedler notation of (2.7), t o simplify the calculation. Starting from c ( 1 ) := I d , we define, for a E ker E ,
C ( a ) := -T[f(a)
+
c C(al1)
(2.24)
f(a12)3 7
assuming C(b) to be already defined, and multiplicative, whenever b has smaller degree than a. By comparing the expansions of A(ab) and (Aa)(Ab), we see that
+
+
+
c ( a b ) l l KJ (ab)12 = KJ b b KJ a c a b ! , KJ b12 bll 8 ab12 + allb 8 a12 + all KJ a12b allbll 8 a12b12.
+
Using the multiplicativity constraint (2.21) and the definition C ( a ) := - - ~ ( z ( a ) we ) , get
C(a)C(b)= T @ ( a ) ]T P ( b ) ]= - T P ( a ) X ( b ) + C ( a ) Z ( b )+ R ( a ) C ( b ) ] = -T[f(a)f(b)
+ C(a)f(b) + f(a)C(b)+ CC(4C(bll)f(bl2)
+ c f(a)C(bl,)f(blz) + C(al1)f(al2)C(b)
c C(aI1)f(al2)f(b)+ Cbll )f(alz)C(bl, -T[f(a)f(b) + C(a)f(b)+ C(b)f(a)+ CC(abldf(bl2) + c C(bldf(abl2) + C(.llb)f(~l,) + c C(.l,)f(al,b) + C(.l,bl,)f(.lablz)] +
=
)f(b12)]
= -T[f(ab)
+ CC((41)f((ab)l2)]= C(ab),
where, in the penultimate line, we have used the assumed multiplicativity of C in lower degrees. b The decomposition R = C * f has a further consequence. Assume that the unrenormalized integrals, although divergent a t E = 0, make sense on the circle S in the complex plane where I&( = Id - DI = T O , say. Evaluation at any d = z defines a character xz:A -+ C of the Laurent-series algebra. Composing this character with f : HQ + A gives a loop of characters of HQ:
y(z) := xz o f ,
for any
z E
S.
Likewise, y-(z) := xz oC and y+(z) := xz o R define characters of HQ -here is where we use the multiplicativity of C and R- and R = C * f entails y+( z ) = y- (z)y( z ) ,or equivalently,
y(z) = y-(z)-l y+(z),
for all
z
E S.
(2.25)
The properties of the subalgebras A+ and A- show that y+(z) extends holomorphically t o the disc Iz - DI < T O , while y-(z) extends holomorphically
46
t o the outer region Iz - DI > TO with y-(m) being finite. Since a function holomorphic on both regions must be constant (Liouville's theorem), we can normalize the factorization (2.25) just by setting y-(m) := 1. The renormalization procedure thus corresponds t o replacing the loop { y(z) : z E S } by the finite evaluation y+(D). The decomposition (2.25) of a groupvalued loop is known as the Birkhoff factorization, and arises in the study of linear systems of differential equations
where A ( z ) is a meromorphic n x n matrix-valued function with simple poles. The solution involves constructing a loop around one of these poles zo with values in the Lie group GL(n,C). We refer to [84, Chap. 81 for an instructive discussion of this problem. Any such loop factorizes as follows:
Y ( Z ) = y-(z)-l X(z> Y+(Z)7 where y+ ( z ) is holomorphic for Iz-zo I < T O , y- ( z ) is holomorphic for I Z - Z O ~ > ro with y- (m) = 1, and { X(z) : Iz - ZOI = TO } is a loop with values in the ntorus of diagonal matrices. The loop X provides clutching functions for n line bundles over the Riemann sphere, and these are obstructions t o the solvability of the differential system. However, in our context, the Lie group GL(n,C) is replaced by the topologically trivial group s ( H + ) ,so that the loop X becomes trivial and the decomposition (2.25) goes through as stated, thereby providing a general recipe for computing finite values in renormalizable theories. 3 3.1
Cyclic Cohomology Hochschild and cyclic cohomology of algebras
We have already discussed briefly, in subsection 2.1, the Hochschild cohomology of associative algebras. Recall that a Hochschild n-cochain, for an algebra over the complex field, is a multilinear map p: An+' + C, with the coboundary map given by (2.3). These n-cochains make up an A-bimodule C" = C"(d,A*); the n-cocycles 2" = { cp E Cn : bcp = 0} and the ncoboundaries B" = { b+ : 1c, E C"-' } conspire to form the Hochschild cohomology module H H n ( A ) := Z n / B n . A 0-cocycle 7 is a trace on d,since T ( U o U 1 ) - T ( O 1 U o ) = b T ( U 0 , Ul) = 0. In the commutative case, when A = C m ( M )is an algebra of smooth functions on a manifold M (we take A unital and M compact, as before), there is a theorem of Connes [22], which dualizes an older result in algebraic geometry due to Hochschild, Kostant and Rosenberg [58], to the effect that Hochschild
47
classes for C "(M) correspond exactly to de Rham currents on M . (Currents are the objects which are dual to differential forms, and can be thought of as formal linear combinations of domains for line and surface integrals within M.) The correspondence [cp] H C, is given by skewsymmetrization of 'p in all arguments but the first:
Dually, Hochschild homology classes on C" ( M ) correspond to differential f o m s on M ; that is, H H k ( C " ( M ) ) 2 d k ( M )for k = 0 , 1 , . . . ,dimM. On the de Rham side, the vector spaces il)k(M) of currents of dimension k form a complex, but with zero maps between them, so that each Hochschild class ['p] matches with a single current C , rather than with its homology class. To deal with the homology classes, we must bring in an algebraic expression for the de Rham boundary. This turns out to be a degree-lowering operation on Hochschild cochains: if 11, E C k ,then B11, E Ck-', given by k-1
B q ( a 0 , . . . , a k - l ) := x ( - l ) j ( k - l ) l l , ( l , a j , . . . ,ak-l,ao,. . . , a j - ~ )
(3.1)
j=O
+ ( - l ) ( j - l ) ( k - l ) + ( a j , . . . ,ak-1, ao, . . . ,aj-1, I ) , does the job. Indeed, if C is a k-current and 'pc is the (already skewsymmetric) cochain
'pc(a0,a l , . . . ,a k ) :=
s,
a0 dal A . . . A dak,
then p c ( a 0 , . . . , ak-1,1> = 0 , and therefore k-1
=EL
daoA...Adak-l = k
j=O
s,,
aodalA...Adak-l,
by using Stokes' theorem; thus B p c = k p a c . Up to the normalization factor k = degC, the algebraic operator B delivers the de Rham boundary. Thus, the algebraic picture for de Rham homology involves a cohomology of algebras which uses both b and B. Dually, the Hochschild homology of algebras supports a degree-raising operator, also called B , which is closely related to the de Rham b
48
coboundary (that is, the exterior derivative). Indeed, if we use the version of Hochschild homology where the chains belong t o the universal graded differential algebra R o d , with b given by (2.1), then B : R'd + Rk+'d is simply k
B ( a o d ~ l. . d
~ :=)E ( - l ) " d a j . .
. dak duo.. . daj-1.
(3.2)
j=O
which mimics the operation w H k dw on differential Ic-forms. In the manifold case, the various daj anticommute, but for more general algebras they do not, so the cyclic summation in (3.2) is unavoidable. From the formula, it is obvious that B2 = 0. One checks easily that b B B b = 0 , too. Exercise 3.1. If e E d is an idempotent element, that is, e2 = e , and k is even, check that
+
b(e ( d e ) k )= e (de)"', B ( e ( d e ) k )= ( k
b((de)'") = (2e - 1 ) (de)k-',
+ 1)(de)'"+',
B ( ( d e ) k ) = 0.
If k is odd, show that instead, b ( e ( d e ) k )= b ( ( d e ) k )= 0
and
B ( e ( d e ) k )= B ( ( d e ) k ) = 0.
Moving back t o cohomology, one can check that b2
bB
=
+ B b = 0 hold there, too. This gives rise t o a bicomplex:
bT c 3
6T c 1
bT
bT
0
0, B2 = 0, and
bT
-- B
c 2
bT
B
c 1
B
co
bT
A co
bT CO Folding this up along the diagonals, we get a "total complex" whose coboundary operator is b B , and whose module in degree n is C" @ c"-2 @ c"-4 @ . . . @ C#",
+
49
where #n = 0 or 1 according as n is even or odd. The cohomology of this total complex is, by definition, the cyclic cohomology HC'(A) of the algebra A. (The letters H C stand for "homologie cyclique": on replacing C k by Rk(d) and running all the arrows backwards, we get a dual bicomplex; the homology HC.(d) of its total complex is the cyclic homology of A.) b There is an alternative description of cyclic cohomology, which in some ways is simpler. Let T be the operation of cyclic permutation of the arguments of a Hochschild cochain:
TP(ao,. . . an) := (P(an1a07 . . . l
%-I).
(3.3)
We say that 'p is cyclic if r'p = (-1)"'p -notice that (-1)" is the sign of this cyclic permutation- and denote the subspace of cyclic n-cochains by Cr = Cr(A) (the notation X = ( - 1 ) " ~ is often used). If ZF(A) and B,"(A) are the respective cyclic n-cocycles and cyclic n-coboundaries, an exercise in homological algebra shows that HC" ( A ) N 2; ( A ) /BY (A). Let us compute HC'(A) for a simple example: the algebra A = @, which is the coordinate algebra of a single point. The module C" is one-dimensional, since ' p ( a 0 , .. . ,a,) = a0 . . . a,cp(l, 1 , . . . ,1); it has a basis element 'p" determined by cp"(l,l,. . . ,1) := 1. Clearly, bcp" = ~ ~ ~ ~ ( - l ) ~ c=p0"or+ 'pn+', ' according as n is even or odd. We also find that Bcp" = 0 or 2ncpn-', according as n is even or odd. The total complex is of the form @
O
@
d
@2
5@ 2 A+@3 2!+@3 3.. .
each d j being injective with range of codimension 1; for instance, dz(cp3,'p') = ('p4,7p2,2 ~ ' ) . The alternative approach, using cyclic n-cocycles, argues more simply that ~ ' p "= cpnl so that Z:(@) = CC or 0 according as n is even or odd, while BY(@)= 0 for all n. Either way, H C n ( @ ) = @ if n is even, and H C n ( @ )= 0 if n is odd. This periodicity might seem surprising: the de Rham cohomology of a onepoint space is C in degree zero, and 0 in all higher degrees. Now we may notice that there is an obvious "shifting operation" S on the bicomplex, moving all modules right and up by one step (and pushing the total complex along by two steps); it leaves behind the first column, which is just the Hochschild complex of A. At the level of cohomology, we get a pair of maps
HC"-2(A)
HCn(A)
HHn(A),
which actually splice together into a long exact sequence: I . . . -% HCn(A)2H H n ( A )5 H C n - l ( A ) 5 HCn+'(A)4 . . .
50
whose connecting homomorphism comes from the aforementioned B at the level of cochains. The detailed calculations which back up these plausible statements are long and tedious; they are given in [70, Chap. 21 for cyclic homology, and in [53, §lO.l] is the cohomological setting. The upshot is that, by iterating the periodicity operator S, one can compute two direct limits, which capture the main algebraic invariants of A. Definition 3.1. The periodicity maps S: HC" -+ HCn+2 define two directed systems of abelian groups; their inductive limits
H P o ( d ) := l@HC2'((A),
HP1(d):= l%HC2k+1(d),
are called the even and odd periodic cyclic cohomology groups of the algebra A. In particular, H P o ( C )= C and H P 1 ( C )= 0. In the commutative case d = Cm(M), it turns out that HC'(d) does not quite capture the de Rham homology of M . The exact result -see [25, Thm. 111.2.21 or [53, Thm. 10.51- is
H C k ( C m ( M ) )21 Z,dR(M)CBH,d_R,(M)CBH:d_R4(M) CB...CBH$t(M), where Z,dR(M)is the vector space of closed k-currents on M , H,dR(M)is the r t h de Rham homology group of M , and #k = 0 or 1 according as k is even or odd. However, one may use S to promote the closed k-currents, two degrees at a time, until the full de Rham homology is obtained, since Z,dR(M) = 0 for k > d i m M ; then we get de Rham homology exactly, albeit rolled up into even and odd degrees:
H P o ( C m ( M ) )= H : ! n ( M ) ,
H P 1 ( C " ( M ) ) N H,ddRd(M).
There is also a dual result, which matches a periodic variant of the cyclic homology of C " ( M ) with the even/odd de Rham cohomology of M . F The importance of this algebraic scheme for de Rham co/homology is that it provides many Chern characters, even for highly noncommutative algebras. Generally speaking, Chern characters are tools to compute algebraic invariants from the more formidable K-theory and K-homology of algebras. The idea is t o associate, t o any pair of classes [z] E K.(d) and [D]E K'(d) another pair of classes ch, z E HC.(d) and ch' D E HC'(d), given by explicit and manageable formulas, so that the index pairing ([z],[D]) can be computed from a cyclic co/homology pairing (ch. z, ch' D ) , which is usually more tractable. We look at the K-theory version first, and distinguish the even and odd cases. Suppose first that e = e2 is an idempotent in A, representing a class [el E Ko(A);we define che := chk e E Revend, where the component
xEo
51
chains are
i)
(2k)! - (de)2k E C12kd, -(e k! It follows from Exercise 3.1 that (b+B)(che) = 0. Next, if u E d is invertible, 00 representing a class [u] E K1(d); we define c h u := CkzO Chk++ u E Cloddd, with components chk e := (-1)
k
Chk+; u := (-l)kk! u-l d ~ ( d ( u - ' ) du)'" = k! (u-' d ~ ) ~ " ' E C12'+ld.
+
Again, one checks that ( b B)(chu) = 0. Actually, it is fairly rare that Ktheory classes arise from idempotents or invertibles in the original algebra d; more often, e and u belong t o MT(d),the algebra of r x T matrices with entries in dl for some T = 1 , 2 , 3 , .. . ; so in the definitions we must insert a trace over these matrix elements; the previous equations must be modified to chke := (-1) ch,,;
k
(2k)! tr((e - 1) 2 (de)2k)E R 2 k d , k!
u := k! t r ( u - l d ~ ) ~ " ' E C12k+1d.
(3.4a) (3.4b)
For instance, tr(edede) = x e , j d e j k deki. The pairing of, say, the 2-chain chl e and a 2-cochain cp is given by
b The Chern character from K-homology t o cyclic cohomology is trickier to define. First of all, what is a K-cycle over the algebra A? It turns out that it is just a spectral triple (A,7 f l D ) , of Definition 1.1: an even spectral triple is a KO-cycle, an odd spectral triple is a K1-cycle. The unboundedness of the selfadjoint operator D may cause trouble, but one can always replace D (using the homotopy D H DIDIPt for 0 5 t 5 1) with its sign operator F := D which is a symmetry, that is, a bounded selfadjoint operator such that F2= 1. The compactness of 1DI-l translates t o the condition that [F,u] be compact for each u E A; in the even case, F anticommutes with the grading operator x,just like D does. The triple (A, 'H, F), satisfying these conditions, is called a Fredholm module; it represents the same K-homology class as the spectral triple (A, 7 f l 0 ) . Although F is bounded, it is analytically a much more singular object than D , as a general rule. For instance, if D = ( 2 ~ i ) - ~ d / dis0 the Dirac operator on the unit circle S1,one finds that F is given by a principal-value integral: 1
Fh(a) = P l i h(a
-
0) cot.rrOd0,
52
which is a trigonometric version of the Hilbert transform on
h(x r
L2(R),
- t)
dt.
This can be seen by writing both operators in a Fourier basis for 3-1 = L 2 ( S 1 ) : ~ ( ~ 2 7 r a k= e ) k e27rak8
,
F(e2*ake)= (sign k ) e27rake1
with the convention that sign0 = 1. This analytic intricacy of F must be borne in mind when regarding the formula for the Chern character of its K-homology class, which is given by the cyclic n-cocycle rF(a0,. . . provided n is large enough that the operator in parentheses is trace-class. (The Fredholm module is said t o be “finitely summable” if this is true for a large enough n.) One can always replace n by n 2, because it turns out that Sr; and r;+’ are cohomologous, so that the Chern character is well-defined as a periodic class. Much effort has gone into finding more tractable “local index formulas” for this Chern character, in terms of more easily computable cocycles: see [35] or [5].
+
b An important example of a cyclic 1-cocycle -historically one of the first t o appear in the literature [l,21- is the Schwinger term of a 1 1-dimensional QFT. In that context, there is a fairly straightforward “second quantization” in Fock space: we recall here only a few aspects of the formalism. In “first quantization”, one starts with a real vector space V of solutions of a Diractype equation (i a/& - D)$ = 0, together with a symmetric bilinear form g making it a real Hilbert space. If E+ and E- denote the orthogonal projectors on the subspaces of positive- and negative-frequency solutions, respectively, the sign operator is F := E+ - E-; moreover, J := iF = iE+ - iE- is an orthogonal complex structure on V (in other words, J2= -l), which can be used to make V into a complex Hilbert space VJ with the scalar product
+
(uI u ):= ~ g(ulv)
+ i g ( J u ,w).
(In examples representing charged fields, V is already a complex Hilbert space with an “original” complex structure Q = i; the construction of the new Hilbert space with complex structure J is equivalent t o “filling up the Dirac sea”, and Q is the charge, a generator of global gauge transformations.) The fermion Fock space .FJ(V)is simply the exterior algebra over V J ; the scalars in AoV are the multiples of the vacuum vector 10). If { u j } is an
53
orthonormal basis for VJ, there are corresponding creation and annihiliation operators on .FJ(V): U!(Ul A . . . A u k ) :=Ui A U l A . . . A U k ,
Ui(U1
A
'
.. A u k )
k := C ( - l ) ' - l ( U i
1Uj)
j
U1
A
. . . A .^j A . . . A u k .
j=1
+
Any real-linear operator B on V can be written as B = B+ B- where B+ := ; ( B - J B J ) gives a complex-linear operator on VJ because it commutes with J , but B- := + ( B J B J ) is antilinear: J B - = -B-J. A skewsymmetric operator B is quantizable, by a result of Shale and Stinespring [98],if and only if [ J ,B] = 2 J B - is Hilbert-Schmidt operator, and the second-quantization rule is B H b ( B ) ,where b(B) is the following operator on Fock space:
+
The rule complies with normal ordering, because (0 I b(B)10) = 0, i.e., the vacuum expectation value is zero. However, this implies that (3.6) is not quite a representation of the Lie algebra { B = -Bt : B- is Hilbert-Schmidt }. The anomalous commutator, or Schwinger term, is given by
[ f i ( A ) , b ( B-) ]b ( [ A , B ]=) -+Tr[A-,B-]. This is a well-known result: see [52]or (53, Thm. 6.71 for a proof. The trace here is taken on the Hilbert space V J ;notice that, although [ A - , B - ] is a traceclass commutator, its trace need not vanish, because it is the commutator of antilinear operators. The claim is that a ( A ,B ) := Tr[A-, B-] defines a cyclic 1-cocycle on the algebra generated by such A and B . For that, we rewrite it in terms of a trace of operators on the complexified space V @:= V @ i V ; any real-linear operator B on V extends to a @-linear operator on V @in the obvious way: B ( u iv) := B ( u ) i B ( v ) . For instance, F := E+ - E- where E+ and Enow denote complementary orthogonal projectors on V @ .Taking now the trace over V @too, , we find that
-a
+
+
a ( A ,B ) = T r ( F [ FA][F, , B]).
(3.7)
To see that, first notice that F[F,B] = B - FBF = - [ F , B ] F ,and so Tr(F[F,A][F,B]) = Tr([F,B]F[F,A =]-)T r ( F [ F , B ] [ F , A ]The ) . right hand
54
side of (3.7) is unchanged under skewsymmetrization: f T r ( F [ FA] , [F,B ] ) = Tr(A-FB-) = Tr(F[A-,B-I). Thus, in turn, equals
-a
fr
-a Tr(F[A-,B-]) -a Tr(E+[A-,B-]E+) + a T r ( E - [ A - ,B-]E-) =
=
-!j
Tr(E+A-E-B-E+
- E+B-E-A-E+) = a ( A ,B ) .
This is a cyclic cochain, since a ( A ,B ) = -a(B, A ) ; and it is a cocycle because
b a ( A , B , C )= i T r ( F [ F , A B ] [ F , C-] F [ F , A ] [ F , B C+]F [ F , C A ] [ F , B ] )
+
+
Tr(FA[F,B][F,C ] - F [ F ,A][F,B]C F C [ F ,A][F,B] F [ F ,C ] A [ FB , ]) = "r(FA[F,B ] [ FC] , - [F,A][F,B ] F C + FC[F,A][F,B] - [F,C]FA[F,B ] )
- 8 -
= 0.
The Schwinger term is actually just a multiple of the Chern character T;, as specified by (3.5), of the Fredholm module defined by F . The ShaleStinespring condition shows that F [ F ,A][F,B] is trace-class, so that, in this case, the character formula makes sense already for n = 1.
3.2
Cyclic cohomology of Hopf algebras
We now take a closer look at the algebraic operators b and B , in the cohomological setting. They can be built up from simpler constituents. First of all, the coboundary b : Cn-' -+ Cn may be written as b = Cy=o(-l)ibi, where
i = 0,1,. . . ,n - 1,
&p(ao,. . . ,an) := v(a0,.. . ,aia2+1,. . . ,a,),
Snv(ao,. . . ,an) := ~ ( a n a o ,... ? an-1).
We also introduce maps
uj
: Cn+'
-, C", for j
= 0,1,. . . , n, given by
~ j ~ ( a o. .,, a . n ) := c~(a07.. . ,aj,l,aj+~,... ,an),
and recall the "cyclic permuter"
7 : C" -+
C" of (3.3):
. an) := v ( a n ,ao,. . .,an-1).
T ~ ( U O , .
Notice that T"+' = 1 on C". The operator B is built from the uj and T , as " k on ~k C" as follows. The "cyclic skewsymmetrizer" N := ~ ~ = o ( - l ) acts n
~ ~ ( a .o. ,an> . = ~ ( a o ,. . ,an>
+ C ( - l ) n k v ( a n - k + l , . . .,an, ao,. . k=l
The formula (3.1) now reduces t o
B = ( - l ) " N ( ~ o F l+ u") : Cn+'
-+
C".
an-k).
55
The algebraic structure of cyclic cohomology is essentially determined by the relations between the elementary maps &, aj and T . For instance, the associativity of the algebra A is captured by the rule 6i+16i = 6: as maps from C”-l t o Cn+’. Here is the full catalogue of these composition rules: SjSi
= biSj-1
if i
ujai = uiuj+l
Uj6i
=
{
biaj-1 L
&-1aj
if i < j, if i = j or j 1, if i > j 1;
T&
= 1 5 - 1 7 : C”-l
TO^
= a j - 1 ~: Cn+’
T”+’
= L on
< j;
if i < j ;
+
3
+
C” for i = 1 , . . . ,n,
4
C” for j = 1 , . . . ,n,
and and
760 = IS,, 2 T ~= OC T ~ T
,
c”.
(3.8)
The first three rules, not involving I-, arise when working with simplices of different dimensions, where the “face maps” Si identify an (n-1)-simplex with the i t h face of an n-simplex, while the “degeneracy maps” aj reduce an (n+l)simplex t o an n-simplex by collapsing the edge from the j t h t o the ( j 1)st vertex into a point. A set of simplices, one in each dimension, together with maps 61 and uj complying with the above rules, forms the so-called “simplicia1 category” A -see [70], for instance- and any other instance of those rules defines a functor from A t o another category: in other words, A is a universal model for those rules. By bringing in the next three rules involving T also, Connes defined a “cyclic category” A which serves as a universal model for cyclic cohomology [21]. Essentially, one supplements A with the maps which cyclically permute the vertices of each simplex (an ordering of the vertices is given). The point of this exercise is its universality, so that any system of maps complying with (3.8) gives a bona-fide cyclic cohomology theory, complete with periodicity properties and so on. Indeed, one can show [53, Lemma 10.41 that if y : C”-l 4 C” is defined by y := C;=l(-l)kklck, then S := (n2 n)-’by defines the periodicity operator on cyclic ( n - 1)-cocycles.
+
+
b Important cyclic cocycles, such as the characteristic classes for the algebras which typically arise in noncommutative geometry, can be quite difficult to compute. This is especially true for crossed product algebras, such as those of subsection 1.3. It is time to discuss how this problem may be addressed by transfer from cyclic cocycles of an associated Hopf algebra which acts on the algebra in question.
56
We recall from subsection 1.3 that such a crossed product algebra A, obtained from the action of local diffeomorphisms on the frame bundle over a manifold, carries an action of a certain Hopf algebra H of differential (and multiplication) operators, where the Hopf action itself codifies the generalized Leibniz rules for these operators. To define characteristic classes in HC'(d), we introduce a new cyclic cohomology for H and then show how to map H-classes to A-classes. This cyclic cohomology for H was introduced in [36] and developed further in [37-391 and also in [40,41]. Its definition will make full use of the Hopf algebra structure, so we proceed in a "categorical" fashion. We shall first assume that the antipode S is involutive, that is, S2 = L H . As indicated earlier, this holds true for commutative or cocommutative Hopf algebras, although not for the Hopf algebra HCMof subsection 1.3; but that case can be handled by making a suitable adjustment later on. To set up the cyclic cohomology of H, we start with the algebras Cn(H):= H@"for n = 1 , 2 , 3 , .. . and Co(H) := C (or IF, if one prefers other kinds of scalars). This looks superficially like the chain complex for associative algebras, but we shall make it a cochain complex by (once again) taking advantage of duality to replace products by coproducts, and so on. The "simplicial" operations are defined by
bo(h1 8 . .. @ V-1) := 1 @ hl
@.
.
'
@v-1,
6i(h1 @ . . . @ h"-') := h1 @ . . . @ . ( h i )
@ - . - @ hn-l,
. @ h"-' @ 1 , ) 8 . .. @ hj @ hj+' uj(hl @ . . . @ h"+') ._ .- ~ ( h j + lh1 6,(h'
@ . . . @ h-1)
i = 1 , . . . ,n - 1,
:= h'
8 . . . @ hn+l.
(3.9) For n = 0, these reduce to &(l) := 1, &(1) := 1, and uo(h) := ~ ( h ) . The relation &+1& = d: of (3.8) expresses the coassociativity of A and the equation A(1) = 1 @ 1; the relations u j S j = u j b j + l = L are equivalent to ( L @ E ) A= ( E @ L ) A= L ; and the remaining relations involving the 6i and the uj only are trivial. To define the cyclic permuter T , we first note that is itself an Hmodule algebra under the "diagonal" action of H :
h . (k'
@.. . @
k") := (A"-lh)(k'
@ . . . @ k") =
C h:lk'
h,2k28 s . . @ h:,k".
We then define
~ ( h8l . .. @ h") := S(hl) . (h2@ . .
h" @ 1) = A"-'(S(h'))(h28 . .. @ h" @ 1)
(3.10)
57
The cyclicity property of T is a consequence of the following calculation. Proposition 3.1. The m a p I - : H@" H@" satisfies
~ " + ' ( h ' @ h ~ @ - - . @ h "=) S 2 ( h 1 ) @ S 2 ( h 2 ) @ . . . @ S 2.( h n )(3.11)
Proof. First we compute ~ ~ (€4hh2l @ . . . €4 h"). The diagonal action of ~ ( ~ ( h 1 h2) " ) = S ( h 2 )S2(h:',)gives T2(h' @ h2 €4
. . . @ h")
= CS(h2")S2(hln)S(hln-l)
h3 €4 S(h2"-1) S2(hln+1)S(hln-2) h4
@ .. . @ S(h22)S2(h12,4) S(h11) @ S(h21) S2(h12,-,).
Observe that C S2(h,2)S ( h , l ) = S(C h:l S(h,2)) = S ( E ( ~1)) = ~ ( h1.) A further simplification is C ~ ( h , 2 ) S ~ ( hS ,(3h ), l ) = C S2(h,2)S ( h : l )= ~ ( h1,) so the terms S2(hln+,)S(hln-k-l) telescope from left to right, leaving
T2(h1@h2@...@hn) = xs(h2")h3@s(h2n-,)h4@,.@S(h22)@s(h212,) s2(h1), where the sum runs over the terms in A"-'S(h2). After n - 1 iterations of this process, we obtain
Tn(hl @ h2 €4 . . . €4 h") =
c S(h%)
€4 S(h%-,) S2(h1) 8 . .. @ S(h%)S 2 ( P 2 @ ) S(h;) S2(hn-l),
and, since A"-l(S(l))
=
1 8 .. . €4 1, the final iteration gives (3.11).
0
This shows that the condition S2 = LH is necessary and sufficient to give Tn+l = L on C n ( H ) . We leave the remaining relations in (3.8) to the reader. b However, it turns out that S2 is not the identity in the Hopf algebra H C M . For instance,
S 2 ( X )= S ( - x
+ XlY) = ( X - XlY) + S(Y) S(X1) = x + [Y,A,] = x + X1.
The day is saved by the existence of a character 6 of HCM such that the "twisted antipode" Ss := 776 * S is involutive. Indeed, since X and X1 are commutators, any character satisfies 6 ( X ) = &(XI) = 0, so any character is determined by its value on the other algebra generator, Y. We set 6(Y) := 1. (Recall that E ( Y )= 0.) Now
W h ) := (776 * S ) ( h )= C6(h:1) S(h:z), so the twisted antipode does satisfy S i = L H . Exercise 3.2. Show this by verifying S i ( X ) = X , Si(Y) = Y , and Si(X1) = A1 directly. 0
58
The relation with the coproduct is given by
A(Sd(h)) = C S(h:2) 8 Sb(h:l), A2(Sg(h)) = C S ( h : 3 ) 8 S(h:2)@ S6(h:1)7 and more generally, A"-l(Sg(h)) = C S(h:,) 8 ... @ S(h:2) 8 S ~ ( h , l ) .It is also worth noting that
C S&(h:i)h:2 = C 6 ( h : i ) S(h:2)h:3 = C b ( h : l ) ~ ( h 31) = 6(h)1. The crossed product algebra A on which HCM acts carries a distinguished faithful trace, given by integration over the frame bundle F with the r-invariant volume form u :
' p ( f U L ) := 0 if II,#
L,
It follows from (1.18) that, for a = fU$ and b
/
fdv.
(3.12)
= g U + , the
equality 'p(ub)=
'p(f) :=
F
cp(ba) reduces to J f(g o $) du = JF(f o $-')gdu, so that the I?-invariance F of v yields the tracial property of 'p. If f E C F ( F ) , it is easily checked that J F ( X f ) d u = 0 and that J F ( Y fdu ) =J , f dv, using integration by parts. Moreover, since XI(!) := h,f from (1.23) and h, = 0, we also get J,(Xlf) du = 0. These identities are enough to confirm that
cp(h.a ) = 6(h)'p(a), for all h E H C M , a E A. It is standard to call a functional p on A "invariant" under a Hopf action if ) Since the character 6 takes the place the relation p ( h . a ) = ~ ( h ) p ( aholds. of the counit here, we may say that the trace cp is a 6-invariant functional. This d-invariance may be reformulated as a rule for integration by parts, as pointed out in [38]: (3.13) ' p ( ( h .a ) b) = 'p(a ( S d h ) . b)). Indeed, one only needs to observe that
c
cp((h.a ) b) = C p((h:l . a ) ~ ( h : 2b)) = P ( ( ~ :. Ia ) (b2 S ( k 3 ) . b ) ) = C 'p(h:i . ( U ( S ( b 2 ). b ) ) ) = S(h:i) ( ~ ((S(h:2) a .b)) = 'p(a
c
. b)).
The cyclic permuter r must be redefined to take account of the twisted antipode Sg, as follows: r(h1 8 . .. 8
h") := Sg(h1) . (h2 8 . .. 8 h" 8 1) = A"-l(Sb(h'))(h28...8h"81) =
c s(h1"p2 8 ~ ( h l " _ , ) h8~ .. .8 s(hl2)h"8 sg(hl1).
59
A straightforward modification of the proof of Proposition 3.1 yields the following identity [40,Prop. 4.41: ~ " + ' ( h ' @ h ~ @ . - . @=h S,2(h')@S62(h2)@...@S62(hn) ") . Thus, ,962 = L H entails
T"+'
= L on C n ( H ) .
b The cyclic cohomology HC,'(H) is now easily defined. The maps b : C"-l(H) + C n ( H ) and B : C"+'(H) + C n ( H ) are given by the very same formulae as before: n
b := E(-l)zdi, B := (-l)"N(aor-'
+ on),
i=O
where N := C i = o ( - l ) " k ~ kon Cn(H). Exercise 3.3. Show that h E H is a cyclic 1-cocycle if and only if h is 0 primitive and 6(h)= 0. It remains to show how HC,'(H) and HC'(d) are related; the trace 'p provides the link. For each n = 0,1,2, ..., we define a linear map yv: C n ( H ) + C"(d,d*)by setting y,(l) := 'p and yv(hl @ . . . @ h") : (ao,. . . , a n ) H 'p(u0 (hl . a1) . . (h" . a,)). f
Following [38], we call yv the characteristic map associated t o 'p. It is easy to check that y, intertwines the maps &, aj and r defined on the two cochain complexes. For instance, if i = 1 , 2 , . . . , n - 1, then
yvbz(hl @ . . . @ h") : (ao, . . . ,a,+1)
H
yv(hl @ . . . @ A(hi)@ . . . @ h"-') (ao, . . . ,a,+l) = p(u0 (hl . a1). . . (hf1 . U i ) ((h!,. U i + l ) .
. . (h" . a,+1)) = 'p(a0 (hl . a 1 ) . . . (hi.( a i a i + 1 ) ) .. . (h" . a,+1)) = &y,(h' @ . . . @ h") (ao,. . . ,a,+1). To match the cyclic actions, we first recall that r(hl @ h2 @ . . . @ h") is S a ( h l ) . (h2 @ . . .
@
h"
@
1) = CS(h12). (h2@ . . - @ h "@ ) Sb(h11).
Write b := (h2 . al). . . (h" . an-l); the "integration by parts" formula yields yv~(hl@ h2 8 . . . @ h") (ao, . . . , a,) =
=
=
C ~ ( a (S(h12) o . b) &(h11) . a,)
c 'p(h11 . S(h12) . b ) a,) c (h11 . c (h11 . ao) (h' . ao) b) (a0
'p(a,
=
'p(an
E(h12)b) = 'p(%
= p ( a , (hl . ao) ( h 2 . a 1 ) . . . (h" . a,-,))
= ~ 3 ; p ( h ' @ h ~ @ . . . @ h " ) (..., a oa,). ,
ao) (h12 S(h13) . b ) )
60
In retrospect, we can see what lies behind the definition of r on Cn(H): on reading the last calculation backwards, we see that the formula for T is predetermined in order t o fulfil Y,+,T = ryV for any &invariant trace cp. b We conclude with two variations on this algorithm for characteristic classes. The first concerns algebras which support a Hopf action but have no natural 6invariant trace. In the theory of locally compact quantum groups [68], another possibility arises, namely that instead of a trace the algebra supports a linear functional ‘p such that p(ab) = cp(b(o . u ) ) where 0 is a grouplike “modular element” of the Hopf algebra. If cp is also S-invariant for a character S such that 6 ( 0 ) = 1, only two further modifications of the elementary maps (3.9) and (3.10) are needed:
This time, the computation in Proposition 3.1 leads to
Thus, the necessary and sufficient condition for rn+l = L is S;(h) = oha-l for all h. See [37] and [53, s14.71 for the detailed construction of the characteristic map in this “modular” case. The other variant concerns the application t o the original problem of finding characteristic classes for foliations, in the higher-dimensional cases, as discussed at the end of subsection 1.3. What is needed is a cohomology theory which takes account of the Hopf algebroid structure, when the coefficient is R = C m ( F )instead of C. The formula (3.12) continues t o define a F-invariant faithful trace on the algebra A. Now, however, instead of seeking a special character 6, the main role is taken by the integration-by-parts formula (3.13). The twisted antipode in that formula is replaced by a map H + H , subject to four requirements: (a) that it be an algebra antihomomorphism; (b) which is involutive, that is, Z2 = L H ; (c) that it exchange the algebroid actions of (1.25), namely, s p = a; and (d) that m ( s & L ) A= P E ~ Connes . and Moscovici show in [39] that a unique map satisfying these properties exists, and with its help one can again build a cyclic cohomology theory for the Hopf algebroid of transverse differential operators, which provides the needed invariants of A.
s:
s
61
4 4.1
Noncommutative Homogeneous Spaces
Chern characters and noncommutative spheres
A fundamental theme of noncommutative geometry is the determination of geometric quantities from the spectra of certain operators on Hilbert space. An early precursor is Weyl’s theorem on the dimension and volume of a compact Riemannian manifold: these are determined by the growth of the eigenvalues of the Laplacian. For spin manifolds, one can obtain the same data from the asymptotics of the spectra of the Dirac operator 8. This phenomenon forms the background for the study of spectral triples. We know, for instance, complying with that a spectral triple (A,‘H, D ) over the algebra A = Cw(M), the seven requirements listed in subsection 1.1, provides a spin structure and a Riemannian metric on M for which D equals 8 plus a torsion term. A question raised in the paper which introduced these seven conits algebra of smooth ditions [26] is whether the manifold itself -or coordinates- may be extracted from spectral data. The key property here is the orientation or volume-form condition: 7r~g(c)= x, with
c E C,(A) such that
bc = 0,
(4.1)
where n is the classical dimension of the spin geometry. In view of the isomorphism between H H , ( C m ( M ) ) ‘v A ” ( M ) ,there is a unique n-form v matched t o the class [c] of the Hochschild n-cycle. It turns out that (4.1) entails that v is nonvanishing on M , so that, suitably normalized, it is a volume form; in fact, it is the Riemannian volume for the metric associated t o the Dirac-type operator D. To see how this works, recall that the standard volume form on the 2sphere S2 is
v
= z d y A dz
+ y d z A da: + zda: A dy E A2(S2).
(4.2)
The corresponding Hochschild 2-cycle in R2(C” (S2))is c := ; (5 (dy dz
-
dz dy)
+ y (dz da: - d 5 d z ) + z (da: dy
-
dy d 5 ) ),
(4.3)
x.
(4.4)
and (4.1) becomes
;(a:“ D,YI, [D,.I1 + Y “ D ,21, [ D ,41 +
“ D ,51, ID,Yll)
=
The algebra A = C”(S2) is generated by the three commuting coordinates z,y,z, subject to the constraint x 2 y2 z2 = 1. It is important to note that one can vary the metric on S2 while keeping the volume form Y fixed; one usually thinks of the round metric g = dx2 dy2 dz2 which is SO(3)invariant, but one can compose g with any volume-preserving diffeomorphism
+ +
+
+
62
of S2 t o get many another metric g’ whose volume form is also v. Therefore, the D in the equation (4.1) is not uniquely determined; it may be a Dirac operator D = pg,obtained from any such metric g’ (the Hilbert space IH is the vector space of square-integrable spinors on S2). On the other hand, one may think of (4.4) as a (highly nonlinear) equation for the coordinates z, y, z . To see how this comes about, we collect the three coordinates for the 2-sphere into a single orthogonal projector (selfadjoint idempotent), (4.5) This is actually the celebrated in the algebra of 2 x 2 matrices, M2(Co0(S2)). Bott projector, whose class [el E K0(C”(S2)) = K o ( S 2 )is nontrivial. It is easy to check the following identity in exterior algebra: tr((e
-
i)de A de) = $v E d2(S2).
Now, up t o normalization and replacement of the exterior derivative by the differential of the universal graded differential algebra W(Cm(S2)),the left hand side is just the term chl e of the cyclic-homology Chern character of [el. Notice that cho e = tr(evanishes also. The cyclic homology computations preceding (3.4) show that, in full generality,
3)
b(ch1 e)
=
-B(cho e ) ,
so that the vanishing cho e = 0 is enough to guarantee that chl e is a Hochschild cycle: b(ch1 e) = 0. F We now switch t o a different point of view. Suppose we wish t o produce examples of spectral triples (A, IH,D,C,x) satisfying the seven conditions for a noncommutative spin geometry. We first fix the classical dimension, which for convenience we shall suppose to be even: n = 2m. Then we start from the orientation condition:
rD(chrn e) = X ,
(4.6a)
subject to the constraints choe=O,
chle=O,
. . . , chrn-le=O,
(4.6b)
which guarantee that ch, e will be a Hochschild 2m-cycle. Consider (4.6) as a system of equations for an “unknown” projector e E MT(d),r being a suitable matrix size. What does this system tell us about the coordinate algebra A?
63
In Connes' survey paper [28], the answer is given in detail for the case n = 2, T = 2: it turns out that (4.6b) forces A to be commutative, and (4.6a) ensures that its character space is the 2-sphere. We summarize the argument, following our (53, §11.A]. First of all, the selfadjointness e* = e and the equation choe = tr(e = 0 allow us to write e in the form (4.5), where x, y, z are selfadjoint elements of A. The positivity of the projector e implies -1 5 z 5 1 (here we are implicitly assuming that A is a dense subalgebra of a C*-algebra). The idempotence e2 = e boils down to a pair of equations
4)
+ x2 + y2 f i[", y] = 2(1 iz ) , (1 z)(x f iy) + (" f i y ) ( l f z ) = 2(" f iy), which simplify to [x,y] = [y,z] = [z,"] = 0 and '2 + y2 + z2 = 1. (1f z)2
Thus, x , y , z generate a commutative algebra A. Moreover, by regarding them as commuting selfadjoint operators in a faithful representation of A, the equation x2 y2 z2 = 1 tells us that their joint spectrum in R3 is a closed subset V of the sphere S2: the C*-completion of A is C(V). This partial description of A has not yet used the main equation (4.6a), whose role is to confirm that V is all of S2. For convenience, we abbreviate du := [D, u] (at this stage, d is just an unspecified derivation on A). Since
+ +
Y
de = -
dz
2 dx+idy
dx-idy -dz
1
'
a short calculation gives
x = tr((e - ?jde ) de) = $ (z [dy,dz] + y [dz, dx] + z [dx, dy]). This is of the form T D ( C ) = x,where c is just the Hochschild 2-cycle of the formula (4.3). The corresponding volume form on V is precisely (4.2): but this volume is nonvanishing on all of S2, so we conclude that V = 9'. The pre-C*-algebra A, generated by x, y, z , is none other than Cw(S2)! b The odd-dimensional case n = 2m+ 1 uses the odd Chern character (3.4b), and its orientation condition is TD(Ch,++ u)= 1, with constraints chk++u = 0 for k = 0 , 1 , . . . , m - 1. The unitarity condition u*u = uu* = 1 may be assumed. For instance, in dimension three, Connes and Dubois-Violette [30] have shown that, under the sole constraint chl/2 u = tr(u-l du) = 0, all solutions of the equation ~rg(ch3/2u)= 1 form a 3-parameter family of algebras; one of these is the commutative algebra Cw(S3),but the others are noncommutative. b Moving on now to dimension 4, we take e = e* = e2 in M4(A),and look for solutions of (4.6)with 2m = 4. In [28], a commutative solution is again
64
found, by using a “quaternionic” prescription reminiscent of the Connes-Lott approach t o the Standard Model (see [25, VI.51 or [73] for the story of how quaternions enter in that approach). One writes e in 2 x 2 blocks: where
12 =
(iy), (-p* ’) q=
a a* .
(4.7) . , Here again, z is a selfadjoint element of A such that -1 5 z 5 1, and e2 = e yields the equalities qq* = (1 - z2) = q*q and [z12, q] = 0. Since qq* = q*q is diagonal, we find that z, a, a*,p, p* are commuting elements of A, subject t o the constraint aa* pp* = 1- z2: these are coordinate relations for a closed subset of S4. Once more, the equation (4.6a) produces the standard volume form supported on the full sphere, and the conclusion is that A = Cm(S4): the ordinary 4-sphere emerges as a solution t o the cohomological equation (4.6) in dimension four. Now, the particular quaternionic form of q in(4.7) is merely an Ansatz, and Landi soon pointed out that one could equally well try
+
q=
(-:p*
$) ,
with
X E CC.
The consequences are worked out in a recent paper by Connes and Landi [34] -see also [29]. One finds that qq* = (1- z 2 ) = q*q and [ z 12, q] = 0 still hold, but these relations now lead t o
ap = F p a , ff*p = Xpa*, aa* + pp* = 1 - z2 = a*a + AX p*p. pp* = p*p,
(4.8a)
The computation of chl(e), carried out in [42], yields chl(e) = i ( 1 - Xi)(z[dp,dp*]+p* [dz,dP] +P[d,B*,dz]), which vanishes if and only if X is a complex number of modulus one. In particular, this scheme parts company with the ever-popular deformations where X = q would be a real number other than f l . Foremost among these are the well-known PodleS spheres Sic, which were originally constructed [82] as homogeneous spaces of the quantum group SUq(2). Other higher-dimensional q-spheres currently on the market are described in [9,15,42,60,100];the C*-algebra construction of S y by Hong and Szymariski [60], in particular, is quite far-reaching. However, none of these arises from a Hochschild cycle in the manner described above. On the other hand, Aschieri and Bonechi [3] have constructed, with R-matrix techniques, a multiparameter family of quantum spaces which yields the spheres described here as limiting cases; see also [4].
65
By assuming
1x1 = 1, X = eaTie from now on, the relations (4.8a) simplify
to
a*p = r;pa*, aa* + pp* = 1 - 22,
ap = Apa, aa* = a*a, pp* = p*p,
(4.8b)
which determines a noncommutative algebra A, baptized C" ( S t ) by Connes and Landi.
4.2
How Moyal products yield compact quantum groups
To construct a spin geometry over A = C"(Sj), we need a representation of this algebra on a suitable Hilbert space. The key is to notice that the relation ap = e 2 ~ i 6P a of (4.8b), for normal operators a and p (that is, aa* = a*a and pp* = p*p), is closely related t o the definition of the noncommutative torus [20, 871. This is a pre-C*-algebra C"(Tf) with two generators u and u which are unitary: uu* = u*u = 1, vu* = v*u = 1, subject only t o the commutation relation uu = e 2 ~ i e uu.
(4.9)
One can then define "spherical coordinates" ( u ,v,4 , $) for the noncommutative space S : by setting a =: u sin II,cos 4,
p =: u sin $ sin 4,
z =: cos $,
(4.10)
where 4 , $ are ordinary angular coordinates. It is clear that this is equivalent t o (4.8), for X = e2.rrie. There is a canonical action of the ordinary 2-torus T2 on the algebra C"(Tf), obtained from the independent rotations u +-+ e2Ti$l u, u H e2xi$2 w which respect (4.9). By substituting these rotations in (4.10), we also obtain an action of T2 on Cm(Si). In the commutative case B = 0, this becomes an action of the abelian Lie group T2 by rotations on the compact manifold S4, and these rotations are isometries for the round metric on S4. Any smooth function on S4 can be decomposed as a generalized Fourier series f = C, f,, indexed by T = ( T I , 7-2) E Z2, where fT satisfies ( e 2 x i 4 i e 2 7 4 z ) . fT
- e2~i(ridl+rz$~)
-
fr.
Indeed, each f, is of the form uT1vT2h(4, II,),in terms of the coordinates (4.10); all such functions form the spectral subspace E, of C"(S4). The same is true of C"(Sj) when 6 # 0.
66
If g, = us1v"Zk(+,$), then gs E E, and e z x i e r z S lf r g , = u ' l + s ~ v r ~ + Sh ~k lies in E,+,, so we may identify the algebra Cw(S$)with the vector space Cm(S4) of smooth functions on the ordinary 4-sphere, gifted with the new product: fr
* gs --._ e 2 x i 6 r ~s1 f r g s ,
(4.11a)
defined on homogeneous elements f r E E r , g, E E,. Since the Fourier series f = f r converges rapidly in the F'rkchet topology of Cm(S4), one can show that this recipe defines a continuous bilinear operation on that space. A more symmetric-looking operation, which yields an isomorphic algebra, is given by
cr
fr
gs := e
~ W ~ ~ s l - - frgs. r ~ s ~ )
(4.11b)
This deserves t o be called a Moyal product of functions on S4. Indeed, suppressing the coordinates yields exactly the Moyal product on Cm(T2), which has long been recognized t o give the smooth algebras Cw(Ti) of the noncommutative %tori [ I l l ] . The only nonobvious feature of the products (4.11) is their associativity. To check it, we generalize a little. Suppose that M is a compact Riemannian manifold on which an 1-dimensional torus acts by isometries (there is no shortage of examples of that). Then one can decompose C m ( M )into spectral subspaces indexed by Z1. A "twisted" product of two homogeneous functions f r and g, may be defined by
+,+
fr
* gs := P ( T , s) f r g s ,
where the phase factors { p(r,s) E U ( 1 ) : T , s E the additive group Z'. The cocycle relation
(4.12)
Z" make up a 2-cocycle on
d.7 s + t ) P ( S l t ) = P ( T , S ) P ( T + s, t )
(4.13)
ensures that the new product is associative. To define such a cocycle, one could take [108]:
p ( r , s) := exp{ -27ri
C j Crkj O j k s k } ,
where 8 = [ O j k ] is a real 1 x 1 matrix. Complex conjugation of functions remains an involution for the new product provided that the matrix 8 is skewsymmetric. (When 1 = 2, it is customary t o replace the matrix 9 by the real number 812 = -821and, rather sloppily, call this number 8, too; but in higher dimensions one is forced to deal with a matrix of parameters.) The product (4.12) defines a C*-algebra which, when M = T', is isomorphic t o that of the noncommutative torus C(Tk) with parameter matrix 8, as we shall soon see.
67
Moreover, we may define a “Moyal product”: fr
x 9 s := u ( r ,s) f r g s ,
(4.14)
by replacing p by its skewsymmetrized version,
u(r,s) := exp{ -ri
1
(4.15) rjejksk}, which is again a group 2-cocycle; in fact, p and u are cohomologous as group cocycles [88], therefore they define isomorphic C*-algebras. b To see why (4.14) should be called a Moyal product, let us briefly recall the real thing. The quantum product of two functions on the phase space It2” was introduced by Moyal [77] using a series development in powers of A whose first nontrivial term gives the Poisson bracket; later, it was noticed [83] that it could be rewritten in an integral form [51]:
(f X J g)(z) where J
=
(ol k)
:=
//
f (z + s)g(z + t ) e2is.Jt’h ds d t ,
is the skewsymmetric matrix giving the standard sym-
plectic structure on (and the dot is the usual scalar product). This is in fact the Fourier transform of the “twisted conv~lution~~ of phase-space functions which goes back to von Neumann’s work on the Schrodinger representation [80]. For suitable classes of functions and distributions on RZm, it is an oscillatory integral, which yields Moyal’s series development as an asymptotic expansion in powers of A [45,109]. This integral form of the Moyal product is the starting point for a general deformation theory of C*-algebras, which was undertaken by Rieffel [89]. He gave it a mildly improved presentation by rewriting it as
(f x J
g)(z) := //f(z
+ J s ) g ( z + t ) e2nis’tds d t ,
taking A = 2 and rescaling the measure on R2”. He then replaced the functions f,g by elements a, b of any C*-algebra A , and the translations f(z) H f ( z t ) by a strongly continuous action LY of R’ on A by automorphisms; and he replaced the original matrix J by any skewsymmetric real I x 1 matrix, still called J , ending up with
+
(4.16) This formula makes sense, as an oscillatory integral, for elements a , b in the subalgebra A” := { a E A : t H crt(a) is smooth}, which is a Frkchet pre-C*algebra (as a subalgebra of the original C*-algebra A ) .
68
We wish to complete the algebra (A", X J ) to a C*-algebra AJ, which in general is not isomorphic to A (for instance, A may be commutative while the new product is not). The task is to find a new norm (1 . ( I J on A" with the C*-property ]la*x J all J = lla11;; then AJ is just the completion of A" in this norm. Rieffel achieved this by considering the left multiplication operators L;I = L J ( a )given by :=
//
+
a,+~,(a)f (x t )e2*is.tds d t ,
where f is a smooth A-valued function which is rapidly decreasing at infinity. A particular "Schwartz space" of such functions f is identified in [89],on which the obvious A-valued pairing (f I g) := Jwl f (x)*g(z)dx yields a Hilbert-space norm by setting lllf1112 := Il(f 1 f ) l l A . It can then be shown that if a E A", LL is a bounded operator on this Hilbert space; l l a l l ~is defined to be the operator norm of L;. Importantly, L J is a homomorphism:
L J ( a x J b)f (x) =
=
I///
11
CY,+J,(~
x J b)f (z
(a)az+v+Js(b) f (z
+ t )e2nis'tds dt
+ v + t') e2?"(S.t'+u''v)dsdt' du' dv (4.17)
so that L J ( a X J b) = L J ( a ) L J ( b ) .The calculation uses only the change of variable t' := t - v, u' := s u,for which s . t u.v = s . t' u' . v. Rieffel's construction provides a deformation A H AJ of C*-algebras which is explicit only on the smooth subalgebra A". This construction has several useful functorial properties which we now list, referring to the monograph [89] for the proofs.
+
+
+
If A and B are two C*-algebras carrying the respective actions a and p of R', and if 4: A + B is a *-homomorphism intertwining them: +at = ,Bt 4 for all t , then 4(A") C B" and the restriction of 4 to A" extends uniquely to a *-homomorphism $ J : AJ 4 B J . 0
The map 4~ is injective if and only if q5 is injective, and if and only if 4 is surjective.
+J
is surjective
69 0
0
0
When A = B and a = 0,we may take q5 = a, for any s, because asat = a,+t = ata, for all t ; thus a ~s :H a as)^ is an action of R' on A J by automorphisms, whose restriction to A" coincides with the original action a. Deforming ( A J , ~ with J ) another skewsymmetric matrix K gives a C*algebra isomorphic t o A J + K . In particular, if K = - J , the second deformation recovers the original algebra A . The smooth subalgebra (AJ)" of AJ under the action a~ coincides exactly with the original smooth subalgebra A" (although their products are different).
When the action a of R' is periodic, so that at = LA for each t in a subgroup L , then a is effectively an action of the abelian group H = R'/L, and H cx Tk x for some k. Suppose that H is compact, i.e., Ic = 1 and H N T'. Then A" decomposes into spectral subspaces { Ep : p E L } where a s ( a p )= e2?ripsapfor a p E Ep. If b, E E, also, one can check [89, Prop. 2.211 that a p x J b, = e - 2 n i ~ . J q UPb,.
On comparing this with (4.14), we see that if A = C(T') and J := 30, then AJ is none other than the noncommutative 1-torus C(TL). Moreover, if A = C(S4) and 0 is a real number, then the rotation action of T2 on S4 and the parameter matrix
*)
Q:=I( 2 -00 define a deformation such that C ( S 4 ) ~ N C(Si) b We now apply this machinery t o the case of the C'-algebra C(G), where G is a compact connected Lie group. The dense subalgebra R(G)is a Hopf algebra: we may ask how its coalgebra structure is modified by this kind of deformation. The answer is: not a t all! It turns out that, for suitable parameter matrices J , the coproduct remains an algebra homomorphism for the new product X J . This was seen early on by Dubois-Violette [43] in the context of Woronowicz' compact quantum groups: he noticed that the matrix corepresentations of C ( S U , ( N ) ) and similar bialgebras could be seen as different products on the same coalgebra. There are many ways in which a torus can act on G. Indeed, any connected abelian closed subgroup H of G is a torus; by the standard theory of compact Lie groups [13,99], any such H is included in a maximal torus, and
70
all maximal tori are conjugate. Thus H can act on G by left translation, right translation, or conjugation. In what follows, we shall focus on the action of the doubled torus H x H on G, given by
(h,k) . z := h z k - l .
(4.18)
The corresponding action on C(G) is [(h,k).f](z) := f(h-'zk). If 9 is the Lie algebra of H , we may pull this back to a periodic action of the b @ on C(G). For notational convenience, we choose and fix a basis for the vector space b 21 R',which allows to write the exponential mapping as a homomorphism e: R' 4 H whose kernel is the integer lattice Z'. If X := e ( l , l , . . . ,l), we may write As := e(s) for s E R'; and the action of b @ b on C(G) becomes
[ a ( st, ) f ( z ):= f(x-szX".
(4.19)
The coefficient matrix J for the Moyal product (4.16) is now a skewsymmetric matrix in Mzl(lw). It is argued in [92] -see also [107, 541- that compatibility with the coalgebra structure is to be expected only if J splits as the direct sum of two opposing 1 x 1 matrices:
J:=(' 0 -Q O )
(4.20)
where Q E M'(R)is evidently skewsymmetric. Here, we accept this as an Ansatz and explore where it leads. The Moyal product on the group manifold G can now be written as ( f x J g)(z) :=
f(x-Q5sx-Q')g(X-"zX") e2?ri(s.u+t'v) d s d t d u d w . (4.21) 4
We remind ourselves that this makes sense as an oscillatory integral provided f , g E Cw(G), since the smooth subalgebra of C(G) for the action (4.19) certainly includes Cw(G); it could, however, be larger, for instance if the torus H is not maximal. In subsection 1.2, the coproduct, counit and antipode for the Hopf algebra R ( G ) are defined by A f ( X , Y ) := f ( X Y ) ,
a)
:=
fU),
Sf(.)
:= .f(.-l).
(4.22)
These formula make sense in C'(G), which includes R(G) since representative functions are real-analytic, or even in C(G). In accordance with the remarks at the end of subsection 1.2, we shall now discard the algebraic tensor product and work in the smooth category. The coproduct may now be regarded as a homomorphism
A : Cm(G) + Cw(G x G),
71
the counit is a homomorphism E : C”(G) 4 C, and the coalgebra relations ( A @ L ) A= ( L @ A ) Aand ( E @ L )= A ( L @ E )= A L continue to hold. Moreover, the antipode S is an algebra antiautomorphism of C”(G). Let us check that all of those statements continue t o hold when the pointwise product of functions in C”(G) is replaced by a Moyal product. The following calculations are taken from [go]; they all make use of changes of variable similar to that of (4.17). First of all,
(Af
=
=
XJ
1.
Ag>(., Y)
zX-
Q~
lg
~ t YX” ~ t ’
-uZX-U” Y
~
e~ 2 ~ i’s(. u + ~ ’ . v ’ )
6 (t”)b(u”)
(f x J 9 ) ( z Y ) = A ( fxJg)(Xt.,Y).
Integrations like Jbe2?rit”.” d v = b(t”) are a convenient shorthand for the Fourier inversion theorem. Next,
I4
f ( x - Q ~ ’ ) ~ ( x v ‘ )e 2 7 r i ( S ’ . U + t . V ’ )
=
f(A-Qs’)g(XV’)
d sI d t d u d v ’
b(s’) 6(v’) ds’dv’ = f ( l ) g ( l ) ,
so ~ ( XfJ g) = E(f)&(g). Finally, if Q is invertible, then
72
where the skewsymmetry of Q has been used. On the other hand, if Q = 0, then f x J g = fg and the calculation reduces to ( S f x J Sg)(x) = f(x-')g(x-') = S(g X J f ) ( x ) ; since we may integrate separately over the nullspace of Q and its orthogonal complement, the relation Sf X J S g = S(g X J f ) holds in general. Exercise 4.1. Show, by similar calculations, that
whenever f E C"(G). 0 The functoriality of Rieffel's construction then lifts these maps t o the C*-level, without further calculation. That is: the maps A , E and S , defined as above on smooth functions only, extend respectively to a *homomorphism A , : C ( G ) J-+ C ( G ) J @ C ( G )(using J the minimal tensor product of C*-algebras), a character E J : C ( G ) J C, and a *-antiautomorphism -+
S J : C ( G ) j -+ C ( G ) j . However, the Moyal product itself on C"(G) generally need not extend t o a continuous linear map from C ( G ) J @ C ( G )tJo C ( G ) J .This may happen because the product map m is generally not continuous for the minimal tensor product. (There is an interesting category of "Hopf C*-algebrasn, introduced by Vaes and van Daele [104], which does have continuous products, but the link with Moyal deformations remains t o be worked out.) The C*-algebras C ( G ) J, arising from Moyal products whose coefficient matrices are of the form (4.20), are fully deserving of the name compact quantum groups. Indeed, they are thus baptized in [go]. They differ from the compact quantum groups of Woronowicz [115] in that they explicitly define the algebraic operations on smooth subalgebras, and are thus well-adapted t o the needs of noncommutative geometry.
4.3
Isospectral deformations of homogeneous spin geometries
The Connes-Landi spheres S t can now be seen as homogeneous spaces for compact quantum groups. The ordinary 4-sphere is certainly a homogeneous space; in fact, it is -almost by definition- an orbit of the 5-dimensional rotation group: thus, S4 M SO(5)/SO(4). Now, SO(5) is a compact simple Lie group of rank two; that is t o say, its maximal torus is T2. By regarding S4 as the orbit of ( O , O , O , O , 1) in R5, whose isotropy subgroup is SO(4), we see that the maximal torus of SO(4) is also T2.We can exhibit this maximal
73
torus as the group of block-diagonal matrices cos41 sin41 - sin 41 cos $1
cos42 sin& - sin 4 2 cos $2
h=[
)
.
When the 4-sphere is identified as the right-coset space SO(5)/SO(4), and the doubled torus T2 x T2 is made to act on SO(5) by left-right multiplication as in (4.18), then the right action of the second T2 is absorbed in the cosets, but the left action of the first T2 passes to the quotient. This is a grouptheoretical description of how the 2-torus acts by rotations on the 4-sphere. The action is isometric since the left translations preserve the invariant metric on the group, and also preserve the induced S0(5)-invariant metric on the coset space. There is an immediate generalization, proposed in [107], which highlights the nature of this torus action. Consider a tower of subgroups
H 5 K 5 G, where G is a compact connected Lie group, K is a closed subgroup of G, and H is a closed connected abelian subgroup of K , i.e., a torus. The example we have just seen reappears in higher dimensions as
T' 5 SO(21) 5 S 0 ( 2 1 +
l),
with
S"
FZ
S0(21+ l)/S0(21).
Odd-dimensional spheres yield a slightly different case:
T'5 S0(21+
1) 5 SO(21t 2),
with
S2'+l FZ S0(21+ 2)/S0(21+ 1).
This time, H is a maximal torus in K but not in G. Since H 5 K, the left-right action (4.18) of H x H on both G and K induces a left action of H on the quotient space M := G/K, since the right action of H is absorbed in the right K-cosets. If we deform C(G), under the action of H x H , by means of a Moyal product with parameter matrix J = Q @ (-Q), the natural thing to expect is that the C*-algebra C(G/K) undergoes a deformation governed by Q only. We now prove this, following [107]. It helps to recall the discussion of homogeneous spaces at the end of subsection 1.2. We are now in a position to replace the generic function space F(G) used there by either C"(G) or C(G), according to need. In particular, the algebra isomorphism <: Cm(G)K + C"(G/K) given by
74
If(%)
:=
f(z) intertwines the coproduct A on C"(G) with the coaction
p : C " ( M ) + Cm(G)8 C"(G/K) defined by p f ( z , y ) := f(-).
We can distinguish three abelian group actions here. First there is action b on C ( G ) ,already given by (4.19). Next, the formula (&h)(%):= h(A-tz) determines an action ,8 of b on C ( G / K ) . Then there is action y of b on C ( G ) Kwhere (ytf)(z) := f(A-tz); it can be regarded as an action of b @ b where the second factor acts trivially, so that y is just the restriction of a to the subspace C ( G ) Kof C(G). Let f,g E C"(G)K be smooth right K-invariant functions. Then (Y
of b @
=
I,
f(A-Qsz)g(A-uz) eZxis'"d s d u = (f X Q g)(z),
where the Q-product comes from the action y on C ( G ) K . On passing t o C"(G/K) with the isomorphism <, which obviously intertwines the actions y and 0,this calculation shows that
<(f X J 9) = Cf
XQ
19 for all f , g E C D S ( G ) K .
In other words, the J-product on C ( G )induces the Q-product, as claimed. The reason for this bookkeeping with actions and isomorphisms is t o be able t o lift everything t o the C*-level, using Rieffel's functoriality theorems. First, since
-
C ( G / K )2 C ( G ) K-+ C ( G ) restricts t o the smooth subalgebras
-
C w ( G / K )N Cm(G)K
C"(G),
and from there extends to an isomorphism and inclusion
C ( G / K ) QN C(G)$ + C ( G ) j . This shows that the deformed C*-algebra C ( G / K ) Qis an embedded homogeneous space for the compact quantum group C ( G ) J .
75
Example 4.1. To get the noncommutative spheres of Connes and Landi, just take G = SO(21 l), K = SO(21) and let H = T' be the maximal torus for both. Then let Q = i 0 , where 0 is any real skewsymmetric 1 x 1 matrix. The resulting deformation of C(S2') is the C*-algebra C(Si'), and its smooth subalgebra (for the T'-action) is just Cm(S;') := Cm(S2')with the Moyal product x Q . The odd-dimensional spheres S21f1 = S0(21+ 2)/S0(21+ 1) may be deformed in like manner, using, say, the maximal torus T' of S 0 ( 2 1 + 1). However, in this case, since this torus is not maximal in the full group S0(21+ 2), one can regard the T'-rotations as an action of the torus T1+l that is trivial in one direction. The algebras C(Sil+'), with their T1-actions and the corresponding deformations of C(SO(21 2)), have recently been discussed extensively by Connes and Dubois-Violette [30] from the cohomological standpoint. In fact, in [30], the noncommutative spheres are constructed in another way, by directly obtaining generators and relations for the corresponding algebras from twistings of Clifford algebras, as already outlined in [34], before checking that those algebras also come from 8-deformations. The advantage of this procedure is that what is obtained is manifestly spherical, in the sense that the homology-sphere condition (4.6), or its odd-dimensional counterpart, is built-in [44]. The simplest nonspherical examples in even dimensions are CP2 N S U ( 3 ) / U ( 2 ) and the 6-dimensional flag manifold F6 = SU(3)/T2. With G = S U ( 3 ) and H = K = T2 and any irrational 8 = 2Q12, one obtains a family of 6-dimensional quantized flag manifolds.
+
+
b We have outlined a general construction of noncommutative algebras, including all the Connes-Landi spheres, which come equipped with dense preC*-algebras. The final step is to build noncommutative spin geometries based on these algebras. This was done by Connes and Landi for their spheres [34] by means of an isospectral deformation. It was observed in [107], and likewise in [30], that their algorithm extends directly t o any of the aforementioned quantum homogeneous spaces, with only notational changes. The compact homogeneous manifold G I K can be regarded as a Riemannian manifold, since it has a G-invariant metric. We shall assume that G I K also has a homogeneous spin structure (this is not always the case; for instance, CP2 is only spinc, while SU(3)/ S0(3) does not even admit a spinC structure [48, fj2.4]), and we let p be the corresponding Dirac operator; it is a selfadjoint operator on the Hilbert space 7-L of square-integrable spinors. As we shall see, in the end we only need that the metric, and the Dirac operator,
76
be invariant under the action of the torus H rather than the full group G. It is important to remark that the action of H by isometries on G I K does not lift directly to the spinor space 7-1 (or, if one prefers, to the spinor bundle S). Rather, in view of the double covering Spin(n) -+ SO(n) where n- = dimG/K, there is a double covering fi -2H and a homomorphism H 4 Aut(S) which covers the homomorphism H 4 Isom(G/K) [30, $131. This yields a group of unitaries { Vz : E E g } on 7-1 which preserve the subspace r M ( G / KS, ) of smooth spinors and cover the isometries { I , : x E H } of G/K. More precisely: if 4, $J E r m ( G / K , S ) and f E C“(G/K), then
Vz(f$J) = L(f)Vz$J1 and
(vz4J)+vZ.1C) = I&+$J),
where x = ~ ( 2 )Consequently, . the Dirac operator p on 7-1 commutes with each V,. Now choose a basis X I , . . . ,Xl of the Lie algebra IJ, and for j = 1,.. . ,1, let p j be the selfadjoint operator representing X j on ‘H; if exp: ti -+ H and Exp: IJ + fi denote the exponential maps, then ~ ( E x p ( t X j ) = ) exp(tXj12) and p j = -id/ dt t,oV~xp(tXj). Therefore, the spectrum of each operator p j lies either in Z or Z+ !j. For each r E R1, we may define a unitary operator (4.23) by formally replacing half of the arguments of the group cocycle (4.15) with the operators p j ; its inverse is the similarly defined operator o ( r , p ) . These operators commute with each other and also with p , but they do not commute with the representation of C”(G/K) on 7-1 (multiplication of spinors by functions). The unitary conjugations T H a ( p , t ) T a ( t , p define ) an action of R1on the algebra of bounded operators on ‘H, which is periodic on account of the half-integer spectra of the p j , and this action gives a grading of operators into spectral subspaces, indexed by Z1. Therefore, any bounded operator T in the common smooth domain of these transformations has a decomposition T = CTEZi TTlwhere the components satisfy the commutation rules a(plr ) T, = T, o ( p
+ s, r )
for r, s E Z1
For any multiplication operator f obtained from the representation of the algebra Cm(G/K) on spinors, this grading coincides with the previous decomposition f = CTEZi fT. The operator Z1-grading allows us to define a “left twist” of T by L(T) :=
C TT~ ( pr ), . TEZ‘
77
on account of (4.14). Therefore, L yields a representation of C M ( G / K ) ~ := (C"(G/K), X Q ) on 3-1. In other words, the Moyal product gives not only an abstract deformation of the algebra C" ( G I K ) , but more importantly, it yields a deformation of the spinor representation of C" ( G I K ) , without disturbing the underlying Hilbert space. The recipe for creating new spin geometries should now be clear: one deforms the algebra (and its representation), while keeping unchanged all the other terms of the spectral triple: the Hilbert space 3-1 together with its grading x if dim(G/K) is even, the operator p , and the charge conjugation C. This deformation is isospectral [34] in the tautological sense that the spectrum in question is that of the operator 8,which remains the same. It remains t o check that the new spectral triple satisfies the conditions governing a spin geometry. First of all, each [ p , L ( f ) ]for , f E C"(G/K), must be a bounded operator; this is ensured by noting that
[87L(f)I =
frI C ( P , r ) = L ( [ @fI>1 , r
since each [P,f] is bounded. The grading operator x is unaffected by the torus action on G / K since the metric is taken to be H-invariant: this implies L ( x ) = x. In view of the previous equation, the orientation equation 7rp(c) = x survives after application of L to both sides. The reality condition is more interesting. The charge conjugation operator C on spinors [53, Chap. 91 commutes with all u ( p , r ) (again, due to H-invariance of the metric). It follows from (4.23) and the antilinearity of C that Cp,C-' = -pj for each j. We can now define a "right twist"
R ( T ) := CL(T)*C-' =
C a ( ~ , pCT,*C-l ) = C CT,*C-' u ( r , p ) . rEZ'
rEZ'
Now, C intertwines multiplication operators from C"(G/K) with their comfro(r,p), plex conjugates: Cf*C-l = f for f . Therefore, R ( f ) = CrEZl from which one can check that R ( f ) R ( g )= R(f X - Q 9);in other words, R gives an antirepresentation of C"(G/K)Q on 3-1. This commutes with the
78
representation L:
since [[p, f r ] , g s ] = 0 in the commutative case (the commutator [p,f T ] is an operator of order zero which commutes with multiplication operators). Regularity and finiteness are straightforward, since the smooth subalgebra C”(G/K) does not grow or shrink under deformations. Poincark duality also goes through, on account of another theorem of Rieffel, t o the effect that the K-theory of the pre-C*-algebras remains unaffected by deformations [91]. The construction is now complete. We sum up with the following Proposition. Proposition 4.1. Let H 5 K 5 G be a tower of compact connected Lie groups where H is a torus, such that G I K admits a spin structure. Let (C”(G/K), ‘FI, p , C, x) denote any commutative spin geometry over C”(G/K) where ‘FI is the spinor space and p is the Dirac operator for an H-invariant metric o n G I K . Then there is a noncommutative spin geometry obtained from it b y isospectral deformation, whose algebra C-(G/K)Q i s that of any quantum homogeneous space obtained from a Moyal product X Q o n C”(G/K). 0
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54. R. Grossman and R. G . Larson, “Hopf-algebraic structure of families of trees”, J. Algebra 126 (1989), 184-210. 55. A. Grothendieck, Produits Tensoriels Topologiques et Espaces Nucle‘aires, Memoirs of the Amer. Math. SOC.16,Providence, RI,1966. 56. G. P. Hochschild, La Structure des Groupes de Lie, Dunod, Paris, 1968. 57. G. P. Hochschild, Basic Theory of Algebraic Groups and Lie Algebras, Springer, Berlin, 1981. 58. G. Hochschild, B. Kostant and A. Rosenberg, “Differential forms on regular affine algebras”, Trans. Amer. Math. SOC.102 (1962), 383-408. 59. M. E. Hoffman, “Combinatorics of rooted trees and Hopf algebras”, math.c0/0201253, USNA, Annapolis, MD, 2002. 60. J. H. Hong and W. Szymaliski, “Quantum spheres and projective spaces as graph algebras”, Newcastle, NSW, 2001; to appear in Commun. Math. P hys . 61. C. Kassel, Quantum Groups, Springer, Berlin, 1995. 62. D. Kastler, “Connes-Moscovici-Kreimer Hopf algebras”, in Mathematical Physics in Mathematics and Physics: Quantum and Operator Algebraic Aspects, R. Longo, ed., Fields Institute Communications 30,Amer. Math. SOC.,Providence, RI, 2001; math-ph/0104017. 63. A. U. Klimyk and K. Schmudgen, Quantum Groups and their Representations, Springer, Berlin, 1997. 64. A. W. Knapp, Representation Theory of Semisimple Groups, Princeton Univ. Press, Princeton, NJ, 1986. 65. D. Kreimer, “On the Hopf algebra structure of perturbative quantum field theories”, Adv. Theor. Math. Phys. 2 (1998), 303-334; q-alg/9707029. 66. D. Kreimer, “On overlapping divergences”, Commun. Math. Phys. 204 (1999), 669-689; hep-th/9810022. 67. D. Kreimer, “Combinatorics of (perturbative) quantum field theory”, Phys. Reports 363 (2002), 387-424; hepth/0010059. 68. J. Kustermans and S. Vaes, “Locally compact quantum groups”, Ann. Sci. Ec. Norm. Sup. 33 (ZOOO), 837-934. 69. H. B. Lawson and M. L. Michelsohn, Spin Geometry, Princeton Univ. Press, Princeton, NJ, 1989. 70. J.-L. Loday, Cyclic Homology, Springer, Berlin, 1992. 71. J.-H. Lu, “Hopf algebroids and quantum groupoids”, Int. J. Math. 7 (1996), 47-70; q-alg/9505024. 72. S. Majid, Foundations of Quantum Group Theory, Cambridge Univ. Press, Cambridge, 1995. 73. C. P. Martin, J. M. Gracia-Bondia and J. C. VBrilly, “The Standard Model as a noncommutative geometry: the low energy regime”, Phys.
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Reports 294 (1998), 363-406; hep-th/9605001. 74. M. Mertens, “Hopfalgebren-Struktur und Quantisierung” , Diplomarbeit, Universitat Mainz, 1996. 75. J. W. Milnor and J. C. Moore, “On the structure of Hopf algebras”, Ann. Math. 81 (1965), 211-264. 76. S. Montgomery, Hopf Algebras and their Actions o n Rings, CBMS Regional Conference Series in Mathematics 82, American Mathematical Society, Providence, RI, 1993. 77. J. E. Moyal, “Quantum mechanics as a statistical theory”, Proc. Cambridge Philos. SOC. 45 (1949), 99-124. 78. G. J. Murphy, C*-algebras and Operator Theory, Academic Press, San Diego, CA, 1990. 79. S. B. Myers and N. Steenrod, “The group of isometries of a Riemannian manifold”, Ann. Math. 40 (1939), 400-416. 80. J. von Neumann, “Die Eindeutigkeit der Schrodingerschen Operatoren” , Math. Ann. 104 (1931), 570-578. 81. F. Panaite, “Relating the Connes-Kreimer and Grossman-Larson Hopf algebras built on rooted trees”, Lett. Math. Phys. 51 (2000), 211-219; math.qa/0003074. 82. P. Podles, “Quantum spheres”, Lett. Math. Phys. 14 (1987), 193-202. 83. J. C. T . Pool, “Mathematical aspects of the Weyl correspondence”, J. Math. Phys. 7 (1966), 66-76. 84. A. Pressley and G. B. Segal, Loop Groups, Clarendon Press, Oxford, 1986. 85. A. Rennie, “Commutative geometries are spin manifolds”, Rev. Math. Phys. 13 (2001), 409-464; math-ph/9903021. 86. A. Rennie, “PoincarQduality and spinCstructures for complete noncommutative manifolds”, math-ph/0107013, Adelaide, 2001. 87. M. A. Rieffel, “C*-algebras associated with irrational rotations”, Pac. J. Math. 93 (1981), 415-429. 88. M. A. Rieffel, “Projective modules over higher-dimensional noncommutative tori”, Can. J. Math. 40 (1988), 257-338. 89. M. A. Rieffel, Deformation Quantization for Actions of Rd, Memoirs of the Amer. Math. SOC.506,Providence, RI, 1993. 90. M. A. Rieffel, “Compact quantum groups associated with toral subgroups”, in Representation Theory of Groups and Algebras, J. Adams et al, eds., Contemp. Math. 145 (1993), 465-491. 91. M. A. Rieffel, “K-groups of C*-algebras deformed by actions of Rd”, J. h n c t . Anal. 116 (1993), 199-214. 92. M. A. Rieffel, “Noncompact quantum groups associated with abelian sub-
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Geometric and Topological Methods for Quantum Field Theory Eds. A. Cardona, H. Ocampo & S. Paycha @ 2003 World Scientific Publishing, pp. 86-156
THE NONCOMMUTATIVE GEOMETRY OF APERIODIC SOLIDS JEAN BELLISSARD Georgia Institute of Technology, School of Mathematics, Atlanta G A 30332-0160 and Institut Universitaire de France
Introduction These notes correspond to a series of Lectures delivered in July 2001 at the Summer School Geometry, Topology and Quantum Field Theory, that was organized at Villa de Leyva, in Colombia. Their purpose is to provide the students with a summary of a longstanding program with aim to describe aperiodic solids and their property with an appropriate mathematic framework. As explained in Section 1, the lack of periodicity in some solids does not allow to use the famous Bloch theory [5]. For this reason, physicists have developped various tools, such as finite scaling and the ,&function of the renormalization group for disordered systems [1,111],or such as the curved space representation for amorphous materials [72], or the cut-and-project method for quasicrystals. However, most of these technics are specific to sub-families of materials. The program presented here intends to give a general theory valid for all kinds of aperiodic solids. The framework provided by Noncommutative Geometry, as proposed by A. Connes [41] since the late seventies, will be shown to be the right tool replacing Bloch’s theory whenever the translation invariance, that occurs in crystals, is broken. The main difference is that the so-called Brillouin zone becomes a noncommutative manifold, with a non trivial topology. In these notes, only the topological aspects and few of its consequences in physics are investigated. The systematic study of the transverse Geometry as well as the N-body problem [lo51 are not treated here and are left for future developments. The main construction is the notion of Hull of an aperiodic solid. This is the content of Section 1 and Section 2. Several examples are proposed, impurities in a semiconductor, quasicrystals, tilings. It will be shown that the Hull is actually determined by the Gibbs thermodynamical ground state of the set of atoms. This Gibbs state also determines in a unique way various thermodynamical properties such as the diffraction pattern, the electronic density of states or the vibrational density of states (phonon modes). Once
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the Hull is constructed, it leads to the construction of the Noncommutative Brillouin zone (NCBZ) and its Geometry. Then the description of electrons in the one-particle approximation, or of the phonons in the harmonic approximation follows easily. No attempt t o account for the large number of results obtained in the eighties and later concerning the spectral properties for both electrons and phonons will be made here. The reader is invited to look at [33,43,86] concerning spectral results on disordered or quasiperiodic systems or at [18] concerning transport properties with anomalous spectra or diffusion. A special emphasis will be put upon recent results obtained to compute the K-theory of the NCBZ, especially in the context of the so-called gap labelling theorem (see Section 3). This theorem was formulated in the early eighties in its most general form [12] and has been given many illustrations in the case of one-dimensional systems during the eighties [16]. It required however another decade t o get precise results for systems in higher dimensions. The present notes will conclude on a short description of what is still today the most spectacular application of the Noncommutative Geometry t o realistic physics, namely the integer quantum Hall effect [22,41].
Acknowledgments: The author wishes to express his warm thanks to the organizers of the Summer School, that was held in one of the most beautiful place in the world that is Villa de Leyva, for giving him the opportunity t o deliver this series of lectures. He also wishes t o thank the Colombian students of this school for their enthusiasm and their willingness t o learn the best of modern Theoretical Physics. They all showed that in a country plagued by terrorism and corruption for decades, there is hope for individuals through these exceptional men and women who are working with competence and courage a t creating a future of peace and progress. The author wishes also to thanks his collaborators past and present without whom this program could not have been developped.
1
Mathematical Description of Aperiodic Solids
In Solid State physics, most of the theory available in textbooks concerns periodic crystal in which the Bloch theory applies. Since the mid-sixties, however, physicists started wondering about what happens for non periodic materials. In this section we propose a formalism developped in various papers since the early eighties [13,16,19,22-241, that is a substitute t o Bloch theory, whenever Bloch theory fails t o apply.
1.1
Examples of Aperiodic Solids
The structure of solids was investigated from the second half of the 19th century mainly from the point of view of their macroscopic properties. Crystals were then the focus of attention. It was not until the first experiment using X-rays by von Laiie in 1911 [64], that the microscopic structure of solids could be observed. For obvious reasons of symmetry, perfect crystals have been the focus of attention until quite recently. Besides, many of the known materials exhibit a microscopic structure that is perfectly periodic. This is the case for metals, such as copper, iron, aluminium, or for many ionic salts, like sodiumchloride, or most oxydes. It is one of the most challenging questions, even nowadays, to understand why perfect crystals are so common in nature.
Figure 1. Band spectrum for a 2 0 Bloch electron in a uniform magnetic field
Even in a perfect crystal, the electronic motion, when submitted t o a magnetic field, is no longer periodic [89]. The lack of periodicity is due t o the quantum phase created by the magnetic field in the electronic wave functions that breaks the translation symmetry. The problem of computing the band spectrum for a Bloch electron in a magnetic field has been one of the most challenging ones in Solid State Physics. The Peierls substitution [89] permits to reduce the problem t o a tight-binding one, the most celebrated such a
89
model being the Harper one [60). It was only in 1976 that the spectrum of the Harper model could be computed by Hofstadter [63]thanks to the arrival of a new generation of computers (see Fig. 1). Another way to describe this aperiodicity is to see the magnetic field acting as an effective Planck constant that makes the ordinary space noncommutative from the point of view of quantum charged particles [13,15,95]. However, there are many materials that are not crystalline microscopically. The various varieties of glass are probably the most common examples. These materials are still a challenge for theoreticians. Less common, but easier to study, is the silicon that exhibits a crystalline phase with diamond lattice, and also an amorphous phase that is more like a glass than a crystal. A rather successful description of amorphous semiconductors was proposed by Sadoc et al. [72] in the early eighties, based upon saying that such a structure looks like a perfect crystal in a curved space, only flattened to accomodate the flat 3 0 space. A serious need for investigating non crystalline solid came from semiconductors due to their importance for the electronic industry. For indeed, when pure, a semiconductor is an insulator due to a large gap at the Fermi level (1041. However, impurities, either created by artificially doping or spontaneously present in nature, provide enough electrons in the conduction band (or holes in the valence band) to make it a conductor as long as the temperature is large enough. At low temperature, however, the charge carriers are trapped in the impurity band and they see only the sublattice of impurities. Hence, from the point of view of charge carriers, the semiconductor looks like a random lattice, completely disordered. This problem became the focus of attention in the late seventies when the technology was ready to produce mesoscopic devices. Several phenomena like the Anderson localization, the enhancement of the magnetoresistance, the universality of quantum fluctuations, the quantum Hall effect, became the basic elements of such physics. One side consequence of the semiconductor technology of this time was the possibility of creating artificial structures, like superlattices, that may mimick situation proposed by theoreticians. For example, 1D chains with potentials varying according to a prescribed rule, such as the Fibonacci or the ThueMorse sequences can be created in this way. In 1984, a new class of materials, called quasicrystals was discovered [102]. The first sample was an alloy of aluminium and manganese. The surprise was that the diffraction spectrum obtained by transmission electron microscopy (T.E.M.) was pointlike, like in a perfect crystal, but exhibited a forbidden 5-fold symmetry (see Fig. 2). The solution proposed to this paradox was that
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Figure 2. Diffraction picture of a quasicrystal
the atomic arrangement was no longer periodic, but rather quasiperiodic in space. This implies a long range order of a new type.
Figure 3. Left: Faceted hole in a n Al-Mn-Pd quasicrystal [ll] Figure 4. Right: Dodecahedra1 single grain Ho-Mg-Zn quasicrystal [52]
Later, a large number of quasicrystalline alloys, mainly made with aluminium and mainly ternary, were produced [62]. The most important are Als2.5C~2~Fe12 (where .~ the iron concentration may not vary more than .5% to stay in the quasicristalline phase), Al70Pd21Mng (see Fig. 3) or Al7oPd2lReg due t o the extreme quality of the samples that can be pro-
91
duced (the concentration of impurities or of defect is now less than These materials may have mostly an icosahedral symmetry, like the H o M g Z n monograin shown in Fig. 4,or a decagonal symmetry. But other compounds have been produced with other symmetries, like %fold or 12-fold symmetries. Besides having their surprising structure, these materials exhibit strange properties. For instance, even though made of good metals, they are mostly insulators at low temperature [62,99]. They are also mechanically hard and fragile, and they exhibit a superplastic transition a t high temperature just below the melting temperature (see for instance [62] for some of these p r o p erties). From the point of view that is developped in this course, the most interesting aspect of these materials is that there is no way of treating the aperiodicity as a perturbation of a periodic structure. Moreover, numerical calculations proved t o be extremely hard and they are not powerful enough t o explain most of the properties described above. There is a need for a new approach and this a typical interesting problem of Mathematical Physics.
1.2
The Hull
The starting point of the theory consists in considering the set of atomic positions in the ideal case where the atoms are fixed at their equilibrium position at zero temperature. This is a set of points in the ambient space Rd. This set will be considered in the idealized situation for which the solid has infinite volume and is homogeneous in space. This is a convenient approximation that fails only for submicronic samples. Let L denote this set. It is clear that atoms cannot be too close to each other, due t o nucleus repulsion at short distance. Moreover, since the system is idealized at zero temperature, except for special situations, like for zeolites, no lacuna is expected t o occur so that there is a maximal size t o holes between these points. This can be axiomatized in the following way [76] Definition 1 i)- A subset L of Rd is uniformly discrete i f there as r > 0 such that every open ball of radius r meets C at most o n one point. Then C will be called r -discrete. 2)- A subset L of Rd is relatively dense i f there is R > 0 such that every closed ball of radius R meets C at least on one point. Then C wall be called R-dense. 3)- A subset C of Rd is a Delone set (or also Delauney) i f at is both uniformly discrete and relatively dense. L will be called ( r ,R)-Delone af it is r-discrete and R-dense. 4)- A Delone set L has finite type whenever C - C is locally finite 5)- A Delone set is a Meyer set whenever L - L is itself a Delone set. Example 1 1. A random subset L of Rd distributed according t o the Pois-
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son distribution with a finite positive density is almost surely discrete, but with probability one, it is neither uniformly discrete nor relatively dense.
2. Let Lo be a lattice in Rd,namely a discrete cocompact subgroup of Rd. Let L be a random subset of Lo distributed according t o the Bernoulli law on each sites. Then with probability one L is uniformly discrete (obvious) but not relatively dense. This situation occurs precisely for the distribution of impurity sites in a lightly doped semiconductor [104], 3. Most solids are described, at zero temperature by Delone sets. This is the case for amorphous materials (silicon), glasses, crystals.
4. The cut-and-project method t o describe quasicrystals [62] shows that the set of atomic sites of such a material is a Meyer set. 0
In order to represent a uniformly discrete set 13, it is convenient to consider its counting measure which is the Radon measure on Rd
dC) =
c
6(. - y ) .
(1)
YEL:
Recall that a Radon measure p on Rd is an element of the dual space to the space Cc(Rd) of continuous functions with compact support. Since it is not the aim of this paper t o give all technical details, the reader should look at [71] t o learn more about the natural topology on this space. Thus p becomes a linear map p : f E Cc(Rd) H p(f) E C which is continuous. Here dC)(f) = f ( y ) and this sum is finite since f has compact support and since L is discrete. The measure dC)is a counting measure that is Definition 2 A counting measure o n Rd is a Radon measure u such that any ball B c Rd has a n integer measure, that i s u ( B ) E N. This measure is rdiscrete i f for any open ball B of radius less than or equal to r then u ( B )5 1. It is called R-dense i f for any closed ball B of radius larger than or equal to R as a measure u ( B ) 2 1. I t i s ( r ,R)-Delone i f it i s both r-discrete and R-dense. It is not difficult to check that there is a one-to-one correspondence between discrete sets and counting measures, the set being the support of the measure, so that each property of such a set can be read on the associated measure and vice versa. Representing the atomic set by a measure is a convenient tool to describe topologies. The space Im(Rd) of Radon measures on Wd will be endowed with the weak* topology over Cc(Rd). This means that a sequence
xyEc
93
(P,),~N of Radon measures converges to p if and only if given any continuous function f with compact support on Rd, then limn-m p,(f) = p(f). Thanks to this language, to say that a sequence L, of discrete sets converges to the discrete set L means that in each open ball B the sets L, n B converges to L n B , say for the Hausdorff distance. Note however, that this convergence needs not being uniform w.r.t. B. So it is convenient to denote by UD,(Rd) the set of r-discrete counting Radon measures on Rd. In very much the same way let Del,,R(Rd) be the set of ( r ,R)-Delone Radon measures on Rd. Then [23]: Proposition 1 1)- UD,(Rd) and D e l , R ( R d )are closed and compact subspaces of m(rwd). 2)- The closure Q D ( R d )of the union UD,(Rd) in m(Rd),is the set of
u,.,o
counting measures. 3)- U D , ( R ~ )is the closure of the union UR,, Del,,R(Rd). Remark 1 1)-means that from any sequence of r-discrete sets it is possible t o extract a convergent subsequence that converges t o an r-discrete set. In particular the limit of a convergent sequence of r-discrete sets is itself rdiscrete. 2)- a measure of Q D ( R d )can be seen as an atomic set in which a finite number of atoms can sit on top of each other. 3)- means that each r-discrete set can be approximated by a sequence of r-discrete Delone sets. 0
Remark 2 In much the same way, a Meyer set is an (r,R)-Delone set such that L - C is (r’,R’)-Delone for some 0 < r < R , O < T’ < R’. If MeyT,R;,l,~?(Rd) denote such a set of measures, it is also compact in the weak* topology. 0 Remark 3 The property for L of having finite type is not preserved under limits. 0
+
Given now a E R d ,the translation T~ : 5 E Rd H z a E Rd acts on Cc(Rd) through T ~ ~ (=xf( ) 5 - a ) whenever f E Cc(Rd). Therefore it also acts on the space of Radon measures through ~ ~ p (=f p) ( T a f ) whenever f E Cc(Rd) and p E m(Rd).It is elementary to check that these maps are continuous and invertible. Hence the translation group Rd acts on m(Rd)in a continuous way and Proposition 2 The spaces Q D ( R d ) , UD,(Rd) and D e l , R ( R d ) are Rd-
invariant. Remark 4 In much the same way Mey,,R;,~,R~(Rd) is translation invariant. 0
94
Remark 5 If L has finite type, then all its translated have finite type. 0 This last result allows to define the Hull of a uniformly discrete set as follows Definition 3 Let L be a uniformly discrete subset of Rd. Then its Hull is the dynamical system (R, Rd, T) where R is the closure of the Rd-orbit of dL) in m(rwd). Remark 6 1)-Since L is uniformly discrete there is T > 0 such that dL)E UD,(Rd). Hence, by Prop. 2 its orbit is contained in UD,(Rd). By Prop. 1 then, R is a compact subset of UD,(Rd). In particular, any measure w E R defines an r-discrete set L,, namely its support. 2)- The closure of the orbit of any point is obviously translation invariant, so that Rd acts on R through T. 3)- If in addition L is (r,R)-Delone, the same argument implies that all 13,'s are (r,R)-Delone. 4)- If L has finite type, then so does any element of the Hull. Actually then L, - 13, c L - L for any w E R. In particular if L is Meyer, so does any element of the Hull. 0 1.3
Properties of the Hull
L be a uniformly discrete set in Rd and let R be its Hull. Then the canonical tmnsversal is the subset of X c R defined by
Let
X = { w E 52 ; O E L w } = {w E 0 ; w ( { O } ) = 1)
(Canonical transversal)
(2) Each orbit meets this transversal on the corresponding atomic set, namely T-"w
E
x
w
XEL,.
