QUANTUM THEORY, DEFORMATION AND INTEGRABILITY
QUANTUM THEORY, DEFORMATION AND INTEGRABILITY
NORTH-HOLLAND MATHEMATICS STUDIES 186 (Continuation of the Notas de Matem&tica)
Editor: Saul LUBKIN University of Rochester New York, U.S.A.
2000 ELSEVIER Amsterdam
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York
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Q U A N T U M THEORY, DEFORMATION AN D I NTEGRABILITY
Robert CARROLL
University of Illinois, Urbana, IL 61801
2000 ELSEVIER Amsterdam
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Oxford
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Contents Preface
ix
1 QUANTIZATION AND INTEGRABILITY 1.1 ALGEBRAIC AND GEOMETRIC METHODS . . . . . . . . . . . . . . . . . 1.1.1 Remarks on integrability .classical KP . . . . . . . . . . . . . . . . . 1.1.2 Dispersonless theory for KdV . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Toda and dToda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Remarks on integrability . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Hirota bilinear difference equation . . . . . . . . . . . . . . . . . . . . 1.1.6 Quantization and integrability . . . . . . . . . . . . . . . . . . . . . . 1.2 VERTEX OPERATORS AND COHERENT STATES . . . . . . . . . . . . . 1.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Heuristics for QM and vertex operators . . . . . . . . . . . . . . . . . 1.2.3 Connections to ( X ,$J) duality . . . . . . . . . . . . . . . . . . . . . . . 1.3 REMARKS ON THE OLAVO THEORY . . . . . . . . . . . . . . . . . . . . 1.4 TRAJECTORY REPRESENTATIONS . . . . . . . . . . . . . . . . . . . . . 1.4.1 The Faraggi-Matone theory . . . . . . . . . . . . . . . . . . . . . . . . 1.5 MISCELLANEOUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Variations on Weyl-Wigner . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Hydrodynamics and Fisher information . . . . . . . . . . . . . . . . .
1 1 4 9 11 18
2
GEOMETRY AND EMBEDDING 2.1 CURVES AND SURFACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 The role of constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Surface evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 On the embedding of strings . . . . . . . . . . . . . . . . . . . . . . . 2.2 SURFACES IN R3 AND CONFORMAL IMMERSION . . . . . . . . . . . . 2.2.1 Comments on geometry and gravity . . . . . . . . . . . . . . . . . . . 2.2.2 Formulas and relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 QUANTUM MECHANICS ON EMBEDDED OBJECTS . . . . . . . . . . . 2.3.1 Thin elastic rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Dirac field on the rod . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 The anomaly in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 WILLMORE SURFACES, STRINGS, AND DIRAC . . . . . . . . . . . . . . 2.4.1 One loop effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 CMC surfaces and Dirac . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Immersion anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Quantized extrinsic string . . . . . . . . . . . . . . . . . . . . . . . . . V
19 21 23 23 31 36 45 49 50 56 5G 57
63 63 63
67 68 72 73 79 80 86 87 88 95 98 98 100 104 108
CONTENTS
vi
2.5 CONFORMAL MAPS AND CURVES . . . . . . . . . . . . . . . . . . . . . .
110
3 CLASSICAL AND QUANTUM INTEGRABILITY 113 3.1 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 114 3.1.1 Classical and quantum systems . . . . . . . . . . . . . . . . . . . . . . 3.2 R MATRICES AND PL STRUCTURES . . . . . . . . . . . . . . . . . . . . . 119 3.3 QUANTIZATION AND QUANTUM GROUPS . . . . . . . . . . . . . . . . . 124 128 3.3.1 Quantum matrix algebras . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Quantized enveloping algebras . . . . . . . . . . . . . . . . . . . . . . 129 130 3.4 ALGEBRAIC BETHE ANSATZ . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.5 SEPARATION OF VARIABLES . . . . . . . . . . . . . . . . . . . . . . . . . 138 3.5.1 r-matrix formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 141 3.5.3 XXX spin chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 HIROTA EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 3.7 SOV AND HITCHIN SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . 147 149 3.8 DEFORMATION QUANTIZATION . . . . . . . . . . . . . . . . . . . . . . . 151 3.8.1 Path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2 Nambu mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 3.9 MISCELLANEOUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3.9.1 Geometric quantization and Moyal . . . . . . . . . . . . . . . . . . . . 160 164 3.10 SUMMARY REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 DISCRETE GEOMETRY AND MOYAL
4.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Phase space discretization . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Discretization and K P . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Discrete surfaces and K P . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Multidimensional quadrilateral lattices . . . . . . . . . . . . . . . . . . 4.1.5 d methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 HIROTA, STRINGS, AND DISCRETE SURFACES . . . . . . . . . . . . . . 4.2.1 Some stringy connections . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Discrete surfaces and Hirota . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 More on HBDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Relations to Moyal (expansion) . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Further enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.6 Matrix models and Moyal . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.7 Berezin star product and path integrals . . . . . . . . . . . . . . . . . 4.3 A FEW SUMMARY REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Equations and ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 MORE ON PHASE SPACE DISCRETIZATION . . . . . . . . . . . . . . . . 4.4.1 Review of Moyal-Weyl-Wigner . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Various forms for difference operators . . . . . . . . . . . . . . . . . . 4.4.3 More on discrete phase spaces . . . . . . . . . . . . . . . . . . . . . . .
167
167 167 171
175 178 188 194 195 198 201 204 212 221 223 227 227 234 236 240 246
5 WHITHAM THEORY 255 5.1 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 5.1.1 Riemann surfaces and BA functions . . . . . . . . . . . . . . . . . . . 256 257 5.1.2 Hyperelliptic averaging . . . . . . . . . . . . . . . . . . . . . . . . . .
CONTENTS
5.2
5.3
5.4
5.5 5.6
5.1.3 Averaging with $J*+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Dispersionless KP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ISOMONODROMY PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 JMMS equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Gaudin model and KZB equations . . . . . . . . . . . . . . . . . . . . 5.2.3 Isomonodromy and Hitchin systems . . . . . . . . . . . . . . . . . . . WHITHAM AND SEIBERG-WITTEN . . . . . . . . . . . . . . . . . . . . . 5.3.1 Basic variables and equations . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Other points of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . SOFT SUSY BREAKING AND WHITHAM . . . . . . . . . . . . . . . . . . 5.4.1 Remarks on susy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Soft susy breaking and spurion fields . . . . . . . . . . . . . . . . . . . RENORMALIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Heuristic coupling space geometry . . . . . . . . . . . . . . . . . . . . WHITHAM, W D W , AND PICARD-FUCHS . . . . . . . . . . . . . . . . . . 5.6.1 ADE and LG approach . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 F’robenius algebras and manifolds . . . . . . . . . . . . . . . . . . . . . 5.6.3 Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations . . . . . . . .
vii
2G2 264 270 275 279 283 293 294 300 302 302 306 309 309 314 314 319 321
6 GEOMETRY AND DEFORMATION QUANTIZATION 325 6.1 NONCOMMUTATIVE GEOMETRY . . . . . . . . . . . . . . . . . . . . . . 325 6.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 329 6.1.2 Spectral triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 The noncommutative integral . . . . . . . . . . . . . . . . . . . . . . . 332 6.1.4 Quantization and the tangent groupoid . . . . . . . . . . . . . . . . . 335 6.2 GAUGE THEORIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 6.2.1 Background on noncommutative gauge theory . . . . . . . . . . . . . . 341 343 6.2.2 The Weyl bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Noncommutative gauge theories . . . . . . . . . . . . . . . . . . . . . 349 353 6.2.4 A broader picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 6.3 BEREZIN TOEPLITZ QUANTIZATION . . . . . . . . . . . . . . . . . . . . Bibliography
363
Index
401
This Page Intentionally Left Blank
Preface Some years ago I wrote a book [143] called Mathematical Physics (not my choice of title) which was designed mainly to be of service in teaching courses in this area to students from engineering backgrounds. It also included some original work on inverse scattering and transmutation following [144, 145, 146] (cf. also [147], which was aimed in somewhat different directions). These books all contained an enormous amount of information in a limited number of pages and the results were at times difficult to read. It has been said that I tend to write for myself and in particular these books exhibit what I needed in order to lecture on the subjects in question without recourse to outside reference, thus making it easy to teach from them. Also I tried to make the books self contained and rich enough in content to justify their purchase. I often attempt to establish meaning through computation and via relations and analogy; this often requires working through the calculations in order to find and appreciate the meaning. At the present time I am retired from teaching and my motivation in writing is primarily to learn and understand, which is partially accomplished by organizing and connecting various subjects, with occasional contributions inserted as they may occur. The hope in this book is to produce a vdhicle to greater understanding and a stimulus-guide for further research. We do not believe it is possible to include all of the relevant background mathematics and leave room for anything else. Thus we will not even try to develop mathematical topics axiomatically and will adopt style, definitions, and notation from reference sources we have found to be especially illuminating (such references are given as we go along). We feel that mathematics is basically easy (but very complicated) and can be self taught (and often developed) quickly, but physics is by nature exploratory and philosophical, and intuition therein seems to require more time (or perhaps discovery of the appropriate mathematical setting). One sees in many places where a few physical assumptions (possibly incorrect as such or suspicious) lead nevertheless to a mathematical thread of reasoning of some elegance and beauty whose consequences are believable. In such cases we do not probe the physical assumptions too carefully but suggest that perhaps the assumptions are more or less correct or, if not, perhaps other related assumptions and/or techniques might lead to similar mathematical conclusions. Thus the mathematical framework is given priority here. The theme of quantization via deformation of algebras is emphasized throughout as are connections to integrability. Some material from my publications is worked into the text and references to publications before 1991 can be found in the books cited (cf. also http://www.math.uiuc.edu/-rcarroll/for a home page listing). We have adopted a writing style that lies close to the discussion format of physics (although there is an occasional S U M M A R Y x.y remark which could often be regarded as a theorem); theorems are generally worked into the text either before or after a proof. This is largely experimental at first writing and will probably not be easily describable in any event; the main point seems to be to know what not to say. As models of good writing we think immediately of [42, 54, 486] which are rich in verbal imagery. With the mathematics we try to avoid excessive generality which might result in axiomatic "trash" without any visible ix
connection to the physics. It seems desirable to discover and develop those mathematical structures which capture some physical behavior but it would seem silly (or premature) at the present state of knowledge to identify "physics" with appropriate generalizations of such structures (despite maxims to the effect that symmetry is the driving engine for physics, and the frequent temptation to say that mathematics is physics - or perhaps nature). In this spirit one might discover mathematics via physics as has occured at various times in history (see e.g. [220] involving a "derivation" of K theory from M theory). In any event at the moment it is of primary concern for me at least to understand what is quantum and what is classical and how all this is logically connected (as one perhaps hopes it must be - only a coherent emergent reality is requested, not an a priori given structure or grand design). We will make no attempt to describe all the marvels now unfolding which relate to string (M) theory, quantum gravity, noncommutative geometry, etc. (many of which are still in a rather embryonic state) but will try to keep in touch with some of this. Some themes will be pursued more diligently than others and some historical matters will be discussed at some length because of their rich content and interaction with other areas. The pace of development today is so rapid that one feels at times privileged to serve even in a reportorial role.