Since L is uniformly discrete, so is Lw, so that there is a minimum distance from one point of X to any other along the orbit. In this sense the orbits are t r ansversa1. Example 2 Let L be periodic with period group (6. If G is a lattice, namely a discrete subgroup of Rd that generates Rd as a vector space, then the Hull is homeomorphic to the torus Rd/(6. Moreover, the transversal is the finite set X = L/G. 0 Example 3 A quasicrystal can be constructed by means of the cut-andproject method (see Section 2.3). For N > d, let R be a lattice in RN and let A be a polyhedral fundamental domain. Let then Ell be a &dimensional subspace of RN meeting R only at the origin. Then project all points of the strip
95
A +€11 on €11 t o get L. By identifying €11 with E X d , C becomes a Meyer set that is a model for all known quasicrystals [62]. Let W = xl(A), where x i denote the projection operator on the orthogonal subspace of €11. Let W be endowed with the coarsest topology such that W n (W r l ( a 1 ) )n . . . n (W r l ( a , ) ) becomes closed and open for any family { a l , . . . ,a,} c R. Then W is home0 omorphic t o the transversal [23]. Associated with the transversal X of the Hull, is a groupoid r ( X ) [96]. This groupoid plays a r6le similar t o the notion of Poincare' map or first return map in the theory of dynamical systems [42,69]. This groupoid is defined as follows. The set of unit r(O)coincides with X . The set of arrows r ( X ) is the set of pairs ( w , a ) E X x Rd such that T - ~ W E X . Then the range, the source and the composition maps are defined by r(w,u) = w ,
s(w,a) =
T-aW,
+
+
(w, a) o ( T - ~ w b) , =
(w, a + b)
The fiber is r-'({w}). Endowed with the topology induced by R x Rd, this is a locally compact groupoid. If P is an Rd-invariant ergodic measure on R, then it induces on r ( X ) a transverse measure represented by a probability measure PtT on X [36]. In the following it will be convenient t o introduce the Hausdorff distance of two sets A , B in Rd, namely d H ( A , B ) = max{supzEAinfYEBI C yI , supyEBinfzEA 12 - yl}. The first property of such a system is given as follows [23]. It is necessary t o recall that a dynamical system is minimal if every orbit is dense Proposition 3 If C is uniformly discrete but not relatively dense, then R admits a &point the orbit of which does not meet X . I n particular the Hull is not minimal. Consequently, C must be Delone t o have a minimal Hull. Definition 4 Let C be a Delone set in Rd. Then L is repetitive i f for any finite subset p c 13, and any E > 0 , there is R > 0 such that any ball of radius R contains a translated of a finite subset p' such that dH(p,p') < E . The following can be found in [23,70,76] Theorem 1 Let C be a Delone set. Then its Hull is minimal i f and only i f it is repetitive. From Def. 1 a uniformly discrete set L has finite type whenever C - L is discrete, thus closed. In such case the following is true Proposition 4 A uniformly discrete subset L of Rd with finite type admits a Cantorian transversal.
96
A characterization of the Hull of a finite type repetitive Delone set has been given in [24] (see Section 2.4). Theorem 2 Let C be a repetitive Delone set with finite type in Rd. Then its Hull is conjugate by homeomorphisms to the projective limit of an inverse sequence of branched oriented flat compact manifolds without boundaries, in which the Rd-action is induced b y parallel transport of constant vector fields. 1.4
Atomic Gibbs groundstates
In realistic solids, the atomic positions are also determined by their thermodynamical properties. This is because atoms can vibrate around their equilibrium position and can also diffuse through the solid. This motion creates both acoustic waves (phonons) and lacuna Nevertheless, the atomic position can still be described by a discrete set, thus an element of Q D ( R d ) . Hence this last space plays the r61e of the configuration space that is needed in Statistical Mechanics. Nevertheless, it is very convenient to describe the atomic motion in solids as a perturbation of the equilibrium position. The acoustic waves are usually treated as phonon degrees of freedom, whereas lacunz can be seen as impurities. Still, the atomic positions can be seen as typical configurations for the Gibbs measure describing the thermal equilibrium of atoms. Neglecting the atomic motion is equivalent to considering the T 1 0 limit P of the Gibbs measure for the atoms. Then P can be seen as a probability measure o n QD(Rd). From the mathematical point of view, there is a difficulty. For indeed, Q D ( R d ) is not a locally compact space, so that the notion of Radon measure is meaningless. Nevertheless, the weak* topology makes this space a Polish space [71]. That is, the topology can be described through a distance for which the space is complete (the choice of such a distance is actually not unique and certainly not canonical). But the beauty of Polish spaces is that there is a genuine theory of probabilities [85] using the Borel approach through a-additive functions on the a-algebra of Borel sets. In particular, since the translation group acts on QD(!Rd)in a continuous way, it transforms Borel sets into Borel sets so that it also acts on the space of probabilities by T ~ P ( A = ) P(T-"A) for A a Borel set. The Prokhorov theorem gives also a very useful criterion for compactness of a family of such probabilities. It turns out that describing the atomic configurations through such probabilities gives rise to several interesting results for physicists [23].
If the solid under consideration is homogeneous, then its Gibbs measure P is expected to be translation invariant. Moreover, standard results of Statistical
97
Mechanics [loo] show that a translation invariant pure phase is described by a Gibbs measure that is ergodic under the translation group. In addition with such considerations, the analysis proposed in Section 1.2 shows that typical configurations of atoms at zero temperature should be at least uniformly discrete, but also Delone unless in very special cases. This is why the following definition can be useful Definition 5 An atomic groundstate is a probability measure P on Q D ( R d ) such that I.
P is Rd-invariant;
2.
P is Rd-ergodic;
3. the space of uniformly discrete sets has P-probability one.
I n addition, P is called Delone (resp. Meyer) i f it gives probability one to the space of Delone (resp. Meyer) sets. Several results have been obtained in [23] from such a definition. Theorem 3 Let P be an atomic groundstate. Then 1. there is r > 0 such that P-almost every atomic configuration C is rdiscrete an not r’-discrete for r’ > r; 2. there is a compact subset R c U D , ( R d ) such that for P-almost all atomic configuration C, the Hull of C is R; R coincides with the topological support of P;
3. i f , in addition, P is Delone (resp. Meyer), there is a unique pair (r,R ) (resp. family (r,R;r’, R’)) such that P-almost every configuration i s ( r ,R)-Delone and not (r”,R”)-Delone for r < r” and R” < R (resp. (r,R;r’, R‘)-Meyer and not (r1,R1;r i , Ri) f o r r < r 1 , R > R1 ,r‘ < ri ,R’ > Ri). The next result concerns the notion of diffraction measure. Let C be a point set representing the position of atoms in the solid. Then, the diffraction pattern seen on a screen, in an X-ray diffraction experiment or in a transmission electronic microscope (T.E.M.), can be computed from the Fourier transform of C restricted to the domain A occupied by the sample in R d . More precisely, the intensity seen on the screen is proportional to
98
where k E Rd represents the wave vector of the diffraction beam, the direction of which gives the position on the screen. The Fourier transform of I A ( k ) is given by the following expression: if f E C,(Rd), with Fourier transform denoted by $, then
where p p ) will be called the finite volume diffraction measure. From eq. (4), it follows that p p ’ E m(Rd)is a positive measure with a Fourier transform being also a positive measure. The main problem is whether such a quantity converges as A Rd. The next theorem gives conditions under which convergence holds
Theorem 4 Let P be an atomic groundstate. Then:
(i) For P-almost every C the family p!$) of measures o n W d converges to a positive measure pp E m(iRd). (ii) The distributional Fourier transform of pp is also a positive measure o n Rd.
In other words, P determines in a unique way the diffraction pattern. 1.5
0
Bloch Theory
If the solid is a perfect crystal, the set C is invariant under a translation group (6. G is a lattice in Rd namely a discrete subgroup generating Rd as a vector space. G is called the period group of C. Bloch theory deals with the Schrodinger equation with a G-periodic potential. More precisely, let 3-1 = L2(Rd) be the Hilbert space of quantum states. The groups G is unitarily represented in 3-1 through U ( a ) + ( x ) = +(x - a ) whenever a E G. The Schrodinger operator is a selfadjoint operator H = H* = -A V with dense domain, where A is the Laplacian on Rd and V is a locally L1, Gperiodic real valued function. In particular
+
U ( a )H U ( a ) - l = H
VaEG.
Therefore H and the U(a)’s can be simultaneously diagonalized. Since G is Abelian, diagonalization of the U(a)’s is performed through its character group G*. Standard results in Pontryagin duality theory imply that G* is isomorphic to the quotient B = Rd */G1of the dual group of Rd (isomorphic t o Rd) by the orthogonal G I of G in this group. It is a well known fact that
99 (6'- is a lattice in Rd (called the reciprocal lattice in Solid State Physics [65]) so that B = Rd*/G1 is a compact group homeomorphic to a d-torus. B will be called the Brillouin zone (strictly speaking this is slightly different from what crystallographers call Brillouin zone). The concrete calculation of B goes as follows: any character of Rd is represented by an element k E Rd*. Since Rd* and Rd can be identified canonically, by using the usual Euclidean structure, k can be seen as a vector k = ( k l , . . . ,kd) E Rd. The corresponding character is given by the map
771,
:x E
Rd H ez(klx)E U(1) 7
In particular q k restricts t o a character of qk' [a: if and only if k - k' E G1, where (6'-
= {b E
(klx) = k l x l (6,
+ . . . + kdXd.
with the condition that
7]k la:
=
Rd ; ( b ( a )E 2 7 ~ 2 ,VU E 6 ) .
Since B is a compact group, the diagonalization of the U(a)'s requires the use of a direct integral decomposition of 'H over B, so that
+
Here, ' H k is the space of measurable functions $ on Rd such that $(z a ) = e z ( k l x ) $ ( x for ) all a E G and that &ddxI$(x)12 = 11$11&, < m, where V = Rd/G. Hk is then the partial differential operator formally given by the same expression as H , but with domain Z)k given by the space of elements $ E ' H k such that & $ / a ~Ei 3ik, for 1 5 5 d, and Ax$ E 'Flk. Then Hk is unitarily equivalent to an elliptic operator on the torus Rd/G = V. (In solid state physics, V is called the Wagner-Seat2 cell, whereas it is called the Voronoi cell in tiling theory.) Consequently, for each k E B, the spectrum of Hk is discrete and bounded from below. If E i ( k ) denotes the eigenvalues, with a convenient labelling i , the maps k E B H E i ( k ) E R are called the band functions. The spectrum of H is recovered as Sp(H) = U i , k E B E i ( kand ) is called a band spectrum. A discrete spectrum is usually liable to be computable by suitable algorithms, since it restricts to diagonalizing large matrices.
This is a short summary of Bloch theory. Strutt first realized the existence of band functions [107],but soon after Bloch wrote his important paper [27]. In 1930, Peierls gave a perturbative treatment of the band calculations [88] and Brillouin discussed the 2 0 and 3 0 cases [30]. The reader is invited to look at [5,65] to understand why this theory has been so successful in solid
100
state physics. Let us simply mention that the first explicit calculations of bands in 3 0 were performed in 1933 by Wigner & Seitz [115] on sodium using the cellular method that holds their names. The symmetry properties of the wave function were explicitly used in an important paper by Bouckaert, Smoluchowski & Wigner [29] leading t o noticeable simplifications of the band calculation.
1.6
The Noncommutative Brillouin zone
In Section 1.2 it has been shown that an aperiodic solid is well described by its Hull (R, Rd,T), namely a dynamical system with group Rd acting by homeomorphisms on a compact metrizable space R. With any dynamical system, there is a canonical C*-algebra namely the crossed product C(R) >a Rd [87]. In a similar way, such a system can also be described through its canonical transversal X, and its related groupoid r ( X ) . With any locally compact groupoid r, endowed with a transverse function [36], and with any module b on r, is associated a C*-algebra C * ( r , b )[96]. In this section, it will be shown that, after a slight modification if the solid is submitted t o a magnetic field, C ( 0 ) xRd is the smallest C*-algebra generated by the electronic Schrodinger operator and all its translated. Moreover, it will be shown that, C * ( r ( X ) )is also generated by the matrix of phonon modes that appears in the equations of motion for phonons or by the effective Hamiltonian derived form the so-called tight binding representation [13,16]. It will also be shown that, for periodic crystals, this algebra is nothing but the set of continuous functions on the Brillouin zone. Given a uniform magnetic field B = (Byp),namely a real-valued antisymmetric d x d-matrix, the C*-algebra C*(n x Rd, B) is defined as follows. Let do be the topological vector space Cc(s2 x Rd) of continuous functions with compact support in R x Rd. It becomes a topological *-algebra when endowed with the following structure
f9 ( w , x) =
Ad
dY f ( w , Y) 9(T-Ywl x
-
Y) e
i?r(e/h)B.zAy 7
(5)
f * ( w , x) = f(T--Zw, -x) , (6) where f,g E Cc(O x Rd), B . x A y = C Bvpx,,yp and w E R, x E Rd. Here e is the electric charge of the particle and h = 27~his Planck's constant. This *algebra is represented on L 2 ( R d )by the family of representations {nu;w E R} given by
nu(f)$J(x) =
1,1
dy f ( T - " w , y - x) e--iT(e'h)B'zAy $J(Y)l
$J E L 2 ( R d 1) (7)
101
where 7rw is linear, 7rw(fg)= 7rw(f)7rw(g)and n w ( f ) *= 7 r w ( f * ) . In addition 7rw( f) is a bounded operator and the representations (7rw),En are related by the covariance condition:
u ( a )n w ( f ) u(a)-' =
XTaw(f)
(8)
1
where the U's are the so-called magnetic translations [119] defined by:
-
where A = ( A l l..., A d ) is a vector potential defined by B,, = a,A, - &A,, a E Rd, 1c, E L2(Rd)and [x- a , x] is the line segment joining z - a to a in Rd. A C*-norm on do is defined by
IlfII
= SUP
Il.w(f)ll
(10)
.
W E 0
Definition 6 The noncommutative Brillouin zone is the topological manifold associated with the C*-algebra d = C*(R >a Rd, B ) obtained by completion of = C,(R x Rd) under the norm 11 . 11 defined by eq. (10). For B = 0 this construction gives the definition of the C*-crossed product C(R) )a Rd [26,87]. In the case of a perfect crystal (see Section 1.5), with lattice translation group 6, the hull R = R d / 6 is homeomorphic to Td (see Example 2) and Theorem 5 /16,98] The C*-algebra C*(lRd/G>a Rd,B = 0 ) associated with a perfect crystal with lattice translation group 6, is isomorphic to C(B) @ lcl where C(B) is the space of continuous functions over the Brillouin zone and lc the algebra of compact operators. Even though the algebra C(B) @ lc is already noncommutative, its noncommutativity comes from lc, the smallest C*-algebra generated by finite rank matrices. It describes the possible vector bundles over B. Theorem 5 is the reason to claim that A generalizes the Brillouin zone for aperiodic systems. The groupoid C*-algebra of r ( X ) can be defined very similarly. Here C,(F(X)) and the structure of *-algebra is given by
fg
(w1.)
=
c
f(w1y) g(T-Ywl. - Y ) e
zrr(e/h) B . x A y 1
f * ( w ,). = f (T--2w7).-
B,
=
(11)
(12)
Setting 5, = L!2(I'(w))l there is a representation Fw on 5, defined by ?w(f)$J
I(.
=
c
yEr(W)
f ( T - " w , y -). e
-irr(e/h) B . z A y
+(Y),
II,E 5,. (13)
102
Giving y = ( w , a ) E r ( X ) , there is a unitary operator U(y) : defined by
H
fi,
such that the covariance condition holds
U(Y) % , - pU(y)-l w(f) =
?W(f).
(15)
A C*-norm on BO is defined by
llfll
=
The C*-algebra B = C*(I'(X),B)is the completion of Bo under this norm. The notion of Morita equivalence, quickly defined below, is defined in Section 3.1.1 (see Definition 13). The main result is the following [98] Theorem 6 1)- The C*-algebras B = C * ( F ( X )B , ) and A = C*(R >a Rd,B ) are Morita equivalent, namely A is isomorphic to B 8 K. 2)- For B = 0 and for a crystal L with period group (6 and transversal X = C / G , the C*-algebra B is isomorphic to C(B) 8 MN((C)i f N is the cardinality of
x.
In order to see the noncommutative Brillouin zone as a manifold, rules of Calculus are required. Integration is provided by using a transverse measure. Let P be an &@-invariantergodic probability measure on Rd and let Pt, be the probability induced on X . Then traces are defined on A and B as follows
%(f) =
@(w)f(w,O),
f
E do,
%(f) =
& and 5 are traces in the sense that % ( f g )
@tr(w) f(u,O)f E Bo ,
(17)
=
'&(sf),and that '&! f f * )> 0.
Whereas 7+ is not bounded, $p is actually normalized, namely %(1) = 1. Moreover, using the Birkhoff ergodic theorem [42,69], these traces can be seen as trace per unit volume in the following sense
where B ( x ;R) denotes the ball of radius R centered at x E Rd. A similar formula holds for provided IB(0;R)I is replaced by the number of points in 13, n B(0;R). In particular in the crystalline case,
5
103
if J(k) denotes the representative of f in C(B)8 K and dk is the normalized Haar measure on B N Td.Therefore, these traces appears as the noncommutative analog of the integration over the Brillouin zone. A positive measure on a topological space defines various spaces of measurable functions, such as the LP-spaces. In much the same way, a positive trace on a C*-algebra defines also LP-spaces [loll. Whenever 1 5 p < 00, LP(A,Ip) is the separation-completion of d o with respect to the seminorm JJAJJ,= '&~{(A*A)p/~}l/pfor A E do. A similar definition is given for B. In particular, L2(d,T p ) coincides with the Hibert space of the GNS-representation T G N S of d associated with Tp [45,108]. Then L"(d, 5)denotes the von Neumann algebra generated by (namely the weak closure of) n G N s ( d ) . Similarly, a differential structure is provided in the following way
eif( w ,z) = zZf(w,z),
f
Ed
o or 230.
(18)
Here Z denotes the vector z E Rd. It is easy to check that defines a *-derivation namely it obeys Leibniz rule a ( f g ) = e ( f ) g f e ( g ) and = Moreover 9 is the infinitesimal generator of a norm pointwise continuous d-parameter group of automorphisms defined by [87]
+
a(f)* a(f*).
f
rlE(f) ( w , z ) = ezz.z f(w,z) 1
E
do or BO.
Let 2 be the position operator on L2(Rd),(resp. on 3 j u ) , defined by d $ ( z ) =
Z+(x). Then nu
(ef) = 43,n,(f)l , f
E do,
zu (Vf)
=
z[2,ii,(f)]
7
f
E
ao,
In the crystalline case, it is easy to show that f? coincides with a/ak in B [16]. Hence appears as the noncommutative analog of the derivation in momentum space.
e
In the philosophy of A.Connes, a noncommutative Geometry for a compact manifold is given by a spectral triplet (U,'H, D), where U is a dense subalgebra of a unital C*-algebra, invariant by holomorphic functional calculus, 7i a &-graded Hilbert space on which U is represented by degree 0 operators
104
and D is a degree 1 selfadjoint operator with compact resolvent, such that [ D , A ] E B ( H ) for all A E U. Then D plays the rjle of a Dirac operator on the noncommutative space associated with U, giving both a differential structure and a Riemannian metric corresponding t o ds2 = D P 2 [41]. Then the dimension spectrum is provided by the set of poles of the <-function
If SO denote the maximum real pole, then the residue a t SO provides an integral over the manifold given by
where TrDixdenotes a Dzxmier trace on 7-1 [41,44] (The definition and properties of the Dixmier trace will be given in Section 4.3). Such a structure is also present on the noncommutative Brillouin zone B. For indeed, let 71,. . . , -fd be an irreducible representation of the Clifford algebra of Rd by Dirac matrices satisfying yP7,, + 7,,7, = 26,,, and 7; = 7, in the finite dimensional Hilbert space Cliff(d). Then Cliff(d) is graded by the matrix 70 = 7172 . . . Y d , so that setting 7-1 = fi, 8 Cliff(d), G = 1 8 7 0 , this gives a graded Hilbert space where d G defines the graduation. Moreover U = 230 and D = C,=, 7,X, give a spectral triplet for which SO = d. In particular, the Dixmier trace TrDix(lDl-d) exist for Pi,-almost every w E X and its common value is given by
where Densp(L,) is the density of L, which exists for IF’-almost all w’s (see [23] see proof of Theorem 1.12). The numerical factor represents the volume of the unit ball of Ed. Moreover, if f E Bo this gives
showing that Connes’ integral and the integral over the Brillouin zone coincide modulo normalization.
105
1.7
Electrons and Phonons
The formalism developped in the previous sections will eventually be useful to describe the quantum motion of electrons and phonons in an aperiodic solid, within the one particle approximation. The interacting case is more involved [lo51 and will not been considered in these notes. The quantum motion of an electron in a monoatomic aperiodic solid represented by a uniformly discrete set C of atomic positions, and submitted to a uniform magnetic field B , is well described by a covariant family of Schrodinger operators of the type
-
where ti is the Planck constant, m is the electron mass, A is a vector potential and v is an effective atomic potential, that represents the binding forces acting on the valence electron. In practice, an aperiodic solid contains more than one species of atoms, so that C must be replaced by a family L1,. . . , C r of uniformly discrete subsets representing the position of each atomic species, supposing that r species are present. Then each species i acts via an effective potential vi. Moreover, the number of valence electrons involved in the conduction may be more than one per atom, so that it should be necessary to consider instead an 1-body problem if I valence electrons per atom are involved. This latter case will not be considered either, even though its treatment does not represent a major difficulty. In these notes the model described in eq. (19) will be sufficient. The Schrodinger operator (19) is not well defined as long as no assumption is given on the nature of the atomic potential v. For the sum over the atomic sites to converge, it is necessary that v decay fast enough at infinity. Also local regularity is necessary. A sufficient condition has been given in [23] (Lemma 2.22), namely let
be the set of integrable K-subharmonic functions on Rd. Then Lemma 1 Let v E LX,,(Rd) and v E U D r ( R d ) . Then v * v E Li$’(Wd) and the map v E Lk,,(Ed) H Y * v E Li$’(Rd) is continuous. The following Proposition is a consequence of [16] Section 2, Theorem 4 and [23] Theorem 2.23
106
Proposition 5 1)- For any 1: an r-discrete subset of Rd, let H L be the Schrodinger operator given by eq. (19). Then, i f v E L k , , ( R d ) the map C E UD,(Rd) H L is strong resolvent continuous. 2)- If R i s the Hull of C , the map w E R H H, is strong resolvent continuous and covariant. As a consequence, thanks to [16] Theorem 6, Theorem 7 Let C be a uniformly discrete set in Rd with Hull 52. Let A = C*(R M Rd, B) be the C*-algebra of the corresponding Noncommutative Brillouin zone. Let {H,; w E R} be the strong resolvent continuous and covariant family of Schrodinger operators defined by eq. (19) with v E Lk,,(Rd). Then there is a holomorphic family z E C \ R +-+ R ( z ) E A such that
-
vw E R . %J(R(z)) = ( Z - H w ) - l 7 As a consequence, it can be said that the Schrodinger operator is affiliated to the C*-algebra A [55,117,118]: Definition 7 A covariant family (A,) of selfadjoint operators is affiliated to A i f , for all f E Co(R), the bounded operator f(A,) can be represented as r,(Af) f o r some A f E A such that the map A : f E Co(R) H A f E A is a bounded * -morphism. The resolvent map R ( z ) is then given by A,.= if r, : s E R l / ( z - s) E C for z E C \ R. Conversely the resolvent map permits to reconstruct the map A through a contour integral. It has been argued in [16] that the C*-algebra A above can be actually reconstructed from the Schrodinger operator H L itself. 'H
In Solid State Physics the conduction electrons are commonly described through the so-called tight binding approxzmation. This is because only those electrons with energy within O(k,T) from the Fermi level do contribute to the current. There is no need then to consider the full range of energy to describe these electrons. In particular, the Schrodinger operator can be replaced by its restriction to an energy interval of size O(k,T) around the Fermi level. However this is not practically accessible in most cases. The method to perform such a reduction goes as follows [13,56,61] for atoms with one valence electron. The single atom Schrodinger equation
has usually a non empty discrete spectrum, described by quantum numbers. The valence electron occupy a given eigenstate, denoted by 4, E L 2 ( R d )corresponding to the energy E,. If the potential v decays fast enough to zero at
107
infinity, 4, decays exponentially fast away from the origin. Then an approximate basis of eigenstate of H, is provided by the family {U(y)&; y E 13,) whenever U is the magnetic translation group. Denoting by P, the projection on the subspace generated by such a family in L 2 ( R d ) ,it is possible to check that P, = 7ru(P) for some projection P E A. Thanks to an orthonormalization procedure, the family U(y)& gives rise to an orthonormal basis {GY;y E C,} indexed by the atomic sites, where each GY is exponentially localized near y. By using either the Schur complement (or Feshbach) method [51] or a Grushin method [57] there exists an effective Hamiltonian in the form of a function z H HeB(z)holomorphic in a neighbourhood of E, with values in covariant matrices over C,. Moreover, the spectrum of the original problem is given by the implicit equation E E SpH @ E E SpH,(E). The advantage of this method is that HeB(z)can be seen as an element of the groupoid C*-algebra B [13,16], with matrix elements decreasing exponentially fast away from the main diagonal. In practice only a finite number of such diagonals are kept to compute the spectrum, giving rise to tight-binding models. Such a method is actually commonly used also in computer software for the purpose of band calculations in crystals. The various ab initio methods of Quantum Chemistry or the functional density calculations (Kohn-Sham method), may lead to an accurate calculation of the parameters involved in such an effective Hamiltonian. It is not the purpose of these notes to explain these methods. However, they lead to effective Hamiltonians described by elements of the groupoid C*-algebra B. Phonons can also be described in a similar way through I3 with zero magnetic field. This is because phonons are the harmonic approximation of the atomic motion around their equilibrium positions. If the atom located at Z E C is moving around 2, let 2 iix(t)be its position at time t. Since x is an equilibrium position for the atom, the potential energy created by its neighbors increases away from 2, hence it is expected that C x ( t ) stay small at all time. Moreover, the lowest order expansion of the potential energy around x gives rise to a quadratic potential, namely harmonic forces, tending to force the atom back to its position x. Therefore, within this harmonic approximation, the classical equation of motion for the atom has the form
+
where M is the atomic mass and K,(x, y) is the d x d matrix of spring constants between atoms located at y and x. So K,(x, y) E Md(R). Phonons are just the
108
quantized degrees of freedom associated with this classical motion. In practice however, the influence of atoms far apart is negligible so that K ~ ( Z y) , decays fast enough as 1-. - yI -+ 00. Moreover, the action-reaction principle implies that ~ ~ ( ( y) 5 , = ~ ~ (x). y ,In addition, since it describes an attracting force, K ~ ( . - y) , is a positive matrix for any (5,y ) . Also, the translation invariance of interactions between atoms leads to the covariance condition KTUw(X f
a ,Y
f
a) =
K w ( z , y).
In very much the same way, the translation invariance of interactions between atoms leads to the continuity of K ~ ( z , with ~ ) respect to w . Hence the map ( w , 3) E r ( X ) H ~ ~ ( .-) 0E Md(Rd) , defines a positive element K of C*(r(z))@ Md. The equation of motion (20) is usually solved by looking at the stationary solutions, namely solutions for which
leading to the eigenmode equation
(21) It is easy to check that this K defines a positive element of B so that the eigenmodes are given by the spectrum of K 1 1 2 . These modes are the plane waves that are allowed to propagate through the crystal. Historically, Einstein [48] was the first to propose a quantized version of these oscillations, assuming that only one mode was propagating. Then he could compute the heat capacity of the solid and show that it saturates at high temperature leading to the Dulong an Petit law. Soon after, Debye ( [64] Section 1.1) introduced a distribution of eigenmodes together with a density of eigenmodes, namely the number of eigenmodes per unit volume and unit of frequency at a given value of 5. He also introduced a cut-off to take into account the uniform discreteness of the crystal, in the form of the Debye Temperature 00. In 1912, Born and von Karman [28] performed the first explicit calculation of the eigenmodes in a cubic crystal and discovered the existence of optical modes, explaining a small discrepancy between the Debye predictions and the measurements of the heat capacity.
In both cases, the density of states (DOS), for electrons, and the vibrational density of states (VDOS), for the phonons, are defined in a similar way. In
109
the phonon case, the relevant operator is K112 13, whereas for electrons it is its Hamiltonian H , namely either the Schrodinger operator, in the continuum case, or the effective Hamiltonian in the tight binding representation. Let then H be the name of such an operator. In both cases it is given by a strong reTolvent continuous family (H,), of selfadjoint operators, bounded from below, indexed either by the Hull s1 or by the transversal X . It acts either on L2(Rd) or on e2(Lu).In both cases, given a bounded box A c Rd, it is meaningful t o restrict H , to A provided boundary conditions are prescribed. Let H,,A be this restriction. Then its spectrum is discrete in both cases so that there is only a finite number of eigenvalues (counted with their multiplicities), below E E R. This allows t o define the integrated density of states (IDS) as
N ( E ) = lim
AtWd
1
- # { E l E s p ( H , , ~ ) ;El
1111
5 E}
The existence of the limit depends upon the choice of an invariant ergodic probability measure P on the Hull so that the limit exists only P-almost surely (resp. Pt,-almost surely) with respect t o w [13,33,43,86]. Let then Np denote this common limit. The Shubin fornula [13,16] expresses it in term of the noncommutative Brillouin zone as =
5 (x{H IE l ) ,
(Shubin formula)
(22)
with a similar formular in the tight-binding case. Here x{. 5 E } denotes the characteristic function of the interval (-m,E] C R. Then x { H 5 E } is the spectral projection of H seen as an element of the von Neumann algebra L m ( d , 7 p ) . It is then clear from the definition, that Np is a non decreasing non negative function of E. It vanishes below EOinf Sp{H}. It is constant on spectral gaps (see Fig. 5). In particular, its derivative d&/dE exists in the sense of a Stieljes-Lebesgue measure and is called the density of states (DOS) (resp. the vibrational density of states). The DOS is usually used t o expressed thermodynamical quantities such as the heat capacity. In the case of phonons, the VDOS is given by
N$(W)=
% ( x ( K 5 5’))
PP
,
(Shubin for phonons) (23)
where pp is the atomic density (in number of atoms per unit volume) in order to get a number of modes per unit volume. Because there are d possible choices for the polarization of the acoustic waves, the trace ‘&(x(K 5 m)) = d. The
110
Figure 5. An example of electronic integrated density of state3 in 1D
Debye formula for the phonon contribution t o heat capacity per unit volume leads to
where LJDis the Debye cut-off frequency of the solid. If T the Dulong-Petit law, namely
Cch Tfm pp k, d .
00,
(Dulong & Petit)
eq. (24) gives
(25)
1. 0, only the low frequencies contribute, so that it is enough to consider the asymptotic of the VDOS at L;r 1. 0. Since low frequencies correspond to large wave lengths, the corresponding phonons do not see the fine structure of the solid and behave like acoustic waves with sound velocity c, for the polarization Q. In particular W+(L;r) behaves like L;rd-'&, so that As T
CEh 0; T d,
as T 1 0 .
For d = 3 this gives the usual T 3 law discovered by Debye. The electronic contribution to the hezt capacity must take the fermionic character of the electrons into account. Then
111
if S is the total entropy in the volume V . The derivative is taken at constant volume and constant electron number. The computation of S is usually performed in the grand canonical ensemble. If Nel denotes the electronic IDS, the chemical potential p is fixed by the condition
Here N is the total number of electrons in volume V and EF is called the Fermi energy. The factor 2 comes from the spin degrees of freedom. Then the entropy S is given by
Here P
=
l/k,T. Setting z = P(E - p ) / 2 this gives
Whenever the IDS is smooth around E = E F , this gives the usual electronic contribution to the heat capacity
C:'ocT,
as T 1 0 .
If, however NJ admits some fractal behavior near the Fermi energy, then the low temperature behavior may be modified. It turns out that such a fractal behavior is hardly seen in practice. This is because the disorder and the interactions as well tend to smooth out the IDS at small energy scales. 2
2.1
Examples of Hulls
Perfect Crystals
If L denotes the set of atoms of a perfect crystal, let (6 be its period group (see Section 1.5). Then clearly the set of translated of L: can be identified with the quotient space Rd/G = V by the very definition of the period group. In particular, G is cocompact. Thus the Hull is the Wzgner-Seitz cell with periodic
112
boundary conditions. Since any cocompact subgroup of Rd is isomorphic t o Zd , it follows that R is homeomorphic t o a d-torus. In very much the same way, the transversal is the finite set X = C/G. The number of points in X is the number of atoms in each unit cell. The groupoid of the transversal r ( X ) is made of pairs (z, b) E X x Rd such that (z - b) mod G E X . Such b’s can be labelled in the following way. If z - b = y mod G an element of r ( X ) can be seen as a triplet (z, y; a ) E X x X x G with the range, source maps and composition given by
r(z,y;a)
= z
Y,
s(s,y;a) =
(z, y; a ) 0 (Y,2; b) = (z, z ; a
+ b) .
Td
Figure 6. The Hull of a periodic crystal
In particular, a typical element of the C*-algebra associated with this groupoid is a matrix valued sequence Fz,y(a)with product and adjoint given by
z E X bEG
In particular, using the Fourier transform (here B is the dual group t o G and has been called the Brillouin zone in Section 1.5)
(mZ,, c =
Fz,y(a)
,
kEB,
aEG
the previous algebra becomes nothing else but the tensor product C(B) @ Ad,(@) if n = is the number of points contained in X . The same argument shows that the C*-algebra of the Hull is isomorphic t o C(B) @ K where K is the C*-algebra of compact operators.
1x1
113
2.2
Disordered Systems
Let us consider now the case of impurities in a perfect monoatomic crystal. The physical example is the silicon diamond lattice in which some of the Si atoms may be replaced by an impurity of the columns IIIA ( B ,All Gal I n or ptype), or VA (P,As, S b or n-type). The finite set of possible doping atoms is denoted by U and will be called an alphabet. For example, if the Si is doped only with boron B or antimony Sb, the alphabet will contain three letters, namely U = { S i , B , S b } . Let L be the underlying lattice of the pure silicon. Then, if z E L let n, E U be the letter denoting the atom sitting at z. A family of the type n = (n,),€L E UL will be called a configuration of impurities. The period group G of L acts on the configuration space UL through
To take into account that most of the lattice sites are occupied by a silicon atom, it is necessary t o define a probability measure P which forces the impurities t o be very rare. The simplest example of such a probability consists in demanding that the n X 1 sbe independent identically distributed random variables with a common distribution p such that p ( S i ) = 1 - c where c = C,,,En,isi) p ( w ) E ( 0 , l ) is the concentration of impurities. Then P(n) = B z E L p ( n , ) is G-invariant. The main result is the following [23,105] Proposition 6 If p ( w ) > 0 for all letter w E U, namely i f each impurity occurs with a nonzero concentration, then (a) for P-almost all configuration 14 of impurities, the transversal X of the Hull of n is homeomorphic to UL.I n particular X i s a Cantor set. (ii) the Hull i s homeomorphic t o the suspension of X by the action of G, namely
s1 N where G acts o n X through
T
x x Rd/G ,
and o n Rd by translation.
The previous result has been generalized to probabilities P with correlations in [23,105]. The condition that each impurity occurs with a nonzero concentration must be then replaced by: given any finite subset A c L then any configuration n A = (n,),EA in Q" have a n o n zero probability. Such a condition is usually satisfied by Gibbs measures describing the thermal equilibrium of the impurities.
114
2.3
Quasicrystals
The simplest example of non-periodic tiling was provided by R. Penrose [go] (see Fig. 7). It is built from two types of tiles in the 2D plane, through inflation rules, and exhibits a fivefold symmetry. It was extensively studied by de Bruijn [31]. But it was recognized only later on by physicists that it is quasiperiodic. However, de Bruijn and also Kramer & Neri [74] built examples of quasiperiodic tilings. Most models describing the quasiperiodic order in quasicrystals, are based upon the so-called cut-and-project method, independently proposed by Duneau & Katz [46,47], Kalugin, Kitaev & Levitov [67], Elser [49] and Levine & Steinhardt [81]. It was not until 1995 that this method was recognized as equivalent to the notion of model sets provided by Meyer [83] in his thesis work [82].
Figure 7. The Penrose tiling
The best models known nowdays t o represent the structure of a quasicrystal are built by using the cut-and-project method (see [62]). It is well illustrated by the construction given in Fig. 8. The idea is the following. Let N = d + n be an integer bigger than the dimension of the physical space. In R N ,the lattice L = Z N defines some periodicity. Then let Ell be a d-dimensional subspace of EXN intersecting L only at the origin (irrationality condition). The idea of the construction is that the physical space is precisely Ell. Then E l will denote
115
Figure 8. The cut-and-project construction
the orthocomplement subspace. The corresponding orthogonal projection on Ell and E l will be denoted by T I I ,TI respectively. Let then W be the polytope obtained by projecting the unit N-cube on E l and let 0 be its center, namely the projection of the center of the N-cube. W will be called the w i n d o w or the acceptance d o m a i n . Given a face of W of maximal dimension n - 1, there is a unique opposite face obtained by the inversion symmetry around 0. For each such a pair of faces, one, called permitted, will be added to W the other one, called forbidden, will be excluded from W . If two ( n- 1)-permitted faces are adjacent, the closed ( n- 2)-face they have in common will be taken in W . Otherwise the same procedure must be applied t o ( n - 2)-faces, then to the ( n- 3)-faces etc. In this way, for each pair of opposite points of the boundary of W one is permitted, and belongs to W , the other one is forbidden. Thus, W is neither closed nor open. Let then CW = W +Ell be the strip parallel t o the physical space and generated by W . The lattice points contained in C w , are then projected on the physical space t o give the atomic sites L,namely (see Fig. 8)
116
Example 4 (The octagonal tiling) The simplest example of such a construction is the octagonal or Ammann-Beenker tiling given in Fig. 9
Figure 9. T h e Ammann-Beenker tiling
To build this model it is enough t o start from Z4. The 8-fold symmetry seen in Fig. 9 can be implemented through the transformation (see Fig. 10)
where { e l , e2, e3, e4) is the canonical basis of It4. The matrix R has integer coefficients, is orthogonal and satisfies R4 = -1. It can be decomposed into the direct sum of two plane rotations of angles x/4 and 3x14. The corresponding invariant planes will be Ell and E l respectively. Since both these planes and the lattice Z4 are R-invariant, and because R commutes with the projections xll , xl , it follows that R leaves the set L: of atomic sites invariant, which is what is seen in Fig. 9. The acceptance domain is the projection of the unit 4-cube on E l . It must be R invariant, so that it must be an octagon.
117
t
Figure 10. The acceptance domain of the Ammann-Beenker tiling
This acceptance domain is given in Fig. 10 together with the projection of the four basis vectors in E l . The acceptance domain can be used to read the properties of the tiling. For instance, le L3 the set of points in 13 having exactly three nearest neighbors in the direction e l , e3, -e4. Such points come from points a E L such that a e1,a - e4,a e3 belong to the strip CW whereas the other a f ei7s fall out of C W . Their projection onto W gives the subdomain of W given by the rectangle triangle with hypothenuse on the boundary of W and located along the direction el from the center. The other subdomains designed in Fig. 10 correspond to the acceptance domains of sites with other possible nearest neighbor environments.
+
+
The transversal of C is nothing but the acceptance domain W provided it is completed for a topology finer than the usual topology in R2. More precisely, given any finite family of vectors ( a l , . . . , a l ) in Z4, the sets W n ( W + r l ( a l ) ) n ... n (W r l ( a l ) ) are closed and open for this new topology. In particular all possible acceptance domain of local environments are closed and open. It is easy to see that W is a Cantor set.
+
To build the Hull, it is enough to pull back the topology of W to R4 by demanding that (WO) be closed and open for each WOc W closed and open. Let R& denote the completion of R4 for this topology. It is a locally compact space on which both Z4 and Ell acts by homeomorphisms. In particular the Hull can be identified with T& = R&/Z4 and IR2 acts by translation by vectors of Ell. 0
118
In general the cut-and-project method can be well described in term of model sets of Meyer. Namely it is defined with spaces and maps as follows
Rd I K 1 R d XR" "2.R" (28)
U
A(M)Z
R
"2.M
where R C Rd x R" is a lattice (a co-compact discrete subgroup) and T I and 7rz are the projections onto Rd and R", respectively. Furthermore xl restricted on R is injective and ~ ~ ( 7is2 dense ) in R". Rd wil be called the physical space and Rn the internal space. Here and 7r2 are the restriction maps on the corresponding coordinate of Rd x R". Therefore the setting of ) . a subset M a cut-and-project scheme is given by the triple ( 7 r l , ~ z , RFor in the internal space R" we define the corresponding point set in the physical space R~ as
A ( M ) = { T ~ ( ~ ) E\ cRA, ~ z ( aE) M } . (29) M is called the acceptance domain of the point set A ( M ) . For a lattice vector a E R
A(M
+ ~ z ( u )=) A ( M ) + 7rl(a).
(30) Definition 8 A point set L in Rd is called a model set i f there exists a bounded set M with non-empty interior such that L = A ( M ) . The next result is proved in [23] Proposition 7 A model set is a Meyer set. Definition 9 Let M be a bounded subset of R" with non-empty interior. M is called admissible i f for every ball B ( ~ , Ewith ) E > 0 and z in ~ ~ ( 7n2M) , there exists a finite family { u l , . . . , u p } in ~~('72) such that M n ( M a1) n . . . n ( M u p ) is a subset of B ( z ,E ) with non-empty interior. A model set L is called admissible i f there exists an admissible set M such that L = A ( M ) . 0 Example 5 A convex polytope is an admissible set. Let L = A ( M ) be an admissible model set and let AM denote the (7'-algebra generated by the set of functions {f 8 ( X M 0 T12(a));a E R,f E C,(Rd)}, where Tz(a) denotes the translation in R" by 7r2(a). Here X M denotes the characteristic function of M . Let R",fd be the set of characters of d m so that, by Gelfand's theorem, AM is isomorphic to Co(R",fd) where is a locally compact space. Since M is admissible C O ( R ~ +is~a) closed subalgebra of A M .By duality there is a surjective continuous map T M : R",fd + Rnfd. Therefore R Z d can be seen as the completion of for a finer topology than the usual one, that will be called the M-topology, in which the sets
+
+
RZd
119
+
Rd x M a , for a E R,are open and closed. By construction, for a E R, the map x E Rd x R" -+ x + a E Rd x R" extends to R Z d by continuity. One sets Tzd= R",fd/R. By construction, for y E Rd x {0}, the map x E Rd x R" + x+ y E Rd x R" extends also by continuity to R z d and commutes with the action of R. Thus it defines a Rd-action T on T %+'. Similarly R& can be defined as the set of characters of the C*-algebra generated by the set { x o~T ! ( = ) ; aE R}. Since it is a set of idempotents, R s is totally disconnected and Tzdis transversally totally disconnected. Definition 10 The dynamical system (Tzd, Rd) is called the pseudo-torus associated with the window M . The following result is proved in [23] Theorem 8 Let L = h ( M ) be a n admissible model set in Rd, and Y = d L ) . Then (Qv, Rd, T ) is topologically conjugated to (Tzd, Rd,?'). This dynamical system is minimal. It is uniquely ergodic, providing M is a Bore1 set in If%". Up to now, M has only been assumed to be admissible. The following property actually holds in quasicrystals: Definition 11 A polytope in R" is said R-compatible zf its vertices belong t o 7rz(R). If the acceptance domain M is an R-compatible polytope, let F l , . . . ,Fp be the hyperplanes of Rn+d parallel to the maximal faces of R" x M . For each j E (1, . . . , p } , let uj E be a unit vector perpendicular to Fj so as to define FT = {x E Rn+d; (ujlx)2 0) and F3T = \ FT. Let then F be the family of affine hyperplanes F3 a with j E (1, .. . , p } and a E 7 r 2 ( R ) . Rd x R" is then endowed with the coarsest topology for which, given any F E F ,the closed half-space F+ is both closed and opeq. It will be called the and F-topology. The same construction can be performed in R". Let R F be the completions of Rd x R" and R" with this topology, respectively. In much the same way, TF+d = RF+d/Ris well defined and can be endowed with an Rd-action. The following proposition is easy to prove: Proposition 8 If M is R-compatible, the M-topology and the 3-topology are equivalent o n R " + ~ .I n particular T F + ~= TM ~' + ~ Remark 7 An alternative description of this pseudo-torus is proposed in 0 [54,80] under very general hypothesis on M . Remark 8 In a Meyer set obtained through the cut-and-project method any bounded pattern repeats itself infinitely often. More generally, each Meyer set is a Delone set of finite type. Thanks to Proposition 4, this implies that the canonical transversal is totally disconnected. This can also be seen from 0 the particular topology described just before.
+
120
2.4
Finite type Tilings
The construction of quasicrystalline lattices given in the previous Section 2.3 cannot be extended to any Delone set of finite type. Nevertheless a construction has been proposed in [24], inspired by earlier papers by [4,103], which gives the description of the Hull in such a case. The consequence is summarized in the Theorem 2 given at the end of Section 1.3. Let this theorem be illustrated with the help of the octagonal tiling. The octagonal tiling shown in Fig 9 can be tiled by 20 tiles modulo translation. Each tile is repeated eight times when the 8-fold rotation R is applied (see Section 1.3). Therefore, modulo R, three tiles are remaining, one is a rhombus with sides of length one and a 45" angle, the two others are rectangle triangles corresponding to half a square of side one. The two half squares are symmetric to each other around the diagonal of the square. These tiles are also decorated by arrows. They are shown in Fig. 11 below. The arrows have to match within the tiling. The octagonal tiling is invariant by an inflation symmetry of ratio 1. The inflated tiles are also shown in Fig. 11. For instance, the inflated rhombus is made of three rhombi and two of each variety of half squares. The inflated square is made of 4 rhombi and 3 of each variety of half squares.
a+
Figure 11. Construction of BOF manifolds for the octagonal tiling
The family of these 20 tiles is made into a compact branched, oriented, flat Riemannian manifold (BOF) by identifying the oriented sides of various tiles whenever they match somewhere in the tiling. The result is the manifold Bo. To obtain B1, it is sufficient to proceed the same way with the inflated tiles.
121
The decoration of the sides of the inflated tiles is now more complicated. But the same rule applies. Due to the inflation symmetry, Bo and B1 are actually diffeomorphic BOF manifolds. However, the inflation changes the metric from ds2 t o (fi 1)2ds2. It is thus possible t o define a map fo : B1 H Bo, with derivative Dfo = 1 (the unit matrix), which identifies each tile of order 0 in an inflated tile with the corresponding tile of Bo (see Fig. 11). It is important to remember that two tiles which differ by a rotation are considered as distinct. Such a map is called a BOF submersion. It is locally the identity map, it is globally onto, but it maps several point of B1 to the same point in Bo. In a sense, fo folds B1 onto Bo. Then, by inflating n times, and proceeding in the same way at each inflation, the BOF manifold B, is constructed. Again, in this particular example, due to the inflation symmetry, B, is diffeomorphic t o Bo. But the metric ds2 is multiplied by (&? l)’, in B,. In very much the same way, as shown in Fig. 11, there is a BOF submersion f, : B,+1 H B, defined by identifying the n-tiles tiling the ( n 1)-tiles in B,+1 with the corresponding n-tile of B,.
+
+
-
+
This construction leads to a projective family . . . fn+l B,+1 5 Bn fzl ... of BOF manifolds. Then the successive unfolding of the B,’s is described through the projective limit
R
=
lim(B,,fn) t
.
(31)
By definition, a point in R is a family w = (x,),€N E n B n such that fn(z,+l) = x, for all n’s. Actually only the large enough n’s do matter to define w , since given x,, all the xj with j < n are uniquely defined. Therefore, given a € It2, there is N large enough so that x, is at distance larger than [la11from the boundary of the tile it belongs to. In particular, x, a is a well defined point in the same tile as 5,. Since DF, = 1, it follows that fn(zn+i a ) = x, a for n large enough. Thus the point T ~ W= (Y,),EN defined by y, = xn a for n > N is well defined in R. Hence R2 acts on R through T. The Theorem 2 says that the dynamical system (R, R2, T) is conjugate t o the Hull of the tiling through a homeomorphism.
+
+
+ +
In the general situation of a Delone set C c Rd that is repetitive with finite type, the procedure is similar. First, a tiling is built, by means of the Voronoi construction. Namely given x E C , the Voronoi cell of x is the open convex polyhedron V, defined as (see Fig. 12)
122
/
I
/
\
I
Figure 12. Construction of a Voronoi cells and a Voronoi tiling
It is known that such polyhedra tile the space. Then, since C has finite type, there is only finitely many Voronoi cells modulo translation. Each such cell will be decorated _ _ by the family of its neighbors: a Voronoi cell V, is a neighbor of V, if V, nV, # 8. A prototile is the equivalence class of a decorated Voronoi cell modulo translations. Then Bo is obtained from the disjoint union of the prototiles by gluing two such prototiles TI and T2 along one of their face, whenever there is a region in the tiling where two tiles equivalent t o TI and T2 respectively, are touching along the corresponding face. The decorated Voronoi tiles of C will be called 0-tiles. In general there is no natural inflation rule in a tiling. But it is possible to proceed as follows. Let one of the prototiles T be chosen. And let LT be the subset of C made of points with Voronoi cell equivalent to T . Thanks to C being repetitive, CT is itself a Delone set of finite type. It is then possible t o build the corresponding Voronoi cells of CT.It will be convenient t o substitute to each such cell, the polyhedron V;') obtained as the union of the 0-tiles intersecting it, with some convention a t the boundary, decorated by the 0-tiles that touch it from the outside. This will be a 1-tile. Then B1 is built in the same way as Bo. Moreover each 1-tile is tiled by a family of 0-tiles, so that the map fo : B1 H Bo is well defined as in the octagonal case. The construction of B1 from Bo, can be repeated to get again a projective family of BOF, as in the previous example, and so t o the space s1 together with the Rd action. And this is the interpretation of the Theorem 2.
3
The Gap Labelling Theorems
The Shubin formula for electrons or phonons, given by eq. (22,23), shows that the integrated density of electronic states or of phonon modes are given in term of the trace of a spectral projection of a self adjoint operator affiliated
123
to the C*-algebra A of interest. In the electronic case, this operator is the Schrodinger Hamiltonian H , which is unbounded, while in the phonon case, it is the mode operator K which belongs to A. There are many situations in which either H or K may have a spectral gap. If this happens, then the spectral projection x ( H < E ) or x ( K < G 2 ) becomes elements of A whenever E or G2 belong to a gap. Since the trace of a projection P E A does not change by a unitary transformation, the value of this trace depends only upon the unitary equivalence class of the projection P . The notion of unitary equivalence is, however, meaningless in general in a C*-algebra since it may not have a unit. This is why it is better to use the von Neumann definition, namely [87] Definition 12 Two projections P and Q of a C*-algebra A are equivalent, and it is denoted b y P M Q whenever there is U E A such that UU* = P and U*U = Q. As a matter of fact, there are not so many such equivalence classes, more precisely [87] Theorem 9 I n a separable C*-algebra , the family of equivalence classes of
projections is at most countable. A consequence is that the trace of a projection belongs to a countable subset of R+. Such numbers will be called gap labels. The question is whether it is possible t o compute this subset. The answer is yes and this is the purpose of the various versions of the gap labelling theorem to do so. The main tool is that the set of equivalence classes of projections may be enlarged in a canonical way to become a discrete abelian group, called Ko(A) or the K-theory group. 9.1
K-theory
This section is devoted to a short review of K-theory and Morita equivalence [26,114]. 3.1.1
The Group KO
The set of equivalence classes of projections in A will be denoted by P ( A ) ,and the equivalence class of P by [PI. Two projections P and Q are orthogonal whenever PQ = QP = 0. Then P Q is a new projection, called the direct sum of P and Q , denoted by P @ Q. Proposition 9 Let A be a separable C*-algebra. Let P and Q be two projections in A. Then the equivalence class of their direct sum, i f it exists, depends only upon the equivalence classes of P and of Q. I n particular, a s u m is defined o n the set 0 of pairs ([PI,[Q])in P ( A ) ,such that there are P' M P and
+
124
+
Q' M Q with P'Q' = Q'P' = 0, by [PI [Q] = [PI @ Q']. This composition law is commutative and associative. The main problem is that the direct sum may not be everywhere defined. To overcome this difficulty, A is replaced by its stabilization A @ K , where K is the algebra of compact operators. A C*-algebra A is stable if A and A@K are isomorphic. For any C*-algebra A, A@K is always stable, because K@K K. Definition 13 Two C*-algebras A and B are Morita equivalent whenever A @ K: is isomorphic to B @ K . Proposition 10 Given any pair P and Q of projections in A 8 K, there is always a pair PI, Q' of mutually orthogonal projections in A @ K such that P' M P and Q' M Q. Therefore the s u m [PI [Q]= [PI @&'I is always defined. In this way, if A is a stable algebra, the set P ( A ) of equivalence classes of projections is an Abelian monoid with neutral element given by the class of the zero projection. If A is not stable, P ( A ) will be replaced by P(d @ K). The Grothendieck construction gives a canonical way t o construct a group from such a monoid. This is a direct generalization of the construction of the group of integers Z from N. The formal difference [PI - [Q]is defined as the equivalence class of pairs ([PI,[Q]) E P(d@K:) x P ( ABK:)under the relation
+
([PI,[QI)
([PI], [&'I)
31571 E P ( A @ x ) ;[PI+ [&'I
+ [SI = [PI]+[Ql + [Sl.
The corresponding quotient is the Abelian group Koo(A) = P(d 8 K ) x P(d @ K)/%. Whenever A is unital, &(A) := Koo(d). Otherwise, A must be enlarged t o A+ obtained by adjoining a unit, so that A becomes a twosided closed ideal of A+. The quotient map 7r : A+ 4 A + / A induces a group homomorphism 7r* : Koo(d+) + Koo(A+/A),the kernel of which being the group Ko(A) (see [26] for details). It leads to: Proposition 11 Let A be a separable C*-algebra.
(i) The set Ko(A) is countable and has a canonical structure of Abelian group. (ii) Any *-isomorphism 'p : A H B between C*-algebras induces a group homomorphism 'p* : Ko(A)H Ko(B), so that K becomes a covariant functor from the category of C*-algebras into the category of Abelian discrete groups. (iii) A n y trace 7 on A defines in a unique way a group homomorphism I , from Ko(A) to R such that i f P is a projection o n A, 7 ( P ) = I , ( [ P ] ) where [PI is the class of P in &(A).
125
(iv) If A and t3 are two Morita equivalent C*-algebras then Ko(A) and Ko(t3) are isomorphic. 3.1.2
Higher K-groups and exact sequences
The explicit computation of K-groups can be performed using the methods developped in homological algebra. The main tools are exact sequences and spectral sequences. However, these methods require introducing higher order K-groups. Let A be a C*-algebra and let GL,(A) be the group of invertible elements of the algebra M,(A). (when A is non-unital, GL,(d) = { u E GL,(A+); u = 1, mod M,(d)}). GL,(A) is embedded as a subgroup ofGL,+l(A) using (GLO",(A)O j )
(with 0, = (0...0)). Let GL,(A) be the inductive limit of GL,(A), namely the norm closure of their union, and let [GL,(A)]o be the connected component of the identity in GL,(A). K1 is defined as follows: K1(A) = GLm(A)/GL,(A)o
= lim{GL,(A)/[GL,(A)lo} +
(32)
If A is separable, then K1(A) is countable, since nearby invertible elements are in the same component. For u E GL,(A), let [u]be its class in KI(A). The relation [u][w] = [diag(u,v)]defines a product in KI(A). Then [26] Proposition 12 KI(A) is an Abelian group. The suspension of A is the C*-algebra SA of continuous functions f : R 4 A vanishing at fca,endowed with point-wise addition, multiplication and adjoint, and the sup-norm. Hence SA Co(R) @ A. Then [26] Theorem 10 KI(A) is canonically isomorphic to Ko(SA). Therefore we can also define higher K-groups by
KZ(A) = KI(SA) = Ko(S2A),. . . ,K,(A) = ... = Ko(S,d). Theorem 11 (Bott Periodicity) Ko(A) Kz(A). More precisely &(A) is isomorphic to the group .Irl(GL,(d)) of homotopy classes of closed paths in GL,(A). Furthermore, zf 7 is a trace on A and i f t E [0,1] -+ U ( t ) is a closed path in GL,(A) [37]:
126
Ki is a covariant functor with the following properties: Theorem 12 Let 3,A, A,, B be C*-algebras and n, i non-negative integers: (i) Iff : A + B is a *-homomorphism, then f induces a group homomorphism f* : &(A) -t Ki(B). Then id, = id, and ( f o g ) * = f* o g*.
(iii) If A is the inductive limit of the sequence (An),,o of C*-algebras then Ki(A) is the inductive limit of the groups Ki(An). (iv) If q5 : J' sequence
4
A , and II, : d
4
23 are *-homomorphisms such that the
O--tJ+A--tB+O be exact, there is a six-term exact sequence of the form:
Ko(J')
-L Ko(A))
IndT
Ki(a))
Ko(B) 1EXP
**
c-Ki(A)
(34)
Ki(3)
In the previous theorem, Ind et E x p are the connection automorphisms defined as follows (whenever A is unital): let P be a projection in B @ K, and let A be a self-adjoint element of A @ K such that 1c, @ id(A) = P. Then II,@ id(e22"A) = e2'"' = 1, so that B = eZaaAE (J' @ K)+ and is unitary in (J' @ K)+. The class of B gives an element of K I ( J ' ) which is, by definition, E z p ( [ P ] ) In . much the same way, let now U be an unitary element of 1 (B@K).Without loss of generality it is the image under $(Bid of a partial isometry W in ( A @ K ) . Then I n d ( [ U ] )is the class of [WW'] - [W*W]in K o ( J ) . These definitions actually make sense.
+
9.1.3
The Connes- Thorn isomorphism
The C*-algebra of a dynamical system introduced in Section 1.6 is a special case of the c*-crossed product construction. Let d be a C*-algebra, G be a locally compact group, and cy be a continuous homomorphism from G into Aut(A) (namely the group of *-automorphisms of A endowed with the topology of point-wise norm-convergence). A covariant representation of the triple (A,G,a ) is a pair of representations (II,p) of A and G on the same Hilbert space such that p(g)II(a)p(g)*= II(a,(a)) for all a E A and g E G. Each covariant representation of (A,G,a ) gives a representation of the twisted
127
convolution algebra C c ( 6 A) , by integration (compare with Section 1.6), and hence a pre-C*-norm on this *-algebra. The supremum of all these norms is a C*-norm, and the completion of C c ( 6 ,A ) with respect to this norm is called the crossed product of A by (6 under the action a , denoted by A > d a 6. The *-representations of A x, G are in natural one-to-one correspondence with the covariant representations of the dynamical system (A, 6,a ) .
Theorem 13 [37] Ki(A >a, R) E Kl-i(A), 3.1.4
for i = 0,l.
The Pimsner 0 Voiculescu exact sequence
The following result can be found in [91] Theorem 14 Let A be a separable C*-algebra, and let (Y be a *-automorphism of A. There exists a six-term exact sequence:
where j is the canonical injection of A into the crossed product. 3.1.5
Morita equivalence
In Section 3.1.1 the notion of Morita equivalence was defined. Namely, C*algebras A, B are called Morita equivalent whenever A @ K N B @ K , if K is the algebra of compact operators. In Prop. l1.iv1 it was shown that if A and B are Morita equivalent, then they have same K-groups. The following theorem will be used here (see for instance [41])
Theorem 15 Let R be a compact metrizable space endowed with an action of Rd b y homeomorphisms. Let X c R be a transversal. Let r ( X ) be the groupoid of the transversal. Then the C*-algebras A = C(R) >a Rd and B = C * ( F ( X ) ) are Morita equivalent. In particular, whenever d = 1, a transversal is called a Poincare' section. The construction of the first return map T, also called Poincare' map, shows that the groupoid r ( X ) is obtained as the action of Z on X defined by T (see Fig. 13). Hence, the C*-algebra of the transversal is nothing but B = C ( X ) x Z.
128
Figure 13. The Poincarb first return map
3.2
Gap Labels
Let be a self-adjoint operator affiliated t o A and satisfying Shubin’s formula (22). Let g be a gap in the spectrum SpH of H . The integrated density of states (IDOS) is constant on this gap. Let n/(g) be its value there. Moreover, x(x 5 E ) E R is continuous if E E g, the characteristic function x E SpH on SpH and does not depend upon the choice of E E g. Therefore the spectral projection x ( H 5 E ) is an element P(g) E A which depends only upon the gap. Hence, it defines an element n(g)= [P(g)]E Ko(A). The Shubin formula implies:
N(B) = %(P(d)= % *(n(e>> ’ namely, the IDOS on gaps is a number that belongs t o the image of the KOgroup by the trace. Since this group is countable such numbers belong to a countable subgroup of R. At last since a projection satisfies 0 5 P 5 1, the trace of a projection satisfies o 5 & ( P ) 5 &(l). Hence: Theorem 16 Abstract Gap Labelling Theorem [12,13] Let H be a self-adjoint operator afiliated t o A and satisfying Shubin’s formula. Then:
(a) For any gap g in the spectrum of H, the value of the IDOS of H o n g belongs t o the countable set of real numbers T,(Ko(A))f[O,‘T(l)]. l
129
t
Spe trum
Figure 14. Conservation rules for gap labels (Theorem 16 (v))
(ii) The equivalence class n(g) = [P(g)]E Ko(d), gives a labelling which is invariant under norm perturbations of the Hamiltonian H within A. (iii) If S c R is a closed and open subset in SpH, then ns = [PSI E Ko(d), where Ps is the eigenprojection of H corresponding to S , is a labelling f o r each such part of the spectrum. Let t E R + H ( t ) be a continuous family of self-adjoint operators (in the norm-resolvent topology) with resolvent in A. (iv) (homotopy invariance) The gap edges of H ( t ) are continuous and the labelling of a gap { g ( t ) } , is independent o f t as long as the gap does not close. (v) (additivity) (see Fig. 14) If f o r t E [to,t l ] ,the spectrum of H contains a clopen subset S ( t ) such that S(0) = S+ U S- and S(1)= S$ U ‘S where S* and 5’; are clopen sets in spec(H(t0)) and spec(H(t1)) respectively, then nS, ns- = nsf ns;.
+
+
Example 6 The first example of such a gap labelling, using K-theory, was the case of the Harper model [12] using the results of Rieffel and Pimsner & Voiculescu [92,97] on the irrational rotation algebra. The Harper model is the Hamiltonian describing the motion of a 2D-electron on the square lattice
130
and submitted to a uniform magnetic field perpendicular to the lattice like in Fig. 15. The Hilbert space of electronic states is t2(Z2). The energy operator is reduced to its discrete kinetic term:
tB m1
Figure 15. A square lattice in a uniform nagnetic field
H = U1+U~1+U2+U~1 where U1, U2 are the magnetic translations in the directions of the two axis [119] defined by
*
U&(*)
= e
te/h +:$
2.dZ
= ( r n l , r n 2 ) E 2 2 , 61 = ( 1 , O )
$,(* - G) >
, z2
=
(0,l).
It is elementary to check that
where 4 is the magnetic flux through the unit cell, whereas = h/e2 is the flux quantum (here h = 27rti). The C*-algebra generated by two unitaries satisfying (36) is denoted by A, and is called the rotation algebra [97]. It is easy to see that the C*-algebra generated by one unitary, say U2, is isomorphic to C(T)if U2 is identified with the trigononietric monomial z E T H eax E @. The commutation rule given in (36) implies that U1 acts on C(T) through this isomorphism as
+
?71fU,-l(z) = f(z 2 7 r a ) .
+
Hence, U1 is the generator of the Z-action a : z E T ++ z 2nc:( mod ax), so that A, is isomorphic to the crossed product C(T)>a, Z. It is clear that
131
+
within the algebra C(T) 8 M,((C) the automorphism a ( F ) ( s )= F ( s 27ra) is homotopic to the identity, so that the action of a on the K-group is trivial so that a , - id = 0. Moreover, the K-groups of the torus are KO 2: Z N K I . The generator of KOis the function f(s)= 1, whereas U2 generates K1. The Pimsner-Voiculescu exact sequence (Theorem 14), splits into two independent short exact sequences
0 H Z h &(A)
a H
ZH 0 ,
i=O,l,
where i, is the canonical injection of C(T)into A, = C(T) X, Zwhereas d is the connecting homomorphism. Since Z is a free group, the only solution is Ki(A) N Z2 for i = 0 , l . In addition, the trace per unit area satisfies
?-(U,nlU,n*) = 0
if
n1
=nz = O ,
7(1) = 1 .
Since 1 is one generator of Ko(A), it is sufficient t o built a projection non equivalent t o 1 t o get another generator. The following result was proved by [92] and can be proved nowadays in several other ways Theorem 17 If a is a n irrational number, and if P E A, is a projection, there i s a unique integer n such that
7 ( P ) = na - [nay] where [XI denote the integer part of x. These integers was recognized by Claro and Wannier [35] in 1978 (see Fig. 16: the horizontal axis corresponds t o the spectrum of HI the vertical axis corresponds t o the value of a ) , on the basis of the numerical calculation by Hofstadter [63] (see Fig. 1). Eventually these integers are the one occurring in the Quantum Hall effect (see Section 4). 3.3
Computing Gap Labels
The first systematic computation of gap labels was performed for 1 D systems in [16,21]. Let R be the Hull of the system and X its transversal. In Section 3.1.5 it has been shown that the K-group of A = C(R) x lR is the same a the one of the groupoid of the transversal C ( X ) x Z where Z acts through the Poincark map T. The Pimsner-Voiculescu exact sequence, defined in Section 3.1.4 allows t o compute the K-group from the topology of X . The simplest case occurs whenever X is a Cantor set.
132
Figure 16. Gap labels for the Harper model: each color corresponds to an integer [84]
Definition 14 Let G be a n Abelian group and let T : G H G be a group isomorphism. The set & = { g E G;3h E G,g = h - T ( h ) }i s a subgroup. The set GT = {g E G ; T ( g ) = g } i s called the group of invariants whereas GT = G/& is called the groups of co-invariants. Theorem 18 [16, 211 Let X be a totally disconnected compact metrizable space, endowed with a Z-action T and let B = C ( X ) >a Z be the corresponding C*-algebra . Let T also denotes the induced action o n the abelian group C ( X ,Z)defined by T ~ ( x=)~ ( T - ~ xThen ).
(i) K , ( B ) is isomorphic to the group of invariant C ( X ,Z ) T . I n particular, i f T is topologically transitive, namely if there is one dense orbit, K1(B) 2
z.
(ii) Ko(B) is isomorphic to the group of co-invariants C ( X , Z ) T . (iii) Let JP be a T-invariant ergodic probability measure o n X and let % be the corresponding trace o n B. Then, the set of gap labels oft?, namely the image of Ko(B) by 3 is the countable subgroup P ( C ( X ,Z))of R.
133
Example 7 The main application is the following. Let U be a finite subset of R that will be called an alphabet. The elements of U will be called letters. A word is a finite sequence of letters. The set of words is denoted by B?. The length of a word w is the number, denoted by IwI, of its letters. If v and w are two words, vw denotes the word obtained by concatenation, namely by associating the list of letters appearing in v followed by the one in w. Then JvwI = 1v1 + IwI. Let now 5%" be the set of doubly infinite sequence of letters 3 = (ui)icz with ui E U for every i E Z.Z acts on Uz through the bilateral shift (T& = ui-1. Given u(*)E !2l let X be the closure in U" of the set { T ~ E ( ' ) n ; E Z} of the shifted of ~ ( ' 1 . X is called the Hull of u(O). A cylinder set is a clopen set of the form Uw,n = { g E X ;U n + i = wi , 1 5 i 5 lull}, for some n E Z and some word w. Cylinder sets form the basis of the topology of X . A cylindrical function f : X H C on X is a function for which there is N E N such that, for 3 E X , f (u)depends only on ( u - N , ~ N - ~ ,,.u . N. - ~ , u N ) . By definition of the product topology, any continuous function f E C ( X ) can be uniformly approximated by a sequence of cylindrical function. As a consequence, it is possible to show that any integer valued continuous function g is a finite sum of characteristic function of cylinder sets. Given any T-invariant ergodic probability measure P on X , the probability P(Uw,n)of the cylinder set Uw,n does not depend on n and is nothing but the occurrence probability of the word w in the sequence a('). Hence, Theorem 18 implies Proposition 13 Let U be a n alphabet and let g(') be a doubly infinite sequence of letters in a, and let X be ats Hull. Let P be a T invariant ergodic probability on X . Then the set of gap labels of the algebra B = C ( X ) >a Z is nothing but the Z-module ZU(o) - generated by the occurrence probabilities of finite words contained in the sequence a('). Let now H be a selfadjoint operator on e2(Z) defined by
HlNn) =
c
h(T-mU, n - m) N m ) 1
mEZ
where, for each n E Z, the map h, : g H h ( g , n ) E C is continuous and satisfies: (i) h(g,n ) = h ( T n g l-n) and (ii) supuEx Ih(g,n)l < 03. Then H E f3 and it IDOS on gaps takes on values ifi Z u-( o ) .
xnEz
A substitution CJ is a map CJ : U ++ B?. It extends t o B? by concatenation, namely if w = ala2...an then ~ ( w = ) a ( a i ) a ( a 2 ) . . . o ( a n ). It will be assumed that (i) 3a E U such that .(a) = vaw,for some non empty words wlw;
134
(ii) u is primitive namely, given any pairs of letters b, c there is an n E N such that u n ( b ) contains the letter c; (iii) u is generating, namely, for any letter b, the length of u n ( b ) diverges asn-oo. Then, the sequence of words u n ( a ) converges in 5%" to an infinite sequence u(O). Such a sequence will be called a substitution sequence. The following is a classical result [94]
Theorem 19 Let be a substitution sequence. T h e n there is a unique T-invariant ergodic probability measure o n its Hull. Let M(a)b,, be the number of occurrences of the letter b in th.e word .(a). Then M ( u ) is the matrix of the substitution u. Because it is a matrix with nonnegative entries, the Perron-F'robenius Theorem implies that it has a simple eigenvalue 8 equal to its spectral radius, called the Perron-Frobenius eigenvalue, with eigenvector v = (Vb)b,a with positive entries and normalized to Ebb$ = 1, called the Perron-Frobenius vector. It follows from the definition of this matrix that vb is the occurrence probability of the letter b in the substitution sequence. Let now 5 % be ~ the set of words of length N . Then, considering 5 % as ~ a new alphabet, u induces on UN a substitution O N defined as follows. If w is a word with N letters and first letter b, let m be the length of u ( b ) and let U(W) = a l , . . a , . Then U N ( W ) = ( a ~ . ~ . a ~ ) ( a ~ . . . a N + l ) . . . ( a , . . . Let then M N ( o ) be the matrix of this new substitution. Then it follows that 8 is also the Perron-F'robenius eigenvalue of M N ( u ) [94], for any N . Moreover [16,21]
Theorem 20 Let u be a substitution satisfying the condition (a), (ii), (iii) above. Let g ( O ) be the corresponding substitution sequence and X be its hull. T h e n the set of gap labels is given by the ZIO-l]-module generated by the components of the Perron-Frobenius vectors of M ( a ) and M z ( u ) , where 8 i s the Perron-Frobenius eigenvalue. Example 8 The generalization of the previous example can be described as follows. Let X be a Cantor set endowed with an action of Zd by homeomorphisms. Let B = C ( X ) x Zd be the corresponding C*-algebra . Then Forrest and Hunton [53], using the Atiyah-Hirzebruch spectral sequences [6] and a result by Adams [2], proved the following theorem Theorem 21 If X is a Cantor set endowed with a minimal action of Zd by homeomorphisms. T h e n the group of K-theory of the corresponding C*algebra B = C ( X ) x Zd is given by
135
where H n ( Z d ,G ) is the group cohomology of Zd with ca$kients in the abelian group G and C ( X ,Z)denotes the abelian group of integer valued continuous functions o n X . Let now P be a Zd-invariant ergodic probability measure on X and let 5 be the corresponding normalized trace on 23. Then it has been proved recently by [25,68] that Theorem 22 If X is a Cantor set endowed with an action of Zd by homeomorphisms and i f P is a Zd-invariant ergodic probability measure on X , then the set of gap labels, namely the image by the trace 7 p induced by P o n L3 = C ( X ) >a Zd, is given by P ( C ( X , Z ) ) .
Example 9 The definition of a finite type repetitive Delone set C given in Section 1.3, is actually the d-dimensional generalization of a bi-infinite sequence of letters. For indeed, thanks to the Voronoi construction (see Section 2.4) the points of L can be seen as tiles instead and the finite type property means that the number of such tiles is finite modulo translations. A prototile is an equivalence class of tiles modulo translation. Let then 2 be the set of prototiles. This is the d-dimensional analog of the alphabet. However, prototiles lead to patches instead of words. A patch is a finite union of tiles of C, but such a notion is too loose. Given r 2 0 an r-patch of 13 is the finite union p of closed tiles of L centered at points of a subset of the form C ( p ) = B(O;r]n (C - z) for some z E C. Here B ( y ; r ]denotes the closed ball of radius r centered at y. Geometrically an r-patch p is a polyhedron. In the following, R will denote the Hull of C and X its transversal. According to the construction given in Section 1.2, any w E R is a Radon measure on Rd supported by a Delone set L,, and giving mass one to each point of L,. Then C, is itself repetitive and has finite type. Since C has finite type, it is easy to check that the prototiles and the patches of C, are the same as for C. It is useful to recall that w E X if and only if 13, contains the origin. Let then p ( r ) be the set of r-patches of C. Since C has finite type, p(r) is finite so that the map r E [O,oo) H p ( r ) is locally constant and upper semicontinuous. Hence there is an infinite increasing sequence 0 = TO < r1 < . . . < T , < r,+1 < ... such that r, 5 r < r,+1 + p ( r ) = p(r,). Let pn denote ' ? J ~ ( T , )and let 9 be the union of the p,'s. In particular, 9 0 = 2. The analogy with the 1 D case is now simple. The alphabet rU is replaced by the set of prototiles 2, the set of words 2D by the set of patches and
136
the bi-infinite sequence is replaced of letters by 13. Then the Hull of by the transversal X of L. Cylinder sets are now given by clopen subsets U ( p ,n) c X , with p E 9, for some n E N, where U ( p ,n) denotes the set of w E X for which the union of the closed tiles centered on points of B(0;r,]nL, is precisely p . In Section 2.4, eq. (31) gives the construction of the Hull from a projective sequence . . . ’ 2 B,+1 ’ 5 B, fn-!. . . of BOF manifolds from the notion of decorated patches. The following result has been proved in [24] which is a an extension of the Forrest-Hunton theorem 21 [53] Theorem 23 Let L be a repetitive Delone set of finite type in Rd. Let R its Hull and let A = C(R) >a Rd be the corresponding Noncommutative Brillouin zone. T h e n (a) the group of K-theory of A is given by d/2
K,(A) N nlim -+
@H
~ (Bn~7 z)+
~
m=O
(ii) The longitudinal homology group group Hd(R) = lim, Hd(b,, R) i s canonically ordered with positive cone Hd(R)+. There is a canonical bijection between Hd(R)+ and the set of Rd-invariant positive measure o n 0. Let now P be an iRd-invariant measure on R. It then defines a canonical probability on X , called the transverse measure induced by P [36,41]. To build this transverse measure, let a box be defined as a subset of R of the form
Z ( p , n ; r ) = {w’ E R ; 3a E I R ~ la,1
< T ,TpaW’
E U ( p , n ) },
where n E N,p is a patch in 9, and r > 0. Then, if L is ro-discrete, the transverse measure is the probability PX on X uniquely defined by
where IAl denotes the Lebesgue measure of A c Rd. If, in addition, P is ergodic, then PX ( U ( p ,n ) ) is nothing but the occurrence probability of the patch p in the tiling associated with 13. The following result was proved in [24] and is complementary to [25,68].
137
Theorem 24 Let L be a repetitive Delone set of finite type in Rd. Let R its Hull and let P be an Rd-invariant ergodic measure on R. Let d = C(s2) >a Rd be the corresponding noncommutative Brillouin zone and Tp the trace induced by P. Then the set of gap labels, namely the image by 7 p of the KO-group of d,is the Z-module generated by the occurrence probabilities of patches.
4
The Quantum Hall Effect
In 1880, E.H. Hall [58] undertook the classical experiment which led t o the so-called Hall effect. A century later, von Klitzing and his co-workers [73] showed that the Hall conductivity was quantized at very low temperatures as an integer multiple of the universal constant e 2 / h . Here e is the electron charge whereas h is Planck’s constant. This is the Integer Quantum Hall Effect (IQHE). This discovery led t o a new accurate measurement of the fine structure constant and a new definition of the standard of resistance [106]. After the works by Laughlin [78] and especially by Kohmoto, den Nijs, Nightingale and Thouless [110] (called TKN2 below), it became clear that the quantization of the Hall conductance at low temperature had a geometric origin. The universality of this effect had then an explanation. Moreover, as proposed by Prange [66,93], Thouless [lo91 and Halperin [59], the Hall conductance plateaus, appearing while changing the magnetic field or the charge-carrier density, are due to localization. Neither the original Laughlin paper nor the TKN2 one however could give a description of both properties in the same model. Developing a mathematical framework able t o reconcile topological and localization properties at once was a challenging problem. Attempts were made by Avron et al. [8] who exhibited quantization but were not able t o prove that these quantum numbers were insensitive to disorder. In 1986, H. Kunz [75] went further on and managed t o prove this for disorder small enough to avoid filling the gaps between Landau levels. However Bellissard [13-151 proposed t o use the Non-Commutative Geometry of Connes [41] to extend the TKN2 argument t o the case of arbitrary magnetic field and disordered crystal. It turned out that the condition under which plateaus occur was precisely the finiteness of the localization length near the Fermi level. This work was rephrased later on by Avron et al. [9] in terms of charge transport and relative index, filling the remaining gap between experimental observations, theoretical intuition and the mathematical framework. The part concerning the localization was later reconsidered by Aizenman and Graf [3] by using more conventional tools. This Section is devoted t o a review
138
of this work which can be found in an extended paper in [22] and in a shorter version in [17].
4.1
Physics
The Hall effect is observed only for very thin flat conductors. Ideally it is a 2 0 effect. The thinnest possible type of material available at the time of Hall were gold leaves that could be produced with thickness of few micrometers. Nowadays, using potential interface between two semiconductors, it is possible to make the electron gas exactly two-dimensional by forcing the quantized transverse motion to its lower energy state. Let then such a conductor be placed in a perpendicular uniform magnetic field (see Fig. 17)
Figure 17. T h e classical Hall effect
If a constant current j’ is forced in the 5 direction, the electron fluid will be submitted to the Lorentz force perpendicular to the current and the magnetic field creating an electric field & along the y axis. In a stationary state, the total force acting on the charge
-
Ftot = q g f j ’ x
I?,
q=ke,
vanishes leading to the relation j’ = u z with u ,called the conductivity tensor, is an antidiagonal antisymmetric 2 x 2 matrix with element &OH given by
139
Here n is the twedimensional density of charge carrier, h is Planck's constant, e is the electron charge and Y is called the filling factor. We remark that the sign of u~ depends upon the sign of the carrier charge. In particular, the orientation of the Hall field will change when passing from electrons to holes. This observation, made already by Hall himself in 1880, was understood only in the late twenties after the work by Sommerfeld on electron theory [64]. It is commonly used nowadays to determine which kind of particle carries the current. 0 3.5,
011
0
1
2
3
2 ,
I
.
4 I
8
6
. ,
,
I
.
1 I
0 ,
1
,
2
1 ,
4
,
,
V
Figure 18. IQHE. left: conductivity (schematic view); right: resistivity (experiment)
The quantity RH = h / e 2 is called the Hall resistance. It is a universal constant with value RH = 25812.800. RH can be measured directly with an accuracy better than lo-' in QHE experiments. Since January 1990, this is the new standard of resistance at the national bureau of standards [106]. As the temperature is lowered to few Kelvins, however, the observation made of semiconductors, like MOSFET [73] or heterojunctions [112], leads to a completely different scenario. As shown in Fig. 18, the conductivity, expressed in units of the Hall resistance, is no longer equal to the filling factor, but is rather a stairlike function with plateaus at integer valued. In heterojunctions, it is even possible to observe plateaus for fractional values of the filing factors (see Fig. 19). The relative accuracy b f l H / f l H of the Hall plateaus can be as low as 10-s-lO-lo depending upon the sample used for the measurement for the integer values. It goes up to for the fraction 113 and higher for other observed fractions. The experiments shows also that the direct conductivity and the direct resistivity as well, vanish on the plateaus and are appreciable only for values of the filling factor corresponding to transition between plateaus. That both the direct resistivity and conductivity vanish at the same time is
140
due to the matrix character of the conductivity tensor, the inverse of which being the resistivity tensor. In particular
MAGNETIC FIELD (Tesla)
Figure 19. FQHE: fractional plateaus of the resistivity (taken from [116])
This experimental fact is explained in term of Anderson localization. Namely for the corresponding the 2 0 electron gas is submitted to a random potential created by the impurities used for doping the system. Without such a potential, the one-particle Hamiltonian would be the Landau model of the free electron in a uniform magnetic field [77]
were q = f e is the charge of the carrier, m, its effective mass, P is its momentum operator while = ( A l , A 2 ) is the vector potential defined by &A2 - &A, = B. It is well known that the spectrum of H L is given by the
A'
Landau levels
141
9B wc = m
n = 0,1,2,...
Each of these levels has an infinite degeneracy in the infinite volume limit, corresponding to a degeneracy per unit area equal to g = eB/h. This number can be intuitively computed as follows: the total number of states available in a large surface of area S is @/Sq50if @ = BS is the total flux through this surface, while q50 = h/e is the flux quantum. Hence the filling factor is the ratio v = n / g of the actual number of electrons et the number of available states in one Landau level. Thus it gives the fraction of the Landau levels that is filled by electrons.
Figure 20. Schematic view of the DOS in the quantum Hall regime
When the disorder potential is turned on, the Landau levels split into large overlapping bands. This gives a density of states (DOS) of the form schematically given in Fig. 20. It shows that the spectrum has no gap. But away from the Landau levels, the states are localized. This has been proved rigorously both for the lattice models [20] and the Landau model with disorder [10,113]. Actually, from the renormalization group technics [l]it is expected that all states are localized in the infinite volume limit, but that the localization length diverges at the Landau levels [34]. Since the electron gas obeys Fermi’s statistics, at zero temperature all states of lowest energy are occupied up to a maximum value EF called the Fermi level. It is possible to vary the relative position of EF either by changing the charge carrier density n, or by modifying
142
the magnetic field B . Since both parameters arises through the filling factor, changing u is equivalent to change E F . Since, in addition there is no spectral gap, the relation EF = f(u) is monotonous. Hence, as long as EF stays in a region of localized states, the current cannot change, explaining why there are plateaux of the conductivity. This explanation must be supplemented by a more serious mathematical proof which is explained in the following Sections. However , this argument does not explain why the plateaus occur precisely at the integers. This is related to a topological invariant that cannot be expressed in term of usual Geometry, but which is the noncommutative analog of a Chern number.
4.2
The Chern-Kubo formula
Since the seminal paper by Laughlin [79], it is generally accepted that the fractional quantum Hall effect (FQHE) is due to interactions between the charge carriers, creating a new type of groundstate. It is also clear from the experimental observations, that interactions can more or less be ignored for Landau level with quantum numbers n > 2. Hence, if the Coulomb interaction between particles is ignored, the charge carrier fermion fluid is entirely described by the one-particle theory. The quantum motion can be derived from the Hamiltonian of the system. A typical example of one-particle Hamiltonian involved in the QHE for spinless particles, is given by
where V,(Z) describes the potential created by disorder in the Hall bar. Here w , which denotes the configuration of disorder, can be seen as a point of the Hull 52 associated with the sublattice of impurities. Then the covariance condition VU(5- 2) = V&,(Z) expresses that moving the sample or changing the reference axis backward are equivalent. Such a model is typical, and is actually accurate for semiconductors. But it may be replaced by others, such as lattice approximants, or particle with spin. In any case, the oneparticle Hamiltonian describing the fermion fluid becomes affiliated to (see Definition 7 ) the C*-algebra A = C(R x R2, B ) (see Definition 6). Standard results in transport theory permit to compute the conductivity in term of the linear response of the fermion fluid under the influence of an external field. This is the famous Green-Kubo formula. In the QHElimit, namely in the limit of (i) zero temperature, (ii) infinite sample size, (iii) negligible collision processes, (iv) vanishingly small electric fields, the direct
143
conductivity either vanishes or is infinite, whereas the transverse conductivity, when defined, is given by [22,110]
It turns out that Ch is nothing but the non commutative analog of a Chern character. Thus Kubo’s formula gives rise to a Chern character in the QHE limit. This is why eq (38) can be called the Kubo-Chern formula, associating Japan with China. The main properties of the non commutative Chern character are the following [41]
(i) homotopy invariance: given two equivalent C1 projections P and Q in A, namely such that there is U E C1(A) with P = U*U and Q = UU’, then Ch(P) = Ch(&). This is actually what happens if P and Q are homotopic in C1 (A).
(ii) additivity given two C1 orthogonal projections P and Q in A, namely such that P Q = Q P = 0 then C h ( P @I Q) = Ch(P) Ch(Q).
+
In particular, the homotopy invariance shows that Ch(PF),when it is defined, is a topological quantum number. One of the main results of Noncommutative Geometry is that this Chern character is an integer provide it is well defined. Thus, thanks t o eq. (38) the Hall conductance is quantized. It will be shown in Section 4.4 that this Chern character is well defined precisely whenever the Fermi level lies in a region of localized states. Moreover changing the value of the filling factor produces the moving of the Fermi level, which does not change the Chern character as long as the localization length stay bounded.
4.3
The Four Traces W a y
In this section four different traces will be defined and used. They are technically needed t o express the complete results of this theory. The first one is the usual trace Tr on matrices or on trace-class operators. The second one, introduced in Section 1.6, eq. (17), is the trace per unit volume Z$ attached to an R2-invariant probability measure P on the Hull. The third one Trs is the graded trace or supertrace introduced in this Section below. This is the first technical tool proposed by A. Connes [38,41] t o define the cyclic cohomology and constitutes the first important step in proving quantization of the Hall conductance [14]. The last one is the Dixmier trace TrDix defined by Dixmier in 1964 [45] and of which the importance for Quantum Differential Calculus
144
was emphasized by A. Connes [39-411. It will be used in connection with Anderson's localization. Let 'H be the physical one-particle Hilbert space of Section 1.6 namely L2(R2). In the language of Noncommutative Geometry, 'H can be seen as the space of sections of a hermitian vector bundle over the noncommutative Brillouin zone. Following Atiyah's proof of the Index theorem, through the Dirac operator [7], it is convenient to introduced a spin bundle (a similar construction has been proposed in Section 1.6). Practically, here, this is done through the new Hilbert space 6 = 'H+ @ 7-l- with 'H* = 'H. A grading operator and the (longitudinal) Dirac operator D are defined as follows:
+
where X = X1 2x2 (here the dimension is d = 2). It is clear that D is selfadjoint and satisfies D2 = X 2 1 . Moreover F is the phase of D,namely F = D1Dl-l. Then F = F* and F 2 = 1. A bounded operator T acting on 6 will said to have degree 0 if it commutes with and of degree 1 if it The graded commutator (or supercommutator) of two anticommutes with operators and the graded differential dT are defined by
g.
[T,T'Is = TT'
-
(-)deg(T)deg(T')T'T
dT
=
[F,T]s.
Then, d2T = 0. The graded trace Trs (or supertrace) is defined by 1 Trs(T)= -Tr,(GF[F, T ] s )= Tr,(T++ - u T - - E ) , 2
(40)
where u = X / l X l and T++ and T-- are the diagonal components of T with respect to the decomposition of 7?. It is a linear map on the algebra of operators such that Trs(TT') = Trs(T'T). However, this trace is not positive. Observables in A will become operators of degree 0, namely A E A will be represented by A , = A, @ A,. Given a Hilbert space 'H, the characteristic values P I , . . . ,pn,. . . of a compact operator T are the eigenvalues of IT1 = ( T T * ) 1 / 2 labelled in the decreasing order. The MaEaev ideals LP+('H)is the set of compact operators on 'H with characteristic values satisfying
145
Let Lim be a positive linear functional on the space of bounded sequences lY(N) of positive real numbers which is translation and scale invariant. For T E L1+(X) its Dixmier trace is defined by
1 n,ix(T) = Lim(In N
N
C pn) . n=l
Clearly, T E L1+ if and only if TrDix(lTI) < 00 and if the sequence I N ( InN pn) converges, then all functionals Lim of the sequence are equal to the limit and the Dixmier trace is given by this limit. From this definition, it can be shown that TrDix is a trace [41,45]. The first important result is provided by a formula that was suggested by a result of A. Connes [39]. Namely if A E C 1 ( d ) and i f ? = (al,a2) [22]:
&( 1 ?A12)
=
1
FTrDix(ldAW12)7
for P-almost all w .
(41)
Let now S denote the closure of C1(d) under the noncommutative Sobolev norm 11A11: = &(A*A) '&(?A*GA). The eq. (41) shows that for any element A E S, dA, belongs t o L2+(7?),P-almost surely. In what follows, L"(d,'&) denotes the weak closure of d in the GNS-representation with respect t o Tp (see Section 1.6).
+
The following formula, valid for Ao, A1, A2 E C'(d), is the next important result proved in [9,14,22,38]:
This formula actually extends to Ai E L"(d, '&) n S. For indeed, the right hand side is well defined by the Cauchy-Schwartz inequality. On the other hand, thanks to eq. (41), dAi,, E L2+ c L3(7?),if L P ( X ) denotes the Schatten ideal of compact operators T on X with trace class I T I P . Then the integra%d occurring under the integral of the left hand side can be written as ~ ~ ( G ~ ~ A o , , d A ~ , , dwhich A ~ , ,is) well defined, thanks to Holder's inequality.
146
Applying these formulzz to the Fermi projection, the Chern character Ch(PF) is well defined provide PF E S and
C ~ ( P F= )
~ ( w ) T r s ( ~ F , w d ~ F , w d .~ F , w )
(43)
The last step is a consequence of the Calder6n-Fedosov formula [32,50] namely the operator PwF+-Ip,~is Fredholm and its index is an integer given by:
n ( w ) = Ind(PwF+-IP,n-) = ns(&,wd&,wd&,w)
.
(44)
It remains t o show that this index is P-almost surely constant. By the covariance condition P ~ a ~ F + - l p ~ ~and , n -PwT(cz)-lF+-T(.)Jp,nare unitarily equivalent, so that they have same Fredholm index. Moreover PwT(a)-lF+-T(a)(p,n- - PwF+-Ip,n- is easily seen t o be compact so that p~a~F+-Ip,,,n- have the same index as PwF+-Ip,n-. In other words, n ( ~is) a R2-invariant function of w. The probability IP being R2-invariant and ergodic, n ( w ) is IP-almost surely constant. Consequently, since F+- = u, if PF E S :
ch(pF) = Ind(PF,wUlPF,_X) E
z
P-almost surely
.
In [9], Avron et al. showed that this index can also be interpreted as
l P F , ~It. The right hand side is called the relative index of u P ~ , ~ u -and represents the variation of the dimension of the projector P F ,when ~ .the unitary transformation u is applied. It turns out that u = X / l X l is exactly the (singular) gauge transformation applied to the original Hamiltonian whenever an infinitely flux tube is pierced at the origin and the flux is adiabatically increased from 0 t o one flux quantum. Laughlin [78] argued that this number is exactly the number of charges that are sent t o infinity under this adiabatic transformation.
4.4 Localization It remains t o show how the condition PF E S is related to the Anderson localization. The easiest way to define the localization length consists in measuring the averaged square displacement of a wave packets on the long run. Let A c R be an interval. Let Pa be the eigenprojection of the Hamiltonian
137
corresponding t o energies in A. If X is the position operator, let xA+,(t) etHwtPA,wXPA,we-tHwt. The A-localization length is defined as:
=
It is shown in [22} that, equivalently,
= sup
c
%(IvPaj12).
(46)
A'EP
where P runs in the set of finite partitions of A by Bore1 subsets. Moreover [22] Theorem 25 If c2(A) < 03, then the spectrum of H , is pure point in A,
P-almost surely. The density of states is the positive measure N on R defined by d N ( E )f ( E ) = &( f ( H ) ) (see eq. (22)) for f a continuous function with compact support. It turns out [22] that if Z2(A) < 03, there is a positive N-square integrable function e on A such that
s,
for any subinterval A' of A. Then, [ ( E )is the localization length at energy
E. Thanks to eq. (46), (47) the finiteness of the localization length in the interval A implies that [22]
(i) PF E S whenever the Fermi level EF lies in A, (ii) EF E A ++ PF E S is continuous (for the Sobolev norm) a t every regularity point of N.
(iii) Ch(PF)is constant on A, leading to existence of plateaus for the transverse conductivity.
(iv) If the Hamiltonian is changed continuously (in the norm resolvent topology), Ch(PF) stay constant as long as the localization length remains finite a t the Fermi level.
148
As a Corollary, between two Hall plateaus with different indices, the localization length must diverge [34,59,75]. The reader will find in [22] how to compute practically the Hall index using homotopy (property (iv)) and explicit calculation for simple models.
149
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Geometric and Topological Methods for Quantum Field Theory Eds. A. Cardona, H. Ocampo & S. Paycha @ 2003 World Scientific Publishing, pp. 157-227
NONCOMMUTATIVE GEOMETRY AND ABSTRACT INTEGRATION THEORY MOULAY-TAHAR BENAMEUR' Institut G. Desargues, universite' L y o n l , 43, Bd du 11-nou-1918, 69622 Villeurbanne Cedex, France, Email:
[email protected] We give an introductory survey of some recent developments in type I1 non commutative index theory. The Dixmier trace is extended to the case of type 11, spectral triples and a local formula for the Hochschild class of the Chern-Connes character is given. We also extend the Calderon formula to the equivariant and type 11, case and we deduce an equivariant von Neumann cyclic cohomology pairing which implements the index map associated with any equivariant von Neumann spectral triple. A brief survey of the classical Atiyah-Singer index theorem is also given.
Introduction The Atiyah-Singer index theorem stands at the junction of many different mathematical trends and is a t the origin of many recent developments. It is also involved in the understanding of innumerable results in some areas of physics. This theorem is, t o the opinion of the author, certainly one of the deepest discoveries of the last forty years. Given a smooth closed manifold M , an elliptic differential operator D on M has finite dimensional kernel and finite dimensional cokernel. Therefore, the Fredholm index of D can be defined by Ind(D) = dim(Ker(D)) - dim(Coker(D)). In the early fifties, Gelfand stated the problem of computation of the index of elliptic operators by topological formulae, i.e. involving topological data of the operator and of the underlying manifold. The solution of this problem has been obtained in many steps but the general result was achieved by Atiyah and Singer in 1963-65 using a new powerful tool: topological K-theory. Beyond the index formula then obtained, the methods developed by Atiyah and Singer cleared up the index problem and produced fundamental K-theory constructions. For spin manifolds for instance, K-theory techniques enable t o reduce the index problem t o the computation of the index of twisted Dirac operators. In this case and for any hermitian vector bundle E over an even spin closed *partially supported by a CNRS grant
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manifold M, we get the famous formula: Ind(d2)
=< Ch(E)A(M),[MI >,
where 82 is the positive part of the self-adjoint Dirac operator twisted by the bundle El Ch(E) is the Chern character of E , A(M) is the A-genus of the tangent bundle T M of M and where [MI is the fundamental class of M . So by Poincar6 duality, this formula becomes: Ind(d,f)
=< Ch(E),A(M) fl [MI >,
where A(M) n [MI E H,(M, Q) is the Poincar6 dual of A(M). Viewing this index as a pairing between E and the operator d [2], we can define the Chern character of the Dirac operator as the homology class Ch(d) := A ( M ) n [MI. The index formula can then be reinterpreted as a homological pairing: Ind(d2) =< Ch(E),Ch(d) > .
(0.1)
In other words, the Atiyah-Singer index theorem computes the so called Atiyah map:
KO ( M ) -+ Z defined by E H Ind(8;). Formula (0.1) is the starting point of the Non Commutative Geometry point of view on index theory. More precisely, A. Connes has succeeded in giving an appropriate framework for index problems, when the algebra C"(M) of smooth functions on the manifold M is replaced by a, non commutative in general, smooth algebra A. The motivating examples came from the study of singular spaces such as the space of leaves of a foliated manifold or the space of irreducible representations of a given discrete group, but many other important examples arose later from the physics of solids or quantum field theory (QFT). The concept of Dirac operator led A. Connes to his formalism of spectral triples. In the presence of any such spectral triple an index problem can be stated and the spectacular Connes-Moscovici local index theorem is a complete solution of this general index problem. The first interesting but highly non trivial example, and that cannot be reached by classical index theory techniques, encodes the Diff-equivariant index theory. It was defined by Connes and Moscovici by using a Diffequivariant signature operator. The identification of the local terms appearing in the Connes-Moscovici index formula for such triple led A. Connes and H. Moscovici to define an appropriate cyclic homology for Hopf algebras [31]
159
where the involved characteristic numbers could live. But the story has just begun and a large program remains to be carried out even in simpler examples. On the other hand, and since the appearance of the Atiyah-Singer index theorem, many generalizations have been obtained [l,11,23,25,26,49,62],encompassing many new geometric situations. Among them, some constructions involve non Fredholm operators and use in a decisive way an extra ingredient: von Neumann algebras and the Murray-von Neumann dimension theory. While the index of elliptic operators on non compact manifolds for instance, does not make much sense, some useful assumptions enable to affiliate them to appropriate type I1 von Neumann algebras and to give a Murray-von Neumann sense to their index. This is true for instance in the examples treated in [1,23,62]. We have thus taken the opportunity of the present notes to explain how von Neumann algebras can fit into the Non Commutative Geometry framework and how the formalism of spectral triples can be generalized to deal with the type I1 index and also with group actions. All the examples listed above become corollaries of a unifying von Neumann approach to the ConnesMoscovici index theorem. The purely non commutative consequences of such results are also interesting directions for the future. The present notes summarize a more detailed course adressed to beginners and we have devoted the whole first chapter to an overview of the fundamental Atiyah-Singer index theorem. We point out on the other hand that some of the results of the second chapter rely on a larger investigation carried out in collaboration with T. Fack. We have chosen to expound the case of foliated flat bundles as an experimental application. In particular, we show how the Atiyah covering index theorem can be generalized to flat bundles and even to general Galois coverings of foliations. This paper is divided in two chapters. The first one is adressed to beginners in index theory and is a tentative introduction to the now classical Atiyah-Singer index theorem. The second chapter on the other hand deals with the type I1 index theory in Noncommutative geometry. Experts in the domain can therefore skip the first chapter. The contents of this paper are more precisely as follows: Section 1 summarizes the index theory for elliptic complexes on smooth closed manifolds and gathers some background material. Section 2 is devoted to the index theorem and some of its corollaries are treated in Section 3. In Section 4, we introduce Dixmier traces and residues of zeta functions in type 11, von Neumann algebras. Section 5 summarizes the index theory with respect to a given von Neumann algebra and explains the constructions in an instructive example. In the last Section, we give some definitions and examples of von
160
Neumann spectral triples with the statement of the associated index problem. The equivariant case is also treated and an equivariant Chern-Connes character is proposed. We assume as known some background material as topological Ktheory, characteristic classes, von Neumann algebras and also some classical properties of pseudodifferential operators on manifolds. However, for the convenience of the reader and because these notes are intended to be understood by a large public, we have tried as much as possible t o recall some of the material or to give the appropriate references.
Acknowledgements. I am indebted to C. Brouder, T. Fack and A. Frabetti for the many helpful remarks on a previous version of the present paper, and to C. Blanchet, J. Bellissard, S. Paycha, J. Varilly, T. Wurzbacher and J. Zanelli, for the several interesting discussions and comments during the school. I also would like to thank A. Cardona, M. Kovacsiks, S. Adarve and again Sylvie Paycha for their kind and warm hospitality in this beautiful small village of Villa de Leyva.
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Chapter I. Review of the classical index theory Index theory has gained ground during the last thirty five years. However, we believe that the original version of the Atiyah-Singer index theorem remains the best way t o introduce this active research subject t o neophytes. In this first chapter, we shall review some of the basic tools needed t o understand correctly the Atiyah-Singer index theorem. The detailed proof is not included in this short review and can be consulted in the literature. 1
Some preliminaries and examples
We begin this section by an overview of the index theorem for Toeplitz operators on the circle. Then we briefly recall the definitions and properties of pseudodifferential operators and state the Atiyah-Singer index theorem with a sketch of proof and some applications.
1.1
The index theorem f o r Toeplitz operators
Toeplitz operators on the circle are an interesting toy model and we begin by describing the meaning of the index theorem for them. We also explain how they can be generalized to higher dimensional odd manifolds. We denote by S1 = R/27rZ the unit circle with its usual normalized Haar measure d0/27r. The Fourier transform enables to identify the Hilbert space L2(S1)of measurable square integrable functions on S1 (identified with L2(0,27r))with the space Z2(2)of square summable sequences of complex numbers. This identification corresponds t o the decomposition of L2-functions with respect to the orthonormal basis (en)nEZ defined by e,(e) = cine. More precisely, we set: tlf E L2(Sl),tlnE
Z,Cn(f)
:=
t o obtain a sequence ( ~ ~ ( f ) which ) ~ ~ belongs z t o 12(Z). The Hilbert subspace 12((n) of 12(Z)corresponds under this Fourier identification to a Hilbert subspace denoted H2(S1),of L2(S1). The Hilbert space H2(S1)is called the Hardy space and corresponds in fact t o L2-functions on the circle which admit a holomorphic extension t o the unit disk. The projection p : L2(S1) H2(S1) is usually called the Szego projection. It corresponds under the Fourier identification to the quasi-trivial projection 12(Z)-+ 12(W) and is implemented by the Poisson kernel. Let q5 be any continuous (for simplicity) function on the circle S1, then pointwise multiplication by q5 determines a bounded operator
162
M4 on L2(S'). Moreover the map M which assigns to each q5 the operator M4 is an isometric representation of the commutative C*-algebra C(S') of continuous functions on S', in the Hilbert space L2(S') "= 12(Z).
Lemma 1. For any continuous function q5 on S', the commutator [p, M4] i s a compact operator o n L2(S1). Proof. The map q5 4 M4 is a continuous homomorphism (in fact an isometry) and the ideal of compact operators in L2(S') is norm-closed. Hence using the Stone-Weierstrass theorem, it remains to show that [p, M4] is compact whenever q5 is a trigonometric polynomial. So we can even assume that q5 = em, with m E Z. But then we can easily see that [p,Mem]coincides with the projection onto the finite dimensional space generated by the family 0 ( e j Min(-m,o)<j<_M,,(-m,o) and SO is compact. Definition 1. For any continuous function q5 on S', we define the Toeplitz operator T+ associated with q5 as the composite operator p o M4 acting on H2(S1).In other words:
Toeplitz operators play a fundamental role in operator theory and the invariant subspace Hilbert's problem. See for instance [34]. With respect to the canonical basis (en),20 of H 2 ( S ' ) , such Toeplitz operators are infinite matrices with entries given by: (T4)i,j = Ci-j(q5).
So these matrices have constant diagonals. Examples. (1)If cn(q5) = 0,Vn < 0 then M+ preserves the Hardy Hilbert space H 2 ( S ' ) and thus T4 = M + l p ( s ~ l . (2) If q5 := e-, with n E N*, then for any m E Z we have:
T+(em)= ~ ( e m - n ) . Hence we obtain:
We can see in the example (2) that the kernel of Te-n is finite dimensional and is the subspace of H 2 ( S ' ) generated by (eo,... ,e,-~), while Te-,, is surjective. Therefore Te-,, is a very simple example of a Fredholm operator on the Hilbert space H2(S1).Its Fredholm index (say dim(Ker) -dim(Coker)) is exactly the integer n. But there is a "topological" way to recover the integer n from q5 = e-,, since it is exactly minus the winding number w(e-,) of the
163
non-vanishing smooth function e-, on the circle. Recall that for any smooth nonvanishing function q5 on the circle S1,the winding number w(4) of q5 is the Cauchy integral:
The above relation between the winding number of e-, and the index of the Toeplitz operator with symbol e-, is a general phenomenon and we have: Proposition 1. Let q5 be a smooth (for simplicity) n o n vanishing function o n S1. T h e n the operator T+ i s a Fredholm operator o n H 2 ( S 1 ) . The index of the Fredholm operator T4 can be computed by the formula: WT4) =-44),
where w(q5) is the winding number of q5. This very simple proposition is in fact the Atiyah-Singer index theorem on the simplest manifold S1. The idea of its proof is already in the spirit of the general proof.
Proof. We have on L 2 ( S 1 ) : PO Ml/4 OPO M4 OP - P
=P
O
[Ml/47PI
0
M 4 O P.
But the operator [ M l / 4 , p ] is'a compact operator on L 2 ( S 1 )by Lemma 1. Therefore the operator p o [ M l / 4 , p ]o M4 is a compact operator on L 2 ( S 1 ) which preserves H 2 ( S 1 ) . Hence it is a compact operator on H2(S1). This shows that is a left inverse of T4 modulo compact operators. It is easy to see that it is also a right inverse of T4 modulo compact operators. Recall that the map q5 H w(q5) classifies the homotopy class of 4 so that two such functions that have the same winding number are necessarily homotopic. Let now q5 be a smooth non-vanishing function on the circle and set m = w(q5). Then q5 is homotopic to the function em by the above property of the winding number. Therefore, the Toeplitz operator T4 is homotopic to the Toeplitz operator T, inside F'redholm operators. Hence the F'redholm index of Tb is equal to the F'redholm index of T,,. This last index is easily seen to be equal to -m as we have already observed. 0 Hence we are able to compute the (analytic) index of the operator T4 as an integral over the manifold S1 of a differential form given by &q5-'d+. To prove this index theorem, we have used that the space of connected components of Cm(S1,C " ) is isomorphic to Z. This is a very simple expression of Bott periodicity. In the general proof of the Atiyah-Singer index theorem, Bott periodicity is also fundamental as we shall see.
164
More generally now, if u is an invertible matrix of smooth functions on S1,say u E GLN(C"(S')), then we can also consider the matrix of 1-forms L 2 iu r - l d u and it is also well known that the integral of the trace of this matrix over S1 is always an integer (see Proposition 2 below) also called the winding number of u, i.e. W(U)
:=
1
ll
Tr(u-ldu) E Z.
To recover this integer as a Fredholm index, one simply considers a generalization of the Toeplitz construction. Hence, we consider the operator Mu associated with u corresponding t o pointwise matrix multiplication by u in L2(S1)Nand define the Toeplitz operator
T, : H2(S1)N -+ H 2 ( S1) N ,T, := ( p @ 1 ~0 Mu ) Again because u is invertible, the same arguments as in Proposition 1 show that T, is a F'redholm operator in the Hilbert space H2(S1)N. Bott periodicity now states that the group of connected components of C m ( S i , G L ~ ( ( C ) is ) isomorphic to Z. For the sake of simplicity, this result is admitted in the following proposition: Proposition 2. ( I ) The winding number w(u) is an integer and the map u -+ w ( u ) is a group homomorphism; (2) The map u H Ind(T,) is a continuous group homomorphism; (3) We have the following index formula: Ind(T,)
=
-w(u).
Proof. The matrix u defines a smooth loop in G L N ( C )and the map u H Ind(T,) is again a continuous function which only depends on the homotopy class of u. On the other hand, UJ is also a continuous function of u for the C1 topology, and it is integer valued. The justification is classical: Set for s E [0,27r], F ( s ) = u-l(t)u'(t)dt,then the derivative of the MN(C)-valued smooth function s H u(s)e-F(S) is everywhere zero. Thus
s;
IN =
u ( 2 7 r ) - 1 ~= ( ~e-F(2r). )
and hence em(F(2fl))= det(eF(2")) = 1.
Therefore 3k
E
Z such that Tr(F(27r)) = 2ik7-r. This shows that W(U)
1 = -Tr(F(27r))E 2.1~
z.
165
So we have two locally constant integer valued functions w and u -+ Ind(T,), which are in fact group homomorphisms. This can be justified for instance as follows: Let us consider u,w E C m ( S 1 , G L ~ ( C )and ) set x := uw @ 1~ E C " ( S 1 , G L ~ ~ ( C ) )Then . the Toeplitz operator associated with x is exactly the direct sum of the Toeplitz operator associated with uv and the identity operator. Hence Ind(T,,) = Ind(T,). In the same way, the winding number of x can be computed as follows: W(X)
= w(uw)
+ W ( 1 N ) = w(uv).
But a classical argument shows that the element x is homotopic t o the element u @ v. In fact we have:
and we notice that:
which enables to construct the allowed homotopy. Hence the two functionals are homomorphisms from the multiplicative group C"(S1, G L N ( C ) )t o the group Z. Now let us admit that for any N 2 1, the homotopy classes of smooth paths in G L N ( C )form a free abelian group isomorphic t o Z, we deduce that w is an isomorphism for any N . This is justified by the fact that w is clearly surjective. Hence if u satisfies w(u) = m E Z, then taking w = diag(e,, ,. . . , e m N ) with C mi = m, we get w ( u ) = w(w) and hence u is homotopic to w in C"(S1,GL~(C)). Therefore and since the index is a continuous integer valued functional, Ind(T,) = Ind(T,). Finally, it remains t o compute the Fredholm index of T,, but this is easily seen to be equal t o mz = -m. 0
cz
We have seen in the above proof a if u belongs t o C"(S1,GL~(C))then changed when u is replaced by u @ 1" under consideration are in fact defined
GLm(C"(S1))
= C"(S1,
first appearance of K-theory. In fact, its index (or winding number) is unfor any N'. Therefore the functionals on the inductive limit
GL,(C))
:= limCm(S1,GLN((C)), +
with respect to the "adding 1 on diagonals" inductive system. Furthermore since these two functionals only depend on the connected components of
166
C"(S1,GL,(C)), on the group
we deduce that these two functionals are in fact defined
where GLE)(C"(S1)) is the connected component of the identity. So the index theorem is the claim that the following two isomorphisms are equal -w,Ind : K1(C"(S1))-+ Z. What we have precisely used in the proof is that K1(Cm(S1))N Z. This is again one of the different versions of Bott periodicity. Let us now anticipate what will be treated in detail in the next sections. Recall that the Dirac operator on S1 is the usual self-adjoint operator D = -a$. It corresponds under the Fourier decomposition to the diagonal operator
D(en) = nen,Vn E Z. Let now F be the Hilbert transform on S1,say the bounded symmetry defined by:
F(e,)
= sign(n)
en if n # 0 and F(e0) = eo.
Then the relation FlDl = D holds and F is equal to the sign of the operator D up to the rank one operator which is the projection onto the kernel of D , i.e. onto the complex line Ceo. The Szego projection p can also be reinterpreted as the projection onto the eigenspace associated with the positive eigenvalues of the Dirac operator D and is equal to (1 F ) / 2 . Thus, given an invertible element u E C"(S1,GL~(C)), the Toeplitz operator can be defined as the composite operator T, = 8 1 ~ o1u o 8 1 ~ 1 In . fact, the operator p = is a pseudodifferential operator with symbol given on the cosphere bundle of S' by (see definitions in the next subsection):
+
[q
[F
a ( p ) ( z ,1) = 1 and a ( p ) ( z ,-1) = 0,
Vz E S'.
Now extending the Toeplitz operator T, to L2(S1) by setting A, := T, + (1 - p) 8 IN we can easily see that the principal symbol of the pseudodifferential operator A, is exactly pointwise multiplication by u on S' x (1) and is the identity on S1 x {-l}. The Chern character of this principal symbol is therefore given by:
167
see for instance [9].Thus the above index formula can be reinterpreted as the homological pairing: Ind(A,)
=< Ch(a(A,)), [S'] >
It is the goal of the Atiyah-Singer index theorem to produce such formula on each smooth closed manifold computing the analytic index of an elliptic pseudodifferential operator as the integral over the manifold of a local topological data of the operator and of the manifold.
1.2
Some properties of pseudodigerential operators
We shall very briefly recall some properties of classical pseudodifferential operators. For more details we refer to the lecture notes of S. Paycha [59]in the same volume. Let X be a smooth closed manifold of dimension n. Let Eo,El be two smooth hermitian vector bundles over X and denote by P m ( X ;Eo,E l ) the space of classical pseudodifferential operators of order m over X acting from the smooth sections of Eo to the smooth sections of E l . The formula for the change of coordinates shows that the top component of the locally defined symbol of P yields a global section o ( P ) := p , of the vector bundle Hom(Eo,E l ) lifted to the cotangent bundle of X . This section is called the principal symbol of P. More precisely, if 7r : T * X \ X + X is the projection, then u ( P )E Coo(T*X\X, Hom(7r*Eo,7r*E1)),and we get a linear morphism:
a : P " ( X ; E', E l ) + C-(T*X
\
X , Hom(7r*Eo,7r*E1)).
The range of this symbol map lies in the m-positively homogeneous sections and therefore is completely determined by its restriction to the cosphere bundle S * X , i.e. the quotient bundle of T * X \ X under the radial action of the multiplicative group R;. In other words, m-homogeneity is carried by a line bundle L , and we have
a : P"(X; E o ,E l ) -+ C m ( S * X ,L, 8 Hom(7r*Eo,7r*E1)). We denote by H " ( X , E i )the s-Sobolev Hilbert space of ( X , E i ) . The pseudodifferential operator P of order m extends for any s E R to a bounded operator
P, : H S ( X EO) , -+ H S - , ( X , E l ) .
-
Moreover the Rellich theorem states that for t < s, the inclusion H " ( X ,E z ) H t ( X , E i ) is a compact operator. Therefore if m < 0 then P induces a bounded operator PO between the Hilbert spaces of L2-sections which is in addition compact. To get similar results in the case of non compact manifolds,
168
one needs to use local Sobolev spaces, this is described for instance in [63].The proof of these classical assertions can be found for instance in [42]. Moreover, we have a more precise relation between the order m < 0 and the ideal of compact operators where P = Po lies. Lemma 2. Let E be a hermitian vector bundle over the closed Riemannian n-manafold X . If P E P " ( X ; E ) with m E 27, then the extended bounded
operator P : L 2 ( x ,E ) + L 2 ( X ,E ) satisfies: (1) If m < -n then P belongs to the ideal L 1 ( L 2 ( XE , ) ) of trace-class operators; (2) If -n 5 m 5 -1 then P belongs to the Dixmier ideal L-nlm,m ( L 2 ( X E , ) ) . In particular, i f P E T n ( X ;E ) then P is in the E)). Dixrnier ideal L1ym(L2(X, The Dixmier ideal L 1 ? " ( L 2 ( X , E ) )is the space of compact operators T whose singular numbers IlTlI = p o ( T ) 2 Pi(T) 2 . . . 2 pk(T) 2 . . . N
satisfy &=Opk(T) = O(Log(N f I)), see [29]. The Dixmier ideal L'>"(Lz(X,E ) ) for r 2 1 is here the space of compact operators T such that (T*T)T/2 belongs t o L1?"(L2(X,E)), where T x is the adjoint operator of T . For a positive operator T E L1>"(H), the Dixmier trace Tr,(T) of T was defined [29]as a renormalized limit:
-
where a = (a,),?o w ( a ) = limw(a,) denotes a state on P ( N ) which vanishes on Q(N) and satisfies lim, a, = lim, a2,. The above lemma shows that the Dixmier trace of any order -n pseudodifferential operator makes sense. We can be more precise in the above lemma: If p k is the decreasing sequence of eigenvalues of the compact operator [PI,then we have [62]: pk
kml".
Assume that P is a pseudodifferential operator on the closed n-manifold X with coefficients in the vector bundle E as before but such that the order of P is exactly -n. Then A. Connes has proved the following formula for the Dixmier trace Tr,(P) of P [24]:
Tr,(P) =
~
n(27rIn
1
S'X
1 T r ( a ( P ) ( z , < ) ) d z d= < - x lim zTr(PoA-'), n Z+O+
(1.1)
169
where A is a Laplace operator and ,(P) is the principal symbol of P , so it is homogeneous of degree -n. This integral is taken with respect to the Liouville measure on $ * X , i.e. the interior product of the symplectic measure on the cotangent bundle T * X by the outgoing radial vector field R on each fibre of S * X . Hence Tr,(P) is independent of the mean w used to define it and is given by the integral over the cosphere bundle of a local residue. The Dixmier trace is therefore a local data of the operator P. Denoting by P" ( X ,E) the set of classical pseudodifferential operators of integral order on the smooth closed manifold X with coefficients in E, we obtain a filtered algebra for the composition of operators [42]. In addition, the formal adjoint of a pseudodifferential operator is also pseudodifferential. We denote by P - " ( X , E) the ideal of regularizing operators defined by:
P-"(X, E )
:= n m E Z P m ( XE). ,
Hence we have an exact sequence of algebras: 0 -+ P-"(X, E)
P " ( X , E) 4 d ( X ;E) -+ 0 ,
The quotient algebra d ( X ;E) is called the algebra of complete symbols on X with coefficients in E . To each pseudodifferential operator P we can associate, using the Schwartz theorem, a distributional kernel k(z,y) E Hom(E,, E,) so that for any u E C " ( X , E),
This kernel has the additional property that it is smooth outside the diagonal and is more and more regular as the order of P decreases. For instance when the order is < -n, this kernel is continuous and the trace of P can then be computed by the formula:
Tr(P)=
s,
Tr(k(z,z))dz.
Operators in P-"(X, E ) have smooth Schwartz kernel and thus are regularizing operators, i.e. They send Sobolev distributional sections to smooth sections. Up to regularizing operators, we can always assume that the Schwartz kernel of the pseudodifferential operator under study is supported in a given neighborhood of the diagonal. Recall that a trace is by definition a cyclic 0-cocycle [25]. As we have observed, any trace on the algebra P " ( X , E) which is trivial on the ideal P-"(X, E) furnishes a local tool for index theory. Moreover, any cyclic cocycle on P " ( X , E) which vanishes on P-"(X, E) also yields a local ingredient.
170
These remarks show the relevance of the computation of the cyclic cohomology of the quotient algebra d ( X ,E ) . For the basic definitions about cyclic (co)homologies, we refer for instance to [25,29,48,53]. Proposition 3. (211 Let d(X,E ) be the algebra of complete symbols o n the smooth closed manifold X with coefficients in the vector bundle E . Then the cyclic cohomology HC*(d(X,E ) ) ofd(X, E ) is given by:
HCk(d(X,E ) ) N @HkV2j(S*Xx S1), j20
where S*X is the cosphere bundle of the manifold X . By direct inspection we deduce that the periodic cyclic cohomology HP*(d(X,E ) ) is given by:
HP'(d(X, E ) ) N @H"'j(S*X
x
S1).
j€Z
We refer the reader to [29,53] for these basic concepts of NCG. Remark 1. The easy consequence of this computation is that the space of traces HCo(d(X,E ) ) is one dimensional. We also deduce that the only trace, called the residual trace and denoted RTr, is up to constant given by:
where u-,(P)is the component of order -n of the complete symbol of P. This trace is the famous Wodzicki residue trace [67]. The above homology computation can be generalized to other geometric situations. A good tool to reach interesting geometric situations is then to use groupoids. This generalization will not be taken up in these notes and we refer the interested reader to [18] for the computation in a general class of groupoids including for instance manifolds with corners as well as foliations. Here we only give the result. Let then D be any longitudinally smooth groupoid with corners and let AQ be the Lie algebroid of Q and A*D the Lie coalgebroid of Q. The sphere bundle of A*Qwill be denoted by S*G. The algebra +"(G) of classical (scalar for simplicity) pseudodifferential operators on Q was defined in [57] and it recovers all the interesting situations. We consider the algebra d L ( Q ) of complete symbols thus obtained but which have rational expansion near the corners. Then the periodic cyclic homology of d L ( Q ) is given by:
HF"(dL(Q))
N
@ H L k-2j
($* Q x
$9,
j€iz
where HE denotes rational de Rham cohomology. See [18] for more details.
171
1.3
The index of elliptic complexes
Definition 2. A pseudodifferential operator P E P m ( X ;Eo,E l ) is elliptic if its principal symbol .(P) is a section over the cosphere bundle S*X of isomorphisms from Eo t o E l , i.e. V(x,E) E S*X,.(x,J)
: E:
4
E; is an isomorphism.
Theorem 1. Let P E P m ( X ;Eo,E l ) be a n elliptic pseudodifferential operator over the smooth closed manifold X . Then there exists a pseudodifferential operator Q E P - m ( X ; E l , E o ) such that: 1 - Q P E P - " ( X ; Eo,E o ) and 1 - P Q E P - " ( X ; E l , E l ) . For the proof of Theorem 1, see for instance [63]. Lemma 3. If P E P m ( X ; E o , E 1 )is elliptic then its nullspace Ker(P) in the Sobolev Halbert space H " ( X ,E o ) does not depend o n s and is built up of smooth sections of Eo.
Proof. Let 1 - Q P = R be regularizing. Let u E Ker(P) then we have R(u) = u and therefore u is a smooth section of Eo since R is regularizing. 0 Definition 3. Let P : C m ( X ,E o ) -+ C " ( X , E l ) be an elliptic pseudodifferential operator. Then we define the analytic index Ind(P) of P as: Ind(P) := dim(Ker(P)) - dim(Coker(P)). In the above definition, we have used that the kernel of the extension of P to Sobolev spaces is the same as the kernel of P acting on smooth sections and is finite dimensional. The cokernel of P can also be defined either as the kernel of the adjoint of the extension to the Sobolev spaces or as the kernel of the formal adjoint acting on the space of smooth sections, the resulting space is the same. In the same way we consider, more generally, complexes of pseudodifferential operators. A pseudodifferential complex ( E ld ) = (EZ, dZ)o
5 C " ( X , E l ) % .. .
C " ( X , Ek) 4 0 ,
"l
of pseudodifferential operators such that d j + l o d j = 0, V j . Such a complex is elliptic if the corresponding sequence of principal symbols 0
4
Y'..*(El)
T*(EO)
0 3 1 )
is exact over the cosphere bundle rr : S*X
.. .
4
a(&-1) -+
X.
rr*(Ek)4 0
172
The case k = 1 corresponds of course t o the previous definition of elliptic operators. The link between complexes and operators is made explicit in the following proposition: Proposition 4. Let ( E , d ) be a pseudodifferential complex as above and fix as before hermitien metrics o n the vector bundles ( E J ) o < j < k . Denote by ( d j ) * the formal adjoint of d j . W e define f o r any j the L i p G c i a n operator A j = (dj)' o d j dj-' o (dj-')* and set:
+
D : C W ( X ,@jEZj)-+ C W ( X ,@jE2 j + l )
+
D ( u o , u ~ , . . .:= ) (dO(U0) (d1)*(u2),d2(u2)+ ( d 3 ) * ( u 4 ) ; . . ) . T h e n we have: (a) D*o D = @A2j and D o D* = @A2j+' ; (ii) T h e pseudodifferential complex ( E ,d ) is elliptic iff the operator D is ellaptic; (iiz) The pseudodifferential complex ( E ,d ) is elliptic z f f the operators A-1 are elliptic; (iv) If the pseudodifferential complex ( E ,d ) is elliptic then the cohomology spaces H j ( E , d ) := Ker(dj)/Im(dj-') of the complex ( E , d ) are finite dzmensional vector spaces, and we have:
Ind(D) =
c(-l)j
dimHj(E,d).
j
Proof. (i) is immediate. The proof of (ii) and (iii) uses the fact that the principal symbol of A j is given by: c ~ ( A j= ) g ( d j - ' ) o g(dj-')*
+ g(dj)* o g(dj).
(iv) It is clear that Ker(D) = Ker(D*D) = @ j Ker(A2j) and Ker(D*) = Ker(DD*) = @j Ker(A2j+l Therefore it remains to show that Ker(Aj) is isomorphic t o the j t h cohomology space of the elliptic complex. Now working on corresponding Sobolev Hilbert spaces, we have:
Ker(dj-l)*
21
Im(dj-l)l,
and hence from the ellipticity of Aj, we see that Ker(Aj) N Ker(dj) n Ker(&l)* is built up of smooth sections and is isomorphic t o the orthogonal of I m ( d j - ' ) in Ker(dj) ( d j o dj-' = 0 ) . This shows that this last orthogonal is finite dimensional and hence that I m ( d j - ' ) is closed and also that 0 K e r ( d j ) / I m ( d j - ' ) is isomorphic t o Ker(Aj).
173
This proposition encompasses for instance the de Rham complex as well as the Dolbeault complexes and enables to treat them as elliptic operators. Definition 4. The Euler-Poincark characteristic (or analytic index) of an elliptic pseudodifferential complex ( E ,d ) on a smooth closed manifold X is the alternate sum:
x ( E , d ) := x ( - l ) j d i m ( H j ( E , d ) ) . jl0
The computation of the Euler-Poincark characteristic of ( E ,d ) is then equivalent t o the computation of the index of one elliptic operator constructed using hermitian metrics on the vector bundles involved. 1.4
Some geometric operators
The first complex one meets in differential topology is the de Rham complex. In this case E j = Aj@(T*X)is the complexified j t h exterior power of the cotangent bundle T * X of X . The smooth sections of Ej are differential j forms on X . The de Rham differential d : C " ( X , E j ) -+ C"(X, Ejfl) enables t o form an elliptic complex. The principal symbol of d is given by
a ( d ) ( z , t ): A~(T,*x) A ~ + ~ ( T , * x ) , ~ ( ~= ) (J-iezt(<), x,J) -+
where ezt(6) is exterior multiplication by the covector t. The Euler-Poincark characteristic of the manifold X is by definition the Euler-Poincark characteristic of the de Rham complex on X. This characteristic is a topological invariant of the manifold and it can be computed in many ways, using triangulations on the manifold, vector fields, Morse theory, . . . etc. The well known Gauss-Bonnet theorem enables t o compute the Euler-Poincark characteristic of X in terms of the Euler class e ( X ) . This is the answer t o the index problem in the case of the de Rham complex. The Euler-Poincark characteristic of odd manifolds is always zero, as a consequence of the index theorem (see below). In the classification of oriented manifolds up to oriented homotopy, we need yet further invariants of the manifold. The most important one is the signature (and its higher versions), it is a relevant observation of Atiyah and Hirzebruch that such invariant is the index of an elliptic operator. The index theorem is then a good tool to compute this oriented homotopy invariant. Moreover we shall see in Section 3 that the signature is, thanks to the index theorem, a rational combination of the Pontryagin numbers of the manifold. Therefore, the index formula for the signature operator gives a partial answer to the following problem:
174 " Find all the rational combinations of the Pontryagin numbers which are oriented homotopy invariants." Pontryagin numbers are not homotopy invariants. Other conjectured homotopy invariants are constructed by using the fundamental group of the manifold. More precisely, if r = 7r1(X) is the fundamental group of the smooth closed manifold X, then there exists (up to homotopy) a continuous map f : X -+ Br, where B r is the classifying space of r, which classifies the universal cover. If now we denote by L(X) the Hirzebruch polynomial in the Pontryagin classes of X, then the push-forward f*(L(X) n [XI) E H*(BI?,Q) is suspected to be an oriented homotopy invariant of the manifold. This is the famous Novikov conjecture. The pairing of this class with the @dimensional cohomology class corresponding to the 1 function on X is in fact equal to the signature of the manifold by the index theorem as we shall see and is therefore an oriented homotopy invariant. Let us now explain more precisely the Atiyah-Hirzebruch construction, which plays a fundamental role in the construction of the Diff-equivariant spectral triple. Let X be a smooth oriented closed manifold with even dimension n = 2m. Then we have a well defined bilinear form on Hm(X,Z) given by cup product. More precisely,
<(Y,p>:=
+
L
sup.
+
If m = 2k 1 so that the dimension of X is n = 4k 2 then the above bilinear form on H2'"+'(X,Z) is antisymmetric and thus its signature does not make much sense. In this case the convention is to take this signature to be zero. In the case m = 2 k , the signature of the quadratic form associated with the above symmetric bilinear form is by definition the signature of the manifold X and is denoted by Sgn(X). Recall that the signature of a symmetric non degenerate bilinear form is the difference between the number of positive eigenvalues and the number of negative eigenvalues of the associated invertible symmetric matrix. Definition 5. Let X and Y be two oriented smooth closed manifolds. We say that X and Y are (strongly) homotopy equivalent if there exist two orientation preserving smooth maps f : X + Y and g : Y + X such that f o g i d y andgo f N i d X . In the above definition ho hl : Z Z' means homotopy equivalence of maps in the sense that there exists a smooth map h : Z x [0,11 + Z' such that h l z x ( 0 ) = ho and h l z x i l ) = hl. If X and Y are homotopy equivalent even oriented closed smooth manifolds, then it is clear that Sgn(X) = Sgn(Y). Let us consider again the de Rham differential d : R*(X) + Q*(X). The
-
N
---f
175
operator associated with the de Rham complex and whose index is exactly the Euler-PoincarB characteristic is then exactly d
+ d* : Q e v e n ( X-+) Rodd(X).
The Euler characteristic thus corresponds to the splitting of d+d* with respect to even and odd forms. In order to take into account the Riemannian structure of X we shall use another splitting of d + d*, i.e. another grading involution of Q * ( X )which anticommutes with d+ d*. So fix a metric on X, this induces metrics on the bundles Aj(X). On each fibre, the inner product can be defined for every fixed j-form p, and by duality, as a wedge product with a form of degree 2m - j denoted *pz,so that:
< ax,,Ox >d The operator
dA
... Adz"
= a, A
*(pz)
* is the usual Riemannian Hodge operator. < a ,p >:=
s,
It enables to write:
a A *p.
We shall need the following equality, which can be easily checked:
* o * = (-1)j Lemma 4.
on
!$(XI.
The Riemannian adjoint d* of d is given by: d* = - * od o *.
Proof. From the symmetry of the scalar product defined above we deduce that for any w E Q j ( X ) and any 6 E Q 2 m - j ( X ) we have:
If now (a,p)E Q j ( X ) x Q j - ' ( X ) , then:
< d*a,P > = < a,dp >
= (-1)j
/
pAd(*a)
X
=
< p, -(*
0d 0
*)(a> ) .
176
J--rj(j-l)+??l
* on O j ( X )@ C. Then when m is odd, r is We set now r = imaginary and it is real only when m is even. In addition we have: T
o r = id.
Hence r enables to split O * ( X )8 C into the eigenspaces associated with the +l and -1 eigenvalues. We denote these subspaces respectively by O + ( X ) and O - ( X ) . Using the expression of d* in Lemma 4, we deduce that the operator D = d + d* anticommutes with r and hence can be written in the following way:
where D- is the formal adjoint of D+ with respect to the above inner product. The operator D2 is the Laplace operator dod* +d* od and is thus a differential operator of order 2 whose principal symbol is given by: x id.
a ( D 2 ) ( x , [= )
Therefore the operators D+ and D- are elliptic operators of order 1. We set:
H+ = Ker(D+) and H- = Ker(D-). These spaces are exactly the intersections of the kernel of the Laplace operator with R+(X) and L ( X ) respectively and are therefore composed of harmonic forms. We set
h+ := dim(H+) and h- := dim(H-). Proposition 5 . The index Ind(D+) = h+ - h- is equal to the signature of the closed smooth oriented manifold X .
Proof. [lo] If 0 5 j < m, we denote Aj stable by r so that it splits into:
= O-&(X)@ Oim-’(X).
Then Aj is
with respect to the involution r. On the other hand when j < m, .(AT) = AT , more precisely we have for any ( w I , w ~ E) fli(X) @ O i m - j ( X ) : T(W1,W2)
Therefore dim(Hi)
W2 =
= f(Wl,W2)
= dim(Hj).
Thus we obtain
Ind(D+) = h y and it remains to treat the case j = m.
-
h?,
&T(W1).
177
If m is odd, then I- = f l c with u real and c2 = -1. The eigenspaces associated with the eigenvalues *&f of c are then conjugated and Ind(D+) = 0. Since in this case the signature of our manifold X is trivial by definition, the proof is complete. If now m is even, say m = 2k, then I - I ~ ~ ~ is( Xnothing ) but the Hodge operator *. Hence on H:k(X,R), the quadratic form giving the signature coincides with the norm:
S
w Aw =
J
w A*w
=< w , w > > 0 ifw f o .
On H5k(X,R) on the other hand, we obtain cxactly the opposite of the norm. This shows that is exactly the number of positive eigenvalues of the matrix associated with the quadratic form w H s w A *w while h;k corresponds t o the number of negative eigenvalues. The conclusion follows.
hlk
Another important example in Non Commutative Geometry is the Dirac operator, which broadly speaking, corresponds t o a "root" of a Laplacian. Let us recall now the definitions and main properties. Recall that if E is a euclidien vector bundle over X then the Clifford bundle of E l Cliff(E) is a bundle of algebras in which we have the relation * 6 - IE121 = 0, for any 6 E E c Cliff(E). The group Spin(lc) is the non trivial twofold cover of S O ( k ) , i.e. the connected twofold cover of S O(k). It can be realized using Clifford algebras as follows:
<
Spin(k):={w=vl*...*v2j,vl
and
Ivl( =1},
* is Clifford multiplication. We then define a group morphism e by: e : Spin(k) -+S O ( k ) , e(w)(z)= w * z * w t , where (v1* . . . * v ~ j = ) v2j ~ * . . . * v1. It is easy to show that e is a surjective
where
homomorphism which is not the trivial cover, see [52]. To study the representations of Spin(2r) we can make it act naturally on Cliff(JR2') by Clifford multiplications on the left. It turns out that this representation enables t o construct the Dirac operator we are looking for. More precisely denote by L : Spin(2r) 4 Cliff&X2') this representation and by L the extended representation t o Cliff@(R2'). Then it is well known [42] that L splits into 2' equivalent subrepresentations A of Cliffc(R2') which are irreducible. The restriction A of A to the spin group Spin(2r) is called the
178
spin representation and it also splits into 2 inequivalent irreducible subrepresentations A = A+ @ A- obtained by using the A-preserving involution ( f l ) r e l * . . . * e2,. of the Clifford algebra. Given a vector bundle E with even dimension 2r and with a spin structure, the spin frame bundle P ( E ) of E , a principal Spin(2r)-bundle, then enables to form the spin bundle
This bundle is in evidence &-graded and its fibres are isomorphic to the typical fibre A = A+ @ A-. So S is a vector bundle of irreducible representations of the complexified Clifford algebra bundle of E and we have more precisely End(S) N Cliff@(E). Clifford multiplication by sections of E interchanges S+ := P ( E ) xspin(zr)A+ and S- := P(E) xspin(zr)A- and we denote this section by: c :E
-+
End(r*(S)), where T : E -i X is the projection.
Let us now take E = T X , the tangent bundle to the even closed spin manifold X and denote again by S the resulting spin bundle. Let Vs be a connection on the bundle S associated with the metric on X . This means that the connection Vs on S is spin, i.e. it satisfies the Clifford compatibility with the Levi-Civita connection V on T X :
VjS(c(u)(w)) = c(Vdu))(w) + C(.)(VjS(W))7
'duE C"(X, T X ) and 'dw E C " ( X , S ) . The construction of the connection Vs is given for instance in [35]. From the definition of a connection we also know that Vs : C"(S) Cm(S 8 T * X ) satisfies the Leibniz rule:
Vs(fu) = f V s ( u )+ u 8 df, V j
E
-+
C o 3 ( X )and 'du E C m ( X , S ) .
Definition 6. The Dirac operator on X is the first order differential operator which is the composite of the above differential operator Vs and the zero order differential operator Z : C " ( X , S 8 T * X ) -+ C " ( X , S ) ,
given by C(u @ w ) ( z ) = c ( w ( z ) ) ( u ( z ) ) .
179
In a local frame ( e l , . . . ,e2r) of T * X around a point show that:
2 E
X , we can easily
2r
D = c c ( e j ) V : where V:(u) =< V s ( u ) , e j > . j=1 In the above formula ej is the dual basis of e3. The symbol of the Dirac operator D does not depend on the base variable 2 and is given by Clifford multiplication:
.(D)(.,E)
= c(E) : Sx
-+
sx.
Hence D is elliptic. Since each c ( e j ) interchanges the fibres S+ and S - , the Dirac operator anticommutes with the grading S = S+ @ S-. We obtain a formally selfadjoint operator which extends t o a unique essentially selfadjoint operator still denoted D on the L2 sections of S. We denote by D+ the Dirac operator acting from C"(X, S+)t o C " ( X , S-). Remark 2. We deduce immediately from the Leibnitz rule that
[D,Mf]:= D o M f
- M f o D = ~(df), Vf E
C"(X).
The operator M f is pointwise multiplication by f and the operator c ( d f ) is pointwise Clifford multiplication by the covector d x f . It turns out that the norm of the bounded operator c(df) acting in L 2 ( X , S ) is exactly the Supremum of the norm of dx f when x runs over X . Therefore we can recover the geodesic distance in X by the following Connes' mean formula [24]:
4 2 , Y ) = sup{lf(.)
-
f ( Y ) I , II[D,flll 51).
If now E is a further hermitian vector bundle over X , then we consider the bundle S @ E and use a hermitian connection on E t o define out of the spin connection on S a connection V E on S @ E . Therefore we can define similarily the Dirac operator with coefficients in E that we denote by D @ E hoping that no confusion can occur. So:
D @ E :C"(X, S
@
E ) -+ C"(X, S
@ E).
The positive part (D 8 E)+ of D @ E acting from C " ( X , S+ @ E ) t o C " ( X , S- @ E ) is then an elliptic operator whose index is important for the purpose of Poincark duality. From the definition of topological K-theory [9], we deduce the following Proposition 6. The map E H Ind[(D @ E)+] induces a well defined morphism from K ( X ) to Z called the Atiyah-Chern character of the Dirac operator
D.
180
To work with a Dirac operator, it suffices to assume the existence of the bundle S of irreducible representations of the Clifford bundle. This assump tion is weaker and is called K-orientation. It corresponds t o the existence of a spinCstructure on X, see [52]. The following Thom isomorphism is valid for K-oriented manifolds and is Poincark duality (in K-theory). Theorem 2. (The Thom isomorphism) The map K(X) + K ( T * X ) which assigns to a vector bundle E the class of the symbol of a Dirac operator with coefficients in E , is a n isomorphism. I n other words, K(T*X) is a free K ( X ) module which is generated by the K-theory class of the symbol of the Dirac ope rat or. This theorem is proved in [6], see also [28,49]. Indeed, working with the dual theory of K-homology, the Dirac operator determines a K-homology class in X, say an element of the K-homology group Ko(X). The above theorem then states exactly that
Ko(X) N Ko(X)whenever X is spin (or even spinc). We point out that the Chern character enables t o read these properties in the usual de Rham homology and cohomology. In the NonCommutative case, the existence of Poincark duality or rather some weaker version is one of the axioms assumed on (non commutative) manifolds in [27]. 2
The Atiyah-Singer index theorem
The first general proof of the index formula was given by -4tiyah and Singer in [9] and heavily rests on topological K-theory. We have already seen that any elliptic complex admits an analytic index also called the Euler-Poincark characteristic of the elliptic complex. This analytic index only depends on the K-theory class of the principal symbol of the complex and yields a group morphism Ind, : K(T*X)+ Z. The study of the properties of this map will completely identify it and prove the index theorem. The content of this section can be generalized to deal with other situations.
2.1
Review of the splitting principle
The splitting principle enables t o handle characteristic classes. The idea that governs this principle is that one can always assume that the complex vector bundles under study are Whitney sums of line bundles. For precise definitions and more details about characteristic classes, we refer to [46,55].
181
Proposition 7. Let E be a complex vector bundle over the manifold X . Then there exists a smooth manifold B E and a smooth fibmtion rr : B E -t X over X such that T * : H * ( X )+ H * ( B E ) is injective and the pull-back vector bundle T * E splits into line bundles
T*E z 11 @ ” .
@ lk,
over B E . A similar result holds for oriented real vector bundles which split into real plane bundles when pulled back t o the right manifold. When the vector bundle E is the complexification of an oriented real vector bundle F , we obtain a splitting of T’E = rr*(F @ C) into T * ( F @ C )N where
(l1@tl)@‘..@(lk@tk),
is the conjugate bundle of
lj.
More precisely, the splitting F
1 :
. . @ Fk of F into oriented plane bundles Fj enables to describe Fj @ C N l j @ T j . These results are now classical and the proof can be found for instance in [52][page 2251. To sum up and in the complex case for instance, the total Chern class c ( E ) of a given complex vector bundle E can always -using the injectivity- be formally decomposed into a product
c ( E ) = nS,1(1+
~ j ) ,
where the x j are the Chern classes of the bundles l j that appear in the above splitting principle. In other words, the j t h Chern class c j ( E ) E H2j(E)can always be thought of as the j t h symmetric function in the formal variables xi. In the same way for an oriented real vector bundle F , the total Pontryagin class p ( F ) can be written
so that p j ( F ) := (-1)jc2j(F @ C) expresses as the j t h symmetric function in the variables xp. F. Hirzebruch proposed in the early sixties t o construct characteristic classes by using the splitting principle together with the notion of multiplicative sequences [46]. Let us explain very briefly this approach. Let Q ( z ) = 1 alz . . . akzk . . . be a formal power series with constant term 1. We compute, depending on the allowed dimension, the product Q ( z ~ ) Q ( z .~. Q ) .( x k ) and one can prove that this product only depends on the symmetric functions in the variables x j . Therefore, this product can be
+
+ +
+
182
expressed in terms of the Chern classes and yields new characteristic classes, i.e. Q(zi)Q(z2). . . Q ( z k )
=1
+ Qi(C1) + Q 2 ( ~ 1 ,~ 2 +) . . . + Qk(C1,.
. . ck) + . . . .
Take for instance X
+
Q(x) := -- 1 z/2 1 - e-"
+ s2/12 + . . . .
The characteristic class that we obtains in this way is called the Todd class and denoted Td or Td(E) for a given complex vector bundle E. A straightforward computation gives
In four dimensions only the first and second classes are non trivial and the computation is complete. Hence for instance, the formula
+
means (1/12) Jx(c2(E) ci(E)). Remark 3. If we compute a product Q(q)Q(z2). . . Q(z,) with n > k then the homogeneous polynomials Q j for j 5 k are unchanged, which simplifies notably the calculations. A similar construction works for real oriented bundles and one gets polynomials in the Pontryagin classes. As an example, take the formal power series Q(z) = 4 2 , which yields the so called A-polynomial and which is given in low dimensions by:
e,
where the Pontryagin classes p j are the symmetric functions in the variables zs. In four dimensions for instance, only the first Pontryagin class is non trivial and we get Jx A ( F ) = (-1/24) J,pl(F). To finish this review, we point out that if E is a complex vector bundle, then the Chern character Ch(E) [55] of E can also be redefined by using the splitting principle: Ch(E) = x e x j = dim(E) 2 1
+ cl(E) + . .
183
2.2
Statement of the index theorem
Let us give the explicit formula which computes the index of any elliptic pseudodifferential complex on a smooth closed manifold. Let ( E , d ) be an elliptic complex on X . Denote by [a(E,d ) ]the compactly supported K-theory class of the principal symbol of ( E ,d ) in K ( T * X ) . Then the Chern character Ch[a(E,d ) ] of this symbol class belongs to H Y e n ( T * X ,Q), the compactly supported even cohomology space of the total manifold T * X . Let E be a real vector bundle over X , the A genus of E , A ( E ) ,is the characteristic class associated with the formal power series -, thus it is a polynomial in the Pontryagin classes of E. Recall also that the Todd characteristic class Td(V) of a complex vector bundle V over X is the characteristic class associated with the formal power series and is a polynomial in the Chern classes of v . Lemma 5. 15.21 Let E be an oriented real vector bundle over X , then we have:
+
Td(E @ C ) = A ( E ) 2 .
Proob If E is even dimensional then we can apply the splitting principle and use a formal decomposition [46] E @ C = (11 ell)@ - . .e(1, Therefore setting x j = ~
( l j ) the ,
eln).
first Chern class of l,, we deduce that:
which coincides with A ( J ? ~ ) ~ .
0
Theorem 3. (The Atiyah-Singer index theorem) Let ( E ,d ) be an elliptic complex over the closed n-dimensional manifold X as above. Then the analytic index of ( E , d ) is given by the AS-formula: Ind,(E, d ) = (-l),
< Ch[a(E,c ~ ) ] A ( X[T’X] )~, >.
where the total A class of X i s pulled back to T * X and where T * X is endowed with the symplectic orientation. The RHS of the index formula is a local topological data of the complex ( E ,d) and of the manifold X. In general an index formula is a formula which computes the index of an elliptic operator by the integral of a local data. Notice that the formula in Proposition 2 was of this type. Locality has a very precise meaning here, see [5] or [30].In the present case of smooth closed manifolds, the Gilkey invariance theory [42]enables to unify the different local
184
formulae one may obtain by different methods and to recover the above one. As an example of a local formula, we indicate the Connes-Moscovici index theorem when we specialize to the manifold case [30]. The proof of Theorem 3 that we summarize below is the K-theory proof due to Atiyah and Singer.
2.3
Sketch of the original K-theory proof
Instead of giving all the technicalities behind the proof of the index theorem, we have preferred to try t o explain the main steps and to be sketchy. The references are intended to clear up the details of the constructions and proofs. Let us first itemize the main ideas of the proof. We start with an elliptic pseudodifferential operator P on a smooth closed manifold X . If X were a sphere, then we would be able to carry out the computation using Bott periodicity. If X is any closed manifold then we can use the Whitney imbedding theorem to assume that X is a submanifold of a euclidian space RN for large N ( N 2 2n 1). The problem is then to construct out of P an elliptic operator on RN which would have an index and whose index would be equal to that of P. If N is a tubular neighborhood of X in RN, then the construction consists first in finding an elliptic operator on n/ and then in extending it to the whole euclidian space RN.The extension of an elliptic operator with index on N to RN preserves the index by a weak version of an excision theorem, so that we are reduced to the case of N + X which can be diffeomorphically identified with the normal bundle of X in RN. Hence given now a vector bundle N 4 X over X , we are led to the problem of constructing in a natural way out of P an elliptic operator on N whose index is equal to that of P. If X is reduced to the point, this problem is equivalent to finding a pseudodifferential operator (or a pseudodifferential complex) on Rq whose index is equal to 1. Such an operator exists and the properties of K-theory enable to produce the general topological construction for any X . Let us now give more details. The idea on Rq is that the problem can be reduced to the case q = 1 thanks to multiplicative properties of the index. Lemma 6. Let b, be the class in K(T*Rq) of the complex of symbols ouer T*(Rq)given by:
+
0 + A°Cq
2 AICq 3 . . . 2 AQ@Q
4
0,
where u,(x,E ) is exterior multiplication by x + it. T h e n the analytic index of by exists and equals 1. The class [b,] in K(T*Ry) i s called the Bott generator of K(T*RQ)= Z.
185
Proof. There exists a sharp product # : K ( T * R )@I. . .@I K ( T * R )-+K(T*Rq) [9] such that from the very definition of this product we have [58]
Thus a multiplicativity property of the analytic index with respect to this sharp product, and that will not be expanded here, shows that we can assume q = 1. The operator whose index we thus need t o compute is the operator 3: d / d x over R. It is a classical result due t o Hijrmander that this operator is surjective. Since its kernel is one dimensional, the conclusion follows. 0
+
Before explaining the construction of the operator on N associated with
P , let us clear up the excision property of the index. The idea is t o construct an analytic index map on open manifolds. So if M is such a non compact manifold and if i : M -+ Y is an open imbedding in a closed manifold Y , then one naively defines the analytic index map as the composite of the index map K ( T * Y )-+ Z with the K-theory extension map i, : K ( T * M ) 4 K ( T * Y ) . One then needs t o check the independence of the imbedding (z,Y).It is an easy exercise to prove that this definition of the index coincides with taking the analytic index of the corresponding operator on M itself so that it does not depend on the imbedding [9]. Hence we have now reduced the index problem to the construction of an appropriate symbol class in K ( T * N )out of the symbol class [o(P)] E K(T*X). The rough idea is t o take a sharp product of the class [o(P)] with a symbol on the vertical tangent bundle to the fibration N X corresponding t o the generator b, on each fibre. This is very similar to the Thom homomorphism in de Rham cohomology and this construction is called the Thom homomorphism (in K-theory). More precisely now, let T : E -+ M be a complex vector bundle over the (not necessarily compact) manifold M . The dimension of the fibres of E is denoted q. The Thom complex of E is the complex X(E) over E given by: -+
o
T * ( ~ o -i? ~ )T
2 . .. A T * ( A Q E )+ 0,
* ( ~ l ~ )
where A(e) = e z t ( e ) is the exterior multiplication by e E E . The above complex is exact off the zero section of E . However when M is not compact it does not define a compactly supported K-theory class of E. If ( F ,d ) is a compactly supported complex over M defining a class in K ( M ) then the sharp product of x * ( F , d ) by X(E) is a complex over E which is compactly supported and thus defines a class in K ( E ) . In summary we have
186
a well defined additive map:
-
X E : K ( M )4 K ( E ) , [F,4 w*P,4#XW The homomorphism XE is called the Thom homomorphism of E . Back to our situation N -+ X , we point out that the normal bundle to T*X in T*RN is isomorphic to w*T*N N K*(N8 C ) if we denote by w : T*X -+ X the projection. Therefore starting with our class [a(P)]E K ( T * X )we construct using A(.,"@) a class in K ( T * N ) .Roughly speaking and formally, the index of this class can then be computed by first evaluating the index in the fibres of T * N -+ T*X to obtain a class in K ( T * X ) and then computing the index of this latter. But because the construction of XE consists exactly in putting the class b, on each fibre, we deduce that the vertical index image of X,.(N@@)( [ a ( P ) ]coincides ) with [ a ( P ) ]therefore , the index is preserved by the Thom homomorphism. We denote by i! : K ( T * X )+ K ( T * R N the ) resulting homomorphism. So we have a commutative diagram:
The commutativity of this diagram is the heart of the index theorem. To give an explicit formula for the index in terms of characteristic classes we then simply apply the Chern character to this diagram and compute. To get more insight into the explicit constructions and proofs we again refer to [9]. We are then reduced to the problem of commutativity of the following diagram
where the horizontal maps are the K-theory Thom homomorphism and the cohomological Thom homomorphism. It turns out that this diagram is not commutative and this is the reason why the Todd class appears in the index formula. The detailed computation using the Chern character is given in [lo]. See also [52].
187
2.4
The heat expansion method
Thanks t o the Gilkey invariance theory, there are many different proofs of the Atiyah-Singer index theorem. The aim as we have explained is t o find a local formula by using only local tools. We have already met local tools of a given elliptic pseudodifferential operator in these notes. The Dixmier trace for instance is a local tool when the operator is in the right Dixmier ideal. More generally, the Wodzicki residue trace is a local tool and has the advantage over the famous homological formula of having a direct generalization t o the Non Commutative setting. Good references are [30] and [20]. The Wodzicki residue is closely related with the heat expansion method, by using the Riemann r function. Let us therefore rather explain how the heat kernel method works and yields the index formula. The first property that we recall is the MacKean-Singer cancellation formula. Let D be an elliptic differential operator of order one on a smooth closed n-manifold X with coefficients in a vector bundle E. Denote A = D' 0 D = 1DI2 and define for any t > 0 the bounded operator eVtAby continuous functional calculus in L 2 ( X ,E ) . Because (1 A)(n+1)12e-tAis a bounded operator for any t > 0, we know that the operator e-tA is a trace class operator. Moreover the Schwartz kernel kt of e-tA is a smooth section over X x X of the bundle Hom(7r;(E),r:(E)) where 7 r j : X x X -+ X is the projection on the j t h factor. More precisely, we have: Lemma 7. For any t > 0, the operator e-tA is a regularizing operator and belongs to P P m( X ;E ) .
+
Proof. We give a formal proof and refer t o [42] for a rigorous justification. To show that e-tA is regularizing, we must prove that it induces a bounded operator between each pair of Sobolev spaces. But recall that the Sobolev spaces can be defined using any positive elliptic operator instead of the Riemannian Laplace operator. So we can take A itself and therefore the proof is completed by observing that the operator Ase-tA is bounded for any s E R. 0 Proposition 8. (MacKean-Singer) Let D be an elliptic differential operator of order 1. Then we have:
Ind(D) = Tr(e-tD*D)- Tr(e-tDD*), W
> 0.
Proof. The operator D furnishes an isomorphism between the eigenspace Eo(X) of D'D and the eigenspace El(X) of DD' corresponding to IXI2 # 0.
188
Therefore their dimensions are equal and we have Tr(e-tD*D)- Tr(e-tDD*) =
e-""*(dirn(Eo(X))
-
dim(EI(X)))
XESP(D)
= dim(Eo(0)) - dim(El(0)) = Ind(D).
The MacKean-Singer formula enables to prove the index theorem. The method is an expansion of Tr(e-tD*D) - Tr(ectDD*) when t 0. I t turns out that the restriction of the kernel kt of e-tD*D to the diagonal of X x X admits an expansion near 0 of the form: N
kt(Z,Z)
- C~*CUY*I-)(Z), i20
where C $ * ~ ( Z ) E End(&) is a local data which defines a global section. The justification of this expansion is tedious but is a straightforward consequence of an application of the pseudodifferential calculus with parameter to produce a parametrix for the resolvent of D , see [42]. If D is a pseudodifferential operator, then one needs to add in this expansion terms involving Log(t). Proposition 9. The index of the operator D on the n-manifold X is given by the local formula Ind(D) =
1
[Tr(c~:'~(z))- T~(cu:~*(z))] dvol(s).
X
The equivalence between this formula and the Atiyah-Singer formula is a corollary of the invariance theory, see (51 and [42]. Let us recall now the link between the heat expansion method and the zeta functions. Recall that the complex powers A' of A can be defined using holomorphic functional calculus. These powers are pseudodifferential operators, this is the Seeley theorem [61]. But then we have from the definition of the Riemann r-function and by using the Fubini theorem, for Re(z) > 0:
The asymptotic expansion of Tr(ectA) given above then shows that the zeta function <=(A)= Tr(A-.) of A, which at first converges only for large Re(z), actually extends t o a meromorphic function in the z-plane. Moreover it shows
189
that the function I'(z)&(A) has at most simple poles a t the points z j ( j - n ) / 2 , j 2 0 , and the residue is given for j # n by:
=
sx
a t ( z ) d v o l ( z ) . This proves that the while Q(A) exists and coincides with index of D can be related with the regularized values [o(D*D) and Q ( D D * ) of the zeta functions. This remark is due t o [5]. Other links with residues of zeta functions appeared later but we will not expand them here since the local Connes-Moscovici index theorem is explained in parallel lectures. 3
Corollaries and examples
The first observation is that the Atiyah-Singer theorem shows that the index of any elliptic pseudodifferential operator from the Kontsevich-Vishik class [51] on an odd dimensional manifold is trivial, this is true for instance for differential operators. The principal symbol of such operators satisfies u(z,-<) = (-l)mu(z,<) and therefore, the homotopy class is unchanged under the involution (z,[) H (z, -0. This shows that the analytic index map, which only depends on the homotopy class of the principal symbol is unchanged under this Z2-action. However, the homology class [ T * X ]which appears in the RHS of the AS-formula is multiplied by (-l)dim(x). This remark shows the importance of the index theorem for Toeplitz operators briefly explained in the first section. Notice that these operators are order zero pseudodifferential operators in the sense of [ 9 ] ,and thus furnish invariants for the odd case. Let us restrict ourselves now to even spin manifolds and denote again by D 8 E the Dirac operator on X that we have twisted by the hermitian vector bundle E. We are led t o the following index problem: Give a local computation of the Chern-character of the Dirac operator. In other words: Find a local representative for the Atiyah map:
K(X)
-
Z,defined by [El
-
Ind[(D 8 E)+].
Theorem 4. Denote by A ( X ) n [XI the Poincare' dual of the A genus of the manifold X . Then the rational homology class A(X)fl [XI satisfies: Ind[(D 8 E),]
=< Ch(E), A ( X ) n [XI > .
190
In other words, the index theorem can be rewritten as: Ind[(D @ E)+]=
L
Ch(E)A(X).
Proof. We give the details of the now classical proof for the convenience of the reader. Recall that the A characteristic class is associated with the generating function -. We must compute the integrand in the Atiyah-Singer index formula when the elliptic pseudodifferential operator P is (D@ E)+.Since X is oriented the index formula in the Atiyah-Singer theorem becomes:
where r! : H*(T*X,Q ) + H*(X,Q) is integration along the fibres and m is the dimension of X . The K-theory class of a ( ( D @ E ) + )in K ( T * X )is represented by (r*(S+@E),n*(S-@E),c)wherec(z,<) : S$@Ex4 S;@Ex isc(<)@IdET and c([) is Clifford multiplication by acting from S$ t o S;. Therefore we have:
<
T ! Ch(a((D
@ E)+))= Ch(E)r! Ch(a(D+))
So we need t o prove that r!Ch(r*(S+),r * ( S - ) ,c) = (-l)mA(X)-l.
The restriction p ( T X ) = i * ( r * ( S + ) , r * ( S - ) , c )t o X is exactly the class p(TX) = [S+] - [S-] in K(X). If T X N F $ F’ where F and F’ are two vector bundles over X then we have in K ( X ) : P(TX) = P(F)P(F’). Therefore we can use the splitting principle t o assume that T X decomposes into El $. . .$Em where each Ej is a 2-plane bundle and Ej @CN l j $ij where Ej N l j as oriented bundles. It follows that the spin bundle S is the tensor product of the corresponding spin bundles. But each Ej is isomorphic as an oriented real 2-plane to the complex line bundle l j . We have assumed that X is a spin manifold, therefore each complex line bundle l j has a spin structure. This is equivalent to the fact that the first Chern class cl(lj) of l j is even. Hence l j admits a square root, i.e. a bundle 1;’ such that 1;’ @ 1;’ N l j . In this case it is easy to see that the spin bundle of 1 is exactly
191
where S+(Ej) = ij 'I2 and S- (E j ) = 1:/2. So
Hence Ch(p(Ej)) = e-xj/2 - exj/2, where x j is the first Chern class of E j . Finally we get: - ez3l2) = (-l)"e(X)A(X)-', Ch(p(TX)) = I111j<m(e-2j/2
where e ( X ) is the Euler class defined by e ( X ) hand,
=
I I l 5 j l m x j . On the other
Ch(p(TX)) = i * c h ( ~ * ( S + ) , ~ * ( S - ) = , c i* ) oi! O T ! C ~ ( T * ( S + ) , T * ( S - ) , C ) , where i! is the Thom isomorphism i! : H*(X,Q) 4 H*(TX,Q) which is the inverse of the integration along the fibres T ! . From the definition of i! we know that
(i* o i!)[w]
= e(X)[w].
Therefore we get Ch(p(TX)) = e(X).rr!Ch(T*(S+),T * ( S - ) ,c ) , and so: T!Ch(T*(S+),T*(S-),C)
=
(-l)mA(X)-l,
as announced. The proof is then complete. If we take for E the spin vector bundle S itself, then the Dirac operator D 8 S with coefficients in S coincides with the operator d + d* acting on differential forms. Indeed, if V is the Levi-Civita connection on the exterior algebra &.(X) then we can recover the de Rham differential by the formula [52][page 1231:
du =
e j A Vjw while d* is given by d*w = i(ej)Vjw, j21 e l with i(ej) the interior product by ej. But recall that Clifford multiplication by e j can be read through the linear isomorphism ( S 8 S 111) Cliff@(X)2 h t ( X ) as: e j * w = e j A w - i(ej)w Let us decompose now the bundle S 8 S into S+ 8 S and S- 8 S exactly as we did for any coefficient bundle E . Then we have:
192
Lemma 8. The operator [D8 S]+is exactly the positive part of the AtiyahHirzebruch Signature operator, and thus: L(X) =
Ind([D 8 S ] + )= Sgn(X) = 2m
L(X),
where L ( X ) and L ( X ) are the L-characteristic polynomial and the Hirzebruch L-polynomial associated respectively with the power series and z. th(x)
&
Proof. The decomposition of S into Sf and S- is obtained by using left multiplication by the involution w@ = imel * ... * ezm which is a globally defined section of Cliffc(X), [52][page 1361. But under the linear isomorphism Cliff@(X)N &(X), left multiplication by w~ corresponds up t o sign to the Riemannian Hodge operator *. This is proved in [52][page 1291. We deduce that the Clifford grading gives in this case exactly the signature grading. Applying the above index theorem for twisted Dirac operators, we obtain: Ind([D
S],) = Sgn(X) =
'I
Ch(S)A(X).
But the Chern character of S can be computed using again the splitting principle. In fact this principle tells us that we can assume that the tangent bundle TX is the direct sum of 2-plane bundles of the form
TX
N $jEj
where Ej 8 C N
l j $ij.
Thus writing the Chern character of each S ( l j ) as the first Chern class c l ( l j ) of l j , we obtain:
+
Ch(S) = IIl<j<m(ezj/2 - e-xj/2).
+ e - x j / 2 where xj is
*
Now with respect t o the same generators, A(X) is by definition the charand thus we acteristic class associated with the generating function have:
The product of the two expressions then gives:
Remark 4. The index formula for the signature operator given above holds even if the manifold is (only oriented and) not spin.
193
The above example of the signature operator is important t o understand the Connes-Moscovici Diff-equivariant spectral triple [30]. One of the corollaries of the index formula concerns integrality [lo].The first immediate but relevent observation is that the characteristic numbers obtained for various operators in the RHS of the AS-formula are integers. Let us give now two examples, probably the most known, t o explain such integrality results. Many other corollaries can be consulted in the literature, for instance in [8,52]and the references therein, many of them rather use the equivariant version of the Atiyah-Singer index theorem. This last version is omitted in these introductive notes. Proposition 10. (1) W e have f o r f o u r dimensional oriented manifolds: Sgn(X) = - 8 L
A(x).
(2) (Rochlin) If X i s a four dimensional spin manifold then an even integer and the signature of X is a multiple of 16.
Proof. ( 1 ) In this case we have only two generators
But p l ( X ) = xy
+ x:
XI
and
22,
sxA ( X ) is
so that:
is the first Pontryagin class of X , therefore we get:
On the other hand, let us compute the signature of X using also the index formula. We have: 4 x1/2 x2/2 = 2(1+ (1/12)x:)(1+ (1/12)x;)= 4(1+ (1/12)pl(X)). t h ( x 1 / 2 )t h ( x 2 / 2 ) Therefore we get:
sx
( 2 ) When X is spin, A ( X ) is the index of the Dirac operator and is integer. In fact it is an even integer and this can be justified as follows: In the four dimenisonal case, Clifford multiplication by the operator J = el * e3 yields a well defined endomorphism of S which preserves the grading. In addition it satisfies J 2 = 1 and preserves the kernel of D. Since J+ is orthogonal to +, we deduce that Ker(D+) and Ker(D-) are even dimensional. The conclusion follows by using ( 1 ) . 0
194
A simple consequence is that an oriented four manifold whose signature is not a multiple of 16 cannot carry a spin structure. A simple example is given by Pz(@),in this case we deduce that:
J,
A(x> = -1/8,
cannot even be an index. We can also apply the index theorem to twisted signature operators. Let E be a smooth complex vector bundle over the smooth closed manifold X. We can extend the operator D+ to the sections of &(X) @ E to get a new elliptic first order differential operator D i , called the twisted (by E ) signature operator. The index theorem for this operator can then be written as: Ind(D2)
=
Ch(E)L(X).
The integrality corollary of this formula yields a simple proof of a classical theorem due to R. Bott: Corollary 1. Let E be a complex vector bundle over the even sphere Szn. T h e n the top Chern number of E as a multiple of ( n - l)!. In other words:
Proof. The cohomology spaces of the sphere S2" are trivial except in dimensions 0 and 2n. Therefore we have: [Ch(E) n L(X)lzn = Chn(E)
+ Ln/2(X).
But the signature of the sphere is trivial since Hn(SZn) = 0, thus Ln,2(X) = 0 and we get:
,s
Other corollaries of the index formula concern some rigidity theorems. Again we refer to [52],for instance from page 280 to page 290. Besides the importance of the Atiyah-Singer index theorem on its own, the methods of proof are ready to be generalized t o more complicated situations. The index theorem admits for instance equivariant versions for actions of groups. The case of the action of a compact Lie group led Atiyah and Segal [8] to prove a Lefschetz fixed point formula. Some references, among others, concerning the equivariant approach are [3,4,8,10]. See also the nice paper [66] where
195
E. Witten shows how such Lefschetz formulae may be applied to the study of loop spaces.
196
Chapter 11. The Murray-Von Neumann index theory The introduction of von Neumann algebras in the framework of Non Commutative Geometry is natural. It unifies many known von Neumann index problems [l,23,621, and opens up the way to the investigation of new non commutative phenomena. Von Neumann spectral triples are also expected to play an important part in a reformulation of local Quantum Field Theory as pointed out to me by C. Brouder. We take the opportunity of the present notes to give some further developments which involve in a fundamental way von Neumann algebras. The experimental example chosen to this end is the case of flat bundles, say Galois suspensions. This choice is motivated by the fact that, first and in many respects, this example already embodies all the typical constructions that give rise to von Neumann algebras, and second because the von Neumann index theorem for measured foliations [23] turned out to be a fundamental tool in solving a physical problem concerning the computation of the range of the truce in the Bellissard approach to quasi-crystals. To end this chapter and in the last section, we show how Connes' Non Commutative Geometry can be rearranged to deal with von Neumann algebras. The results of Section 4 and Section 5 were obtained in collaboration with T. Fack [16]. The author wishes to thank him for allowing these new results to be expounded in this school. 4
Dixmier traces in von Neumann algebras
In this section, we shall denote by M a von Neumann algebra acting on a Hilbert space H . For more information about von Neumann algebras, we refer to [32] but also to [45] where a nice survey, motivated by local quantum field theory, is given. The von Neumann algebras considered here will be type 11, von Neumann algebras.
4.1
Dixmier trace and residue of zeta functions
To show the existence of a non normal trace on the algebra B ( H ) of all bounded operators on an infinite dimensional separable Hilbert space H , J. Dixmier [33] constructed in 1966 a unitarily invariant state Tr, on the ideal L1>"(H)of all compact operators T on H whose singular numbers p o ( T ) 2 p I ( T ) 2 ... satisfy n
p k ( T ) = O(Log(n)) when n + m. k=O
197
The Dixmier trace of such operator was recalled in 1.2. Let A be the Riemannian Laplacian operator with principal symbol for a given Riemannian metric on M , then Formula 1.1 a(A)(x,E) = gives:
IM
f duo1 =
n x (27r)" UOl(S")
x Tr,(fA-"),
'df E C " ( M ) .
This led A. Connes t o reinterprete the Dixmier trace Tr, as the correct operator theoretical substitute for integration of infinitesimals of order one in Non Commutative Geometry. We shall now extend the Dixmier trace t o the case of semi-finite von Neumann algebras. Let us denote by M an infinite semi-finite von Neumann algebra acting on a Hilbert space H and equipped with a faithful normal semi-finite trace T . For any 7-measurable operator T in H , denote by
p t ( T ) = Inft>o{llTEII,E = E* = E2 E M , T ( -~ E) 5 t } the tth generalized s-number of T . An element T E M is called T-compact if lirnt-" p t ( T ) = 0. The set of all T-compact elements in M is a norm closed ideal of M that we shall denote by K ( M , T ) .By [37][page 3041, the ideal K ( M , T )is the norm closure of the ideal R ( M , T )of all elements X in M whose final support .(X)= Supp(X*) satisfies T ( T ( X ) )< 03. Definition 7. An element T E M is called a T-Dixmier operator if:
The set of .r-Dixmier operators is a vector space that we shall denote by
L 1 l o ( M , ~ )It. is clea.rly a Banach space for the norm:
and an ideal in M which contains L1(M,.r)n M . We have for any T L1i" ( M ,T ) and any t > 0:
which shows that for any
L 1 ( M , 7 )n M
E
> 0:
c L1?"(M,7)c L1+'(M,7)n M (cK ( M , T ) ) .
E
198
Let w be a state on the C*-algebra Cb(R+) vanishing on Co(R+). Such states do exist thanks to the Hahn-Banach theorem. For any bounded function f in R+, let M ( f ) be the mean defined by
1 M ( f ) ( x )= Logo
1
f(t),
dt
where a
> e is fixed.
Since M ( f ) E c b ( R + ) we can set as in [30]:
lu(f>= w ( M ( f ) ) . Definition 8. For any state w as above, the Dixmier trace T,(T) of a positive operator T E L1pm(M,7)is defined by: r,(T) := Z,(t
i
Ut(T)
Log(t
).
+ 1)
We point out that this definition depends on the choice of the state w . We shall fix such w for the rest of this subsection. As in the classical case (cf [30]) one may prove that we have for any positive operators T ,S E L1ym(M,7):
r,(T
+ S ) = T,(T)+ T ~ ( S )
This additivity property then enables to extend ru to a positive linear form on the Dixmier ideal L1>"(M,7).We also easily check that T,(ST) = T ~ ( T S ) for any T E L1>"(M,T) and any S E M . The proof used in the classical case (cf [29]) clearly extends to our setting. Note that T,(T) = 0 whenever T E M n L 1 ( M ,r ) . Definition 9. Let M be as before an infinite semi-finite von Neumann algebra acting on a Hilbert space H and equipped with a normal faithful positive trace r. A positive self-adjoint operator T on H is called r-discrete if (T X ) - l E K ( M , T )for any X > 0. It may be proved (cf [60][page481) that T = XdEA is 7-discrete if and only if one of the two following properties holds: (i) VX E R, ~ ( E A<)+m; (ii) 3x0 > 0 such that (T XO)-' E K ( M ,7).
+
";s
+
For such positive r-discrete operator T , the function NT(X) := T(EA)
is well defined on R; moreover, it is non decreasing, positive and right continuous. This is the state density of T .
199
Definition 10. Let T = s,'" AdEx be a positive self-adjoint 7-discrete operator with spectrum in [ E , + - ~ o )where , E > 0. The zeta function
l+"
A"dNT(A),
for any complex argument z such that the above integral converges. Since we have for any R > 1:
lR
1
LodR)
A"dNT(A) =
e z t d a ( t ),
where a(t) = N T ( e t ) , we know from the classical Laplace-Stieltjes transform theory that the integral
lim
AzdNT(A)
converges for R e ( t ) < -dT and diverges for R e ( z ) > -dT, where:
Log(a(t)) = Gx--t+" Log(NT (A) dT := limt-++a Log(A) . Moreover,
E
L1(M1.) for R e ( z ) > p } .
Indeed, it follows from the normality of the trace
T
that:
rR
In particular:
dT < +-00
3x 5 0 such that T"
E
L1( M ,T ) .
Sometimes, dT is called the quantum .r-dimension of the operator T . Theorem 5 . (161 Let T be a positive r-discrete operator with spectrum in [ E , -too),E > 0. If 0 < dT < +-00, the following conditions are equivalent: (2) (z d T ) < T ( z ) 4 A when z 4 - d T , Z E (--00, -dT[; (ii) T-dT E L'?"(M,r) and
+
T w ( ~ - d= ~ )
lim
t++m
Log(1
+t)
1'
p s ( T - d T ) d s= - A / d T .
(4.1)
200
The proof of this theorem is given in [16]. The Dixmier trace is strongly related to the asymptotics of the spectrum. For instance we have the following Proposition 11. [60] Let T be as in Theorem 5. Assume that there exists S > 0 such that CT admits a meromorphic extension to { R e ( z ) < -dT S> with a simple pole at z = -dT. Then we have:
+
This proposition generalizes the Guillemin spectral estimate t o a more general class of pseudodifferential operators. Some of the above properties have been explicited. In the case of measured foliated manifolds in [16]. In particular, there exists a generalization of the Connes formula expressing the Dixmier trace by a local residue. 5
Index theory in von Neumann algebras
We define in this section r-Fredholm operators and their T-index in type I1 von Neumann algebras. We also show how the famous Calderon formula can be generalized to the setting of von Neumann algebras, the main technical difficulty being that the spectrum of a r-compact operator is not discrete in general. This formula will be used in the next section t o give a polynomial formula for the Chern-Connes character of a von Neumann spectral triple. In the present section, we shall also explain how von Neumann algebras are involved in the index theory of measured suspensions. 5.1
Definitions and properties
As before, let M be a von Neumann algebra in a Hilbert space H , equipped with a semi-finite normal faithful trace T . Lemma 9. For any 7-compact projection e E M , we have T ( e ) < $00.
Proof. Since e = e* = e2, we have p t ( e ) E (0,l). But p t ( e ) -+ 0 as t by hypothesis, so that there exists t o such that pt(e) = 0, and hence T ( e ) =
s,”u pt(e)dt < t o o .
-+
+oo
for t 2 t o , 0
Definition 11. An operator T E M is called .r-Fredholm if there exists S E M such that 1 - ST and 1 - T S are r-compact. Proposition 12. If T E M is r-Fredholm, the projections p~ and p p onto Ker(T) and Ker(T*) are .r-finite.
20 1
Proof. Let S be as in Definition 11. The projections p~ = (1 - s T ) p ~and 0 p p = (1 - TS)*pp are 7-compact, and Lemma 9 gives the result. Definition 12. The index Ind,(T) of a 7-Fredholm operator T is defined by: Ind,(T) := 7 ( P T ) - 7 ( P T * ) ,
(5.1)
where p~ and p p are the projections onto the kernel of T and T* respectively. The following proposition generalizes the Calderon formula and computes Ind,(T) from the powers of 1- ST and 1- TS in the setting of semi-finite von Neumann algebras. This formula will be used to get a polynomial formula for the Chern-Connes character (see 6.2). Proposition 13. Let M be a semi-finite von Neumann algebra with a normal faithful semi-finite (positive) trace T and T E M . Assume that there exists p 2 1 and a n operator S E M such that: 1 - ST E L P ( M , 7 )and 1 - TS E LP(M,T).
Then T is 7-Fredholm and we have for any integer n 2 p: Ind,(T) = ~ ( ( 1 ST)") - ~ ( ( 1 TS)").
(5.2)
Proof. [16] The operator T is 7-Fredholm because LP(M,7 ) n M C K ( M ,T ) . To prove the proposition, we may assume that n = 1. Indeed, let S E M be such that:
A = 1 - ST and B
=
1 -TS
are in Lp(M,T) (and hence in L " ( M , T) for any n S' = S ( l + B
2 p ) and set:
+ B2 + ... + B"-').
We have: 1 - TS' = Bn and 1 - S'T
=
A".
Replacing S by S', A by A" E L 1 ( M , 7 ) and B by B" E L 1 ( M , 7 ) ,we are thus reduced to the case where n = 1. When A = 1 - ST and B = 1 - T S are in L1(M,T ) , we get from the relations APT = p~ and p p B = p p the equality: Ind,(T) = APT) - ~ ( pBp) . To prove that Ind,(T) = T ( A )- T ( B ) ,it thus suffices to show that:
.r(eAe) = T ( f B f ),
202
where e = 1 - p~ and f have:
=
1 - p p . To this end, set V := fTe. We clearly
TeA = T A = BT = B f T , and hence:
V(eAe)= ( f B f ) V . If the intertwining operator V from e ( H ) t o f ( H ) were invertible (the inverse would then be automatically in e M f by the bicommutant von Neumann theorem), we would get by cyclicity of the trace:
.r(eAe)= . r ( f B f1,
(5.3)
and Calderon’s formula would be proved. Although V is not necessarily invertible here, it is injective with dense range from e ( H ) to f ( H ) . It turns out that this is enough to prove (5.3) (see [16][Lemma 21)’ and Calderon’s formula follows. 0 5.2
An experimental example: Suspended actions
The index theorem in the case of non compact manifolds has been actively studied under different assumptions. The first significant result was obtained by Atiyah in [l]. His method proved t o be very fruitful in the applications of von Neumann algebras t o index theory on the one hand, and in the non commutative harmonic analysis of discrete groups on the other hand. The novelty of Atiyah’s approach was the meaning that he gave t o the index in the sense of the Murray-von Neumann dimension theory described in the previous subsections. In many geometric situations, Atiyah’s approach helped t o perform new strategies t o attack singular index problems. One of the famous examples with immediate applications t o physics is the Shubin computation of a von Neumann index for almost periodic operators and the reader is encouraged to read Shubin’s paper [62] which will not be expounded in these notes. We point out on the other hand that Connes’ measured index formula holds for any closed foliated manifold which admits a holonomy invariant transverse measure [23]. One of the applications of such theorem is t o the study of the so called solenoid where the transverse structure is modelled on a Cantor space [56][page 421. Before embarking in the flat bundles world, we point out that the index theory for foliations was the first important application of non commutative C*-algebras t o geometry and it led A. Connes t o introduce his famous non commutative foliation C*-algebra as a topological counterpart for the singular (in general) space of leaves.
203
Let T be a smooth (for simplicity) closed manifold of dimension q and let M be a Galois covering over the smooth closed connected pmanifold M with group r. Let e : r + Diff ( T )be a representation of r and denote by V the suspended smooth closed manifold V = M X y T , where M X y T is a usual notation for the quotient manifold of 2 x T under the free and proper action of r on the right given by:
7r
: M -+
(5, t ) g = ( Z g , p(g-')t),
V ( Z lt ) E A? x T and V g E r.
Then we have a commutative diagram of natural projections:
M-M Hence V fibres over M with fibres diffeomorphic t o the manifold T . The foliation F of V that we are interested in, is the one whose leaves are the quotients in V of the manifolds M x { t } when t runs over T . We assume that there exists a (positive for simplicity) measure p. on T which is invariant under the action of the group through e. So T is a measure space which we have assumed smooth for simplicity and on which the group I? acts by measure preserving diffeomorphisms. We fix a Lebesgue measure on the manifold M and lift it to a Lebesgue measure v to & This I . measure is defined more precisely by choosing a fundamental domain U in M, i.e. a subset U of M such that
U c1 g U = 8 , V g
E
r \ (1) and M
= UgErUg,
and setting for any Bore1 subspace A of M:
v[7r(Ag n U ) ] ,where
v ( A )=
7r
:
k+M
is the projection.
9-
Note that U has non empty interior i n t ( U ) and because we have taken a Lebesgue measure v on we know for instance that v(int(U))= v ( U ) . Furthermore we also deduce that:
M,
v(M \ U , E r i n t ( U ) g )
= 0.
The leaves of V are quotients of M by isotropy (in T ) subgroups of I? and since the measure v on h;r is I?-invariant it also induces a well defined measure still denoted v on each leaf of (V,F ) , and hence a Lebesgue measure on the
204
leaf manifold 3. The leaf manifold 3 of ( V , F ) is the pmanifold which is the discrete union of the leaves. Again we need to be more precise about the construction of the measure u on each leaf: This is easy since we just choose again a fundamental domain Ur, for the corresponding subgroup rl of r and set for any Borel subset A in a leaf L (which is diffeomorphic to M / I ' 1 ) of
(V,F ) : v ( A ) = v(rT1(A)n Ur,), where r1 : M
4
L is the projection.