About 4 years ago a prominent string theorist was quoted as saying that it might be possible to understand quantum mechanics by the year 2000. Sometimes new mathematical developments make such understanding appear possible and even close but on the other hand increasing lack of experimental verification make it seem to be further distant. In any event one seems to arrive at new revolutions in physics and mathematics every year. I hope to be able to convey some of the excitement of this period here but will adopt a relatively pedestrian approach designed to illuminate the relations between quantum and classical. There will be some discussion of philosophical matters such as measurement, uncertainty, decoherence, etc. but philosophy will not be emphasized; generally we want to enjoy the fruits of computation based on the operator formulation of QM and quantum field theory. In Chapter 1 connections of QM to deterministic behavior are exhibited in the trajectory representations of Faraggi-Matone [90, 315, 317] (these papers give a deep theoretical foundation for such approaches and we have made some contact in [153, 155, 169]). Chapter 1 also includes a review of KP theory and some preliminary remarks on coherent states, density matrices, etc. and more on deterministic theory. We develop in Chapter 4 relations between quantization and integrability based on Moyal brackets, discretizations, KP, strings, and Hirota formulas and in Chapter 2 we study the QM of embedded curves and surfaces illustrating some QM effects of geometry. Chapter 3 is on quantum integrable systems, quantum groups, and modern deformation quantization. Chapter 5 involves the Whitham equations in various roles mediating between QM and classical behavior. In particular, connections to Seiberg-Witten theory (arising in iV" = 2 supersymmetric (susy) Yang-Mills (YM) theory) are discussed and and we would still like to understand more deeply what is going on (cf. [148, 149, 150, 151, 152, 154] for some forays in this direction which often indicate some things I don't know and would like to clarify). Thus in Chapter 5 we will try to give some conceptual background for susy, gauge theories, renormalization, etc. from both a physical and mathematical point of view. In Chapter 6 we continue the deformation quantization theme by exhibiting material based on and related to noncommutative geometry and gauge theory. We would also like to say a few words about pedagogy in the information age. There is so much new material arising today in mathematics and physics, much of it on electronic bulletin boards, that students may find it hard to cope with, let alone master, anything. In
xi 1995-1997 for example, before retiring from teaching, I found that, with a small group of thesis level students (from mathematics, physics, and engineering departments), it was possible in a course on mathematical physics to build some foundations and then to go directly to the net where I would select current papers and work through them in class. I also wrote out lecture notes from such papers, some of which are incorporated into this book, and this accounts in part for the style of the book. I believe that what made this approach possible. and reasonably successful, was the presence in class of two or three students with ample chutzpah and sufficient understanding to push me to explain (when I could) so that everyone benefits. This is probably a classical recipe for transmission and expansion of knowledge of course but in any case I am pleased to report that it seems to work at a distance using the net. In any event the net has made it possible for people in the provinces to be much closer to the main centers of research activity and one can use this material in a pedagogical spirit as indicated, in addition to enhancing personal research activity. A few musings may not perhaps be inappropriate. Thus we do not present here a coherent deductive structure of any physical theory. Rather, topics have been selected to describe many interacting aspects of quantum mechanics, integrability, noncommutative geometry. deformation quantization, classical mechanics, string theory, etc., for which any complete physical theory must account. There are many ideas and themes with general features such as tau functions, Hirota formulas, prepotentials, deformations, gauge transformations, index theorems, scaling, discretization, extremal principles, analyticity, scattering, moduli. and various algebraic structures acting as glue and language. It is in this spirit that we suggest the book's possible usefulness as a guide and stimulus for further research. One might inquire into the development of discretization methods with deformations as a supplement to noncommutative geometry for example (cf. however Section 4.4.3). Another theme possibly devolves from the appearance of K theory and noncommutative algebraic geometry in recent work on string, brane, and M theory (and from related ideas in noncommutative geometry); namely a lot of apparently fundamental physics seems to be intimately connected with basic mathematical objects such as natural numbers and various structures on finite (or infinite) sets, albeit via category theory, schemes, modular functors, etc. Again discretization and combinatorics, along with appropriate algebraic structure and deformations thereof, seem to arise naturally, leading one to question calculus as an appropriate directive language for quantum mechanics (or nanotechnology). One remarks also that discretization is compatible with digital computing and algorithmic thinking which implements such algebra; however we hope that the converse does not become prevalent in the guise of channeling discovery through computability (we omit quantum computation from discussion here since it remains to be realized and its realization should involve a lot of discovery).
I have given many invited talks (in Australia, Brazil, Canada, China (PRC), England, France, Germany, Greece, India, Italy, Japan, Mexico, The Netherlands, Russia, Scotland. South Africa, Taiwan, Turkey and the USA) and would like to acknowledge fruitful conversation and/or correspondence with many people. A list of names would seem excessive so let me simply express thanks for opportunities and information. I am also very grateful to my wife and traveling companion Joan for over 21 years of love, support, patience, and rationality. The book is dedicated to grandchildren: Bradley, Christopher, Emilee, Geoffrey, and Jonathan from Joan's side and Annette and Katherine from mine.
Chapter 1
QUANTIZATION AND INTEGRABILITY 1.1
ALGEBRAIC
AND GEOMETRIC
METHODS
Notations such as (15A), (16A), ( 1 S t ) , and (21A) are used. We begin with a more or less classical treatment of some main features in the Moyal-Weyl-Wigner (MWW) theory. More general material on deformation quantization appeaars in Section 3.8 for example. We refer here to [12, 18, 31, 68, 186, 138, 218, 404, 410, 458, 522, 536, 537, 538, 539, 540, 611, 647, 750, 753, 931, 901, 987, 997] for perspective and will follow here mainly [192, 203, 204, 308, 310, 311, 312, 313, 343, 638, 781, 930]. We will be essentially formal here and exhibit various formulas from the classical theory without regard for domains of validity; proofs will often be sketched, deferred, or omitted. Thus following [203, 204] for SchrSdinger wave functions ~b(x), Wigner functions (WF) are defined via 1 /
f(x,p) = -~
( h )
dye2* x -
7y
When r is an energy eigenfunction (He = Er H, f = f ,H=Ef; f ( x , p ) 9g(x,p) - f
exp(-iyp)r
( h )
(1.1)
x + 7y
then f satisfies
,~exp
(O~Op-
OpO~
;
(1.2)
( x + -~irz~COp,p- -~i~0 x ) g(x,p)
The WF's are real and satisfy -(2/h) _< f < (2/h); they may be negative. Time dependence for WF's is expressed via i h O t f ( x , p , t ) = H 9f ( x , p , t ) - f ( x , p , t ) , H
(~.3)
As h ~ 0 this reduces to O t f + {f, H} = 0 (standard P=Poisson bracket). The 9exponential is determined via V . ( x , p , t ) = exp.
= 1 + -~g(x,p) +
H. H +...
(1.4)
and generally one has f(x,p,t) = U.l(x,p,t)
9f ( x , p , O ) 9U . ( x , p , t )
(1.5)
2
CHAPTER 1. Q UANTIZATION AND INTEGRABILITY
The dynamical variables evolve classically via
=
x,H-H,x ih
= OpH; p =
p,H-H,p ih
= -OxH
(1.6)
so that the quantum evolutions (16A) x(t) = U, 9x , U, 1 and p(t) = U, , p , U, 1 turn out to flow along classical trajectories. We note also that given an operator A(2, i5) one has < .4 > = / / d x d p f ( x , p ) A ( x , p ) w h e n
/ /dxdpf(x,p)=l
(1.7)
Now one can always write (for suitable f ) (16B) f(x,p) = f f dadb]exp(iax)exp(ibp) and further exp(iax)exp(ibp) = exp,(iax)exp,(ibp) with
9 = exp, [X + - ~i~ -'~p] ez,bp 9 = e _iax _iax 9ez,bp e, , ei,bPe_ihab/2
(1.8)
Consequently any W F can be written as
f (x, p) = f ] dadb](a , b)eihab/2 e,iax e,ibp
(1.9)
so that by inserting U, 9U,1 pairs
f ( x , p , t ) - / f dadbeiabh/2eiau*l*x*U* *e ibU*l*p*U* -
-- / /dadb](a, b)eiabh/2eiax(-t) , ei,bp(-t)
(1.10)
The steps cannot generally be reversed to yield an integrand ](a, b)exp(iax(-t))exp(ibp(-t)). Consider now a generalized form of this from [278, 930] where the motivation arises in quantum statistical mechanics with kinetic equations of Lindblad type. The classical picture for phase space functions A(q, p) is operators
matrix elements
position (t < xl~lx' > = ~ ( x - ~') momentum < xl~l~' > = -i~Ox~(X- x') general fI < xlAIx' >= A(x,x') density matrix ~ < xl~lx' > = p(x,x') =< x,x'l~ >
(1.11)
with Fourier transforms
1/ dqdpe-i(rlq+(P)A(q,p); A(q,p) = ~1/ drld~ei(rlq+(P)ft(rl,~)
.7t(rl,~) -- ~
One will write z = (q,p), ~, = (0,/5), and o" = (rl,~). defined via
(1.12)
The Weyl transform of A(q,p) is
flw" A(q, p)--, •w(A)= fi = 1 f daft(a)ei~ ~
(1.13)
and the Wigner transform of .4 is
W" A ~ W(A) = A(q, p) = 2 / dte -pt/2 < q From the identities
e i(v4+~) -- ei~O/2ei~peiT?O/2;
tlAlq + t >
(1.14)
(1.15)
1.1. A L G E B R A I C A N D G E O M E T R I C M E T H O D S
3
< xlei('70+~P)lx' > = ei'7(x-it~)5(x ' - x + 27ri#~) (for # = ih/2) one obtains (16C) W(f~w(A)) = A and f~w(W(J)) = J so W ~ f~wI From (1.13) one has the Weyl ordering
n(:)
aw(qnp m) = ~
ok~mo n-k
(1.16)
0 whereas (1.14) implies that for any t5 and J (16D) T r ( ~ J ) = 1 f pAdpdq so p ~ ~wl(t)) and A = f~wl(j.). In fact p/27rh is the Wigner function associated with the density matrix ~. The 9 product is defined now via 1
1
f * g = f~wl(f~w(f)~w(g)); { f , g } u = -~#[f,g]M = -~p(f * g -- g * f)
(1.17)
where { , }M Moyal bracket. The corresponding phase space formulation of quantum statistical mechanics follows from the above via (16D). Thus (z* = 5 and flwl(A) = Aw(q,p)) ~
9 (i) J = J t
~
Aw=A~
9 Tr(A)= 1 r
f Awdqdp= 27rh
. fl;~(Ok) = qk, ~ ; l ( $ k ) = pk 9 If J = ~ = Ir > < @l is a pure state then
/,.(q..)n. = 2nlr
(1.18)
=
A substantial and natural generalization of (I.13), which includes many useful operator orderings distinct from (I.16), arises from
1/
f~" A(q,p) - , ~t(A) = ~
dafl(a)J(a)e i ~
(1.19)
One assumes the weight function fl to be an entire analytic function with no zeros. If e.g.