For any t E T we define Ut to be the projection in V of U x { t } and we set
u,
:= UtETUt
cv
Using the measure u on the leaf manifold together with the I?-invariant positive measure p, we can define a positive measure uw on the whole manifold V . This is true on each foliated manifold but in our case this is trivial because we can define this measure in the interior of UV as the tensor product measure and set for any Borel subset A of V :
u,(A)
:= u,(A
n i.t(Vv)).
So in the measure sense the situation is quasi-trivial. We denote by
M := [L"(T,p)
.))Ir
@ B(L2(M,
the algebra of B ( L 2 ( M u))-valued , p-a.e. bounded functions on T which are I?-equivariant. The algebra M can also be described as the commutant in the von Neumann algebra L"(T, p ) @ B ( L 2 ( a )of) the diagonal left representation of r in L 2 ( M ,v) @ L2(T,p). Therefore it is a von Neumann algebra which acts on the Hilbert space
7-f := L 2 ( T ,p) @ L 2 ( M ,Y )
L 2 ( T ,p) @ L2(U,u ) @ Z2(r).
Let 1c, be the characteristic function of U and denote also by 1c, the operator in L2(h;r)corresponding to multiplication by $.I Then we can define a positive functional on M by setting:
Let T E L w ( T , p ) @ B ( L 2 ( M u, ) ) . Then for p-a.e. t E T , Tt admits a Schwartz kernel kt. It is clear from the definition of k that T E M iff for any g E r and any (5,jj) E M , we have:
k-l(t)(5cg,Gg) = kt(5,Y).
205
Hence if T E M , then k induces a kernel ko which is a distribution on the quotient groupoid
G :=
MxMxT
r
defined by
ko([%zl, tl) = kt([4,[$'I). One can show that the functional Trr,, is a trace [l]. Let us recall some useful terminology. A Trr,,-measurable operator T is a Hilbert-Schmidt operator in the Hilbert space 1-I if T'T is a Trr,,-trace class operator. So if Ict is the Schwartz distributional kernel of Tt defined for p a.e t E T , then T is a (r,p)-Hilbert-Schmidt operator iff
J
Ikt(2,~)12du(z)du(~)dp(t)
TxUxU
If T is a Trr,,-measurable operator in 1-I, then the polar decomposition T = UlTl of T defines a partial isometry of 1-I which lives in M and the positive operator (TI is a Trr,,-measurable operator in 1-I [22]. Therefore a (I',@)-trace class operator is exactly the product of two (I?, p)-Hilbert-Schmidt operators. Assume that for p-a.e. t, the kernel kt of Tt is a smooth function on fi x fi and that the induced kernel ko is compactly supported in then T is of ( p , r)-trace class and we have:
w,
0
Trr,,(T) =< up,k >=
ko([?,2,t])du,([2,t]),
where up is the measure on V induced by the Lebesgue measure on the leaves and the invariant measure p on T . The justification of this assertion is a rephrasing of the proof given in [l]with the extra term T here. The trace T r y , p is obviously faithful and semi-finite. Remark 5. An operator A in the Hilbert space L2(&f) whose kernel k is integrable on the diagonal is not necessarily trace-class. Let E be a continuous longitudinally smooth vector bundle over V of dimension N . We shall denote by E the pull back of E to T x &'. The restriction of E t o each {t} x h ? enables to define $Jm(Tx E ) as the space of continuous families (Tt)tETof classical pseudodifferential operators on the manifold M. Thus $Jm(Tx M , E ) fibres continuously over T with typical fibre the space $Jm(fi, C N ) (where $Jm(Tx 8)is endowed with the topology of kernels, see [57]). Note that the group I' acts on $Jm(Tx k;l,E) so that this fibration is equivariant.
u,
M,
206
Definition 13. (1) A longitudinal pseudodifferential operator P of order m on the foliated manifold V with coefficients in the smooth vector bundle E is a smooth section of the I?-equivariant bundle $"(T x B) over T ,
a,
P :T
+ qhrn(Tx
fi,E),
which is r-equivariant. (2) The principal symbol of the operator P is defined as the family a ( P ) = ( a t ( P ) ) t Esuch ~ that at(P) is the principal symbol of the operator Pt on Hence a ( P ) is a continuous I?-equivariant section of the bundle
M.
C E ( ( T * ( M\)
M)x T,Hom(n*(E))),
\M)
where 7r : ( T * ( M ) x T 4 M x T is the projection and C z means smooth in sections which are positively m-homogeneous in the sense that
a ( t ,5,A t )
= A W a ( t , 5,E ) ,
VA
> 0.
A pseudodifferential operator of order m is elliptic if for any t E T , a ( P ) ( t5, , 6) is an automorphism of the vector space Et,z. We point out that again by using the Schwartz kernel of P on each M inside T x M, we can construct an operator Po on the quotient manifold V. It is (longitudinally) elliptic if P is elliptic. Let us briefly define now the von Neumann index in the foliated bundle V. The groupoid of interest in V is the groupoid G := which is the quotient of the family indexed by T of groupoids x M x {t} by the diagonal action of I?. The v,-measure of any smooth compactly supported kernel k in G is then finite. The space of such kernels will be denoted by CF(G). We shall only consider longitudinal pseudodifferentialoperators on V with compactly supported distributional kernel and we denote by +"(V(T;E ) the space of those operators. As usual we set: +OO(VIT;E ) = UrnEz+rn(VIT; E ) and +-OO(VIT;E ) = nmEz+rn(VIT;E ) . The Schwartz theorem enables to identify +-OO(VIT;E) with the ideal C F ( G , End(E)) of longitudinally smooth continuous sections. Thus we have a short exact sequence of algebras: 0
-+
C:>'(G,End(E))
~f
+ " ( V J T ; E )-+ A ( V ) TE ; ) -+ 0 ,
and A(V1T;E) is called the algebra of longitudinal complete symbols on (V,F ) . This is a filtered algebra by the order of the pseudodifferential operators. The symbol map induces an isomorphism between the graded algebra of A(VIT;E ) and the graded algebra of I?-equivariant homogeneous sections of End(E) over T x T * ( ~\ )M . It turns out that the parametrix theorem recalled in 1.3 is true in the foliation case and we have:
207
Proposition 14.
If P E +"(VIT; E ) i s elliptic then its class in d(VIT;E )
is invertible. Proof. The proof is classical. For any t E T , Tt is an elliptic operator on the non compact (in general) manifold A?. Therefore we can find a parametrix St which is a pseudodifferential operator on M such that id-StoTt and id-TtoSt are regularizing operators in the manifold M . From the construction of the parametrix, we easily deduce that we can take the family (St)tET I?-invariant. It remains then to check that S can be chosen so that it induces a compactly supported kernel in the quotient groupoid ( M x f i x T ) / r . But the parametrix S, being constructed locally we know that we can take it almost local in the sense that its support in &f x &f is contained in a small neighborhood of the diagonal. Therefore the Schwartz kernel of the quotient operator of S can be taken compactly supported in the groupoid (Gx M x T ) / r as allowed, see also [I]. CI In other words, compactly supported elliptic pseudodifferential operators admit parametrices modulo the algebra CF(G, End(E)). We point out, but this will not be used here, that this last algebra is (algebraically) Morita equivalent to the algebra CF(G) and the latter is also Morita equivalent to the algebra C"(T) xalg I?, the algebraic crossed product algebra of the algebra C"(T) of smooth functions on T by the action of the discrete group I?. Therefore, the above exact sequence enables to define a I?-index as in [40], i.e. an index which belongs to the K-theory group of the algebra C m ( T )xalgI?. Proposition 15. Let P be a pseudodifferential operator in $ J - ~ ( V I Twith ) k 2 0. Then P extends to a bounded operator which belongs to M and we have: (i) If k < -p/2 then P is a (I?, p)-Hilbert Schmidt operator; (ii) If k < -p then P is a (I?, p)-trace class operator in M ; (iii) If k = -p, then the operator P belongs to the Dixmier ideal associated with M and the trace o n M induced by p . The Dixmier trace T r : , , associated with the trace Trr,Mo n M and any state w on the C*-algebra Cb(RW;)/Co(R;), is then given by:
where o(Pt)is the principal symbol of Pt, so it is positively (-p)-homogeneous in and $J is again the characteristic function of the fundamental domain U . Proof. (i) and (ii) are proved in [23].(iii) is proved in [16].
0
208
Let us recall now the construction of the Ruelle-Sullivan morphism associated with the I?-invariant measure p on the foliated bundle V . Smooth differential forms in the leaf direction of V can be identified with r-invariant differential forms in the M-direction of M x T . So a longitudinal differential k-form w on V is a smooth map w : T + a’(&?),
which is r-equivariant. We shall denote by R’yo(V/IT)the space of longitudinal k-forms on V . The longitudinal differential de can then be defined as the composition with the de Rham differential on M . We get in this way a longitudinal de Rham complex [19]: de : R”O(V1T) 4 52k+1*o(VIT).
The cohomology of this longitudinal complex will be denoted by H*>O(VIT). Using again the characteristic function $ of a fundamental domain U in we define:
a,
Because we are working with invariant differential forms, the map Cp,rinduces a well defined map on Hp1’(VIT)[19]. There is another way to define the Ruelle-Sullivan current in the foliated manifold V itself. This is classical and uses a partition of unity construction. It turns out that this Ruelle-Sullivan map C, on V when evaluated at a longitudinal differential form w is exactly equal to the evaluation of the above map Cp,r at the r-invariant longitudinal form 3 which is the lift of w t o M x T . This is easy t o check and the justification is then omitted. Let now be any (scalar for simplicity) order one elliptic longitudinal differential operator on M x T which is r-invariant. Denote by D the induced operator on the quotient closed foliated manifold V . Then D is a longitudinally elliptic scalar differential operator. The projections onto the kernel and cokernel, denoted respectively fi and fi’, of fi belong t o the von Neumann algebra M . We have already observed that the regularizing operators are Trr,, class operators in the von Neumann algebra M , therefore the operator fi is a Trr,p Fredholm operator in the sense of Section 4. Hence fi and @’ are Trr,p finite projections and we define the (r,p)-index of D by:
D
Indr,,(D) := Trr+(fi) - n r , p ( P’)E R. Indeed the operators fi and @‘ have p a.e bounded kernels which are smooth x T . In the smooth closed foliated in restriction to each leaf M x { t } of
209
manifold V, we can also define the p-index Ind,(D) of D using partitions of unity in V and the holonomy invariant transverse measure p [23]: Theorem 6. Let D be any r-invariant differential operator o n M x T which is elliptic in the M direction. Then the operator D admits a won Neumann (I?,p)-index Indr,,(D) with respect to the won Neumann algebra M and this real number coincides with the p-index of the underlying longitudinal operator D on the foliation of V : Indr,,(D) = Ind,(D) This theorem generalizes Atiyah’s result on coverings and the proof is an easy generalization of Atiyah’s method, see [l]. The triviality of the foliated manifold M x T enables to deduce the measured index theorem for many geometric operators from the Atiyah covering theorem without foliations. The main property of these operators is that they are independent of the T-component and are already I?-invariant on M. Assume for simplicity that M is a spin even dimensional manifold. Let D be the operator obtained as the tensor product of the Dirac operator on &’ by the identity operator on T. So for any t E T, Dt is independent o f t and is just the Dirac operator of M constructed out of the Dirac operator on M . The operator b is obviously a r-invariant differential operator which is elliptic in the &f direction. For any t E T, the I?-index of the operator Dt is then equal to the spin genus of M thanks to the covering index theorem [l]. One then immediately checks that the (I?, p)-index of fi is simply the mean, with respect to the measure p of the I?-indices. Rewriting the spin genus of M as an integral over the fundamental domain U of the A genus of M, we deduce the measured index theorem in this case. Note that most of the geometric operators are constructed in this way, so the index theorem for them is easy to prove in this case. However, when we consider for instance the Dirac operator along the leaves of V twisted by a vector bundle E over V , the lift depends on the T-variable and the index theorem is not obvious. Nevertheless, the same result holds in the general case and is a consequence of the analysis carried out by A. Connes in [23]. So Connes’ measured index theorem [23] provides information on the solutions of longitudinal elliptic equations on the non compact manifold M x T , whenever the operator which is involved is I?-invariant as a family operator over T . Corollary 2. Under the above assumptions, we have the following index formula: Indr+(fi) =< Ch(&)A(M),[Cr,,] > .
210
Proof. We apply Connes’ measured index theorem together with Theorem 6 and obtain: Indr,,(fi) =< Ch(E)A(F), [C,]>, where F is the longitudinal bundle of V and [C,] is the Ruelle-Sullivan current associated with the invariant measure p on (V,F ) . But in our case, it remains to compute the integral of the top-dimensional component of the differential form Ch(E)A(F) on U and then integrate the result against p on T . This is easily justified by the fact that V \ ( I n t ( U )x T ) is Lebesgue negligible. Thus the proof is complete. 0
6
Type I1 non commutative geometry
The data proposed by A. Connes to define a “geometry” is a triple (A,1-1, D ) , where d is a *-algebra represented in a Hilbert space 1-1 and D is an unbounded densely defined self-adjoint operator with a summability condition. To work with such a spectral triple, A. Connes imposes some constraints on the interaction of D with A. This formalism has been very useful especially in exploring index theory for singular spaces. We extend in this section some known results in non commutative geometry to the setting of von-Neumann algebras. The main idea is simple, we replace the algebra B(1-1)of all bounded operators in 1-1 by a von-Neumann algebra M , i.e. by the commutant of some unitary group in B(1-1). In the 11, factor case, the comparability of projections involves a semi-finite trace which is not the usual trace on B(1-1),and which can be used to define the index of von Neumann Fredholm operators as in the previous sections. Any pdimensional von Neumann spectral triple (A,M , D ) defines an analytic index map: Indo,, : K,(d)
+
R,
which is essentially given in the even case by the formula:
PKW( D ~ ))*= dim, (Ker(De)) -dim, (KdDe)’). This index map may be computed by using the natural pairing < ., . > between cyclic cohomology and cyclic homology of the algebra d . More precisely, we shall see that there exists a natural cyclic p-cocycle Ch,(M, 0 ) such that:
)) (-. Indo,, ( [el) := 7(her(o,
Indo,,(z) =< Ch,(M, D ) ,Ch(z) >, where < ., . > denotes the natural pairing and Ch(z) is the Chern-Connes character of z as defined in [25] or in [41].
21 1
6.1
Von Neumann spectral triples
In view of polynomial formulae, we shall restrict ourselves t o finite dimensional spectral triples. The general case can be treated similarly by extending the definition of 0-summability. Definition 14. By a von Neumann p-summable spectral triple we mean a triple ( A , M , D ) where M c B(3-1)is a von Neumann algebra faithfully represented in a Hilbert space 3-1 and endowed with a (positive) normal semifinite faithful trace 7 , A is a *-subalgebra of the von Neumann algebra M , and D is a 7-measurable self-adjoint operator such that: (i) b'a E A, the operator a ( D i)-' belongs t o the Dixmier ideal
+
LP+'(M, 7 ) ; (ii) Every element a E A preserves the domain of D and the commutator
[D,a] belongs t o M ; (iii) For any a E A, the operators a and [ D , a ]belong to nnENDOm(bn), where 6 is the unbounded derivation of M given by b(b) = [IDI,b]. When H is &-graded with A even and D odd, we say that the von Neumann-spectral triple is even and denote by y the grading involution assumed to belong t o M . In (i) the ideal L p , o ( M , ~ is ) the ideal composed of operators T E M such that (2' 0 T)PI2belongs t o the Dixmier ideal L1?"(M,7). Examples. (1)Let M be a smooth closed (Riemannian) spin manifold of dimension n. Let D be the Dirac operator associated with the spin structure. Then ( A = C"(M),M = B ( X ) , D )where 3-1 is the L2-space of spinors, is a n-summable von Neumann spectral triple. As pointed out by A. Connes, it is possible t o recover the Gauss-Riemann calculus on M from (A,M , D ) [29]. (2) Let (V, F ) be a (closed) foliated manifold with a holonomy invariant transverse measure A and p 1 1 the dimension of the leaves of (V, F ) . Let D be a generalized Dirac operator along the leaves of (V, F ) acting on sections of a hermitian vector bundle E. Denote by W i ( V ,F ; E ) the von Neumann algebra associated with A and E . The holonomy invariant transverse measure A gives rise t o a semi-finite trace 7~ on W i ( V ,F ; E ) given by the formula: 7 A ( T ) :=
-
k,F
n(TL)dA(L).
Then ( A = C"(V), W i ( V ,F ; E ) ,D )is a p-summable von Neumann spectral triple which is not a type I spectral triple [16]. (3) Let I? --+ M be a Galois covering over a closed manifold M of dimension n. Let 2, be the I? covering of a generalized Dirac operator D on M , and consider the von Neumann algebra M of bounded r-invariant operators
212
as defined in [l]with its von Neumann trace Trr. Then ( A = C m ( M ) ,M , D ) is a n-summable von Neumann spectral triple as can be deduced easily from the results of [l]. (4)Let D = ai(s)D$ + b(s) be a first order uniformly elliptic differential operator with almost periodic coefficients on R". As proved by Shubin in [62], even if not written in these terms, the index of such an operator can be defined by looking at a spectral triple (A,M , B) that we shall briefly describe. The operator fi is a direct integral over the Bohr compactification W; of operators Dz defined by:
xi
So B acts on L2(RE x R"). The algebra A is the algebra CAP"(R") of smooth almost periodic functions on R" , and M is the von Neumann crossed product algebra L"(R") x R~iiscrete which is a 11, factor. 6.2
The analytic Chern-Connes character
As usually, we shall replace D by sgn(D) the sign of D. Let
D
= FIJDI
be the polar decomposition of the self-adjoint operator D. To get rid of the possible non invertibility of F1, we replace the Hilbert space H = Ker(D)* @ Ker (D) by
IH
= H @ Ker(D)
E
HI @ H2 @ H3,
(6.1)
where H1 = Ker(D)*, H2 = Ker(D) and H3 is an extra copy of Ker(D). Denote by e l , e2 and e3 the projections onto H I , H2 and H3 respectively. According to the splitting (6.1) of E , set:
F=
(!!:)
(iy :) 000
= Fl+
V where V =
We thus define F, V in the semi-finite von Neumann algebra
M
=M
@ ~ D MC ~B(IH), D
which is equipped with the trace ? a -+ a @ 0. We have: Lemma 10. (1) F = F* and F 2 = 1; (2) Va E A, [ F l u ]E L p + ( f i , ? ) ;
= T @
r. Finally, embed A in M by
213
(3) Va E A, aFa = aF1a; (4) Va E A, a ( F - F1) E LP>"(fi,?) and hence a ( F - F1) E K ( M l ? ) .
Proof. ( 1 ) Trivial. (2) Note first that apD (and hence and pDa) belongs to L P @ ( M , T ) , for any a E A. Indeed, we have:
apD
= a(D
+ i)-'(D + i ) p =~ i a ( D + i ) - ' p ~E L p @ ( M l
T).
On the other hand we have [Fl,a] E L P , " ( M 1 ~ )for any a E A. Indeed we get by easy computations:
[FlI a] = [FlI a](Fl +PO) (Fl +PO) = [Fl,a]Fl(Fl +pD)+[Fl I a]pD(Fl+PO) = [Fl,a]Fl(Fl+ P O ) +Fiapo(Fi + P O ) , where FlapD(F1+PO) E LP~"(M1T ) by the preceding observation. Moreover, we have by straightforward computation:
+
[ ~ l , a ]=~[ lD , ~ ] ( Di ) - l + Z
+ i)-l F ~ [ ~ D I , ~ ] ( D + i)-',
[ F ~ , ~ ] F ~ ( D
-
and hence [F1,a]Fl E L p ~ " ( M 1 ~ )It. follows that [Fl1a]F1(F1 +PO) E L P @ ( M ,T ) and finally,
[Fl,a] E LP>"(M T ) for any a E A, as announced. But we have:
[F,a]= [ F l 1 a +[Vial. ] where:
[V,a]= V p D a ( 1 - e3) - ( 1 - e 3 ) a p ~ V e 3 , by straightforward computation. Since [Fl,a]I apo and pDa are in the Dixmier ideal LP>" ( M T),we finally get:
[ ~ , aE]LP+(M,?) and (2) is proved. (3) is obvious. (4)We have a ( F - F1) = aV and hence
a V ( a V ) * = aVV*a* = apDa. It follows that:
p S ( a v )= p s ( ( a ~ ) ( a ~ ) * )=' /p2s ( ( a m ) ( a p o ) * ) 1 /=2 pS(apD),for any s > 0 , and the result follows since we know that apD E LP+(M,
T).
0
214
We are now in position to define the index map Indo,, : K,(d)-+ R. Assume for simplicity that the spectral triple (A,M , D) is even and denote by y the grading involution. For any self-adjoint idempotent e E M,(d), the operator:
T = e o ( F @ I,) o e = e o ( F I BI,) o e anticommutes with y and satisfies:
T 2 - e = e o [ F @l n , e ] 0 ( F @1,)e = e o [ F @ l n , e ] o [ F @l n , e ] ,
(6.2) since F2 = 1. It follows that T 2 - e E LP/'y0"(e(&f @ End(C"))e,? @ Tr) and hence T is a (? @ Tr)-Fredholm operator in the von Neumann algebra e(&f @ End(Cn))e acting on e(7-l"). Denote by Indo,,(e) the (?@Tr)-indexof the positive part of T acting from e ( H T ) to e ( H 2 ) . Now if e , e' are two self-adjoint idempotents representing a class [el - [el]in Ko(A),then the number Indo,,(e) - I n d ~ , ~ ( e 'only ) depends on the class of [e)- [e'] E Ko(A). The 7-index map thus induces a group morphism: IndD,T : Ko(A) -+ R.
So, any even von Neumann spectral triple map: IndD,, : &(A)
(A,M , D) gives rise to 4
an index
R,
that we shall describe in Theorem 7 as a pairing with some (polynomial) cyclic cocycle over A. Note that the odd case can be treated similarly, see [16].Let 4 be a cyclic k-cocycle on the algebra A. As in [25],we shall denote for any N 2 1, by +/j Tr the cyclic k-cocycle on M N ( A ) given by:
(4#Tr)(ao@ A o , . . ., a k @ A A ":= ) $ ( a o , . . . ,U')T~(A'-..A~)), for any (a',... ,aA")E A"' and any ( A ' , . . . , A A "E) M N ( C ) . Theorem 7. Let (A,M , D ) be an even von Neumann-spectral triple of dimension p with grading involution y. Let F be the symmetry associated with D as above so that F2 = 1. Then the formula: 2k
) = ( - l ) k ~ ( y a o [ F , a ... l ] [F,a2']); defines, for k > p / 2 , a 2k-cyclic cocycle on the algebra A and we have for any projection e in M N ( A ) : 42k(ao, ..., a
IndD,T(e) := Ind,((eFe)+) = ( 4 2 k # n ) ( e , . . ., e ) , independently of the choice of k .
215
Proof. The proof follows the lines of [25]. That 4 2 k is cyclic is evident. Let (a', . . . ,u ~ ~ +E' d2k+2. ) Then we have: b(42k)(ao,. . . ,a2k+') = (-l)'T(y[a'[F,
a'] . . . [F,a2'], aZk+'])
= (-l)kT([Cio[[F,a'] . f . [F,a2'],yCL2'+'])= 0.
Therefore the cochain 452k is a cyclic cocycle on A. To compute the r-index of (e o ( F 8 id^) o e)+, we apply the Calderon formula. From the relation (6.2), we deduce that the operator T = (e o ( F @ id^) o e)+ is 7-Fredholm in e ( M 8 End((CN))e with parametrix given by S = (e o ( F 8 id^) o e)-. Moreover, e - ST as well as e - T S are in Lk(e(M 8 End((CN))e,78 Tr). Therefore Proposition 13 gives: Ind,.((eFe)+) = (~#Tr)(yo (e - e o ( F @ Z d N ) o e)')'). Computing (e-eo(F@idN)oe)2)kand using the relation eo[F@lN,e]oe = 0 , we obtain [13]: (e - e o ( F 8 ZdN) o el2)' = (-l)ke o [F 8 I N ,
0
and hence the conclusion.
So associated with any von Neumann spectral triple, there is an index problem which can be stated as follows: "Give a local formula for the traced index map &(A)
--+ R."
Using Theorem 7, we see that the index problem can be completely stated in the the cyclic cohomology world. It can be stated as follows: "Find a local even cyclic cocycle 11, on A such that: Ve E Ko(A),< 11,, e >=< 4, e >, where 4 is the cyclic cocycle defined in Theorem 7." This index problem reduces to the index problem solved by A. Connes and H. Moscovici in [30] if one takes the usual von Neumann algebra of all bounded operators in a Hilbert space with the usual trace. Examples. In the examples listed after Definition 14, the index problem becomes: (1) In the first example of Riemannian geometry, we recover the classical index problem which was solved by Atiyah and Singer in [9,29] and which was recalled in the first chapter of these notes. More precisely, we have already observed that the Dirac operator D on a spin closed Riemannian manifold, yields a spectral triple (A,B(7f),0). The index map in this case thus coincides
216
with the Atiyah map of Section 1 and the local computation of this map is precisely given by the Atiyah-Singer index theorem. ( 2 ) In the case of measured foliations we recover the measured index problem which was solved by A. Connes in [23]. (3) In the case of Galois coverings, we recover the index problem which was solved by M.F Atiyah in [l]. (4) For almost periodic operators, we obtain the Shubin index problem that was solved in [62]. The index map yields here a morphism: Indo,, : Ko(RE) + R, where RE is the Bohr compactification of R". For further details we refer to [17] where a new proof of Shubin's index theorem is derived. Up t o normalization constants, the sequence 4" of Theorem 7 can be arranged t o represent a periodic cyclic cocycle on A [25], i.e. up t o appropriate constants, we have:
S(4n)= &+2. The periodic cyclic class obtained is called the Chern-Connes character of the von Neumann spectral triple. In [17] we give a local formula for this ChernConnes character using Wodzicki residues and following the method of [30]. This local formula unifies all the examples listed above and gives a complete solution t o the von Neumann index problem. To end these notes we give the local formula for the Hochschild class of the Chern-Connes character. This formula is proved in [16]. Theorem 8. Let I : HC*(d) -+ HH*(A) be the natural m a p which corresponds t o forgetting the cyclicity of the cocycles. T h e n the pairing of the Hochschild class I ( C h ( M , 0 ) )of the Chern-Connes character C h ( M , 0 ) of the even (2r)-von Neumann spectral triple is given by the following local formula:
< I ( C h ( M , D ) ) , z a : @ a t ~ ? ~ . . . @ a>= T C, C.u(yoa~o(D,aflo...o[D,aTT], a
i
where y i s the grading involution, C, is a constant depending only o n r and C ,a! @ a,' @ . . . @ a? i s any Hochschild cycle over A. 6.3
The equivariant case
In this last subsection, we briefly explain how the constructions of the previous subsections can be generalized t o deal with group actions. For simplicity, we shall only give the details for bounded Fredholm modules, the case of
217
equivariant spectral triples can be easily carried out in a similar way. We point out the close relation of the equivariant von Neumann non commutative geometry with the Lefschetz fixed point theory, see for instance [13-151. Let A be as before a (unital) *-algebra over C. Let I' be a given group which acts on A by automorphisms. Definition 15. A I?-invariant psummable von Neumann Fredholm module over (A,I?) is a couple (MIF ) such that: 0
M is a von Neumann algebra of operators in a Hilbert space H , with a positive faithful normal trace
0
7;
The couple (A,I?) admits a faithful covariant *-representation
(T,
U ) in
M; 0
The operator F belongs to M and satisfies: (i) F* = F and F 2 = 1; (ii) For any a E A, the commutator [F,.(a)] belongs to the Shatten space
LP(M,~); (iii) F is I?-invariant, i.e.
[F,U(Y)l = 0,
V-r E
The von Neumann Fredholm module (MIF ) is even if in addition the Hilbert space H admits a &-grading automorphism cy which lives in M and if A and U ( r ) are even for the grading while F is odd for the grading. We have assumed in the above definition that V-y E I?, the unitary operator U ( y ) belongs to the von Neumann algebra M . This assumption simplifies our discussion and can be lightened. Note then that any I?-invariant von Neumann nedholm module on A gives rise to a von Neumann F'redholm module over the discrete crossed product algebra A x r, given by the same couple (MIF ) . We shall concentrate on the even case, say the case of even I?-invariant von Neumann Fredholm modules. The notation ( M ,F ) hides the Hilbert space H and the trace T on M and these data will then be implicit in the sequel. Since the operator [ F , r ( a ) ]in the above definition is also bounded, Condition (ii) implies in fact that the commutator is a T-compact operator. The representation ( n , U ) is assumed covariant. This means that the action of I? on M c B ( H ) preserves the algebra A and that the induced action coincides with the original one on A, i.e. V(y) o T ( U ) o V(7-l) = T ( $ U ) )
Vy E I? and tla E
d,
Let now (MIF ) be a fixed even r-invariant von Neumann F'redholm module which is psummable and denote by cy e M the Z2 grading involution.
218
Then since (M,F) is a von Neumann Fredholm module over the discrete crossed product algebra A x I?, we can define its Chern-Connes character as a cyclic cocycle on A >a I? by [25]: Ch(M , F )(bo, . .
,bp) := (- 1 ) P / 2 ~ ( ao bo o [F,b'] o .. . o [F,b p ] ) , E ( A x r)P+'.Note that Ch(M, F ) ( b o ,. . . , P)is trivial
for any (bo, . . . ,bp) unless p is even. Thus we assume that p is even. Note also that in restriction to A, Ch(M, F ) induces a cyclic cocycle on A. Assume €or simplicity and for the rest of this subsection that r is a finite group and denote by R(F) the representation ring of I?. We can state: Theorem 9. Let V : I? + End(X) be a finite dimensional unitary representation of r and let e be a r-invariant projection in A 8 End(X). Then: (i) For any y E I?, the formula
< ChY(M,F),[e1>:= (Chr(M,F)ttn)(eo[U(y)@ V ( Y ) ] , ~ , e. ). ,. defines a pairing with the equivariant K-theory of A; (ii) The pairing of (i) respects the prime ideal I-, = {x E R ( r ) ,x(y) = 0) of R ( r ) associated with the conjugacy class of y in J? and induces an additive map: ChY(M,F ) : K;(A)-,+ C;
(aii) The mapping y H< ChY(M,F ) , [el > is a central function o n r and hence the Chern-Connes character furnishes an additive map:
< Chr(M, F ) ,. >: K:(d)
4
k(I?),
where R(r)is the space of central functions o n the finite group r. The proof of this theorem is given in [13]. See also 6.4. Let us define now the equivariant index map associated with any equivariant von Neumann Fredholm module. Let V : r 4 End(X) be a finite dimensional unitary representation of the finite group I?. To any I?-invariant projection e E A @End(X), we associate the operator e F e := e o ( F @1 ~o e) acting on the I?-Hilbert space e ( H @ X ) . Recall that a T-Fredholm operator in a sub-von Neumann algebra N of M is an operator in N which is invertible in N modulo .r-compact operators in N [16]. Lemma 11. The r-invariant self-adjoint operator e o ( F @ 1 ~ o )e is a TFredholm operator in the won Neumann algebra e(M @ End(X))e.
Proof. We have with obvious notations: (eFel2 - e = e[F,e ] ~=e e [ F ,e ] [ el. ~,
219
Therefore the operator e F e is 7-Fredholm with inverse (modulo 7-compact operators) given by e F e itself. Finally we point out that e F e is r-invariant, since e and F are r-invariant. 0 The positive part (eFe)+ of the operator e F e admits a von Neumann I?-index which lives in the ring k(r)of complex central functions on F. More precisely, if p e , and ~ pL,F are respectively the projections onto the kernel and cokernel of (eFe)+, then the von Neumann F-index Indr((eFe)+) : -+ C of (eFe)+ is defined by: Indr((eFe)+)(?) := (T#D)([U(Y) 8 V(Y)l OPe,F) - (7#Tr)([U(Y) @ V(Y)l
OPL,,).
That this is a central function is obvious from the trace property of 7 and the I?-invariance of p e , and ~ P:,~. Theorem 10. With the above notations we have the following polynomial equivariant index formula in R(r): Indr((eFe)+) =< C h r ( M , F ) , [el > .
Proof. Let e E A 8 End(X) be any r-invariant projection with V : r -+ End(X) a finite dimensional unitary representation of r as above. Denote by P the r-invariant 7-Fredholm operator:
P
:= [e(F 8 idx)e]+ : e(X+ 8
X) 4 e(X- 8 X).
Denote by Q the parametrix of P which is the negative part of the operator e ( F 8 idx)e. Then we have: e - QP = -e
o
[ F 8 l x , el o [ F 8 l x , el,
and the corresponding result for e - PQ. Therefore: e - QP E LP12(e(M8 End(X))e,7#Tr) and e - PQ E LP12(e(M8 End(X))e,~ # ” r ) . Hence we can apply the equivariant Calderon formula (See Lemma 5.2 below) to compute the equivariant von Neumann index of P. This gives: Indr((eFe)+)(Y) = l n d r ( p ) ( r ) =
(d~ ) ( [ U ( Y8) V(Y)]0 (1 - QP)””)
-# .(
~ ) ( U ( Y0) (1 - PS)”I2). Using the supertrace defined by S 7 ( T ) := 7 ( ao T ) where (Y is the involution associated with the &-grading of H , we can rewrite the r-index as: Indr((eFe)+)(y)= ( S T # T T ) ( U ( Y0)(e - [ e ( F @ i d ~ ) e ] ~ ) * / ~ )
220
The above computation of
then gives: 0
and hence the conclusion.
Lemma 12. (Equivariant Calderon formula) Let U : 'r + U('H) be a unitary representation of the compact Lie group I' an a Hilbert space 'H. Assume that conjugation by the elements of I' in the algebra B(X) of bounded operators in 'H preserves a von Neumann algebra N admitting a faithful normal trace I-. Let P, Q be two r-invariant operators in N such that 1- QP and 1 - PQ belong the Schatten ideal L"(N,r ) for some n 3 1. T h e n V y E I',Indr(P)(r) = r ( U ( y )0 [l - QP]")- .(V(y)
0
[l - PQ]").
Proof. The proof of this lemma is classical. It is given in [16] in the non equivariant case but still works in the equivariant setting with minor modifications. 0
Corollary 3. Let ( B ( H ) , F ) be a r-invariant even Fredholm module over the I?-algebra A, with the usual trace. T h e n for any y E I', we have: (ChY(B(H),F ) , Kr(A)) This is an integrality theorem. When Z,, the above corollary becomes: (ChY(B(H),F ) ,K r ( d ) )
6.4
c W). is for instance the cyclic group
c Z[eZiT/"].
Back to the equivariant periodic homology
Let again B be a C-algebra with unit. The complex of Hochschild homology of B is defined as the complex (C*(B), b ) , where Ck(B) := B @ (B/C)@band: k-1
b(bo @ . . . @ b k ) = c ( - l ) j b o 8 . .. @ bj-' 8 bjvfl 8 . .. @ bk+ j=O
(-l)'bkbo
@ b' @ .
. . @ bk-'
In the same way we can define the operator B : C k ( B ) + Ck+l(B) as the transpose of the cohomological operator B. Then again we have b2 = 0, B2 = 0 and bB + Bb = 0. This enables t o define Hochschild, cyclic and periodic cyclic homologies of B [53].
22 1
B is endowed with an action of the finite group
Assume in addition that
r. The subspace of C,(B) generated by differences y(x) x where y runs over r and x runs over C,(B) and the action of r on C,(B) is the diagonal one, is -
preserved by the operators b and B. We can therefore consider the quotient bicomplex that we denote by (Cf(B), b, B ) . The homologies thus obtained will be denoted by HHf(B), HCff(B) and HPf(B) and called respectively, the co-invariant Hochschild, cyclic and periodic cyclic homology of B. Assume now that B = A >a r is the discrete crossed product algebra considered in the previous sections. Denote by C,""""(B) the quotient of the Hochschild chains by the subspace generated by differences of the type
bo @ . . . @ V U ( y )@ bj"
@
. . . @ bk - bo @ . . . @ V
@ y ( V + ' ) U ( y )@
. . . @ bk.
Again the operators b and B induce well defined operators on this quotient space and the homologies will be denoted by HHfqUiW(B), HC:qUiv(B) and HpfWi" (B) respectively and called the co-equivariant homologies. We have a natural chain map
c,' (B) + cyzv (B), which induces morphisms between the corresponding homologies. If V : r 4 End(X) is a finite dimensional representation of I?, then we can consider the discrete crossed product algebra [ A@ End(X)] x F. If bo @ . . . @ bk is a Hochschild chain on [A@ End(X)] >a r, then we define a Hochschild chain on A >a F as the centre projection < bo @ . . . @ bk > of bo 8 . . . @ bk defined using an orthonormal frame of X by:
< b o @ . . . @ b k >:=
C 20,".
b:oil @ . . . @ b f k i o
,ik
Lemma 13. The projection
< . >: C,([d@End(X)] x r)
--f
C,(A XI I?),
i s a chain map between the periodic bicomplexes and it induces well defined maps between the co-invariant bicomplexes and the co-equivariant bicomplexes. Proof. That < . > is a chain map is classical [53]. Let us compute the expression:
< b o [ U ( y )@ V(y)] @ b'
. . ' 8 bk > < bo @ y ( b ' ) [ U ( y )@ V(y)] @ b2 . . . @ bk > . ( 6 . 3 )
@
222
We notice that: V(y)iky(b:,)V(y-')hj,
Y(bl)ij = k,h
Thus we obtain that the difference (6.3) is equal to: io 1'i0 9 i 1, ... 7 ik
byo,; 8 V(y)ibi,y(bl)i,i,v(y) B b:2i3 B . . . @ bfkio]. Therefore the operator < . > induces a well defined operator between the co-equivariant quotient bicomplexes. A similar computation gives the result for the co-invariant homologies. 0
Proposition 16. Assume that V : I' -+ End(X) is a finite dimensional representation of I'. Let e be a r-invariant projection in A @ End(X). We define the sequence (Ch,(e, y)),>o by: Ch,(e,y) := (-l),-
(2n)! < (e-l/2)[U(y)@V(y)]@e@...@e> E Cz,(dxr), n!
Then we have: FOTany y E r, the co-equivariant class [Ch(e,y)] of Ch(e,y) := (Ch,(e, y)),>o is ( b B ) closed; (ii) For any [el E KF(A) and for any n 2 0, the map: (2)
+
-., c
y i u
(A x I?) defined by y ++ [Ch,(e,y)]
E
C g y ( d x I?),
is a central function on I?. (iii) For any y E I?, the co-equivariant periodic cyclic homology class of Ch(e,y) only depends on the equivariant K-theory class of e in A; Proof. (i) The computation of b(Ch,(e, y)), gives:
The computation in C:quiu(A>a I?) of B([Ch,(e, y)]) gives: (2n)! e B([Ch,(e,y)]) = ( 2 n + l ) ( - l ) , ~ [ < U ( y ) @ e @ . . . @ >] Hence we obtain:
223
(ii) Let (71,72) E I?' and let us compare Ch,(e, We have:
7172)
with Ch,(e, y2y1).
Since we are using the co-equivariant classes, we can move U(72) @ V(72) to the right in this expression and use the r-invariance of e to move U(y~)@V(72) until we obtain:
<(~(Yz)
V(yz))(e - 1 / 2 ) ( U ( x )@ V(YI))@ e @ . . . @ e >
Again using the I?-invariance of e we finally deduce that: Chn(e, 7172) = Chn(e, 7271) (iii) The proof given in [53] still works in the co-equivariant setting.
0
The above Chern-Connes character is actually defined on the I?equivariant K-theory of the crossed product algebra A x I?. This is obvious from the proof. Note that if the algebra A can be endowed with a Banach norm, then the above Chern-Connes character belongs to the equivariant entire cyclic homology of d as defined in [29] or as defined in [41]. The central function which assigns to each y E I? the class of Ch(e,y) in HPZ4"a"(A x I') is called the equivariant Chern-Connes character of the projection e and denoted by Chequiv(e).It belongs to k(r)@HPZqui"(d x I?). Note that Ch(e,y) is not a cycle in the co-invariant periodic cyclic homology. In summary, we have: CheWiv . K r ( d ) + k(r)B HP:Q~~"(Ar), and for any equivariant von Neumann Fredholm module (M,F ), we have seen that C h r ( M , F ) E HPF(d x I?), is the image of an element Ch,,,,,(M,
F ) in HP&(d
x
r).
References 1. M. F. Atiyah, Elliptic operators, discrete groups and won Neumann algebras., Astkrisque 32/33, SMF, (1976).
224
2. M. F . Atiyah, Global theory of elliptic operators, Proceedings of the International conference in functional analysis and related topics, university of Tokyo Press (1970), 21-29. 3. M. F. Atiyah, Elliptic operators and compact Lie groups., Lecture Notes in Math. 401, (1974). 4. M.F. Atiyah and R. Bott, Notes o n the Lefschetzfixed point theorem for elliptic complexes, Harvard University (1964). 5. M.F. Atiyah, R. Bott and V. Patodi, O n the heat equation and the index theorem, Inv. Math. 19, (1973), 279-330. 6. M.F. Atiyah and F. Hirzebruch, Spin manifolds and group actions, Essays on topology and related topics, Mkmoires dkdiks A Georges De Rham, (1970). 7. M.F. Atiyah, V. Patodi and I. Singer, Spectral asymmetry and Riemannian geometry I, Math. Proc. Camb. Phil. SOC.77,(1975), 43-69. 8. M.F. Atiyah and G. Segal, The zndex of elliptic operators, 11, Anna. Math. 87 (1968), 531-545. 9. M.F. Atiyah and I. M Singer, The index of elliptic operators, I, Anna. Math. 87 (1968), 484-530. 10. M.F. Atiyah and I. M Singer, The index of elliptic operators, 111, Anna. Math. 87 (1968), 546-604. 11. M.F. Atiyah and I. M Singer, The index of elliptic operators, IV, Anna. Math. 93 (19), 484-530. 12. J. Bellissard, Lecture notes, Villa de Leyva 2001. 13. M. Benameur, O n the equivariant Chern-Connes character in N o n Commutative Geometry, preprint. 14. M. Benameur, Cyclic cohomology and the family Lefschetz theorem, t o appear in Math. Annalen. 15. M. Benameur, A higher Lefschetz formula for flat bundles, t o appear in Trans. AMS. 16. M. Benameur and T. Fack, O n von Neumann spectral triples, preprint. 17. M. Benameur, T. Fack and V. Nistor, work in progress. 18. M. Benameur and V. Nistor, Homology of complete symbols and non Commutative geometry, in Landsman, N. P. et al. (ed.), Quantization of Singular Symplectic Quotients. Birkhuser. Prog. Math. 198, 21-46 (2001). 19. M. Benameur and H. Oyono-Oyono, Computation of the range of the trace for quasi-crystals, preprint. 20. N. Berline, E. Getzler and M. Vergne, Heat kernels and Dirac operators, Grundlehren Math. Wiss., Vol. 298, Springer-Verlag, (1992). 21. J.-L. Brylinski and E. Getzler, The homology of Algebras of Pseudodif-
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Fredholm module, J. Funct. Analysis 84 (1989), 343-357. 42. P. B. Gilkey, Invariance theory, the heat equation and the Atiyah-Singer index theorem, second edition, CRC Press, (1995). 43. D. Guido and T. Isola, Singular traces on semifinite von Neumann algebras, J. funct. Anal. 134 (1995), 451-485. 44. D. Guido and T. Isola, Noncommutative Riemann integration and Novikov-Shubin invariants f o r open manifolds, J. funct. Anal. 176 (2000), 115-152. 45. R. Haag, Local quantum physics, Springer-Verlag, Berlin-Heidelberg-New York, 1992. 46. F. Hirzebruch, Topological methods in algebraic geometry, SpringerVerlag, Berlin-Heidelberg-New York, 1966. 47. A. Jaffe, A. Lesniewski and K. Osterwalder, Quantum K-theory, Comm. Math. Phys. 118 (1988), 1-14. 48. M. Karoubi, Homologie cyclique et K-the'orie, AstGrisque 149 (1987). 49. G. G. Kasparov, The operator K - b n c t o r and extensions of C*-algebras, Math. USSR Izv. 16 (1981), 513-572. 50. C. Kassel, Le re'sidu non commutatif [d'aprks Wodzicki], s6minaire Bourbaki 4lhme ann. 708 Astkrisque, (1989). 51. M. Kontsevich and S. Vishik, Determinants of elliptic pseudodifferential operators, MPI preprint, (1994). 52. H. B. Lawson and M-L. Michelson, Spin Geometry, Princeton Mathematical series 38,Princeton University Press, (1989). 53. J . L. Loday, Cyclic homology, Springer-Verlag 301, (1991). 54. J . Milnor, Introduction to algebraic K-theory, Ann. of Math. Stud., 72, Princeton Univ. Press (1971). 55. J. Milnor and D. Stasheff, Characteristic classes, Ann. of Math. Stud., 76, Princeton Univ. Press (1974). 56. C. C. Moore and C . Schochet, Global analysis o n foliated spaces, SpringerVerlag 1988. 57. V. Nistor and A. Weinstein, and Ping Xu, Pseudodifferential operators on groupoids, Penn State Preprint No. 202 (1997). 58. R. S . Palais, Seminar o n the Atiyah-Singer index theorem., Ann. of Math. Stud., 57 (1965), Princeton University Press, Princeton. 59. S. Paycha, Lecture notes, Villa de Leyva 2001, in the web-site http: //wwwlma.univ-bpclermont .fr/-paycha/. 60. R. Prinzis, Traces re'siduelles et asymptotique du spectre d 'oph-ateurs pseudo-diffe'rentiels., Thhse de Doctorat de l'Universit6 de Lyon, 1995. 61. R. T. Seeley, Complex powers of an elliptic operator, Proc. Symp. Pure Math. 10, (1967), 288-307.
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Geometric and Topological Methods for Quantum Field Theory Eds. A. Caxdona, H. Ocampo & S. Paycha @ 2003 World Scientific Publishing, pp. 228-264
INTRODUCTION TO QUANTUM INVARIANTS OF 3-MANIFOLDS, TOPOLOGICAL QUANTUM FIELD THEORIES AND MODULAR CATEGORIES CHRISTIAN BLANCHET LMAM, Universite' d e Bretagne-Svd, BP 573, F-56017 Vannes, France, Christian.Blanchet @univ-ubs.fr
Introduction In 1988 E. Witten introduced the notion of Topological Quantum Field Theory. This was the starting point of a fascinating interaction between mathematics and theoretical physics, deeply relating many domains such as knot theory, Von Neumann algebras, Hopf algebras, Lie algebras and quantum groups, Chern-Simons theory, conformal field theory. A key point in Witten' ideas is the interpretation, in term of Quantum Field Theory, of a famous knot invariant discovered by V. Jones in 1984. A Topological Quantum Field Theory includes topological invariants of 3-dimensional manifolds. Following Witten' ideas, a rigorous construction of such invariants was first obtained from knot theory and quantum groups by Reshetikin and Turaev. It was then shown by Turaev that the relevant algebraic structure was that of a modular category. From a modular category one can derive a Topological Quantum Field Theory and in particular, invariants of links acd 3-manifolds. Our purpose here is to give an introduction to the topological aspect of the subject. In the first part we focus on the construction of knots and links invariants. We discuss the diagrammatic description of knots and links. We introduce the notion of a solid link and use it to describe 3-manifolds; this description is so called surgery presentation of 3-manifolds. We also develop a categorical formalism which will be used further and give indications for the construction of specializations of the Homfly polynomial invariant of links, including the famous Jones polynomial. In the second part we give the axiomatic definition of a Topological Quantum Field Theory. Such a theory associates to a genus g surface a finite dimensional vector space which corresponds to the space of so called conformal
229
blocks in conformal field theory. In the third part we show that a Topological Quantum Field Theory can be derived from a modular category, which is a braided category with some additional algebraic features. In the last part we sketch examples. These fundamental examples can be obtained either by skein theory (some linearization of links) or by using quantum groups. In conformal field theory, the dimension of the space of conformal blocks is the Verlinde formula. It depends on a simple simply connected complex Lie group and an integer called the level. The interested reader will be able t o recover the combinatorics of the S U ( N )Verlinde formula from the TQFTs associated with the Homfly polynomial, and will find in the literature the material for the other classical series. These are lecture notes of a course given at the Summer School Geometr i c and Topological Methods f o r Quantum Field Theory in Villa de Leyva, Columbia, July 2001. We thank Sylvie Paycha and Sergio Adarve, the CIMPA, the Universidad de 10s Andes and ECOS-Nord for organizing and supporting this Summer School. 1
LINKS, 3-MANIFOLDS AND THEIR INVARIANTS
We work in the smooth category. For 3-dimensional manifolds, the topological and smooth classifications coincide. This means that a topological 3-manifold admits a unique C" structure, and that two 3-manifolds are homeomorphic if and only if they are diffeomorphic. It may be useful t o consider manifolds equipped with a Riemannian metric, for example if we need t o consider the bundle of (oriented) orthonormal frames; we point out that whenever we do so, the choice of the metric is irrelevant. 1.1
Knots, links and diagrams
For an elementary introduction t o knot theory, see [12,28,19,17]. An oriented knot is an embedding of the circle S1 in the Euclidean space R3 or in its one point compactification, identified with the sphere S3. An oriented link with m components is an embedding of a disjoint union of m copies of S1. We are interested in knots and links up t o the equivalence given by ambient isotopy. Definition 1.1. Two embeddings ko, kl : S1 4 S3 are ambient isotopic if
230
and only if there exists a smooth 1-parameter family of diffeomorphisms
h : S3 x [0, 11 + S3 (z,t) h ( z , t )= such that (for each t , ht is a diffemorphism and)
ho = Ids3 and
Icl
= hl o Ico
.
This relation includes oriented reparametrization. A knot (resp. link) is an embedding of the circle S1 (resp. disjoint union of copies of S') up to (oriented or not) reparametrization. Two isotopic knots are represented in figure 1, called Reidemeister moves. An immersed closed curve in the plane
Figure 1. Isotopic knots
R2 is generic if and only if any multiple point is a double point with non colinear tangent vectors. A regular diagram in the plane is a closed generic immersed curve, and a choice, for each double point, of the arc which passes over. Such a diagram gives a link in R3 or S3 well defined up to isotopy. Any link can be obtained this way, and the well known Reidemeister theorem describes the equivalence relation on diagrams induced by isotopy of links. Theorem 1.2 (Reidemeister theorem). Two diagrams represent isotopic links i f and only i f they are related by isotopy in the plane and a finite sequence of the moves in figure 3. We will mainly consider here thick knots and links: oriented embeddings of solid tori D2 x S1, up to isotopy. (This is equivalent to considering framed knots and links.) An oriented diagram in the plane gives a thick link by using a blackboard convention. The core of the link is given by the diagram, and the image of
23 1
Figure 2. A (trivial) solid knot
the longitude 1 x S1 follows the normal vector pointing outside the blackboard. The Reidemeister theorem for solid knots and links in R3 or S3 holds if we replace the first Reidemeister move (Rl) by (Rl’) in figure 4. In the Reidemeister moves one has to consider the various choices of orientations. Exercise 1.1. We define the weight of a crossing t o be f l for the first crossing below (positive crossing), and to be -1 for the second one.
a) Let ( K ,K’) be a 2 components link given by a diagram. Show that the sum of the weights over all mixed crossings is an invariant of the link. Show that it is an even integer. This invariant is twice the linking number. For further use, we denote the linking number by Zk(K,K’). b) Let K be a solid knot. Show that the sum of the weights over all self crossings is an invariant of the solid knot. This invariant will be called the framing coefficient of the solid knot.
232
/ \ Figure 3. Reidemeister moves
Figure 4. Reidemeister move (Rl’)
233
1.2
Surgery o n links and K i r b y calculus
Let us consider a solid knot in S3
g : D2 x S1 L-) S3 . We use it t o build a new 3-manifold as follows. We remove the interior of the embedded solid torus, and we glue back another solid torus -S1 x D2 using the restriction g l s l x s ~(the minus sign here is for reversing orientation). This process is known as surgery. The manifold only depends on the isotopy class K of the solid knot represented by g; we denote it by S 3 ( K ) . Exercise 1.2. a) Show that S3 is a union of two solid tori with common boundary. b) Show that S2x S1 is also a union of two solid tori with common boundary. c) Show that the manifold S2 x S1 is obtained by surgery on the solid knot represented by the diagram in figure 5 (0-framed unknot).
Figure 5. Surgery diagram yielding S2 x S'
More generally, we will do surgery on a solid link with m components whose isotopy class is denoted by L = (L1,. . . ,Lm);we get a surgered manifold denoted by S3( L ) . A theorem of Lickorish and Wallace asserts that any closed oriented 3manifold is homeomorphic to S 3 ( L )for some L. As a consequence, we have that any oriented closed 3-manifold can be represented by a diagram in the plane. A famous Kirby theorem describes the equivalence relation on diagrams induced by oriented homeomorphism of 3-manifolds. Theorem 1.3. The equivalence relation on diagrams induced b y oriented homeomorphism of oriented 3-manifolds is generated by isotopy in the plane, the Reidemeister moves (Rl'), (R2), (R3), reversing the orientation of one component, and b y the two Kirby moves ( K l ) , (K2) described in figure 6. The (Kl) move is called the handle slide; here the two arcs belong t o distinct components of the link. The (K2) move is called stabilization.
234
Figure 6. Kirby moves
Exercise 1.3. a) Show that the diagrams in figure 7 represent the projective manifold RP3, and the same manifold with reversed orientation: -RP3. b) Use Kirby calculus to show that RP3 and -RP3 are positively homeomorphic oriented manifolds (i.e. there exists an oriented homeomorphism between R P 3 and -RP3).
Figure 7. Projective space with positive and negative orientation
235
1.3
The tangle category
The main idea in the following is t o consider invariants of links obtained by associating some algebraic quantities t o small pieces of diagram, and algebraic operations t o elementary gluings of the pieces. In order t o do so, we consider horizontal slices of solid links. Suppose that we have a solid link in R3 which intersects the planes IK2 x 0 and R2 x 1 transversally. Then the intersection of the core of the solid link with these planes gives a finite family of framed points in each plane. Here a framing at a point is a trivialization of the tangent bundle t o the plane at this point; the orientation of this framing coincides or not with the orientation of the plane, depending if the orientation of the core of the link induced by the embedding goes upwards or downwards at this point. We define a category* 7 in which objects are finite sets of framed points in the disc D 2 , and morphisms are pieces of solid links in the cylinder D2 x [O; 11, with boundary the framed points in the given source and target objects, up t o isotopy which is identity on the boundary of the cylinder. Composition is given by stacking the cylinders. Such a morphism is called a tangle and 7 is called the tangle category. People often call tangle category the full sub-category of 7 build up with objects whose points have standard positions along the real axis and whose framings are the standard positive or negative one as above. This sub-category is equivalent t o 7;we will denote it by 7. and call it the strict tangle category. Morphisms in this category are represented by diagrams in [-1,1] x [O; 11, and the Reidemeister theorem holds. The proof in [12] can be adapted; see [14] for a detailed presentation of the tangle category. An example of tangle is depicted in figure 8. In such a figure the time parameter goes upwards, so that the morphism gf will be depicted with g lying above f ; if the string at a boundary point is upwards, then it is understood that the framing is the standard one in the horizontal plane, and if the string is downwards then the framing is obtained * A category C consists of a class of objects Ob(C), for each pair of objects (V,W ) a set of morphisms Homc(V,W ) ,for each triple of objects (U, V,W ) a composition Homc(U,V) x Homc(V, W )
--f
Homc(U,W )
which is associative and such that any object V admits an identity morphism l v E En& ( V ) = Homc (V,V ).
236
Figure 8. A tangle
from the standard one by applying [ z H -Z]. We define a tensor product on the category 7 by using
-
j =j-l H j l : D2H D2 L) D 2 ,
+ iz.
where, for E = f l , j , : D2 D2 is the embedding sending z t o 5 Equipped with this tensor product, 7 is a monoidal category [20] (the associativity constraints are the obvious ones) and 7"is a strict+ monoidal g is category. If the morphisms f and g are given by diagrams, then f 18 represented by putting these diagrams next t o each other , with f on the left and g on the right. The strategy in the following is t o use representations of the strict tangle category. We would like a monoidal functor with source the strict tangle category I", and target a category where computations can be done, for example where set of morphisms are finite dimensional vector spaces.
1.4
Yang-Baxter operator
The Yang-Baxter equation was motivated by physical considerations (statistical physics, integrable systems ...) and played a key role in the foundation works on quantum groups. A Yang-Baxter operator is a solution of this famous equation. Let V be a finite dimensional vector space over the field k. Here any field k could be used; in most examples one can use complex numbers C. When we introduce a parameter s it may be useful t o work with the field of rational functions Q ( s ) (generic s). In order to get a 3-manifold invariant we need t o tThis means that the objects ( m@J n)@ p and m @J (ng p ) are equal; in a general monoidal category the associativity constraint only gives an isomorphism.
237
specialize this parameter to a root of unity, and then the convenient scalar field would be a cyclotomic field. Definition 1.4. A Yang-Baxter operator is an isomorphism T : V g 2P2 i such that
( r @l v ) ( l v @ r ) ( r @I v ) = ( l v @ r ) ( r @I v ) ( l v @ r )
(1)
We denote by B, the Artin braid group. We may define B, as the group generated by the invertible tangles a1,. . . ,an-l represented in Figure 9. This 2
i t 1
Figure 9. Braid generator ui
group has a presentation with generators
{
uia2+1ai uiaj
01,.
= ai+10iua+1
= ajai
. . ,~
~ - and 1 ,
relations
1 . .. n - 2, if li - j l 2 2 .
i
=
Note that a braid t E B, defines a element of the tangle category. In fact we have B, c E n d p ( n ) . If r is a Yang-Baxter operator, then we can define a representation @i-1
8,-i-1
B, -+ GL(V@,) by sending a i to 1, . We would like an @ r @ 1, extension to the strict tangle category, i.e. we want a functor from the tangle category 7"to the category of vector spaces. Generators of the tangle category 7"are depicted in Figure 10. The two first ones are braidings and will be sent to Yang-Baxter operators. The remaining ones should be understood as evaluations and co-evaluations. Our functor will send these generators to duality morphisms satisfying certain compatibility conditions. In order to do so, we pick p E G L ( V ) , and use the isomorphisms End(V) M Hom(V* @ V,k) M Hom(k,V* @ V ) given by the (standard)
238
duality t o define the homomorphisms corresponding t o p:
e v : V * @ V + k ;b v * : k - + V * @ V . Here ev(f @ x) = f(p(x)) and bv- is the transposed of ev. The two homomorphisms below are defined similarly using p-'.
e v * : V @ V * + k ;b v : k + V @ V * . We define r$.,v E Hom(V*@V,V@V*) and r$,v. E Hom(V@V*,V*@ V ) by the formulas below. T
f
r:,v.
~
*=,
(ev ~ 8 I v w * ) ( ~ v *8 rT
I v - ) ( ~ v * ~ v bv) ,
= ( l V * B v 8 e v * ) ( l v -@ TT 8 ~ v - ) ( b v * 8 IVSV*)
.
Definition 1.5. The triple (V,T , p ) is called an enhanced Yang-Baxter operator if and only if i) p @ p commutes with r ; ii) (ev 8 I v ) ( ~ v - ~ r ) ( b v@- I V ) ( ~ V@ I V ) ( ~ V *8 r - )(bv. @ l v ) = l v ; iii) T f~ , ~ , T =~ lV.@v. . , ~ This terminology was introduced in [31] with a slightly more restrictive definition. The theorem below is established by using a presentation of the tangle category. As a monoidal category, it is generated by the six morphisms in Figure 10, with the relations described in Figure 11.
Figure 10. Generators for the strict tangle category 7'
239
w
Figure 11. Diagrammatic description of relations in the strict tangle category 7'
For a proof, see for example chapter 12 in [14]. Note that our tangles here involve the framing, so that the relation (2.6) there has to be replaced by the one given by the Reidemeister move (Rl'). Theorem 1.6. If the triple ( V , r , p ) is an enhanced Yang-Bmter operator then there exists a unique monoidal functor from the strict tangle category 7' to the category of vector spaces whose value o n elementary morphisms are given in figure 12. Exercise 1.4. (See [31,30,24].) Fix an integer N 2 2. Let V be the k-vector space with basis vi, i E {-N + 1,-N + 3,. . . ,N - 3, N - 1). Let s be a non
240
Figure 12. Functor associated with an enhanced Yang-Baxter operator
zero element in k;denote by s4 a square root of s. We suppose that s4 is in
k. Let E E End( V @ V ) be defined by
E.ei
Ez"j"ek@ el, with
@ ej = k.1
!
ifi<j, k=iandl=j; s-l if i > j , k = i and 1 = j ; 1 if i # j, k = j and 1 = i; 0 otherwise. s
Ek! = 'j
Let r = s l v @ v - E , and p : V --+ V be defined by p('ui) = s:vi. Note for further use that r-l = s - l l ~ ~ -E v , and r - r-l = ( s - s - ' ) l v @ v . 1. Show that E 2 = ( s
+ s-I)E.
2. Show that
Deduce that r is a Yang-B%xteroperator.
3. Show that (V, r, p ) is an enhanced Yang-Baxter operator
24 7
The invariant of links associated with this enhanced Yang-Baxter operator is the rank N specialization of the Homfly polynomial. This terminology will be explained further in relation with quantum groups. The case N = 2 gives the famous Jones polynomial.
I .5
Ribbon category
Let C be a monoidal category with trivial object 0. A (left) duality on C associates t o any object V a n object V* and morphisms
ev : V * 8 V
4
0 ,bv : 0 4 V 8 V *
such that
(ev @ I v * ) ( ~ v * 8 bv) = I V -
.
A braiding in C is a natural family of isomorphisms
cv,w : vc3 w + w @ v
,v,w E O b j ( C ) ,
such that C V , W @ W ~=
(1 w 8 CV,WJ)(CV,W @ 1 W J ),
CVBV’,w = (.V,W
c3 IVJ)(lV c3 C V J , W ).
(4)
(5)
A twist in C compatible with the duality and the braiding is a natural family of isomorphisms
ev : v -+ v ,v E obj(c) , such that
(0,
€3
Iv*)bv = (lv c3 Bv*)bv ,
d V @ W = cw,vcv,w(~v BOW) .
(6)
(7)
Definition 1.7. A ribbon category is a monoidal category C with duality, braiding and compatible twist. The tangle category is a ribbon category. It is universal in the following sense.
242
Theorem 1.8. Let C be a strict ribbon category, and V an object in C , then there exists a unique couariant functor from the strict tangle category 7”to C , which sends the standard object with one point, denoted by 1, to the object V , and preserves tensor product, twist and duality. Such a functor is called a ribbon functor. Here strict means that the tensor product is strictly associative. For a proof see Theorem 1.2.5 in [32]. Note that a link is an endomorphism of the trivial object, hence we have that this construction provides an invariant of links with values in E n d c (0). Exercise 1.5. Associate a ribbon category t o an enhanced Yang-Baxter operator. The functor in the above theorem has various extensions; in view of our construction of TQFT in the next sections, we introduce a slightly modified version of the colored ribbon graphs considered in [32], which we call solid graphs. Roughly speaking, a solid graph is a 3-dimensional thickening of a finite graph. A solid graph is a 3-dimensional manifold with boundary, which is decomposed into parametrized copies of cylinders D2 x [0, 11, and solid tori 0’ x S1. The cylinders are decomposed in two parts which are called respectively the (solid) edges, and the (solid) vertices. The tori are called the closed components. The bottom and top of each solid vertex have (positively or negatively) framed points and so are objects in the category 7; each of these points is glued t o the end of a solid edge, using the framing. A coloring of a solid graph over a ribbon category C, is a labelling of the closed components and the solid edges by objects in C, and of the solid vertices by morphisms in C, which is compatible with the gluings. We define the category of C-colored solid graphs similarly as we did for tangles; here objects are finite sets of framed points in the disc, labelled with objects of C, and morphisms are embedded colored solid graphs whose free ends of edges are conveniently glued to the source and target points. We denote by & this category, and by Gz the strict sub-category (i.e. the full sub-category built up with objects whose points have standard position along the real axis and have standard positive or negative framing). Here also, morphisms in this category are represented by diagrams. An example of a diagram representing a solid graph is given in Figure 13. Edges are represented by arcs; solid vertices are represented by boxes, and for each of them top and bottom are given by the diagram. A colored solid vertex will
243
Figure 13. A diagram of a solid graph
be called a coupon. The following theorem is a reformulation of Theorem 1.2.5 in [32]. Theorem 1.9. Let C be a strict ribbon category, then there exists a unique monoidal functor f r o m the strict C-colored solid graphs category G z t o C , which respects duality, braiding and twist, and sends a colored positively framed standard point to its label, and a coupon to the morphism given by its label. Note that this theorem allows to use the graphical calculus for morphisms in the ribbon category. Whenever we do so, the functor is implicit. In a ribbon category there is a notion of trace of morphisms and dimension of objects. The trace of a morphism f is denoted by (f).
The dimension of an object V is the trace of I v ; we will use the notation ( V )as well as (1 v ) . We often say quantum trace and dimension to distinguish from the usual trace and dimension in vector spaces. 1.6
The skein method
The tangle category 7 is in a natural way a ribbon category. It will be useful to switch to a category in which the objects are the same, but morphisms are
244
linear combinations of tangles with coefficients in some ring k. This category is denoted by k [ q ; note that composition and tensor product are bilinear. The skein method builds ribbon categories by quotienting the linear tangle category k[7Jby local relations; the local relations generate a tensor ideal in the monoidal category k [ q , and the quotient is a ribbon category. Example 1.1 (Simplest example). The ring is Z[q*'], the local relation is given below.
0 -
1
Example 1.2 (Homfly skein theory). Here k is a ring which contains u*', uf', s*I, -. The local relations are given in figure 14.
0
v-1-v
=g--Q-l
Figure 14. Homfly relations
This example with specific specializations of the parameters will play a key role in our constructions. The ribbon category defined with these relations
245
is called the Hecke category; we denote it by H . The trivial object is denoted by 0, the standard object with n points is denoted by n, the module E n d H ( n ) is denoted by H,. Exercise 1.6. We suppose that k is a field, N 2 2, v = s - ~ a, = 1 in k, and that si is in k. Show that the functor in exercise 1.4 induces a functor from the category H to the category of k-vector spaces. Theorem 1.10. Ho i s isomorphic to k . This theorem gives an invariant of links ( L ) E k, which is the Homfly polynomial (framed version); here we fix the isomorphism with the normalization (8) = 1. The Homfly polynomial was obtained by several authors [13,26]. A proof based on exercise 1.4 can be found in [31],as well as a proof of the following result. Theorem 1.11. The algebra H , i s isomorphic to the quotient of the Artin braid group algebra k[B,] by the qugdratic relations a-lai = s - s-l. This algebra is the Hecke algebra; it is a deformation of the symmetric group algebra.
sot-'
1.7 Hopf algebra and quantum groups We saw in exercise 1.4 that some r E E n d ( V @ V ) satisfies the Yang-Baxter equation. This can be explained by the fact that V is a module over the quantum group U,sl(N). We give an outline of this approach (see [14]for a basic exposition). Definition 1.12. A bialgebra over a field k is an algebra (with unit) A equipped with two algebra morphisms
A :A + A @ A
c:A+k such that A (the comultiplication) is coassociative, and c (the augmentation) is a counit. A Hopf algebra$ (A, A , E , S ) is a bialgebra with an antipode S , i.e. an endomorphism S of A such that (here m : A @ A -+ A denote the multiplication) for any x E A
( m0 ( S @ I d A ) 0 A)(X) = ( m0 ( I d A @ S ) 0 A ) ( X )
=c(X)lA
.
$See Section 1 in the lecture notes Hopf algebras in noncommutative geometry by Joseph C . VBrilly in this book.
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As a fundamental example, we have the universal enveloping algebra Ug of a Lie algebra 0. Consider the category of finite dimensional representations of a Hopf algebra A , denoted by Rep(A). This category has a tensor product, defined by using the coproduct on A, making Rep(A) into a monoidal category. Moreover, by using the antipode, we can provide Rep(A) with a duality. Denote the coproduct A by a’ 8 a’’ ,
A(a)= (a)
then the A-module structure on a tensor product and on the dual are defined by
a.(v 8 w ) =
C a‘.w 8 a”.w , (a)
( a . f ) ( v )= f(S(a).w) . A universal R-matrix on A is an invertible element in A 8 A such that VU E A , AoP(a) = RA(a)R-l
(A‘8 IdA)(R)
R13R23
.
(8)
(10)
The element w in the center of A is then a universal twist if and only if
A(W) = PA(R)R(W8 W) , Here PA : A 8 A
4
f(W)
= 1 , s ( V ) = 2, .
(11)
A 8 A is the flip, AoP(a) = PAA(u)and
R12 = R 8 1~ , R23 = 1~ C3 R
,
R13 = ( I d A ‘8 p ~ ) ( R i 2 .)
Definition 1.13. A ribbon Hopf algebra is a Hopf algebra with universal R-matrix and twist. Proposition 1.14. If A is a ribbon Hopf algebra, then Rep(A) is a ribbon category. The universal enveloping algebra Ug of a finite dimensional simple Lie algebra admits a deformation U,g which is almost a ribbon Hopf algebra. Here the almost means that the universal R-matrix lives in a suitable completion. If q is a generic parameter, the representation category (which is a ribbon
247
category) is very similar to the classical case, and (type 1) irreducible representations are deformations of the classical ones. There exists an invariant for a link whose components are colored with irreducible representations. For g = s l ( N ) , if the vector representation is used (deformation of the standard representation C"), then one can show [30] that the corresponding invariant is the specialization of the Homfly invariant obtained in Exercise 1.4. 2
TOPOLOGICAL QUANTUM FIELD THEORY
In this section, we introduce the axiomatic definition of a TQFT in dimension 3. This concept first appeared in [35,4]. For basic references, see [32,5].
2.1
Axiomatic definition
A TQFT in dimension 3 will provide the following assignments. C
V ( C )7
where C is a surface (possibly with additional structure) and V ( C )is a finite dimensional k-vector space;
M
++
V ( M ) E H o m ( V ( C ) , V ( C ' ) ),
where M is a 3-dimensional cobordism between C and C', i.e. a 3-manifold with boundary -C LI C'. This has to satisfy axioms which are conveniently expressed in the categorical language. The cobordism category C2 is defined as follows. Objects are closed oriented surfaces; morphisms are cobordisms up to equivalence. A morphism between CO and C1 is a 3-manifold M together with a diffeomorphism j between the boundary -CO LI C1 and a M . The appropriate notion of equivalence on the cobordisms is orientation preserving diffeomorphism rel. boundary. The cobordisms ( M ,j ) and (MI,j') are equivalent if and only if there exists an oriented diffeomorphism 4 : M -+ M' such that j'=(boj.
Disjoint union makes this category into a monoidal category. The surface with opposite orientation is a left and right dual; the duality morphisms are given by the product of the surface with the interval [0;1]; the evaluation ex is equal to C x [O; 11, viewed as a cobordism from -C LI C to 0, and the
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coevaluation bc is equal t o C x [O; 11, viewed as a cobordism from 0 t o ELI -C. We get a (symmetric) ribbon category. Definition 2.1. A TQFT in dimension 3 is a monoidal functor V from C2 to the category of finite dimensional vector spaces. LFrom the definition, we have that V(0) = k. Hence for a closed 3manifold M we have an invariant V(M) E End(V(0))= k. If aM = C, then we have a vector V(M) E Hom(k,V(C))21 V(C). Exercise 2.1. Use the duality cobordisms ex and bc to construct inverse isomorphisms between V(-C) and V(C)*. Exercise 2.2. Show that the dimension of V(C) is equal t o the invariant of
Cx
2.2
s1.
Framings
In most cases, we have to consider a cobordism category where surfaces and bordisms are equipped with additional structure. Many kinds of structures may be considered; see [29] for an extensive exposition on cobordism with structure. Note that gluing structures along a surface and changing orientation of the surface have t o be well defined. Here are some examples. Framed manifolds. The surfaces are equipped with a trivialization of the oriented stable tangent bundle; cobordisms are equipped with an extension of the given trivialization on the boundary, up to homotopy rel. boundary. Here one has t o fix the identification on the boundary. Our convention is 'outgoing normal vector first' for the oriented boundary. The discussion of existence of trivialization of an oriented bundle, and parameterization up t o homotopy, is a piece of obstruction theory (see e.g. [22]). The first homotopy groups of SO(n) are: r l ( S O ( n ) )= 2 / 2 for n 2 (3), 7rz(SO(n))= O for all n and r 3 ( S O ( n ) )= Z for n 2 5. In the dimensions we are interested in (up t o dimension 4), obstruction theory and parameterization up t o homotopy will rest on the Stiefel-Whitney class W Z , and the Pontryagin class p l . If we do so, we have to consider spin structures [lo]; the theory has interesting features, but for a basic exposition it is better t o work in a simpler context. Complex framing. It happens that in most cases, because of a so called framing anomaly which will be explained further, some kind of framing should be considered. A weak version of framing was proposed by Atiyah to solve this question; this was called 2-framing. The key point is that obstruction theory in dimension up t o 4 rests exclusively on the Pontryagin class p l ,
249
or equivalently on signatures of 4-manifold, by the Hirzebruch formula. An equivalent notion called pl-structure was considered in [9]. We would like to give here a geometric definition. The operation of adding a trivial line bundle to a vector bundle is called stabilization. If we do this repeatedly on a real vector bundle, then we will get a R" vector bundle, where Roo is the inductive limit lim R". If we do so on 71"'
the tangent bundle of a manifold, then we get the stable tangent bundle. If the manifold is oriented, then the complexification of its stable tangent bundle is a SU bundle. Definition 2.2. A complex framing of an oriented manifold M is a trivialization of the complexified stable tangent bundle (as a SU bundle). The first obstruction in the existence problem is the second Chern class of the complex bundle c2 = - p l ( M ) . Here pl ( M ) is the first Pontryagin class of the manifold M . Milnor and Stasheff book [22] is an excellent reference for what is needed here about characteristic classes. We deduce the following. Proposition 2.3. a) A n oriented surface C admits a complex framing which is unique up to homotopy. b) A closed oriented 3-manifold M admits a complex framing and homotopy classes are afinely isomorphic to H 3 ( M ,Z). c) A closed oriented 4-manifold W admits a complex framing i f and only i f its signature is zero and homotopy classes are afinely isomorphic to H 3 ( W ,Z). We may also consider the extension problem. If we fix a complex framing on the boundary of an oriented manifold, we are interested in extensions of it to the whole manifold up to homotopy rel. boundary. Proposition 2.4. a) For an oriented 3-manifold M , a complex framing of the boundary extends, and relative homotopy classes of extensions are afinely isomorphic to H 3 ( M ,a M , Z). b) For an oriented 4-manifold W , a complex framing a on the boundary aW = M extends, if and only i f some relative obstruction cz(a) = -p1(a) E H 4 ( W ,awlZ)vanishes; moreover another complex framing a' on M is homotopic to a i f and only zf p1(a) = p l ( a ' ) . The lemma below will be the key point for further use. Lemma 2.5. Let M be a connected closed oriented 3-manifold. There exists a bijection a H .(a) between complex framings on M up t o homotopy and Z,
250
such that for any oriented W with a W
=M
one has
a(a) = 3signature(W) - ( p l ( a ) ,[ W ] ). The integer .(a) will be called the sigma invariant of a.
2.3
Colored solid graphs
Let C be a ribbon category. We have already considered (Section 1.5) C-colored solid graphs. It will be useful in the construction t o extend the cobordism category with embedded C-colored solid graphs. The cobordism category Cob(C) is defined as follows. Objects are oriented surfaces with a trivialization of the complexified stable tangent bundle, and a finite set of points inside with trivialized neighborhood labelled with objects in C; in brief such an object will be called a C-marked surface. Morphisms are cobordisms with embedded C-colored solid graphs, and extension of the given complex framings, up to the equivalence given by homeomorphism rel. boundary, isotopy of the graph rel. boundary and homotopy of the complex framing rel. boundary.
2.4
The universal construction
In this subsection we fix a ribbon category C, and we work in the cobordism category Cob(C). The whole TQFT can be derived from a convenient invariant of closed 3-manifolds. Our method follows the construction in [9]. Here a closed 3-manifold M = ( M ,a , G) is an oriented closed 3-manifold M with a complex framing a and a C-colored solid graph G. Recall that the complex framing cy can be defined by its sigma invariant a(&); we denote simply by S3 and S 2 x S1the corresponding 3-manifold equipped with empty graph and the complex framing a0 whose a-invariant is zero. Let 2 be a multiplicative (for disjoint union) invariant of closed 3manifolds, with value in a field k. We say that 2 satisfies the surgery axioms provided (SO), (Sl) and (S2) below are satisfied. In (Sl) and (S2), the surgery gives a four dimensional cobordism between the two closed 3-manifolds M and M‘. We fix the (homotopy class of the) complex framing on M’ from the given one on M , by extending t o this cobordism.
25 1
(Sl) There exists a scalar 2)such that, if M‘ is obtained from M by an index one surgery, then Z(M’) = D Z ( M ) . (S2) There exists a linear combination of objects of C, u,such that, if M’ is obtained from M by an index two surgery on a solid knot K , then Z ( M ’ ) = Z ( M ,K ( u ) ) . Here the component K is colored by a linear combination of objects in the ribbon category C; the corresponding invariant Z ( M ,K ( w ) ) is obtained by expanding linearly. Exercise 2.3. Suppose that Z satisfies the surgery axioms, then show that D Z ( S 3 )= 1 and Z(S2 x Sl) = 1. Given an invariant Z which satisfies the surgery axioms, then we associate t o each object C of the category Cob(C) vector spaces V ( C ) and V(C) as follows. The vector space V ( C )is freely generated by morphisms with boundary C, i.e. by Homcob(c)(QJ,C). The vector space V(C) is the quotient of V ( C ) by the right kernel of the natural bilinear map
V (-C) @ V ( C )-+ k , defined on the basis by evaluating the invariant Z on the glued manifold. From the construction we have that V is a functor from the cobordism category Cob(C) t o the category of vector spaces. However, we will need additional hypothesis in order to have finite dimension and multiplicativity. The role of the surgery axioms is given by the following important lemma ([9], Proposition 1.9). Lemma 2.6 (Connexion lemma). Let N (resp. N ’ ) be a connected oriented manifold with boundary the surface underlying the object C (resp. -C), then a) V(C) i s generated by bordisms M with underlying manifold equal to N . b) A linear combination C XiMi of bordisms is zero in V(C) i f and only if, f o r any bordism M‘ with underlying manifold N’, one has 1XiZ(Mi UCM’) = 0. 3
3.1
MODULAR CATEGORIES Pre-modular and modular categories
Let k be a field. A monoidal category C, with trivial object 0, is said t o be kadditive if the Hom sets are k-vector spaces, composition and tensor product are bilinear, and E n d ( @ ) = k. An object X of a k-additive monoidal category
252
Figure 15. A transparent object p
is said to be simple if the map u H u l x from k = End(O) t o E n d ( X ) is an isomorphism. Note that this simplicity condition is stronger that the usual one in module theory. Definition 3.1. A modular category [32], over the field k, is a k-additive ribbon category C in which there exists a finite family r of simple objects X satisfying the axioms below.
Domination axiom For any object X of the category there exists a finite decomposition lx = f i l ~ ~ with g ~ ,X i E ? ! for every i.
xi
Non-degeneracy axiom The following matrix is invertible.
S = (SXp)X,pa-, where S,, E k is the endomorphism of the trivial object associated with the (A, p)-colored, 0-framed Hopf link with linking +l. It follows that r is a representative set of isomorphism classes of simple objects; note that the trivial object 0 is simple, so that we may suppose that 63 is in r. If we remove the last axiom, we get the definition of a pre-modular category [ 111. If the category C has direct sums and a zero object, then it is pre-modular if and only if it has finitely many isomorphism classes of simple objects, and any object splits as a direct sum of simple objects. Note. Direct sums may be added formally. By using the domination property, one can show that any pre-modular category with direct sums is abelian. Definition 3.2. An object p of a pre-modular category A is called transparent, if for any object V in A the equality in Figure 15 holds. Such an object is also called a central object. It is enough t o have the above equality for any V in a representative set of simple objects. Note that a category containing a non trivial transparent simple object cannot be modular,
253
simply because the row in the S-matrix corresponding to this transparent object is collinear t o the row of the trivial one. In the next subsection we show that the absence of non trivial transparent simple objects implies (under a mild assumption) that the category is modular.
3.2
Properties of pre-modular categories
We will first give some general facts about pre-modular categories. Let C be a pre-modular category over the field k and let r ( C ) be a representative set of isomorphism classes of its simple objects. A morphism f E H o r n c ( X , Y ) is called negligible if for any g E Hornc(Y, X ) one has ( f g ) = 0. We suppose that C has no non trivial negligible morphism (we quotient out by negligible morphisms if necessary). We denote by R the Kirby color, i.e. R = CxEr(c)(X)X. Here and above ( p ) is the dimension of p. Proposition 3.3. (Sliding property) Let u be an object in the pre-modular category C , then the two colored solid graphs depicted below define the same morphism in Endc ( u ).
Here the dashed line represents a part of the closed component colored by R. This part can be knotted or linked with other components of a colored solid graph representing the morphism. Note that the morphism is unchanged if we reverse the orientation of this closed component.
Proof. For A, p , u E r ( C ) we have a canonical isomorphism Hornc(X 8 v,p ) N Hornc(p,X 8 u)’. (Note that the obvious pairing is non singular, because we have no negligible morphisms.) We denote by ai, i E I i v , a basis in Hornc(p, X g u ) , and by bi the dual basis in Homc(X@v,p ) . By the domination property, we get that a basis in Hornc(X 8 u, X 8 u ) , is given by the bjai, i , j E I,””, p E r(c).
2%
By writing the identity of X @ u in this basis, we get the following decomposition formula (fusionformula)
x
cc P
ZE I p
x
# u
u
We denote by a: E H o m c ( p @3 u*,X) (resp. b: E H o m c ( k P @3 V*>) the element corresponding to ai (resp. to bi) via the isomorphism given by duality. We also have a fusion formula.
I
The calculations below establish the sliding property.
255
In the first and third equalities we use the fusion formula, the second equality holds by isotopy. 0
Lemma 3.4. (Killing property) Suppose that (0)is nonzero. Let X E I’(C), then the following morphism is nonzero in C i f and only zf X is transparent.
Proof. If X is transparent, then this morphism is equal to ( 0 ) l ~ which , is nonzero. Conversely, if this morphism is nonzero, it is equal to CIA for some
256
0 # c E k. Then, for any u E I’(C), we have
The second equality holds by the sliding lemma.
0
Proposition 3.5. A pre-modular category C with (0)# 0 which has no nontrivial transparent sample object is modular.
Prooj We have to check the non-degeneracy axiom. Let us denote by the matrix whose (A, p ) entry is equal t o the value of the 0-framed Hopf link with linking -1 and coloring of the components A, p. Then we have that
We deduce that the (A, p ) entry of the matrix S s is equal to the invariant of the colored link depicted below.
257
By using (12) and the killing property we obtain the formula
ss = (R)1 , where I is the identity matrix, which proves the invertibility of the S matrix.
0 3.3
A 3-manifold i n v a r i a n t
If L = (L1,.. . ,L,) is a solid link in S3,let BL = ( b i j ) be its linking matrix; for i # j , bij is the linking number lk(Li,L j ) (see Exercise 1.1) and bii is the framing coefficient of Li . We denote by b+ (resp. b-) the number of positive (resp. negative) eigenvalues of BL. The signature s i g n ( L ) = b+ - b- is the signature of the 4-manifold given by surgery on L , and whose boundary is S 3 ( L ) . For E = 51, we denote by U, the unknot with framing epsilon. From the sliding property we get the following theorem (here R is the Kirby color). Theorem 3.6. Let C be a pre-modular category over a field k such that (U1(0))(U-1(0))is non zero. Then there exists an invariant &(Ad) of closed oriented 3-manifolds such that for any link L = (L1,. . . , L,) in S 3 , one has
This invariant can be extended to closed 3-manifolds with a C-colored graph K inside by the formula
In view of the index 2 surgery axiom (S2), we note that a surgery on U E l E = f l does not change the %manifold. We would like t o represent this surgery by inserting some Ue(w). A renormalization w =
258
m.
We extend k if necessary, and fix K. such that K . ~= Let V = K - ~ ( U ~ ( S=~~)~) ( U - , ( f l note ) ) ; that V 2= (a). We define the normalized invariant of connected closed 3-manifolds M equipped with complex framing a and colored graph K by Zc(M,a,K) = VD-lK.a(”)Bc(M,K).
(15)
We extend multiplicatively Zc to non connected closed 3-manifolds. Proposition 3.7. The invariant ZC satisfies the surgery axioms (SO), ( S l ) and (5’2) with w = V - I f l .
3.4
TQFT associated with a modular category
As explained in Section 2.4, we use the invariant Zc to construct a functor Vc from the category Cob(C) to the category of vector spaces. Exercise 3.2. Show that: 1. Vc(S2) = k.
2. If A is a non trivial simple object, then V(S2,A) = 0. 3. If A, v are simple objects, then Vc(S2,A, v*) is equal to k if A and v are isomorphic and zero else. 4. If A, p , v are simple objects, then Vc(S2,A, p, v*) = Hornc(A @I p , v) = Vc(S2,A, p, v*) andVc(S2,A,p*, v*) = Hornc(A, p@Iv)Vc(S2,A, p * , v*).
Theorem 3.8. If C is a modular category, then the functor Vc is multiplicative and for any C-marked surface C the vector space Vc(C) has finite dimension. Corollary 3.9. The functor VC defines a Topological Quantum Field Theory. In the theorem above, multiplicativity is shown by using the surgery axioms. The description of the vector space Vc(C) reduces to those in exercise 3.2 by using the splitting formula below. Lemma 3.10 (Splitting formula). Let y be a simple closed curve on a connected surface C such that C cut along y has two connected components C’ and C”. Then one has an isomorphism
259
where C i i s obtained from C' by closing with a standard disc whose origin is colored by A, and xC" is obtained by closing C" with a copy of the same disc with reversed orientation. Exercise 3.3. Let y be a simple closed curve on a connected surface C such that C cut along y is a connected surface C'. Show that there exists an isomorphism
where 3.5
is obtained by closing as above.
Verlinde formula
The Verlinde formula gives the dimension of the TQFT modules. Proposition 3.11. Let C be a modular category with r as a representative set of simple objects, then for a closed genus g surface C g without marked point, the dimension of the TQFT module Vc(C,) is
xr This formula is proved in [32],Corollary 12.1.2.It follows from the computation of the trace of the identity, which is equal to Zc(C, x S'). Exercise 3.4. The 3-manifold C, x S' is obtained by surgery on the borromean link with 29 + 1 components represented in figure 16. Show 3.11. (Hint: use the fusion formula 12 and the killing lemma 3.4.) 4
4.1
EXAMPLES
The skein method
We give a sketchy description of the whole procedure. This was first applied to the Kauffman bracket skein theory. It was the possible to achieve the construction for the Homfly skein theory [36],[8], and for the skein theory associated with the 2 variables Kauffman polynomial [33,7]. The relation between these constructions and quantum groups is as follows. The Kauffman bracket theory corresponds to UqsZ(2). The Homfly skein theory corresponds to U,sl(N), n22.
260
Figure 16. Borromean link: surgery diagram yielding C, x S'
+
The Kauffman skein theory corresponds to Uqso(2n-tl), Uqsp(2n 1) and Uqso(2n 2), n 2 1.
+
Step 1 Consider a multiplicative (for disjoint union) invariant of (framed) links in the sphere with values in some field k.
Step 2 Define a k-linear ribbon category by considering linear combinations of framed tangles, and quotienting out by negligible morphisms rel. t o the invariant given in step 1. Show that the set of morphisms are finite dimensional vector spaces.
Step 3 Add the idempotents of the category as objects and show the (strong) semi-simplicity. Step 4 Study simple objects. If isomorphism classes of simple objects form a finite set, and there is no non trivial transparent object, then the category is modular. Otherwise, by considering transparent simple objects apply
26 1
a modularization criterion and procedure developed by Bruguihres [ll]; see also [25]. Exercise 4.1. Construct a TQFT based on the simplest skein theory in Example 1.1 with q a N-th root of unity if N is odd and a 2N-th root of unity if N is even (cf 1231). Exercise 4.2. We consider the Homfly skein theory as defined in Example 1.2. Recall that in this theory H, denotes the algebra of endomorphisms of n positively oriented points (Hecke algebra). 1. Suppose that the parameters are generic and k is a field. Show that there exists a unique idempotent f n E H , such that for i = 1,. . . , n - 1, a i f n = f n O i = asf,, and a unique idempotent gn E H, such that i = 1 , . . . ,n - 1, trig, = g , a i = -as-lg,.
2. Construct a TQFT based on Homfly skein theory with parameters s = a P 2 ,v = s - ~ ,with a a 41-th root of unity. 3. Construct a TQFT based on Homfly skein theory with parameters s a-2, v = s - ~ , with a a 21-th root of unity, 1 odd.
4.2
=
The quantum group machinery
Let g be a simple Lie algebra, then its universal enveloping algebra admits a deformation U,g which is called a quantum group. It is a Hopf algebra, and we are interested in the category of its finite dimensional modules. In the generic case, this category is a semi-simple ribbon category, but has infinitely many simple objects, indexed as in the classical case by the fundamental weights. If q is a root of unity, the representation category is much more complicated, however one can obtain the following facts, for an appropriate version of the quantum group (see [32,16,5]). 1. It is a ribbon category.
2. There exists a finite set of simple objects (indexed by the affine Weyl alcove) which, after quotienting by negligible morphisms, is closed under decomposition of tensor products and generates a modular category. The required decomposition of tensor products is established in [2,3]. The construction was first done in [27]. See (33,21,15,18] for further developments.