A(q,p) - qnpm then (1.19) gives an ordering different from (1.16). Setting ( 1 6 E ) w ( z ) (1/2~) f daexp(iaz)(1/fl(a)) the inverse ~t transformation exists and is given by fl-1.
1 j __. A(q, p) = -~ f dq'dp'dtw(q - q', p - p')e -(p't/t~) < q' - tlA]q' + t >
(1.20)
The proof is straightforward and by (1.14) one obtains
fl-l(J)
1 /
dq'dp'w(q-
= ~
q,
1
, p - p ' ) A w ( q ' , p ' ) = -~-~w@ Aw
(1.21)
where | ~ convolution (in fact any linear transformation of quantum operators to phase space functions which is phase space translation invariant is of this form). Consequently f | g = (27rfg) and (16F) = A(q,p) =~ ft(a) = (1/ft(a))ftw(a) with
ft-l(J) I
- 2 i # / dqe -i~q < q + ip~lftlq - i#~ >
(1.22)
This leads then to
f~-l(.j,.)
= _itt
dq, drld~
ei[~(q-q')+(p]
fl(~,~)
< q, + i#~lJlq'
ip~ >
(1.23)
4
CHAPTER 1. Q UANTIZATION AND INTEGRABILITY
We obtain then in analogy to (1.17) a nonabelian associative algebra structure f *n g = fl-l(fl(f)fl(g));
1 1 ~_~[f, g] = ~_~(f *a g - g *n f)
(1.24)
This is called a generalized Moyal product corresponding to the associated generalized Weyl and Wigner transformations. Now
(1.24)
can be given an explicit form as follows. From (1.19) one has
1 fdada,f(a)~(a,)f~(a)n(a,)e(i~)e(i~,~) f~(f)f~(g) = (27r)2
(1.25)
Using (1.15) one obtains (16G) exp(ia~,)exp(ia'~,) = exp[p(a' A a)]exp[i(a + or')2] where a' A a = (~', ~') A (rl, ~) = rl'~ -- ~?~'. This leads to
(f , a g)(z) = (2 1)2 f dada'](a)[7(a')f~(a)f~(a')eU(G'A~)f~-l[ei(a+~')~]
(1.26)
However by (1.19) (16H) f~[exp(iaz)] = exp(ia~,)f~(a) and hence
(f .~ g)(z) =
1 / dada' f(a)(?(a') f~(a)f~(a') eU(a,A~)ei(a+a,)z (2 )2 +
(1.27)
Some further manipulation gives also --21 g](z) = 1 f dada' f~(a)a(a') sinh[#(a' A a)] 2 , [f' (27r) 2 a(o" + a') #
(1.28)
One can then show that from (16I) Uf(z) = f do~(o)f(o')exp(ioz) there follows U(f,ng) = Uf 9Ug. This shows that the 9 and ,n products define isomorphic algebras. In fact if in (1.27) the kernel [f~(a)f~(a')/f~(cr+~')]exp[p(a'Aa)] is replaced by B(a, a'), and associativity is required, then (1.27) is the only possibility; this is related to the uniqueness of the Moyal algebra (cf. [343]). In particular one has now (16J) (1/2#) [f, g] = U -1 ((1/2#)[UI, Ug]M) and some further calculation shows that if f~ is # independent then the Moyal bracket is the only *n product whose associated Lie bracket tends to the P bracket as # ~ 0. We note finally the generalization of (16D) in the form
Tr(a(f)a(g)) = ~a(O) h f f *a gdz 1.1.1
Remarks
on integrability-
(1.29)
classical KP
We will be concerned with various aspects of classical and quantum integrability with special interest in relations between dispersionless hierarchies and deformation quantization. Some references of interest here are in particular [22, 134, 158, 160, 175, 308, 309, 374, 409, 422, 423, 424, 536, 537, 615, 617, 695, 696, 752, 891, 892, 902, 910, 911, 912, 913, 973] (let us cite also [57, 82, 127, 481, 527, 528, 623, 649]). We go first to a review of dKP following [158, 160, 161, 555, 902]. A brief sketch of this appears in Section 5.1.4 but we want to include now the "twistor" formulation of Takasaki-Takebe (cf. [160, 902, 911,912]), a more detailed examination of the dKdV situation as in [160, 153], and a treatment of dispersionless Toda (dToda) as in [32, 87, 760, 902, 911,916]. Thus there will be some overlap with Section 5.1.4 but each section is self contained.
1.1. A L G E B R A I C A N D G E O M E T R I C M E T H O D S
5
We follow here [147, 163] at first and simply list a number of formulas arising in KP theory; the philosophy is discussed at greater length in [147] for example or in other books such as [222, 558] (cf. also [450, 639]). One begins with a Lax operator L 0-~-U20-1 -~0 + F_,~ Un+l O-n (with u2 = u) and a gauge operator P = 1 + E ~ w~ O-n determined via L = P O P -1 where /) ~ cox. For wave functions ~ = Pexp(~) with ~ = E ~ Xn An where x ,,~ Xl one has ( . ) L r = )~r with Om~2 = B m r for B m = L'~ (here + ~ En>0 and "~ ~ n < 0 in an obvious notation where ~ means "corresponds to"). One knows that (8) OnBm- OmBn = [Bn, Bm] (Zakharov-Shabat (ZS)equations) and O n L - [Bn, L] (Lax equations). The symbol 0 "1 can mean e.g. f__x , _ fxoo, or ( 1 / 2 ) ( f x ~ - fx~ in particular contexts but generally we think of it algebraically as simply f. In particular =
.
.
.
.
-
B2=0 2+2u;
B3=0 a+3u0+~ux+
0 -102u; (1.30)
03u = 403u + 3uux + 3 0 - l O ~ u
The last equation is the KP equation for 02 "~ Oy and 03 ~ Or. One writes also (t b) ~P* = ( P * ) - l e x p ( - ~ ) with L*r = Ar and One* = - B n r where B n = (L*)+ n and 0* ~ - 0 . The operators Bn can also be considered as arising from "dressing" procedures (On- Bn)P-
P(On
-on);
(i)nP)P -1 = B n -
(1.31)
p i ) n p -1 = - L n_
(cf. [147, 805]) and the equation ( . . ) OuR = -Ln___P is called the Sato equation. More generally, following [559], given differential operators A, B in c9 one considers flows (&&) OAP = -(PAp-1)_P or equivalently for W = p - l , OAW -- W ( W - 1 A W ) - . Here OA ~ i)/OXA and one thinks of L and P as fixed with the coefficients wi of P determined via the u~ in L (cf. [147, 701]). Thus define LA = P A P -1 and note that OAPP -1 + Pi)AP -1 - 0 to obtain i)AL B = [L~, LB] + P ( O A B In particular for r
(1.32)
Pexp(~) one has (r ~ Baker-Akhiezer = BA function)
oo
r
[A, B ] ) P -1
(x)
= P(~-~ kxk)~k-1)exp(~) = M e ; M = P(~"~, k x k O k - 1 ) P -1 = p M p - 1 1
and [L, M ] -
(1.33)
1
1 (this operator is also discussed later). One can write also
Amn - l~ni)m; Lmn -- MaLta; ( i ) m n - L+mn)P- P ( O m n - Amn); (OmnP) --oo
= -(MnLm)-P;
Mlm
:
Am+l,1
--
]~0m+
1
= ~
kxk Oqk + m ; PMlm P - 1
-
(1.34)
MLm+I
1
One notes that (tbtb) OnMlm = [0n, Mlm] = nO n+m and hence (1.32) becomes for LB = L and LA = 0m+l,1 ~ A = Mlm (1.35)
Om+l,lL = [(MLm+I)+,L] + L m+l
where one takes Om+l,l(i)) = 0 with which it can be seen also that the flows based on/)~ and 0m+1,1 commute. Now for some background on the tau function one recalls the vertex operator equation
(VOE)
r
A) = X()QT = _e~G_(~) 1 - = T = e~(Z,~) -7_ T
T
T
(1.36)
6
CHAPTER 1. Q UANTIZATION AND I N T E G R A B I L I T Y - - ~
(
; G-t-(A)--exp(i~
T
A-I,~
)
; ~=
and ~(A -1, c5) = ~ Oj/jA j. The Hirota bilinear identity (18A) 0 = f c r A)r A)dA leads to various Hirota equations which describe tau functions via bilinear formulas, where the notation
Ot}
a(t,b(t')
=
(1.37)
t=t'
=
~s~
a(tj + s j ) b ( t j - sj) sj=O
is used. In particular, with pj ,.. Schur polynomials, one has
Epn(--2y)pn+l(~)exp
yjOj T " T = 0
(1.38)
0
The bilinear form of K P is included in (1.38) in the form the yj are "free" parameters in (1.38) and the coefficients r = Pexp(~) = (1 + ~ WnA-n)exp(~) = @exp(~) and r note also r = X*(A)~-/T = (1/7)exp(--~)G+w. Then (cf. -
wj =
,
(04 + 302 - 401i)3)~'.T = 0 (note are set equal to zero). Write now = (P*)-lexp(-~) = tb*exp(-~)" [701])
1
pj(-O)T; wj = Tpj(O)T
(1.39)
and u = 02log(T). Relations between the ui in L and the wi in P (resp. the w~ in ( p , ) - l ) can be obtained directly from writing out L = POP -1 and its adjoint. Using in addition some of the other relations above one can also obtain formulas of the form u3 = (1/2)(O02--03)log(T) etc. We list some further formulas of interest now. In particular (18B) 0 = L + E ~ a)L-J and the a~ can be computed directly via L = 0 + ~ u/+l 0 - / or via 7 as indicated below. Using L r = Ar with ( 1 8 B ) one obtains ( 1 8 C ) 0 r = Ar + E ~ ( a l / A J ) r so Olog(r A + a ~ a 1/Aj). In this spirit one obtains a l s o oo
Bm = L M = L m + ~ a~L-J; Omr = Amr + amr
(1.40)
1
oo 1 0n(71 : O n O l o g ( r determines a conservation law via ( . . . ) fo_~ O(Onlog(r On $ a l d x = 0 (given suitable b o u n d a r y conditions). Further from [701] ( 1 8 D ) 0 7 = OmPj(--O)log(T) and OnOIOg(T) = -- naln -- E ~ - - I Ojan_ 1 j. The proof can be obtained by
Thus
recursion via the Schur polynomial formula
Pn(Y) = ~
~ylkl) ~yk2~
\-~lv. [-~2v.] "'" (~-~jkj = n)
(1.41)
Finally one notes the important formulas
(X) r
" --
E
0
n 8n A - n ; 8n --
E
0
*
WjWn_ j
(1.42)
1.1. A L G E B R A I C AND GEOMETRIC METHODS
7
Using a subset of the Hirota equations one finds in fact
1
8n -- - ~ P n (
c~)
1
T --- 0n_10--1U
" T = ~T2OOn_IT"
(1.43)
since On-lOlog(T) = On-lOT" T/2T 2 and u = 02log(T). The Sn represent conserved densities since Sn = On-lOU and the basic symmetries for KP can be exhibited via the sn via K0 = O(1) = 0; K1 = 0(0) = 0; K2 = Ou; K3 = O2u;
(1.44)
3 i 02u; ... KN = OSn K4 = K = ~1 03 u + 3uOu + -~Oleading to flows On-l U = OSn = Kn. Next we extract a few formulas from [7] which are often useful. From the Fay trisecant identity (cf. [7, 147, 902]) one can derive
r
1
=, =
(._ 1
o(X(x,)~,p)T(x))
j!