262
References
1. A.K. Aiston, H.R. Morton, Idempotents of Hecke algebras of type A , Journal of Knot Theory and Ram. 7 (1998), 463-487. 2. H. Andersen, Tensor product of quantized tilting modules, Comm. Math. Physics 149 (1991), 149-159. 3. H. Andersen, Paradowski J. Fusion categories arising from semi-simple Lie algebras, Comm. Math. Physics 169 (1995), 563-588. 4. M. Atiyah, Topological quantum field theories, Publ. Math. IHES 68 (1989),175-186. 5. B. Bakalov, A. Kirillov, Lectures o n Tensor Categories and Modular Functors, Univ. Lect. Series Vol. 21, AMS. 6. A. Beliakova, C. Blanchet, Skein construction of idempotents in the Barman-Murakami- Wenzl algebras, Math. Ann. 321 (2001) 2, 347-373. 7. A. Beliakova, C. Blanchet, Modular categories of types B,C and D, Comment. Math. Helv. 76 (2001) 467-500. 8. C. Blanchet, Hecke algebras, modular categories and 3-manifolds quantum invariants, Topology, Vol. 39 (2000), 193-223. 9. C. Blanchet, N. Habegger, G. Masbaum and P. Vogel, Topological Quant u m Field Theories derivedfrom the Kauffman bracket, Topology 34, No 4 (1995), 883-927. 10. C. Blanchet, G. Masbaum, Topological Quantum Field Theories f o r Surfaces with Spin Structure, Duke Math. Journal, Vol. 82, No. 2, 229-267. 11. A. Bruguikres, Catkgories prkmodulaires, modularisations et invariants des variitis de dimension 3, Math. Ann., Vol. 316 (2000), No. 2, 215236. 12. G. Burde, H. Zieschang, Knots, De Gruyter Studies in Math. 5, 1985. 13. P. F'reyd, D. Yetter, J. Hoste, W. Lickorish, K. Millet, A. Ocneanu, A new polynomial invariant of knots and links, Bull. AMS 12 (1985), 239-246. 14. C. Kassel, Quantum groups, Graduate Texts in Math. 155, SpringerVer lag. 15. A. Kirillov, O n inner product in modular tensor categories, Journal AMS 9 (1996), 1135-1170. 16. C. Kassel, M. ROSSO,V. Turaev, Quantum groups and knots invariants, Panoramas et Synthkses No 5, SMF (1997). 17. L. H. Kauffman, Knots and Physics, World Scientific (1991).
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18. T. Le, Quantum invariants of 3-manifolds: integrality, splitting, and perturbative expansion, math.QA/0004099 . 19. W. B. R. Lickorish, An introduction to Knot Theory, Springer-Verlag (1997). 20. S. Mac Lane, Categories for the working mathematician, Springer-Verlag (1971). 21. G . Masbaum, H. Wenzl, Integral modular categories and integrality of quantum invariants at root of prime order, Journal fur die Reine Ang. Math. 505 (1998) 209-235. 22. J. Milnor, J . Stasheff, Characteristic classes, Annals of Math. Studies 76, Princeton Univ. Press. 23. H. Murakami, T. Othsuki, and M. Okada, Invariants of 3-manifolds derived from linking matrices of framed links, Osaka J. Math. 29 (1992), 545-572. 24. H. Murakami, T. Ohtsuki, S. Yamada Homfly polynomial via an invariant of colored planar graphs, Ens. Math. 44 (1998), 325-360. 25. M. Muger, Galois theory for braided tensor categories and the modular closure, Advances in Math. 150 (2000), 151-201. 26. J. Przytycki, P. Traczyk, Invariants of kinks of Conway type, Kobe J. Math. 4 (1987), 115-139. 27. N. Reshetikhin, V. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Inv. Math. 103 (1991), 547-597. 28. D. Rolfsen, Knots and Links, Publish Perish (1976). 29. R. Stong, Notes on cobordism theory, Princeton Univ. Press (1968). 30. V. Turaev, The Yang-Baxter equation and invariants of links., Invent. Math. 92 (1988), 527-533. 31. V. Turaev, Operator invariants of tangles and R-matrices, Math. USSR Izvestia Vol. 35 (1990), 411-443 32. V. Turaev, Quantum invariants of knots and 3-manifolds, De Gruyter Studies in Math. 18, 1994. 33. V. Turaev, H. Wenzl, Quantum invariants of 3-manifolds associated with classical simple Lie algebras, Int. J. of Math., Vol. 4, No. 2 (1993) 323-358. 34. V. Turaev, H. Wenzl, Semi-Simple and modular categories from link invariants, Math. Ann., Vol. 309 (1997), 411-461. 35. E. Witten, Topological Quantum Field Theory, Comm. Math. Phys. 117
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Geometric and Topological Methods for Quantum Field Theory Eds. A. Cardona, H. Ocampo & S. Paycha @ 2003 World Scientific Publishing, pp. 265-311
AN INTRODUCTION TO DONALDSON-WITTEN THEORY MARCOS MARINO* Jefferson Physical Laboratory, Harvard University, Cambridge M A 02138, USA In these lecture notes a pedagogical introduction to Donaldson-Witten theory is given. After a survey of four-manifold topology, some basic aspects of Donaldson theory are presented in detail. The physical approach to Donaldson theory is based on topological quantum field theory (TQFT), and some general properties of TQFT’s are explained. Finally, the T Q F T underlying Donaldson theory (which is usually called Donaldson-Witten theory) is constructed in detail by twisting N = 2 super Yang-Mills theory.
Contents Introduction 1 Basics of four-manifolds
267 268
1.1 Homology and cohomology
268
1.2 The intersection form
269
1.3 Self-dual and anti-self-dual forms
271
1.4 Characteristic classes
272
1.5 Examples of four-manifolds
273
2 Basics of Donaldson invariants
274
2.1 Yang-Mills theory on a four-manifold
274
2.2 SU(2) and SO(3) bundles
276
2.3 ADS connections
277
2.4 Reducible connections
279
2.5 A local model for the moduli space
281
2.6 Donaldson invariants
283
*Email: marcosmlorentz .harvard. edu
266
3
N = 1 supersymmetry
287
3.1 The supersymmetry algebra
287
N = 1 superspace and superfields 3.3 Construction of N = 1 Lagrangians
289
3.2
291
4 n/ = 2 super Yang-Mills theory
294
5 Topological field theories from twisted supersymmetry
296
5.1 Topological field theories: basic properties
296
5.2 Twist of N
299
=2
supersymmetry
6 Donaldson-Witten theory
300
6.1 The topological action
301
6.2 The observables
302
6.3 Evaluation of the path integral
304
7 Conclusions and further developments
A Appendix: Conventions for spinors References
306 307 309
267
Introduction Donaldson-Witten theory has played an important role both in mathematics and in physics. In mathematics, Donaldson theory has been a fundamental tool in understanding the differential topology of four-manifolds. In physics, topological Yang-Mills theory, also known as Donaldson-Witten theory, is the canonical example of a topological quantum field theory. The development of the subject has seen a remarkable interaction between these two different approaches -one of them based on geometry, and the other one based on quantum field theory. In these lectures, we give a self-contained introduction to DonaldsonWitten theory. Unfortunately, we are not going to be able to cover the whole development of the subject. A more complete treatment can be found in [30]. The organization of the lectures is as follows: in section 2, we review some elementary properties of four-manifolds. In section 3, we present Donaldson theory from a rather elementary point of view. A more detailed and rigorous point of view can be found in [13]. In section 4, we give a quick introduction to N = 1 supersymmetry. In section 5, we present the physical theory behind Donaldson-Witten theory, i. e. N = 2 supersymmetric Yang-Mills theory. In section 6, we give a general overview of topological field theories and we explain the twisting procedure. In section 7, we construct Donaldson-Witten theory in detail and show that its correlation functions are in fact the Donaldson invariants introduced in section 3. Finally, in section 8, we give a very brief overview of recent developments. Acknowledgements. First of all, I would like to thank Greg Moore for initiating me in the u-plane integral approach to Donaldson-Witten theory, for discussions and explanations, for making available to me his notes on Donaldson theory and four-manifolds, and for the fruiful collaborations on these and other topics during the last four years. I would also like to thank L. Alvarez-Gaurn6, J.D. Edelstein, M. G6mezReino, J.M.F. Labastida, C. Lozano, J. Mas, G. Peradze and F. Zamora for collaboration and discussions on the topics discussed in these lectures. I would like to thank the organizers of the Summer School “Geometric and topological methods for quantum field theory” held at Villa de Leyva (Colombia) in july 2001 for giving me the opportunity of presenting these lectures. Last but not least, I would like to thank all the students and people in Villa de Leyva for such a wonderful school. Special thanks to Alejandro, Ana Maria, Boris, Fernando, Hernbn, Juan Camilo, Juan Pablo, Luis, Marcela, Maria Paula, Mario Fernando, Marta, Milena, and Oscar. This work has been supported by DOE grant DEFG02-96ER40959 and by NSF-PHY/98-02709.
268
1
Basics of four-manifolds
The purpose of this section is to collect a series of more or less elementary facts about the topology of four-manifolds that will be used in the rest of these lectures. We haven’t made any attempt to be self-contained, and the reader should consult for example the excellent book [20] for a more complete survey. The first chapter of [13] gives also a very good summary. A general warning: in these lectures we will assume that the four-manifolds under consideration are closed, compact and orientable. We will also assume that they are endowed with a Riemannian metric. 1.1
Homology and cohomology
The most basic classical topological invariants of a four-manifold are the homology and cohomology groups Hi(X, Z), H i ( X , Z). These homology groups are abelian groups, and the rank of Hi(X, Z) is called the i-th Betti number of X , and denoted by bi. Remember that by Poincark duality one has
H i ( X , 2) N H,-((X, Z).
(1.1)
and hence bi = b,-i. We will also need the (co)homology groups with coefficients in other groups like Zz. To obtain these groups one uses the universal coefficient theorem, which states that Hi(X,G)
N
Hi(X,Z)@z G@Tor(Hi-1(X1Z),G).
(1.2)
Let’s focus on the case G = Z,. Given an element z in Hi(X, Z), one can always find an element in Hi(X, Z,) by sending x + z @ 1. This in fact gives a map:
which is called the reduction mod p of the class x. Notice that, by construction, the image of (1.3) is in Hi(X, Z)@Z,. Therefore, if the torsion part in (1.2) is not zero, the map (1.3) is clearly not surjective. When the torsion product is zero, any element in H i ( X , Z,) comes from the reduction mod p of an element in Hi(X, Z). For the cohomology groups we have a similar result. Physicists are more familiar with the de Rham cohomology groups, HLR(X) which are defined in terms of differential forms. These groups are defined over R, and therefore they are insensitive to the torsion part of the singular cohomology. Formally, one has HhR(X) E (Hi(X,Z)/Tor(Hi(X, Z))) @ Z R. Remember also that there is a nondegenerate pairing in cohomology, which in the de Rham case is the usual wedge product followed by inte-
269
gration. We will denote the pairing of the cohomology classes (or differential form representatives) a, P by ( a ,P). Let’s now focus on dimension four. PoincarC duality gives then an isomorphism between H2(X1Z) and H 2 ( X ,Z). It also follows that b l ( X ) = b3(X). Recall that the Euler characteristic x ( X ) of an n-dimensional manifold is defined as n
x ( X )= x ( - l ) ? I i ( x ) . i=O
For a connected four-manifold X , we have then, using PoincarC duality, that
+
x ( X )= 2 - 2bl(X) bz(X).
1.2
(1.5)
The intersection form
An important object in the geometry and topology of four-manifolds is the intersection form,
Q : H 2 ( X ,Z ) x H 2 ( X ,Z ) -+ Z
(1.6)
which is just the pairing restricted to the two-classes. By Poincark duality, it can be defined on H2(X1Z) x H2(X1Z ) as well. Notice that Q is zero if any of the arguments is a torsion element, therefore one can define Q on the torsion free parts of homology and cohomology. Another useful way to look at the intersection form is precisely in terms of intersection of submanifolds in X . One fundamental fact in this respect is that we can represent any two-homology class in a four-manifold by a closed, oriented surface S: given an embedding
i : S r X ,
(1.7)
we have a two-homology class i + ( [ S ]E)H2(X1Z ) , where [ S ]is the fundamental class of S. Conversely, any a E H 2 ( X 1Z) can be represented in this way, and a = [Sa][20]. One can also prove that
&(a,b) = S a U s b , where the right hand side is the number of points in the intersection of the two surfaces, couted with signs which depend on the relative orientation of the surfaces. If, moreover, 7s-, vsb denote the PoincarC duals of the submanifolds S,, SI, (see [ S ] ) ,one has
270
If we choose a basis { U * } ~ = ~ , . , . , ~ ~for ( X )the torsion-free part of H2(X, Z), we can represent Q by a matrix with integer entries that we will also denote by Q. Under a change of basis, we obtain another matrix Q 4 CTQC, where C is the transformation matrix. This matrix is obviously symmetric, and it follows by Poincari! duality that it is unimodular, i.e. it has det(Q) = f l . If we consider the intersection form on the real vector space Hz(X,R), we see that it is a symmetric, bilinear, nondegenerate form, and therefore it is classified by its rank and its signature. The rank of Q, rk(Q), is clearly given by bZ(X), the second Betti number. The number of positive and negative eigenvalues of Q will be denoted by bz(X), bi(X), respectively, and the signature of the manifold X is then defined as
a(X) = bz(X) - b,(X).
(1.8)
We will say that the intersection form is even if &(a, a ) = 0 mod 2. Otherwise, it is odd. An element z of H2(X, Z)/Tor(Hz(X, Z)) is called characteristic if Q(z, a) = Q(a,a ) mod 2
(1.9)
for any a E H2(X, Z)/Tor(H2(X, Z)). An important property of characteristic elements is that Q(z,z) = o(X) mod8.
(1.10)
In particular, if Q is even, then the signature of the manifold is divisible by 8. Examples. (1) The simplest intersection form is: n(1) @ m(-1) = diag(1,. . . ,1,-1,. . . ,-l),
(1.11)
which is odd and has b l = n, b, = m. (2) Another important form is the hyperbolic lattice, (1.12)
which is even and has b$ = b, = 1.
27 1
(3) Finally, one has the even, positive definite form of rank 8 21000000 12100000 01210000 00121000 00012101 00001210 00000120 00001002
(1.13)
which is the Dynkin diagram of the exceptional Lie algebra Es. Fortunately, unimodular lattices have been classified. The result depends on whether the intersection form is even or odd and whether it is definite (positive or negative) or not. Odd, indefinite lattices are equivalent to p(l)@q(- l ) , while even indefinite lattices are equivalent to p H @ q&. Definite lattices are more complicated, since they involve “exotic” cases. Clearly the intersection form is a homotopy invariant. It turns out that simply connected smooth four-manifolds are completely characterized topologically by the intersection form, i. e. two simply-connected, smooth four-manifolds are homeomorphic if their intersection forms are equivalent. This is a result due to Freedman. The classification of smooth four-manifolds up to diffeomorphism is another story, and this is the main reason to introduce new invariants which are sensitive to the differentiable structure. But before going into that, we have to give some more details about classical topology. 1.3
Self-dual and anti-self-dual f o r m s
The Riemannian structure of the manifold X allows us to define the Hodge star operator *, which can be used to define an induced metric on the forms by:
1c, A *e = ($J,d)dP, where dp is the Riemannian volume element. Since *2 = 1,the Hodge operator has eigenvalues f l , hence it gives a splitting of the two-forms C12(M)in selfdual (SD) and anti-self-dual (ASD) forms, defined as the f l eigenspaces of * and denoted by C12i+(X)and C129-(X),respectively. Given a differential form 1c, E C12(M),its self-dual (SD) and anti-self-dual (ASD) parts will be denoted by @. Explicitly, 1 $* = 5(1c,f
(1.14)
272
The Hodge operator lifts t o cohomology and in four dimensions it maps
* : H2(X) + H2(X). (1.15) The number of +l eigenvalues of * in H2(X)is precisely b,f, and the number of -1 eigenvalues is b,. This means that we can interpret b: as the number of self-dual harmonic forms on X. This interpretation will be useful in the context of gauge invariants.
1.4
Characteristic classes
An important set of topological invariants of X is given by the characteristic classes of its real tangent bundle. The most elementary ones are the Pontriagin class p ( X ) and the Euler class e(X), both in H4(X,Z) N Z. These classes are then completely determined by two integers, once a generator of H4(X,Z) is chosen. These integers will be also denoted by p(X), e(X), and they give the Prontriagin number and the Euler characteristic of the four-manifold XI so e(X) = x. The Pontriagin number is related t o the signature of the manifold through the Hirzebruch theorem, which states that: p ( X ) = 3a(X).
(1.16)
If a manifold admits an almost-complex structure, one can define a holomorphic tangent bundle T(llo)(X). This is a complex bundle of rank T = dim(X), therefore we can associate t o it the Chern character c ( T ( ~ ~ ~ ) which (X)) is denoted by c(X). For a four-dimensional manifold, one has c(X) = 1 q ( X ) ca(X). Since q ( X ) is a two-form, its square can be paired with the fundamental class of the four manifold. The resulting number can be expressed in terms of the Euler characteristic and the signature as follows:
+
+
Cq(x) = 2X(X)
+ 3a(X).
(1.17)
Finally, the second Chern class of X is just its Euler class: c2(X) = e(X). If the almost complex structure is integrable, then the manifold X is complex, and it is called a complex surface. Complex surfaces provide many examples in the theory of four-manifolds. Moreover, there is a very beautiful classification of complex surfaces due to Kodaira, using techniques of algebraic geometry. The interested reader can consult [4,6]. There is another set of characteristic classes which is perhaps less known in physics. These are the Stiefel-Whitney classes of real bundles F over X, denoted by w i ( F ) . They take values in Hi(X, Z2), and a precise definition can be found in [20,31], for example. The Stiefel-Whitney classes of a fourmanifold X are defined as wi(X) = wi(TX). The first Stiefel-Whitney class of
273
a manifold measures its orientability, so we will always have wl(X) = 0. The second Stiefel-Whitney class plays an important role in what follows. This is a two-cohomology class with coefficients in Z2, and it has three important properties. If the manifold admits an almost complex structure, then
q ( X ) = w2(X) mod2,
(1.18)
i.e. wz(X) is the reduction mod 2 of the first Chern class of the manifold. This is a general property of w2(X) for any almost-complex manifold. In four dimensions, w 2 ( X ) satisfies in addition two other properties: first, it always has a integer lift to an integer class [21] (for example, it the manifold is almost complex, then cl(X) is such a lift). The second property is the Wu formula, which states that (w2(X),a ) = ( a ,a ) mod 2,
(1.19)
for any a E H2(X,Z). The 1.h.s can be interpreted as the pairing of a with the integer lift of w2(X). A corollary of the Wu formula is that an integer two-cohomology class is characteristic if and only if it is an integer lift of w2(X).
1.5
Examples of four-manifolds
(1) A simple example is the four-sphere, S4. It has bl = b2 = 0, and therefore x = 2, 0 = 0, Q = 0. (2) Next we have the complex projective space CP2. Recall that this is the complex manifold obtained from C3 - ((0,O)) by indentifyhg zi N Xzi, i = 1,2,3, with X # 0. CP2 has bl = 0 and b2 = 1. In fact, the basic two-homology class is the so-called class of the hyperplane h, which is given in projective coordinates by z1 = 0. It is not difficult to prove that h2 = 1, so QcPz = (1). Notice that h is in fact a CP1, therefore it is an embedded sphere in CP2. -The projective plane with the opposite orientation will be denoted by CP2, and it has Q = (-1). (3) An easy way to obtain four-manifolds is by taking products of two Riemann surfaces. A simple example are the so-called product ruled surfaces S2x C,, where C , is a Riemann surface of genus g. This manifold has bl = 2g, b2 = 2. The homology classes have the submanifold representatives S2 and C,. They have self intersection zero and they intersect in one point, therefore Q = H , the hyperbolic lattice, with b,f = b2 = 1. One then has x = 4(1- g). (4) Our last example is a hypersurface of degree d in CP3, described by a homogeneous polynomial z: = 0. We will denote this surface by S d . For d = 4,one obtains the so-called K3 surface.
xf=l
274
Exercise 1.1. 1) Compute c ~ ( S d and ) c2(Sd). Deduce the values of x and 0. 2) Use the classification of unimodular symmetric, bilinear forms to deduce Q K (for ~ help, see [20]). 2
Basics of Donaldson invariants
Donaldson invariants can be mathematically motivated as follows: as we have mentioned, Freedman’s results imply that two simply-connected smooth manifolds are homeomorphic if and only if they have the same intersection form. However, the classification of four-manifolds up to diffeomorphism turns out to be much more subtle: most of the techniques that one uses in dimension 2 5 to approach this problem (like cobordism theory) fail in four dimensions. For example, four dimensions is the only dimension in which a fixed homeomorphism type of closed four-manifolds is represented by infinitely many diffeomorphism types, and n = 4 is the only dimension where there are “exotic” R”’s,ie. manifolds which are homeomorphic to R” but not diffeomorphic to it. One has to look then for a new class of invariants of differentiable manifolds in order to solve the classification problem, and this was the great achievement of Donaldson. Remarkably, the new invariants introduced by Donaldson are defined by looking at instanton configurations of nonabelian gauge theories on the four-manifold. We will give here a sketch of the mathematical procedure to define Donaldson invariants, in a rather formal way and without entering into the difficult parts of the theory. The interested reader can consult the excellent book by Donaldson and Kronheimer [13]. Other useful resources include [16-181, on the mathematical side, and [9,34] on the physical side. The reference [15] gives a very nice review of the mathematical background.
2.1
Yang-Mills theory on a four-manifold
Donaldson theory defines differentiable invariants of smooth four-manifolds starting from Yang-Mills fields on a vector bundle over the manifold. The basic framework is then gauge theory on a four-manifold, and the moduli space of ASD connections. Here we review very quickly some basic notions of gauge connections on manifolds. A more detailed account can be found for example in [lo, 151. Let G be a Lie group (usually we will take G = SO(3) or S U ( 2 ) ) . Let P + M be a principal G-bundle over a manifold A4 with a connection A , taking values in the Lie algebra of G, g. Given a vector space V and a representation p of G in GL(V), we can form an associated vector bundle E = P X G V in the standard way. G acts on V through the representation
275
p. The connection A on P induces a connection on the vector bundle E
(which we will also denote by A ) and a covariant derivative VA. Notice that, while the connection A on the principal bundle is an element in O1(P, g ) , the induced connection on the vector bundle E is better understood in terms of a local trivialization U,. On each U,, the connection 1-form A , is a gl(V) valued one-form (where gl(V) denotes the Lie algebra of GL(V)) and the transformation rule which glues together the different descriptions is given by:
where gap are the transition functions of E . Recall that the representation p induces a representation of Lie algebras p* : g .+ gl(V). We will identify p , ( g ) = g , and define the adjoint action of G on p * ( g ) through the representation p. On M one can consider the adjoint bundle g E , defined by:
gE =p
XG
g,
(2.2)
which is a subbundle of End(E). For example, for G = SU(2) and V corresponding to the fundamental representation, g E consists of Hermitian, tracefree endomorphisms of E . If we look at (2.1), we see that the difference of two connections is an element in f l l ( g E ) (the one-forms on with values in the bundle g E ) . Therefore, we can think about the space of all connections d as an affine space with tangent space at A given by TAd = n l ( g E ) . The curvature FA of the vector bundle E associated to the connection A can be also defined in terms of the local trivialization of E . On U,, the curvature F, is a gl(V)-valued two-form that behaves under a change of trivialization as:
x
Fp = g;;F,g,P,
(2.3)
and this shows that the curvature can be considered as an element in f12(gE). The next geometrical objects we must introduce are gauge transformations, which are automorphisms of the vector bundle E , u : E -+ E preserving the fibre structure ( i . e . , they map one fibre onto another) and descend to the identity on X . They can be described as sections of the bundle Aut (E). Gauge transformations form an infinite-dimensional Lie group G, where the group structure is given by pointwise multiplication. The Lie algebra of G = r(Aut(E)) is given by Lie(G) = O0(gE). This can be seen by looking at the local description, since on an open set U , the gauge transformation is given by a map u, : U, -+ G, where G acts through the representation
276
p. As it is well-known, the gauge transformations act on the connections as
u*(A,) where V A U ,= du,
+~(VAU,)U,~,
= U,A,U;'
+ idu,u,l
+ i[A,,u,],
and they act on the curvature as:
= A,
u*(Fa)= u,F,u,1. 2.2
(2.4)
(2.5)
S U ( 2 ) and SO(3) bundles
In these lectures we will restrict ourselves t o the gauge groups S U (2 ) and S 0 ( 3 ) , and E corresponding t o the fundamental representation. Therefore, E will be a two-dimensional complex vector bundle or a three-dimensional real vector bundle, respectively. S U ( 2 ) bundles over a compact four-manifold are completely classified by the second Chern class Q ( E ) (for a proof, see for example [161). In the case of a SO(3) bundle V, the isomorphism class is completely classified by the first Pontriagin class Pl(V) = -Cz(V 63 C ) , and the Stiefel-Whitney class wz(V) E H 2 ( X ,22). classes are related by wz(V)' = pl(V) mod4.
(2.6) These characteristic
(2.7)
SU(2) bundles and SO(3) bundles are of course related: given an S U ( 2 ) bundle, we can form an SO(3) bundle by taking the bundle g E in (2.2). However, although an SO(3) bundle can be always regarded locally as an SU(2) bundle, there are global obstructions to lift the SO(3) group to an S U ( 2 ) group. The obstruction is measured precisely by the second StiefelWhitney class wz(V). Therefore, we can view SU (2) bundles as a special case of SO(3) bundle with zero Stiefel-Whitney class, and this is what we are going to do in these lectures. When the SO(3) bundle can be lifted t o an S U ( 2 ) bundle, one has the relation: PIP) = -4dE).
(2.8)
Chern-Weil theory gives a representative of the characteristic class p l (V)/4 in terms of the curvature of the connection:
277
where FA is a Hermitian, trace-free matrix valued two-form. Notice that Hermitian, trace-free matrices have the form:
(
a
I = ib+c
-ib+c -a
)
1
alblcER1
(2.10)
so the trace is a positive definite form:
Tr12= 2(a2 + b2 + c2) = 21
k = --
8T2
I
E SU(~).
LTrFi.
(2.11)
(2.12)
Notice that, if V has not a lifting t o an SU(2) bundle, the instanton number is not an integer. If V lifts to E l then k = cz(E). The topological invariant wz(V) for SO(3) bundles may be less familiar to physicists, but it has been used by 't Hooft [39] when X = T4,the fourtorus, t o construct gauge configurations called torons. To construct torons, one considers S U ( N ) gauge fields on a four-torus of lengths ap, p = 1, . . . 4. To find configurations which are topologically nontrivial, we require of the gauge fields to be periodic up to a gauge transformation in two directions: A , ( U l , .2)
= ~1(.2)Ap(O1.2)1
Ap(z1,m) = R z ( z l ) A p ( ~ lO), ,
(2.13)
where we have denoted by R A the action of the gauge transformation R on the connection A. Looking a t the corners, we find the compatibility condition %(az)R2(0) = ~ 2 ( a l ) Q l ( o > z l
(2.14)
where Z E C ( S U ( N ) )= Z N is an element in the center of the gauge group. We can allow a nontrivial Z since a gauge transformation which is in the center of S U ( N ) does not act on the S U ( N ) gauge fields. This means that when we allow torons we are effectively dealing with an S U ( N ) / Z N gauge theory. For SU(2), this means that we are dealing with an SO(3) theory, and the toron configurations are in fact topologically nontrivial SO(3) gauge fields with nonzero Stiefel-Whitney class.
2.3
A S D connections
The splitting (1.14) between SD and ASD forms extends in a natural way to bundle-valued forms, in particular t o the curvature associated to the connection A, FA E R2(gE). We call a connection ASD if
FA+ = 0.
(2.15)
278
It is instructive to consider this condition in the case of X = R4 with the Euclidean metric. If {dxl, dxp,dxg, dx4) is an oriented orthonormal frame, a basis for SD (ASD) forms is given by: { d ~Al dxp fd ~ A3d ~ 4dxl , A dx4 fdxp A d23, d ~A ld ~ f 3d ~ A4dzl}, (2.16) with f for SD and ASD, respectively. If we write F = +F,,dxp A dx”, then the ASD condition reads: F1p
F14 F 1 3
+ + +
F34 = 0 Fp3 = 0 F42
= 0.
(2.17)
Notice that the second Chern class density can be written as
V F i ) = {lFA+I2- IFit2}dP, where dp is the volume element and the norrn is defined as: 1
I+I2
=
p(+ A *+I.
(2.18)
(2.19)
We then see that, with our conventions, if A is an ASD connection the instanton number k is positive. This gives a topological constraint on the existence of ASD connections. One of the most important properties of ASD connections is that they minimize the Yang-Mills action (2.20) in a given topological sector. This is so because the integrand of (2.20) can be written as IFZ12 IFiI2,therefore
+
1 SYM =2
S,
+
lFA+lpdp 8?k,
(2.21)
which is bounded from below by 8r2k. The minima are attained precisely when (2.15) holds. The ASD condition is a nonlinear differential equation for non-abelian gauge connections, and it defines a subspace of the (infinite dimensional) configuration space of connections A This subspace can be regarded as the zero locus of the section
’.
s : A --+
R2’+(gE)
(2.22)
~
‘In the following, when we refer to the space of smooth connections we will proceed on a purely formal level, and we will avoid the hard functional analysis which is needed in order to give a rigorous treatment. We refer t o [13,16]for details concerning this point.
279
given by
s(A) =FA+.
(2.23)
Our main goal is to define a finite-dimensional moduli space starting from s-’(O). The key fact to take into account is that the section (2.22) is equivariant with respect to the action of the gauge group: s(u*(A))= u*(s(A)). Therefore, if a gauge connection A satisfies the ASD condition, then any gauge-transformed connection u*(A)will also be ASD. To get rid of the gauge redundancy in order to obtain a finite dimensional moduli space, one must “divide by Q” i.e. one has to quotient ~ ~ ‘ (by 0 )the action of the gauge group. We are thus led to define the moduli space of ASD connections, M A S D as , follows: (2.24) where [A]denotes the gauge-equivalence class of the connection A. Notice that, since s is gauge-equivariant, the above space is well-defined. The fact that the ASD connections form a moduli space is well-known in field theory. For example, on R4SU(2) instantons are parameterized by a finite number of data (which include, for example, the position of the instanton), giving 8k - 3 parameters for instanton number k 1221. The moduli space M A S Dis in general a complicated object, and in the next subsections we will analyze some of its aspects in order to provide a local model for it.
2.4
Reducible connections
In order to analyze M A S Dwe , will first look at the map Qxd-+d
(2.25)
and the associated quotient space d/Q.The first problem we find when we quotient by Q is that, if the action of the group is not free, one has singularities in the resulting quotient space. If we want a smooth moduli space of ASD connections, we have to exclude the points of A which are fixed under the action of Q. To characterize these points, we define the isotropy group of a connection A, r A , as = { U E QIU(A)= A } ,
(2.26)
which measures the extent at which the action of Q on a connection A is not free. If the isotropy group is the center of the group C(G), then the action is free and we say that the connection A is irreducible. Otherwise, we say that the connection A is reducible. Reducible connections are well-known in
280
field theory, since they correspond to gauge configurations where the gauge symmetry is broken to a smaller subgroup. For example, the SU(2) connection
A = ( a0 -a O )
(2.27)
should be regarded in fact as a U(1) connection in disguise. It is clear that a constant gauge transformation of the form u03 leaves (2.27) invariant, therefore the isotropy group of A is bigger than the center of SU(2). We will denote the space of irreducible connections by A*. It follows from the definition that the reduced group of gauge transformations = G/C(G) acts freely on A*. By using the description of as a section of Aut(E) and the action on A given in (2.4), we see that = {U E r ( A U t ( E ) ) I V A U =
o},
(2.28)
ie. the isotropy group at A is given by the covariantly constant sections of the bundle Aut(E). It follows that is a Lie group, and its Lie algebra is given by Lie(rA) = {f E flo(gE)IVAf = 0).
(2.29)
Therefore, a useful way to detect if is bigger than C(G) (and has positive dimension) is to study the kernel of VA in flo(gE). Reducible connections correspond then to a non-zero kernel of -VA
: n o ( g E ) --$ f l l ( g E ) .
(2.30)
In the case of SU(2) and S 0 ( 3 ) , a reducible connection has precisely the form (2.27), with isotropy group rA/C(G) = U(1). This means, topologically, that the SU(2) bundle E splits as:
E = L @L - l ,
(2.31)
with L a complex line bundle, while a reducible SO(3) bundle splits as
V=R@T,
(2.32)
where R denotes the trivial rank-one real bundle over X . The above structure for V is easily derived by considering the real part of Sym2(E). Notice that, if V admits a SU(2) lifting E, then T = L2. There are topological constraints , to have these splittings, because (2.31) implies that c2(E) = - c ~ ( L ) ~and (2.32) that Pl(V)=c m 2 .
(2.33)
When E exists, the first Chern class X = cl(L) is an integral cohomology class. However, when w ~ ( E#) 0, then it follows from (2.7) that L does not
28 1
exist as a line bundle, since its first Chern class is not an integral class but lives in the lattice 1 (2.34) H 2 ( X ,Z) -W2(V). 2 In particular, one has that
+
cl(T) = wz(V) mod2.
(2.35)
Therefore, reductions of V are in one-to-one correspondence with cohomology classes (Y E H 2 ( M ;Z) such that a2 = p l ( V ) . In the following, when we study the local model of MASD, we will restrict ourselves to irreducible connections. 2.5
A local model for the moduli space
To construct a local model for the moduli space means essentially t o give a characterization of its tangent space at a given point. The way t o do that is to consider the tangent space at an ASD connection A in A, which is isomorphic to 0' ( g E ) , and look for the directions in this vector space which preserve the ASD condition and which are not gauge orbits (since we are quotienting by G). The local model for M A S Dwas first obtained by Atiyah, Hitchin and Singer in [2]. Let us first address the second condition. We want to find out which directions in the tangent space a t a connection A are pure gauge, i.e. we want to find slices of the action of the gauge group 4. The procedure is simply t o consider the derivative of the map (2.25) in the G variable at a point A E A* to obtain
C : Lie(G)
-
TAA,
(2.36)
which is nothing but (2.30) (notice the minus sign in VA, which comes from the definition of the action in (2.4)). Since there is a natural metric in the space R*(gE), we can define a formal adjoint operator: Ct : 0Yg.E)
R0(gE)
(2.37)
given by Ct = VL. We can then orthogonally decompose the tangent space at A into the gauge orbit Im C and its complement:
R 1( g E ) = I m C @ K e r Ct.
(2.38)
Here we used the fact that I M C is closed as a consequence of its Fredholm property, which follows from the injectivity of the leading symbol of C. (2.38) is precisely the slice of the action we were looking for. Locally, this means that the neighborhood of [A] in A * / G can be modelled by the subspace of TAA
282
given by Ker oh. Furthermore, the isotropy group r A has a natural action on sll(g~)given by the adjoint multiplication, as in (2.5). If the connection is reducible, the moduli space is locally modelled on (Ker va)/rA (see [13,16]). We have obtained a local model for the orbit space d*/Q, and now we need to enforce the ASD condition. Let A be an irreducible ASD connection, verifying FA+ = 0, and let A+a be another ASD connection, where a E n'(gE). The condition we get on a starting from FA+a = 0 is p + ( V ~ a a A a ) = 0, where p+ is the projector on the SD part of a two-form. At linear order we find:
+
p+VAa = 0.
(2.39)
Notice that the map ~ + V is A nothing but the linearization of the section s, ds: ds : T A d
-
(2.40)
R2'+(gE)
The kernel of d s corresponds to tangent vectors that satisfy the ASD condition at linear order (2.39). We can now give a precise description of the tangent space of M A S Dat [ A ] :we want directions which are in Kerds but which are not in Im VA. First notice that, since s is gauge-equivariant, Im VA c Ker ds. This can be checked by direct computation:
P + V A V A=~IF,+,41 = 0,
6 E n0(gE),
(2.41)
since A is ASD. Taking now into account (2.38), we finally find:
T [ A ~ M A= S D(Ker d s ) n (Ker V l ) .
-
(2.42)
This space can be regarded as the kernel of the operator D = ~ + V @ A VL:
D : nl(gE)
-
Since Im VA c Ker d s there is a short exact sequence: 0
nO(gE)
(2.43)
OO(PE)€9 n2>+(gE).
3nl(gE)p-+VA W+(gE)
-
0.
(2.44)
This complex is called the instanton deformation complex or Atiyah-HitchinSinger ( A H S ) complex [2], and gives a very elegant local model for the moduli space of ASD connections. In particular, one has that T[A]MASD = HA 1 ,
(2.45)
where 23; is the middle cohomology group of the complex (2.44): (2.46)
283
The index of the AHS complex (2.44) is given by ind = dim HA - dim H i
-
dim H i ,
(2.47)
where HZ = Ker and H i = Coker ~ + V AThis . index is usually called the virtual dimension of the moduli space. When A is an irreducible connection (in particular, KerVA = 0) and in addition it satisfies H 2 = 0, it is called a regular connection [13]. For these connections, the dimension of T[A] MASD is given by the virtual dimension. This index can be computed for any gauge group G using the Atiyah-Singer index theorem. The computation is done in 121, and the result for SO(3) is: 3 dim MASD = -2pl(V) - -(x 2
+a),
(2.48)
where pl(V) denotes the first Pontriagin number (ie. the Pontriagin class (2.6) integrated over X) and x, are the Euler characteristic and signature of X ,respectively. Exercise 2.1. Dimension of instanton moduli space. Compute M A ~ using D the index theorem for the twisted Dirac operator. Hint: use that S2'(X) N S+ @ S- , and 02i+si S+ @ S+. The conclusion of this analysis is that, if A is an irreducible ASD connection, the moduli space in a neighborhood of this point is smooth and can be modelled by the cohomology (2.46). If the connection is also regular, the index of the instanton deformation complex gives minus the dimension of moduli space. Of course, the most difficult part of Donaldson theory is t o find the global structure of M A ~ DIn. particular, in order t o define the invariants one has to compactify the moduli space. We are not going t o deal with these subtle issues here, and refer the reader t o the references mentioned at the beginning of this section. 2.6
Donaldson invariants
Donaldson invariants are roughly defined in terms of integrals of differential forms in the moduli space of irreducible ASD connections. These differential forms come from the rational cohomology ring of A*/G = B*,and it is necessary t o have an explicit description of this ring. The construction involves the universal bundle or universal instanton associated t o this moduli problem, and goes as follows: if the gauge group is SU(2), we consider the SO(3) bundle g E associated t o El and if the gauge group is SO(3) we consider the vector bundle V . We will denote both of them by g E , since the construction is the same in both cases. We then consider the space A* x g E . This can be
284
regarded as a bundle:
A*
X gE
+A*
X
x
(2.49)
which is the pullback from the bundle 7r : g E + X . The space A* x g E is called a family of tautological connections, since the natural connection on A* x g E is tautological in the g E direction and trivial in the A* direction: at the point ( A , p ) , the connection is given by A,(.rr(p)) (where we have chosen a trivialization of g E as in section 3.1, and ~ ( p E) ua).Since the group of reduced gauge transformations 6 acts on both factors, A* and g E , the quotient P=A*x0 - gE
(2.50)
is a G/C(G)-bundle over B* x X. This is the universal bundle associated t o E (or V ) . In the case of G = SU(2) or S 0 ( 3 ) , the universal bundle is an SO(3) bundle (since S U ( 2 ) / Z 2 = SO(3) and SO(3) has no center). Its Pontriagin class p l ( P ) can be computed using Chern-Weil theory in terms of the curvature of a connection on P. One can construct a natural connection on P, called the universal connection, by considering the quotient of the tautological connection (see [9, 131 for details). The curvature of the universal connection will be denoted by Kp. It is a form in fi2(B*x X , g E ) , and splits according t o the bigrading of R*(B* x X ) into three pieces: a two-form with respect to B*,a two-form with respect to X , and a mixed form (one-form on B* and one-form on X ) , all with values in g p . The Pontriagin class is:
(2.51) and defines a cohomology class in H4(B*x X) . By decomposing according to the bigrading, we obtain an element in H * ( B * )8 H * ( X ) . To get differential forms on a*, we just take the slant product with homology classes in X (i.e. we simply pair the forms on X with cycles on X ) . In this way we obtain the Donaldson map: p :Hi(X)
-
H4-2(B*).
(2.52)
One can prove [13] that the differential forms obtained in this way actually D generate the cohomology ring of B*. Finally, after restriction to M A ~ we obtain the following differential forms on the moduli space of ASD connections:
o(z) E H 4 ( M ~ s ~ ) ,
Ho(x) 6 E Hl(X)
zE
Il(6) E H3(MASD)7
s E ~ 2 ( x )12(s)E H ~ ( M A s D ) . +
(2.53)
285
There are also cohomology classes associated to three-cycles in X, but we will not consider them in these lectures. In the next lecture we will see that the Donaldson map arises very naturally in the context of topological field theory in what is called the descent procedure. In any case, we can now formally define the Donaldson invariants as follows. Consider the space
A(X) = Sym(Ho(X) @ H2(X))c3 A*Hl(X),
(2.54)
with a typical element written as xeSil ... SZpSj1 ... 6 j q . The Donaldson invariant corresponding t o this element of A(X) is the following intersection number:
v y (V)qxeSi,
. . . sip 63, . . . b j J
=
where we denoted by M A S D ( W ~ ( V k)) ,the moduli space of ASD connections specified by the second Stiefel-Whitney class wz(V) and the instanton number k . Notice that, since the integrals of differential forms are different from zero only when the dimension of the space equals the total degree of the form, it is clear that the integral in (2.55) will be different from zero only if the degrees of the forms add up t o d i m ( M ~ s ~ ( W 2 ( V ) , k )It ) . follows from (2.55) that Donaldson invariants can be understood as functionals:
DF(V)’k: A(X) -+
Q.
(2.56)
The reason that the values of the invariants are rational rather than integer is subtle and has t o do with the fact that they are rigorously defined as intersection numbers only in certain situations (the so-called stable range). Outside this range, there is a natural way to extend the definition which involves dividing by 2 (for more details, see [17]). It is very convenient t o pack all Donaldson invariants in a generating function. Let {bi}i=1,...,bl be a basis of one-cycles, and {si}i=~,._., bz a basis of two-cycles. We introduce the formal sums bi
b2
s =~
b = -y
V i S i , i=l
(2.57)
where ui are complex numbers, and
(2.58) k=O
286
where in the right hand side we are summing over all instanton numbers, i.e. we are summing over all topological configurations of the SO(3) gauge field with a fixed w2(V). This gives a formal power series in p,
Dwz(v): Sym(H:!(X))
+
Q.
(2.60)
The basic goal of Donalson theory is the computation of the generating functional (2.58) (or, in the simply-connected case, of the Donaldson series (2.59)). Many results have been obtained along the years for different four-manifolds (a good review is [36]). The major breakthrough in this sense was the structure theorem of Kronheimer and Mrowka [24] (see also [19]) for the Donaldson series of simply-connected four-manifolds with b l > 1 and of the so-called Donaldson simple type. A four-manifold is said t o be of Donaldson simple type if
($
- 4)Z2$)(p1
S) = Q,
(2.61)
for all choices of 202(V). When this holds, then, according t o the results of Kronheimer and Mrowka, the Donaldson series has the following structure: P
D"~(")(s) = exp(s2/2)
C ase(nsis),
(2.62)
s=l
for finitely many homology classes 6 1 , . . . ,I C ~E H2(X,Z) and nonzero rational numbers al, . . . , ap. Furthermore, each of the classes ~i is characteristic. The classes I C ~are called Donaldson basic classes. A simple example of this situation is the K 3 surface. In this case, the Donaldson-Witten generating functional is given by (2.63)
287
In this expression, 2x0 is a choice of an integer lifting of wz(E). The overall factor ezTix; gives a dependence on the choice a such a lifting, and this is due to the fact that the orientation of instanton moduli space depends on such a choice [13]. From the above expression one can deduce that for example for W ~ ( V=)0, one has (2.64) and so on. Notice that in the first integral in (2.64) we integrate over the moduli space M A S Dwith instanton number k = 2, and in the second one we have k = 6. According to (2.63), K 3 is of simple type, and the Donalson series is simply given by: (2.65) which satisfies indeed the structure theorem of Kronheimer and Mrowka and shows that K 3 has only one Donaldson basic class, namely K. = 0.
3
N = 1 supersymmetry
In this section, we give some useful background on supersymmetry. Since our motivation is the construction of topological field theories, our presentation will be rather sketchy. The standard reference is [40]. A very useful and compact presentation can be found in the excellent review by Alvarez-GaumB and Hassan [l],which is the main source for this very quick review. We follow strictly the conventions of [l],which are essentially those in [40], although there are some important differences. Some of these conventions can be found in Appendix A. Another useful reference, intended for mathematicians, is [ll].
3.1
The supersymmetry algebra
Supersymmetry is the only nontrivial extension of PoincarB symmetry which is compatible with the general principles of relativistic quantum field theory. In R1l3one introduces N fermionic generators
Qu =
($:)
288
where u = l , . . .,N. The superPoincar6 algebra extends the usual Poincar6 algebra, and the (anti)commutators of the fermionic generators are:
{Qau, Qg,) = 2tuvg$Pp
Pp,Qav1 =0 [MptQ a U l = - ( o p ) o P Q a u {Qaut Qpv
1=~ J Z W Z , ~
[qJL.&] =0 [M,,,Q""]
= -(i?,,)~bQpu
(34
where u,v = l , . . ., N , and Mpv are the generators of the Lorentz group SO(4) = S U ( 2 ) + x S U ( 2 ) - . The terms Z,, are the so-called central charges. They satisfy
z,,= and they commute with all the generators of the algebra. When the central charges vanish, the theory has an internal metry: Qav
4
uvwQ,w
Qg
4
UV+"Qd;",
U(N)sym(3.4)
where U E U ( N )is a unitary matrix. This symmetry is called in physics an R-symmetry, and it is denoted by U ( N ) z .The generators of this symmetry will be denoted by B,, and their commutation relations with the fermionic supercharges are:
[Sou,B a ] = ( b a ) v w Q a w where b, = b:. generators
B a ] = -Qv(ba)v"'
(3.5)
The central charges are linear combinations of the U ( N )
Z,, = d,,"B,.
(3.6)
If the central charges are not zero, the internal symmetry gets reduced to USp(N), formed by the unitary transformations that leave invariant the 2form (3.6) in dimensions. The U ( ~ ) of R the internal symmetry (3.4), with generator R, gives a chiral symmetry of the theory,
(3.7) This symmetry is typically anomalous, quantum-mechanically, and the quantum effects break it down to a discrete subgroup.
289
3.2 N = 1 superspace and superfields In order t o find a local realization of supersymmetry, one has t o extend the usual Minkowski space t o the so-called superspace. In this section we are going t o develop the basics of N = 1 superspace, which is extremely useful to formulate supersymmetric field multiplets and supersymmetric Lagrangians. Therefore, we are going t o construct a local realization of the supersymmetry algebra (3.2) when we have two supercharges Q a , Q". The superspace is obtained by adding four spinor coordinates OQ,8d: to the four space-time coordinates z,. The generator of supersymmetric transformations in superspace is - -. -iCaQa - i<"Q" (3.8)
c"
where ta, are (fermionic) transformation parameters2. Under this generator, the superspace coordinates transform as xfi
-+
x,
+ ie&
- i[&
e-,e+e,
e-te+c.
(3-9)
The representation of the supercharges acting on the superspace is then given by
and they satisfy {Qa, Q"} = -2i& a,. Since P, = -id,, this gives a representation of the supersymmetry algebra. It is also convenient to introduce the super-covariant derivatives
which satisfy {Da, D"} = -2iaE" a, and commute with Q and Q. A superfield is just a function on the superspace F(x,B,B). Since the 8-coordinates are anti-commuting, the Taylor expansion in the fermionic coordinates truncates after a finite number of terms. Therefore, the most general N = 1 superfield can always be expanded as
*This is the only case in which we do not follow the conventions of 111: their susy charges are - 2 times ours.
290
Under a supersymmetry transformation (3.8), the superfield transforms as 6F = (EQ+fQ)F, and from this expression one can obtain the transformation of the components. The generic superfield gives a reducible representation of the supersymmetry algebra. Therefore, in order to obtain irreducible representations one must impose constraints. There are two different N = 1 irreducible supermultiplets: a) Chiral multiplet: The N = 1 scalar multiplet is a superfield which satisfies the following constraint:
D&@= 0
(3.13)
and it is called the chiral superfield. The constraint can be easily solved by noting that, if yp = x p + i8ap8, then D,yp = 0,
D,eo
=0.
(3.14)
Therefore, any function of (y, 0) is a chiral superfield. We can then write @(Y,O>= (b(Y>
+ &e"+a(Y) + 02F(Y)
7
(3.15)
and we see that a chiral superfield contains two complex scalar fields, (b and F , and a Weyl spinor +a. In a similar way we can define an anti-chiral superfield by D,@t = 0, which can be expanded as
+ Jz@(yt) + e2Ft(yt) ,
@t(yt,G) =
(3.16)
where, ypt = x!' - i d o p 8 . Exercise 3.1. Show that, in terms of the original variables, @ and @t take the form 1
qz,e, G) = ~ ( z+) i e # i j a , ~ - ~ P P V ~+Ahe$(,) @t(z, 8, G) = At(.)
-
-eea,$c,ij+
eeqz) ,
Jz 1 - ieopGapAt - -02G2V2At + fi8$(z) 4 4
i -_
+ -ee ecw,$ + G W ( z ) .
Jz
(3.17)
Here, V2 = d,W. b) Vector Multiplet: this is a real superfield satifying V = V t . In components, it takes the form
v ( O,G) ~ , = c + iex
-
a 2
i~+ x -e2(M
+ ie2e(i + ia.apx) 2
+ ilv) - ?P(M 2
- iGZe(x
- ilv) - ecp8A,
1 i + -ewpx) + %P(o - -v~c). 2 2 2 (3.18)
29 1
+ +
By performing an abelian gauge transformation V -+ V A At, where A (At) are chiral (antichiral) superfields, one can set C = M = N = x = 0. This is the so called Wess-Zumino gauge, where
v = -eapBA, + ie2eX - @ex + .2t e 2 p D .
(3.19)
In this gauge, V 2= iA,Afi82e2 and V 3 = 0. The Wess-Zumino gauge breaks supersymmetry, but not the gauge symmetry of the abelian gauge field A,. The Abelian field strength is defined by 14
1 4
W&= - - D 2 D & V ,
W, = --D2D,V,
and W, is a chiral superfield. In the Wess-Zumino gauge it takes the form
W,
= -iX,(y)
a - -(aC”iiu8), F,,
+ 8,D
2
+ 82(~”a,x),.
(3.20)
The non-Abelian case is similar: V is in the adjoint representation of the gauge group, V = VAT., and the gauge transformations are
e-2v where A
= AATA.
e-iAt
~
e -2VeiA
The non-Abelian gauge field strength is defined by
w, = lD2e2VDae-2V 8
and transforms as
W,
-+
W: = e-iAW,eiA.
In components, it takes the form
W, = T” (-zAz
+ 8,D”
i
- -(afia“”8),F;v
2
)
+ 82upV,Xa
(3.21)
where
F;, = a,Az - &A; 3.3
+ f abCALAE,
V,Xa = a,X”
+ fabcALXc.
Construction of N = 1 Lagrangians
In the previous subsection we have constructed supermultiplets of = 1 supersymmetry. The next step is to construct manifestly supersymmetric Lagrangians. Again, this is easily done in superspace.
292
The most general N = 1 supersymmetric Lagrangian for the scalar multiplet (including the interaction terms) is given by
L
=
1
+
d4BK(@,Qt)
s
+
d2BW(@)
J
d2efi(@+),
(3.22)
We are following here the usual rules of Grasmmannian integration, and the i3integrals pick up the highest component of the superfield. In our conventions, S d 2 0 0 2 = 1 and Sd2$g2 = 1. The kinetic term for the scalar fields Ai has the form gaj a,AiaP
A;
(3.23)
where (3.24) is in general a nontrivial metric for the space of fields @. This has the form of a Kahler metric derived from a Kahler potential K ( A i , AS). For this reason, the function K ( @ , @ tis) referred to as the Kahler potential. The simplest Kahler potential, corresponding to the flat metric, is
i= 1
which gives the free Lagrangian for a massless scalar and a massless fermion with an auxiliary field which can be eliminated by its equation of motion:
L=
102g2=
a,AfdPAi
+ F/Fi - z&CPap$i.
(3.25)
i
Exercise 3.2. Show that 1 1 1 leZez = --4 A ~ V -~ -4Av ~~ A ~ AF ~, ~ F+~-2 ~ , A ~ P A ~ a i (3.26) - - $jop++& -a 2 2’13 and from this derive (3.25). The function W ( @in ) (3.22) is an arbitrary holomorphic function of chiral superfields, and it is called the superpotential. It can be expanded as,
+
+
~((a,) = W
( A+~h e $ ,
dW = W(Ai) dAi
+
+ ew,)
::(
+ ee
-F.
1 a2w 2 dAiAj
---
293
Supersymmetric interaction terms can be constructed in terms of the superpotential and its conjugate. Finally, we have t o mention that there is U ( ~ ) R symmetry that acts as follows:
Under this, the component fields transform as
A
q
f
+
e2i7212
A,
e2i(n-1/2)a $ 7
F
+
e2i(n-l)QF.
(3.29)
Let us now present the Lagrangian for vector superfields. The super Yang-Mills Lagrangian with a &term can be written as
' ( J
C = -1m 8n
7Tr
d28W"Wa
)
+
where r = 8/2n 4ni/g2, and F a P w = +cPVaPFao. Exercise 3.3. Using the normalization TrTaTb= dab, show that n ( W a W a lee)
=
1 -2iXaaPVPXa DaDa - -FaP"Fi,, 2
+
i + -cPvpUFa P" F a 4
PO'
(3.31) and from here derive (3.30). Now we can present the general Lagrangian that describes chiral multiplets coupled to a gauge field. Let the chiral superfields CPi belong to a given representation of the gauge group in which the generators are the matrices TG. The kinetic energy term C P i is invariant under global gauge transformations a' = eci"CP. In the local case, t o insure that CP' remains a chiral superfield, A has to be a chiral superfield. The supersymmetric gauge invariant kinetic energy term is then given by The full N = 1 supersymmetric Lagrangian is
C=
81r
Im (rTr
1
de W QWa)
+
d20d2#CPte-2vCP
+
S
d28W
+
S
d2eW. (3.32)
294
Exercise 3.4. Expand (3.32) in components to obtain
L
= --Fa 1
492
FaP"
- -i ~ " # V , ~ "+ -DaD" 1 + L F " Fa,"
,"
3 2 ~ ~
P"
g2
+ (d,A - i A ; T a A ) t ( P A- iAapTaA)- i &?(a,$ -
DaAtTaA- i&AtT"X"$
2g2 - iA;Ta$)
+ ih$T"AX" + FJFi
In (3.33), the auxiliary fields F and D" can be eliminated by using their equations of motion. The terms involving these fields, thus, give rise to the scalar potential (3.34) i
4
n/ = 2 super Yang-Mills theory
To construct topological field theories in four dimensions, we are actually interested in models with two supersymmetries ( i . e . with eight supercharges). In this section we will present N = 2 supersymmetric Yang-Mills theory in some detail, following the conventions in [27,32]. We will use the N = 1 superspace formalism of the previous section as our starting point, and then we will write the supersymmetric transformations in a manifest N = 2 supersymmetric way. N = 2 Yang-Mills theory contains a real superfields V , and a massless chiral superfield 'P, both in the adjoint representation of the gauge group G. The action, written in n/ = 1 superspace, is a particular case of the general action (3.32) with W = 0 , and it reads:
1
d42d2QTr(W2) +
J
d42d28Tr(Wt2)+
J
d42d28d28Tr('Ptke2"~'(p'). (4.1)
In this equation, v k l = TaklVa,where T" is a Hermitian basis for the Lie algebra in the adjoint representation, and the real superfield is in the WZ gauge, with components A,, X1, = X Z a and D (all in the adjoint representation of the gauge group): 1 2
V = -Qu'"~A,- i82QX1 - iQ2ex2+ -Q2e2D.
295
-x2,.
Notice that the conjugate of A1, is 51, = in the adjoint representation, has components
The chiral superfield CP, also = -XI, and F :
4, A2,
xl,.
and the conjugate of A2, is 52, = We can now write the action in components. First, we redefine the auxiliary field D as D -+ D [+,4t]. The action then reads:
+
This action is not manifestly N = 2 supersymmetric, since the N = 2 supersymmetric algebra has an internal su(2)Rsymmetry, which is not manifest in (4.4).su(2)Rinvariance is easily achieved: the scalars q5 and the gluons A, are singlets, while the gluinos A, w = 1,2 form a doublet. The auxiliary fields form a real triplet:
Dvw
=
iD (JiZDF @)’
(4.5)
and the su(2)Rindices are raised and lowered with the matrices E,,, cvw. Notice that D”” = D:,. Finally, by covariantizing the N = 1 transformations, one finds the N = 2 transformations:
296
5
Topological field theories from twisted supersymmetry
In this section, we introduce topological field theories and we give a brief general overview of their properties, focusing on the so-called theories of the Witten or cohomological type. We then explain the twisting procedure, which produces topological field theories from N = 2 theories, and put it in practice with the examples of the previous section. General introductions to topological field theories can be found in [7,9,25], among other references.
5.1
Topological field theories: basic properties
Topological field theories (TFT's) were first introduced by Witten in [43]. A quantum field theory is topological if, when put on a manifold X with a Riemannian metric gPV,the correlation functions of some set of operators do not depend (at least formally) on the metric. We then have
b -(oil kl,"
. . . c?&) = 0,
297
where Oil, . . . ,Oi,, are operators in the theory. There are two different types of TFT’s: in the TFT’s theories of the Schwarz type, one tries t o define all the ingredients in the theory (the action, the operators, and so on) without using the metric of the manifold. The most important example is Chern-Simons theory, introduced by Witten in [45]. In the TFT’s of the Witten type, one has an explicit metric dependence, but the theory has an underlying scalar symmetry 6 acting on the fields in such a way that the correlation functions of the theory do not depend on the background metric. More precisely, if the energy-momentum tensor of the theory T,, = ( 6 / 6 g p u ) S ( 4 i )can be written as
T,, = -ibG,,
(5.2)
where G,, is some tensor, then (5.1) holds for any operator 0 which is 6invariant. This is because:
In this derivation we have used the fact that 6 is a symmetry of the classical action S ( 4 %and ) of the quantum theory. Such a symmetry is called a topological symmetry. In some situations this symmetry is anomalous, and the theory is not strictly topological. However, in most of the interesting cases, this dependence is mild and under control. We will see a very explicit example of this in Donaldson theory on manifolds of b t = 1. If the theory is topological, as we have described it, the natural operators are then those which are &invariant. On the other hand, operators which are &exact decouple from the theory, since their correlation functions vanish. The operators that are in the cohomology of 6 are called topological observables: Ker 6 Im6 ’ The topological symmetry b is not nilpotent: in general one has OE-
a2 = 2
(5.4)
(5.5)
where 2 is a certain transformation in the theory. It can be a local transformation (a gauge transformation) or a global transformation (for example, a global U(1) symmetry). The appropriate framework t o analyze the observables is then equivariant cohomology, and for consistency one has to consider only operators that are invariant under the transformation generated by Z (for example, gauge invariant operators). Equivariant cohomology turns out
298
t o be a very natural language t o describe TFT's with local and global symmetries, but we are not going t o explore it in these lectures. The interested reader should look a t [9,26]. The structure of topological field theories of the Witten type leads immediately to a general version of the Donaldson map [43]. Remember that, starting with the curvature of the universal bundle, this map associates cohomology classes in the instanton moduli space to homology classes in the four-manifold. One can easily see that in any theory where (5.2) is satisfied, one can define topological observables associated t o homology cycles in spacetime. If (5.2) holds, then one has:
P,
= TO,=
-ibG,,
(5.6)
where
G,
Go,.
(5-7) In the theories that we are goint t o consider, b is essentially given by a supersymmetric transformation, and therefore it is a Grassmannian symmetry. It follows that G, is an anticommuting operator. On the other hand, from the point of view of the Lorentz group, they are a scalar and a one-form, respectively. Then, topological field theories of Witten type violate the spinstatistics theorem. Consider now a &invariant operator q5(O) (.) The descent operators are defined as q5,1,z...,n(~) (n) = G,,G,, ...G,nq5(o)(x), n = 1 . . . d 7 1 1 (5.8) where d is the dimension of the spacetime manifold. Since the GPi anticommute, q5(n) is antisymmetric in the indices P I , . . . ,pn, and therefore it gives an n-form: A
dxpn.
As an immediate consequence of (5.6) and the b-invariance of the following descent equations: dq5W = b$("+l),
(5.9) one has
(5.10)
where d is the exterior derivative and we have taken into account that P, = -2,. The descent equations can be also obtained by considering the cohomology of the operator d 6 , see [5,9] for more details. Using now (5.10) it is easy t o see that the operator
+
(5.11)
299
where T~ E H , ( X ) , is a topological observable:
(+$'
=
I@
= Jdd("-l'
=
Yn
"In
I
$(,-I)
= 0,
(5.12)
%n
since dy, = 0. Exercise 5.1. Homology and observables. Show that, if 7 , is trivial in homology ( i e . , if it is &exact), then W:,' is &exact. Therefore, given a (scalar) topological observable, one can construct a family of topological observables
W,($ (7.1,
2,'
= 1 , . . . ,b,;
72
= 1,. . . , d ,
(5.13)
in one-to-one correspondence with the homology classes of spacetime. This descent procedure is the analog of the Donaldson map in Donaldson-Witten and Seiberg-Witten theory. Notice that any family of operators q5(n) that satisfies the descent equations (5.10) gives topological observables. The explicit realization (5.8) in terms of the G operator can then be regarded as a canonical solution to (5.10). 5.2
Twist of N
=2
supersymmetry
In the early eighties, Witten noticed in two seminal papers (41,421 that supersymmetry has a deep relation to topology. The simplest example of such a relation is supersymmetric quantum mechanics, which provides a physical reformulation (and in fact a refinement) of Morse theory [42]. Other examples are N = 2 theories in two and four dimensions. In 1988 Witten discovered that, by changing the coupling to gravity of the fields in an N = 2 theory in two or four dimensions, a theory satisfying the requirements of the previous subsection was obtained [43,44]. This redefinition of the theory is called twisting. We are now going to explain in some detail how this works in the four-dimensional case. The N = 2 supersymmetry algebra (with no central charges) is:
Here, the indices v ,w E {1,2}. The twisting procedure consists of redefining the coupling to gravity of the theory, i e . in redefining the spins of the fields.
300
To do this, we couple the fields to the SU(2)+ spin connection according to their isospin. This means that we add to the Lagrangian the term
(5.15) where J;1” is the sU(2)R current of the theory, and w: is the SU(2)+ spin connection. We then have a new rotation group K‘ = SU‘(2)+ 8 S U ( 2 ) - , where SU’(2)+ is the diagonal of SU(2)+ x s U ( 2 ) ~ .In practice, the twist means essentially that the s U ( 2 ) ~indices w ,w become spinorial indices ir, and we have the change Oa, -+Qaa and Q,, -+ &,a. It is easy to check that the topological supercharge
6,
-
Qii
(5.16)
is a scalar with respect to K‘. This topological supercharge will provide the topological symmetry 6 that we need for a topological theory. The N = 2 algebra also gives a natural way to construct the operator G, defined in (5.7). In fact, define:
G p-
Using now the {Q,
i 4
(5.17)
a}anticommutator one can show that (5.18)
This means that the supersymmetry algebra by itself almost guarantees (5.2), since it implies that the momentum operator P, is G-exact. In the models that we will consider, _ (5.2) - is true (at least on-shell). Finally, notice that from the anticommutator {Q, Q} in (5.14) follows that the topological supercharge is nilpotent = 0 (in the absence of central charge). Our main conclusion is that by twisting N = 2 supersymmetq one can construct a quantum field the0 y that satisfies (almost) all the requirements of a topological field theory of the Witten type.
a2
6
Donaldson-Witten theory
Donaldson-Witten theory (also known as topological Yang-Mills theory) is the topological field theory that results from twisting N = 2 Yang-Mills theory in four dimensions. Historically it was the first TFT of the Witten type, and as we will see it provides a physical realization of Donaldson theory,
30 1
6.1
The topological action
Remember that N = 2 super Yang-Mills theory contains a gauge field A,, two gluinos A,, and a complex scalar 4, all of them in the adjoint representation of the gauge group G. In the off-shell formulation, we also have auxiliary fields D,, in the 3 of the internal su(2)R. The total symmetry group of the theory is
'H = s u ( 2 ) + x s u ( 2 ) - x su(2)R x U(1)R. Under the twist, the fields in the content as follows:
A , (1/2,1/2,0)O (1/2,0,1/2)-1 X,& (0,1/2,1/2)1 4 (o,0,0)-2
A,,
4t (o,o,o)2
D,, (070,
N
-+
-+ -+
+
+
+
(6.1)
= 2 supermultiplet change their spin
A,
QP,1/21°,
$aa P P 7 1/211, x.
q (0,0)-1, (1,0)-1, 4 (O,O?i 4t (0, 0l2, '
(6.2)
D&b
where we have written the quantum numbers with respect to the group H before the twist, and with respect t o the group 7-i' = SU(2)'+ 8 SU(2)- 8 u ( 1 ) after ~ the twist. In the topological theory, the u ( 1 ) charge ~ is usually called ghost number. The q and x fields are given by the antisymmetric and symmetric pieces of X&a, respectively. More precisely:
From the N = 2 action it is straightforward to find: S=
J
d4~&Tk{V,4Vp4t - i$PaV'u~pd. - i$&,Vbov - -FPyFMV 1 4
where V"" = @'V,. The -transformations are easily obtained from the = 2 transformations:
N
302
In (6.5), $J, = oPab$J"b and F?. = t?"FPu is the selfdual part of FPu. It is a0
.rP
not difficult t o show that the action of Donaldson-Witten theory is !-exact up to a topological term, i.e.
where
V=
i 1 1 d42Tr { - X & ~ ( F+ " D") ~ - -q[4,q] + -J!Ia&V'aq5t s 4 2 2&
}.
(6.7)
As we will see in a moment, this has important implications for the quantum behavior of the theory. Exercise 6.1. The Lagrangian of Donaldson- Witten theory. Derive (6.4)and (6.6). One of the most interesting aspects of the twisting procedure is the following: if we put the original N = 2 Yang-Mills theory on an arbitrary Riemannian four-manifold, using the well-known prescription of minimal coupling t o gravity, we find global obstructions t o have a well-defined theory. The reason is very simple: not every four-manifold is Spin, and therefore the fermionic fields A,, are not well-defined unless wz(X)= 0. However, after the twisting, all fields are differential forms on X, and therefore the twisted theory makes sense globally on an arbitrary Riemannian four-manifold. 6.2
The observables
The observables of Donaldson-Witten theory can be constructed by using the topological descent equations. As we have emphasized, these equations have a canonical solution given by the operator (5.17). Using again the N = 2 supersymmetry transformations, one can work out the action of G, on the different fields of the theory. The result is:
PW41 = &$J,, [ G i q = 0, * Vq, [Gv,A,I = i S p v V - iXpu7 [G,F+] = iVx (6.8) {G,x} = -3 y * 03, [G,q] = - 9 V 4 , {G,, J!Iv} = -(P'iv D,',),[G,D] = *Vq ~ O X .
+
--?
+
+
We can now construct the topological observables of the theory by using the descent equations. The starting point must be a gauge-invariant, !-closed operator which is not a-trivial. Since 41 = 0, the simplest candidates are the operators
[a,
0, = TI-(+,), n = 2,...
7
N.
(6.9)
303
Here we are going t o restrict ourselves t o SU(2), therefore the starting point for the descent procedure will be the operator, 0 = Tr(42).
(6.10)
It is easy t o see that the following operators satisfy the descent equations (5.10):
(6.11) Notice for example that
{Q, O(l)} = 2Tr(4Vp4)dxp = d 0 .
(6.12)
so the first descent equation is satisfied. This is, however, not the canonical solution to the descent equations provided by G, which in this case is a little bit more complicated. Exercise 6.2. Descent equations in topological Yang-Mills theory. Show that (6.11) satisfy the descent equations. Compare with the canonical solution. The observables
(6.13) correspond t o the differential forms on the where 6 E HI(X),S E Hz(X), moduli space of ASD connections that were introduced in (2.53) through the use of the Donaldson map (2.52) (and this is why we have used the same notation for both). Notice that the ghost number of the operators in (6.11) is in fact their degree as differential forms in moduli space. The operators (6.11) are naturally interpreted as the decomposition of the Pontriagin class of the universal bundle (2.51) with respect to the bigrading of Q*(B* x X ) . In fact, the Grasmannian field qhp can be interpreted as a (1,l)form: a one-form in spacetime and also a one-form in the space A. The operator Q is then interpreted as the equivariant differential in A with respect to gauge transformations. This leads t o a beautiful geometric interpretation of topological Yang-Mills theory in terms of equivariant cohomology [23] and the Mathai-Quillen formalism [3],which is reviewed in detail in [9].
304
6.3 Evaluation of the path integral We now consider the topological theory defined by the topological Yang-Mills action, STYM = {G,V}, where V is defined in (6.7). The evaluation of the path integral of the theory defined by the Donaldson-Witten action can be drastically simplified by taking into account the following fact. The (unnormalized) correlation functions of the theory are defined by (6.14) where
41,. . .
, 4n are generic fields, and
g is the coupling constant.
Since
STYM is - e x a c t , one has: (6.15) where we have used the fact that g is a symmetry of the theory, and therefore the insertion of a - e x a c t operator in the path integral gives zero. The above result is remarkable: it says that, in a topological field theory in which the action is a-exact, the computations do not depend on the value of the coupling constant. In particular, the semiclassical approximation is exact! 1431. We can then evaluate the path integral in the saddle-point approximation as follows: first, we look at zero modes, i.e. classical configurations that minimize the action. Then we look at nonzero modes, ie. we consider quantum fluctuations around these configurations. Since the saddle-point approximation is exact, it is enough to consider the quadratic fluctuations. The integral over the zero modes gives a finite integral over the space of bosonic collective coordinates, and a finite Grassmannian integral over the zero modes of the fermi fields. The integral over the quadratic fluctuations gives a bunch of determinants. Since the theory has a bose-fermi symmetry, it is easy to see that the determinants cancel (up to a sign), as in supersymmetric theories. Let us then analyze the bosonic and fermionic zero modes. A quick way to find the bosonic zero modes is to look for supersymmetric configurations. These are classical configurations such that {&,fermi} = 0 for all Fermi fields in the theory, and they give minima of the Lagrangian. Indeed, it was shown in [46] that in topological field theories with a fermionic symmetry & one can compute by localization on the fixed points of this symmetry. In this case, by looking at x} = 0 , one finds
{a,
F+
=
D+.
(6.16)
But on-shell D f = 0, and therefore (6.16) reduces to the usual ASD equations. The zero modes of the gauge field are then instanton configurations.
305
In addition, by looking at 4 field,
{ g ,$} = 0, we find the equation of motion for the vA4 = 0.
(6.17)
This equation is also familiar: as we saw in section 3, its nontrivial solutions correspond to reducible connections. Let us assume for simplicity that we are in a situation in which no reducible solutions occur, so that q5 = 0. In that case, (6.16) tells us that the integral over the collective coordinates reduces to an integral over the instanton moduli space MASD. Let us now look at the fermionic zero modes in the background of an instanton. The kinetic terms for the $, x and 7 fermions fit precisely into the instanton deformation complex (2.44). Therefore, using the index theorem we can compute:
N,J - N x = dim MASD,
(6.18)
~ the number of zeromodes of the corresponding fields, and where N , J ,denotes we have used the fact that the connection A is irreducible, so that 77 (which is a scalar) has no zero modes (in other words, VAQ= 0 only has the trivial solution). Finally, if we assume that the connection is regular, then one has that Cokerp+VA = 0 , and there are no x zero modes. In this situation, the number of $ zero modes is simply the dimension of the moduli space of ASD instantons. If we denote the bosonic and the fermionic zero modes by dai, d$i, respectively, where i = 1,. . . ,D and D = dimMAsD, then the zero-mode measure becomes: n
(6.19) i=l
This is in fact the natural measure for integration of differential forms on MASD, and the Grassmannian variables $i are then interpreted as a basis of one-forms on MASD. We can already discuss how to compute correlation functions of the operators 0 , I z ( S ) ,and II(6). These operators contain the fields $, A , and 4. In evaluating the path integral, it is enough to replace $ and A , by their zero modes, and the field q3 (with no zero modes) by its quantum fluctuations, that we then integrate out at quadratic order. Further corrections are higher order in the coupling constant and do not contribute to the saddle-point approximation, which in this case is exact. We have then to compute the one-point function (4"). The relevant terms in the action are
306
since we are only considering quadratic terms. We then have t o compute
(4%))
=
/ wnh#fw
exp 4 ( 4 7 $+I.
(6.21)
If we take into account that - Y),
(4"(z)4bt(Y))= -"(a:
(6.22)
where Gab(z- y) is the Green's function of the Laplacian V,Vc", we find: (6.23) This expresses 4 in terms of zero modes. It turns out that this is precisely (up to a multiplicative constant) the component along a* of the curvature Kp of the universal bundle (see for example [13], p. 196). This is in perfect agreement with the correspondence between the observables (6.11) and the differential forms on moduli space (2.53) constructed in Donaldson theory. The main conclusion of this analysis is that, up t o possible normalizations, -
(oe12(si,) .. . sM,,,oe . ..
. . . Il(Sj,)) 11(6.j1)A . . . A
12(Sip)1l((5j1
(6.24)
A 12(sip)A
11((5.jq), (6.25)
A 12((sil) A
ie. the correlation function of the observables of twisted N = 2 Yang-Mills theory is precisely the corresponding Donaldson invariant. The requirement that the differential form in the r.h.s. has top degree (otherwise the invariant is zero) corresponds, in the field theory side, t o the requirement that the correlation has ghost number equal t o dimMAsD, i e . that the operator in the correlation function soaks up all the fermionic zero modes, which is the well-known 't Hooft rule [38]. (6.24) was one of the most important results of Witten's seminal work [43], and it opened a completely different approach t o Donaldson theory by means of topological quantum field theory. 7
Conclusions and further developments
What we have covered in these lectures is just the beginning of a very beautiful story that we can only summarize a t this concluding section. Since we have a quantum field theory realization of Donaldson-Witten theory, one could imagine that knowledge of the physics of this theory would be extremely useful in learning about the deep mathematics of Donaldson invariants. A first step in that direction was taken by Witten in [47], but a much more ambitious picture appeared after the classical work of Seiberg and Witten on the lowenergy effective action of N = 2 super Yang-Mills theory [35]. This led to the
307
introduction [48] of the Seiberg-Witten monopole equations and the SeibergWitten invariants. It turns out that the unknown constants a , in (2.62) are essentially Seiberg-Witten invariants, and the basic classes of Kronheimer and Mrowka can be reformulated in terms of a finite set of Spin, structures called Seiberg-Witten classes. Moreover, one can evaluate using quantum field theory techniques the Donaldson-Witten generating functional and write it in terms of some universal functions and Seiberg-Witten invariants [32]. In one word, quantum field theory leads t o a complete solution of the basic problem in Donaldson theory (the evaluation of Donaldson invariants). To learn about this, we recommend the references [14,28,33] (from a mathematical point of view) and [12,29,37] from the quantum field theory viewpoint. A systematic account of these developments can be found in [30].
Appendix A
Conventions for spinors
In this appendix we collect our conventions for spinors (both in Minkowski and Euclidean space). We follow almost strictly [ 11. The Minkowski flat metric is vPV = diag (1,-1, -1, -1). We raise and lower spinor indices with the antisymmetric tensor cap , E,B: $a = Eap$,p,
where the
E
$a
= Eap?lP,
tensor is chosen as follows: €21
= €12 =
-EI2
=
-p
=
1,
Contractions satisfy the perverse rule:
$“4a
=
We define the matrices: (Ocl)a&
= (L4,
where a‘ are the Pauli matrices, and after raising indices we find
1’
(0 - P ffa = EaP,ao
( O q P b = (1, -a‘).
308
The continuation from Minkowski space is made via xo = -ix4, po = ip4. The conventions for Euclidean spinors are as follows: aaB P -- ( i , Z ) ,
= ( i ,-0‘).
(IP+
The following identities are useful: cP0”
+ c”uP = -2gP”6ab,
(eP)bqOP)yp =
-2spys5.
The (A)SD projectors are 0P”
-
ffP”
where
0;;
is ASD, while
1 4 1
= - (&V
-
&q,
= -(aP 0y - ( T V O P ) ,
4
is SD. We have, explicitly:
We finally define:
This implies that, if we consider a self-dual tensor
‘zy=
O a b c ( --a b - c 0 0c -ba ) -c
one has
b -a
0
309
References 1. L. Alvarez-Gaum6 and S.F. Hassan, “Introduction to S-duality in N = 2 supersymmetric gauge theories,” hep-th/9701069, Fortsch. Phys. 45 (1997) 159-236. 2. M.F. Atiyah, N.H. Hitchin and I.M. Singer, “Self-duality in four dimensional Riemannian geometry,” Proc. R. SOC.Lond. A 362 (1978) 425461. 3. M.F. Atiyah and L. Jeffrey, “Topological Lagrangians and cohomology,” J. Geom. Phys. 7 (1990) 119-136. 4. W. Barth, C . Peters and A. van den Ven, Compact complex surfaces, Springer-Verlag, 1984. 5 . L. Baulieu and I.M. Singer, “Topological Yang-Mills symmetry,” Nucl. Phys. Proc. Suppl. 5 B (1988) 12-19. 6 . A. Beauville, Complex algebraic surfaces, 2nd edition, Cambridge University Press. 7. D. Birmingham, M. Blau. M. Rakowski and G. Thompson, ‘(Topological field theory,” Phys. Rep. 209 (1991) 129-340. 8. R. Bott and L. Tu, Differential forms in algebraic topology, SpringerVerlag, 1982. 9. S. Cordes, G. Moore and S. Rangoolam, “Lectures on 2D Yang-Mills theory, equivariant cohomology and topological field theories,” hepth/9411210, in Fluctuating geometries in statistical mechanics and field theory, Les Houches Session LXII, edited by F. David, P. Ginsparg and J. Zinn-Justin, Elsevier, 1996, p. 505-682. 10. M. Daniel and C.M. Viallet, “The geometrical setting of gauge theories of the Yang-Mills type,” Rev. Mod. Phys. 52 (1980) 175-197. 11. P. Deligne et al., editors, Quantum fields and strings: a course for mathematicians, volume 1. AMS, 1999. 12. R. Dijkgraaf, “Lectures on four-manifolds and topological gauge theories,” Nucl. Phys. Proc. Suppl. B 45 (1996) 29-45. 13. S.K. Donaldson and P.B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, 1990. 14. S. Donaldson, “The Seiberg-Witten equations and four-manifold topology,” Bull. Amer. Math. SOC.33 (1996) 45-70. 15. T. Eguchi, P.B. Gilkey and A.J. Hanson, “Gravitation, gauge theories, and differential geometry,” Phys. Rep. 66 (1980) 214-393. 16. D. Freed and K. Uhlenbeck, Instantons and four-manifolds, second edition, Springer-Verlag, 1991. 17. R. Friedman and J.W. Morgan, Smooth four-manifolds and complex sur-
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faces, Springer-Verlag, 1994. 18. R. Friedman and J.W. Morgan, editors, Gauge theory and the topology of four-manifolds, IAS/Park City Mathematics Series, American Mathematical Society, 1998. 19. R. Fintushel and R. Stern, “Donaldson invariants of 4-manifolds with simple type,” J. Diff. Geom. 42 (1995) 577-633. 20. R. Gompf and A. Stipsicz, Four-manifolds and Kirby calculus, American Mathematical Society, 1999. 21. F. Hirzebruch and H. Hopf, “Felder von Flachenelementen in 4dimensionalen Mannigflatigkeiten,” Math. Ann. 136 (1958) 156-172. 22. R. Jackiw and C. Rebbi, “Degrees of freedom in pseudoparticle systems,” Phys. Lett. B 67 (1977) 189-192. 23. H. Kanno, “Weil algebra structure and geometrical meaning of BRST transformations in topological quantum field theory,” Z. Phys. C 43 (1989) 477-484. 24. P.B. Kronheimer and T.S. Mrowka, “Recurrence relations and asymptotics for four-manifold invariants,” Bull. Am. Math. SOC. 30 (1994) 215-221; “Embedded surfaces and the structure of Donaldson’s polynomial invariants,” J . Diff. Geom. 41 (1995) 573-734. 25. J.M.F. Labastida and C. Lozano, “Lectures in topological quantum field theory,” hep-th/9709192, in Trends in theoretical physics, La Plata 1997, edited by H. Falomir et al., American Institute of Physics, 1998, pp. 54-93. 26. J.M.F. Labastida and M. Mariiio, “Twisted n/ = 2 supersymmetry with central charge and equivariant cohornol~gy,’~ hep-th/9603169, Comm. Math. Phys. 185 (1997) 37-71. 27. C. Lozano, Teon’as supersim&icas y teorias topoldgicas, Tesiiia, Universidade de Santiago de Compostela, 1997. 28. M. Marcolli, Seiberg-Witten gauge theory, Hindustan Book Agency, 1999. 29. M. Mariiio, “Topological quantum field theory and four-manifolds,” hepth/0008100, to appear in the Proceedings of the 3rd European Congress of Mathematics. 30. M. Mariiio, “Lectures on topological field theory and four-manifolds,” to appear. 31. J. Milnor and J. Stasheff, Characteristic classes, Princeton University Press, 1974. 32. G. Moore and E. Witten, “Integrating over the u-plane in Donaldson theory,” hep-th/9709193, Adv. Theor. Math. Phys. 1 (1998) 298-387. 33. J.W. Morgan, The Seiberg- Witten equations and applications to the topology of smooth four-manifolds, Princeton University Press, 1996.
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34. C. Nash, Differential topology and quantum field theory, Academic Press, 1991. 35. N. Seiberg and E. Witten, “Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory,” hepth/9407087, Nucl. Phys. B 426 (1994) 19-52. 36. R. Stern, “Computing Donaldson invariants,” in Gauge theory and the topology of four-manifolds, IAS/Park City Math. Ser., American Mathematical Society, 1998. 37. G. Thompson, “New results in topological field theory and abelian gauge theory,” hep-th/9511038, in 1995S u m m e r School o n High Energy Physics and Cosmology, edited by E. Gava et al., World Scientific, 1997. 38. G. ’t Hooft, “Computation of the quantum effects due to a fourdimensional pseudoparticle,” Phys. Rev. D 14 (1976) 3432-3450. 39. G. ’t Hooft, “A property of electric and magnetic flux in non-abelian gauge theories,” Nucl. Phys. B 153 (1979) 141-160. 40. J. Wess and J. Bagger, Supersymmetry and supergmvity, Second edition, Princeton University Press, 1992. 41. E. Witten, “Constraints on supersymmetry breaking,” Nucl. Phys. B 202 (1982) 253-316. 42. E. Witten, “Supersymmetry and Morse theory,” J. of Diff. Geom. 17 (1982) 661-692. 43. E. Witten, “Topological quantum field theory,” Comm. Math. Phys. 117 (1988) 353-386. 44. E. Witten, “Topological sigma models,” Comm. Math. Phys. 118 (1988) 411-449. 45. E. Witten, “Quantum field theory and the Jones polynomial,” Commun. Math. Phys. 121 (1989) 351-399. 46. E. Witten, “The N matrix model and gauged WZW models,” Nucl. Phys. B 371 (1992) 191-245. 47. E. Witten, “Supersymmetric Yang-Mills theory on a four-manifold,” hepth/9403193, J. Math. Phys. 35 (1994) 5101-5135. 48. E. Witten, “Monopoles and four-manifolds,” hep-th/9411102, Math. Res. Lett. 1 (1994) 769-796.
Geometric and Topological Methods for Quantum Field Theory Eds. A. Cardona, H. Ocampo & S. Paycha @ 2003 World Scientific Publishing, pp. 312-371
(SUPER)-GRAVITIES BEYOND 4 DIMENSIONS JORGE ZANELLI" Centro de Estudios Cientificos, Casilla 14 69, Valdivia, Chile These lectures are intended as a broad introduction t o Chern Simons gravity and supergravity. The motivation for these theories lies in the desire to have a gauge invariant action -in the sense of fiber bundles- in more than three dimensions, which could provide a firm ground for constructing a quantum theory of the gravitational field. The case of Chern-Simons gravity and its supersymmetric extension for all odd D is presented. No analogous construction is available in even dimensions.
LECTURE 1 GENERAL RELATIVITY REVISITED In this lecture, the standard construction of the action principle for general relativity is discussed. The scope of the analysis is t o set the basis for a theory of gravity in any number of dimensions, exploiting the similarity between gravity and a gauge theory as a fiber bundle. It is argued that in a theory that describes the spacetime geometry, the metric and affine properties of the geometry should be represented by independent entities, an idea that goes back to the works of Cartan and Palatini. I it shown that he need for an independent description of the affine and metric features of the geometry leads naturally to a formulation of gravity in terms of two independent 1-form fields: the vielbein, e a , and the spin connection w:. Since these lectures are intended for a mixed audiencelreadership of mathematics and physics students, it would seem appropriate to locate the problems addressed here in the broader map of physics. 1
Physics and Mathematics
Physics is an experimental science. Current research, however, especially in string theory, could be taken as an indication that the experimental basis of physics is unnecessary. String theory not only makes heavy use of sophisticated modern mathematics, it has also stimulated research in some fields of *Lectures given at the 2001 Summer School Geometric and Topological Methods for Quantum Field Theory, Villa de Leyva, Colombia, June 2001.
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mathematics. At the same time, the lack of direct experimental evidence, either at present or in the foreseeable future, might prompt the idea that physics could exist without an experimental basis. The identification, of high energy physics as a branch of mathematics, however, is only superficial. High energy physics in general and string theory in particular, have as their ultimate goal the description of nature, while Mathematics is free from this constraint. There is, however, a mysterious connection between physics and mathematics which runs deep, as was first noticed probably by Pythagoras when he concluded that that, at its deepest level, reality is mathematical in nature. Such is the case with the musical notes produced by a violin string or by the string that presumably describes nature at the Planck scale. Why is nature a t the most fundamental level described by simple, regular, beautiful, mathematical structures? The question is not so much how structures like knot invariants, the index theorem or moduli spaces appear in string theory as gears of the machinery, but why should they occur at all. As E. Wigner put it, “the miracle of the appropriateness of the language of mathematics f o r the formulation of the laws of physics i s a wonderful gift, which we neither understand nor deserve.” [l]. Often the connection between theoretical physics and the real world is established through the phenomena described by solutions of differential equations. The aim of the theoretical physicist is to provide economic frameworks to explain why those equations are necessary. The time-honored approach t o obtain dynamical equations is a variational principle: the principle of least action in Lagrangian mechanics, the principle of least time in optics, the principle of highest profit in economics, etc. These principles are postulated with no further justification beyond their success in providing differential equations that reproduce the observed behavior. However, there is also an important aesthetic aspect, that has to do with economy of assumptions, the possibility of a wide range of predictions, simplicity, beauty. In order t o find the correct variational principle, an important criterion is symmetry. Symmetries are manifest in the conservation laws observed in the phenomena. Under some suitable assumptions, symmetries are often strong enough t o select the general form of the possible action functionals. The situation can be summarized more or less in the following scheme:
314 Feature
Theoretical predictions
1
I Ingredient I I Symmetry I I group I
1 1 1 Symmetries Variational Principle Dynamics
Phenomena Experiments
Examples
Translations, Lorentz, gauge I Action I 6I = b [ ( T - V ) d t Functional = . . Field F = ma‘, Equations Maxwell eqs. Solutions Orbits, states trajectories I Data I Positions. times
o,:
1
T Theoretical conatruction
I
Theoretical research proceeds inductively, upwards from the bottom, guessing the theory from the experimental evidence. Once a theory is built, it predicts new phenomena that should be confronted with experiments, checking the foundations, as well as the consistency of the building above. Axiomatic presentations, on the other hand, go from top t o bottom. They are elegant and powerful, but they rarely give a clue about how the theory was constructed and they hide the fact that a theory is usually based on very little experimental evidence, although a robust theory will generate enough predictions and resist many experimental tests.
1.1
Renormalizability and the Success of Gauge Theory
A good example of this way of constructing a physical theory is provided by Quantum Field Theory. Experiments in cloud chambers during the first half of the twentieth century showed collisions and decays of particles whose mass, charge, and a few other attributes could be determined. From this data, a general pattern of possible and forbidden reactions as well as relative probabilities of different processes was painfully constructed. Conservation laws, selection rules, new quantum numbers were suggested and a phenomenological model slowly emerged, which reproduced most of the observations in a satisfactory way. A deeper understanding, however, was lacking. There was no theory from which the laws could be deduced simply and coherently. The next step, then, was to construct such a theory. This was a major enterprise which finally gave us the Standard Model. The humble word “model”, used instead of “theory” , underlines the fact that important pieces are still missing in it. The model requires a classical field theory described by a lagrangian capable of reproducing the type of interactions (vertices) and conservation laws observed in the experiments a t the lowest order (low energy, weakly interacting regime). Then, the final test of the theory comes from the proof of its
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internal consistency as a quantum system: renormalizability. It seems that Hans Bethe was the first to observe that non renormalizable theories would have no predictive power and hence renormalizability should be the key test for the physical consistency of a theory [2].A brilliant example of this principle at work is offered by the theory for electroweak interactions. As Weinberg remarked in his Nobel lecture, if he had not been guided by the principle of renormalizability, his model would have included contributions not only from S U ( 2 ) x U(1)-invariant vector boson interactions -which were believed to be renormalizable, although not proven until a few years later by 't Hooft [32]- but also from the S U ( 2 ) x U(1)-invariant four fermion couplings, which were known to be non renormalizable [4]. Since a non renormalizable theory has no predictive power, even if it could not be said to be incorrect, it would be scientifically irrelevant like, for instance, a model based on angels and evil forces. One of the best examples of a successful application of mathematics for the description of nature at a fundamental scale is the principle of gauge invariance, that is the invariance of a system under a symmetry group that acts locally. The underlying mathematical structure of the gauge principle is mathematically captured through the concept of fiber bundle, as discussed in the review by Sylvie Paycha in this school [5]. For a discussion of the physical applications, see also [6]. Three of the four forces of nature (electromagnetism, weak, and strong interactions) are explained and accurately modelled by a Yang-Mills action built on the assumption that nature should be invariant under a group of transformations acting independently at each point of spacetime. This local symmetry is the key ingredient in the construction of physically testable (renormalizable) theories. Thus, symmetry principles are not only useful in constructing the right (classical) action functionals, but they are often sufficient to ensure the viability of a quantum theory built from a given classical act ion.
1.2
The Gravity Puzzle
The fourth interaction of nature, the gravitational attraction, has stubbornly resisted quantization. This is particularly irritating as gravity is built on the principle of invariance under general coordinate transformations, which is a local symmetry analogous to the gauge invariance of the other three forces. These lectures will attempt to shed some light on this puzzle. One could question the logical necessity for the existence of a quantum theory of gravity at all. True fundamental field theories must be renormaliz-
316
able; effective theories need not be, as they are not necessarily described by quantum mechanics at all. Take for example the Van der Waals force, which is a residual low energy interaction resulting from the electromagnetic interactions between electrons and nuclei. At a fundamental level it is all quantum electrodynamics, and there is no point in trying to write down a quantum field theory to describe the Van der Waals interaction, which might even be inexistent. Similarly, gravity could be an effective interaction analogous to the Van der Waals force, the low energy limit of some fundamental theory like string theory. There is one difference, however. There is no action principle to describe the Van der Waals interaction and there is no reason to look for a quantum theory for molecular interactions. Thus, a biochemical system is not governed by an action principle and is not expected to be described by a quantum theory, although its basic constituents are described by QED, which is a renormalizable theory. Gravitation, on the other hand, is described by an action principle. This is an indication that it could be viewed as a fundamental system and not merely an effective force, which in turn would mean that there might exist a quantum version of gravity. Nevertheless, countless attempts by legions of researchers -including some of the best brains in the profession- through the better part of the twentieth century, have failed to produce a sensible (e.g., renormalizable) quantum theory for gravity. With the development of string theory over the past twenty years, the prevailing view now is that gravity, together with the other three interactions and all elementary particles, are contained as modes of the fundamental string. In this scenario, all four forces of nature including gravity, would be low energy effective phenomena and not fundamental reality. Then, the issue of renormalizabilty of gravity would not arise, as it doesn’t in the case of the Van der Waals force. Still a puzzle remains here. If the ultimate reality of nature is string theory and the observed high energy physics is just low energy phenomenology described by effective theories, there is no reason to expect that electromagnetic, weak and strong interactions should be governed by renormalizable theories at all. In fact, one would expect that those interactions should lead to non renormalizable theories as well, like gravity or the old four-fermion model for weak interactions. If these are effective theories like thermodynamics or hydrodynamics, one could even wonder why these interactions are described by an action principle at all.
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1.3
Minimal Couplings and Connections
Gauge symmetry fixes the form in which matter fields couple to the carriers of gauge interactions. In electrodynamics, for example, the ordinary derivative in the kinetic term for the matter fields, a,, is replaced by the covariant derivative,
V, = a,
+ A,.
(1)
This provides a unique way to couple charged fields, like the electron, and the electromagnetic field. At the same time, this form of interaction avoids dimensionful coupling constants in the action. In the absence of such coupling constants, the perturbative expansion is likely to be well behaved because gauge symmetry imposes severe restrictions on the type of terms that can be added to the action, as there are very few gauge invariant expressions in a given number of spacetime dimensions. Thus, if the Lagrangian contains all possible terms allowed by the symmetry, perturbative corrections could only lead to rescalings of the coefficients in front of each term in the Lagrangian. These rescalings can always be absorbed in a redefinition of the parameters of the action. This renormalization procedure that works in gauge theories is the key to their internal consistency. The “vector potential” A, is a connection 1-form, which means that, under a gauge transformation,
A(x) 4 A(x)’ = U ( ~ ) A ( X ) U ( S > +-u~( ~ ) d U - ~ ( x ) ,
(2)
where V ( x )represents a position dependent group element. The value of A depends on the choice of gauge U(x) and it can even be made to vanish at a given point by an appropriate choice of U ( x ) . The combination V, is the covariant derivative, a differential operator that, unlike the ordinary derivative and A itself, transforms homogeneously under the action of the gauge group,
v,
+
v; = U(x)V,.
(3)
The connection can in general be a matrix-valued object, as in the case of nonabelian gauge theories. In that case, V, is an operator 1-form,
V=d+A = dd’(8,
+ A,),
(4)
Acting on a function 4(x),which is in a vector representation of the gauge group ($(x) -+ @(x) = U(x) . 4(x)),the covariant derivative reads
V4 = d4 + A A 4 .
(5)
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The covariant derivative operator V has a remarkable property: its square is not a differential operator but a multiplicative one, as can be seen from (5)
0 0 4 = d(A4) + Ad$ + A A A4
(6)
(~A+AAA)+ = F+ =
+
The combination F = dA A A A is the field strength of the nonabelian interaction. This generalizes the electric and magnetic fields of electromagnetism and it indicates the presence of energy. One can see now why the gauge principle is such a powerful idea in physics: the covariant derivative of a field, VIP, defines the coupling between and the gauge potential A in a unique way. Furthermore, A has a uniquely defined field strength F, which in turn defines the dynamical properties of the gauge field. In 1954, Robert Mills and Chen-Nin Yang grasped the beauty and the power of this idea and constructed what has been since known as the nonabelian Yang-Mills theory [7]. On the tangent bundle, the covariant derivative corresponding t o the gauge group of general coordinate transformations is the usual covariant derivative in differential geometry,
+
D=d+r = dxqad,
+ rp),
(7)
where I' is the Christoffel symbol, involving the metric and its derivatives. The covariant derivative operator in both cases reflects the fact that these theories are invariant under a group of local transformations, that is, operations which act independently at each point in space. In electrodynamics U(x) is an element of U(1), and in the case of gravity U(x) is the Jacobian matrix (dx/dx'), which describes a diffeomorphism, or general coordinate change,
x
4
1.4
x'. Gauge Symmetry and Diffeomorphism Invariance
The close analogy between the covariant derivatives V and D could induce one to believe that the difficulties for constructing a quantum theory for gravity shouldn't be significantly worse than for an ordinary gauge theory like QED. It would seem as if the only obstacles one should expect would be technical, due t o the differences in the symmetry group, for instance. There is, however, a more profound difference between gravity and the standard gauge theories that describe Yang-Mills systems. The problem is not that General Relativity lacks the ingredients to make a gauge theory, but that the right action for
319
gravity in four dimensions cannot be written as that of a gauge invariant system for the diffeomorphism group. In a YM theory, the connection A, is an element of a Lie algebra whose structure is independent of the dynamical equations. In electroweak and strong interactions, the connection is a dynamical field, while both the base manifold and the symmetry group are fixed, regardless of the values of the connection or the position in spacetime. This implies that the Lie algebra has structure constants, which are neither functions of the field A , or the position z. If G"(z) are the gauge generators in a YM theory, they obey an algebra of the form
where CFb are the structure constants. The Christoffel connection l"&, instead, represents the effect of parallel transport over the spacetime manifold, whose geometry is determined by the dynamical equations of the theory. The consequence of this is that the diffeomorphisms do not form a Lie algebra but an open algebra, which has structure functions instead of structure constants [8]. This problem can be seen explicitly in the diffeomorphism algebra generated by the hamiltonian constraints of gravity, 'Hl(z),'Hi(z),
'2::").
where 6(y, z),i = Here one now finds functions of the dynamical fields, g i j ( z ) playing the role of the structure constants CFb,which identify the symmetry group in a gauge theory. If the structure "constants" were t o change from one point to another, it would mean that the symmetry group is not uniformly defined throughout spacetime, which would prevent an interpretation of gravity in terms of fiber bundles, where the base is spacetime and the symmetry group is the fiber. It is sometimes asserted in the literature that gravity is a gauge theory for the translation group, much like the Yang Mills theory of strong interactions is a gauge theory for the S U ( 3 ) group. We see that although this is superficially correct, the usefulness of this statement is limited by the profound differences a gauge theory with fiber bundle structure and another with an open algebra such as gravity.
320
2
General Relativity
The question we would like t o address is: W h a t would you say i s the right action for the gravitational field in a spacetime of a given dimension? On November 25 1915, Albert Einstein presented t o the Prussian Academy of Natural Sciences the equations for the gravitational field in the form we now know as Einstein equations [9]. Curiously, five days before, David Hilbert had proposed the correct action principle for gravity, based on a communication in which Einstein had outlined the general idea of what should be the form of the equations [lo]. This is not so surprising in retrospect, because as we shall see, there is a unique action in four dimensions which is compatible with general relativity that has flat space as a solution. If one allows nonflat geometries, there is essentially a one-parameter family of actions that can be constructed: the Einstein-Hilbert form plus a cosmological term,
where R is the scalar curvature, which is a function of the metric gpu, its inverse gp', and its derivatives (for the definitions and conventions we use here, see Ref. [ll]). The expression I[9]is the only functional of the metric which is invariant under general coordinate transformations and gives second order field equations in four dimensions. The coefficients a1 and a1 are related to the gravitational constant and the cosmological constant through
1 A 161rG 8nG ' Einstein equations are obtained by extremizing this action (10) and they are unique in that: (i) They are tensorial equations (ii) They involve only up t o second derivatives of the metric (iii) They reproduce Newtonian gravity in the weak field nonrelativistic approximation. The first condition implies that the equations have the same meaning in all coordinate systems. This follows from the need t o have a coordinate independent (covariant) formulation of gravity in which the gravitational force is replaced by the nonflat geometry of spacetime. The gravitational field being a geometrical entity implies that it cannot resort t o a preferred coordinate choice or, in physical terms, a preferred set of observers. The second condition means that Cauchy conditions are necessary (and sufficient in most cases) t o integrate the equations. This condition is a concession t o the classical physics tradition: the possibility of determining the a1
= -, c Y 2 = -
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gravitational field at any moment from the knowledge of the positions and momenta at a given time. This requirement is also the hallmark of Hamiltonian dynamics, which is the starting point for canonical quantum mechanics. The third requirement is the correspondence principle, which accounts for our daily experience that an apple and the moon fall the the way they do. If one further assumes that Minkowski space be among the solutions of the matter-free theory, then one must set A 5 0 , as most sensible particle physicists would do. If, on the other hand, one believes in static homogeneous and isotropic cosmologies, then A must have a finely tuned nonzero in some “natural” value. Experimentally, A has a value of the order of units [12]. Furthermore, astrophysical measurements seem to indicate that A must be positive [13]. This presents a problem because there seems t o be no theoretical way t o predict this “unnaturally small” nonzero value. As we will see in the next lecture, for other dimensions, the EinsteinHilbert action is not the only possibility in order t o satisfy conditions (i-iii). 2.1
Metric and A m n e Structures
We conclude this introduction by discussing what we mean by spacetime geometry. Geometry is sometimes understood as the set of assertions one can make about the points in a manifold and their relations. This broad (and vague) idea, is often interpreted as encoded in the metric tensor, gpd/(x),which provides the notion of distance between nearby points with slightly different coordinates,
This is the case in Riemannian geometry, where all objects that are relevant for the spacetime can be constructed from the metric. However, one can distinguish between metric and f i n e features of space, that is, between the notions of distance and parallelism. Metricity refers t o lengths, areas, volumes, etc., while affinity refers t o scale invariant properties such as shapes. Euclidean geometry was constructed using two elementary instruments: the compass and the (unmarked) straightedge. The first is a metric instrument because it allows comparing lengths and, in particular, drawing circles. The second is used t o draw straight lines which, as will be seen below, is a basic affine operation. In order t o fix ideas, let’s consider a few examples from Euclidean geometry. Pythagoras’ famous theorem is a metric statement; it relates the lengths of the sides of a triangle: Affine properties on the other hand, do not change if the length scale is changed, such as the shape of a triangle or, more generally, the angle between
322
Figure 1. Pythagoras theorem: c2 = (a - b)2
+ 4[ab]/2
two straight lines. A typical affine statement is, for instance, the fact that when two parallel lines intersect a third, the corresponding angles are equal, as seen in Fig.2.
Figure 2 . Affine property: L
1) L‘ e a = a’
= 6 = 6’ , p = p’ = y = y’
Of course parallelism can be reduced to metricity. As we learned in school, one can draw a parallel to a line L using a right angled triangle (W) and an unmarked straightedge (R): One aligns one of the short sides of the triangle with the straight line and rests the other short side on the ruler. Then, one slides the triangle to where the parallel is t o be drawn, as in Fig.3. Thus, given a way to draw right angles and a straight line in space, one can define parallel transport. As any child knows from the experience of stretching a string or a piece of rubber band, a straight line is the shape of the shortest line between two points. This is clearly a metric feature because it requires measuring lengths. Orthogonality is also a metric notion that can
323
IL Figure 3. Constructing parallels using a right-angled triangle (W) and a straightedge (R)
be defined using the scalar product obtained from the metric. A right angle is a metric feature because we should be able t o measure angles, or measure the sides of triangles1. We will now show that, strictly speaking, parallelism
does not require metricity. There is something excessive about the construction in Fig.3 because one doesn’t have to use a right angle. In fact, any angle could be used in order to draw a parallel to L in the last example, so long as it doesn’t change when we slide it from one point to another, as shown in Fig.4. We see that the essence of parallel transport is a rigid, angle-preserving wedge and a straightedge to connect two points. There is still some cheating in this argument because we took the construction of a straightedge for granted. What if we had no notion of distance, how do we know what a straight line is? There is a way to construct a straight line that doesn’t require a notion of distance between two points in space. Take two short enough segments (two short sticks, matches or pencils would do), and slide them one along the other, as a cross country skier would do. In this way a straight line is lThe Egyptians knew how t o use Pythagoras’ theorem to make a right angle, although they didn’t know how to prove it. Their recipe was probably known long before, and all good construction workers today still know the recipe: make a loop of rope with 12 segments of equal length. Then, the triangle formed with the loop so that its sides are 3, 4 and 5 segments long is such that the shorter segments are perpendicular to each other [14].
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Figure 4. Constructing parallels using an arbitrary anglepreserving wedge (W) and a straightedge (R) .
generated by parallel transport of a vector along itself, and we have not used distance anywhere. It is this a f i n e definition of a straight line that can be found in Book I of Euclid’s Elements. This definition could be regarded as the straightest line, which does not necessarily coincide with the line of shortest distance. They are conceptually independent. In a space devoid of a metric structure the straightest line could be a rather strange looking curve, but it could still be used to define parallelism. Suppose the ruler R has been constructed by transporting a vector along itself, then one can use it to define parallel transport as in Fig.5. There is nothing wrong with this construction apart from the fact that it need not coincide with the more standard metric construction in Fig.3. The fact that this purely affine construction is logically acceptable means that parallel transport needs not be a metric concept unless one insists on reducing affinity to metricity. In differential geometry, parallelism is encoded in the affine connection mentioned earlier,I’&(z), so that a vector u at the point of coordinates z is said to be parallel to the vector ii at a point with coordinates z dz, if their components are related by “parallel transport”,
+
qz + dz) = r;7dzW(z).
(13)
The affine connection I?&(,) need not be logically related to the metric tensor gpv(z). Einstein’s formulation of General Relativity adopted the point of view
325
Figure 5. Constructing parallels using any angle-preserving wedge (W) and an arbitrary ruler (R). Any ruler is as good as another.
that the spacetime metric is the only dynamically independent field, while the affine connection is a function of the metric given by the Christoffel symbol,
This is the starting point for a controversy between Einstein and Cartan, which is vividly recorded in the correspondence they exchanged between May 1929 and May 1932 [15]. In his letters, Cartan insisted politely but forcefully that metricity and parallelism could be considered as independent, while Einstein pragmatically replied that since the space we live in seems to have a metric, it would be more economical to assume the affine connection to be a function of the metric. Einstein argued in favor of economy of independent fields. Cartan advocated economy of assumptions. Here we adopt Cartan’s point of view. It is less economical in dynamical variables but is more economical in assumptions and therefore more general. This alone would not be sufficient argument to adopt Cartan’s philosophy, but it turns out to be more transparent in many ways and to lend itself better to make a gauge theory of gravity.
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3
First Order Formulation for Gravity
We view spacetime as a smooth D-dimensional manifold of lorentzian signature M , which at every point x possesses a D-dimensional tangent space T,. The idea is that this tangent space T, is a good linear approximation of the manifold M in the neighborhood of x . This means that there is a way to represent tensors over M by tensors on the tangent space2.
9.1
The Vielbein
The precise translation (isomorphism) between the tensor spaces on M and on T, is made by means of a dictionary, also called "soldering form" or simply, "vielbein". The coordinate separation dxp, between two infinitesimally close points on M is mapped to the corresponding separation dz" in T,, as
dz" = eE(x)dzp
(15)
The family {eE(x),a = 1, ..., D = d i m M } can also be seen as a local orthonormal frame on M. The definition (15) makes sense only if the vielbein eE(x) transforms as a covariant vector under diffeomorphisms on M and as a contravariant vector under local Lorentz rotations of T,, SO(1, D - 1) (we assumed the signature of the manifold M to be Lorentzian). A similar one to one correspondence can be established between tensors on M and on T,: if II is a tensor with components IIp1...pnon M , then the corresponding tensor on the tangent space T, is3
p"l..."*(x)
@ : ( x ) .. . ez(x)npl...pn(x).
1
(16)
An example of this map between tensors on M and on T, is the relation between the metrics of both spaces, gpV )(.
=
' E (x)eL (x)%b.
(17)
This relation can be read as to mean that the vielbein is in this sense the square root of the metric. Given e;(x) one can find the metric and therefore, all the metric properties of spacetime are contained in the vielbein. The converse, 2Here, only the essential ingredients are given. For a more extended discussion, there are several texts such as those of Refs. [6], [16] and [17] . 3The inverse vielbein e g ( z ) where e:(z)eb,(z) = Sg, and e:(z)e;(z) = S;, relates lower index tensors, Pal...an(z) = eg:(z)...eg:(z)~pl...pn(z).
327
however, is not true: given the metric, there exist infinitely many choices of vielbein that reproduce the same metric. If the vielbein are transformed as eE(z)
-
(18)
eC(z) = &(z)e;(z),
where the matrix A(z) leaves the metric in the tangent space unchanged,
A: (z)A: (z)
b
(19)
= 7]cd,
then the metric g,,(z) is clearly unchanged. The matrices that satisfy (19) form the Lorentz group S O ( 1 , D - 1). This means, in particular, that there are many more components in e: than in g,,,. In fact, the vielbein has D 2 independent components, whereas the metric has only D ( D 1)/2. The mismatch is exactly D ( D - 1)/2, the number of independent rotations in D dimensions.
+
3.2
The Lorentz Connection
The Lorentz group acts on tensors a t each T, independently, that is, the matrices A that describe the Lorentz transformations are functions of z. In order to define a derivative of tensors in T, , one must compensate for the fact that at neighboring points the Lorentz rotations are not the same. This is not different from what happens in any other gauge theory: one needs to introduce a connection for the Lorentz group, w$,(z), such that, if @(z) is a field that transforms as a vector under the Lorentz group, its covariant derivative, D,V(.)
=
a,P(z) + W $ , ( 4 4 b ( z ) l
(20)
also transforms like a vector under SO(1,D - 1) at z. This requirement means that under a Lorentz rotation A:(z), w t P ( z ) changes as a connection [see (2)] w$,(z)
+
w ’ & ( ~ )= A:(z)A:(z)w‘&,(z)
+ A:(z)d,A;(z).
(21)
In physics, w t P ( z )is often called the spin connection, but Lorentz connection would be a more appropriate name. The word “spin” is due t o the fact that w$, arises naturally in the discussion of spinors, which carry a special representation of the group of rotations in the tangent space. The spin connection can be used t o define parallel transport of Lorentz tensors in the tangent space T, as one goes from the point t o a nearby point z dx. The parallel transport of the vector field @(z) from the point z to z dx,is a vector at z dz, q$(z d z ) , defined as
+ +
+
+
328
Here one sees that the covariant derivative measures the change in a tensor produced by parallel transport between neighboring points,
dz'D,$"(x)
= $;(z
+dz)
-
@(x).
(23)
In this way, the afine properties of space are encoded in the components
w$'(z), which are, until further notice, totally arbitrary and independent from the metric. The number of independent components of w& ' , is determined by the symmetry properties of w": = rlbcw%, under permutations of a and b. It is easy to see that demanding that the metric rlab remain invariant under parallel transport implies that the connection should be antisymmetric, waL = -wb:. We leave the proof as an exercise to the reader. Then, the number of independent components of w": is D 2 ( D- 1)/2. This is less than the number of independent components of the Christoffel symbol, D 2 ( D 1)/2.
+
3.3
Differential Forms
It can be observed that both the vielbein and the spin connection appear through the combinations
e a ( z )= e t ( z ) d z p ,
w $ ( z ) = w$,(x)dx',
(24) (25)
that is, they are local 1-forms. This is not an accident. It turns out that all the geometric properties of M can be expressed with these two 1-forms and their exterior derivatives only. Since both ea and w$ only carry Lorentz indices, these 1-forms are scalars under diffeomorphisms on M , indeed, they are coordinate-free, as all exterior forms. This, means that in this formalism the spacetime tensors are replaced by tangent space tensors. In particular, the Riemann curvature 2-form is4
where R$," = eaae$R5,v are the components of the usual Riemann tensor projected on the tangent space (see [ll]). The fact that w $ ( z ) is a 1-form, just like the gauge potential in YangMills theory, Aab = A$,dzfi, suggests that they are similar, and in fact they 4Here d stands for the 1-form exterior derivative operator dxpa,A
329
both are connections of a gauge group'. Their transformation laws have the same form and the curvature R"i,is completely analogous to the field strength in Yang-Mills,
There is an asymmetry with respect to the vielbein, though. Its transformation law under the Lorentz group is not that of a connection but of a vector. There is another important geometric object obtained from derivatives of e" which is analogous to the Riemann tensor is another, the Torsion 2-form,
T" = de" +w"i, A e b ,
(28)
which, unlike R$ is a covariant derivative of a vector, and is not a function of the vielbein only. Thus, the basic building blocks of first order gravity are e", w"i,, Rab,T". With them one must put together an action. But, are there other building blocks? The answer is no and the proof is by exhaustion. As a cowboy would put it, if there were any more of them 'round here, we would have heard ... And we haven't. There is a more subtle argument to rule out the existence of other building blocks. We are interested in objects that transform in a controlled way under Lorentz rotations (vectors, tensors, spinors, etc.). Taking the covariant derivatives of e", R"i,,and T", one finds always combinations of the same objects, or zero:
De" = de" + w: A eb = T" DR"b = dR"i,+w: A RCb+wC, AR: DT" = dT" + w$ A T b = R$ A e b .
(29) =0
(30) (31)
The first relation is just the definition of torsion and the other two are the Bianchi identities, which are directly related to the fact that the exterior derivative is nilpotent, d2 = dpd"dxp A dx" = 0. We leave it to the reader to prove these identities. In the next lecture we discuss the construction of the possible actions for gravity using these ingredients. In particular, in 4 dimensions, the Einstein action can be written as
51n what for physicists is fancy language, w is a locally defined Lie algebra valued 1-form on M , which is also a connection on the principal S O ( D - 1, 1)-bundle over M .
330
This is basically the only action for gravity in dimension four, but many more options exist in higher dimensions.
LECTURE 2 GRAVITY AS A GAUGE THEORY As we have seen, symmetry principles help in constructing the right classical action. More importantly, they are often sufficient to ensure the viability of a quantum theory obtained from the classical action. In particular, local or gauge symmetry is the key t o prove consistency (renormalizability) of the field theories we know for the correct description of three of the four basic interactions of nature. The gravitational interaction has stubbornly escaped this rule in spite of the fact that, as we saw, it is described by a theory based on general covariance, which is a local invariance quite analogous t o gauge symmetry. In this lecture we try t o shed some light on this puzzle. In 1955, less than a year after Yang and Mills proposed their model for nonabelian gauge invariant interactions, Ryoyu Utiyama showed that the Einstein theory can be written as a gauge theory for the Lorentz group [18]. This can be checked directly from the expression (32), which is a Lorentz scalar and hence, trivially invariant under (local) Lorentz transformations. Our experience is that the manifold where we live is approximately flat, four dimensional Minkowski spacetime. This space is certainly invariant under the Lorentz group S 0 ( 3 , 1 ) , but it also allows for translations. This means that it would be nice to view S O ( 3 , l ) as a subgroup of a larger group which contains symmetries analogous t o translations, SO(3,l) L) G. (33) The smallest nontrivial choices for G -which are not just S O ( 3 , l ) x Go-, are:
G=
{
S O ( 4 , l ) de Sitter S 0 ( 3 , 2 ) anti-de Sitter I S O ( 3 , l ) Poincark
(34)
The de Sitter and anti-de Sitter groups are semisimple, while the PoincarC group, which is a contraction of the other two, is not semisimple. (This is a rather technical detail but it means that, unlike the PoincarC group, both S O ( 4 , l ) and S 0 ( 3 , 2 ) are free of invariant abelian subgroups. Semisimple groups are preferred as gauge groups because they have an invertible metric in the group manifold.) Since a general coordinate transformation Z Z
+ Zi
+ti,
(35)
331
looks like a local translation, it is natural t o expect that diffeomorphism invariance could be identified with the local boosts or translations necessary to enlarge the Lorentz group into one of those close relatives in (34). Several attempts t o carry out this identification, however, have failed. The problem is that there seems t o be no action for general relativity, invariant under one of these extended groups [19-221. In other words, although the fields wab and ea match the generators of the group G, there is no G-invariant 4-form available constructed with the building blocks listed above. As we shall see next , in odd dimensions (D = 2n- 1), and only in that case, gravity can be cast as a gauge theory of the groups SO(D, l ) , S O ( D - 1,2), or I S O ( D - 1, l),in contrast with what one finds in dimension four.
4
Lanczos-Lovelock Gravity
We turn now to the construction of an action for gravity using the building blocks at our disposal: ea, w%, Rab,T a . It is also allowed t o include the only two invariant tensors of the Lorentz group, ?lab, and t o raise, lower and contract indices. The action must be an integral over the D-dimensional spacetime manifold, which means that the lagrangian must be a D-form. Since exterior forms are scalars under general coordinate transformations, general covariance is guaranteed by construction and we need not worry about it. The action principle cannot depend on the choice of basis in the tangent space since Lorentz invariance should be respected. A sufficient condition t o ensure Lorentz invariance is t o demand the lagrangian t o be a Lorentz scalar, although, as we will see, this is not strictly necessary. Thus, we tentatively postulate the lagrangian for gravity t o be a D-form constructed by taking linear combinations of products of the above ingredients in any possible way so as t o form a Lorentz scalar. We exclude from the ingredients functions such as the metric and its inverse, which rules out the Hodge *-dual. The only justification for this is that: i) it reproduces the known cases, and ii) it explicitly excludes inverse fields, like eE(z),which would be like A;' in Yang-Mills theory (see [23] and [24] for more on this). This postulate rules out the possibility of including tensors like the Ricci tensor R,, = r l a , e ~ e ~ Ror ~ ,R,~R,,Ra~Bul , etc. That this is sufficient and necessary t o account for all sensible theories of gravity in D dimensions is the contents of a theorem due to David Lovelock [25], which in modern language can be stated thus:
Theorem [Lovelock,l970-Zumino1l986]: The most general action for gravity that does not involve torsion, which gives a t most second order field
332
equations for the metric and is of the form
p=o
where the a p s are arbitrary constants, and L(DJ’)is given by
Here and in what follows we omit the wedge symbol in the exterior products. For D = 2 this action reduces t o a linear combination of the 2-dimensional Euler character, x 2 , and the spacetime volume (area),
=
(?R+
K /
= a;. x 2
2a1) d2x
+ a; . v 2 .
This action has only one local extremum, V = 0, which reflects the fact that, unless other matter sources are included, 1 2 does not make a very interesting dynamical theory for the geometry. If the geometry is restricted to have a prescribed boundary this action describes the shape of a soap bubble, the famous Plateau problem: What is the surface of minimal area that has a
certain fixed closed curve as boundary?. For D = 3, (36) reduces t o the Hilbert action with cosmological constant, and for D = 4 the action picks up in addition the four dimensional Euler invariant x 4 . For higher dimensions the lagrangian is a polynomial in the curvature 2-form of degree d 5 D/2. In even dimensions the highest power in the curvature is the Euler character XD. Each term L(DiP ) is the continuation to D dimensions of the Euler density from dimension p < D [23]. One can be easily convinced, assuming the torsion tensor vanishes identically, that the action (36) is the most general scalar D-form that be constructed using the building blocks we considered. The first nontrivial generalization of Einstein gravity occurs in five dimensions, where a quadratic term can be added t o the lagrangian. In this case, the 5-form
eabcdeRabRcdee =
+ R2]d5x
[ R a p y g R a B r b- 4 R a P R a p
(39)
can be identified as the Gauss-Bonnet density, whose integral in four dimensions gives the Euler character x 4 . In 1938, Cornelius Lanczos noticed that this term could be added t o the Einstein-Hilbert action in five dimensions [as].
333
It is intriguing that he did not go beyond D = 5. The generalization to arbitrary D was obtained by Lovelock more than 30 years later as the Lanczos -Lovelock (LL) lagrangians,
p=o
These lagrangians were also identified as describing the only ghost-free effective theories for spin two fields6, generated from string theory at low energy [23,27]. From our perspective, the absence of ghosts is only a reflection of the fact that the LL action yields at most second order field equations for the metric, so that the propagators behave as a k - 2 , and not as ak-' +PICp4, as would be the case in a higher derivative theory.
4.1
Dynamical Content
Extremizing the LL action (36) with respect to ea and wab, yields
S I =~
s
[bea&,
+ 6Wab&&]= 0,
(41)
modulo surface terms. The condition for I D to have an extreme under arbitrary first order variations is that the coefficients &, vanish identically. This condition is the geometry satisfies the field equations
[?I &, =
1a p ( d
-
2p)&?) = 0,
(42)
p=o
and
where we have defined
6Physical states in quantum field theory have positive probability, which means that they are described by positive norm vectors in a Hilbert space. Ghosts instead, are unphysical states of negative norm. A lagrangian containing arbitrarily high derivatives of fields generally leads to ghosts. Thus, the fact that a gravitational lagrangian such as 40 leads t o a ghost-free theory is highly nontrivial.
334
These equations involve only first derivatives of ea and w$, simply because d2 = 0. If one furthermore assumes, as is usually done, that the torsion vanishes,
Eq. (45) is automatically satisfied and can be solved for w as a function of derivative of e and its inverse w = w ( e , a e ) . Substituting the spin connection back into (44) yields second order field equations for the metric. These equations are identical t o the ones obtained from varying the LL action written in terms of the Riemann tensor and the metric,
I D [ g ]= / d D x &
[a: + a',R
+ aL(RaPrsRap,6- 4RaPR,p + R 2 )+ . . .] .
(47) This purely metric form of the action is the secalled second order formalism. It might seem surprising that the action (47) yields only second order field equations for the metric, since the lagrangian contains second derivatives of gpv. In fact, it is sometimes asserted that the presence of terms quadratic in curvature necessarily bring in higher order equations for the metric but, as we have seen, this is not true for the LL action. Higher derivatives of the metric would mean that the initial conditions required t o uniquely determine the time evolution are not those of General Relativity and hence the theory would have different degrees of freedom from standard gravity. It also means that the propagators in the quantum theory develops poles a t imaginary energies: ghosts. Ghost states spoil the unitarity of the theory, making it hard to interpret its predictions. One important feature that makes the LL theories very different for D > 4 from those for D 5 4 is the fact that in the first case the equations are nonlinear in the curvature tensor, while in the latter case all equations are linear in Raband in T". In particular, while for D 5 4 the equations (45) imply the vanishing of torsion, this is no longer true for D > 4. In fact, the field equations evaluated in some configurations may leave some components of the curvature and torsion tensors indeterminate. For example, Eq.(43) has the form of a polynomial in Rabtimes T", and it is possible that the polynomial vanishes identically, leaving the torsion tensor completely arbitrary. However, the configurations for which the equations do not determine Raband T" completely form sets of measure zero in the space of geometries. In a generic case, outside of these degenerate configurations, the LL theory has the same D(D - 3)/2 degrees of freedom as in ordinary gravity [28].
335
4.2
Adding Torsion
Lovelock's theorem assumes torsion t o be identically zero. If equation (46) is assumed as an identity, means that ea and wt are no longer independent fields, contradicting the assumption that these fields correspond t o two equally independent features of the geometry. Moreover, for D 5 4, equation (46) coincides with (45), so that imposing the torsion-free constraint is, in the best case, unnecessary. On the other hand, if the field equation for a some field 4 can be solved algebraically as 4 = f ( $ ) in terms of the other fields, then by the implicit function theorem, the reduced action principle I [ 4 ,$1 is identical to the one obtained by substituting f ( $ ) in the action, I [ f ( $ ) , $ ] . This occurs in 3 and 4 dimensions, where the spin connection can be algebraically obtained from its own field equation and I [ w ,el = I ( w ( e ,ae), el . In higher dimensions, however , the torsion-free condition is not necessarily a consequence of the field equations and although (45) is algebraic in w , it is practically impossible to solve for w as a function of e. Therefore, it is not clear in general whether the action I [ w ,el is equivalent to the second order form of the LL action, I [ w( e lae>, el.
Since the torsion-free condition cannot be always obtained from the field equations, it is natural t o look for a generalization of the Lanczos-Lovelock action in which torsion is not assumed to vanish. This generalization consists of adding of all possible Lorentz invariants involving T a explicitly (this includes the combination DT" = Rabeb). The general construction was worked out in [29]. The main difference with the torsion-free case is that now, together with the dimensional continuation of the Euler densities, one encounters the Pontryagin (or Chern classes) as well. For D = 3, the only new torsion term not included in the Lovelock family is
e"Ta while for D = 4, there are three terms not included in the LL series,
(48)
eaebRab, TaTa,RabRab. (49) The last term in (49) is the Pontryagin density, whose integral also yields a topological invariant. It turns out that a linear combination of the other two terms is also a topological invariant related to torsion known as the Nieh-Yan density [30] N4 = TaTa- eaebRab.
(50)
The properly normalized integral of (50) over a 4-manifold is an integer [31].
336
In general, the terms related t o torsion that can be added to the action are combinations of the form
e,, Ra&Rai3. . . R"a",--'ean ,even n 2 2 BZn+l = T,, R"d,R"a",' ' . R2n-1ean,any n 2 1 C2n+2 = T,, R"a',R"a', . ' . Rzn-'Tan,odd n 2 1 AZn =
(51) (52) (53)
+
+
which are 2n, 2n 1 and 2n 2 forms, respectively. These Lorentz invariants belong t o the same family with the Pontryagin densities or Chern classes,
RZ3 . . . R:,
Pzn=
even n 2 2.
(54)
The lagrangians that can be constructed now are much more varied and there is no uniform expression that can be provided for all dimensions. For example, in 8 dimensions, in addition t o the LL terms, one has all possible 8-form made by taking products among the elements of the set (A4 i A 8 B3 7 B5 i B7, c 4 ,c 8 I p4, p8). They are (B3BB)i (A4C4)1
(c4)2,c8, (A4P4)1
(c4p4), (p4)2ip8.
(55)
To make life even more complicated, there are some linear combinations of these products which are topological densities. In 8 dimensions these are the Pontryagin forms
Pa = R"a',R",', . . . R2, , ( ~ 4= ) (RabRba)2, ~
which occur also in the absence of torsion, and generalizations of the Nieh-Yan forms, (N4)2 = N4P4 =
(T"T, - eaebRab)2, (T"T, - eaebRab)(RcdRd,),
etc. (for details and extensive discussions, see Ref. [29]). 5
Selecting Sensible Theories
Looking at these expressions one can easily get depressed. The lagrangians look awkward, the number of terms in them grow wildly with the dimension7. This problem is not only an aesthetic one. The coefficients in front of each term in the lagrangian is arbitrary and dimensionful. This problem already 7As it is shown in [29], the number of torsion-dependent terms grows as the partitions of D / 4 , which is given by the Hardy-Ramanujan formula, p(D/4) exp[xJD/6]. N
&
337
occurs in 4 dimensions, where the cosmological constant has dimensions of [length]-4 , and as evidenced by the outstanding cosmological constant problem, there is no theoretical argument t o fix its value in order t o compare with the observations. There is another serious objection from the point of view of quantum mechanics. Dimensionful parameters in the action are potentially dangerous because they are likely t o give rise t o uncontrolled quantum corrections. This is what makes ordinary gravity nonrenormalizable in perturbation theory: In 4 dimensions, Newton’s constant has dimensions of length squared, or inverse mass squared, in natural units. This means that as the order in perturbation theory increases, more powers of momentum will occur in the Feynman graphs, making its divergences increasingly worse. Concurrently, the radiative corrections t o these bare parameters would require the introduction of infinitely many counterterms into the action t o render them finite [32]. But an illness that requires infinite amount of medication is also incurable. The only safeguard against the threat of uncontrolled divergences in the quantum theory is t o have some symmetry principle that fixes the values of the parameters in the action and limits the number of possible counterterms that could be added to the lagrangian. Thus, if one could find a symmetry argument to fix the independent parameters in the theory, these values will be “protected” by the symmetry. A good indication that this might happen would be if the coupling constants are all dimensionless, as in Yang-Mills theory. As we will see in odd dimensions there is a unique combination of terms in the action that can give the theory an enlarged symmetry, and the resulting action can be seen to depend on a unique constant that multiplies the action. Moreover, this constant can be shown to be quantized by a argument similar t o Dirac’s quantization of the product of magnetic and electric charge [38].
5.1
Extending the Lorenta Group
The coefficients ap in the LL lagrangian (40)have dimensions 1 D - 2 P . This is because the canonical dimension of the vielbein is [ea] = 11, while the Lorentz connection has dimensions that correspond t o a true gauge field, [wab]= 1’. This reflects the fact that gravity is naturally only a gauge theory for the Lorentz group, where the vielbein plays the role of a matter field, which is not a connection field but transforms as a vector under Lorentz rotations. Three-dimensional gravity is an important exception t o this statement, in which case ea plays the role of a connection. Consider the simplest LL
338
lagrangian in 3 dimensions, the Einstein-Hilbert term L3 = fabcRabeC.
Under an infinitesimal Lorentz transformation with parameter forms as
while ec, Rab and
&bc
(56) Xab,
wab trans-
transform as tensors,
Combining these relations the Lorentz invariance of L3 can be shown directly. What is unexpected is that one can view ea as a gauge connection for the translation group. In fact, if under "local translations" in tangent space, parametrized by A", the vielbein transforms as a connection,
6e" = DX" = dXa
the lagrangian
L3
+ wabXb,
changes by a total derivative,
6L3
= d[fabcRabXc].
(59)
Thus, the action changes by a surface term which can be dropped under standard boundary conditions. This means that, in three dimensions, ordinary gravity can be viewed as a gauge theory of the Poincare group. We leave it as an exercise to the reader to prove this. (Hint: use the infinitesimal transformations be and 6w to compute the commutators of the second variations to obtain the Lie algebra of the Poincare group.) The miracle also works in the presence of a cosmological constant A = Now the lagrangian (40) is
$&.
L,A~'
a eb e c ), (60) 312 and the action is invariant -modulo surface terms- under the infinitesimal transformations, 1 Swab = [dXab waCXcb wbcXaC]T [e"Xb- Xaeb] (61) 1 6ea = [Xabeb] [dX" -k W"bXb]. (62)
= fabc(RabeC f -e
+ +
+
339
These transformations can be cast in a more suggestive way as
This can also be written as
+
bWAB= dWAB+ WA,ACB WB,AAC, where the 1-form W A Band the 0-form AAB stand for the combinations
+
where a , b, .. = 1,2, ..D, while A , B , ... = 1,2, ..,D 1. Clearly, W A Btransforms as a connection and AAB can be identified as the infinitesimal transformation parameters, but for which group? A clue comes from the fact that AAB = -ABA. This immediately indicates that the group is one that leaves invariant a symmetric, real bilinear form, so it must be one of the SO(r,s) family. The signs ( z t ) in the transformation above can be traced back to the sign of the cosmological constant. It is easy t o check that this structure fits well if indices are raised and lowered with the metric IIAB=
Pab 1 0 *O1
(65)
'
so that, for example, W $ = I I B c W ~Then, ~ . the covariant derivative in the connection W of this metric vanishes identically,
DwIIAB = dIIAB + WA,IIcB + WB,IIAC= 0.
+
(66)
Since I I A B is constant, this last expression implies W A B W B A= 0, in exact analogy with what happens with the Lorentz connection, wab+wba = 0, where W a b - bc a - 7 w c . Indeed, this is a very awkward way to discover that the 1-form W A Bis actually a connection for the group which leaves invariant the metric I I A B . Here the two signs in IIAB correspond t o the de Sitter (+) and anti-de Sitter (-) groups, respectively. Observe that what we have found here is an explicit way t o immerse the Lorentz group into a larger one, in which the vielbein has been promoted to a component of a larger connection, on the same footing as the Lorentz connection.
340
The Poincark symmetry is obtained in the limit 2 instead of (61, 62) one has
--+
00.
In that case,
bwab = [dXab+ waCXcb+ wbcXac] bea = [ ~ " ~ e ~[ ] d + ~w " ~ x ~ ] .
(67) (68)
+
In this limit, the representation in terms of W becomes inadequate because the metric TABbecomes degenerate (noninvertible) and is not clear how to raise and lower indices anymore. 5.2
More Dimensions
Everything that has been said about the embedding of the Lorentz group into the (A)dS group, starting at equation (61) is not restricted to D = 3 only and can be done in any D. In fact, it is always possible to embed the Lorentz group in D dimensions into the de-Sitter, or anti-de Sitter groups,
so(D
- " ')
L-)
S O ( D ,l), IIAB = diag(vab,+1) S O ( D - 1 , 2 ) , ITAB = diag(vab,-1)
'
(69)
with the corresponding Poincark limit, which is the familiar symmetry group of Minkowski space.
S O ( D - 1,l)L) I S O ( D - I, 1).
(70)
Then, the question naturally arises: can one find an action for gravity in other dimensions which is also invariant, not just under the Lorentz group, but under one of its extensions, S O ( D ,l), SO(D - 1 , 2 ) , ISO(D - 1,l)? As we will see now, the answer to this question is affirmative in odd dimensions. There is always a action for D = 2n - 1, invariant under local SO(2n - 2,2), SO(2n - 1,l) or ISO(2n - 2 , l ) transformations, in which the vielbein and the spin connection combine to form the connection of the larger group. In even dimensions, however, this cannot be done. Why is it possible in three dimensions to enlarge the symmetry from local SO(2,l) to local SO(3, l),S 0 ( 2 , 2 ) , ISO(2, l)?What happens if one tries t o do this in four or more dimensions? Let us start with the Poincarb group and the Hilbert action for D = 4, ab c d Lq = EabcdR e e .
Why is this not invariant under local translations be" = dXa simple calculation yields
bLq = 2EabcdRabec6ed = d(2EabcdRab ec d )
+ 2EabcdRabTCXd.
(71)
+ WabXb?
A
(72)
34 1
The first term in the r.h.s. of (72) is a total derivative and therefore gives a surface contribution to the action. The last term, however, need not vanish, unless one imposes the field equation T a = 0. But this means that the invariance of the action only occurs on shell. On shell symmetries are not real symmetries and they need not survive quantization. On close inspection, one observes that the miracle occurred in 3 dimensions because the lagrangian contained only one e. This means that a lagrangian of the form
L2n+1 = €al.,.,azn+lp 1 a z . . . ~ a z ~ - i a z ~ ~ a z ~ + i (73) is invariant under local Poincark transformations (67, 68), as can be easily checked out. Since the Poincark group is a limit of (A)dS, it seem likely that there should exist a lagrangian in odd dimensions, invariant under local (A)dS transformations, whose limit for 1 -+ co (vanishing cosmological constant) is (73). One way to find out what that lagrangian might be, one could take the most general LL lagrangian and select the coefficients by requiring invariance under (61, 62). This is a long, tedious and sure route. An alternative approach is to try to understand why it is that in three dimensions the gravitational lagrangian with cosmological constant (60) is invariant under the (A)dS group. If one takes seriously the notion that W A Bis a connection, then one can compute the associated curvature, F A B = dWAB+ WA,WCB, using the definition of W A B(63). It is a simple exercise to prove
If a, b run from 1 to 3 and A , B from 1 to 4, then one can construct the 4-form invariant under the (A)dS group, which is readily recognized as the Euler density in a four-dimensional manifold whose tangent space is not Minkowski, but has the metric IIAB =diag (vab,f l ) . E4 can also be written explicitly in terms of Rab,T a ,and e a , E4 =
4€,bc(Rab f l-2eaeb)l-1Ta
(76)
which is, up to constant factors, the exterior derivative of the threedimensional lagrangian (60), 4 E4 = -dL,AdS. (77) 1
342
This explains why the action is (A)dS invariant up to surface terms: the 1.h.s. of (77) is invariant by construction under local (A)dS, so the same must be true of the r.h.s., 6 ( d L f d S )= 0. Since the variation (6) is a linear operation,
d ( 6 L t d S )= 0, which in turn means, by Poincarb's Lemma that, locally, 6 L f d S = d(something). That is exactly what we found for the variation, [see, (59)]. The fact that three dimensional gravity can be written in this way was observed many years ago in Refs. [33,34]. The key to generalize the (A)dS lagrangian from 3 to 2n - 1 dimensions is now clear'. First, generalize the Euler density (75) to a 2n-form,
Ezn = c A I . . . ~ 2 , F A.1. A . FA2,-lAzn. 2
(78)
Second, express E2n explicitly in terms of Rab,T", and ea, and write this as the exterior derivative of a (2n- 1)-form which can be used as a lagrangian in (271 - 1 ) dimensions. Direct computation yields the (2n - 1)-dimensional lagrangian as
p=o
where L(DJ') is given by (37) and the coefficients 'Yp are no longer arbitrary, but they take the values
where n is an arbitrary dimensionless constant. It is left as an exercise to the reader to check that dL$?$ = Ezn and to show the invarianceof Lgl_d,s under the (A)dS group. In five dimensions, for example, the (A)dS lagrangian reads 2 a b e de e e e R f - e e ae be ee d e . (81) L y ) d S= IF. ' 313 515
1
The parameter 1 is a length scale -the Planck length- and cannot be fixed by other considerations. Actually, 1 only appears in the combination
8The construction we outline here was discussed by Chamseddine [35],Muller-Hoissen [36], and Bafiados, Teitelboim and this author in [40].
343
which could be considered as the "true77dynamical field, which is the natural thing t o do if one uses W A Binstead of uaband ea separately. In fact, the lagrangian (79) can also be written in terms of W A Band its exterior derivative, as
where all indices are contracted appropriately and the coefficients a3, a5, are all combinatoric factors without dimensions. The only remaining free parameter is K . Suppose this lagrangian is used t o describe a simply connected, compact 2n - 1 dimensional manifold M , which is the boundary of a 2n-dimensional compact orientable manifold R. Then the action for the geometry of M can be expressed as the integral of the Euler density E2, over 52, multiplied by K . But since there can be many different manifolds with the same boundary M , the integral over R should give the physical predictions as that over another manifold, R'. In order for this change t o leave the path integral unchanged, a minimal requirement would be
The quantity in brackets -with right normalization- is the Euler number of the manifold obtained by gluing R and 0' along M , in the right way to produce an orientable manifold, x[R u R'j, which can take an arbitrary integer value. From this, one concludes that K must be quantized [38], K
= nh.
where h is Planck's constant.
5.3
Chern-Simons
There is a more general way t o look a t these lagrangians in odd dimensions, which also sheds some light on their remarkable enlarged symmetry. This is summarized in the following Lemma: Let C ( F ) be an invariant 2n-form constructed with the field strength F = dA A2,where A is the connection for some gauge group G. If there exists a 2n - 1 form, L , depending on A and dA, such that dL = C , then under a gauge transformation, L changes by a total derivative (exact form). The (2n - 1)-form L is known as the Chern-Simons (CS) lagrangian. This lemma shows that L defines a nontrivial lagrangian for A whichis not invariant under gauge transformations, but that changes by a function that only depends on the fields at the boundary.
+
344
This construction is not only restricted to the Euler invariant discussed above, but applies to any invariant of similar nature, generally known as characteristic classes. Other well known characteristic classes are the Pontryagin or Chern classes and their corresponding CS forms were studied first in the context of abelian and nonabelian gauge theories (see, e. g., [6,39]). The following table gives examples of CS forms which define lagrangians in three dimensions, and their corresponding characteristic classes, Lagrangian Lij°r L'<°r Tu(i) ^3
TSU(N)
L
3
CS formL
dL
u b
j}b R abH a
a
TaTa - eaebRab FF tr[FF]
In this table, R, F, and F are the curvatures of the Lorentz connection u)ab , the electromagnetic ( U ( l ) ) connection A, and the Yang-Mills (SU(N)) connection A, respectively.
5.4
Torsional CS
So far we have not included torsion in the CS lagrangians, but as we see in the third row of the table above it is also possible to construct CS forms that include torsion. All the CS forms above are Lorentz invariant up to a closed form, but there is a linear combination of the first two which is invariant under the (A)dS group. The so-called exotic gravity, given by T Ll
Exotic _ T Lor ,
a
— -k.i
^ Tor
T T TT-ks
i
(84)
is invariant under (A)dS, as can be shown by computing its exterior derivative, j r Exotic r>a jyb \ (rnarrt a Jo T> aLz —K bKa± — (l la- e e Kab T?A 7-1 = t Rf
B
A-
This exotic lagrangian has the curious property of giving exactly the same field equations as the standard dLAds, but interchanged: the equation for ea form one is the equation for uiab of the other. In five dimensions there are no
345
new terms due to torsion, and in seven there are three torsional CS terms, Lagrangian L p
CS formL w(dw)3 ..
dL
+. +
$w7 R$R~~RC,R$ LioTRtRba(wtdwb, ?jw$w:w:)R$Rb, (RZR~,) (TaTa- eaebRab)R",Rb, L;'o, R$ R; eaTaR$Rba
+
In three spacetime dimensions, GR is a renormalizable quantum theory [34]. It is strongly suggestive that precisely in 2+1 dimensions this is also a gauge theory on a fiber bundle. It could be thought that the exact solvability miracle is due t o the absence of propagating degrees of freedom in three-dimensional gravity, but the final power-counting argument of renormalizability rests on the fiber bundle structure of the Chern-Simons system and doesn't seem to depend on the absence of propagating degrees of freedom. 5.5
Even Dimensions
The CS construction fails in 2n dimensions for the simple reason that there are no characteristic classes C ( F ) constructed with products of curvature in 2n + 1 dimensions. This is why an action for gravity in even dimensions cannot be invariant under the (anti-) de Sitter or Poincark groups. In this light, it is fairly obvious that although ordinary Einstein-Hilbert gravity can be given a fiber bundle structure for the Lorentz group, this structure cannot be extended to include local translational invariance. In some sense, the closest one can get to a CS theory in even dimensions is the so-called Born-Infeld (BI) theories [37,40,41]. The BI lagrangian is obtained by a particular choice of the apsin the LL series, so that the lagrangian takes the form
where Rabstands for the combination
With this definition it is clear that the lagrangian (85) contains only one free parameter, 1. This lagrangian has a number of interesting classical features like simple equations, black hole solutions, cosmological models, etc. The simplification comes about because the equations admit a unique maximally symmetric configuration given by Rab = 0, in contrast with the situation when all apsare arbitrary. As we have mentioned, for arbitrary aps, the field equations do not determine completely the components of Rab and
346
T a in general. This is because the high nonlinearity of the equations can give rise to degeneracies. The BI choice is in this respect the best behaved since the degeneracies are restricted to only one value of the radius of curvature (Rabf &eaeb = 0). At the same time, the BI action has the least number of algebraic constrains required by consistency among the field equations, and it is therefore the one with the simplest dynamical behavior [41]. Equipped with the tools to construct gravity actions invariant under larger groups, in the next lecture we undertake the extension of this trick to include supersymmetry.
LECTURE 3 CHERN SIMONS SUPERGRAVITY The previous lectures dealt with the possible ways in which pure gravity can be extended by relaxing three standard assumptions of General Relativity: i) that the notion of parallelism is derived from metricity, ii) that the dimension of spacetime must be four, and iii) that the action should only contain the Einstein Hilbert term &R. On the other hand, we still demanded that iv) the metric components obey second order field equations, v) the lagrangian be an D-form constructed out of the vielbein, eat the spin connection, w:, and their exterior derivatives, vi) the action be invariant under local Lorentz rotations in the tangent space. This allowed for the inclusion of several terms containing higher powers of the curvature and torsion multiplied by arbitrary and dimensionful coefficients. The presence of these arbitrary constants was regarded as a bit of an embarrassment which could be cured by enlarging the symmetry group, thereby fixing all parameters in the lagrangian and making the theory gauge invariant under the larger symmetry group. The cure works in odd but not in even dimensions. The result was a highly nonlinear ChernSimons theory of gravity, invariant under local Ads transformations in the tangent space. We now turn to the problem of enlarging the contents of the theory to allow for supersymmetry. 6
Supersymmetry
Supersymmetry is a symmetry most theoreticians are willing to accept as a legitimate feature of nature, although it has never been experimentally observed. The reason is that it is such a unique and beautiful idea that it is commonly felt that it would be a pity if it is not somehow realized in nature. Supersymmetry is the only symmetry which can accommodate spacetime and
347
internal symmetries in a nontrivial way. By nontrivial we mean that the Lie algebra is not a direct sum of the algebras of spacetime and internal symmetries. There is a famous no-go theorem which states that it is impossible to do this with an ordinary Lie group, closed under commutator (antisymmetric product, [., .I). The way supersymmetry circumvents this obstacle is by having both commutators and anticommutators (symmetric product, { ., .}), forming what is known as a graded Lie algebra, also called a super Lie algebra or simply, a superalgebra. For a general introduction t o supersymmetry, see [43,44]. The importance of this unification is that it combines bosons and fermions on the same footing. Bosons are the carriers of interactions, such as the photon, the graviton and gluons, while fermions are the constituents of matter, such as electrons and quarks. Thus, supersymmetry predicts the existence of a fermionic carriers of interaction and bosonic constituents of matter as partners of the known particles, none of which have been observed. Supersymmetry also strongly restricts the possible theories of nature and in some cases it even predicts the dimension of spacetime, like in superstring theory as seen in the lectures by Stefan Theisen in this same volume [42]. 6.1
Supemlgebm
A superalgebra has two types of generators: bosonic, Bi, and fermionic, F a . They are closed under the (anti-) commutator operation, which follows the general pattern
[Bi,Bj] = CZBk
(87)
[Bi,Fa] = CfaFp
(88)
(89) The generators of the Poincark group are included in the bosonic sector, and the Fa’s are the supersymmetry generators. This algebra, however, does not close for an arbitrary bosonic group. In other words, given a Lie group with a set of bosonic generators, it is not always possible t o find a set of fermionic generators to enlarge the algebra into a closed superalgebra. The operators satisfying relations of the form (87-89), are still required t o satisfy a consistency condition, the super-Jacobi identity, P a , F p ) = C,pB,
[G,I [GI/,GxI&
+ (-)‘(yX~)[Gv, [Gx,G,1& +
(_)+W)
[GA,[G,, GI/]+]+= 0. (90) Here G, represents any generator in the algebra, [R,S]* = RS i~SR, where this sign is chosen according the bosonic or fermionic nature of the opera-
348
tors in the bracket, and u ( v X p ) is the number of permutations of fermionic generators. As we said, starting with a set of bosonic operators it is not always possible t o find a set of N fermionic ones that generate a closed superalgebra. It is often the case that extra bosonic generators are needed to close the algebra, and this usually works for some values of N only. In other cases there is simply no supersymmetric extension at all. This happens, for example, with the de Sitter group, which has no supersymmetric extension in general [44]. For this reason in what follows we will restrict t o AdS theories. 6.2
Supergmvity
The name supergravity (SUGRA) applies t o any of a number of supersymmetric theories that include gravity in their bosonic sectors. The invention/discovery of supergravity in the mid 70’s came about with the spectacular announcement that some ultraviolet divergent graphs in pure gravity were cancelled by the inclusion of their supersymmetric partners [45]. For some time it was hoped that the nonrenormalizability of G R could be cured in this way by its supersymmetric extension. However, the initial hopes raised by SUGRA as a way taming the ultraviolet divergences of pure gravity eventually vanished with the realization that SUGRAs would be nonrenormalizable as well [46]. Again, one can see that the standard form of SUGRA is not a gauge theory for a group or a supergroup, and that the local (super-) symmetry algebra closes naturally on shell only. The algebra could be made t o close off shell by force, at the cost of introducing auxiliary fields -which are not guaranteed to exist for all d and N [47]-, and still the theory would not have a fiber bundle structure since the base manifold is identified with part of the fiber. Whether it is the lack of fiber bundle structure the ultimate reason for the nonrenormalizability of gravity remains t o be proven. It is certainly true, however, that if G R could be formulated as a gauge theory, the chances for its renormalizability would clearly increase. At any rate, now most high energy physicists view supergravity as an effective theory obtained from string theory in some limit. In string theory, eleven dimensional supergravity is seen as an effective theory obtained from ten dimensional string theory at strong coupling [42]. In this sense supergravity would not be a fundamental theory and therefore there is no reason to expect that it should be renormalizable. In any case, our point of view here is that there can be more than one system that can be called supergravity, whose connection with the standard theory is still not clear. As we have seen in the previous lecture, the CS
349
gravitation theories in odd dimensions are genuine (off-shell) gauge theories for the anti-de Sitter (A)dS or Poincar6 groups.
6.3
From Rigid Supersymmetry t o Supergmvity
Rigid or global SUSY is a supersymmetry in which the group parameters are constants throughout spacetime. In particle physics the spacetime is usually assumed to have fixed Minkowski geometry. Then the relevant SUSY is the supersymmetric extension of the Poincare algebra in which the supercharges are “square roots” of the generators of spacetime translations, { Q , Q} F . P . The extension of this to a local symmetry can be done by substituting the momentum P, = ia, by the generators of spacetime diffeomorphisms, MI,, and relating them to the supercharges by { Q , Q } I’ . W. The resulting theory has a local supersymmetry algebra which only closes on-shell [45]. As we discussed above, the problem with on-shell symmetries is that they are not likely to survive in the quantum theory. Here we consider the alternative approach of extending the Ads symmetry on the tangent space into a supersymmetry rather than working directly on the spacetime manifold. This point of view is natural if one recalls that spinors are naturally defined relative t o a local frame on the tangent space rather than to the coordinate basis. In fact, spinors provide an irreducible representation for S O ( N ), but not for G L ( N ) ,which describe infinitesimal general coordinate transformations. The basic strategy is to reproduce the 2+1 “miracle” in higher dimensions. This idea was applied in five dimensions [35], as well as in higher dimensions [48-501. N
N
6.4
Assumptions of Standard Supergmvity
Three implicit assumptions are usually made in the construction of standard SUGRA: (i) The fermionic and bosonic fields in the Lagrangian should come in combinations such that they have equal number of propagating degrees of freedom. This is usually achieved by adding to the graviton and the gravitini a number of fields of spins 0,1/2 and 1 [45]. This matching, however, is not necessarily true in Ads space, nor in Minkowski space if a different representation of the Poincark group (e.g., the adjoint representation) is used [43]. The other two assumptions concern the purely gravitational sector and are dictated by economy: (ii) gravitons are described by the Hilbert action (plus a possible cosmological constant), and,
350
(iii) the spin connection and the vielbein are not independent fields but are related through the torsion equation. The fact that the supergravity generators do not form a closed off-shell algebra can be traced back to these assumptions. The argument behind (i) is closely related to the idea that the fields should be in a vector representation of the Poincark group. This assumption comes from the interpretation of supersymmetric states as represented by the in- and out- plane waves in an asymptotically free, weakly interacting theory in a Minkowski background. Then, because the hamiltonian commutes with the supersymmetry generators, every nonzero mass state must have equal number of bosonic and fermionic states: For each bosonic state of energy, IE > B , there is a fermionic one with the same energy, IE > F = Q IE > B ,and vice versa. This argument, however, breaks down if the Poincark group in not a symmetry of the theory, as it happens in an asymptotically Ads space, and in other simple cases such as SUSY in 1+1, with broken translational invariance [51]. Also implicit in the argument for counting the degrees of freedom is the usual assumption that the kinetic terms and couplings are those of a minimally coupled gauge theory, a condition that is not met by a CS theory. Apart from the difference in background, which requires a careful treatment of the unitary irreducible representations of the asymptotic symmetries [52], the counting of degrees of freedom in CS theories is completely different from the counting for the same connection 1-forms in a YM theory (see Lecture 4 below). 7
Super Ads Algebras
In order to construct a supergravity theory that contains gravity with a cosmological constant, a mathematically oriented physicist would look for the smallest superalgebra that contains the generators of the Ads algebra. This was asked -and answered!- many years ago, at least for some dimensions D = 2 , 3 , 4 mod 8, [54]. However this is not all, we would also want to see an action that realizes the symmetry. Constructing a supergravity action for a given dimension that includes a cosmological constant is a nontrivial task. For example, the standard supergravity in eleven dimensions has been know for a long time [55], however, it does not contain a cosmological constant term, and it has been shown to be impossible to accommodate one [56]. Moreover, although it was known to the authors of Ref. [55] that the supergroup that contains the Ads group in eleven dimensions is S0(32)1),no action was found for almost twenty years for the theory of gravity which exhibits this symmetry. An explicit representation of the superalgebras that contain Ads algebra
351
so(D - 1 , 2 ) can be constructed along the lines of [54], although here we consider an extension of this method which applies t o the cases D = 5, 7, and 9 as well [49]. The crucial observation is that the Dirac matrices provide a natural representation of the Ads algebra in any dimension. Then, the Ads connection W can be written in this representation as W = eaJa i W a b J a b , where
+
Jab=
[
i(rab);
0
0 01
Here ra,a = 1,..., D are m x m Dirac matrices, where m = 2ID/'] (here [r]denotes the integer part of r ) , and r a b = $",&,I. These two class of matrices form a closed commutator subalgebra (the AdS algebra) of the Dirac algebra D,obtained by taking antisymmetrized products of r matrices I , r a , r a , a z , ...,ralaz...ao,
(93)
where I'alaz...ak = m(I'alI'a2 . . . rakf [permutations]). For even D these are all linearly independent, but for odd D they are not, because r12...~ = cI and therefore half of them are proportional t o the other half. Thus, the dimension of this algebra is m2 = 22[D/21and not D2 as one could naively think. This representation provides an elegant way to generate all m x m matrices (note however, that m = 2[Ol21 is not any number). 1
7.1
The Fermionic Generators
The simplest extension of the matrices (91, 92) is obtained by the addition of one row and one column. The generators associated t o these entries would have one on spinor index. Let us call Qr the generator that has only one nonvanishing entry in the y-th row of the last column, Qr =
[
0 -crp
sy* 0
]
(94)
Since this generator carries a spinorial index, we will assume it is in a spin 1/2 representation of the Lorentz group. The entries of the bottom row will be chosen so as to produce smallest supersymmetric extensions of Ads. There are essentially two ways of reducing the representation compatible with Lorentz invariance: chirality, which corresponds t o Weyl spinors, and reality,
352
for Mujorunu spinors. A Majorana spinor satisfies a constraint that relates its components to those of its complex conjugate, (95)
= cap$@
The charge conjugation matrix, C = (Cap)is invertible, CapCpr = bya and therefore, it can be used as a metric in the space of Majorana spinors. Since both F a and obey the same Clifford algebra = 2qab), there could be a representation in which the is related t o F a by a change of basis up t o a sign,
({ra,rb}
( f a ) T = qCI'aC-l
with
v2 = 1.
(96) The Dirac matrices for which there is an operator C satisfying (96) is called the Majorana representationg. This last equation is the defining relation for the charge conjugation matrix, and whenever it exists, it can be chosen t o have definite parity,
CT = XC,withA = *l.
(97)
It can be seen that with the choice (94), Majorana conjugate of Q is Q ' = : C"0
Qp
0 =[-ha
7.2
cay
0
]
Closing the Algebra
We already encountered the bosonic generators responsible for the Ads transformations (91, 92), which has the general form required by (87). It is also straightforward t o check that commutators of the form [J,Q] turn out to be proportional t o Q, in agreement with the general form (88). What is by no means trivial is the closure of the anticommutator {Q, Q} as in (89). Direct computation yields
9Chirality is defined only for even D , while the Majorana reality condition can be satisfied in any D , provided the spacetime signature is such that, if there are s spacelike and 1 timelike dimensions, then s - t = 0,1,2,6,7 mod 8 [43,44](that is D = 2,3,4,8,9, mod 8 for lorentzian signature). Thus, only in the latter case Majorana spinors can be defined unambiguously.
353
The form of the lower diagonal piece immediately tells us that unless C x, is antisymmetric, it will be necessary to include at least one more bosonic generator (and possibly more) with nonzero entries in this diagonal block. This relation also shows that the upper diagonal block is a collection of matrices M,x whose components are
(M,x)F
=
-(6,"Cxp
W,d.
i-
Multiplying both sides of this relation by C , one finds (CM,x),p
= -(Ca,CXP
+ CaXC,P),
(101)
which is symmetric in (
[email protected] means that the bosonic generators can only include those matrices in the Dirac algebra such that, when multiplied by C on the left (CI,Cr,, CI',,,,, ..., C r a l a z . . . aturn D ) out to be symmetric. The other consequence of this is that, if one wants to have the AdS algebra as part of the superalgebra, both Cr, and Crab should be symmetric matrices. Now, multiplying (96) by C from the right, we have = Aver,,
(102)
which means that we need AQ = 1.
(103)
It can be seen that =
-Acr,b,
which in turn requires A = -1 = Q. This means that C is antisymmetric (A = -1) and then the lower diagonal block in (100) vanishes identically. However, the values of A and 17 cannot be freely chosen but are fixed by the spacetime dimension as is shown in the following table (see Ref. [50] for details)
m 3 -1 -1 5 -1+l 7 +1-1 9 tltl 11 -1 -1
and the pattern repeats mod 8. This table shows that the simple cases occur for dimensions 3 mod 8, while for the remaining cases life is a little harder. For
354
D = 7 mod 8 the need to match the lower diagonal block with some generators can be satisfied quite naturally by including several spinors labeled with a new index, t,bZp, i = 1,..A,and the generator of supersymmetry should also carry the same index. This means that there are actually N supercharges or, as it is usually said, the theory has an extended supersymmetry ( N 2 2). For D = 5 mod 4 instead, the superalgebra can be made to close in spite of the fact that 77 = +l if one allows complex spinor representations, which is a particular form of extended supersymmetry since now Qr and Q' are independent. So far we have only given some restrictions necessary t o close the algebra so that the Ads generators appear in the anticommutator of two supercharges. In general, however, apart from J, and Jab other matrices will occur in the r.h.s. of the anticommutator of Q and Q which extends the Ads algebra into a larger bosonic algebra. This happens even in the cases where there is no extended supersymmetry ( N = 1). The bottom line of this construction is that the supersymmetric extension of the Ads algebra for each odd dimension falls into three different families: D = 3 mod 8 (Majorana representation, N 2 l), D = 7 mod 8 (Majorana representation, even N ) , and D = 5 mod 4 (complex representations, N 2 1 [or 2N real spinors]). The corresponding superalgebraslowere computed by van Holten and Van Proeyen for D = 2 , 3 , 4 mod 8 in Ref. [54], and in the other cases, in Refs. [49,50]:
8
CS Supergravity Actions
The supersymmetric extension of a given Lie algebra is a mathematical problem that has a mathematical solution, as is known from the general studies of superalgebras [57]. A particularly interesting aspect of these algebras is their representations. The previous discussion was devoted to that point, of which some cases had been studied more than 20 years ago in Ref. [54]. What is ~~
'OThe algebra osp(p1q) (resp. usp(plq)) is that which generates the orthosymplectic (resp. unitary-symplectic) Lie group. This group is defined as the one that leaves invariant the quadratic form G A B L ~=Zg a~b z a x b TmpOm08,where gab is a p-dimensional symmetric (resp. hermitean) matrix and 7ayap is a q-dimensional antisymmetric (resp. anti-hermitean) matrix.
+
355
not at all trivial is how to construct a field theory action that reflects this symmetry. We saw in the previous lecture how to construct CS actions for the Ads connection for any D = 2n 1. The question is now, how to repeat this construction for the connection of a larger algebra in which AdS is embedded. The solution to this problem is well known. Consider an arbitrary connection one form A, with values in some Lie algebra g, whose curvature is F = dA A A A. Then, the 2n-form
+
+
@zn - < F A . . . A F > , (104) where < . . . > stands for an invariant trace, is invariant under the group whose Lie algebra is g. Furthermore, is closed: dC2, = 0, and therefore can be locally written as an exact form, @2n = a 2 n - 1 .
The (2n-l)-form I L s ~ -is~a CS lagrangian, and therefore the problem reduces to finding the invariant trace < . . . >. The canonical -and possibly uniquechoice of invariant trace with the features required here is the supertrace, which is defined as follows: if a matrix has the form
where a , b are (bosonic) tensor indices and a , /3 are (fermionic) spinor indices, then STr[M]= T r [ J ]- T r [ S ]= J," - SE. If we call 6~ the generators of the Lie algebra, so that A = Gn/lA', F = G M F ~then ,
is an invariant tensor of rank n in the Lie algebra. Thus, the where gMl...M, steps to construct the CS lagrangian are straightforward: Take the supertrace of all products of generators in the superalgebra and solve equation (105) for IL2n-1. Since the superalgebras are different in each dimension, the CS lagrangians differ in field content and dynamical structure from one dimension to the next, although the invariance properties are similar in all cases. The action
is invariant, up to surface terms, under the local gauge transformation
356
where A is a zero-form with values in the Lie algebra 0 , and V is the exterior covariant derivative in the representation of A. In particular, under a supersymmetry transformation, A = 2Qi - Qici, and
where D is the covariant derivative on the bosonic connection,
Two interesting cases can be mentioned here: A. D=5 SUGRA In this case the supergroup is U ( 2 , 2 ( N ) .The associated connection can be written as 1 A=eaJa - W a b J a b A K T ~ (drQr AZ, (109) 2 where the generators J , , .Jab, form an AdS algebra ( s o ( 4 , 2 ) ) ,T K ( K = 1,.. . N 2 - 1) are the generators of s u ( N ) , Z generates a U(1) subgroup and Q, Q are the supersymmetry generators, which transform in a vector representation of S U ( N ) . The Chern-Simons Lagrangian for this gauge algebra is defined by the relation dL = iSTr[IF3],where IF = dA A2 is the (antihermitean) curvature. Using this definition, one obtains the Lagrangian originally discussed by Chamseddine in [35],
+
+
+
a'&) +
+
+
L = LG(uab,ea)+ L , , ( N ) ( A ~ ) L,(q(uable a , A ) + Lp(uab,ea, A:, A , + r ) , (110)
with
= &bc& [RabRcdee/l + $Rabecedee/13+ &eaebecedee/15] LG L,,(N) = -Tr [ A ( d A ) 2+ 3A3dA + ?A5] L,(1) = ($ - h)A(dA)3 $-[TaTa- Rabeaeb- l2RabRab/2]A , ( 1 1 1 ) +~F;F;A Lf = ;?i g N [+'RV+T GS~lv+r]C.C.
+
+
+
where A: = A K ( T ~ )is: the s u ( N ) connection, F,' is its curvature, and the bosonic blocks of the supercurvature: R = ~ T a l ? , + ~ ( R a b + e a e b ) r a b + ~ d A I 3;= F,' SdAb; - $@+s. The cosmological constant is -1-2, and the AdS covariant derivative V acting on is
i+s$s,
+
+,.
357
where D is the covariant derivative in the Lorentz connection. The above relation implies that the fermions carry a ~ ( 1“electric” ) charge given by e = The purely gravitational part, LG is equal t o the standard Einstein-Hilbert action with cosmological constant, plus the dimensionally continued Euler density”. The action is by construction invariant -up t o a surface term- under the local (gauge generated) supersymmetry transformations ~ A = A - (dA+ [A,A]) with A = FQ,. - Q r c T , or
(2 h).
bea = 1 (ma$,. - @ye,.) bWab = _ _ (Trab+,.- + r r a b e T ) bA‘, = -i (FgSb$,. = -VET 61c1’ = - v 7 bA = -i - Per).
;
As can be seen from (111) and (112), for N = 4 the U(1) field A looses its kinetic term and decouples from the fermions (the gravitino becomes uncharged with respect t o U(1)). The only remnant of the interaction with the A field is a dilaton-like coupling with the Pontryagin four forms for the Ads and S U ( N ) groups (in the bosonic sector). As it is shown in Ref. [58], the case N = 4 is also special a t the level of the algebra, which becomes the superalgebra ~ ~ ( 2 , 2 1 with 4 ) a ~ ( 1 central ) extension. In the bosonic sector, for N = 4, the field equation obtained from the variation with respect to A states that the Pontryagin four form of Ads and S U ( N ) groups are proportional. Consequently, if the spatial section has no boundary, the corresponding Chern numbers must be related. Since IT4(SU(4)) = 0, the above implies that the Pontryagin plus the Nieh-Yan number must add up t o zero. B. D = l l SUGRA In this case, the smallest AdS superalgebra is osp(3211) and the connection is
“The first term in LG is the dimensional continuation of the Euler (or Gauss-Bonnet) density from two and four dimensions, exactly as the three-dimensional Einstein-Hilbert Lagrangian is the continuation of the the two dimensional Euler density. This is the leading term in the limit of vanishing cosmological constant (1 ---t cm),whose local supersymmetric extension yields a nontrivial extension of the Poincare group [48].
358
where Aabcdeis a totally antisymmetric fifth-rank Lorentz tensor one-form. Now, in terms of the elementary bosonic and fermionic fields, the CS form in lL2,- 1 reads j y11 P(3211)
(4= LI;(32’(n) + L f ( % $4,
= $(eara + ;Wabrab + $Aabcderabcde)
where bosonic part of (114) can be written as
=2-
L ”11 P(32)
(114)
is an sp(32) connection. The
6L G A d1 S1 ( W , e ) - Z L T 1 1A(dW S 7e)+L~1(A,w7e),
where Lid; is the CS form associated to the 12-dimensional Euler density, and L$’: is the CS form whose exterior derivative is the Pontryagin form for SO(10,2) in 12 dimensions. The fermionic Lagrangian is
Lf = 6 ( 4 R 4 W - 3 [ ( W W+ (4R+)] (W2W -3 [(4R3+)+ P 4 R 2 W J )(]4 W + 2 + (4R+I2+ ( $ R + ) ( m m J ) ] where R = df-2 + n2is the sp(32) curvature. The supersymmetry transfor-
[(mw
(4m%
mations (108) read
6ea =
S+
pa+
= DE
bWab
= -;crab+
6Aabcde =
1-
SEr
abcde
+’
Standard (CJS) eleven-dimensional supergravity [55]is an N=l supersymmetric extension of Einstein-Hilbert gravity that cannot admit a cosmological constant [56,64]. An N > 1 extension of the CJS theory is not known. In our case, the cosmological constant is necessarily nonzero by construction and the extension simply requires including an internal so(N) gauge field coupled to the fermions. The resulting Lagrangian is an osp(32JN)CS form [59]. 9
Summary
The supergravities presented here have two distinctive features: The fundamental field is always the connection A and, in their simplest form, they are pure CS systems (matter couplings are discussed below). As a result, these theories possess a larger gravitational sector, including propagating spin connection. Contrary to what one could expect, the geometrical interpretation is quite clear, the field structure is simple and, in contrast with the standard cases, the supersymmetry transformations close off shell without auxiliary fields.
359
Torsion. It can be observed that the torsion Lagrangians, LT, are odd while the torsion-free terms, LG, are even under spacetime reflections. The minimal supersymmetric extension of the Ads group in 4k - 1 dimensions requires using chiral spinors of SO(4k) [60]. This in turn implies that the gravitational action has no definite parity and requires the combination of LT and LG as described above. In D = 4k 1 this issue doesn't arise due t o the vanishing of the torsion invariants, allowing constructing a supergravity theory based on LG only, as in [35]. If one tries t o exclude torsion terms in 4k - 1 dimensions, one is forced to allow both chiralities for SO(4k) duplicating the field content, and the resulting theory has two copies of the same system [61].
+
Field content and extensions with N>1. The field content compares with that of the standard supergravities in D = 5 , 7 , 1 1 in the following table, which shows the corresponding supergravities
ID IStandard suDernravitvI CS suDereravitv I Algebra I 5 e; wEb A , ASp $15$&, , i, j = 1, ...N usp(2,2IN) e; $;I $ a p 7 e; Ap] aLJ A" q5 $7 e; WEb A;? $,';I i , j = 1, ...N = 2n osp(NI8) ea wab Afbcded? i. I = 1. ...N 11 osv(321N) ef Arqi d?
.
Standard supergravity in five dimensions is dramatically different from the theory presented here, which was also discussed by Chamseddine in [35]. Standard seven-dimensional supergravity is an N = 2 theory (its maximal extension is N = 4), whose gravitational sector is given by Einstein-Hilbert gravity with cosmological constant and with a background invariant under OSp(218) [62,63]. Standard eleven-dimensional supergravity [55] is an N = 1 supersymmetric extension of Einstein-Hilbert gravity with vanishing cosmological constant. An N > 1 extension of this theory is not known. In our construction, the extensions t o larger N are straightforward in any dimension. In D = 7, the index i is allowed t o run from 2 t o 2s, and the Lagrangian is a CS form for osp(2s18). In D = 11, one must include an internal s o ( N ) field and the Lagrangian is an osp(321N ) CS form [49,50]. The cosmological constant is necessarily nonzero in all cases. Spectrum. The stability and positivity of the energy for the solutions of these theories is a highly nontrivial problem. As shown in Ref. [53], the number of degrees of freedom of bosonic CS systems for D 2 5 is not constant throughout phase space and different regions can have radically different dynamical content. However, in a region where the rank of the symplectic form is maximal the theory may behave as a normal gauge system, and this condition would be stable under perturbations. As it is shown in [58] for D = 5, there exists a nontrivial extension of the AdS superalgebra with a central extension
360
in anti-de Sitter space with only a nontrivial U(1) connection but no other matter fields. In this background the symplectic form has maximal rank and the gauge superalgebra is realized in the Dirac brackets. This fact ensures a lower bound for the mass as a function of the other bosonic charges [65]. Classical solutions. The field equations for these theories, in terms of the Lorentz components ( w , e, A , A, $), are the different Lorentz tensor components for < I F n - l G ~>= 0. It is rather easy t o verify that in all these theories the anti-de Sitter space is a classical solution , and that for $ = A = A = 0 there exist spherically symmetric, asymptotically AdS standard [37], as well as topological black holes [66]. In the extreme case these black holes can be shown t o be BPS states [67].
Matter couplings. It is possible to introduce minimal couplings to matter of the form A.Jezt. For D = 5, the theory couples t o an electrically charged U(1) 0 brane (point charge), t o SU(4) -colored 0 branes (quarks) or t o uncharged 2-brane, whose respective worldhistories couple t o A,, A 7 and wEb respectively. For D = 11, the theory admits a 5-brane and a 2-brane minimally coupled to AZbcdeand wEb respectively. Standard SUGRA. Some sector of these theories might be related to the standard supergravities if one identifies the totally antisymmetric part of wEb in a coordinate basis, kPyA, (sometimes called the contorsion tensor) with the abelian 3-form, Ap]. In 11 dimensions one could also identify the totally antisymmetrized part of AFbcdewith an abelian 6-form Ais], whose exterior derivative, dAp1, is the dual of F[41 = dAp1. Hence, in D = 11 the CS theory may contain the standard supergravity as well as some kind of dual version of it. Gravity sector. A most remarkable result from imposing the supersymmetric extension, is the fact that if one sets all fields, except those that describe the geometry -ea and wab- t o zero, the remaining action has no free parameters. This means that the gravity sector is uniquely fixed. This is remarkable because as we saw already for D = 3 and D = 7, there are several CS actions that one can construct for the Ads gauge group, the Euler CS form and the so-called exotic ones, that include torsion explicitly, and the coefficients for these different CS lagrangians is not determined by the symmetry considerations. So, even from a purely gravitational point of view, if the theory admits a supersymmetric extension, it has more predictive power than if it does not.
36 1
LECTURE 4 EPILOGUE: DYNAMICAL CONTENT of CHERN SIMONS THEORIES The physical meaning of a theory is defined by the dynamics it displays both at the classical and quantum levels. In order t o understand the dynamical contents of the classical theory, the physical degrees of freedom must be identified. In particular, it should be possible -at least in principle- to separate the propagating modes from the gauge degrees of freedom, and from those which do not evolve independently at all (second class constraints). The standard way to do this is Dirac's constrained Hamiltonian analysis and has been applied t o CS systems in [53]. Here we summarize this analysis and refer the reader to the original papers for details. It is however, fair t o say that a number of open problems remain and it is a area of research which is at a very different stage of development compared with the previous discussion. 10
Hamiltonian Analysis
From the dynamical point of view, a CS system can be described by a Lagrangian of the form12
where the (2n and
+ 1)-dimensional spacetime has been split into space and time, Ka
1 -
.
2nnYaal....a,
.
€21 , . . 2 2 n
F2",:, .
The field equations are fiyb(A9
- D j A : ) = 0,
Ka = 0, where
12Note that in this section, for notational simplicity, we assume the spacetime to be (2n+l)dimensional.
362
is the symplectic form. The passage t o the Hamiltonian has the problem that the velocities appear linearly in the Lagrangian and therefore there are a number of primary constraints
Besides these, there are secondary constraints Ka combined with the $s into the expressions
Ga
-Ka
M
0, which can be
+ Diy,.
(120)
The complete set of constraints forms a closed Poison bracket algebra,
{&,4);
= Qyb
{&,Gb) = f i b 4 : {Ga,Gb) = f i b G =
7
where f&, are the structure constants of the gauge algebra of the theory. Clearly the Gs form a first class algebra which reflects the gauge invariance of the theory, while some of the 4s are second class and some are first class, depending on the rank of the symplectic form 0.
10.1
Degeneracy
An intriguing aspect of Chern-Simons theories is the multiplicity of ground states that they can have. This can be seen from the field equations, which for D = 2n 1, are polynomials of degree n which in general have a very rich root structure. As the symplectic form is field-dependent, the rank of the matrix f l z b need not be constant. It can change from one region of phase space to another, with different degrees of degeneracy. Regions in phase space with different degrees of degeneracy define dynamically distinct and independent effective theories [68]. If the system reaches a degenerate configuration, some degrees of freedom are frozen in an irreversible process which erases all traces of the initial conditions of the lost degrees of freedom. One can speculate about the potential of this phenomenon as a way t o produce dimensional reduction through a dynamical process. This issue was analyzed in the context of some simplified mechanical models and the conclusion was that the degeneracy of the system occurs at submanifolds of lower dimensionality in phase space, which are sets of unstable initial states or sets of stable end points for the evolution [68]. Unless the system is chaotic, it can be expected that generic configurations,where the rank of is maximal, fill most of phase space. As it was shown in Ref. [68], if the system evolves along an orbit that reaches a surface of degeneracy,
+
363
C, it becomes trapped by the surfece and loses the degrees o freedom that correspond t o displacements away from C. This is an irreversible process which can be viewed as mechanism for dynamical reduction of degrees of freedom or dimensional reduction. A process of this type is seen to take place in the dynamics of vortices in a fluid, where two vortices coalesce and annihilate each other in a n irreversible process.
10.2
Generic Counting
There is a second problem and that is how t o separate the first and second class constraints among the 4s. In Ref. [53]the following results are shown: is 2 n N - 2 n , where N is the number of genThe maximal rank of erators in the gauge Lie algebra. There are 2 n first class constraints among the 4s which correspond to the generators of spatial diffeomorphisms ('Hi). The generator of timelike reparametrizations 7 - l ~is not an independent first class constraint. Putting all these facts together one concludes that, in a generic configuration, the number of degrees of freedom of the theory (Ccs) is
cCs = (number of coordinates)
- (number of 1st class constraints) 1 --(number of 2nd class constraints) 2 1 = 2nN - ( N 2n) - -(2nN - 2n) 2 = nN - N - n .
+
This result is somewhat perplexing. A standard (metric) Lovelock theory of gravity in D = 2 n 1 dimensions, has
+
= (2n
+ l)(n
-
1)
propagating degrees of freedom [28].A CS gravity system for the Ads group in the same dimension gives a much larger number,
cCs = 2n3 + n2 - 3 n - 1. In particular, for D = 5 , cCs = 13, while
(123) = 5. The extra degrees
of freedom correspond t o propagating modes in wab, which in the CS theory are independent from the metric ones contained in ea.
364
As it is also shown in [53], an important simplification occurs when the group has an invariant abelian factor. In that case the symplectic matrix flybtakes a partially block-diagonal form where the kernel has the maximal size allowed by a generic configuration. It is a nice surprise in the cases of CS supergravities discussed above that for certain unique choices of N , the algebras develop an abelian subalgebra and make the separation of first and second class constraints possible (e.g., N = 4 for D = 5, and N = 32 for d = 11). In some cases the algebra is not. a direct sum but an algebra with an abelian central extension (D = 5). In other cases, the algebra is a direct sum, but the abelian subgroup is not put in by hand but it is a subset of the generators that decouple from the rest of the algebra ( D = 11). 10.3
Regularity Conditions
The counting discussed in [53] was found t o fail in the particular example of CS supergravity in 5 dimensions. This is due t o a different kind of difficulty: the fact that the symmetry generators (first class constraints) can fail t o be functionally independent at some points of phase space. This is a second type of degeneracy and makes it impossible to approximate the theory by a linearized one. In fact, it can be seen that the number of degrees of freedom of the linearized theory is larger than in the original one [58]. This is the subject of an ongoing investigation which will be reported elsewhere [69].
Final Comments 1. Everything we know about the gravitational interaction at the classical level, is described by Einstein’s theory in four dimensions, which in turn is supported by a handful of experimental observations. There are many indications, however, that make it plausible to accept that our spacetime has more dimensions than those that meet the eye. In a spacetime of more than four dimensions, it is not logically necessary t o consider the Einstein-Hilbert action as the best description for gravity. In fact, string theory suggests a Lanczos-Lovelock type action as more natural [27]. The large number of free parameters in the LL action, however, cannot be fixed by arguments from string theory. As we have shown, the only case in which there is a simple symmetry principle to fix these coefficients is odd dimensions and that leads t o the Chern-Simons theories. 2. The CS theories of gravity have a profound geometrical meaning that relates them t o topological invariants -the Euler and the Chern or Pontryagin classes- and come about in a very natural way in a framework where the affine
365
and metric structures of the geometry are taken t o be independent dynamical objects. If one demands furthermore the theory t o admit supersymmetry, there is, in each dimensions essentially a unique extension which completely fixes the gravitational sector, including the precise role of torsion in the action. 3. The CS theories of gravity obtained are classically and semiclassically interesting. They possess nontrivial black hole solutions [40] which asymptotically approach spacetimes of constant negative curvature (Ads spacetimes). These solutions have a thermodynamical behavior which is unique among all possible black holes in competing LL theories with the same asymptotics 1701. These black holes have positive and can therefore always reach thermal equilibrium with their surroundings. These theories also admit solutions which represent black objects, in the sense that they possess a horizon that hides a singularity, but the horizon topology is not spherical but a surface of constant nonpositive Ricci curvature [72]. Furthermore, these solutions seem to have a well defined, quantum mechanically stable ground states [67] which have been shown t o be BPS states of diverse topologies. 4. We have no way of telling a t present what will be the fate of string theory as a description of all interactions and constituents of nature. If it is the right scenario and gravity is just a low energy effective theory that would be a compelling reason t o study gravity in higher dimensions, not as an academic exercise as could have seemed in the time of Lanczos, but as a tool t o study big bang cosmology or black hole physics for instance. The truth is that a field theory can tell us a lot a bout the low energy phenomenology, in the same way that ordinary quantum mechanics tells us a lot about atomic physics even if we know that is all somehow contained in QED. 5 . Chern-Simons theories contain a wealth of other interesting features, starting with their relation t o geometry, gauge theories and knot invariants. The higher-dimensional CS systems remain somewhat mysterious especially because of the difficulties to treat them as quantum theories. However, they have many ingredients that make CS theories likely models t o be quantized: They carry no dimensionful couplings, the only parameters they have are quantized, they are the onIy ones in the Lovelock family of gravity theories that give rise t o black holes with positive specific heat [70] and hence, capable of reaching thermal equilibrium with an external heat bath. Efforts t o quantize CS systems seem promising at least in the cases in which the space admits a complex structure so that the symplectic form can be cast as a Kahler form [71]. However, there is a number of open questions that one needs t o address before CS theories can be applied t o describe the microscopic world, like their Yang-Mills relatives. Until then, they are beautiful mathematical models and interesting physical systems worth studying.
366 6. If the string scenario fails to deliver its promise, more work will still be needed to understand the field theories it is supposed to represent, in order to decipher their deeper interrelations. In this case, geometry is likely to be an important clue, very much in the same way that it is an essential element in Yang Mills and Einstein’s theory. One can see the construction discussed in these lectures as a walking tour in this direction.
It is perhaps appropriate to end these lectures quoting E. Wigner in full
PI:
”The miracle of appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it, and hope that it will remain valid for future research, and that it will extend, for better or for worse, to our pleasure even though perhaps also to our bafflement, to wide branches of learning”.
ACKNOWLEDGMENTS It is a pleasure for me to thank the organizers and the staff of the school for the stimulating discussions and friendly atmosphere in the charming colonial city of Villa de Leyva. I am especially thankful for the efforts of M. Kovacsics to make sure that a vegetarian would not only survive but be treated with best fruits of the land. I wish to thank R. Aros, J. Bellissard, C. Martinez, S. Theisen, C. Teitelboim and R. Troncoso for many enlightening discussions and helpful comments, and F. Mansouri for pointing out to me a number of interesting references. I am especially grateful to Sylvie Paycha for carefully proof reading and correcting many mistakes in the original manuscript, both in style and in mathematics. I wish also thank the Albert Einstein Institute for Gravitational Physics of the Max Planck Institute at Potsdam for the hospitality, encouragement and resources to write up these notes. This work was supported in part by FONDECYT grants 1990189, 1010450, 1020629 and by the generous institutional support to CECS from Empresas CMPC. CECS is a Millennium Science Institute.