0
=
(1.45)
T
where X ( x , )~, #)T(X) = ezp[~ Zn(.n--)~)]exp[E()~-~--.-n)O~/n]r(x) and (18E) W~ (T) = ~ - c ~ "~-n-JwJn (T). The Wn3 are the generators of the W I + ~ algebra (see below) and from (1.45) results also V~2*(X' A ) 0 ; - I ~ ( x ' ~ ) : 0 ( ~ - ~ - ~ ) ~ - n - u W) ~ ( TT)
(1.46)
Here one can write (n > 0) n
c~
0
1
W 1 = On = j1; W2n = j2n _ (n + 1)J 1 = y ~ C~jGgn_j -}- 2 E J X j t ~ n + j - - ( n + 1)On = 2nn - ( n + 1)On
(1.47)
where Ln are the Virasoro generators with [Ln, Lm] = ( n - m)Ln+m for n, m > 0 (we will disregard On and tn for n < 0 most of the time). In particular this leads to (cf. [163]) r162 * = ~--~P~+l,1)~-n-2; Rn+l,1 = ~ 0
T
;Kn+l =
T
(1.48)
Further one notes that ( 1 8 F ) S n + l = O(WI(T)/T) (n >_ --1). Additional symmetries (in particular conformal symmetries) related to this framework are developed in [160, 163] for example (see also the references there). Now with a view toward dKP one can think of fast and slow variables with ,ex = X and
etn = Tn so that On --* eO/OTn and u(x, tn) --+ (t(X, Tn) to obtain from the KP equation (1/4)uxxx + 3uux + (3/4)0 -102u = 0 the equation OT5 = 3(tOX5 + (3/4)0 -1 (02(t/OT~) when e ~ 0 (0 -1 --~ (1/e)0-1). In terms of hierarchies the theory can be built around the pair (L, M) in the spirit of [160, 163, 902]. Thus writing (tn) for (x, tn) (i.e. x ~ tl here) consider (x)
L~=eO+EUn+l(e,T)(eo)-n; 1
oo
oo
M~=EnTnLn-i+Evn+l(e,T)L[n-i 1
1
(1.49)
8
CHAPTER 1. Q UANTIZATION AND I N T E G R A B I L I T Y
Here L is the Lax operator L = 0 + ~ Un+lO - n and M is the Orlov-Schulman operator defined via r = M e . Now one assumes Un(e,T) = Un(T)+ O(e), etc. and sets (recall L r = Ar
r
[l+O(~)]exp(~TnAn)
= e x p ( 1 Se ( T ' A ) + O ( I ) )
T=exp(~F(T)+O(1))
(1.50)
We recall that OnL = [Bn, L], Bn = L~_, O n M = [Bn, M], [L,M] = 1, L r = Ar Ox~ Me, and r = ~-(T- (1/n;~n))exp[E7 Tn:~n]/~-(r). Putting in the e and using On for O/OT,~ now, with P = Sx, one obtains (X) )~ -" P + E
(X) Un+lp-n;
1
CXD
(1.51)
P = )~ - E Wi)~-l; 1
(X)
.h,4 = E nTnAn-1 + E Vn+lA-n-1; OnS = Bn(P) =:~OnP = 9Bn(P) 1
1
where 0 ~ Ox + (OP/OX)Op and M ~ + I ( T ) + O(e)). Further for Bn = ~ Bn = L n + ~ a2L-J). We list a few [160]); thus, writing {A, B} = O p A O A -
+ Ad (note that one assumes also Vi+l(e,T) bnmOTM one has ~n -- ~-~ bnmP m ~ A~_ (note also additional formulas which are easily obtained (cf. OAOpB one has
OnA = {Bn, A}; OnAd = {Bn,A/I}; {A, A J } - 1
(1.52)
Now we can write S = ~ T n A n + ~ Sj+IA-J with Sn+l = -(OnF/n), OmSn+l = (Fmn/n), Vn+l - -nSn+l, and OAS = Ad. Further
Sn
(9O = /~n -4- E OnSJ +l/~-j; OSn+l ~'~ 1
OVn+l ~
OOnF n
n
(1.53)
We sketch next a few formulas from [555] (cf. also [160] and Section 5.1.4). First it will be i m p o r t a n t to rescale the Tn variables and write t ' = ntn, T~n = nTn, On = nO~n - n(O/OT~n). Then <S:
)~A; 0In ~ -- {~n,)~ } ( ~ n - n
~n);
(1.54)
n
O~np = OQn = OqQn -I- O p ~ n O P ; Oq~nQm- O ~ Q n - {Qn, Q m } Now think of (P, X, Tn~), n _> 2, as basic Hamiltonian variables with P = P(X, T~). Then - Q n (P, X, Tn~) will serve as a Hamiltonian via
ptn = dP I dX dT~n = OQn; fCn = dT~n = - 0 p Qn
(1.55)
(recall the classical theory for variables (q, p) involves 0 = OH/Op and/5 = -OH/Oq). The function S(A, X, Tn) plays the role of part of a generating function S for the Hamilton-Jacobi theory with action angle variables (A,-~) where
P d X + QndT~n = -~dA - KndT~n + dS; Kn = - R n = - - - ; d~ d%
= 2n =
= 0;
d~
= ?n =
=
-
~n-1
(1.56)
1.1.
ALGEBRAIC
AND GEOMETRIC
METHODS
9
(note t h a t J~" = 0 ~ 0"A = { Qn, A}). To see how all this fits t o g e t h e r we write OP d X d P = O ' P -~. . . . dT" O X dT"
OQn +
OP _. , OQn + OPOpQn + O F f { , -'~Xn = n
(i.57)
This is c o m p a t i b l e with (1.55) and H a m i l t o n i a n s - Q n . F u r t h e r m o r e one has SA = ~; S x = P; 0 " S
:
~n
-
(1.58)
Rn
a n d from (1.56) one has I
P d X + Q n d T n = -~d)~ + R n d T " + S x d X
/
"
!
+ S;~d)~ + O n S d T n
(1.59)
which checks. We note t h a t O ' S = ~ n : ~ n / n and S x = P by c o n s t r u c t i o n s and definitions. Consider S = S - ~ AnT'n/n. T h e n S x = S x = P and S'n = S ' n - Rn = Q n - Rn as desired with ~ - S), = S ) , - ~ T" )~n - i . It follows t h a t ~ ~ A / / - ~ T'n)~n - i = X + ~ Vi+IA - i - 1 If W is t h e gauge o p e r a t o r such t h a t L = W O W -1 one sees easily t h a t M~2=W
kxkO k-1
W-Ir
"-
(
G+EkXk/~ 2
k-1
)
~
(1.60)
from which follows t h a t G - W x W - i ---+~. This shows t h a t G is a very f u n d a m e n t a l object a n d this is e n c o u n t e r e d in various places in t h e general t h e o r y (cf. [160, 163]). 1.1.2
Dispersonless
theory
for KdV
Following [147, 160, 165] we write L 2 = L~_ = 02 + q = 02 - u (q = - u = 2u2); qt - 6qqx - qxxx = 0; B=403+6q0+3qx;
L2
[B, L2]; q = - v
(i.6i)
2-vz~vt+6v2vx+vxxx=O
(v satisfies t h e m K d V equation). Canonical formulas would involve B ~ B3 - L 3 as i n d i c a t e d below b u t we retain t h e B m o m e n t a r i l y for c o m p a r i s o n to o t h e r sources. K d V is Galilean invariant (x' - x - 6At, t' - t, u' - u + A) and c o n s e q u e n t l y one can consider L + O2 + q - ~ = (O + v ) ( O - v, q - )~ = - v x - v 2, v = ~bz/~, and - ~ x x / ~ 2 = q - )~ or r162 = /kr (with u' = u + / ~ ~ q' = q - A ) . T h e v e q u a t i o n in (1.61) becomes t h e n vt = 0 ( - 6 ) w + 2v 3 - vxx) a n d for A = - k 2 one e x p a n d s for .~k > 0, Ikl - , oc to get ( . ) v ~ ik + Y : . ~ ( V n / ( i k ) n ) . T h e Vn are conserved densities and with 2 - A = - v ~ - v 2 one obtains n-1
p
=
-2Vl;
2Vn+l
=
-
Vn-mVm-
E
!
!
Vn; 2 v 2 - - - v 1
(1.62)
1
N e x t for r 1 6 2 = - k 2 r write r ~ e x p ( i i k x ) as x ---+ • Recall also t h e t r a n s m i s s i o n a n d reflection coefficient formulas (c.f. [147]) T ( k ) r - R(k)r + r and T~+ RLr + r W r i t i n g e.g. r = exp(ikx + r with r oc) - 0 one has r + 2 i k r + (r = u. T h e n r 1 6 2 = ik + r = v with q - A - - v x - v 2. Take t h e n
r
r 1
(2ik) ;
r ~ ik +
v. =
+ Z
(ik)
=> Cn = 2nvn
(1.63)
F u r t h e r m o r e one knows oo
l o g T = - y~. C2n+i 1 / _ ~ k2nlog(1 _ iRi2)d k 0 k2n+l; C2n+1 = ~ e~
(1.64)
C H A P T E R 1. Q U A N T I Z A T I O N A N D I N T E G R A B I L I T Y
10
(assuming for convenience t h a t there are no b o u n d states). Now for C 2 2 - - R L / T and c21 = 1 / T one has as k ~ - o o (.~k > 0) the behavior r ---, c22exp(-2ikx)+c21 ---+c21. Hence exp(r ---, c21 as x ---, - o o or r 1 6 2 = - l o g T which implies
?