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Rev.D53 (1996) R593-596; The Dynamical Structure of Higher Dimensional Chern-Simons Theory, Nucl. Phys.B476 (1996) 611-635. 54. J.W. van Holten and A. Van Proeyen, N=l Supersymmetry Algebras in D=2, D=3, D=4 mod-8, J. Phys. A15 (1982) 3763-3779. 55. E. Cremmer, B. Julia, and J. Scherk, Supergravity Theory in Eleven Dimensions, Phys. Lett. B76 (1978) 409-412. 56. K.Bautier, S.Deser, M.Henneaux and D.Seminara, N o cosmological D = l l Supergravity, Phys.Lett. B406 (1997) 49-53. 57. V. G. Kac, A Sketch of Lie Superalgebra Theory, Comm. Math. Phys. 53(1977) 31-64. 58. 0. Chandia, R. Troncoso and J. Zanelli, Dynamical Content of ChernSimons Supergravity, Second La Plata Meeting on Trends in Theoretical Physics, Buenos Aires, (1998), H. Falomir, R.E.Gamboa Saravi and F.A.Schaposnik, editors, American Institute of Physics (1999) hepth/9903204. 59. R. Troncoso, Supergravedad en Dimensiones Impares, Doctoral Thesis, University of Chile, Santiago (1996). 60. M. Gunaydin and C. Saclioglu, Oscillator-like Unitary Representations of Noncompact Groups with a Jordan Structure and the Noncompact Groups of Supergravity, Comm. Math. Phys. 87 (1982) 159-174. 61. P. Horava, M Theory as a Holographic Field Theory, Phys. Rev. D59 (1999) 046004. 62. P. K. Townsend and P. van Nieuwenhuizen, Gauged Seven-Dimensional Supergravity, Phys. Lett. 125B(1983) 41-45. 63. A. Salam and E. Sezgin, SO(4) Gauging of N=2 Supergravity in Seven Dimensions, Phys. Lett. 126B (1983) 295-300. 64. S. Deser, Uniqueness of D = 11 Supergravity, in Black Holes and the Structure of the Universe, C. Teitelboim and J. Zanelli, editors, World Scientific, Singapore (2000), hep-th/9712064. 65. G. W. Gibbons and C. M. Hull, A Bogomolny Bound for General Relativi t y and Solitons in N = 2 Supergravity, Phys. Lett. B109 (1982) 190-196. 66. S. Aminneborg, I. Bengtsson, S. Holst and P. Peldan, Making anti-de Sitter Black Holes, Class. Quantum Grav.13 (1996) 2707-2714. M. Baiiados, Constant Curvature Black Holes, Phys. Rev. D57 (1998), 1068-1072. R.B. Mann, Topological Black Holes: Outside Looking In, e-print gr-qc/9709039. 67. R. Aros, C. Martinez, R. Troncoso and J. Zanelli,Supersymmetry of Gravitational Ground States, Jour. High Energy Phys. JHEP 05 (2002) 020035. 68. J. Saavedra, R. Troncoso and J. Zanelli, Degenerate Dynamical Systems,
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Geometric and Topological Methods for Quantum Field Theory Eds. A. Cardona, H. Ocampo & S. Paycha @ 2003 World Scientific Publishing, pp. 372-434
INTRODUCTORY LECTURES ON STRING THEORY AND THE ADS/CFT CORRESPONDENCE
ARI PANKIEWICZ AND STEFAN THEISEN* Max-Planck-Institut fur Grauitationsphysik, Albert-Einstein-Institut, A m Muhlenberg 1,D-1&76 Golm, Germany E-mail: apankie,
[email protected] The first lecture is of a qualitative nature. We explain the concept and the uses of duality in string theory and field theory. The prospects t o understand QCD, the theory of the strong interactions, via string theory are discussed and we mention the AdS/CFT correspondence. In the remaining three lectures we introduce some of the tools which are necessary to understand many (but not all) of the issues which were raised in the first lecture. In the second lecture we give an elementary introduction t o string theory, concentrating on those aspects which are necessary for understanding the AdS/CFT correspondence. We present both open and closed strings, introduce D-branes and determine the spectra of the type I1 string theories in ten dimensions. In lecture three we discuss brane solutions of the low energy effective actions, the type I1 supergravity theories. In the final lecture we compare the two brane pictures - D-branes and supergravity branes. This leads to the formulation of the Maldacena conjecture, or the AdS/CFT correspondence. We also give a brief introduction to the conformal group and Ads space.
Lecture 1: Introduction There are two central open problems in theoretical high energy physics: the search for a quantum theory of gravity and the solution of QCD at low energies. The first problem is apparent if one considers the Einstein equations which couple the dynamics of the gravitational field to that of matter and radiation. *Based on lectures given in July 2001 at the Universidad Simon Bolivar and a t the Summer School on ”Geometric and Topological Methods for Quantum Field Theory” in Villa de Leyva, Colombia.
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Since matter and radiation follow the laws of quantum mechanics this must also be the case for the gravitational field. If one applies the methods of perturbative quantum field theory, which have been very successful for the electromagnetic, the weak and the strong interactions, t o the theory of gravity as formulated by Einstein in his general theory of relativity, one gets stuck at a problem which is often stated as the non-renormalizability of a quantum field theory of gravity. What is meant by the second problem is that we have no analytic tools t o prove e.g. the existence of a mass gap in QCD and the phenomenon of quark confinement, i.e. the fact that a t low energies there are neither massless gluons, the gauge particles of the strong interactions, nor free quarks, but rather there are massive Lcolourless’mesons and baryons; for a precise description of the problem, see [l]. Solutions t o both problems have been proposed which stay within the realm of theoretical concepts which have proven successful in the past. There is e.g. the approach of Ashtekar to the canonical quantization of gravity (see e.g. [2] for reviews) and there are lattice simulations of QCD [3]. We will not say anything about these approaches. Instead we take the (unproven) point of view that in order t o solve these two problems, we have to go beyond known and established theoretical frameworks and introduce new ones. This would be in line with the history of physics where apparent conflicts forced completely new lines of thought upon us. As an example consider the special theory of relativity which arose from reconciling discrepancies between the predictions from Newtonian mechanics and electromagnetism, or quantum field theory which combines special relativity and quantum mechanics. We consider this an optimistic point of view since, if true, it would eventually provide new and exciting theories along with their ramifications and implications. Our basic assumption for these lectures is that string theory is the solution to both problems. We will, however, not discuss at any length why string theory is believed to be a consistent (perturbative) theory of quantum gravity. Our goal is rather t o provide the background for understanding the evidence for a duality between
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a s h h g theory
Duality means an exact quantum equivalence of the two theories, which thus really represent only one theory, albeit in very different guises.
To establish such a duality, we must (1)identify the pair of theories which are proposed to be dual to each other and (2) find the duality map H.
A duality between two theories A and B is most useful if we can learn about the non-perturbative behavior (strong coupling) of one theory from the computable perturbative behavior (weak coupling)of the other. Schematically
perturbative region of A
perturbative region of B "couplhg constant space of A"
Of course, t o establish such a duality is very difficult and one will be mostly, at least for the time being, working at the level of conjectures. However, in concrete examples one has gathered compelling evidence for the duality conjecture and one has performed non-trivial tests.
A well known example is the particle-wave duality of quantum mechanics:
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depending on the experiment, either the particle or the wave aspect of light or matter gives the simpler description. An example from (two-dimensional) quantum field theory is the duality between the sineGordon model and the massive Thirring model (41. More recent examples are the Olive-Montonen duality of n/ = 4 supersymmetric Yang-Mills theory [5] and various (perturbative and non-perturbative) string dualities [6]. Except for the first example, the duality is between two theories of the same kind, e.g. a duality between two string theories. The duality t o be discussed in these lectures is not of this type but a duality between a gauge field theory and a string theory:
(wcakiy coupled)
in large N limit
This dual pair was first conjectured by Maldacena in 1997 [7]. For reasons that will be become clear later, it is known as the AdS/CFT correspondence. The duality map was constructed by Gubser, Klebanov and Polyakov [8] and by Witten [9]. Note that the gauge theory does not contain gravity whereas the type IIB string does not contain gauge degrees of freedom in its (perturbative) spectrum. Another feature of the AdS/CFT correspondence is the fact that the two theories which are dual t o each other are formulated in different numbers of dimensions: a four-dimensional gauge theory in Minkowski space and a string theory compactified on Ads5 x S5. One implication of this duality is that all information of the string theory is encoded in the lower-dimensional field theory. With reference t o a similar phenomenon in optics, this is called holography. We will return to it below. A word of caution to prevent confusion: in these lectures we do not discuss gauge theory as a low-energy effective field theory of, say, the heterotic string. The relation between string theory and gauge theory which is implied by the AdS/CFT correspondence is more subtle.
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The idea that a gauge theory has a description as a string theory is in fact an old one. At low energies, QCD is a confining theory. This means that one sees neither free gluons nor quarks but mesons (qq) and baryons (qqq). There is a linear - rather than Coulomb like - potential between a quark and an anti-quark, V ( L ) = aL. The chromoelectric flux lines are confined t o a flux tube or string. a is the tension of the QCD string.
9
t
9
chromoelectric flux line
The hope is now that this can be described by a string theory. That this hope is not completely futile can be seen if one considers a gauge theory with gauge group S U ( N ) for large N . Rather than making a perturbation series in a small coupling constant, which does not exist for QCD at low energies where it is strongly coupled, one makes an expansion in powers of 1/N. This was first done by 't Hooft [lo] (for reviews, see [11,12,13]) who showed that all Feynman diagrams which contribute t o a given order in 1/N can be drawn (without any lines crossing) on a Riemann surface whose Euler number x = 2 - 29 (g being the genus) is precisely the power of N t o which the diagram contributes. 2 g - 2 holes
We will see in lecture 2 that this is very much like the perturbation series of string theory where the expansion parameter is gs, the string coupling constant, rather than 1/N. The Riemann surface is the world-sheet of the
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string. In the figure on the previous page we have shown only surfaces without boundary which are the only ones occurring in a pure gauge theory without quarks in the fundamental representation. In the string theory this corresponds t o a loop expansion of the vacuum amplitude. One can indeed show that fundamental strings reproduce some of the features of the physics of strong interactions. The excitations of the string satisfy a linear relation between their = M 2 and their spin J : M 2 = J/a' const, where Q' is related t o the string tension which becomes a fundamental dimensionful parameter in the theory. This was in qualitative agreement with the Regge trajectories for hadronic resonances which were found in experiments, provided a' was chosen 1/&? 100 MeV, which is the typical energy scale of strong interaction physics:
+
N
Soon after these results were obtained it was realized that string theory could not correctly reproduce the high energy behaviour of hadronic scattering amplitudes. In addition it was observed that the spectrum of the string contains a massless spin two particle with many of the properties of the graviton, the exchange particle which mediates the gravitational force. At around the same time, quantum-chromodynamics (QCD) was developed as a gauge theory of the strong interaction, with gauge group S U ( 3 ) . For these, and other reasons, string theory was abandoned as a theory of the strong interactions and was elevated t o a candidate for a theory of quantum gravity. The natural energy scale is now l/&? 1019 GeV, the Planck energy. This was around 1975. N
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If one attempts t o quantize string theory in four-dimensional Minkowski space-time one finds a dependence on the size of the world-sheet, and this dependence enters as a new field which can be interpreted as an additional space-time coordinate. This means that we need a t least five dimensions for a consistent quantization of string theory [14]. If we want all coordinates to span a d-dimensional Minkowski space-time, we find d = 26 for the bosonic string and d = 10 for the fermionic string. These are the critical dimensions. We now return t o the concept of holography. One interesting physical system where it is realized is a black hole. The Schwarzschild solution of the vacuum Einstein equations, Rpv = 0, is the simplest example. This solution depends on one parameter, the mass M of the black hole. Classically, black holes are black, but due to quantum processes they emit so-called Hawking radia. tion. They radiate like a black body with temperature TBH= As radiating systems black holes are expected t o obey the laws of thermodynamics. If one defines the black hole entropy, as first proposed by Bekenstein and Hawking, as S = 47rM2 , one indeed verifies e.g. the second law d(Mc2) = TBHdS. A quantum theory of gravity should provide the framework for a microscopic computation of the black hole entropy. In the search for such a theory one might turn the logic around and start from the expression for the black hole entropy and try t o deduce certain properties that the quantum theory of gravity must possess in order to lead t o such an entropy formula. The simple fact that S = : A , where A = 47r ( 2 : ? N ) 2 is the area of the black hole horizon, leads to the concept of holography. The information contained inside the region enclosed by the horizon is represented as a hologram on the horizon: all the information about the inside is stored on the holographic screen.’ This is in sharp contrast with what we expect from statistical mechanics and local quantum field theory where the entropy is an extensive quantity and thus should be proportional to the volume of the
& [&]
[w]
[HI
‘More generally, the holographic principle asserts that the information contained in some region of space can be represented as a ‘hologram’ - a theory which ‘lives’ on the boundary of that region. It furthermore asserts that the theory on the boundary of the region of space in question should contain at most one degree of freedom per Planck area h. The black hole precisely satisfies that bound. See [15,16] for reviews of the holographic principle.
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system. The lesson we learn from this is that the nature of the degrees of freedom of quantum gravity is quite different from that of a local quantum field theory. In fact, string theory provides a microscopic account of the states and thus entropy of certain (extremal and near-extremal) black holes [17,18].
-
Let us now suppose that a d-dimensional quantum field theory ‘lives’ on the horizon of a (d 1)-dimensional black hole with entropy SQFT Ad. One wonders how quantum gravity in (d 1) dimensions can be related to a local QFT in d dimensions. The AdSd+l/CFTd correspondence gives an answer t o this question. This correspondence goes far beyond the matching of entropies. It is conjectured t o be a n exact duality in the sense described before.
+
+
In the remaining three lectures we provide the necessary background t o understand the conjecture. This requires a crash course in string theory and supergravity branes. Much more can be said about any of the issues that we touch upon. Good referencs for string theory are [19,20,21,22,23]. Dbranes are reviewed in [24,25,26,27],brane solutions of supergravity theories in [28,29] and the AdS/CFT correspondence in [30,31,32,33,34,35].It migth also be fun t o brouse through the ‘official string theory web-site’ [36]. Lecture 2: Elementary introduction to string theory
In this lecture we provide some aspects of string theory needed t o introduce the AdS/CFT correspondence. We first discuss the bosonic string and then the fermionic string. There are many aspects of string theory which we will not mention a t all. Some of the major omissions are conformal field theory, heterotic and type I strings, compactifications, string dualities, orientifolds, etc. For this we have to refer to the literature [20,23,19,21,22,27]. We start by comparing the classical mechanics of a zero-dimensional object - a relativistic point particle - and a one-dimensional object - a string - moving in D-dimensional Minkowski space-time with metric vpv = d i a g ( - , + , + , . . . ,+), p , v = O , ..., D - 1 . As a particle moves through space-time it sweeps out a one-dimensional world-line C whose embedding in space-time is specified by D functions
380
X p ( r ) , r being an arbitrary parameterization of the world-line. The simplest PoincarBinvariant action that does not depend on the parameterization is
s,,
=
-m.J,
dr
d z z ,
(1)
the integrand being the infinitesimal path length, X p = & X p , X . X = X ~ X ” Q ~ ,c, is, the speed of light and m is the particle’s mass as can be seen from the nonrelativistic limit. Analogously, a one-dimensional object sweeps out a two-dimensional world-sheet C in space-time and its embedding is described by D functions X P ( a , r ) . Again, physics must depend only on the embedding and not on the parameterization of the world-sheet. The simplest invariant action - the Nambu-Goto action - is (X’ = a o X )
SNG= -Tc
I=
drda
d
x = -Tc]=
d r d a d ( X .X ’ ) 2 - X2 . X I 2 .
(2) T is the string tension, a new fundamental constant of nature of dimension massllength and hap is the induced metric on the world-sheet
(2) is the straightforward generalization of (1)to an extended object. It is the area of its world-sheet. One distinguishes between open and closed strings. The world-sheet of a free closed string has the topology of a cylinder, that of a ’There are three fundamental constants of nature which, in pre-string physics, are the speed of light c, the gravitational constant G N ( N for Newton) and Planck’s constant fi. For a discussion about the number of fundamental constants, see [37]. The limit fi -+ 0 is called the classical limit, c -+ 00 the non-relativistic limit and GN + 0 the decoupling of gravity. From these three constants one can construct the so-called Planck-units, namely ), (tp = and mass three fundamental scales, for length ( l p = J G ~ f i / c ~ time ( m p = J-). In string theory the tension replaces GN as a fundamental constant with the relation (valid in four dimensions) GN c’/T. One also introduces the string scale a’ = 1: via the relation T = h/(2.rrca’).1, is the typical length scale in string theory At infinite tension, the instead of l p . The characteristic energy scale is E, = extension of the string becomes zero. The limit T -+ 00 or, equivalently, a’ -+ 0 is called the point particle or field theory limit.
d m )
N
m.
38 1
free open string the topology of a strip. Interactions are taken into account by considering topologically non-trivial world-sheets. For instance, the decay of one closed string into two, will correspond to the the following world-sheets, where we have indicated only the first two terms in an infinite perturbation series (time runs from left to right):
If we denote the strength of the basic closed string interaction, which is given by the left diagram, by gs, we see that the second diagram has strength 92. where x is its Euler In general, a given world-sheet is weighted by gs number and no is the number of external open strings. gs is called the string coupling constant. We will say a little more about it later. Due to the <, the Nambu-Goto action is difficult to deal with. One can remove the square root at the expense of introducing an additional (auxiliary) field on the world-sheet which, however, should not introduce new dynamical degrees of freedom. This field is the world-sheet metric yap (with signature (-, +)). The resulting action - the Polyakov action - couples the D massless world-sheet scalar fields X P t o two-dimensional gravity yap3
The Polyakov and Nambu-Goto actions are in fact classically equivalent. To see this one uses the equation of motion for yap, i.e.4
to eliminate T a p from Sp. One then obtains SNGback. Note that the equations Tap = 0 impose constraints on the dynamical variables X p . In the
=
3From here on we set c 1 and y = det(y,o) 4Ta/3is the energy-momentum tensor which measures the response of the action to a change of the metric.
382
following we will use the Polyakov action. We now discuss the symmetries of the Polyakov action. In addition t o global D-dimensional Poincar6 invariance (X” -+ A’,,X” uplAqAT = Q), A, a constant, Sp has the following local symmetries
+
(I) (world-sheet) diffeomorphism invariance:
for new coordinates
6‘”= [’“(c).
(11) 2-dimensional Weyl invariance:
for arbitrary “‘(7, u). The Weyl invariance, a local rescaling of the world-sheet metric is an extra redundancy of the Polyakov formulation and has no analog in the NambuGoto form. Weyl invariance guarantees that rffDTap= 0, which is easily checked. The local symmetries can be used t o simplify the action and the equations of motion. The diffeomorphism symmetry can be used t o go t o conformal gauge in which the metric has the form
Due t o (classical) Weyl invariance w(7, 0)decouples from the action which is now sp =
T.2 J dfYdT(d,X. a,x - a,x aux) ’
(9)
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The equations of motion for the scalars X” are then obtained as usual
/ 1
L
=T
d7
d o OXp6X, -T boundary term
where L is the length of the string. Thus, the equation of motion for X’” is the two-dimensional wave equation
ox’”= (4 +; a:)x’”= 0 ,
(11)
subject to the vanishing of the boundary term. Here we can distinguish two cases: closed string =+ no boundary term but X’” must be L-periodic in o,i.e.
x’”(7, 0 + L ) = x’”(7,O)
(12)
and there is no boundary term. The general solution t o the wave equation (11) with the periodicity condition (12) is
(13) The powers of T are required for dimensional reasons ( X has dimension length). The numerical coefficients have been chosen to make the canonical commutation relations which we will discuss below free of numerical constants. The solution (13) is the sum of the center-of-mass motion (given by the first two terms) and left- and right-moving waves with amplitudes a: and 6: respectively. The c.0.m. position is
and the total momentum of the string is
1
L
~ ’ ” ( 7= )
P ( To) , =T
384
the momentum density Pfi(u,T ) = Td,Xp(u, 7 ) is the variable conjugate t o X p ( g ,T ) . Reality of X p requires that cxcn = (a:)* and analogously for 5:. In the second case open string
+ boundary term
we have t o impose boundary conditions
a u X p 6 X p= 0 ,
at u = 0 and u = L .
(16)
There are two possibilities t o satisfy the open string boundary conditions (b.c.) (N) Neumann: (D) Dirichlet:
auXplJbndy =0, 6XpIbndy= 0 .
These can be imposed independently for each space-time direction p and each of the two ends of the open string. It is important t o understand the physical meaning of these different boundary conditions. The total momentum L p p ( ~= ) d a P ( Tg) , is conserved for Neumann boundary conditions which place no restriction on the position of the endpoint of the open string. On the other hand, it is easy to see that the space-time momentum is not conserved for Dirichlet boundary conditions. This should not come as a surprise since Dirichlet b.c. fix the endpoints of the open string t o lie on hypersurfaces in space-time; these necessarily break translational invariance. The non-conservation of momentum in directions with Dirichlet b.c. is thus t o be expected. Where does the momentum flow to? The only candidates are the hyper-surfaces on which the open strings end, so these themselves have to be dynamical objects which absorb the open string momentum. These dynamical objects are called Dirichlet-branes or D - b r a n e ~ .More ~ specifically, if we have NN b.c. along (say) p = 0 , . . . , p and DD b.c. along p = p 1,.. . , D - 1 the end-point of the string lies on a Dp-brane. The two ends of an open string can,
so
+
5We will see later that they are not only the loci in space-time where open strings can end but they are also the source for modes of the closed string.
385
of course, lie on two different D-branes, say one end on a Dp and the other end on a Dp' brane. This is illustrated in the figure below for p = p' = 2. No end of an open string can end in 'free space'. It must lie on a D-brane, which can move through space-time.
Notice that D-branes have appeared in two ways: (1) as hyper-surfaces in space-time on which open strings end and (2) as dynamical objects with which open strings exchange momentum. This latter fact is illustrated in the figure by the 'wiggly' shape of the branes. Finally, we list the solutions to O X = 0 with the various possible boundary conditions. For simplicity of notation and without loss of generality, we have set L = T (recall that we have defined a' = &):
(DD) XIr=,
= qi
, X J u = T= qf
(the positions of two D-branes)
386
Notice that only NN boundary conditions allow for a cent,er-of-massmotion. In contrast to the closed string the open string has only one set of oscillators; left- and right movers get tied up through the boundary conditions: a leftmoving wave is reflected and returns as a right-moving wave, and vice versa. Depending on the type of boundary condition the wave is reflected with phase shift 0 (N) or 7r (D). We observe that for mixed b.c. the oscillators have halfinteger moding. Factors of i are chosen such that the reality condition on X translates to a-Yn= (aYn)* and a-,. = (a,)*.
So far the discussion was entirely classical. This means that Planck's constant ti has not appeared yet. It will enter through quantization. There are various ways to subject a given classical system to the laws of quantum mechanics, the most familiar one being canonical quantization. Here the Poisson bracket { q , p } = 1 between a coordinate q and its canonically conjugate momentum p is replaced by the commutator of the position and momentum operators (which we also denote by q and p ) [q,p]= ih. Part of the quantization procedure consists of specifying the Hilbert space on which the operators act. The probabilistic interpretation of the Schrodinger wave-function, which is a vector in the Hilbert space, requires that the Hilbert space be positive definite (this is the requirement of unitarity of the quantum theory). Naive canonical quantization of the string leads, as a consequence of the indefiniteness of the Minkowski metric, to the existence of negative norm states. One then has to ensure that these unphysical states decouple from the theory. This can be shown to be a consequence of the constraint equations Tao = 0. For our purposes the so-called light-cone quantization is most appropriate. In this scheme only physical degrees of freedom appear and it offers the quickest route to the excitation spectrum of the closed and open string. The disadvantage is that explicit space-time Lorentz-covariance is lost. A treatment of covariant quantization procedures can be found in the references on string theory.
387
In light-cone quantization the constraints Tao = 0 are taken care of by solving them explicitly. This becomes possible by choosing the so-called lightcone gauge for the local symmetries of the Polyakov action. In conformal gauge the constraints can be expressed as a+X.a+X = 0 and a - X . d - X = 0 where are derivatives with respect to cr* = cr f r. We observe that even after going to conformal gauge, the diffeomorphism invariance is still not completely fixed. The easiest way to see this is to express the metric in the coordinates u * , in which ds2 = eZwdo+dcr-. Under coordinate transformations cr+ -+ f + ( o + )and cr- -+ f-(o-) the only change in the metric is a change of the conformal factor e2w, i.e. we are still in conformal gauge, provided f* are arbitrary (non-constant) solutions of the wave equation which respect the b.c. or periodicity condition. We can now use this freedom to identify r with any one of the fields X p which solve the wave equation. We choose X + = 2a’p+7- + q+ where X* X o f X D - l are called light-cone coordinates. The remaining D - 2 coordinates are the transverse coordinates. If we insert this into the action and perform a Legendre transformation we find the light-cone Hamiltonian
HI.^. contains only the transverse coordinates Xi.The virtue of the choice X + o( T rather than, say, X o 0; r is that X - does not appear in HI,^. and that it can be expressed, via the constraint equations T&&= 0, up to an integration constant q - , in terms of the X i . In particular one finds that p - = T dad,X- = The dynamical degrees of freedom are thus p+, q- and the complete X i (zero-mode and oscillator parts).
HI,^,.
Once we are in light-cone gauge, we can go ahead with canonical quantization. Imposing standard equal time commutators on the X i and their momenta these become operators in the single-string Hilbert space and so do the 42, p i , ah and 5;. For these the commutation relations are (from now on we set h = 1)
388
All other commutators vanish. Of course, for the open string we have only one kind of oscillators. They satisfy the same commutation relation as, say, the a:, where n is now either integer or half-integer, depending on the boundary conditions. The mass spectrum of the vibrating string is obtained from the eigenvalues of the mass operator which is defined in the usual way as m2 = p+p- CEq2(pi)2.With p+p- = HlJa’ this is m2 = - C(pi)’. This expression is correct for both, open and closed strings. In the case of open strings the sum extends only over the directions with NN b.c.’s; in the other directions there is no c.0.m. momentum.6 Straightforward calculation yields
HI.^.
a’m2 = 2(N + fi) + a
+6
(closed strings)
(19)
and
a’m2 = N”+DD
a‘ + NND+DN + a + (27ra’)2 (Aq)2 ~
(open string)
(20)
where 0-2
n > O i=l
measures the total occupation number of a excitations and likewise for N, etc. Several explanations are in order: (i) It is easy to verify, using the canonical commutation relations, that e.g. for the left-movers of the closed string, m2ak = ak(rn2- %n), which tells us that the a: for n > 0 lower the mass of a state whereas the modes with n < 0 raise its mass. The Fock vacuum 10) is defined to be the state which is annihilated by all lowering operators. Although in the classical expressions the ordering of the oscillators is irrelevant, this is not true in the quantized theory. In the expressions (19) and (20) we have brought the operators to normal ordered form, i.e. we have moved all lowering operators to the right of all raising operators (note that the raising and the lowering operators commute among themselves). This means that the vacuum state 10) is the state with the lowest mass which is in 6Note that when going to the light-cone gauge we need t o assume that we have NN b.c. in the Xf directions.
389
fact given by the normal ordering constants a and 6 , which we will determine is the contribution to the mass below. (ii) The last term in (20), from the stretching of the open string whose endpoints lie on two D-branes which are separated by the distance Aq. We now discuss the normal ordering constants. For each transverse directions they are proportional to C,,,,(n - + v).where v = 0 for integer moded oscillators and v = f for half-integer moded oscillators. Clearly these are divergent sums which can, however, be given a precise mathematical meaning (see e.g. [38]) as follows. For Re(s) > 1 the generalized Riemann zeta function [(s, v ) can be written as the infinite sum C(s, v ) = CT=,(n v)-’. For other values of s it is defined by analytic continuation. We then define the normal ordering constant as the value of the [-function at s = -1 and thus obtain
+
(22) This leads to the following normal ordering constants: (closed string) : (open string) :
0 - 2 a = a = -12 0 - 2 24
a = --
(23)
+ -16d
where d = #(ND+DN directions). The way we have fixed the normal ordering constant looks like a mathematical trick. The procedure of making sense of infinities in a quantum field theory, such as infinite zero-point energies, is called regularization. We can subtract the infinity by modifying the Polyakov action through the addition of a ‘cosmological constant’ proportional to S d a d r f i . The only choice which is consistent with Weyl invariance is the one which leads t o the result above. We are now ready to determine the mass spectrum of the string. First, consider the open string with only NN b.c.’s. Then the ground state has mass m2 ( 24a, 0 - 2 ) , 1.e. . it is tachyonic. The first excited states are with mass
m2
=
126-D --
a’
24
.
390
There are precisely ( D - 2) of these states and they transform as a vector under SO(D-2), the rotation group in the transverse space, which is the little group for massless particles. What this means is the following. Consider first a massive particle moving through D-dimensional Minkowski space-time. Since any massive particle necessarily moves with a speed less than c, we can make a Lorentz boost and go to its rest frame. In this frame the particle’s momentum is f l = (m, 0,. . . ,0) with -p2 = m2 whose invariance subgroup (isotropy group, little group) is S O ( D - 1). This means that massive particles can be classified by representations of SO(D-1). For a massless particle the situation is different. Since they necessarily move at the speed of light and satisfy p2 = 0, we can choose a frame in which its momentum is p” = ( E ,0,. . . , 0, E ) . The invariance subgroup of this vector is E(D - 2), the group of motions in ( D - 2)-dimensional Euclidean space. Massless string states form, however, finite dimensional representations of a SO(D - 2) subgroup. Back t o the string spectrum: since the first excited states form a vector of S O ( D - 2) and since there are no other states of the same mass, these states must form a massless vector. Otherwise Lorentz invariance will be broken. Requiring the mass of these states to vanish fixes D = 26 which is the so-called critical dimension of the bosonic string. We have thus shown that Lorentz invariance of the quantized bosonic string theory requires that space-time has dimension 26. The classical theory was Lorentz invariant for any D but this symmetry is not preserved by the quantization, i.e. there is a Lorentz anomaly except in the critical dimension. This fact holds for general quantization schemes: the absence of anomalies requires the critical dimension D = 26. We have only considered the ground state and the first excited state of the open string with all directions being NN. It is straightforward to consider higher excited states and different boundary conditions. Any state which can be reached by acting with an arbitrary number of creation operators on the vacuum is allowed. Since these states will not be of interest t o us, we will leave their exploration as an exercise. We only want t o make two remarks: (1) The masses of the massive states are n/a’, n = 1 , 2 , . . . . In the field theory limit where a’ 3 0 or, equivalently, T + 00, they become infinitely massive and
39 1
decouple. (2) For, say, all directions NN, the states at each mass level must arrange themselves into representations of SO(25). In the presence of a Dpbrane, for instance, massless (massive) states must come in representations of SO(p - 1 ) (SO(p))since the Dp-brane breaks S 0 ( 1 , 2 5 ) t o SO(1,p). We have remarked that the end-points of open strings must lie on Dbranes. In the presence of several D-branes we must label the open string states by two additional labels to indicate on which of the D-branes the two end-points of the open string lie. If the open string is oriented we can distinguish its two end-points from each other and we denote the states by IN;p ; i,$, where N denotes the oscillator numbers, p the c.0.m. momentum of the string and i and J are the so-called Chan-Paton (CP) indices of the a = 0 and a = 7r endpoints, re~pectively.~ We write these states in the form
a
Later on we will be interested in the case of N parallel and coincident Dpbranes. In this case one can show, by looking at the interactions’ of excited strings, that the allowed matrices Aj: generate the group U ( N ) .The massless 7We will not consider unoriented strings in these lectures, even though they are also of interest. The possible gauge groups are then SO and U S p . They require, in addition to D-branes, also so-called orientifold planes. For a recent review, see [27]. sFbughly speaking, interactions are taken into account by looking at topologically nontrivial world-sheets, as we have alluded to before. More concretely, there is a correspondence between states and operators, the so-called vertex operators and one computes the interaction of strings with given excitation as correlation functions of the corresponding vertex operators. The vertex operators for closed string states are inserted in the interior of the world-sheet, those for open string states at the boundary. As long as the external momenta are small compared to the masses of the massive string excitations, one can reprcduce the scattering amplitudes of the massless states by a low-energy effective field theory action. In the case of the gauge bosons, this action is, to lowest order in a’, the U ( N ) Yang-Mills action in p 1 dimensions with gauge coupling gCM c ~ ~ ( a ’ ) ( P - ~ ) In / ~ .the same way the other massless modes that we will encounter below are identified. The fields corresponding to excitation modes of the open string are confined to the world-volume of the brane on which the string ends whereas those corresponding to the modes of the closed string can propagate anywhere in the bulk, but also interact with the open string modes. Correspondingly the low-energy effective field theories are formulated on the branes and in the bulk, respectively.
+
-
392
states 11;p ; u ) = Ci,j=l Q1 10; p ; i j ) X : j are identified as the gauge bosons of an U ( N ) gauge symmetry if m labels a direction along the brane and as massless scalars transforming in the adjoint representation of the gauge group if m labels a direction transverse t o the brane (we use the normalization T r ( X u X b ) = bab). If we separate the N D-branes (but keep them parallel) into two stacks of Nl and Nz branes, the (TAq)2term in the mass formula (20) contributes to the mass of the excitations of those strings whose endpoints lie on separated branes. The corresponding states become massive and the gauge symmetry is broken t o U ( N 1 ) x U ( N 2 ) . This is a brane realization of the familiar Higgs effect. The extra degrees of freedom needed t o give mass to the gauge bosons which lie in the coset U ( N ) / ( U ( N l )x U ( N 2 ) )are provided by the scalar fields. The figure illustrates the situation for U ( 2 ) . The massless excitations of the four possible oriented open strings represent the four gauge bosons. If the two branes are separated, as in the figure, two gauge bosons are massive, their mass being the string tension times the separation of the two branes. The unbroken gauge group is U(1) x U(1). N
We now discuss the closed string spectrum. It is very similar to the discussion of the open string spectrum, without the complication of different boundary conditions and CP factors. However, there is one further constraint we have t o impose on the closed string states. This comes about as follows. After going to conformal gauge we have used the remaining diffeomorphisms t o go to light-cone gauge. In the case of the open string this fixes the diffeomorphisms completely. However, in the closed string we are still allowed t o make constant shifts in a + afao. Using the mode expansion and the commutation relations it is not difficult t o show that the operator U,, = exp(iao(N - fi)) satisfies
393
Uo0X(7,u)U&' = X ( T ,a+ao). We thus have to impose the following physical state condition (level matching condition) on closed string states:
( N - #)]state) = 0 .
We can now determine the spectrum of the closed bosonic string. Again, the ground state is tachyonic. The first excited states are ( Y ? ~ ( Y ~ _ ~of~ which O) there are ( D - 2 ) 2 . They transform as reducible rank-two tensor representation of SO(D - 2 ) . Its irreducible components are the symmetric traceless, the antisymmetric and the trace parts. The same group theoretical argument as for the open string requires that these states are massless and hence we conclude that also for the closed bosonic string D = 26. The irreducible components of the massless states can be identified, via their interactions as the graviton Gij, an antisymmetric tensor particle Bij and the so-called dilaton @:
As we argue below, @ is related to the string coupling constant gs via gs = e " O where @O is the vacuum expectation value of the dilaton. Again, we will not discuss the massive spectrum. Notice that the space-time fields corresponding to the excitations of the open string are confined to live on the world-volume of the D-branes on which the open strings end. There is no such restriction for the excitations of the closed string. D-branes interact with each other via the exchange of closed strings. The figure shows the emission of a closed string from a D-brane. In a time-reversed process the closed string can be absorbed by another D-brane.
394
The figure also illustrates that while open strings are attached t o D-branes, closed strings can move in 10-dimensional space-time. One can easily generalize the Polyakov action in such a way that all the massless string modes appear:'
+ C " ~ ~ , X ~ ~ ~ ~ X ~+Ba ',R,((y X) @) ( X ) }.(29) This describes the motion of the string through a space-time with metric G,, and a background anti-symmetric tensor and dilaton field. In the last term R ( y ) is the Ricci scalar for the metric y. If we now separate @ into a constant background value @o and a fluctuating piece, @ = @O 4, then the contribution of @O is proportional t o the Einstein action for the metric y which, in two dimensions, is proportional t o the Euler number of the worldsheet (for open strings there are additional boundary terms)." This means that in a path integral evaluation of string scattering amplitudes a world-sheet with Euler number x carries a weight e - X a 0 f g;X.
+
Note that unless G,,, B,, and @ are constant, i.e. independent on X , the world-sheet action is that of an interacting field theory which can only be gThere is also a term over the boundary of the world-sheet which contains the massless gauge bosons of the open string spectrum, but we will not write it. 'OThis makes sense if we perform a Wick rotation and change the signature of the worldsheet from (-,+) to (+,+).
395
quantized perturbatively. In the quantum theory we have to make sure that the local symmetries of (4), which allowed the elimination of the degrees of freedom contained in -yap, are still present. This requirement imposes severe restrictions on the background fields GP,, B,, and 4. The conditions they have to satisfy are in fact equivalent t o the equations of motions of these fields which follow from the low energy effective action.
So far we have dealt with the bosonic string whose world-sheet description involves only bosonic fields and whose excitations all transform in tensor representations of the little group (we have shown this explicitly for the massless states, but this is also true for the massive states) and they are thus spacetime bosons. Clearly, for a realistic theory of nature we also need space-time fermions. They will appear in the spectrum of the fermionic string which we will now discuss. The world-sheet action of the fermionic string contains bosons and fermions.ll In light-cone gauge the action i d 2
The Q iform, as the Xi, a vector of S O ( D - 2). p a are 2-dimensional Dirac ~ the ~ .basis po = iz),p1 = the matrices obeying { p a , p p } = - 2 ~ In components of QT = ($,$) can be chosen to be real (Q is thus a Majorana
(7
(9 A )
___
"We are using the NSR (Neveu-Schwarz-Ramond) formulation. In the Green-Schwarz formulation, one introduces additional world-sheet scalars which are, however, space-time spinors. "One can derive this x t i o n by starting with a generalization of the Polyakov action. While in the bosonic case this is a two-dimensional field theory coupled to gravity, we would now consider a two-dimensional field theory with bosons and fermions which couples to supergravity. q p is the supersymmetry partner of X p . The partner of the world-sheet metric y is a world-sheet vector-spinor, the gravitino. The gravitino can be gauged away via the local fermionic symmetries (supersymmetry and super-Weyl). This defines the superconformal gauge in which the equations of motion for the gravitino are $.a+X = 4 . a - X = 0 which have to be imposed as constraints. The remaining gauge freedom again allows to go to zero and the components $to light cone gauge which sets the components $+ and and can be expressed in terms of the $a and respectively. The resulting action is
4-
(30).
4+
di,
396
spinor). Then % = Qtpo = QTpo. The equations of motion are the wave equation for the X iand the massless Dirac equation for the Qi:
Again we have t o distinguish between the open and the closed string. We start with the open string. For the open string the fermions are subject t o the boundary conditions
which couple the left- and right moving fermions. The relative sign between II,and is a matter of convention, which we choose such that
4
which solves (32). The two choices r] = f l define two sectors of the theory, the Neveu-Schwarz (NS) sector for r] = -1 and the Ramond (R) sector for r] = + l . As we will see below, fields corresponding to states having excitations in the NS-sector are space-time bosons, whereas excitations in the R-sector lead t o space-time fermions. The solution of the equations of motion which respect the b.c. i d 3
T =C
&e i
q i =
-ir(T+u) 9
{;;+
CII,;e-ir(T-u) with
NS sector,
R sector. (34) The fact that there is only one set of fermionic oscillators is due to the boundary conditions. For the contribution of the fermions t o the light-cone Hamilr
r
13When associating (ha1f)integer moded fermionic oscillators for QZ with the (NS) R sectors, we have assumed that the the bosonic field X 2 has integer moded oscillators, i.e. satisfies NN or DD boundary conditions. The precise definition is that in the R sector the moding of bosons and fermions is the same whereas in the NS sector they are different. This means in particular that for ND and DN b.c.'s the (half) integer moded fermions belong to the N S (R) sector.
397
tonian one finds
with T integer (half-integer) in the R (NS) sector. The fermionic oscillators are quantized by imposing anti-commutation relations:
{G:,
= hr+s,o
Pi.
(36)
They also contribute t o the mass of a state. The mass operator for the open string in either sector is now
The normal ordering constants arise from putting the fermionic oscillators in normal ordered form (again, the positive modes are lowering and the negative modes are raising operators). They are (c.f. (22)) CLR= -(D - 2)<(-1,0) = in the R-sector and U N S = -(D-2)C(-11 1/2) = in the NS-sector. Note that the total zero-point energy vanishes in the R-sector. Here we have assumed that we have only NN or DD b.c.’s. If we have d ND plus DN b.c.’s, the normal ordering constant is again zero in the R sector and $ in the NS sector.
9
-9
-9+
The NS-sector has a unique ground state which is tachyonic. It is a spacetime boson and so are all excited states in the NS sector which can be reached n E Z+on the NS ground state. by acting with creation operators $in+1,2, The first excited states in the NS-sector is the SO(D - 2) vector
$?,lo)
with mass a’m2 =
3( 10 - D ) 48
and, therefore, the critical dimension of the fermionic string is D = 10. One can show that a t each positive mass level the states combine into (reducible) tensor representations of SO(9). Note that when enumerating all states in the NS sector, one has t o take into account that = 0 for each a and T . This is the Pauli exclusion principle for the world-sheet fermions. Note that even though we are discussing world-sheet fermions here, the excited string
($t)2
398
states they create are space-time bosons. This will be different in the R-sector which we discuss next. In the R-sector we have the zero modes ment. They satisfy the Clifford algebra
($16,
$;} = b i j
,
$6
which require special treat-
i , j =1, ..., 8 .
(39)
The representation of (39) is essentially unique14 and given in terms of Dirac matrices, i.e. $6 = &I?, ri being the 16 x 16 Dirac-matrices of SO(8). The $6 commute with the mass operator which means that the ground state in the R-sector is degenerate. In fact it has zero mass since the zero-point energy vanishes in the R-sector. The different ground states are transformed into each other via the action of $6. But this means that the ground-state in the R-sector, which we will denote by (A), A = 1 , . . . ,16, transforms as a spinor of SO(8) and that
We can reach excited states by acting on the ground states with oscillators $ 1 5 ~with n E Z+. Of course, the Pauli exclusion principle ($:)2 = 0, V n and i, has to be taken into account when enumerating the states. Since the oscillators carry a SO(8) vector index, all states in the R-sector transform in a spinor representation of SO(8) and are thus space-time fermions. The 16-dimensional spinor representation of SO(8) is reducible, its irreducible components being the eight-dimensional chiral spinors which span the subspaces with eigenvalue f l of the chirality operator r9 = I'1. . . r8 which satisfies (rg)2 = 1. (Spinors with definite r9eigenvalue are called Weyl spinors.") To distinguish the two irreducible components we split the spinor '*In even dimensions the Clifford algebra has only one inequivalent irreducible representation whereas in odd dimensions it has two. This representation is in terms of Dirac matrices of dimension 2[D/21 where [ D / 2 ] is the integer part of D / 2 . The two representations for D odd differ by the sign of rD0: rl. . . A proof of this statement and many other useful properties of Dirac matrices in arbitrary dimensions can be found in [39]. "Weyl spinors exist in all even dimensions. For D = 272 we define r2nf1 =a riwith the phase (L chosen such that = 1. Then Weyl spinors are eigenspinors of r2nf1. In odd dimensions they do not exist since there ri o( 1.
n:r'=",
n,"=,
399
index as follows: A = ( a ,u ) and thus IA) = la) @ J u )with r91a) = +la) and r g l h ) = -1u). The two eight-dimensional spinor representations are often denoted as 8, and &. SO(8) has a third eight-dimensional representation - the vector on which SO(8) acts as a rotation - often denoted as 8,.
For the closed string we have to impose periodicity conditions. In the case of real fermions there are two options: periodic or anti-periodic. Both options leave the action invariant.16 This leads to the following options:17 @(T,
c
+
T)
=~
$J~a (T ),
and
$(T, 0
+ n) = i j @ ( ~ a, )
with 7 ,i j = f l .
(41)
If we now make a mode expansion we need to introduce two sets of oscillators, $: and $:, where, depending on the choices for 7 and i j , the mode number T is either integer (periodic) or half-integer (anti-periodic). This gives four possible sectors (space-time 0 (NS,NS): 7 = i j = -1 , (R,R) : 7 = i j = 1 bosons), (NS,R): 7 = - i j = -1, (R,NS): = -ij = 1 (space-time fermions). Quantization proceeds as for the open string only that we now have two sets of fermionic oscillators, each contributing to the Hamiltonian as in (35) and each satisfying the anti-commutation relations ( 3 6 ) . Also each set of fermionic oscillators contributes to the zero point point energy and the level matching condition (27) now involves the number operators for bosonic and fermionic oscillators. Note that the level-matching condition forbids e.g. a tachyon in the (NS,R) sector. (R,R)-sector ground states are bispinors IA)L
8 I B ) R = b ) L @ Ib)R a3 b ) L 8 l b R @ lU)L
@ Ib)R I
CJd b ) L 8 I h R .
(42)
~~~
“The interpretation of X i as space-time coordinates does not allow for more complicated periodicity conditions of the bosons if we want to describe a string moving in Minkowski space-time, even though other possibilities are compatible with the reality of X and the invariance of the action. When one considers compactifications of string theory such possibilities are, however, considered. 17We have to impose the same conditions for all z if we want to preserve S O ( D - 2) invariance.
400
As for the open string the zero-point energy vanishes in the R-sectors and these states give rise t o massless bosonic fields in space-time (cf. below).
It turns out that the theory we have constructed is not consistent. One sign of the inconsistency is the appearance of the tachyonic ground state in the (NS,NS)-sector. A more severe inconsistency is the lack of modular invariance of the one-loop partition function. We do not intend to elaborate on this very much but try to convey the main point of the argument and then simply state the consequences. We had discussed diffeomorphism invariance of the Polyakov action. Together with Weyl invariance it is necessary t o ensure that with the introduction of the world-sheet metric no new degrees of freedom are added, or in other words, that the three degrees of freedom of yap can be gauged away. This is a non-trivial requirement for the quantized theory and, in fact, quantization often breaks symmetries which the classical action possesses. One then speaks of anomalies. In string theory one must ensure that diffeomorphism and Weyl invariance are still present after quantization. One can show that this requirement also fixes the critical dimension to the values we found. However there are additional restrictions which one encounters when one studies the diffeomorphism invariance of correlation functions on world-sheets of higher genus, e.g. the closed string zero-point function at one loop, i.e. with the world-sheet being a torus. This is also called the closed string partition function and it can be shown t o be Tr(e2Xi7Hl.c.e-2TZTHI =.). The trace is over all states of the closed string, not necessarily satisfying the physical state condition. We have split the contributions from left and right movers t o the Hamiltonian and have denoted the two contributions by HI,=,and Hl.,.. T is the modular parameter of the torus. We can define the torus as R2/A, where A is a two-dimensional lattice. This lattice can be specified by fixing a point T = TI 272, 7 2 > 0 in the upper half of the complex plane. The generators of the lattice are then the T Z ) , where we have used Weyl invariance to scale two vectors ( 1 , O ) and (71, the length of the first generator to one. One now uses the fact that not all choices for T in the upper half-plane lead to diffeomorphically different tori: with if T and T' are related by a PSL(2,Z) transformation, i.e. if r' = a , b, c, d E Z and ad - bc = 1, the two tori defined by r and T' are diffeomor-
+
3
401
phic. In fact, the two lattices are the same, only the choice of generators is different. The diffeomorphism cannot be smoothly deformed t o the identity; it is a so-called large diffeomorphism. Nevertheless, the partition function should be invariant under these diffeomorphisms or, in other words, it must be modular invariant (PSL(2,Z)is the modular group). This condition, namely that the oneloop partition function be invariant under modular transformations, will not be satisfied if one sums over all states of the closed fermionic string. To get a modular invariant expression, one has t o truncate the spectrum, or, in other words, one has to introduce a suitable projection operator P and compute ~ ( P e 2 x i r H l . c . e - 2 xThe ~~~ necessity ~~c~). for such a projection to arrive at a consistent string theory was first realized by Gliozzi-Scherk-Olive and is called GSO-projection. For the closed fermionic string there are two possible GSO projections which lead t o a tachyon free spectrum. We will not describe them in any detail but simply state the resulting massless spectra. In both cases the (NS,NS)-sector contributes a graviton Gij, an antisymmetric tensor B,j and a dilaton @. In contrast to the bosonic theory these states are created from the (NS,NS) vacuum with fermionic oscillators, -.
i.e.
3 $JZ-1/2$J-1/2
I~)NS,NS.
The (R,R)-sectors of the two theories are different. The two consistent choices are:18 type IIA with (R,R) ground state l u ) @I ~ l
b)~
(not chiral),
type IIB with (R,R) ground state l u ) @I ~ l b ) ~ (chiral). The statement about the chirality means the following: in the type IIA theory the part of the spectrum with r9 eigenvalue +l is identical to the part with eigenvalue -1. For the type IIB theory this is not true. This is already obvious from looking a t the (R,R)-ground states. The (R,R) ground states transform under reducible components of S 0 ( 8 ) , namely as 8, x 8, and 8, x 8, for type IIA and IIB, respectively. To extract 18
l u )@ ~ Ib)R and l h ) @ ~ l b ) are ~ equivalent to these two choices.
402
the irreducible pieces we make a short aside and discuss the index structure of Dirac matrices. An arbitrary Dirac matrix F can be decomposed into block r a b Fa& f01-m:'~r A B = . We now define the anti-symmetrized products of
( r&) ,
,
rab
Dirac matrices:
r21...zp - r[il-.ipl
3
L ( r i l . . . r i p f permutations). For p = 0 P!
this is the charge-conjugation matrix which is also used t o raise and lower spinor indices. One can choose a basis in which either the two diagonal or the two off-diagonal blocks of each of these matrices is zero. More concretely, one finds: D = 4n: For p even the blocks with mixed indices vanish and for p odd the blocks with the same indices vanish. D = 4n 2: here the situation is opposite w.r.t. p . In this basis Yo+'
=
(i -:I. \
+
For D = 2n
+ 1 there is
I
no chirality and hence no distinction between dotted and un-dotted indices. Using these results, which can be proven by simple SO(n) group theory or by explicitly constructing the Dirac matrices, we get the following decompositions into irreducible components:
) .1 Type IIB: ) . 1
Type IIA:
@
16) = ribli)@ F Y t l i j k )
8 ~ b= ) C a b [ . ) @ ryblij)CD rytllijki)
Another way of writing this is S,@S, = 6,@ SS and S, @ 6,= 1.@ 28 @ 35+ , where %+ denotes the self-dual fourth rank tensor representation of SO(8). is anti-self-dual. is self-dual whereas In fact, one can show that
I'yt'
To summarize: the massless bosonic excitation spectra of type I1 theories are
(NS,NS) (R,R)
IIA/IIB: IIA: IIB:
G,,, B,, and 1-form A,, 3-form A F y p scalar x, 2-form BI,, 4-form A,VPo with self-dual field strength F = dA
We have written the space-time fields in covariant form. The light-cone components are not directly visible in the light-cone gauge, but in the critical 'gProperties of Dirac matrices are discussed in some detail in the lectures by J. Zanelli in these volume.
403
dimension Lorentz-invariance is preserved and all bosonic fields must transform as tensors of the full Lorentz group SO(1,9). Nevertheless, the number of physical degrees of freedom they present is given by the counting in lightcone gauge. The reduction is due t o gauge invariances and the equations of motion these space-time fields satisfy.
So far we have only discussed the bosonic degrees of freedom. The remaining two sectors, (R,NS) and (NS,R) contain space-time fermions. In fact, one can show that the resulting spectrum has N = 2 space-time supersymmetry. Supersymmetry (SUSY) is a generalization of Poincark symmetry. In addition t o having only bosonic generators which transform in tensor representations of the Lorentz-group (Lorentz-transformations, rotations and translations) the supersymmetry algebra also has fermionic generators, called supercharges. The algebra is a Z’ graded Lie-algebra, since the supercharges satisfy anti-commutation relations.” The corresponding transformation parameters E are fermionic. They transform in spinor representations and transform bosons into fermions and vice versa. Schematically 6,B = E F and 6, F = EVB. Saying that we have N = 2 supersymmetry in D = 10 means that we have altogether 32 supercharges which form two Majorana-Weyl spinors of SO(1,9) which we will call Q L and QR, where the subscript means that one originates from the left and the other from the right-moving sector of the closed string theory. The distinction between type IIA and IIB is that in the former case the two Majorana-Weyl spinors have opposite chirality whereas they have equal chirality in the latter case. The same holds for the supersymmetry parameters which we will call EL and E R . One of the hallmarks of linear representations of supersymmetry is that each of its irreducible representations contains the same number of bosonic and fermionic degrees of freedom; see, however, [40] and J. Zanelli’s contribution in this volume. Since we will not need the fermionic part of the spectrum in the subsequent discussion, we will not discuss it. It can be reconstructed from the bosonic part of the spectrum via the supersymmetry algebra. ”More details on Supersymmetry and Supergravity and on the supersymmetry algebra can be found in the lectures by J. Zanelli.
404
Supersymmetric field theories possess a supersymmetric spectrum and the action is invariant under the supersymmetry algebra. If the SUSY parameters are constants, one deals with global supersymmetry. If they depend on space-time, the theory necessarily contains gravity and one has a supergravity (SUGRA) theory. In the case of the type I1 string theories, one finds that their low-energy effective actions are in fact the type IIA and type IIB =2 supergravity theories in ten dimensions. One can show that the space-time supersymmetry of the type I1 string theories is a consequence of a world-sheet supersymmetry of the Polyakov action for the fermionic string. This is the generalization of the statement that the absence of anomalies of the local world-sheet symmetries in the bosonic string leads t o the critical dimension which also guarantees space-time Lorentz symmetry. Here anomaly freedom of the world-sheet supersymmetry leads to space-time supersymmetry. So far we have discussed the supersymmetry of the closed string sector. If we add D-branes we get theories with open and closed strings. We have seen that in the closed string QL and Q R are associated with the left- and right-moving sectors of the world-sheet theory. Since they are coupled by the open string boundary conditions, one gets a reduction of the number of independent supercharges from 32 to 16. One can show that in the presence of a Dp-brane whose world-volume fills the xo,. . . ,xp directions, the surviving SUSY generators are ELQL CRQR where EL,R are related as
+
The sign choice distinguishes between a brane and an anti-brane. We multiply both sides of this equation by the SO(1,9) chirality matrix = ro. . . r9 and commute r on the r.h.s. through the ( p + 1) ri's. This produces a factor (-l)p+'. If we now use that r E L = r E R for IIB and r e L = - r & R for IIA, we find that SUSY preserving Dp-branes exist in type IIB for p odd and in type IIA for p even. In the other cases one finds that the spectrum contains
405
tachyons so that e.g. a D3 brane in type IIA is unstable.21 One can also work out the condition under which different species of branes, either for different p and p‘ and/or for different orientations in space-time, preserve some supersymmetry. One finds e.g. that for non-parallel branes one can preserve at most eight SUSY charges. SUSY preserving branes are so-called BPS 22 configurations. They can be characterized by the representation theory of the SYSU algebra. We will illustrate this on a simple example, which is relevant t o supersymmetric quantum mechanics, but the idea generalizes t o field theory. Consider the algebra generated by two bosonic generators H and Z and two fermionic generators Q1 and Q2 (supercharges). The only non-zero (anti)commutators are (01, = H f Z , (Q2, = H - 2 . In particular, since [ H ,Z] = 0, they can be diagonalized simultaneously. Consider an eigenstate [$). Its eigenvalues h and z satisfy
QI}
Qb}
the inequality h T = ($l{Qi,a,Q!,z)l$) = IIQ!,zl$)IV + llQi,21$)1I2 2 0, 2.e. h 2 121. This is the BPS-bound. For h > 121, an irreducible SUSY multiplet consists of four states: the rescaled generators q1,2 = Q l , z / G satisfy the algebra of two fermionic oscillators, and the multiplet consists of the following four states: lo), qiIO), &lo), qiq110). The vacuum 10) satisfies 41,210) = 0. If, however, h = IzI, one of the two supercharges decouples and we are left with just one fermionic oscillator. The multiplet then consists of two rather than four states. The states of these ‘short multiplets’ are called BPS states. E.g. for h = 2 , Qz decouples and both Q2 and annihilate the eigenstate I$). In the context of our discussion of branes in lecture 3, h is their mass and z their (R,R)-charge.
QL
Given a SUSY preserving brane, we can always add more branes of the same type without breaking SUSY further. These branes do not have to be coincident. As long as they are parallel one also obtains a BPS configuration. These configurations are, as a consequence of SUSY, stable, i.e. no net force acts between the branes. However, a brane-anti-brane system breaks all supersymmetries and is unstable. This is also true for generic configurations of Dp- and Dp’-branes. The equations (43) have no solution. 21The fate of tachyonic theories has been much discussed recently, see e.g. [41]. 22 Bogomolnyi-Prasad-Sommerfield
406
The low-energy effective actions of the massless excitations of the open string are, t o lowest order in a’, Super-Yang-Mills (SYM) theories on the world-volumes of the branes where the gauge group is determined by the brane configuration. For instance, in the case of N parallel D3 branes one obtains a four-dimensional JV= 4 SYM theory with gauge group U ( N ) . At the end of our discussion of the bosonic string we briefly discussed the generalization of the Polyakov action which incorporates background values for the massless space-time fields. The same generalization also holds for the massless fields of the (NS,NS) sector of the type I1 string theories. However, within the NSR formulation that we have been using, no such coupling t o fields in the (R,R) sector is known.
Lecture 3: Branes from supergravity In the previous lecture we have seen that Dirichlet boundary conditions of the open string ends implies the existence of D-branes and we have argued that they are dynamical objects of the theory. One might wonder whether they are a necessity. After all, one might decide t o impose only Neumann boundary conditions. This would correspond t o the presence of space-time filling D9 branes which can have no dynamics. There are, however, various ways to show that lower dimensional branes must also be considered. One is based on T-duality, which we will not discuss here, except for saying that this is a symmetry of string theory which changes the boundary conditions from Neumann t o Dirichlet and vice versa. Another argument is based on the low-energy effective action for the massless string excitations where one finds brane solutions as solitonic solutions of the classical equations of motion. This is the route we will follow. The discussion in this section mainly involves the bosonic fields. For the type I1 theories they were summarized above. Their low energy dynamics is governed by a low energy effective action (Zeea). In the limit gs -+ 0, which suppresses string loop effects and a’ -+ 0, which renders all massive string modes infinitely heavy and they thus decouple, the Zeea’s are the type
407
IIA and IIB supergravity theories. They involve only terms with at most two derivatives. Higher derivative terms, such as R2would be multiplied by additional powers of a'. They are suppressed in the low energy limit where all external momenta satisfy k << l/JCr'. We will not write down the complete leea; the interested reader may find it e.g. in [23]. We will write down the relevant terms which are needed for finding brane solutions. But first we want to understand which fields such a solution will involve. For this purpose we clarify the relevance of the anti-symmetric tensor fields which appear in the massless spectra. Recall that a charged particle in four dimensions couples to a background vector (1-form) potential A, via the term
I,
(44)
A
in the action. Here qe is the (electric) charge of the particle and C1 its worldline through space-time. The charged particle is also a source for the field. If F = d A is the total field strength (including the one generated by the particle), qe can be determined by integrating the dual field strength * F over a 2-sphere surrounding the particle (but no other source for F ) r
qe =
J,,
*F
(45)
One can modify Maxwell theory to allow magnetically charged objects. In 3 1 dimensions these are also point particles (magnetic monopoles) and their magnetic charge is
+
qrn=l2F. The electric and magnetic charges are not independent from each other. They satisfy the Dirac quantization condition qeqm = 27rn ,
n E Z.
(47)
This can be derived by requiring single-valuedness of the wave-function of an electrically charged particle in the presence of a magnetic monopole.
408
There is a two-fold generalization t o the above, namely going from four to D dimensions and introducing higher-dimensional objects [42,43]. A pdimensional (electrically) charged object couples t o a ( p 1)-form potential via
+
where Cp+, is the object's world-volume. As an example, consider the coupling of the fundamental string t o the (NS,NS) 2-form B-field
B=
1
dTdo Ea~a,xvpxV~,, .
(49)
Here B,p = &XfiapXVB,, is the pull-back of the B-field from space-time to the string world-sheet. We have encountered this coupling in eq.(29). The objects which couple to the (p+l)-form potentials originating from the (R,R)sector are called p-branes. The electric charge of a p-brane is
,Dp-brane
I The position of a p-brane is given by a point in the ( D - p - 1)-dimensional space transverse to it. In this transverse space it can be surrounded by a (large) ( D - p - 2)-sphere. This expression for the electric charge corresponds t o an action of the form S = - 1 q P + 2 ) ! J Ffp+2)+ q e ,,J ,, A(p+'). ~tleads to the equation of motion d * F = qebll, which, upon integration over the transverse space and use of Stokes's theorem gives (50); 611 is the delta function with support along the world-volume of the brane. Given the electrically charged branes, what are the dual magnetically charged objects? Their charges should be given by
409
Since a (p+2)-sphere surrounds a (D-p-4)-dimensional object, we have an electric-magnetic duality between p-branes and (D-p-4)-branes. Again, their charges must obey the Dirac quantization condition. A brief comment: in four dimensions, in addition t o electrically charged particles and magnetically charged monopoles one can also have dyons, which carry electric and magnetic charge. For higher dimensional objects this is in general not possible as the dual objects carrying electric and magnetic charge generally have different dimensions. The same holds for self-dual objects which couple t o a potential with self-dual field strength. In D = 10 the three brane of type IIB string theory is self-dual. From the above discussion we conclude that a p-brane solution of the equations of motion should contain a non-trivial (R,R) A(P+’) background field c o n f i g ~ r a t i o n In . ~ ~addition it must contain the space-time metric which couples to the energy-momentum of the A(P+’). We now write those terms of the leea which contain the metric and A ( P + ’ ) :
where F&+2) = F,o...,p+l F,o.-Pp+l and K~ = i(27r)7a’4g:. We have chosen the action in the Einstein frame in which the metric g,, is related t o the string-frame metric G,, via G,, = e3@g,,. The equation of motion for the dilaton, V24 = &$! e ~ ~ F ~ + implies 2 ) ,that for non-vanishing Ftp+,) the dilaton cannot be constant unless p = 3. This is in fact the case we will eventually be interested in. However, for p = 3 the above action is not valid since F& = 0 for self-dual F(5). Nevertheless, the equations of motion are the same as those following from (52) if one multiplies the F 2 term by 1/2 and imposes the self-duality condition separately [44]. For p = 3 we give the solution in (62). It is now easy t o set up the complete system of equations of motion. To solve them is an entirely different matter altogether. For this we have t o make an ansatz which reflects the symmetries of the solution we are looking for. 23For the fundamental string and its dual object, the NS five-brane, we would need a non-trivial (NS,NS) B-field background. We will not discuss them here.
410
We are interested in brane solutions which extend in say, the (xo,x l , . . . ,x p ) directions, which we call x p . The ansatz for the metric then has to respect Poincark symmetry along the brane or, in other words, the complete solution should not depend on x p . In addition we assume spherical symmetry in the transverse space with coordinates ym, m = 1,.. . , 9 - p . This leads to the following ansatz for an electrically charged brane at y = 0:24 ds2 = A
( T ) V ~ , ~+XB(r)Gmndymdyn ~~X~
7
(53)
e4 = C ( r ) ,
(54)
A p 0 . . . p p = E p 0 . . . p p m - )7
(55)
with r2 = Gmnymyn. Inserting this ansatz into the equations of motion leads to a system of second order ordinary non-linear differential equations. The simplest non-trivial solution which approaches ten-dimensional empty Minkowskispace for T -+ 0;) and which is valid for all T > 0 is (details can be found e.g. in [28, 29]):25326
A(r)= f ( r ) V ,
B ( T )= f ( ~ ) * ,
C ( T )= f ( r ) v , D ( T )= f ( ~ ) - ' ,
(56) where
is a solution to the Laplace equation in the transverse (9 - p)-dimensional space and
24For the magnetic dual we would make an ansatz for * F and for p = 3 an ansatz which leads to a self-dual field strength. 25Since we are looking for supersymmetry preserving solutions, it is in fact simpler to analyze the SUSY condition. This amounts to requiring that the supersymmetry transformation of any fermionic field F vanishes, i.e. 6,F = 0. Here E parametrizes the unbroken supersymmetries. This leads to first order differential equations from which the second order equations derived from the effective SUGRA action can be recovered by iteration. The SUSY preserving solutions are in fact the simplest ones; they have the highest symmetry. 26This solution is valid for p < 7. For p = 7, (f - 1) o( ln(r).
41 1
is the volume of S5-P.27 rp is the tension (mass per unit volume) of the brane which is defined via
Q5-p
(59) The integral is over the transverse space. Here € I , , is energy-momentum pseudo-tensor of the system which is defined as follows. Expand the metric h,, and define R,, = which we found above around flat space as g,, = 77,” R$ C3(h2).28Then 8,, is defined as 72;; - $q,,R( 1 ) = K 2@,, (R(1)=
+
+
One can read this as the wave equation for a spin-two particle (the graviton) with source given by the energy of the gravitational field and matter. For the metric (53) 2n2&,0 = pOA(r) + (8 - p ) O B ( r ) where 0 = a; + What we have computed here is the ADM (for Arnowitt-DeserMisner) tension of the brane. For further reading, see e.g. (45,461. If one computes the electric charge density of the brane, c.f. eq.(50) one finds qe = N ~ ~ K This T ~follows . most easily from the following observation. A(P+l) q W & )1.)
?ar.
satisfies a , ( f i e v 4 F p 0 1 - - - p= ) 2 ~ ~ ~ , N d ( ~ -i.e. ~ ) (there y ) , is a source for A(P+’), the brane.2g In fact, one can incorporate the source term into the action by adding Sbrane =
-NTp
J
dP+’<ee4d-
+ NTp
%+1)
J
A(P+l).
(60)
%+l)
where GPv is the induced metric on the brane which, in static gauge, where we identify E’” = x,, is simply gPv = A(r)qPv.The first term is the analogue of the Nambu-Goto action for the fundamental string: it is the ‘area’ of the world-volume in string frame (expressed in the Einstein frame metric). The crucial observation of Polchinski [53] was that the SUSY preserving p-brane solutions of the SUGRA equations of motion are the same objects as the Dp-branes. Before we discuss the implications (only closed strings in type 11, no explicit gauge fields, etc) let us briefly review the arguments in 27od
=
27~(~+’)’’ r((d+1)/2) ’
28Explicitly, this is R ~ = J +(apa,,hP, t y a ~ h , , - aPaph,,.
+ apa,hp,
- aPaph,, -
29The definition of electric charge that follows from (52) is qe =
a,,a,hpp)
and
-&ss~-p e+$ 3-
~ ( l = )
* F.
412
favour of this identification. D-branes are BPS states. One consequence is that a stack of parallel D-branes is stable. The p-brane solution (56) can be easily generalized to a stack of parallel branes at gi by the substitution f -+ 1 &. This is a static solution which can be shown t o be stable. From the results of the previous lecture it follows that the allowed dimensions of half SUSY preserving D-branes are correlated with the massless (R,R) states. This is also true for the SUGRA branes which carry (R,R) charge. It thus remains to be shown that D-branes also carry (R,R) charge and that the ratio between their charge and tension is the same as for the p-brane. We do not give details of this calculation which was first performed by Polchinski but we will try t o convey the idea. In order to compute the tension of the brane we need t o find the strength with which it couples t o a graviton and in order to find its charge we need the coupling t o the (R,R) fields. The string diagrams which are responsible for these couplings are a disk with a graviton or (R,R) p-form vertex operator inserted.
+ zzl
insertion of graviton or
)
RR vertex operator into the
The boundary of the disc is stuck on the D-brane, as shown in the picture on the right. In the type I1 string theories D-branes are incorporated as so-called boundary states. The subtlety with the calculation is to fix the absolute normalization of the boundary states which is needed in order t o get the correct coupling. The way one does the calculation is to compute the interaction between two parallel D-branes. They interact via exchange of closed strings. The relevant diagram is shown below.
413
--*>-** I
- - ---
.
I
interaction between two parallel D-branes
In this closed string tree-level exchange diagram world-sheet time runs along the axis of the cylinder. However, we may equally well consider this diagram as an open string one-loop diagram, with world-sheet time running around the cylinder. This is just the open string partition function which can be computed straightforwardly. One finds a zero result which can be understood from the fact that this brane configuration preserves half of the supersymmetry. The attractive force mediated by the exchange of dilatons and gravitons and the repulsive force due to anti-symmetric tensor exchange cancel. This is the BPS or no-force property. One can separate these two contributions as they belong t o different sectors ((NS,NS), us. (R,R)). Comparing these two contributions t o the amplitude to a field theoretic calculation one finds for the tension of a single Dp-brane
Furthermore, one finds the same ratio between charge and tension as for the SUGRA p-brane solutions. Up to numerical factors the expression for the tension can be easily understood: the powers of 1, = are needed for dimensional reasons. In natural units, where ti = c = 1,the tension, which is rnassluolume has dimension (length)-(P+'). The dependence on the string coupling constant, rp l / g s follows from the fact that the tension is computed from a disk diagram with Euler number + l . At weak coupling, gs 3 0, the D-branes are very heavy and are not visible in the perturbative excitation spectrum of the type I1 string theories. N
Following Polchinski, we have argued that the D-branes, which have a microscopic description in open string theory, are in fact the same objects
414
as the classical p-brane solutions of the low energy effective SUGRA theories which know nothing about open string modes. The two descriptions are good in different regimes of the parameter space. The D-brane picture is good at
weak string coupling where the string is perturbative. In the presence of N D-branes, the effective coupling is Ng,, which must stay small. Also, we have developed the D-brane picture in Minkowski space-time. This assumes that we can neglect the back reaction of the brane on the background geometry. This is justified if the number of branes which carry energy-momentum, is small. The SUGRA picture also requires weak string coupling since that was assumed in the construction of the leea. In addition, the curvature of spacetime must stay small everywhere (in string units). This requires that g,N is large since this is the condition that the characteristic length scale in the solution is bigger than the string scale. An important difference of the two descriptions is that while the D-branes couple t o open strings and carry a gauge theory on their world-volume, there are no signs of open strings and gauge fields in the p-brane picture. Nevertheless, if the two descriptions are ‘dual’ t o each other, they should describe the same physics. The AdS/CFT correspondence, which we will discuss in the final lecture, establishes the relation between the two pictures. The solutions we have discussed here are extrernal p-brane solutions. In appropriately chosen units they satisfy the equality mass = charge (i.e. the coefficients in front of the two terms in (60) are equal). This equality is known as the Bogomolnyi bound and general solutions satisfy m a s s 2 charge. The solutions which do not satisfy the bound are called non-extremal. They break all 32 supercharges. The nomenclature here is the same as for charged (Reissner-Nordstroem) black holes who are characterized by two parameters, their mass and charge. The extremal solution has a degenerate horizon a t r = 0, gtt has a double zero there. The non-extremal solutions have an inner and an outer horizon. The curvature blows up at the inner horizon but the singularity disappears in the extremal limit. The construction of non-extremal solutions can be found in [28,29].
41 5
Lecture 4: The AdS/CFT correspondence In lectures two and three we have provided background material of string theory and classical solutions of the supergravity equations of motion, which describe the dynamics of the massless string excitations at low energies. One of the main results was the identification of D-branes and brane solutions in supergravity as two descriptions of the same objects. In this lecture we will, after providing additional background material, formulate the AdS/CFT correspondence, also known as Maldacena conjecture. This provides a precise identification between supergravity on the one side and gauge theory (on the brane) on the other side. Of course, this assumes that we are in a particular corner of parameter space, which we will specify. From now on we only consider the case p = 3, the self-dual three-brane solution of type IIB supergravity with a four-dimensional world-volume. For N coincident three-branes the solution is
ds2 = f ( r ) - 1 / 2(-dt2
F5 = (1+ *)df-'
A
+ dx2) + f ( ~ - ) (dr2 l / ~ + T2dfig) , dxo A ds'
A
dx2 A dx3
= 40= const.
f ( r )= 1
R4 +,
R4 = 47~g,(r12 N .
7-4
dfi; is the length element on S5 and
* the Hodge star. The constant term in
f ( r ) is an integration constant and it was chosen such that
as r + 00. We can also look at the limit of the metric as r near-horizon limit). Then
R4 r4
f ( r )= 1 + -
and
(
d
~
-
r2 ~ -)(-dt2~ R2
-
R4
7,
R2 + dx2) ~ ~+ -dr2 ~ ~+ R2dRE. 7.2
0 (i.e. the
416
After the change of variables p
= $ this metric is
which may be recognized as the metric on the product-space Ads5 x S5. Before saying more about anti-de Sitter space we make some remarks. As shown above, the 3-brane metric interpolates between 10-dimensional Minkowski space, being the asymptotic space-time for r -+ co and the nearhorizon geometry Ads5 x S5 at r -+ 0. The geometries in the extreme regions have higher symmetries (bigger isometry groups and more supersymmetry) than the brane solution has. Specifically, while (62) preserves 16 supercharges, both Minkowski space and Ads5 x S5 are invariant under 32 supercharges. The Minkowski space is of course just the type IIB vacuum which is an exact perturbative ground state of string theory, i.e. t o all orders in a' and g s . The same can be shown to be true for Ads5 x S 5 , but it is not true for the interpolating solution (62) [47]. Both AdSd and Sd are symmetric spaces with curvature tensors3'
The upper (lower) sign is for anti-de Sitter space (the sphere). They are conformally flat, i.e. their metrics are proportional t o the flat Minkowski (Euclidean) metric.31 They thus have have vanishing Weyl-tensor. They are solutions of the Einstein equations with cosmological constant derived from the action ( d - 2)(d - 1) S = ddzJ-g(R+R), .A= R2
.I
3 0 0 ~ conventions r for the curvature tensors are [Vm,V,]Vp = -RmnpqVpr R,, = RPmpn,R = g m n R m n . 31This is obvious for AdS from (66). For S d , defined as = R2 one sees this after defining the stereographic coordinates (say, on the Northern hemisphere) xi =
i = 1,.. . , d and xd+l = R ( l The metric is ds2 = Ctz:(dxi)2
A)which solves the defining equation for = gijdyidyj with g"23
-
7623' .
2)
=1
$ for
+ A.
417
For the AdS/CFT correspondence both anti-de-Sitter space and the conformal group play a central r6le. We will return to Ads space after the following brief introduction to the latter. The Poincarh group is familiar from introductory physics courses as the invariance group of (length)2 in Minkowski space, i.e. the invariance group of (ds)2= q,,dx~dx’, p, v = 0,1,. . . ,D - 1. The Poincark transformations are xfi 4 Apyxv a p where the constant matrix A,” satisfies q p , , A ~ p A u=o qpu and a, is a constant vector. A generates Lorentz transformations and a translations. The conformal group is the invariance group of the lightcone, i.e. of a11 transformations which leave ( d ~=) 0~invariant. Clearly this group contains the PoincarC. group as a subgroup, but it is strictly bigger. For instance, constant rescalings xp exx, and inversion xp + x,/x2 also leave the light-cone invariant. If we follow an inversion by a translation by b and a second inversion, we arrive at the special conformal transformations ,+ xp+xZbp 1 + 2 b , x + b Z x T which, in contrast to the inversion, can be expanded around the identity transformation.
+
---f
~
We will now proceed as follows. We will show that infinitesimal translations P,, Lorentz transformations L,, , rescalings (D) and special conformal transformations K p generate the i ( D + 1)(D 2) parameter conformal group in D-dimensional Minkowski space. We will then show that the conformal group is isomorphic to SO(D,2). It is clear from the form of the special conformal transformations that the conformal group acts non-linearly on Minkowski-space. But being isomorphic to SO(D,2) it acts linearly on RD+2 endowed with a metric with signature ( ( + ) D , ( - ) 2 ) . We will then define anti-de Sitter space as a hypersurfaco in this space on which S O ( D , 2 ) acts isometrically.
+
+
Ep(x) the Under infinitesimal diffeomorphisms x, --$ x’p = xp Minkowski metric changes as q,, --$ qp, +ap& This is proportional to the original metric if ape, + a,(, = fq,, for some function f(x). Taking the trace of this equation gives f = $ ( a . which leads to the conformal Killing equation 2 a,<“ a u t , = -p.O?,”. (69)
c)
+
418
+
One may show that the most general solution to this equation is < p = a p +Xxp - 2(b.x)xp+x2bp where X is a constant, ap and b p are constant vectors and wpv = -wVp a constant antisymmetric matrix (wp, = ~ppw,,).
X and bfi parametrize infinitesimal translations, Lorentztransformations, rescalings and special conformal transformations of xp, respectively. They are generated by Pp = ap, L,, = xpa, - x y d p , D = x.a and Kp = -2xpx.d+x2ap whose algebra is easily worked out. The non-vanishing commutators are ap, wp,,
[D, Ppl
= -pp
[DlKPI = [PpyKv]= [LP”,Ppl = L ” 7 KPl = [ L p v lLpul =
,
Kp, - 2 ~ p v D+ 2Lpv
-VppPv -VppK”
+ V”PPP + VVPK,
1 7
-VppLvu - V v u L p p + VpuL,,
+ V,pLpu
.
(70)
+
1 - Kp) and L , , D + ~ = -:(Pp Kp) If one defines L D , D + ~= -Dl L,D = z(Pp the above commutation relations can be combined into the following single relation:
[ L M N l LPQ]
= - 7 M P L N Q - V N Q L M P -k V M Q L N P
+VNPLMQ
(71)
+
where M , N , . . . = O , l , . . . , D 1 and ~ M N= diag(-1, + 1 , . . .,+1,-1) is the invariant metric of S O ( D , 2 ) . This establishes the isomorphism of the conformal algebra of D-dimensional Minkowski space with so(D,2 ) , the Lie algebra of SO(D,2).32 SO(D, 2 ) acts linearly on RD+2 with metric ( d ~ =) ~ v p y d y p d y ” f d ~- (~ d )~ ~~ + ’ ) ~ .
+
We can now identify D-dimensional Minkowski space as a subspace of
RD+2 and describe the non-linear action of the conformal group on it. To this end, consider the subspace defined by the constraint vpVypy” = uv where we have defined u = yD+l + yD and v = yDf’ - y D . Note that this is 32Here we assume D > 2. In D = 2 the conformal algebra is infinite dimensional. This is in fact very relevant for the world-sheet aspects of string theory and two-dimensional conformal field theories in general. We did not encounter this in the lecture on string theory since we fixed conformal transformations when going to light-cone gauge.
419
a D-dimensional cone inside IRD+2. Firstly, this equation constitutes one constraint. Secondly, if y is a solution of this constraint, then so is Xy for any non-zero real A. We can use this rescaling freedom to set CLl(yi)2 = 1= (yo)2 (yD+1)2which shows that this cone has the topology (SD-’ x S1)/Z2, where the Z2 accounts for the fact that rescaling by f X are equivalent. This is in fact the conformal compactification of Minkowski space on which the conformal group acts properly (we need to compactify since the inversion transformation maps the origin to infinity, which is not part of Minkowski space).
+
We now solve the constraint locally (in a patch with u coordin&es xp via
# 0) by
defining
The linear SO(D,2 ) transformations on yM induce conformal transformations on x”. Specifically, we find the following relation between linearly acting SO(D,2 ) transformations of y and conformal transformations of x :
(73) Each of these matrices Mn/iN satisfies M M ~ M N Q=~~ ~M Q N Consider . now the following hypersurface in D
-(yo)2
+ C(Y*)~ - (yD+1)2 = ~ p u y p y ”
- UZI=
-R 2 .
(74)
i=l
+
This (D 1)-dimensional hypersurface, together with the induced metric, defines A ~ S D +(~D, 1)-dimensional anti-de Sitter space. This constraint
+
420
can be solved for v. Doing this and introducing the coordinates xp = find for the induced metric
we
u
-+ 00 corresponds t o the boundary of Ads, which is just compactified Minkowski space as discussed before. This can be seen by looking at the constraint equation (74) (after dividing by u2 the r.h.s. vanishes for u -+ co whereas the 1.h.s. stays finite) or by looking at the Ads metric in the form (75). If we introduce the coordinate p = R2/u the metric becomes
+
ds2 = -(dp2 R2 qpydxpdx”) P2 which coincides with the first part in (66). The boundary is now a t p = 0. Simultaneously rescaling u -+ u/X and xp -+ Xxp leaves the Ads metric invariant but induces a Weyl transformation of the metric on the boudary: Vpv
+
X2%”.
After having provided some background on AdS space and the conformal group, we will now return t o branes, gravity and gauge theory. Take a single D3-brane. The fields living on its world-volume arise from the excitations of open strings ending on the brane. At low energies, lower than the string scale l/&, only the massless string states can be excited and their dynamics is governed by a low energy effective action on the world-volume. The massless open string states are the NS gauge field Am and its fermionic superpartner, the gaugino, a Majorana-Weyl spinor in ten dimensions. Together these fields form the N = 1, d = 10 Yang-Mills supermultiplet. The brane breaks the ten-dimensional Lorentz invariance SO(1,9) -+ S0(1,3)x SO(6) t o Lorentz transformations along the brane and rotations in the transverse space. The gauge field 10 decomposes as 10 = ( 4 , l ) (1,6): A M = ( A p , @ ) .The six scalars describe the fluctuations of the brane in the directions transverse to it. (They are the Goldstone bosons associated t o the spontaneously broken translation symmetry.) The gaugino decomposes into four Weyl spinors which transform as 4 of SO(6) and their complex conjugates. (They are the Goldstinos of the sixteen, in the presence of the brane, spontaneously broken supercharges.) Altogether we get one N = 4 U(1) vector multiplet on the
+
42 1
four-dimensional world-volume of the D3-brane. The generalization from one to N coincident D3 branes is straightforward leading to the gauge group U ( N ) (c.f. the discussion in lecture 2). All fields are in the same supermultiplet and hence they all transform in the adjoint representation. The meaning of the U(1) C U ( N ) factor is as above. The scalar in the U(1) multiplet corresponds to the center-of-mass motion of the branes and those in the adjoint of S U ( N ) to their relative motion. The action of N = 4 SYM theory is highly restricted by the large amount of supersymmetry. In particular there is only one coupling constant. Another important consequence of the large amount of supersymmetry is that bosonic and fermionic contributions to divergences in Feynman diagram calculations cancel and the quantization procedure does not require introducing a scale into the theory. This in particular means that the beta function vanishes. This means that as a quantum theory, Af = 4 SYM is conformally invariant. In other words, the conformal symmetry exhibited by the classical theory is not broken in the process of quantization. In fact, the theory is invariant under local U ( N ) gauge transformation and under global super-conformal transformations which generate the supergroup33 SU(2,2)4)3 SU(2,2) x SU(4)77,N S0(4,2) x SO(6)z.
(77)
On the r.h.s. we have written the bosonic subgroup. The S0(4,2) factor is the conformal group in d = 4 while S o ( 6 ) is ~ called R-symmetry group. It can be understood from the brane picture as the rotation group of the transverse space and we have seen how the various fields in the SYM multiplet transform under it. The fermionic generators are the sixteen supercharges Q of the N = 4 supersymmetry algebra plus sixteen special supersymmetries S which arise in the commutator between Q and the special conformal transformations K . The generators of the R symmetry appear in the {Q, S}anti-commutators. The dynamics of the massless fields can again be described by a lowenergy effective action. For the gauge fields this is the Born-Infeld action 33The corresponding superalgebra is discussed in the lectures by J . Zanelli.
422
[49]:
The integral is over the brane world-volume, Tr the trace over group (ChanPaton) indices34 and gap the induced metric on the world-volume. We have omitted fermions, transverse scalars and Wess-Zumino couplings, as they are not relevant for our discussion. If we extract the O(F2)term and compare with the usual gauge kinetic term - & F ~ , , F a ~ v we are led t o the identification 2
SYM =
(79)
Of course we also have perturbative closed string excitations in the bulk and the closed string modes interact with the open string modes which are localized on the brane. The complete effective action for all massless modes has the form
where Sbulk contains only closed string modes, &,,an, only open string modes and Sint interaction terms between them. The coupling constant is proportional to R , c.f. below. In the decoupling limit a' + 0 all higher derivative corrections as well as the interactions between closed and open strings can thus be neglected and we are left with pure four-dimensional N = 4 SYM on the world-volume of the brane and free type IIB supergravity in the bulk (i.e. free gravitons and their SUSY partners) with no coupling between these two theories. In the previous lecture we have collected evidence that the D-branes of string theory and the p-brane solutions of supergravity are complementary descriptions of one and the same object. We have just seen that the a' + 0 limit leads in the D-brane picture t o two decoupled systems: SYM theory on the branes and free supergravity in the bulk. 34This is unambiguous to O ( F 2 ) . At higher orders one has to be more specific about the prescription how to perform the trace; see [49].
423
The next step will be to find the correct decoupling limit in the SUGRA picture and to compare to the above. The following analysis, first performed in [50], will give an important clue.
If a dilaton hits the D-brane, it can be absorbed, thus exciting the Dbrane. The quantum excitations of the D-brane are the open string modes. Indeed, as we see from the Born-Infeld action, the dilaton couples to the gauge bosons.
hits D-brane and decays into two gluons
At lowest order, this is the cubic coupling (c.f. (78)) $ 4 F 2 . To find the strength of this coupling one should normalize the fields such that they have canonical kinetic energies. This means that we have to rescale the gauge bosons by and the dilaton by IC. This leads to a coupling constant o( tc (which does vanish in the decoupling limit). The (tree level) cross-section for a dilaton is then u 0: n 2E3N2as we shall now explain. The I C ~is clear as the cross-section involves the square of the amplitude which is 0: IC; N 2 because there are that many gluons into which the dilaton can decay. The cross-section for the scattering of a point-particle from a three-dimensional object in nine space dimensions has dimension the only dimensionful quantity t o fix the dimension is E l and the factor E3 indeed arises from the kinematics of the scattering process. A careful calculation gives for the absorption crosssection [50] of a dilaton incident at right angle (i.e. its momentum has no component parallel to the D3-brane)
U D = ~
T4 2~ 6 g2s a1 4E 3 N 2 = -E3R8
8
,
R4 = 4.rrg,Na'2.
424
On the supergravity side one solves the wave equation for the dilaton in the s-channel in the brane geometry. This exhibits a t low energies E << 1/R a potential barrier separating the two asymptotic regions r << R and r >> R, where r is the distance from the brane. One then obtains the absorption cross section from the tunneling probability through the barrier[51]. This calculation was also performed in [50] with the result (7D3 = USUGRA.
(82)
This also works with other SUGRA particles, e.g. the graviton and antisymmetric three-form tensor; their absorption cross sections agree as well. Eq.(82) is a very interesting result: in the D-brane picture a particle incident from infinity produces excitations of the gauge theory on the brane; in the SUGRA description of the brane a particle tunnels from the region r >> R to the region r << R and produces an excitation there. The two & priori unrelated processes occur at exactly the same rate. One is tempted to identify the N = 4 SYM theory with gauge group U ( N )with the excitations in the near horizon region, r << R, of the brane geometry, which we already know is Ads5 x S5. This gets further support from the following identification of two types of low-energy excitations, as measured by an observer at infinity (for this observer the coordinate t appearing in (62) is the time coordinate as g t t ( r = m) = -1). Due to the energy dependence of the cross section, 0 0: E 3 , low-energy SUGRA modes in the region r >> R decouple from the near-horizon region. At r >> R we thus have a free35 SUGRA theory. On the other hand, the energy of an excitation in the near horizon region appears redshifted for an observer a t infinity, Em = [gtt(r)/gtt(oo)]1/2Er= S E T . They cannot penetrate the energy barrier which separates the two asymptotic regions.
So, in the D3 and in the SUGRA picture we get two decoupled systems. In both cases the system in the bulk is free type IIB SUGRA. But then, if the D-brane and the SUGRA brane describe the same object, we should identify 35The gravitational interaction is negligible at^ low energies. The dimensionless coupling constant is nE4, where E is the typical energy of the interaction. For E << l/aand K 0: this is << 1.
425
the two other systems: N = 4 SYM theory with gauge group U ( N )and type IIB string theory on A d s 5 x S5. Before pursuing this further, we need to clarify two points. Firstly, we still need to be more specific about the precise form of the near horizon limit, which zooms into the region of the three-brane SUGRA solution which we want to identify with the gauge theory of the D3 picture. This limit should involve a' +. 0, as this was the decoupling limit for the D3-brane, and is defined as follows: r a'+O, r4 0 such that U = - fixed. (83) a' In this limit a' scales out of the metric which becomes U2 ds2/o' = (-dt2 d x 2 ) y d U 2 J m d R g .
&G3
+
+
+
(84)
-
The limit is taken such that for the observer at infinity the string excitations in the horizon region, which have energies EO l/& and which are redshifted to Em a E ~ U stay finite. This observer sees two decoupled systems: free SUGRA in the asymptotic region and type IIB string theory compactified on A d & x S5.
-
-
The second point we need to clarify is the region of validity of the two calculations of the absorption cross section. In both pictures we have assumed that E << /laotherwise ; massive string modes can be excited and their effect has t o be taken into account. On the supergravity side we also have to require that (i) the typical length scale R of the geometry is large compared to the string scale, i.e. R >> 6; otherwise we have to take higher derivative corrections ( ~ y ' ) ~ Rto~the + lsupergravity action into account, which we did not. (ii) We also need gs << 1 since we have neglected string loop effects. With the help of R4 gsNat2 and gCM gs we can translate these restrictions to the following conditions on the gauge theory parameters:
-
-
-
R4
g$MN = 2 ~ N g = , ->> 1 2a'2
and, since gs << 1,
N
-+ 0 0 .
(85)
This specifies a large N YM theory at strong 't Hoop coupling A. X is the effective coupling constant and loop counting parameter in the large N limit of YM theories.
426
We should thus identify the excitations in the near-horizon regions of the three-brane geometry, which, as we have seen above, is A d s 5 x S 5 , with the excitations of N = 4 U ( N ) SYM theory at large 't Hooft coupling and in the limit N + 0 0 . ~Based ~ on the analysis presented above, this conclusion was first drawn by Maldacena in his famous paper [7] and is called Maldacena conjecture, or, since it involves A d S space on the one hand and a conformal field theory (N = 4 SYM) on the other, the AdS/CFT correspondence. In the weak form the conjecture states that N = 4 SYM with U ( N ) gauge group at large 't Hooft coupling and in the limit N + 00 i s equivalent (dual) t o type IIB supergravity compactijied on A d s 5 x S5. Note that even in this weak form the conjecture has far reaching implications. It relates a classical weakly coupled supergravity theory with a strongly coupled quantum field theory. This is a duality pair of the type we discussed in the beginning. The perturbative regimes of the two theories, g$,N g,N R4/cd2 << 1 for the gauge theory and R4/aI2 >> 1 for the supergravity theory, do not overlap. The good news is that such a duality is very useful for exploring the strongly coupled gauge theory; this is done by performing computations in a classical gravity'theory. The bad news is that it is extremely difficult to prove such a duality conjecture. N
-
Further support for this conjecture comes from comparing the symmetries. The isometries of A d s 5 x S5 are SO(4,2) x SO(6). But these are precisely the bosonic (non-gauge) symmetries of N = 4 SYM theory (c.f.