r
= logT =
517
(2ik)n
(1.65)
Hence f r = 0 and C2m+1 = - f r 2m+1. The C2n+l are related to Hamiltonians H2n+l = anC2n+l as in [155, 145] and thus the conserved densities Vn ~ Cn give rise to Hamiltonians Hn (n odd). There are action angle variables P = klog[T[ and Q = 7 a r g ( R L / T ) with Poisson structure { F , G } ~ f(hF/hu)O(hG/hu)dx (we omit the second Poisson structure here). Now look at the dispersionless theory based on k where A2 ~ (+ik) 2 = - k 2. One obtains for P = S x , p2 + q = _ k 2, and we write P = ( 1 / 2 ) P 2 + p = (1/2)(ik) 2 with q ~ 2p ~ 2u2. One has Ok/OT2n = {(ik)2n, k} = 0 and from ik = P(1 + qp-2)1/2 we obtain
ik = P
1+
qmp-2m 1
)
(1.66)
(cf. (1.51) with u2 = q/2 - we use +ik for convenience here but - i k will be needed later). T h e flow equations become then
02n+lP = 0~2n+1; 02n+l(ik) = {~2n+1, ik}
(1.67)
_,3/2 = 03 + (3/2)q0 + (3/4)qx = B3 Note here some rescaling is needed since we want (02 + q)+ instead of our previous B3 .-~ 403 + 6qO + 3qz. Thus we want Q3 = ( 1 / 3 ) P 3 + (1/2)qP to fit the n o t a t i o n above. The Gelfand-Dickey resolvant coefficients are defined via Rs(u) = (1/2)Res(O 2 - u) s-(1/2) and in the dispersionless picture Rs(u) ~ ( 1 / 2 ) r s _ l ( - u / 2 ) (cf. [160]) where rn
__ R e 8 ( _ k 2 ) n + ( 1 / 2
) _
--
( n + (1/2)) n + 1
qn+l _ (n + 1 / 2 ) ' ' ' (1/2 n+l -(n + iii q ;
20qrn = (2n + 1)rn-1
(1.68)
T h e inversion formula corresponding to (1.51) is P = ik - E ~ Pj(ik)-J (again ik ---, - i k arises later) and one can write 0'
2n+l (p2
+ q) = 0'
2n+l(--k2);
0' 2r n 2n+lq = ~2nO + 1
2
= 2n + 10qrnqx - q x r n - 1
(1.69)
Note for example r0 = q/2, rl = 3q2/8, r2 = 5q3/16, ... and O~q = qxro = (1/2)qqx (scaling is needed here for comparison). Some further calculation gives for P = i k - ~ Pn(ik) -~
Pn "0 -Vn ~ - ~Cn -;
C2n+l = ('-- 1)n+l . I ?
P2n+l(Z)dX
(1.7o)
oo
T h e development above actually gives a connection between inverse scattering and the d K d V t h e o r y (cf. [153, 158, 160, 168] for more on this).
1.1. A L G E B R A I C
1.1.3
Toda
AND GEOMETRIC METHODS
11
and dToda
We will follow the formulation of [902, 916] (dToda can also be subsumed in a general formulation of d K P as in [32]); thus one exploits the Orlov-Schulman operator M --. A4 directly in creating the embellished d K P hierarchy which will have an extension to dToda (indicated below); this provides a richer structure from the beginning. Let us begin with [916] and write the ordinary Toda hierarchy in the language of difference operators in a continuous variable s with spacing unit e. Thus OL
OL = [Bn L] e
a t = [B., t]
e-~Zn
where n -
;
~
= [/~n L]
(1.71)
at = [B t]
O~n
'
1, 2 , . - . and L=e
(X)
(2
o
1
e__0 o, _~_EUn(e,,z,~;,8)e-ne~ t - - E ~ t n ( e , , Z , ~ , 8 ) e n e ~
(1.72)
Here Bn = (Ln)>o and /~n = ( L - n ) < - 1 where one is thinking of projections onto linear combinations of exp[ne,(O/Os)] with n >_ 0 or n < - 1 . We note that exp[ne,(O/Os)]f(s) ^ f ( s + ne,). One will also have Orlov-Schulman type operators M and M satisfying 0M 0M e,-~ZnZ n = [Sn, M]; e,~n-n = [J~n, M]; i) M = [B n 1~/I] e,
~~z~
'
0M
'
=
(1.73)
[/3n, A~]
for n = 1, 2 , . . - w h e r e [n, M] = E,L, [L, A?/] = E,L and (X)
M-
(X)
EnznLn
+ s
1
+ EVn(e,,z,z, 8)n-n; 1
O0
(X)
= --EnZn 1
(1.74)
~ - n at- 8 -~- E ~ ) n ( e , , z , fi.,s)Ln 1
The dispersionless hierarchy arises as e, --* 0 upon positing Un(e,, z, ~, s) = u no (z , e, ~) + o(~) along with similar expressions for Vn, ~tn, and ~)n. To arrive at the limiting forms it is perhaps clearest to introduce "gauge" operators O0
W=
(X)
1 + EWn(e.,z,~,s)e-he~
~r=
1
(1.75)
E(Vn(e.,z,~,s)enE~ o
where the Wn and Wn are singular as e, ~ 0 (unlike the Un, Vn, etc.). Then one has L = W.e
M = W
nz~o
~o
e__0
0~ 9W - l ; L = 12d.e 0~ . l ~ -1",
+ ~ w_l; M = W
_ ~n~n
_~
1
)
(1.76)
+ ~ W-1
The Lax equations can be converted into Sato type equations OW We~ ~ OW e,-~ZnZn -- Bn W ; e,-~ZnZn : JBn W ;
(1.77)
CHAPTER
12
0w
1. Q U A N T I Z A T I O N
0w
AND INTEGRABILITY
w~-n~
One then introduces WKB type wave functions via
r =
1 + ~-~wn(e,z,2, s)A -n 1
ZnA n and 2(A -1) = ~
where z(A) = ~
(1.78)
e x p [ e - l ( z ( A ) + slogA)];
~,nA-n. The equations (1.76) then imply that {1.79/
Further
O~b
a~b = Bn%b; e a~b = Bn~b; E--=O~
/)n~
(1.80)
and we note that the coefficients Vn and Vn of M and M can be read off via
=Enz"An + S + E V~n ) _ n ;
e A 0 l o g_______~ ~
1
1
l og ~
o0
cc
C~A
1
1
(1.81)
Now one thinks of asymptotic forms as e --* 0
~) -- eXF[E-1S(z,z.,,8,)k) + O(1)]; ~ - eXp[E-1S(z, 2,8,)k) + O(1)]
(1.82)
where S = z(A) + slogA + Ec~ Sn(z, 2, s)A -n and S = 2(A -1) + slogA + Ecff Sn(z, 2, s)A n. Similarly tau functions are defined via
T(E, Z -- E[,~--I], 2, 8) T-~,7~;8 i eXp[E--I(z()k) + slogA)] = r
z, 2, A);
(1.83)
T(E,Z,Z.- E[)k], 8 -~-E)eXp[E_l(~()k_l)-~- 810g~)] = ~(E, Z, 2, )k)
7-(E,z,~,s)
where [A] = (A, A2/2, .-. , An~n, .. .). One sees easily that log'r(e, z, 2, s) = e - 2 F ( z , ~,, s) + O(e -1) as e ~ 0 for some function F and it is immediate that
Sn =
1 OF 1 OF OF --nOz----n ; ~n . . . n. 02n ; S 0 = as
(1.84)
Hence in analogy with the KP situation one refers to F as a "free energy" and writes F = logTdToda. It will be convenient to distinguish between the A's in S and S now and write
.OS(z,
exp[
-O-s
] = p = exp[
bs
]
(1.85)
(a corresponding replacement is also appropriate in (1.82) and (1.83)). Then some calculation as in the dKP theory yields for (3O OO )k"'P-Jr - E?.tn(Z, 2)p--n; ~"" E U n ( Z , Z . , 8 ) p n ; ~n ''~ ()in)>0; ~n = (~--n)<--I (1.86) 0 1
1.1.
ALGEBRAIC
with
the equations
AND GEOMETRIC
METHODS
OA OB {A(p, s), B(p, s)} = p Op Os
oA OZn
= {Bn A}; '
OA
13
OA OB Os p Op
= {/~n,A};
(~2n
0 i = {B~,i};
0i = {& i}
OZn
~
(1.87)
(1.88)
'
and for j~A = ~ - ~ nznA n + s + ~ , ~ Vn(Z,Z, 8)~ - n one has 0.A~ __~ {~n ~/~}, 0./~ __ {~n .A~}; OZn ' OZn '
0 M = {B~ ~ } ;
0M = {&,~}
C~Zn
02;n
'
(1.89)
where A~ = - ~ n2nA - n + s + ~ ~n(Z, 2, s)A n. Further one has from [n, M] = eL and [L, ~:/] = eL the equations {A, 3d} = A; {A, A74} = A (1.90) and
eo eo [e b-;, s] = ee o, ~ {p, s} = s
(1.91)
Some of these equations can be combined to give a simple expression dA A dad d i A dA~ A -- w = i ;
(1.92)
w -- - - A d8 -'i- E dl~n A dzn "1" E dBn A d2;n P 1 1
W-infinity symmetries for Toda and the corresponding W-infinity symmetries for dToda are formulated in [916] in bosonic and fermionic language and we will refer to this later as needed. The theory is also developed in [902] for example in a slightly different manner and with somewhat different notation and it will be expedient to indicate this. Thus one starts with (1.71) - (1.72) where Zn ~ tn, ~n ~ tn and OO
e__ o L = e os + E u n + l ( C , t , t , s ) e - n e ~ 0
(1.93)
O0
~ - 1 __ U0(E, t, t~ 8)e -cO + E Un+l(s t, t-)eneO 0
Then upon defining Bn = (Ln)_>0 a n d / ? n = ( L - n ) < - i one has
~
OL OL = [B~, L]; ~ - : - = [&, L]; Ot~ Ot~ OL = [B., t]
~ 0 ? . = [B., t]
(1.94)
C H A P T E R 1. Q U A N T I Z A T I O N A N D I N T E G R A B I L I T Y
14
This notation yields certain advantages in subsequent formulas and is accompanied by M exactly as in (1.74) with M = - E ~ ~ n t n L n + s + E ~ ~n L-n" Thus i, ~ ~ - 1 in these formulas. Note also that OnL = [Bn, L] ~ On(L -1) = [Bn, L-1]. In this context the wave functions have the form (1.82) with r --~ ~, S --. S, and A ~ ~ -~ ~. The rest is straightforward and one notes t h a t the Sato type equations (1.77) take a somewhat simpler form which will be exhibited if needed. The embellished dToda hierarchy then has the form 0,~ = {Bn,,k}; aM
= (B~,M};
Ot~
oM
0~
= {/3n,)~} -
= (S~,M}
~
0~ oX4
-
= (B~ M};
; Ot~ dA AdA/t
= w=
'
O0
= {/3n X}
oX4
(1.95) -
= {~n,M}
O~n
d~ A dX4 -
OO
d~ = Md log~ + ~o~~
0~
= {Bn X}
(1.96) OO
1
1
O0
O0
+ ~ e ~ t ~ + ~ Bn~n 1
~
1
O0
O0
S = E t n ) ~ n n u slog)~- E v---~n)~-n; S = E t n X - n 1 1 n 1
+ s l o g i nt-r
E Vnx n 1 n
where r is a "potential" such t h a t ,k = ezp[ad(r r = ~ o Cn(t,{,s)p-n which we will not discuss unless it is needed later. Note t h a t for logT(e, t, t, s) -- e-2F(t, t, s) + O(e -1) one gets o _ OF - o = OF OF Vn - a t ~ ; v~ a ~ ; s0 = r = ~ s (1.98) Other formulas will also be displayed as needed. For completeness we add some remarks on Toda and W h i t h a m following [87] (cf. also Sections 5.3.1 and 5.3.2 for references and discussion). Thus consider Toda equations in the form ak = bk+l -- bk; bk = bk(ak -- ak-1) (k E Z) (1.99) These can be written as compatibility equations for the equations L r = Ar
Ct = B e
(1.100)
where L = A + u0 + Ul A-1 and B = A + u0. Here A is the shift operator A = exp(O/Ok) so t h a t A f ( k , t ) = f ( k + 1, t) and we take u o ( k , t ) = ak(t) with u l ( k , t ) = bk(t). T h e n (1.99) becomes Ouo = ( A - 1)ul; 0Ul = Ul(1 /k-1)U0 (1.101) ot -~ Note t h a t B - (L)+ = L - (L)_ where ( )+ denotes the polynomial part of L in A. Set then -
OO
OO
A = L - ~ GiL-i; (L)_ = ~ FiL -i 0 1
(1.102)
where Fi - Fi(uo, ul) and Gi = Gi(uo, Ul) and some elementary calculation gives
~log
=
~log A- ~
0
c ~ A -~
=
(1.1o3)
1.1. ALGEBRAIC AND GEOMETRIC METHODS Ct -~
=(A-l)
15
=(A-1)(A-~Fi
A-i)
1
One can then calculate the coefficients Fi and Gi in terms of u0 and Ul as in [54] to obtain
(u(-k) = A-ku)
60 "- UO; 61 "-- Ul; G2 = UlU(o-1) ; G3 = Ul [u(-1)] 2 nu Ul U~-1) ; ' " ; F1
=
=
=
(1.104)
+
Now in (1.103) ( A - 1)f(k,t) = [f(k + 1, t ) - f(k,t)] corresponds to a derivative in the k variable (withfk ~ ~ k ) so that (1.103) represents a system of conservation laws with conserved densities given via log
= log(h - Z
1 ( G1 2A 2 G 0 + ~ + ' "
1(
a~ h-~) = logh - X
)2
+ ....