(77)). This discussion can be extended to the full supergroup SU(2,214). This is the maximally extended supersymmetry algebra on Ad&. It is realized as global symmetry of gauged N = 8 supergravity on A d & which can be obtained as Kaluza-Klein reduction of type IIB SUGRA on S5. What is gauged is the isometry group of S5. But SU(2,214) is also the maximally extended superconformal symmetry in four-dimensional Minkowski space, i.e. the invariance group of N = 4 SYM. Furthermore, the boundary of A d s 5 is four-dimensional Minkowski space on which the isometries of A d & x S5 act 36Earlier we had mentioned that in the presence of N D-branes the effective coupling constant is N g s .
427
as conformal and %symmetry transformations. This leads to the statement that the field theory which is dual to the string theory ‘lives’ on the boundary of AdS5. Recall the observation made above that if we simultaneously rescale U 4 XU, ( t ,x) -+ X - ’ ( t , x) the metric does not change. This leads to an interpretation of U as the energy scale in the field theory: large U corresponds to the UV region and small U to the IR region (the boundary is at U = w).
The conjectured duality has several very remarkable features. First, it is a duality between a gravity theory and a field theory. In addition, these theories live in different numbers of dimensions and have completely different degrees of freedom: gravity us. gauge degrees of freedom. There is the notion of the master field for large N QCD (see e.g. [ll,121). One can show that fluctuations of gauge invariant observables vanish in the N 4 w limit. This is then analoguous to the classical limit h + 0 in which the functional integral is dominated by classical paths. In the same way, there should be a master field such that all Green functions are given by their value at the master field. What the Maldacena conjecture suggests is that this master field for =4 SYM is in fact a gravity theory in ten dimensions. Also, the QCD string of these theories is simply the fundamental type IIB string which lives, however, not in four-dimensional space-time, but in ten-dimensional Ad& x S5. This is most convincingly demonstrated by the SUGRA computation of the gauge theory Wilson loop [52]: a static well separated qtj pair must be viewed as the endpoints of an open type IIB string at the boundary of Ad&. In order to minimize its length (and hence it energy) the string follows a geodesic. The geodesic does not lie in the boundary, which it would if the space were flat, but extends deep into the Ads space. The qq potential is then given by the (regularized) length of this geodesic. One might wonder whether the duality only holds for the parameter region specified in (85), which corresponds to neglecting all O ( a y ’ / R 2and ) all string loop effects or whether it can be extended. Correlation functions of Ad& x S5 type IIB string theory will have a double expansion in powers of gs and a‘/R2 N X - l l 2 . In the large N limit we can write this as an expansion in powers of 1/N N gs(a’/R2)2where each coefficient has an expansion in powers of X-1/2. Clearly, a term at some power in the gs expansion has the
428
same power in the 1/N expansion. Since the type I1 string theory includes only closed oriented strings, we find the same general structure as in the field theory where each correlation function has an expansion in powers of 1 / N 2 , each coefficient being some function of A. The functions of X have different expansion from the point of view of string theory and of field theory. A stronger version of the AdS/CFT correspondence states that both expansions give rise to the same function of X at each power of 1/N2. The fact that one expansion is in powers of X whereas the other is in powers of X-1/2 reflects the fact that the AdS/CFT duality is a strong/weak coupling duality. In the strongest version of the conjecture the two theories are considered as exactly identical for all values of N and gs. Here the corrections distort the space-time which is only required t o be asymptotically Ad& x S5. The gauge theory then effectively sums over all such space-times. As we have remarked in the first lecture, after identifying a pair of theories of which we have indications that they are dual t o each other, we still need t o find the map between them. This amounts t o giving an explicit prescription of how to compute gauge theory correlation functions in the dual supergravity theory. This was done in [8,9]. We will not review the results of these papers nor will we present any of the many applications. They can be found in the reviews on the AdS/CFT correspondence. There the interested reader may also find extensions of Maldacena’s conjecture t o other theories - in dimensions different from four and to theories with other gauge groups, less supersymmetry, not conformally invariant theories, theories a t finite temperature (the latter involve the non-extremal brane solutions37) and theories on non-commutative Minkowski space (the latter require, in addition to the metric and (R,R) four-form potential, also a non-trivial (NS,NS) two-form). 371f one computes the Bekenstein-Hawking entropy and compares it t o the entropy of N = 4 SYM at temperature T B Hone , finds that the field theory value is bigger by a factor 4/3. The fact that there is a discrepancy does not come as a surprise, since the SUGRA calculation is valid for g s N + 00 whereas the field theory calculation assumed weak coupling, i.e. g,N -+ 0. One should thus view the SUGRA result as a prediction for the strongly coupled field theory.
429
Maldacena’s original paper has ignited a storm of activity.38 Hopes were high that extensions of his conjecture would lead to new insight into realistic, i.e. non-supersymmetric QCD. However, all attempts to find its supergravity dual (at large N ) have been futile. More optimistically, the AdS/CFT correspondence has pointed in a very interesting direction, namely the possible connection between gauge theories and gravity (string) theories. In fact, this might also provide insight into string theory in non-trivial backgrounds. The reason why the AdS/CFT correspondence has so far been mainly checked in its weak form is our inability to quantize string theory in an Ads background. However, besides flat Minkowski space-time and Ads5 x S5 there is one other maximally supersymmetric background, which is the secalled pp-wave [55,56]. In this background, which can be obtained from Ads5 x S5 by a limiting process, called Penrose contraction, the string can be quantized exactly (in the Green-Schwarz light cone formulation) [57]. Recently a correspondence between the string theory in the pp-wave background and supersymmetric Yang-Mills has been conjectured 1581. This is presently being explored.
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gauge theory and anti-de Sitter supergravity”, Eur. Phys. J. C 22, (2001) 379-394, hep-th/9803001; J. Maldacena, “Wilson loops in large N field theories”, Phys. Rev. Lett. 80 (1998) 4859-4862 (1998), hep-th/9803002. 53. J. Polchinski, “Dirichlet branes and RR charges”, Phys. Rev. Lett. 75 (1995) 4724-4727, hep-th/9510017. 54. G.T. Horowitz and A. Strominger, ”Black Strings and p-Branes”, Nucl. Phys. B360 (1991) 197-209. 55. R. Gueven, “Plane wave limits and T-duality”, Phys. Lett. B 482(2000) 255-263, hep-th/0005061. 56. M. Blau, J. Figueroa-O’Farrill, C. Hull and G. Papadopoulos, “A new
maximally supersymmetric background of IIB superstring theory” JHEP 0201, 047 (2002), hep-th/0110242; “Penrose limits and maximal supersymmetry”, hep-th/0201081; M. Blau, J. Figueroa-O’Farrill and G. Papadopoulos, “Penrose limits, supergravity and brane dynamics” l hepth/0202111. 57. R. R. Metsaev, “Type IIB Green-Schwarz superstring in plane wave
434
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Geometric and Topological Methods for Quantum Field Theory Eds. A. Cardona, H. Ocampo & S. Paycha @ 2003 World Scientific Publishing, pp. 435-449
GROUP CONTRACTIONS AND ITS CONSEQUENCES UPON REPRESENTATIONS OF DIFFERENT SPATIAL SYMMETRY GROUPS. MAURICIO AYALA-SANCHEZ Departamento de Matemciticas, Universidad de Los Andes, Bogotci - Colombia. E-mail:
[email protected] RICHARD W. HAASE Departamento de Fisica, Uniuersidad Nacional de Colombia, Bogotci - Colombia. Centro Internacional de Fisica, A.A.4948, Bogota' - Colombia. Email:
[email protected]. edu.co We investigate the group contraction method for various space-time groups, including [ s o 3 + &I, [ S 0 3 , 1 --t GS], [SOs-h,h -+ p 3 , 1 ( h = 1,2)], and its consequences for representations of these groups. Following strictly quantum mechanical procedures we specifically pay attention in the asymptotic limiting procedure employed in the contraction [G 4 G'], not only t o the respective algebras but t o their representations spaces spanned by the eigenvectors of the Cartan subalgebra and the eigenvalues labelling these representation spaces. Where appropriate a physical interpretation is given t o the contraction procedure.
1
Introduction
The group contraction method was initially introduced by Inonu and Wigner in 1953 [l],[2], [3]. The main interest resulted from the study of the transition of relativistic to nonrelativistic quantum mechanics in the asymptotic limit when velocities are small compared t o the velocity of light. Under this limit the Lorentz group becomes the Homogeneous Galilei group. The group contraction method continues t o be a subject of active research in particularly as applied to quantum groups. In general this method determines under a change of parameter scale a limiting process on the group generators and its algebra producing as a consequence a new group with corresponding algebra. The main difficultly in a general approach is the construction of the sequence of limiting representation spaces. In this paper we focus on this aspect of the problem by reviewing some well-know applications to space-time symmetry groups. We employ a direct approach by which we apply the limiting procedure t o the standard quantum mechanical treatment of the construction of representation spaces, ie. as an eigensystem problem of the generators of the algebra, ci la SO3 of angular momentum theory. Since a Lie group is uniquely determined by its algebra about the identity,
436
it is possible and easier to discuss contraction with regard to Lie algebra. From this point of view, group contraction is defined as follows: Definition 1 Inonu- Wagner Contraction (Hermann [4]): Given a Lie algebra L associated with G, and a basis Xu(a= 1,2, ..,n)of the vector space of L ,
satisfying n
[x,,x,]= C 0 : b x c
a , b = 1,...,n.
c= 1
where D:, are the structure constants of L with respect to this basis, the Jacobi identity imposes the following condition o n the structure constants n
+
c ( D i d D Z c D&Dic
+ D:,DiC) = 0.
(1)
c= 1
For the basis of L, we suppose that with a continuous, directed, contraction parameter X
the infinite sequence [X,]’and the corresponding structure constants [D:,]’ are known, the lim’-m
[D3’= [D:,lm exists for all a , b, c, and
eqn.(l) is consistent under the limit. Then as a consequence [D:Jm generates a new algebra L’ called the contraction of L. With this as a starting point we investigate in the following sections the consequences of the group contraction method for representation spaces of various well-known spacetime symmetry groups, including the de Sitter groups contracted t o the Poincari: group. 2
2.1
Contraction of SO3
-+
Ez
The Algebras
With the main purpose of establishing our procedure we begin with the algebra contraction so3 + e2. The rotation group SO3 in R3 has three generators { J1, J z , 53) corresponding to infinitesimal rotations and satisfying the associated algebra
[J,., J s ] = ieisJt
r , s , t = 1,2,3,
(2)
437
ets is a rank-(2,l) tensor antisymmetric in the lower indices with & = 1. To initiate the contraction, we introduce a positive real parame-
where
ter R and construct the following sequence of elements defined on the so3 algebra, with J s , = ~ J s ( R ) (See Appendix). In principle we don’t assume the linear relation J = R A P of classical mechanics because it actually is a consequence of the following contraction process.
Jo
JS,R IIs -= -
53,
K,
R
= R+m lim
II,
s = 1,2,
(3)
for which the resulting algebra takes the form [nl,n2] =
i
jiZJ0,
On taking the limit R
-+ 00
[ K i ,K2] = 0,
[n2, JOI
= in1,
[JO,nll = znz.
(4)
in eqn.(4), we obtain the algebra
[Jo, Ki] = iK2.
[K2,Jo] = i K i ,
These relations define the Euclidean algebra of ‘translations’ and rotations, denoted as e2 = t 2 @ s02. Here the generator J3 associated with rotations about the z-axis remains unchanged, while the generators J1 , J2, determining rotations about the 2 and y axes respectively, are transformed under the limit into generators of ‘translation’ in directions y, 2. Interpreting this one requires the symmetry so3 about the centre P N ~= S (O,O, * R ) of the sphere of radius R is locally seen as an approximate symmetry T2 >a SO2 of the tangent plane at the pole or the plane at infinity when R --+ 00 (see [5] for a generalization to [On+l En]). Taking the Casimir of SO3 +
J‘J,
= J;
+ 522 + 532 = J.J,
and applying eqn.(3), we obtain
+
JR.JR = R2(IIf II;)
+ J:.
On taking the contraction limit define the Casimir for
E2
as
Physically these Casimirs reflect the conservation laws of angular momentum and linear momentum, the one being the limit of the other under contraction. In the standard treatment of SO3 it is usual to transform to the spherical tensor basis of the so3 algebra
J+
= J1 + iJ2,
J-
3
J1 - iJ2,
J*+ = J T l
438
where the generators satisfy the algebra
[J3,J+] = +J+,
[J+,J-] = 2J3,
[J3,J-]
=
-J-,
and the Casimir is now expressed as 1
J.J = -(J+J2
+ J- J+) + J i .
Applying the contraction to J*, we obtain K* in
K+ = K1+ iK2, K-
E K1-
iK2,
E2
defined as
K*t = K,,
J*,R
K* = lim R+CQ R ’
with commutators
[K+,K-]= O ,
[Jo,K+] =+K+,
[Jo,K-]= - K - ,
(5)
where the associated Casimir takes the form
2.2
The Representation Spaces
We now investigate the effect of contraction on the representations space of SO3. The bases of the representation spaces, labelled by half-integers j and m, are spanned by the eigenvectors Ijm) for J.J and 5 3 (see [ 6 ] )
+
J. J l j m ) = Ijm)j(j l), J+Ijm) = lj, m + q a j m , J - I j m ) = Ij,, - l)bjm,
with -j 5 m 5 j J3ljm) = Ijm)m, Jlajm((’= j ( j 1) - m(m 1) llbjm112 = j ( j + 1) - m(m - 1).
+
+
The eigenvectors and corresponding eigenvalues of K.K, Jo in as
K.Klkm) = Ikm)k2,
E2
are given
Jollcm) = 1km)m.
(7)
From eqns.(7) and (5) we require
+
JoK+Ikm) = K+Ikm)(m 1) JoK-Ikm) = K-Ikm)(m - 1)
+ K+lkm) = Ik,m + l)6jm
(8)
=+ K-Ikm) =
(9)
Ik,m - l ) b j m
-
Evaluating 6 j m , b j , using eqns.(6,7) and (8,9) we obtain
K.KIkm) = K+K-Ikm) = Ikm)bjm6jm = Ikm)k2
=+
bjmiijm
= k2.
In addition using the adjoint property of the operators
(h(K-K+(h= ) (km(km)ii?,Gjm2 0
+ k2 = Ti*
jm
6I*.
>0
439
From this we deduce that
In order to establish a connection under group contraction between the two groups we need to correlate their respective group actions. Since JO is unchanged by contraction its group action also remains unchanged. The Casimirs provide a different situation and therefore the eigenvalues j , and hence eigenvectors, are dependent on R in such a way that IimR,, =k (see [7,8,9]). Thus we introduce j~ and eigenvectors I j ~ mwith ) action
5
n + I j ~ m=) I j ~ , m + 1)ajRm - 1)bjRm
JJ-IjRm) = IjR,
2 - .
+
+
IlajRmII = . I R ( ~ R 1) - m(m 1) l l b j ~ m 1 1 ~ j R ( j R f 1) - m(m - I),
and satisfying the conditions R lim j=k
R+CQ
R
lirn I j ~ m= ) Ikm).
R+C=2
Therefore we obtain that JR.J R
K . K J k m ) = R+oo lirn R2 bRm)
We then see that from eqns.(6) and (10)
= Ikm)k2.
as required. From eqns.(3) and (10) we can infer that j~ Rk + hjR R(hk) = Rp, in analogy to the standard classical angular momentum definition j’= r‘ x @’.Note that as R increases, j~ takes large values corresponding to the regime of classical physics. N
3
3.1
N
Contraction of S03,1-+ G3
The Algebras
As our second application let us consider the homogeneous Lorentz group S03,1, for which we have six generators J T , B, ( r = 1,2,3) obeying the
440
following algebra
where B, represents the Lorentz boosts, and J, spatial rotations which clearly form a subgroup of S03,1. The Casimirs of the proper Lorentz group are found to be
Ci'3= J'J, - B'B, = J.J - B.B,
C,1'3= J'B,
+ B'J,
= 2J.B.
(12)
In order to perform the contraction in eqn.(ll), we define the following sequence with B , , = B T ( c ) G,
JT,
=
lim O,,
T
c-00
= 1,2,3,
where we introduce the velocity of light c to parametrize the contraction procedure. For convenience the imaginary unit appears solely to obtain positive valued Galilei group Casimirs defined below. In taking the limit c -+ 00, the relations given by eqn.(ll) take the form [Gr,Gs] =07
[Jr,Gs] =icksGt7
[JrTJs]
=ic:,Jt.
(14)
Obviously these relations define locally the homogeneous Galilei group G3 = K3 x SO3, which contains the transformations associated with spatial rotations JT and a change of inertial reference system G,. Clearly this group is isomorphic with the Euclidean group in 3-Dim E3 = T3 >a SO3 but its physical content is distinct - one is the symmetry of 3-space and the other the symmetry group of the laws of Newtonian motion. We find on using the substitutions given by eqn.(l3) in the expressions for the Lorentz Casimirs given by eqn.(l2), redefining the Casimirs and then taking the limit
C;
E
C,113, lim -= lim (JTJT/c2+ O'flT) = G'G,
c+oo
c2
C,113,
c+oo lim
c+oo
= c+oo lim 2(JTR,) = 2(JrG,) ac
= G.G,
= 2J.G.
(15) (16)
But while the former remains a Casimirs for the Galilei group, the latter is not as [ J ,J.G] # 0. Naturally, in the nonrelativistic limit c -+ 00 (or physically small velocities -+ 0) the homogeneous Lorentz group is transformed into the homogeneous Galilei group. The usual analysis of the Lorentz group employs the isomorphism between its algebra and that of z 2 x SL2 where the overscore signifies that the two
441
SL2 are strictly related by complex conjugation. We redefine the generators of the Lorentz group as
with A:+ = A:, J,! = J, and B! = -BTl and we obtain
[A;, A;] = icysA;,
[A:, A;] = 0 ,
[A:, A:] = if,&+.
(17)
The Casimirs of each sl2 algebra are now
&.A-
= A- ,A;,
A+.A+ = A+ 'A+ T '
while the Lorentz Casimirs can be expressed as
C,133 = 2(A-.d-
+ A+.A+),
C,1'3= 2i(A-.A-
- A+.A+).
As the operators { A ; } and {A:} form two sets of independent generators of SL2 (see [ l o ] ) , and therefore can be separately redefined in terms of spherical vector operators A$ and A; with p = (+, 0, -), as in section (2.2)
A?
A$ = A ; + i ~ & 3.2
E
A: - i ~ $ ,
A;
A;.
The Finite Representations
Before developing the representation theory of the Lorentz and Galilei groups, we can rewrite the Lorentz generators as A, E A, @ I and A: f I @ A,. to emphasis the fact that we are dealing with two commuting SL2 algebras. These generators act on the direct product space Vj, @ Vj, and correspond to two independent vector spaces under transformations of Lorentz group. Since SL2 SO3 locally, we can use the results of the previous section, and as a consequence the representations of the Lorentz group can be labelled as [ j l @ j 2 ] with dimension ( 2 j l 1)(2j2 1). N
+
+
+
The two SL2 Casimirs AF.A+ have eigenspectra j l ( j l + 1) and j 2 ( j 2 1 ) respectively, which establishes the eigenspectra of the two Lorentz Casimirs as c,>3
.
.2[jl(jl 1 3
1) + h ( j 2
C4' : 2 i [ j l ( j l
+ I ) ]= 2 ( j 1 + j 2 + l ) ( j l + j z )
+ 1)-
j2(j2
+ l ) ]= 2 i ( j l
+j2
-
4j1jz1
+ l)(jl-j2).
(18) (19)
The basis of the space Vj, @ Vj, is constructed as a direct product of vector bases of so3 with vectors lj1m1,j2m2). The eigenspectra of and A0 are
442
ml and m2 respectively, while we also get - L I j m , j 2 m 2 ) = ljlml,j2m2 f 1)Jjdjl A+ljlml,j2m2) = ljlml,j2m2 f 1 ) J j 2 ( j 2
+ 1) - ml(m1 f l), + 1) - mz(m2 f1).
To determine the action of J, and B, on Ijlml,jzm2) we follow the development for the so3 algebra in section (2.2). We have
J, = d;
+ d:
B, = i(d; - d:).
and therefore
+
JTIjimi, j2m2) = A, Ijimi, jzm2) Ijimi, j2mz), BrIjlm1, j2m2) = i(A; Ijlm1, j2m2) - d z l j l m l , j ~ m z ) ) . In particular for 53 y B3, we have
J31jlm1, j2m2) = ljlml, j2m2) (ml f m2), B3ljlmi7jzmz) = Ijiml,j2mz)i(ml - mz).
(20)
Exploiting the various isomorphisms, all the analysis for SO3,1 is similar to the above analysis for SO3. A separate analysis of G3, which allows us to construct an eigenbasis appropriate to G.G, J3 and G3
G31g7m, s> = 19,m, S)s, J319, m, s ) = 19,m, s)m, G.GIg, m, 4 = 19,m, 4g2.
(21)
The action of the other generators can be obtained using the commutator relations but as we do not need them here we shall omit them. With these results we now establish the connection between the two groups under the group contraction SO3,1 4 G3. In analogy to SO3 we begin with the eigenvalues of the respective Casimirs and we introduce eigenvalues j , and m, with eigenvectors (j~,cml,c,j2,cm2,c). Using eqns.(l5) and (18, 21) lim G.GJg,m, s ) = c+m which yields
J . J - Bc.Bc c2 ljl,cml,c,j2,cm2,c)
443
However using the quartic Casimir of eqns.(l6) and (19) we find
Here we have rewritten the quartic Casimir eigenvalue as 2i(jl,c
+j2,c + W l , c - j2,C)
j?,c+ j;,c + j1,C + j 2 , c
=2c(
iC
C2
- 2 j;J
+j2,c C2
for which the first term on the right yields in the limit the eigenvalue of the quadratic Galilei Casimir g2. To avoid the appearance of infinities in the limit we must then have l 2 1.1m j= lim j2 =9 c C W ’ c 2’ The limiting process for the quadratic Casimir eigenvalues takes form C+W
L
J
while that of the quartic Casimir of eqn.(l6) and (19) we obtain lim
+ j 2 , c + l ) ( j ~- j, ~~ ,=~0. )
2(j1,~
C
C’W
and as a consequence 2 J.Glg, m, s) = 0. Similarly we obtain
+
lim m ~ m2,c , ~ = m,
c-00
lim ’C
00
m1,c - m2,c
= s,
C
and 19,m,
4=
I~l,cml,c,~2,c~2,c)
by observing that under the contraction limit J3Ig,m,s) = W lim C ’
J3Ijl,cml,c,j2,cm~,c)r
444
and hence we have
Introducing the linear expressions ml,, = alc solutions lim C"o0
+
(a1
+ a2)c + (bl + b 2 ) = m,
al=-a2=-
S
2
bl
lim
+ b l , m2,, = a2c + b2, we find (a1 - a2)c
+ ( b l - b z ) = s,
C
c-00
+ b2 = m,
and this draws the connection between the labelling of the representation spaces of the Lorentz and Galilei groups. The action of generators J, and R, on these new eigenvectors is as for the Lorentz group action except for the substitution of j for j , and m for m, in all the matrix elements. From all of the above we can pass over without difficulty to the contraction of the Poincar6 group to the inhomogeneous Galilei group, ie. P3,l = T3,l x SO3,1 G3,l = T3 G3, given the fact that the contraction only affects the generators of Lorentz boosts, and thus contracts SO3,1 4 G3 which has just been shown. --f
dS/AdS groups and Their Contraction to
4
4.1
P3,1
The Algebras
The two SO&h,h groups with h = 1 or 2 correspond to maximal group of symmetries in De Sitter/Anti-de Sitter ( d S / A d S ) spaces respectively, and represent uniform and curved space-time manifolds that are compatible with an expanding universe. The universe can be described as a hypersurface inside a 5-dimensional spacetime of signature (4,1) or (3,2). If we use coordinates y i with i = 1, .., 5, the hypersurface with curvature a is defined as
(y1)2
+ ( y 2 ) 2+ ( y 3 ) 2
(y412 - ( y 5 ) 2 = qij y i y j = -a-'
with
a=
1
-.
R
This hypersurface is invariant under linear transformations that preserve the metric qij = diag(+++k-). These transformations comprise the Sitter/Antide Sitter groups ( S & - h , h ) whose algebra consists of 35(5-1) = 10 generators,
445 J i j representing generalized rotations in Es. These generators satisfy the algebra [ J i j , Jkl]
= i(JikVj1 - Jilqjk
+ Jjlvik
- Jjk’%l),
(23)
withi,j,k,l,m=1,..,5. The two Casimirs of the de Sitter/Anti-de Sitter groups are
C2
-1 J .23. J23’ .,
c, Ez wiwa,
where Wi represent a 5-dimensional vector defined as w . 2
E
= -1- E . . J + J J ~ ~ , j i j = jk l r l i k + . ajklm
is tensor totally antisymmetric with 5 index. The contraction is defined by
II
1
lim
=-Jzp,~,
,-R with J z , , ~= Jz,(R).
= K,,
with z = 1 or 5.
R+CU
(26)
Using eqn.(26) we reexpress eqn.(23) in a 4-dimension spacetime notation as [J,”,
P,
Jpol
= i(J,Prl”U
- J,Url”P
+ J”U77,P
- Jvprl,u)r
(27)
i
Jpol
In,, n u 1 = 5J,”,
= i(nprl,o - norl,p)l
(28)
with p , v,p, 0 = 1 , 2 , 3 , 4 (see also [ l l ] ) .Using eqn.(26) in eqns.(27,28) we obtain the following algebra [Jpv, Jpol
[KplJpo l
= qJ,prl”o
- J,crll”P
+ JYUl7,P
= i(Kprl,o - Korlpp),
- J”Prl,U),
IKp, K”] = 0.
(29) (30)
Kp denotes the translation operator in flat space-time, where a rotation on either the surface (z~,z,) or (z,,z:~) transforms as a space-time translation in the limit of curvature zero ( R -+ co). The generators K,, J,, along with eqns.(29,30) define the Poincark group in the contraction. Rewriting the ‘de Sitter’ Casimir invariants in (24) using eqn.(26) we have
446
We define the first Casimir invariant of PoincarC in the limit R
4
00
as
Using eqn.(26) in eqn.(25), the second ‘de Sitter’ Casimir invariant takes the form ‘ 1 C ~ ,= R Wi,RWaR= -R2€Xz~pvn’J’v~Xz’’~’v’ITp~ Jp~v~ 4 1 4 J~~~p~ J~~~~ JP‘~’. (33)
+
The second term of the right-hand in eqn.(33) vanishes in the limit of zero curvature, and therefore the second PoincarC invariant takes the form
4.2
AdS Unitary Representations
Following Nicolai [12], the generators Jij E SO3,2 permit a spinorial representation in terms of the gamma-matrices (..I J i j l . . ) = rij, this matrix set is written as
rij = rt. Y = -l?lj
for J,,,
with
J45
T,S
= 1,2,3,
(35) We see that not all the J i j s are unitary, in accordance with the fact that finitedimensional representations of a non-compact group are not unitary representations. To obtain a Hermitian representation of the generators Jij = Jjjl a infinite-dimensional representation is required. From the eqn. (35) we distmguish the compact generators J45 and J,., from the non-compact generators Jr5 and J d T . The operators J,, and J45 generate a maximal compact subalgebra associated with so3 x S 0 2 . We now define the following operators in so3,2 as rij
M‘, with
E iJ4T T
+ J5,
,
for
MF
J4,, JT5,
= iJ4, - J5,,
where M , = -(M,‘)+,
(36)
= 1,2,3. The associated algebra is given as
[M,‘7Mi]
= 2 ( 6 T S J 4 5 -kiJTS)>
M,‘,
[M,‘,M$] =
-MF.
[‘FIMi]
=O,
(37) The laddering operators M$ and M,- respectively raise and lower the energy eigenvalues in unit of one when applied to the eigenstates of the energy [J45,M,‘] =
[J45,MF]=
447
operator J45. Choosing J 4 5 , J'J, and J 3 as our set of mutually commuting operators we identify our eigensystem as follows:
where (...) denote a non-specified set of labels. The C2 Casimir can be written as: 1 2
C2 = (J45)2 + -J'" Jrs+ J4,Jqr+ J5'Jgr
Note that $J'"J,, = J'J,, J4'J4,, = -3(J4r)2, J5rJ5r= -3(J5,)2 and {M:,MF} = -2(J& J,,.). Taking the representations for which exists a state set that annihilates Mr-, where the energy spectrum has a lower bound. If we denote the lowest energy eigenvalue by EOand the angular momentum values by s, the ground state consists of (2s 1) states ((Eo,s)Eo, s, m ) , m = -s, -s 1,...,s. To evaluate C2 on the ground state IEo, s), we use
+
+
+
Using eqns.(26, 38) we introducing EO,Rin the contraction and obtain
and using EO,R= Rk with k = mc/h ( m + rest-mass), the contraction is given as
448
Appendix Since the classic angular moment is defined as J’ = r‘ A @, an infinitesimal rotation is generated when --f co,with 6 t h e angle and ;arc length, where
Without loss of majority, in the poles we obtain that
In the quantization process we obtain a R- dependent operator J = R A P , with R and P the position and momentum operators respectively, and where an infinitesimal rotation is rewritten as d
+
X
J.0 = R A P.- = ( e A P).Z, IRI with IRJ the norm of the operator R. In the limit JRI -+ co we obtain ‘the momentum operator’. In the previous sections we defined Ki = e A nil with Pa = h i
Acknowledgments. The first author was partially supported by the Fundacidn Mazda para el Arte y la Ciencia and the Universidad de 10s Andes. He would like to express his gratitude t o the organizers of the Summer School on Geometric and Topological Methods for Q.F.T. - 2001, for the invitation to take part in the school. References 1. Inonu E., Wigner E. P., Proc. Natl. Acad. Sci.,USA 39-510 (1953) and 40-119 (1954). 2. Inonu E., “Contractions of Lie groups and their representations” in Group theoretical concepts in elementary particle physics, F.Giirsey ed., Gordan and Breach, pp.391-402 (1964). 3. Gilmore R., Lie Groups, Lie Algebras, and some of their applications, J.Wiley, New York, 1974. 4. Hermann R., Lie Groups for Physicists, Amsterdam, W. A. Benjamin, 1966.
449
5. Izmestkv A. A., Pogosyan G. S., Sissikian A. N., “Contractions on the Lie algebras and separation of variables. The n-dimensional sphere”, J. Math. Phys. 40 1549-1573 (1999). 6. Greiner W., Muller B., Quantum Mechanics Symmetries, New York, Springer, 1994. 7. U. Cattaneo and W. Wreszinski, “On contraction of Lie algebra representations”, Commun. Math. Phys. 68, 83-90 (1979). 8. E. Weimar-Woods, “Contraction of Lie algebra representations”, J. Math. Phys. 32, 2660-2665 (1991). 9. Dooley, A. H.; Rice, J.W., “On contractions of semisimple Lie groups”, Trans. Amer. Math. SOC.289, no. 1, 185-202 (1985). 10. Ayala M., Haase R., “OSp(N14)group and their contractions to P ( 3 , l ) x Gauge”, hep-th/0102030. 11. El Gradechi A. M., and De BiSvre S., “Space Quantum Mechanics on The Anti De Sitter Space-Time and its Poincart. Contraction”, Annals Phys. 235 (1994) 1-34 (hep-th/9210133). 12. Nicolai H., LLRepresentations of supersymmetry in anti-de Sitter space”, Supersymmetry and Supergravity ’84, Proceedings of the Trzeste Spring School. World Scientific, 1984.
Geometric and Topological Methods for Quantum Field Theory Eds. A. Cardona, H. Ocampo & S. Paycha @ 2003 World Scientific Publishing, pp. 450-467
PHASE ANOMALIES AS TRACE ANOMALIES IN CHERN-SIMONS THEORY ALEXANDER CARDONA’ Laboratoire de Mathe‘matiques Applique‘es Universite‘ Blaise Pascal (Clernont II) Complexe Universitaire des Ce‘zeaux 631 77 Aubidre Cedex, France. The classical Chern-Simons action functional, which is metric invariant of the underlying manifold, gives rise -through Schwarz’s Ansatz- to a partition function which depends on the metric by a phase factor. We show that this “phase anomaly” in odd-dimensional Chern-Simons models can be understood as a mathematical anomaly, coming from the regularization procedure used in order to define traces of operators acting on infinite-dimensional spaces.
Introduction In the %on-perturbative approach” to Quantum Field Theory path integrals are modelled by regularized determinants. The general form of a path integral is
Z ( F )=
20
s,
F(4)exP {-S(4)1 P41,
(1)
where CP is the space of configurations of the fields 4 (sections of fibrations on a Riemannian or Mikowskian manifold M ) , F ( 4 ) a functional on CP, [D4] a formal Lebesgue-type measure on CP and S : CP -+ 6!the classical action of the theory under consideration. Here Zo denotes the partition function of the theory, a normalization factor given by the integral on the right hand side in (1) when F ( 4 ) = 1. Since CP is typically an infinite-dimensional manifold, the formal Lebesgue-type measure [D4] is generally ill-defined. The basic idea is to interpret a partition function as a “regularized” determinant (and get from it the path integral by stationary phase approximation) imitating the Gaussian integral identity
,-is(.)
dv = (detT)-;,
*Present address: DBpartement de MathBmatique, UniversitB Libre de Bruxelles, C P 218, Campus Plaine, B-1050 Bruxelles, Belgique. Email:
[email protected].
451
valid for Lebesgue-type integrals on a finite-dimensional euclidean vector space V , where S ( w ) = (Tv, w) is a symmetric and positive quadratic form defined on V , and dv denotes the Lebesgue measure. In the case of action functionals defined by elliptic differential operators with positive order acting on infinitedimensional spaces of sections, the ordinary determinant on the right hand side of the previous equality is replaced by the <-determinant, defined by Ray and Singer through <-function regularization [21]. Combining this with the Faddeev-Popov procedure led A. Schwarz in [24] t o a definition for the partition function associated with a degenerate action functional. The latter mimicks Milnor’s definition of the Reidemeister torsion of a complex of vector spaces, and therefore yields a relation with the Ray-Singer torsion, a secondary topological invariant used to classify topological spaces with the same homotopy type. Important examples of partition functions in this approach gave the first Topological Quantum Field Theories1, born from pioneering work of A. Schwarz and E. Witten in the late 70’s and the ~ O ’ S which , became an active topic of research in both physics and mathematics (see [9][27], lectures [7][16][30] and references therein). Using this approach in his study of Chern-Simons theories, Schwarz showed that (the modulus of) the partition function of the Abelian ChernSimons model (which can be used to describe the L‘semiclassical)) limit of the non-Abelian theory) can be described in terms of the Ray-Singer torsion of the underlying manifold, a metric invariant. In [27] Witten showed that this partition function must contain (a phase given by) the r]-invariant defined by Atiyah, Patodi and Singer in order to state an index theorem for manifolds with boundary, which corresponds to a <-function regularization of the “signature” of an operator (41. Thus, the partition function associated to the metric invariant action functional describing classically the three-dimensional Abelian Chern-Simons theory, gives rise to a partition function which fails to be metric invariant, because of the presence of a phase (see Section 1). This loss of symmetry in the (quantum) description can be interpreted as an anomaly associated to the quantization process, which we call phase anomaly. In [27] Witten computed the corresponding anomalous term in order to add a counterterm to the metric dependent part in the partition function, so as to get a topological model. Our purpose here is to show that ‘Here topological means that the partition function of the theory is an invariant of the metric on the space-time manifold.
452
this phase anomaly -coming from the quantization of a classical theory- can be understood as a mathematical anomaly, coming from the regula,rization procedure used in order to make sense of path integrals. Regularization techniques used to define determinants of elliptic positiveorder differential (or pseudo-differential) operators can be also used t o regularize other ill-defined extensions of finite dimensional concepts, such as traces. Here we consider [-function regularization, which gives rise t o <-regularized determinants and weighted traces, that extend the usual determinant and trace of finite-rank operators (see Section 2). However, the use of [-function regularization leads t o discrepancies; well-known facts that hold for determinants and traces in finite dimensions fail t o hold true for regularized determinants and traces. For example, the classical identity det(AB) = det(A) det(B ) -that holds for finite-dimensional determinantsbreaks down for [-regularized determinants, giving rise t o the so-called “multiplicative anomaly” for [-determinants (151, and the fundamental tracial identity tr(AB) = tr(BA) -which holds for ordinary matrices- breaks down for <-regularized traces, giving rise to a “tracial anomaly”. Weighted traces of classical pseudo-differential operators were investigated in2 [12][20], as well as the tracial or weighted trace anomalies associated t o them. These give rise t o Wodzicki residues, and hence have some locality features (see Section 2.1), that have been studied in connection with index theorems and QFT anomalies in [ l l ] [ l O ] . In this paper we consider the relation between a (mathematical) trace anomaly [ll]-coming from the regularization procedure- and the (physical) Chern-Simons phase anomaly [27], both given by a local term that can be computed by means of the the Atiyah-Patodi-Singer index theorem. The paper is organized as follows. In Section 1, following [24][27][3][11],we review the construction of the partition function of Chern-Simons models, and we define the phase anomaly from a quantum field-theoretical point of view. In Section 2, following [20][12][11]we define the weighted traces and weighted trace anomalies, stating the results we intend t o use in Section 3 in order t o state our main result. For a more detailed exposition (containing background material, complete proofs and further applications) we refer t o [lo][ l l ] .
’They have been implicitly used, both in theoretical physics and mathematics, under different names, see e.g. [19][2].
453
1
Phase Anomalies in Chern-Simons Models
In this section we build the (regularized) partition function of the ChernSimons model and the phase anomaly. We begin by recalling the definition and basic properties of <-determinants and 17-invariants, and we introduce the geometrical setting underlying the physical model.
Notation. Let E be a vector bundle above a smooth n-dimensional closed Riemannian manifold M , and let CZ(E) denote the algebra of classical pseudo-differential operators acting on smooth sections of E. Let ElZ(E) and EZl*(E) denote the set of elliptic and invertible elliptic operators acting on sections of E , respectively, and dd(E) the subset of EZl*(E) containing the invertible admissible elliptic classical pseudo-differential operators which have positive order (see [25] for background on pseudo-differential operators). 1.1
<-Regularized Determinants and the 17-Invariant
Let us recall here the definition and some basic properties of <-determinants of admissible operators [21] (see also [22]). Let A E EZZ*(E)be a self-adjoint elliptic (classical) pseudo-differential operator with positive order, then the spectrum of A is discrete and contained in the real line, { X k } k E z c R.The (-function of A is defined as the trace of the operator A-', i.e. < A ( z ) = tr
C
(A-") =
A-",
(2)
XEspecA
which is an analytic function for z E G with X ( z ) >> 0. It extends by analytical continuation to a meromorphic function on G, regular at z = 0. The <-determinant of A , denoted by detc A , is the complex number given by detcA
= exp {-<>(O)}.
(3)
(-determinants extends of the usual determinant in finite-dimensional vector spaces, i.e. if A is a finite-rank operator (a matrix) then
n N
detc A =
Xi,
(4)
i=l
where Xi, 1 5 i 5 N , are the eigenvalues of A .
If A E EZl*(E) is self-adjoint but not necessarily positive, its spectrum contains negative eigenvalues but its <-function can still be defined by (2), taking X i " to be IXkI-ze-iTZ if Xk is negative. In [4] Atiyah, Patodi and
454
Singer defined, for large rrZ(z), the 7-function of A as the trace of the operator
AIAI-"-l, i.e. QA(Z)
= tr(AIAI-"-l)
=
1(signXk)XL".
(5)
kE Z
They showed that this function extends meromorphically to the whole z-plane and, moreover, that ~ A ( z is ) finite at z = 0 (its value at z = 0 measures the asymmetry of the spectrum of A). Following 141 we define the 7-invariant of A by
v ~ ( 0=) f.p.l.=otr (signAIAl-") ,
(6)
where the sign of A is the clasical pseudo-differential operator defined by sign(A) = A1AI-l. It is well-known that the phase of the [-determinant of a self-adjoint operator can be expressed in terms of its 7-invariant. Let A E EZZ*(E)be any self-adjoint elliptic pseudo-differential operator with positive order, then (see e.g. [26] [l])
As a consequence of this, using the fact that [ l ~ l ( O )
= 0 when A is a differential operator acting on sections of some vector bundle based on an odd-dimensional closed manifold [23][26], it follows that
detc A = detc JAl. e$'oA(o),
(8)
whenever n is odd.
1.2
The Partition Function in Chern-Simons Models
In the following we shall consider abelian Chern-Simons theories over closed odd-dimensional Riemannian manifolds (see [l][24][27]).
+
Let M be a Riemannian manifold of odd dimension n = 2k 1, and V, the Hermitian vector bundle over M with flat connection VP defined by a representation p of the fundamental group of M . Consider the acyclic de Rham complex 0
-
00
5 . . . Ok-1
dk-l
Rk
5ak+l
d,+!
. . . 52"
-
0,
(9)
where flk = C"(AkT*M @ V,) and dk denotes the restriction to 52'" of the exterior differential on twisted forms given by d, = d @ 1 @ 1 @ VP. Let
455
+
Ak = didk dk-ldi-l be the Laplacian operator on complex (9) implies a Hodge decomposition
Rk. Acyclicity of
Rk = R i 69 R;, where
Qi = Imdk-1
the (10)
= kerdk and 0: = kerdi-, = Imd;-l.
+
Restricting the operator A k = didk dk-ldi-l t o 0; and R2;f, we get invertible operators Ag = dk-ldi-lln; and A; = didkln;’, and the <-function techniques can be used t o define detC(A;) and detc(A,,l). Then, detC(Ag) = exp C ( - l ) k - i l o g d e t ( Ai { ik= O
1
,
with the convention that d*_, = d, = 0. In odd dimensions the square of the Hodge star operator * : RP 4 Rn-P is the identity map and, setting n = 2k 1, the operator *dk : Rk -+ Rk is a formally self-adjoint elliptic differential operator of order one. Restricting *dk t o 0; gives us an invertible self-adjoint elliptic differential operator *d;, which has a well-defined <-determinant.
+
+
The Chern-Simons model in dimension n = 2k 1 is described, in the “weak coupling limit” , in terms of the metric invariant action functional given by
where M is a ( 2 k S1)-dimensional manifold. Since S:” ( W k ) = SF”(w:) where = w i @w[ in the decomposition (lo), this action presents a degeneracy on i l k . To deal with this type of degeneracy, we use Schwarz’s Ansatz [24] for the corresponding partition function, which yields
wk
]’
(12)
Then, from the fact that I detc(*d;)l = det, I *d2;f(=
d w 7it follows that
ZFs(*dk) =
[:In
Using Hodge duality we have
//
(detc A, )
(-l)k-[+l
det, (*d‘,‘)-; .
456
where T ( M ) is the Analytic Torsion of the manifold M , defined by
T ( M ) = exp
{f
g(-1)*k
Since T ( M )is a metric invariant of M [21], the modulus of the partition function z F s ( * d k ) , associated to the metric invariant action sf”(uk), is metric invariant. From equation (7) it follows that
Using the fact that
1=0
and
<~*d;’l(O)
= 0 if the dimension of
zFs(*dk) Notice that, for k as shown in [27].
1.3
=
=
M is n = 2k + 1, we find
ee - i ; q * d ; ’ ( o )
(T(M))
2
1, this yields back the fact that IZCS(*dl)l =
(14)
dm,
Phase Anomalies
In Quantum Field Theory an anomaly occurs when a transformation in the fields, leaving invariant the action functional, changes the corresponding path integral (see [6] for an exposition containing extensive references). In particular, when the classical action is quadratic, i.e. S(q5) = (Tq5,+)in (l),transformations in the path integral can be read off the transformations of the regularized determinant associated to the corresponding partition function Z = (detC T ) - i . In the path integral approach to anomalies, these are interpreted as a lack of invariance in the formal measure appearing in the corresponding path integral, and measured by the logarithmic variation of the partition function under the corresponding transformation. The “anomaly” is then defined to be the logarithmic variation of the corresponding regularized determinants. The classical Chern-Simons action functional (1 1) defining the Chern-Simons model in demension n = 2k 1 is metric invariant. Since q,d;(O) depends on the metric on M (through the dependence on the metric of the Hodge duality operator), the partition function (14) is metric-dependent, a fact
+
457
that can be interpreted as an anomaly in the quantization process. We define the phase anomaly of Chern-Simons model as the the logarithmic variation of the partition function under variations in the metric. Let us state this more precisely. Recall that dk = d p l R k , where d, = d 181 €9 1 63 VP is the twisted exterior differential on V,-valued forms on M . Then, taking a smooth path { g t } t E [ o ,of ~ ~Riemannian metrics on M and a fixed connection on V,, we can build a smooth family { * d k , t } t E [ o , l ~of self-adjoint operators. These give rise as above t o a smooth family {*dl,t}tE[o,l~of invertible self-adjoint operators, and as in (14) t o a smooth family of partition functions { 2 f S ( * d k , t ) } t E ~ o , 1 1 . Moving along the path of Riemannian metrics { g t } t E [ o , l ~leaves invariant both the classical Chern-Simons action functional SFs and the modulus of the partition function IZfS(*dk,t)l, but changes the phase of the partition function. From (14) it follows that the logarithmic variation of ZFs(*dk,t), between t = 0 and t = 1, is given by
We will show in the last section that the phase anomaly given by equation (15) can be written as an integral of a trace anomaly. 2
Weighted Traces of Pseudodifferential Operators, Regularized Determinants and Tracial Anomalies
In this section, following [20][12], we define weighted traces of pseudodifferential operators, weighted trace anomalies, and we relate them with the Wodzicki residue. In what follows by a weight we shall mean an element of d d ( E ) , often denoted by Q, and by q we shall denote its order. We shall very often take complex powers Q-" of operators Q E d d ( E ) ,which involves a choice of spectral cut for the operator Q. However, in order t o simplify notations, we shall drop the explicit mention of the spectral cut.
2.1
The Wodzicki Residue and Weighted Traces
Let E be a vector bundle above a smooth n-dimensional closed Riemannian manifold M . For Q E d d ( E ) and A E Cl(E),the map z H tr(AQ-") is meromorphic with a simple pole at zero (see e.g. [15][28]).Given Q E d d ( E ) , the Wodzicki residue of A E Cl(E)is defined by [28] res(A) = q Res,,o
(tr(AQ-"))
,
(16)
458
where q denotes the order of Q. The definition of res(A) is independent of the choice of Q. Among the many remarkable properties of the Wodzicki residue (for a review see [14]), let us point out:
1. Raciality. If M is connected and dimM > 1, the Wodzicki residue is (up to a constant) the only trace on the algebra Cl(E),i.e. res( [A, B ] )= 0, for any A, B E Cl(E).
2. Locality. The Wodzicki residue of a classical pseudo-differential operator A can be described as an integral of local density on M , namely
where n is the dimension of M , p~ the volume measure on M .
3. Triviality o n finite-rank operators. If A is of finite rank, or if its order is less than -71, then res(A) = 0.
(19)
Since it vanishes on finite-rank operators, the Wodzicki residue is not an extension of the finite-dimensional trace. We therefore consider instead other linear functionals -called weighted traces (see [20][12])- which, even if no longer tracial on Cl(E),extend the usual trace on finite-rank operators (matrices). Nevertheless, Wodzicki residues and their properties, in particular traciality and locality, will play a very important role in the study of weighted traces.
Definition 1 Let Q be a weight and A in Cl(E). W e call Q-weighted trace of A the expression trQ(A) = f.p.l,,o
(tr(AQ-"))
.
It follows from the definition that weighted traces extend usual finitedimensional traces, i.e. trQ(A) = tr(A)
(21)
whenever A is a finite rank operator. Remark. Note that the r]-invariant of A, given by (6), can be written
r ] ~ ( 0=) trA(signA).
(22)
459
In order to relate weighted traces to <-regularized determinants, we extend weighted traces to logarithms of pseudo-differential operators. Given A, Q E Ad(E) with commun spectral cut, the map z H tr ((log A)Q-") is meromorphic on the complex plane with a simple pole at the origin [15].
Definition 2 Given A, Q E Ad(E) we define the Q-weighted trace of log A by trQ(logA):= f.p.Iz=o (tr(logAQ-'))
.
Remark. Note that the <-determinant of A, given by equation (3), can be written detcA = exptrA(logA).
(23)
Thus, for a self-adjoint pseudo-differential operator A, formula ( 8 ) reads
[A[)+ iEtrlAi(signA)] 2 . 2.2
Weighted Trace Anomalies
Unlike Wodzicki residues, weighted traces are not tracial and depend on the weight Q. As a matter of fact both trQ([A,B])and trQ'(A) - trQz(A), for Q , & I , Q 2 E Ad(E) and A, B E CZ(E),can be expressed in terms of Wodzicki residues (see Proposition 1 below). Moreover, for families of operators parametrized by smooth manifolds, the weighted trace fails to commute with the differentiation operator. This is the price we pay for having left out divergences when taking the finite part of otherwise diverging expressions, and we call these obstructions weighted trace anomalies. The following proposition shows that these tracial anomalies (trQ([A,B ] )# 0, trQ1(A)-trQZ ( A ) # 0 and [d, trQ](A) # 0) can be written in terms of Wodzicki residues3 : Proposition 1 1. Given A , B E Cl(E), Q E Ad(E) with positive order 4, we have [12] [18] 1 trQ([A, B ] )= - -res (A[logQ , B ] ). 4
(24)
3Recall that although the logarithm of a classical pseudo-differential operator is not classi- !%& of two such logarithms, and d log Q cal, the bracket [logQ, A ] , the difference 92 lie in C l ( E ) 1121.
460
2. For QI, Q2 E d d ( E ) with positive orders g1,q2 we have [12] trQ1(A) - trQ2(A)
=
-res
I.(F
-logQ2)]
92
.
(25)
c d d ( E ) be a smooth family of weights, with constant positive order9 and common spectral cut, and { A x } x E ~c Cl(E)a smooth family in Cl(E), both parametrized by a smooth manifold X . Then [12]
3. Let {Q,},Ez
1
(dtrQ)(A) - (trQ)(dA) = --res(AdlogQ), 9
(26)
where d denotes differentiation with respect to the parameter x. An important observation is that by (18) all these weighted trace anomalies, being Wodzicki residues of some operator, can be expressed in terms of integrals on the underlying manifold M of local expressions involving the symbols of that operator.
3
Tracial Anomalies and Phase Anomalies in Chern-Simons Models
In this section we use the results of section 2.2 to express phase anomalies in odd dimensions -coming from logarithmic variations of C- determinants of Dirac operators- in terms of weighted trace anomalies, thus giving an a priori explanation for the locality we expect from these anomalies.
3.1
Logarithmic Variations of Regularized Determinants and Tracial Anomalies
In this section we relate logarithmic variations of regularized determinants of certain families of admissible operators with tracial anomalies, thus giving an a priori explanation for the locality of these variations. For families of geometric operators these variations can be expressed, via index theorems, as integrals of characteristic forms on the underlying manifold. Let A1 and A0 be two invertible self-adjoint elliptic operators, and consider a family of self-adjoint elliptic operators {As}xEIO,l~ interpolating them. The eta invariant v ( A , ) = ~ A , ( O ) -which is part of the phase of the C-determinant of A,- varies smoothly in IC modulo integers, i.e. except for jumps coming from eigenvalues of A, “crossing” the zero axis. Indeed, since v(A,) = trlA,I(signA,), it gives the difference between the number of positive
46 1
eigenvalues and negative eigenvalues of A,, and if one of the eigenvalues of A, passes from positive t o negative r](A,) jumps two by two. The spectral flow of the family {Az}tEIO,l~,denoted @({A,}), measures the net number of times the spectrum of the family crosses the zero axis, i.e. the net change in the number of negative eigenvalues of A, as x varies between 0 and 1; it was introduced in [5] in order t o study the LLnon-continuous77 part of the r] invariant4. Let as before (see [ 5 ] ) , at any point x E X for which A, is invertible, V A = ( Z )= tr(AslAsl-z-'),
which we recall is a meromorphic function on the complex plane. It follows from [5] (Proposition 2.10) that for % ( z ) >> 0,
dr]A,(z) = -z tr(dA,~A,~-z-l)l
(27)
and (c.f. Proposition 2.11) res where U, = AzlA,l-'
(&u,)
= 0,
= IA,IAgl = signA,.
Lemma 1 Let {A,},E[o,l~ be a smooth family of elliptic self-adjoint operators of positive constant order a. Then, at any point x E X for which A, is invertible, we have --res a
[
' 1
lA,l-l-Az dx
= --res
where irAz = [ &, trAz].
:
[
d A;'-ia,\] d x
=
dA"' (signA,)
= trA" (signA,),
(29)
Proof. From (27)
*Making this definition precise requires some care since there might well be an infinite see also [ll]). number of crossings of the zero axis (here we follow [17],
462
from which the first equality follows. On the other hand, from Proposition 1, using the fact that [U,,lAzl] = [A,, lAzl] = 0,
dA"' (SignA,)
where a
- frA" (signA,)
= ord(A,),
=
a
so the last equality in (29) follows from the first one,
0
The following theorem relates the variation of the continuous part of the eta invariant t o an integrated tracial anomaly:
Theorem 1 [ll] Let A0 and A1 be two elliptic self-adjoint operators and {A,},EIO,l~ a smooth family of elliptic self-adjoint operators of constant order interpolating them. Then,
d trAZ]. where @({A,}) denotes the spectral flow of the family and t.r A, = [x,
Proof Since the difference q(A1) - q(A0) is invariant under the shift A, H A, + Q . Id, Q E IR,we can reduce the proof of (30) t o the case of a family of invertible operators (for details see [Ill,Section 3). In order t o show (30) in that case, let us first to show that trlAzl -signA, [fx
"L
= --res a On the other hand,
A;'-lA,l]
I
= 0.
+ trlAzl
463
But (28) implies that
hence d [trlAzI(signAZ)]= f.p.l,,o
[-ztr
(
""-lA,l-z-l)]
dx
dx
,
[$A,la,l-']
= --res 1 U
where a = ordA,. The first equality in (29) combined with (32) yields equality (31). Now, q(A1) - q(A0) = trlA1l(signA1)- trlAol(signAo)by definition (see equation (22)), so
Putting (31) into the first equality in (32) gives
1
1
q(A1) - q(A0) =
dAzI(signA,) d x ,
(34)
so, by the last equality in equation (29), the result follows. 0 The following result gives an application of this theorem to the calculation of variations in the phase of regularized determinants:
Corollary 1 Let { A s } z E ~ Obe , ~ a~ smooth family of self-adjoint elliptic operators with vanishing spectral flow and constant order a , such that A0 and A1 are invertible. Let 4(A,) = $ (VA,(O) -
1
1
= -
trA"(sign(A,)) d x
-L l ' r e s 2a
dx
(35)
464
Proof. From the equality (7), relating the (-determinant of a (non necessarily positive) self-adjoint elliptic operator and its 7 invariant, it follows that log detc A = logdetc [A( ( q ~ ( 0-) < l ~ l ( O ) ) . Thus, given that
4
trlA=I
* A
(signA,) = tr =(signA,) =
a
the result follows from Theorem 1. 0 Hence, under the above assumptions, the logarithmic variation of the c-determinant is expressed as a weighted trace anomaly and is therefore local. Although the assumptions of the previous corollary seem strong, they are fulfilled for a wide family of examples where the equality between the index of an elliptic differential operator and the spectral flow of an associated family of operators can be made explicit (see [8] [29] and references therein). The assumption cIAl(0) = 0 is fulfilled on an odd-dimensional manifold. In the case of Chern-Simons models -as we will see in the next section- the phase anomaly is quit similar to the phase difference appearing in the above corollary.
3.2
The Phase Anomaly in Chern-Simons Model as a Weighted Trace Anomaly
Recall that the (classical) action functional (11) used to model the abelian Chern-Simons theory in odd dimensions is metric invariant, but its associated (quantum) partition function (14) is not, since it contains a phase which depends on the metric on M . Given a smooth family of metrics { g t } t E [ o , lon ] M and a fixed connection on the exterior bundle V,, let { Z C S ( * d k , t ) } t E p l ] be the associated family of partition functions. Let us show that the phase anomaly described in (15), given by the difference of phases of the <-determinants of dy,t in the partition functions at t = 1 and t = 0, is given by a Wodzicki residue coming from an integrated tracial a n ~ m a l y . ~ 5Another family of partition functions can be built from a smooth family of connections { V f } t E [ ~on , l the ~ exterior bundle V,, keeping fixed the metric on M . This is the case considered in [27] where, for the three-dimensional model, in order to build a metric invariant partition function, Witten adds to the partition function (14) a local counter-term. For this he proceeded in two steps, first fixing the metric and measuring the dependence of the phase on the choice of connection and then -whenever the manifold M is paralelisablefixing the connection and measuring the dependence of the phase on the choice of metric. Both these dependences can be measured in terms of tracial anomalies along the lines of our results (see [ I l l ) .
465
From the family of operators {*d$,,}tE[o,l]let us construct the elliptic operator given by d dt which acts on sections of the bundle Rk x [0,1]. Since the family of operators {*dE,,}tE[O,l]is induced by a smooth family of Riemannian metrics interpolating 90 and 91,the connection on V, being left fixed, it follows from [5] (see also [29] [S]) that Ak = *dk,t 4- -, I/
ind& = @({*di,,}). Furthermore, since the signature of the manifold M x [0,1] is zero, then indAk = 0 and hence the spectral flow of the family {*d’&} vanishes. Thus, it follows from Theorem 1 that V(*di,l) - V(*di,o) =
Jd
1
. *dc,,
tr
(sign(*di,,))dt,
(36)
so that the difference of the eta invariants is given by a tracial anomaly. Finally, since the analytic torsion is a metric invariant, it follows from (15) that
= 2%
Jd’
res
[I * d&/-l-(*di,t)] d dt
dt.
Being an integrated tracial anomaly, and hence a Wodzicki residue, the phase Zk(*d; anomaly log- Zk(*dk,o) is the integral of a local term on the base manifold6. We summarize this in the following
Theorem 2 Ill][lo] The Chern-Simons phase anomaly between two Riemannian metrics go and 91 is an integrated weighted trace anomaly, i.e. phase anomaly = integrated weighted trace anomaly
I
1
61n the three-dimensional case the anomaly corresponding to a smooth family of connections on the bundle V, gives rise, via the Atiyah-Patodi-Singer theorem [4] to the familiar ChernSimons term t r ( A A d A $ A A A A A ) arising in topological quantum field theory in dimension 3 (cfr. formula (2.20) in [27]), see e.g. [ll].
,s
+
466
Acknowledgments. I wish to thank J. Mickelsson, T. Wurzbacher, M. Lesch, K. Wojciechowski and S. Rosenberg for many helpful discussions, and the Mathematisches Forschungsinstitut (Oberwolfach) for giving me the o p portunity to present parts of this work. The results presented here are part of the my Ph.D. thesis [lo], under supervision of S. Paycha, whom I thank warmly. I am also indebted to the participants of the Summer School on Geometrical Methods in Quantum Field Theory, Villa de Leyva (Colombia), July 2001, for the stimulating atmosphere and the opportunity I was given there to discuss and present parts of his work. References 1. Adams, D. and Sen, S. Phase and Scaling Properties of Determinants Arising in Topological Field Theories. Phys. Lett. B353,pp. 495-500 (1995) 2. Arnlind, J. and Mickelsson, J. Trace extensions, determinant bundles, and gauge group cocycles, preprint hepth/0205126. 3. Atiyah, M. The Geometry and Physics of Knots, Cambridge University Press, 1990. 4. Atiyah, M., Patodi, V. and Singer, I.M. Spectral asymmetry and Riemannian Geometry I, Math.Proc.Camb.Phil.Soc. 77 pp. 43-69 (1975) 5. Atiyah, M., Patodi, V. and Singer, I.M. Spectral asymmetry and Riemannian Geometry III, Math.Proc.Camb.Phil.Soc. 79 pp. 71-99 (1976) 6. Bertlmann, R. Anomalies in Quantum Field Theory, 2nd. edition, Oxford University Press, 2001. 7. Blanchet, C. Introduction to Quantum Invariants of Pmanifolds, Topological Quantum Field Theories and Modular Categories, in these proceedings. 8. Booss-Bavnbek, B. and Wojciechowski, K. P. Elliptic Boundary Problems for Dirac Operators, Birkhuser, Basel, 1993. 9. Birmingham, D., Blau, M., Rakowski, M. and Thompson, G . Topological Field Theory. Phys.Rep. 209,pp. 129-340 (1991) 10. Cardona, A. Geometry of Families of Elliptic Complexes, Duality and Anomalies. Ph.D. thesis, Mathematics Department, Universitk Blaise Pascal, 2002. 11. Cardona, A., Ducourtioux, C. and Paycha, S. From tracial anomalies to anomalies in Quantum Field Theory, preprint math-ph/0207029. 12. Cardona, A., Ducourtioux, C., Magnot, J-P. and Paycha, S. Weighted traces on algebras of pseudo-diflerential opertors and geometry on loop groups, Inf. Dim. Anal. Quant. Prob. and Related Topics, to appear.
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13. Itzykson, C. and Zuber, B. Quantum Field Theory, MacGraw-Hill, 1988. 14. Kassel, C. Le rksidu non commutatif (d'apr?s M. Wodzicki). [The noncommutative residue (after M. Wodzicki)] SCminaire Bourbaki, Vol. 1988/89. AstCrisque No. 177-178, Exp. No. 708, pp. 199-229 (1989) 15. Kontsevich, M. and Vishik, S. Determinants of elliptic pseudo-differential operators, Max Planck Institut Preprint, 1994. 16. Mariiio, M. An introduction to Donaldson- Witten theory, in these proceedings. 17. Melrose, R. The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics Vol 4, A K Peters, Ltd., Wellesley, MA, 1993. 18. Melrose, R. and Nistor, V. Homology of pseudo-diflerential operators I. Manifolds with boundary, funct-an/9606005, june 1999. 19. Mickelsson, J. Wodzicki residue and anom3lies o n current algebras in "Integrable models and strings" ed. A.Wekseev and al., Lecture Notes in Physics 436, Springer 1994. 20. Paycha, S. Renonnalized traces as a looking glass into infinitedimensional geometry, Inf. Dim. Anal., Quant. Prob. and Related topics, Vol 4, pp. 221-266 (2001) 21. Ray, D.B. and Singer, I.M. R-Torsion and the Laplacian on Riemannian Manifolds. Adv. Math. 7,pp. 145-210 (1971) 22. Rosenberg, S. The Laplacian o n Riemannian Manifolds. Cambridge University Press, 1997. 23. Seeley, R. T . Complex powers of a n elliptic operator, in Proc. Sympos. Pure Math., Vol. 10, pp. 288-307, Amer. Math. SOC.,Providence, 1967. 24. Schwarz, A. The Partition Function of a Degenerate Action Functional. Comm. Math. Phys. 67, pp. 1-16 (1979) 25. Shubin, M.A. Pseudodifferential Operators and Spectral Theory. Second Edition, Springer-Verlag, 2001. 26. Singer, I.M. Families of Dirac operators with applications to physics, AstCrisque (hors sGrie), pp. 323-340 (1985) 27. Witten, E. Quantum Field Theory and the Jones Polynomial. Comm. Math. Phys. 121, pp. 351-399 (1989) 28. Wodzicki, M. N o n commutative residue, in Lecture Notes in Mathematics 1289, Springer Verlag, 1987. 29. Wojciechowski, K. P. Heat Equation and Spectral Geometry, in Geometric Methods for Quantum Field Theory. Ocampo, H., Paycha, S. and Reyes A. Eds., pp. 238-292, World Scientific, 2001. 30. Zanelli, J. (Super)- Gravities Beyond 4 Dimensions, in these proceedings.
Geometric and Topological Methods for Quantum Field Theory Eds. A. Cardona, H. Ocampo & S. Paycha @ 2003 World Scientific Publishing, pp. 468-482
DELIGNE COHOMOLOGY FOR ORBIFOLDS, DISCRETE TORSION AND B-FIELDS ERNEST0 LUPERCIO University of Wisconsin, 480 Lincoln Dr., Madison W I 53706, USA E-mail: 1upercioQmath. wisc. edu BERNARD0 URIBE University of Wisconsin, 480 Lincoln Dr., Madison W I 53706, USA E-mail: [email protected] In this paper we introduce the concept of Deligne cohomology of an orbifold. We prove that the third Deligne cohomology group H 3 ( G , 2 ( 3 ) s )of a smooth &ale groupoid classify gerbes with connection over the groupoid. We argue that the B-field and the discrete torsion in type I1 superstring theories are special kinds of gerbes with connection, and finally, for each one of them, using Deligne cohomology we construct a flat line bundle over the inertia groupoid, namely a Ruan inner local system [ll] in the case of an orbifold.
1
Introduction
D-brane fields in type I1 superstring theory can be interpreted as a global object in K-theory, as has been argued by Witten [14]. In his work he studies the proper quantization condition for the D-brane fields. He shows that the fields have a charge in integral K-theory in much the same way in which the electromagnetic charge of a U(1)-field is the first Chern class, namely an element in integral second cohomology. When we incorporate the Neveu-Schwarz B-field with 3-form field strength H and characteristic class [HI E H 3 ( M ,Z), the corresponding statement is that the recipient of the charge is a twisted K-theory group K [ H I ( M ) . Witten has posed the question of generalizing this framework to the orbifold case. We have constructed the appropriate recipient of the charge for types IIA and IIB superstring theories in a previous paper [ 6 ] . In that paper we argue that the corresponding version of K-theory is a twisting via a gerbe of the K-theory of the stack associated to the orbifold. We have done this for a general orbifold not only for orbifolds known as global quotients, that is to say orbifolds of the form X = M / G with G a finite group. In order to do so we have used extensively the notion of &tale groupoid. This will be explained in more detail below. The appearance of gerbes is natural even in the smooth case. Sharpe has argued [13]that orbifold string theories can detect the stacky
469
nature of the orbifold. This is consistent with the fact that the orbifold Euler characteristic [3] is a property of the stack associated to the orbifold (actually of its inertia stack [7]).This point of view also fits nicely with our construction of the orbifold K-theory twisted by a gerbe. In this paper we argue that both discrete torsion and the B-field can be interpreted as classes in the Deligne cohomology group H 3 ( X , Z(3)E). For this we first introduce Deligne cohomology for orbifolds (in fact we do this for foliations). Then we show that the Deligne cohomology group H 3 ( X , Z(3)E) classifies gerbes with connection over the orbifold (we have introduced the concept of gerbe with connection over an orbifold in a previous paper [ S ] ) . The basic idea of interpreting the B-field and the discrete torsion as Deligne cohomology classes is related to the I<-theory twisting in the following way. We have showed that every twisting can be interpreted as a gerbe over the orbifold, and that gerbes are classified by the third integral cohomology of the classifying space H 3 ( B X , Z). In particular we showed how to produce such a gerbe for the case of discrete torsion. In fact in the orbifold case the cohomology class [HI corresponding to the one used by Witten in his twisting is now a class [HI E H 3 ( B X , Z ) . But if we want to actually consider the forms themselves up to isomorphisms, t,hen both the B-field and the discrete torsion will then be classes in the enhanced cohomology group H 3 ( X , Z(3)g). The mathematical expression of the exact relation between the actual B-field B E H 3 ( X , Z(3)E) and its class [HI is given by a forgetting map H 3 ( X , Z(3)g) + H 3 ( ~ X Z),. Dixon, Harvey, Vafa and Witten [3] have shown that from the point of view of string theory the right notion of the loop space for the global quotient orbifold X = M / G should include the twisted sectors, namely LX = L g X / C ( g ) where the disjoint union runs over all the conjugacy classes ( 9 ) of elements g E G, and C ( g ) is the centralizer of g in G. We have generalized this argument to the case of an arbitrary orbifold [7] (even one that is not a global quotient) by means of the concept of the loop groupoid LX. Even in the case of global quotients the loop groupoid has the virtue that it is invariant of the representation chosen for the orbifold. For example if X = Ml/G1 = M2/G2 as orhifolds then the loop groupoid will be the same for both representations. We have also shown that the loop groupoid LX has a circle action, and that the S1 invariant loops in LX correspond exactly with the twisted sectors of the orbifold that we will write as AX (we call these the inertia groupoid [7]of the orbifold X ) . In the case of a global quotient X = M / G and a discrete torsion a E H3(G,Z)we constructed a gerbe L, over X (corresponding to the cohomology map H 3 ( G , Z ) -+ H 3 ( B X , Z ) , ) and then from here by using the holonomy of
u(g)
470
this gerbe we produced a line bundle over AX (corresponding to the restriction of the holonomy line bundle over LX of the gerbe La to AX). In this paper we prove that such line bundle admits a natural flat connection. In fact we prove something more interesting and general. For any gerbe with a connection over X we will construct a flat line bundle over the twisted sectors AX. We will do this using Deligne cohomology in the following way. We will prove that H3(X,Z(3)E) classifies gerbes with connection over X, that H 2 ( Y ,Z(2)E) classifies line bundles with connection over Y , and finally that there exists a natural holonomy map
H3(X, Z(3)g) --+ H 2 ( A X , 2 ( 2 ) g ) (1) whose image lands in the family of flat line bundles over AX. To finish this introduction let us mention that the mathematical results of this paper can be approached from a more general point of view that includes all the Deligne cohomology groups (not only the second and the third) and their geometric interpretations [8]. 2
Preliminaries
In this section we will review the basic concepts of groupoids, sheaves over groupoids and the cohomologies associated to them. For a more detailed description we recommend to see Haefliger [4], Crainic and Moerdijk [2] and Lupercio and Uribe [7]. 2.1
Topological groupoids
The groupoids we will consider are small categories G in which every morphism is invertible. By G1 and Go we will denote the space of morphisms (arrows) and of objects respectively, and the structure maps
G1 t x, G1-
m
G ~ A G ~ t
where s and t are the source and the target maps, m is the composition of two arrows, i is the inverse and e gives the identity arrow over every object. The groupoid will be called topological (smooth) if the sets GI and Go and the structure maps belong to the category of topological spaces (smooth manifolds). In the case of a smooth groupoid we will also require that the maps s and t must be submersions, so that G1 t X, G1 is also a manifold. A topological (smooth) groupoid is called &ale if all the structure maps are local homeomorphisms (local diffeomorphisms). For an Btale groupoid we
47 1
will mean a topological Btale groupoid. In what follows, sometimes the kind of groupoid will not be specified, but it will be clear from the context to which one we are referring to. We will fix notation for the groupoids and they will be denoted only by letters of the type G, H, S. Orbifolds are a n special kind of Btale groupoids, they have the peculiarity that the map ( s ,t ) : G I -+ Go x Go is proper, groupoids with this property are called proper. Whenever we write orbifold, a proper Btale smooth groupoid will be understood. A morphism of groupoids 8 : H -+ G is a pair of maps 8i : Hi -+ Gi i = 1 , 2 such that they commute with the structure maps. The maps !Pi will be continuous (smooth) depending on which category we are working on. The morphism 8 is called Morita if the following square is a Cartesian square
If the Morita morphism is between Btale groupoids and the map 5 7 ~ 2: Ho qox t GI -+Go is a n Btale surjection (local homeomorphism, diffeomorphism) then the morphism is called ktale Morita; for orbifolds we will require the Morita morphisms to be Btale. Two groupoids G and H are Morita equivalent if there exist another K 3 H. groupoid K with Morita morphisms G For an &ale groupoid G , we denote by Gn the space of n-arrows 50
3 5 1 4 * . . 9xn
The spaces Gn (n 2 0 ) form a simplicia1 space:
that together with the face maps di : Gn di(gi,.-.,gn)=
{
+ G,-1
ifi=O ( 9 2 , . . . ,g n ) (gi,...,gigi+i,...,gn) i f O < i < n ifi=n (91,. . .,gn-1)
form what is called the nerve of the groupoid. Its geometric realization [12] is the classifying space of G , denoted BG. A Morita equivalence H 7G induces a weak homotopy equivalence BH 7BG.
472
2.2
Sheaves and cohomology
From now on, we will restrict our attention to the case where G is an &ale groupoid and smooth when required. A G-sheaf 3is a sheaf over Go, namely a topological space with a projection p : 3 -+ Go which is a local homeomorphism on which GI acts continuously. This means that for a E 3, = ~ - ' ( I I : ) and g E GI with s(g) = 2, there is an element ag in 3t(g) depending continuously on g and a. The action is a map 3 , x , GI -+ 3 . All the properties of sheaves and its cohomologies of topological spaces can be extended for the case of &ale groupoids as is done in Haefliger [4] and Crainic and Moerdijk [2]. For 3 a G-sheaf, a section a : Go + 3 is called invariant if o(x)g = a ( y ) for any arrow II: 4 y. rinW(G,3)is the set of invariant sections and it will be a n abelian group if 3 is an abelian sheaf. For an abelian G sheaf 3,the cohomology groups H"(G,3) are defined as the cohomology groups of the complex:
rinw(G,P) -+ rinw(G,'T1) -+ ... T1 + ... is a resolution of 3 by
where 3 -+ 'T' -+ injective G-sheaves. When the abelian sheaf 3 is locally constant (for example 3 = Z) is a result of Moerdijk [9] that
H*(G,3)% H * ( B G , 3 ) where the left hand side is sheaf cohomology and the right hand side is simplicial cohomology. There is a basic spectral sequence associated to this cohomology. Pulling back 3 along E,
: G,
-+
Go
(2)
En(g1, . . . ,gn) = t ( g n > it induces a sheaf .$3on G, (where the G action on G, is the natural one, i.e. (91,. . . ,gn)h = (91,. . . ,gnh); G, becomes in this way a G-sheaf) such that for fixed q the groups Hq(G,, $3)form a cosimplicial abelian group, inducing a spectral sequence:
HpHq(G.,3)3 H p + q ( G , 3 ) So if 0 + 3 -+ 30 -+ 3l -+ . . . is a resolution of G-sheaves with the property that ,$3q is an acyclic sheaf on G,, then H*(G, 3 ) can be computed from the double complex I? (G, ,~ ; 3 ~ ) .
473
We conclude this section by introducing the algebraic gadget that will allow us to define Deligne cohomology. Let 3' be a cochain complex of abelian sheaves, then the hypercohomology groups JHIn(G, 3)are defined as the cohomology groups of the double complex I'inv(G,7.) where 3' -+ 7 'is a quasi-isomorphism into a cochain complex of injectives. 3
Deligne Cohomology
In what follows we will define the smooth Deligne cohomology of a smooth Btale groupoid; we will extend the results of Brylinski [l]to groupoids and will follow very closely the description given in there. We will assume all through out this paper that the set of objects Go of our groupoid G has an open cover by subsets which are each paracompact, Hausdorff, locally compact and of bounded cohomological dimension depending on Go. If not specified explicitly, when working on a groupoid we will have always in mind its description given by this cover; in other words, we will think of Go as the disjoint union of this cover and G1 its respective space of arrows that will keep the groupoid in the same Morita class. Deligne cohomology is related to the De Rham cohomology. We will consider the De %am complex of sheaves and we will truncate it at level p ; what interests us is the degree p hypercohomology classes of this complex. To be more specific, let Z ( p ) := ( 2 7 r G ) P . iZ be the cyclic subgroup of C, AP(Go)@ := Ap(G0) @ C the space of complex-valued p forms on Go and d& the G-sheaf of complex-valued differential pforms; as G is a smooth Btale groupoid the maps s and t are local diffeomorphisms, then the action of G into the sheaf over Go of complex-valued differential pforms is the natural one given by the pull back of the corresponding diffeo. Let Z ( ~ ) be G the constant Z(p)-valued G-sheaf, and i : Z ( ~ ) -+ G d& the inclusion of constant into smooth functions. Definition 3.0.1. Let G be a smooth itale groupoid. The smooth Deligne complex Z ( p ) g is the complex of G-sheaves:
Z ( ~ ) -% G dE,c A d& + . . * d
5dPp1 G,@
The hypercohomology groups WQ(G,Z(p)g) are called the smooth Deligne cohomology of G. Formally, this description will do the job, but we would like to have a more concrete definition of this cohomology. This is done using a Leray description of the groupoid and then calculating the cohomology of the respective eech double complex. For a manifold M , a Leray cover U is one on which all the open sets and its intersections are contractible. With this cover we can
474
calculate the De Rham cohomology of M by calculating the cohomology of the Cech complex with real coefficients; for the De Rham cohomology of the open sets in the cover is trivial. The same idea can be applied to the hypercohomology of the Deligne complex on a manifold, but in this case we obtain a Cech double complex. This will be explained in more detail in the next section. Definition 3.0.2. Let H be a smooth &ale groupoid. A Leray description of H is a n Wale Morita equivalent groupoid G , provided with a n e'tale Morita morphism G + H, o n which all the sets Gi, for i 2 0, are diffeomorphic to a disjoint union of contractible sets. The existence of such a groupoid for orbifolds (namely proper &ale smooth groupoids) can be proved using the results of Moerdijk and Pronk [lo]. In any case, this is clear for most relevant examples. In order to make the calculations clearer, where are going to work with a quasi-isomorphic complex of sheaves to the Deligne one, which is a bit simpler. Definition 3.0.3. Let CX( p ) be ~ the following complex of sheaves:
It's easy to see that there is a quasi-isomorphism between the complexes (2~-)-~+' . z ( p ) g and C x (p)G[-l] (this fact is explained in Brylinski [l] page 216)
hence there is an isomorphism of hypercohomologies:
W'-l(G,
(Cx
( p ) ~?)2(2~-)-'+'
W'(G,
2k)g)
(3) Now let G be such Leray description of the groupoid. We are going to define the Cech double complex associated to the G-sheaf complex ( C x ( p ) ~ . Consider the space
c"' = d;(Gk, &,)
*
:= r ( G k , €id&)
of global sections of the sheaf E ; d L , @ over G k as in (2). The vertical differential Ck$'+ Ckyl+' is given by the maps of the complex C x ( p ) ~and the horizontal differential Ckil -+ Ck+'i' is obtained by 6 = C(-l)Z6i where for 0
E r ( G k , ti&,@)
475
( b ) ( g l , . . . ,gk+l) =
for i = k + 1 . . ,5%) . Sk+l a h , . . . , g m + l , ,. . ,grc+l) for 0 < i < k + 1 for i = 0 4 9 2 , . . . ,%+d
i
4 1 , .
Definition 3.0.4. For G a Leray description of a smooth &ale groupoid, let's ) total complex induced by the double complex denote by C(G,Cx ( p ) ~the
The 2ech hypercohomology of the complex of sheaves Cx ( p ) is ~ defined as the cohomology of the tech complex C(G,Cx ( p ) ~ ) : B * ( G ,CX( p ) ~:= ) H * C ( G ,Cx( p ) ~ ) . Due to all the conditions imposed to the Leray description, the previous cohomology actually matches the hypercohomology of the complex C ( p ) ~ , so we get Lemma 3.0.5. Let H be a smooth itale groupoiduand G a Leray description of it. Then the cohomology of the tech complex C(G,Cx ( p ) ~ is) isomorphic to the hypercohomology of C (p) H
k * ( G , C X ( p ) G )5 W * ( G , C X ( p ) ? ~ ) W*(H,CX(p)~) M
where the second isomorphism is induced by the map G + H. We will postpone the proof of this result to a forthcoming paper [8]. This Cech description will be the one that will allow us understand the relationship between the B-field and the discrete torsion and also gives us an inside view of what the hypercohomology calculates.
476
We conclude this section by observing that with the definition of gerbe with connection over an orbifold given in our previous work [6] we can easily prove the following Proposition 3.0.6. For G a Leray description of a smooth &ale groupoid, a ) , is, a gerbe with connection is a 2-cocycle of the complex C ( G , C X ( 3 ) ~that and h E e(G2,C:) triple ( h , A , B ) with B E e(Go,d&),A E e(G1,d&) that satisfies SB = dA, SA = dlogh and 6h = 1. Two such gerbes with connection are isomorphic if they lie in the same cohomology class, hence they are classified by W3(G, Z(3)g). 4
B-field and Discrete Torsion
4 .I
Manifolds
A B-field over a manifold M (see Hitchin [5]) is a choice of gerbe with connection which in terms of a Leray cover {U,} of M is described by a collection of 2-forms B, over each U,, 1-forms A,p over the double intersections Uap := U, n Up and CX-valued functions hap, over triple overlaps U,ar satisfying B,
-
Bp
= dA,p
+
Aag Ap-y - A,, = d log haor h~,&h,pvh&hhp,v = 1. If we consider the description as a groupoid of M given by the Leray cover pa):
Mo := U U ,
and
01
MI := u U , p 4
and we collect the information of these functions as sections of the sheaves defined in the previous section, in other words B E C(Mo,d&), A E e(M~,dh,~) and h E e ( M 2 , C i ) , it's easy to see that those equations become: SB = dA
SA = d log h Sh = 1 then the triple ( h , A , B ) determines a cocycle in the Cech complex CX( 3 ) ~ (see ) theorem 5.3.11 [ 5 ] ) ;hence, by lemma 3.0.5 we obtain
e(M,
477
Theorem 4.1.1. A choice of B-field in a manifold M determines a cocycle (h,A , B ) of the complex C(M, C X( 3 ) ~ and ) vice versa. Moreover, the isomorphism classes of choices of B-field are classified by the cohomology class of (h,A , B ) in the hypercohomology group W 2 ( MC$ , dlog da,,). Then, in view of the isomorphism 3 we conclude Corollary 4.1.2. The choice of B-field over a manifold M is classified by the third Deligne cohomology group W 3 ( M ,Z(3)F) of the manifold. Here we should point out that we [6] have generalized the picture described by Hitchin [5] to the case of orbifolds, and hence the previous statements remain true for the orbifold case.
dh,e
4.2
Discrete torsion
In what follows we will argue that over an orbifold the discrete torsion is just another choice of gerbe with connection, as is the B-field. In other words, the discrete torsion and the B-field are both extreme cases of the same picture, namely gerbes with connection; while the B-field only takes into account the differentiable structure of the orbifold, the discrete torsion one only considers the extra structure added by the action of the groups. In the case of a global quotient M / G with G a finite group acting via diffeomorphisms over M , we can take one &ale groupoid that models it IS], i.e. Xo := M and XI := M x G with the natural source and target maps: s ( z , g ) = z and t ( z , g ) = zg; denoting by G := * x G =! * the groupoid associated to G we have a natural morphism
x+G.
(4)
As G is formed by a discrete set of points we have that
H ~ ( Gex ,
W2(G, c;
d -+
as there is a natural monomorphism induced by (4)
-5
-+ W2(X,
c;
we get a map
H2(G,ex)-+ W2(X, C;
‘3di,e
d;,,).
Theorem 4.2.1. For the orbifold X = [M/G]the homomorphism H2(G,CX) -+ W2(X,@G
dlog di,c
d;,,)
478
is injective. Therefore the choice of discrete torsion is a subgroup of the equivalence classes of gerbes with connection over the groupoid, which is classified by the third Deligne cohomology of the orbifold, namely (2 E
J-'T)-2.
W3(X, Z(3)E).
Let c : G x G + Cx be a 2-cocycle and c : M x G x G t Cx , c ( x ,g , h) := c(g,h) its image under the morphism. If (Z, 0,O) = 0 in W2(X, C x ( 3 ) ~ then ) there exist a map f : M x G + Cx such that Sf = C and df = 0. As
(sf)(., 9, h ) = f(.,
g>f(., g h ) - l f ( x , h ) = c(x,9, h) = 4 9 , h)
we get that the cocycle c is also exact; take for any fixed x , then 6a = c. 5.
0
: G t C x with
a ( g ) := f ( x , g )
Gerbes with Connection and the Inertia Groupoid
In this section we are going to construct the holonomy bundle of a gerbe with connection, which in the case of a groupoid will be a flat line bundle over the inertia groupoid. To do this we need to recall some definitions. 5.1
Line bundles with connection
From theorem 2.2.12 of Brylinski [l]we know that the group of isomorphism classes of line bundles with connection over a manifold M is canonically isomorphic to its second Deligne cohomology, namely ( 2 7 r d q - l . W 2 ( M ,Z(2)Z) z WyM, C&
dlog
The same result can be extended to cover smooth &ale groupoids, let's explain the idea. For G a Leray description of a groupoid G , a line bundle with connection over it is a morphism of groupoids p : G ?r @x and a 1-form A over Go such that
s*A- t*A = dlogpl. But p is a morphism of groupoids if and only if 6pl = 1; considering p1 also as an element of e(Gl,C,X)(recall that p1 : G I + Cx); i.e.
( h ) ( g 1 , g 2 )= ~
1 ( ~ 2 ) ~ 1 ( ~ 1 ~ 2 ) -= 1 ~1.1 ( ~ 1 )
Proposition 5.1.1. The line bundle with connection (p, A) over G represents ) its isomorphism class is classified a I-cocycle in the complex e(G,Cx ( 2 ) ~ and by the respective element in W1(G,C: d$g A;).
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The inertia groupoid
5.2
The inertia groupoid AG is defined in the following way:
r\Go = {U E Glls(a) = t ( a ) }
r\G1
= { ( a ,b) E Gala E
/\Go}
with s(a, b) = a and t ( a ,b) = b-lab. Here we consider the case in which this groupoid is smooth. We know (see theorem 6.4.2 [7]) that a gerbe over a n &ale groupoid 6 : G 2 -+ C x with 66 = 1 determines a line bundle over the inertia groupoid p : AG + (cx with
(we recommend the reader to see our previous paper [7] to get acquainted with the terminology). Now we want to extend the previous result to define a line bundle with connection over the inertia groupoid from a gerbe with connection. Lemma 5.2.1. Let G be a Leray description of a smooth &tale groupoid and (h,A , B ) a 2-cocycle of C(G,Cx (3)G) (a gerbe with connection). Then the pair ( p , V ) where
is a 1-cocycle of the induced complex e ( A G ,@'(2)&). We just need to prove that 6V = dlogp. We have that in e ( A G ,CX( ~ ) A G ) :
( b V ) ( a ,b) = V ( s ( a b, ) ) . b - V ( t ( a ,b ) ) = V ( a ). b - V(b-lab) and in
C(G,ex(3)G):
+
(bA)(a,b) = A(u) . b - A ( d ) A(b) = dlog h ( a ,b) (bA)(b,b-lab) = A(b) - A(ab) + A(b-lab)
= d l o g h ( b , b-lab).
By definition V ( a ) = A ( a ) and V(b-lab) = A(b-lab), so we get
( b V ) ( a ,6) = (bA)(a,b) - (dA)(b, b-lab) = dlog
h(a7b, = dlogp(a, b). h(b, b-lab)
0 In the same way as before, 1-cocycles in 6'(G,CX(3)G)induce 0-cocycles in 6 ( A G ,Cx (2)AG),so we get that there is a morphism
H2C(G,C x (3)G) HIC(r\G,(Cx ( 2 ) A G ) that extends to a morphisms in hypercohomology:
(5)
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Theorem 5.2.2. Let G be a smooth &ale groupoid, then there exists a holonomy homomorphism
which to every gerbe with connection assigns a line bundle with connection over the inertia groupoid AG. Moreover, this line bundle is flat. Let's assume G is the Leray description of such a groupoid, from lemma 3.0.5
and in the same manner as proposition 1.3.4 of Brylinski [l],there is a canonical homomorphism
which together with the morphism (5) implies (5.2.2). Using the notation of lemma 5.2.1 we know that SB = d A , or in other words, B ( s ( g ) )- B ( t ( g ) )= d A ( g ) for g GI. Then is clear that for a E A G ~ d A ( a ) = 0 , hence d V = 0. The induced line bundle with connection over the groupoid is flat. 0 In the case that G is an orbifold, these induced flat line bundles over the inertia groupoid are precisely what Ruan [ll]denoted by inner local systems (see the last section of our previous paper [7]), then we can conclude Proposition 5.2.3. A gerbe with connection over an orbifold G determines an inner local system [ll]over the twisted sectors CTG. The homomorphism of theorem 5.2.2 can be generalized to all the Deligne cohomology groups. So we get the following result which will be proved in a forthcoming paper [8] Theorem 5.2.4. Let G be an smooth &ale groupoid, then there exist a natural morphism of complexes
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which induces f o r all p > 0 a morphism of cohomologies WQ(G,CXb)G)
1-
(anJ--T)-P+l . WQ+1 (G,ZS(p)g)
W Q - l ( ~ GC, x ( p - l),,G)
-
1-
( 2 7 ~ & i ) - ~ + ~W. Q ( A G , Z (-~ 1)g)
Observe that these facts provide a generalization of the concept of inner local system for the twisted multisectors of Ruan. It is reasonable t o predict that this will be relevant in the full theory of Gromov-Witten invariants on orbifolds.
Acknowledgments. We would like to thank conversations with M. Crainic, I. Moerdijk and Y. Ruan regarding several aspects of this work. The first author would like to thank Paula Lima-Filho for some conversations that motivated his interest in Deligne Cohomology. The second author would like to express its gratitude to the organizers of the Summer School on Geometric and Topological Methods for Quantum Field Theory, and specially to its sponsors the Centre International de Mathbmatiques Pures et Appliqukes (CIMPA) and the Universidad de 10s Andes, for the invitation to take part in the school where some of the previous results were presented. References 1. J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization, Progr. Math., 107, Birkhauser Boston, Boston, MA, 1993 2. M. Crainic and I. Moerdijk, A homology theory for &ale groupoids, J. Reine Angew. Math. 521 (ZOOO), 25-46 3. L. Dixon et al., Nuclear Phys. B 274 (1986), no. 2, 285-314 4. A. Haefliger, Differential cohomology, in Differential topology (Varenna, 19761, 19-70, Liguori, Naples, 1979 5. N. Hitchin, Lectures on Special Lagrangian Submanifolds, arXiv: math.DG/9907034 6. E. Lupercio and B. Uribe, Gerbes over Orbifolds and Twisted K-theory, arXiv: math.AT/0110207 7. E. Lupercio and B. Uribe, Loop Groupoids, Gerbes, and Twisted Sectors on Orbifolds, arXiv: math.AT/0110207
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8. E. Lupercio and B. Uribe, Deligne cohomology of &ale groupoids, in preparation. 9. I. Moerdijk, Proof of a conjecture of A. Haefliger, Topology 37 (1998), no. 4, 735-741 10. I. Moerdijk and D. Pronk, Simplicia1 cohomology of orbifolds, Indag. Math. (N.S.) 10 (1999), no. 2, 269-293 11. Y. Ruan, Stringy geometry and topology of orbifolds, arXiv: math.AG/O011149 12. G. Segal, Classifying spaces and spectral sequences, Inst. Hautes Etudes Sci. Publ. Math. No. 34, (1968), 105-112 13. E. Sharpe, Discrete Torsion, Quotient Stacks, and String Orbifolds, arXiv: math.DG/0110156 14. E. Witten, D-branes and K-theory, arXiv: hep-th/9810188 v2