logA
)
ao + -;
Go
A
(1.105)
+ ....
1 ( _ ~ )
A2 GI +
+'"
Then setting log(Ar162 = l o g A - E ~ ( P i / A i) one has conserved densities of the form P1 = u0; P2 = Ul + lu2; P3 = U l u ~ - l ) + U0Ul nt- 3u3; ...
(1.106)
and the conservation laws are given by (OPi/Ot) = ( A - 1)Fi. Now it is known that all periodic solutions of the Lax equations for Toda systems of the form (1.99) will be of finite zone type, associated to a hyperelliptic Riemann surface Eg of genus g of the form 2g+2 y2 = 1-[ ( A - Ai) = Rg(A) (1.107)
1
(cf. here [268, 608, 609]); we will take the Ai to be real with A1 < A2 < --- < A2g+2 for convenience. Hence we will assume the solutions of the Toda equations to be slowly modulating g-phase wavetrains. The oscillations of the wavetrain are fast compared to the modulations and one averages out the fast oscillationsl This is discussed in [499] in connection with quantum mechanics and it is indicated how the procedure of averaging over fast fluctuations of angle variables (giving some effective slow dynamics on the space of integrals of motion) corresponds to quantization or quasiquantization (in the first W K B approximation). Indeed quantum wave functions appear from averaging along the classical trajectories, very much in the spirit of ergodicity theorems, and that corresponds to the averaging involved here (more on this later). Thus suppose F is a flux or density and write < F > for the spatial average over the fast variables of F (keeping the slow variables fixed). Then < F > ~ limL__,oo(1/2L) f~_LF(x, t)dx corresponds here for a discrete system to < F > = limN__,oo(1/2N) ~ N N F(n, t). Since the systems are integrable there exist infinitely many commuting flows which can be viewed as translations on the g-torus; generically each flow will cover the torus ergodically (assuming incommensurate characteristic frequencies) so one can replace e.g. spatial averages by
1 j02~"'" f02~F ( 0 1 , . . . ,
< F > = (27r)g
Og)H g dOi 1
(1.108)
16
CHAPTER 1. QUANTIZATION AND I N T E G R A B I L I T Y
An interesting observation in [87] indicates that such an integral in 0i variables corresponds to an integral over the cycles on the associated Riemann surface and thus one expects the averaged quantities to correspond to differentials on Eg. One now breaks up the dynamics into fast and slow scales. Let to and no denote the fast time and spatial variables and let T = et and X = en be slow variables. Then one can write
Similarly O/Ot = (O/Oto) + e(O/OT). One uses this in the equation above for (OP~/Ot) to get the Whitham equations 0 0 OT < Pi > = ~ < Fi > (1.110) and we now want to express these equations in terms of differentials on Eg as in Sections 5.1.2, 5.3.1, and 5.3.2 (cf. [148, 338, 722]) Now in [87] one takes a Baker-Akhiezer (BA) function (or wave function) ~b(n, t, A)=roeitfAAod~l+infAtd~~
(1.111)
where d&0 and d&l are normalized Abelian differentials of third and second kind respectively such that, for A ~ ec,
d&o(A) = [I~ (A - ai) dA v/Rg(A )
dA
--~ + O
(dA) ~
(1.112) , .
d&l(A) =
1
dA +
(hg+X-lolAg--[-"),lhg-l-[-...--[-'~g)dA 2r
~ dA + O
(dA) ~
(1.113)
and the normalization is ($) ~a~ &0 = ~a~ ~bl = 0 (j = 1 , . ' . , g). The aj cycles correspond here to cycles over (A2j, A2j+I) and the coefficients ai and 7i in (1.112) - (1.113) are uniquely determined by (:[:). Further one defines here 2g+2
crl- Z
1
Ai; a 2 - ~-~AiAj; a 3 -
i<j
Z
i<j
AiAjAk
(1.114)
(a2 and a3 will come up later). Here F is periodic (~ Riemann theta function) and (1.111) actually corresponds to a symmetric form of (1.100) but this will make no difference in the calculations of interest. (1.111) is a special version of the ~b in [760] for example which will be developed later. Note that r0 can be given an explicit (complicated) form which is not needed here. Now from (1.111) we obtain
1 OF d logr = itddvl + inddvo + --fi--~dA
(1.115)
which implies
d&o = -i-~nO (dlogr + -~n
; ddal = -~-~ (dlogr + it-~
~
(1.116)
Upon averaging one gets then 0
0
d&o = -i-ff-~ < dlogr >; &bl = - i ~ - ~ < dlogr >
(1.117)
1.1. ALGEBRAIC AND GEOMETRIC METHODS
17
It is interesting to compare this with formulas for hyperelliptic situations with KdV and one sees that the quasi-momentum term dp corresponds to &0. One notes that averaging (dF/F)(t) in t for example (via (1/2T) fT_r gives generically a bounded function log F divided by factors of T ~ oc and hence there is no contribution. On the other hand terms involving logr are not necessarily bounded so they contribute (cf. Section 5.1.2). In the notation of Section 5.1.2 one has p(A) = -i(logr for example so heuristically dp ,,~ -i(dlog~2)x ,,~ d&o. Now the integrability condition for (1.117) gives the Whitham equations
Od&o Odwl = OT OX
(1.118)
Further one can regard (1.118) as an averaged form of the conservation laws (1.103). To see this let us write from above (cf. (1.103)- (1.105)), A - L - E ~ Gi L-i, and log ( ~ 1 -
logA- ~
PiA -i which gives dlog
= --~ + ~ iPiA-i-ldA
(1.119)
1
But from (1.111) one has
log
=i
d&o :=~d log
= id&o
(1.120)
o
Now write (1.103) in the form OtA = ( A - 1)B ~ cgxB where A = l o g A - E F Pi A-i and B = A - ~ FiA -i. We can average directly now to get
OT < (dlog ( ~ )
>= iOTd~o = Ox < dB >
This implies from (1.118) that in fact < dB > = idr dA(
d&0(A)=~ 1+ A
+
(1.121)
and 2 A2
+...
}
;
(1.122)
d & l ( A ) = d A ( I + < F I > A + 2 < F 2 >2 A + . . . ) Explicit values can also be determined via < P1 > - <
3
2
1
1
1 uo > - - ~G1 - #1; < P2 > = < z
1
-- i-~Crl -- ~ff2 -- ~71]-tl -}- ~ # 2 ; ' ' "
; < F1 > - - <
la3_l
1
1 Ul-~- ~z u v2 > =
1 2
(1.123) 1
ltl > - - -i--~l - ~r
1
1
-}- ~ 1 ;
1
where the ai have been defined above and #1 = ~g ak with #2 = ~k
Od&o __ _ ~-~g dr Oai ~-2g+2x.., d&o OAk OT 1 A - a~ OT 2 ( A - Ak) OT
(1.124)
18
CHAPTER 1. QUANTIZATION AND INTEGRABILITY 2g+2 &idA OxAk + OXd~l -- E 2 ( t - hk) 2V~g 1
-
)
AgOxal + Ag-lcqx~l -+-"--+- OXq/g
and we denote by Ak the 2g + 2 branch points of the Riemann surface Eg ~ y2 = Rg of (1.107). Evaluating the residues in (1.124) at A = Ak one obtains then from (1.118)
OTAk = s(Ak)OxAk;
(1.125)
Ag+l - lalAg + 71Ag-1 + "'" + 7g ~(A) = 21_igl(A_ ~ ) These are the celebrated equations (from KdV) for the branch points Ak as Riemann invariants and the s(Ak) play the role of characteristic velocities. In fact from [87] one has s(Aj) > s(Ak) for Aj > Ak and Os(Ak)/OAk > O. 1.1.4
Remarks
on integrability
At this juncture it is appropriate to make a few more comments about integrability (cf. in particular [456, 615, 966]). There is some possible confusion in dealing with classically integrable systems of infinite dimension such as KdV which are phrased in terms of partial differential equations (PDE). How is this related (if at all) to quantization schemes (.) { , } ~ (i/h)[, ]? Such questions were treated in [374, 423, 536, 537, 891, 910, 913] for example and this will be examined at various places (cf. also Section 3.1.1). We recall several background facts. First from the Groenwald-vanHove theorem (cf. [426, 630]) there is no reasonable map from C~162 to a Lie algebra of unbounded operators on a Hilbert space which realizes (.) in a general sense (beyond quadratic polynomials- cf. also [122]). Now what about quantized integrability? Classical integrability for a Hamiltonian system involves e.g. N commuting constants of motion In such that {H, I~} = 0. For infinite dimensional systems such as KdV one expects N ---. ec (cf. (1.4) for { , }1,2). Then one could anticipate that quantum integrability should involve N commuting quantum constants of motion such that [/~,/~n] = 0 and [/~n,Im] = 0. However one knows by a theorem of vonNeumann that for any number of commuting selfadjoint operators/~n there exists a selfadjoint operator 2 such that /~n = fn(Z) (recall observables correspond to selfadjoint operators). This shows in particular that it is impossible to fix unambiguously the number of elements comprising a commuting set of selfadjoint operators. Thus one cannot use this definition of quantum integrability to separate quantum systems into two classes of integrable and nonintegrable systems. Some attempts to deal with such problems occur in [127, 671, 507, 509, 82, 481, 455, 456, 615] for example where a main feature of standard versions of quantum integrability is to ignore the Hilbert space context (and the reliance on selfadjoint operators) and simply to map the P algebra (9(T'M) to Z)(M) = algebra of differential operators on M (cf. [127]). For example if a classically integrable system is specified by N algebraically independent P commuting functions In then one looks for N algebraically independent commuting differential operators On arising from the In. This can then be generalized in the context of algebraic geometry, differential Galois theory, 73 modules, etc. as in [127] (cf. also [82, 481]). In [615] (cf. also [507, 509, 671]) one looks at the preservation of the Lax equation framework under a 9 product (1hA) f 9g = fg + E ~ ~rQr(f, g) for suitable deformations (15B) f , g = ~(eS/s!)(Osf/OpS)(OSg/Oq s) and ( 1 5 C ) = (8.6)of Section 3.8 for e -- ih/2 (el. ( 1 5 D D ) below and also [336]). The deformation (15B) is a standard symbol multiplication as with pseudodifferential operators (PSDO - cf. Section 1.1.1). One
1.1.
ALGEBRAIC
AND GEOMETRIC
METHODS
19
writes Lax equations now in the form, e.g. OmL = [L~', L] = [L, L TM] where L = En__~ uip ~ and L m ,,~ L *m with Un - 1 and Un-1 = 0 (while Om "~ O/Otto); the ui are functions of q (s+l) and 9 is extended to such objects L with 0 ~ Oq (one writes i)(ul s)) = u i ). This gives rise to e deformed integrable soliton equations in standard situations, e.g. with the Moyal product (8.6) (modulo a minus sign) L = p2 + u ~ ut = [L+3/2, L] = 3cuu' + 2c3u'"; L = p + y ~ A i p - i - 1 ~ Lt -0
I
(
~uyy = ut
~L+, L
a n d Ly
2E3 ,,,
~u
=[1. 1 ,L
)
(1.126) --~
+ 2~uu'
X
for u ~ A0. Note here that standard KP with multiplication (15B) involves (&) (1/e)Uyy (ut 4- (e3/12)u ' ' - euu')x while KP itself is Uyy = (ut + (1/12)u'" - uu')x. Using e t T, ex = X , and ey = Y with 0t = e0T etc. one obtains then dKP forms e2uyy ~.(6.UT 4- ( 1 / 1 2 ) e 3 u x x x - e u u x ) x or u y y = (u T 4- ( 1 / 1 2 ) e 2 u x x x - u u x ) x ---+ u y y = ( U T - UUX)X. Note in [615] one is only scaling the x variable which corresponds to u y y = e(UT + ( 1 / 1 2 ) e 3 u x x x -- e u u x ) x . We see that (&) and (1.126) appear to be different (in fact scaling of y and t would also be needed as e ~ 0) but regarded as a Benney system (15D) Ai,t = Ai+l,x + i A i - l A i , x they are equivalent. One shows that all the systems of Lax equations OmL = [L*+m, L] are Hamiltonian. Thus in some sense quantizations such as (15D) satisfying (15DD) a . b = ab for a,b E ,4, with p~ 9 pC' _ pe+e' and ap t * bp ~' = abp ~+~' 4- O(< t~4- t~~) parametrize realizations of integrable Hamiltonian systems (A is a differential Q algebra with derivation 0 ~ p).
In [456] one again avoids the problem of selfadjoint operators and the vonNeumann theorem by using differential operators (no boundary conditions, no Hilbert space, no domain restrictions) and suggests that the transition from classical to quantum integrability seems to be always possible if one adds correction terms to the Hamiltonian and the In proportional to h 2. Situations of an excess of commuting (algebraically related) differential operators are discussed and many questions appear unanswered or unclear. Relaxing "regularity" of the functions In apparently requires relaxing the idea of functional independence (polynomial relations should be allowed) 1.1.5
Hirota
bilinear difference equation
We give some discussion here of the Hirota bilinear difference equation (HBDE) based on [7, 102, 154, 561, 740, 757, 830] (cf. also Sections 3.6, 4.1.2, 4.1.3, and 4.1.4). The original form of Hirota was (cf. [830]) af(A 4- I, #, ~,)f(A - 1, #, v) 4-/3f(A, # + 1, ~,)f(A, p - 1, ~,)+
+~/f(A, #, v - 1)f(A, # , , + 1) -- 0
(1.127)
where a + 3 + ~/= 1. It was shown by Miwa (cf. [740]) that every KP tau function satisfies (1.127) using variables determined by ( 2 1 A ) j x j = E~pka~ where e.g. the VOE (1.36) relates T and r with variables x and (21B) : r A)r A)dA = 0 (Hirota bilinear ident8 tity). Here one picks triples (p, q, r) from among the Pk and (a, b, c) from the corresponding t8 a k to get a(b - C)T(; + 1, q, r)T(p, q + 1, r + 1) + b(c - a)T(p, q + 1, r ) T ( p + 1, q, r + 1)+
C H A P T E R 1. Q U A N T I Z A T I O N A N D I N T E G R A B I L I T Y
20
+ c ( a - b)~-(p,q,r + 1)T(p + 1,q + 1, r) = 0
(1.128)
To see the equivalence to (1.127) one writes (21C) f ( q + r + 1, p + r + 1, p + q + 1) = ~-(p, q, r) with a = a ( b - c ) , 13 = b ( c - a ) and -~ = c ( a - b ) . Thus solving to the HBDE (1.127) enables one to study various 3-D projections of the infinite dimensional space spanned by the xi on which the KP hierarchy is defined. Note here that (6.1) in Section 3.6 and (1.49) in Section 4.3.2 have the form (a +/3 + 7 = 0)
ag(n,g + 1, m)g(n,e, m + 1) + ~g(n, e, m ) g ( n , e + 1, m + 1)+ +Tg(n + 1,i + 1, m ) g ( n -
1,e,m + 1) - 0
(1.129)
which is another variant on (1.127) (cf. also [757] and we refer to [993] for a survey of f o r m s - cf. also remarks after (2.37) to connect (1.127) and (1.129)). Somewhat more general formulations can be developed via [102] and (1.120) in Section 4.1.2. Note with the normalization there X ~ 1 / ( A - #) one is working with Cauchy kernel type BA functions r (CBA). Thus in Section 4.3.4 take G to be a unit disc with g(x, A) = exp(~C~ x~A-i) and glg21 ~ g(x,v)g-l(x',v)
= exp
(Xi
-- X~)V -~
= V;
~
-- a
(1.130)
where a, b are arbitrary complex parameters in G (cf. Section 4.3.4 for notation and background). Since log(1 - ~ ) = ~ ( e i / i ) one obtains (21D) x~ - xi = (hi~i) - ( b i / i ) and putting (1.130) in (1.120)-(1.121)yields #
b X(A,#,x+[a])-
A-_b
X(A,#,x+[b])+ (1.131)
+(b-~)x(~,b,z+[~])x(b,,,x+[b])=o(~#,) (here x + [hi = (xi + [a]i with [a]i = ai/i. This leads to
(~- ,)x(~,,,z) = (~-~)x(~,a,x(#
-- a)x(#,
a, x
-
[a] + [,]) = [a] + [#])
( a - , ) x ( z , , , x + [ ~ ] - [~]) = ( a - A)x(a , A, x + [ a ] - [A])
(1.132)
Note also that for b = 0 (1.131) takes the form r where x i - xi
#, x + [hi) - r
#, x) = he(A, 0, x + [a])r
#, x)
(1.133)
(1/i)a i. Evaluating now (1.132) at # = 0 for some a = a0 gives )~X()~, O, x) = (A - a)X(A, co, x - [a0]) Z(,X,x) (-~X-~,a,x-[a0]) = Z(0, x)
(1.134)
where Z ( , X , x ) = ()~-ao)x()~,ao, x - [ a 0 ] ) . Putting (1.134)in (1.132) one has then
( . x - u)x(.x. ,. z) = z(~x. x + [,])
Z ( # , x + [#])
(1.135)
and setting R(a, b, x) = Z ( a , x + [a] + [b]) this becomes (~ - , ) x ( ~ ,
~, ~) = R ( ~ , ~, x - [~]) R(,,,,
9 - [,])
(1.136)
1.1.
ALGEBRAIC
AND GEOMETRIC
METHODS
21
Further calculation (using (1.131)) shows that a general R must satisfy T(X + [b]) R ( a , b, x) = F s ( x + [a] + [b])T( x + [a])
(1.137)
for arbitrary Rs and T. Putting this in (1.136) gives finally a VOE type equation in the CBA context 1 T(X--[A] + [p]) X(A,#,x) = A _ # T(x) (1.138) Here T is an arbitrary function giving the general solution of various functional equations arising from (1.131). Now put (1.138) in (1.131) to obtain a HBDE (a - p ) ( A - b ) r ( x + [a] + [#])r(x + [A] + [b])+
(1.139)
) ~ ) ( b - #)T(X + [A] + [a])T(X + [b] + [#])+
+(a-
+ ( a - b)(# - )~)r(x + [b] + [a])r(X + [A] + [#]) = 0
which is nothing but the simplest addition formula for the tau function (which corresponds to the Fay trisecant formula). 1.1.6
Quantization
and integrability
In view of the Section 1.1.4 we go now to [374, 406, 891, 892, 910, 913]. First in [891] the Moyal bracket deformation of KP is related to dKP. The idea is to develop a deformed differential calculus based on an associative star product and obtain a description of the KP hierarchy in terms of vector fields and differential forms (rather that PSDO). In particular one wants a natural passage from KP with Lax structure to a dKP form, and moreover without scaling, in order to facilitate the study of geometry for KP. Thus let M be a P manifold with a bilinear skew symmetric P bracket {u, v} = ~ij ( O u / O x i ) ( O v / O x j ) satisfying the Jacobi identity in the form {{f,g},h}
+ {{g,h},f}
+ {{h,f},g}
(1.140)
= 0
Assume M is symplectic (i.e. a P manifold for which w ij is of maximal rank) and consequently d i m ( M ) - 2 N . Assume further that wiJ is constant with a basis chosen such that wiJ =
(
0
--IN
IN)
anddefine
0
u
9 ~ =
~xp
~s
-
2w
'~
Ox i O~cJ
..
~- E ~~'131...(.ijisjs 0
2 ss!
~(x)~(~)
OSu
~=x
=
O~v
(1 141)
Ox il . . . Ox i~ OxJl . . . OxJs
(cf. Chapter 3, (8.1)). The Moyal bracket is then defined as {u, V}M The following properties are straightforward
"
- - (ll, $ V -- V *
lim,~__,ou 9 v = uv; c 9 u = cu; lin~{u,_+V } M = { U , V }
lt)/t'C.
(1.142)
while 9 is associative with {U,V}Mbilinear, skew symmetric, and satisfying the Jacobi identity (~ = - i h for Moyal, Fedosov, etc.). A different non-associative product is also introduced via c~
u ov = ~
o
~s
22S(2s + 1)v
w iljl
9
"
" w i28j2s
02s u
02S v
Oxi~ . . . Oxi2s OxJl . . . OxJ2~
(1.143)
C H A P T E R 1. Q U A N T I Z A T I O N A N D I N T E G R A B I L I T Y
22 This product satisfies
uov=vou;
cou=cu;
lim~__,ouov=uv;
d 2--~(~uov)=u,v+v,u
(1.144)
This will lead to a deformed differential calculus satisfying the key property
Ou Ov ~*-- o ~ = {~, ~}M Ox r Ox*
(1.145)
We list here some properties for possible use later. Thus one defines tangent and cotangent bundles via (Oi ~ O/Ox i)
< (~i, dx j > = (5{; < X , w > = < xi(~i, wjdx j > - X i o w j < ~)i,dx j >= X i OWl; Z f = X i o Oif ; < X , df > = X f; df = Oif dx i
(1.146)
while for A = Ail...ipdx~ 1 A . . . A d x ip and B = Bjl...jqdxJl A . . . A d x jq one writes (1bE) A A B = A[il...ip o Bjl...jq]dx il A ... A dx ip A dx j~ A ... A dx aq. Thus e.g. for df = f x d x + fydy and dg = gxdx + gydy one has df A dg = (fx o gy - fy o gx)dx A dy = {f, g } M d x A dy. Note that A A ( B A C ) ~ ( A A B ) A C . The Lie derivative is defined via (15F) s = X I and ( s i= x J o (OYi/Ox j) - F J o (Oxi/OxJ); one writes s = [Z, Y] and d ( s - s for a p-form w. The normal relations
[X,Y]f-X(Yf)+Y(Xf)=O;
[[X,Y],Z]+[[Y,Z],X]+[[Z,Y],X]=O
(1.147)
do not hold for the deformed definitions but for (local) Hamiltonian vector fields (15G) X f = wiJOifOj one has a version of this, namely
Xlh = {f,h}M;
[xl, X~]h =
X~(X~h) -
[Zf,Xg] = Z{f,g}M;
X~(X~h), [[X~, X~], Xh]
(1.148)
+ [[X~, Xh], X~] + [[Xh, X~], X~] = 0
For two dimensional manifolds M such Hamiltonian vector fields generate the Lie algebra S d i f f , ~ ( M ) of area preserving diffeomorphisms of M (area element dx A dy) and the composition of two Hamiltonian vector fields is defined as the Lie bracket. Explicitly X I generates infinitesimal transformations x --+ x - efy and y --+ y + efx (here e ~,, ~). The KupershmidtManin bracket (15B) defines a bracket { f , g } ' ( f 9g - g , f ) / e and a corresponding o product is (e ~,, ~) fog=
cc ~s as-roam f a ys-m O;m g ~ ( s + l)! -x vy o m=O
(1 149)
satisfying r (OV/OXs) --- {U, V}' SO a deformed geometry based on (1.149) can be written down as above. The importance of these new products comes from their relationship to the algebra of PSDO. Thus for PSDO P = ~Nc~ aj(x)O j E • with (15H) i)ma aO m + ~ [ m ( m 1 ) . . . ( m - k + 1)/k!]OkaO m-k one has s y m ( P Q ) - s y m ( P ) *~=1 s y m ( Q ) for 9 as in ( 1 5 B ) so there is a natural relation to the standard KP hierarchy (cf. [615] and earlier remarks). Now let M = M | T where T ~,, times {tn} and for some 9 product on M write
u(x, t ) , v(x, y) = exp ( ~ z ij 0 0 ) u(x, t)v(2, t) Ox i O~J ~=x
(1.150)
(note X I = w ij (Of(x, t)/Oxi)(o/~)x j) will be time dependent). Let L = 0 + ~ Un(X, t)O -n be the Lax operator for KP (cf. now Section 1.1.1). One then applies the geometrical
1.2. V E R T E X O P E R A T O R S A N D C O H E R E N T STATES
23
framework to obtain a Moyal KP hierarchy KP,~, based on deformation of dKP, which is equivalent to the Sato hierarchy based on PSDO. Similar calculations apply to Toda and dToda, KdV and dKdV, etc. Further the geometrical picture can be phrased in the S d i f f 2 format w i t h / 2 , AzI etc. as in Sections 1.1.1 - 1.1.3. It is in this geometrical context that the o product and associated geometrical ideas are fruitful (details are omitted h e r e - cf also [910, 913]). It seems from this that if one starts with dKP as a basic Hamiltonian system with Hamiltonians Bn and standard P brackets then KP,~ can be considered as a quantization of d K P is some sense with quantum integrals of motion Bn (n) which for n = 1 say is equivalent to KP (or n = 1/2 in [374]). The Bn (n) would perhaps have to be extracted from KP after establishing the isomorphism (cf. [374]) and we turn briefly to this approach now. Thus in [374] one writes the Sato KP hierarchy via (v-2 = 1, v-1 = 0)
co OraL = [L'~,L] (m >_ 1); L = ~
(1.151)
Vn(X)Ox n-1
-2
for ~ = (x, t 2 , " ") while the Moyal KP hierarchy is written via (u-2 = 1, u_ 1 = 0)
(x) A - - E Un(:~))~-n-1; GQmA- {A~,A}M (m ~ 1) -2
(1.152)
where A~ ~ (A'm)+ with (cf. (1.13), (1.19))
leading to
{ f ' g } ~ = ~ o (2s + 1)'. j=o ( - 1 ) J
(1.153)
(O~o~-Jf)(o~-JOJ~g)
f * g = ~ 0 ~.w j~0 ( - 1 ) :
1)
j
,'-'x'-'),
s,(vx
-JoY, g)
(1.154)
Note l i m ~ o { f , g}~ = {f, g} = f),gx - fxg)~ so ( K P) M ~ d K P as n ~ 0, namely/)mA = {A~,A} with A m ~ A - - . A . The isomorphism between (KP)sato and ( K P ) M is then determined by relating Vn and un in the form (n - 1/2) n
Un "-- E 2-j 0
J j Vn_
(1.155)
where n = 0, 1,.-. and vJ = O~vo.
1.2
VERTEX
OPERATORS
AND COHERENT
STATES
We extract here from [153, 159] with considerable reorganization and clarification. Notations (A) and ( A B ) are used. 1.2.1
Background
We will give first a refinement of some ideas and formulas developed in [159]. We begin by sketching some definitions and formulas. One could start with Bose operators a ~ 0 - O/i)z
CHAPTER
24
1. Q U A N T I Z A T I O N
AND INTEGRABILITY
and a t = z with e.g. z E S 1 (but z E C is also permissible unless otherwise stated). T h e r e is a v a c u u m I0 > with hi0 > - 0 and one creates states in a Fock space via
In > - - ~(at)nlO > aln > = x / ~ l n - 1 >; atl n > - v/n + lln + 1 > ;
(2.1)
with (A) [a, a t] = 1 and ataln = nln > while < rain > = 5mn. For example one could choose z e S 1 with a ~ Oz, a t = z, and In > = zn/v/-~, with < h i m > = (1/2~ri) ~ z n 2 m ( d z / z ) . T h e operators ~ and 15 can be defined via
1 (a+a t
it=
1 ( a - a t ) ; [ih,O ] = - i ; ih~-iOq; (t"~q
(2.2)
and to bring q u a n t u m mechanics (QM) into the picture we write (h ~ h = h/27r)
qh -- v/hq; Ph -- v/-hP; Ph -- -iv/-hOq -- - i
hA Oqh;
~h, O h ] - - i h
(2.3)
while coherent states are defined for z E C via c~
zn
Iz > = D(z)10 > = e-(1/2)lz12 ~--~ ~n.WIn >; D ( z ) = ezath-2ah;
0
1
z = - - ~ , ( q + ip); D ( a ) D ( ~ ) = e2i~(a$)D(13)D(a)-- - v' An
(2.4)
Note t h a t one can write
ah = - ~1( ( t h
+ i~h) = a; ath = - ~1( q h
-- i~h) -- a t
(2.5)
while zath -- 5ah = (i/h)(pOh -- qPh). We set also (B) U(p, q) = exp(i/h)(pOh - qDh) "-' D ( z ) and Iz > ~ IP, q >- Here we record also that the coherent states Iz > are defined via alz > = ahlz > = zlz > and one has
Dr(z) = D - l ( z ) = D ( - z ) ; D t ( z ) a h D ( z ) = ah + z; D(z)l ~ > = ei~(Z;)lz + ~ >;
(2.6)
D ( z ) = e-(1/2)lzl2eza~e-~ah = e (1/2)lzl2e-zaheza~ ; < zlz I > - e-(1/2)lzl2+2z'-(1/2)lz'j2 In addition (cf. [591, 592]) one observes t h a t (2.4) results from writing Iz > = E ~ Cnl n > and using ahlz > = zlz > to get v/n + lcn+l = ZCn from which Cn can be d e t e r m i n e d as in (2.4). Note also from (2.6) that D(z)I0 > = exp[-(1/2)lzl2]exp(za~)lo >. We see also from (2.6) t h a t coherent states are not orthogonal but they are overcomplete with
I = f d # ( z ) I z > < zl; d# = l d ( ~ z ) d ( ~ z ) 7r J
(2.7)
A n o t h e r useful fact is Zn
< nlz > = -----~.e-(1/2)1zl2 :=~< OIz > - e -(1/2)1zl2 v'n~
(2.s)
Now in [452] one showed the convergence of quasiclassical operators (~ ~ x/~0 = Oh and t5 = v/-~i5 = ihh to classical quantities ~ and r (i.e. (~ --~ ~ and P --~ ~) when h --~ 0. Thus
1.2.
VERTEX OPERATORS AND COHERENT STATES
in some sense quantum fluctuations are smoothed order to determine a classical object. For example
~.2
H = ~
25
out by a multiplicative factor of v/h in consider a classical system
+ V(~); m~ = 7r; # = - V ' ( ~ ) ; 1
~ ( a , 0 ) = ~; ~ ( a , 0 ) = ~; a = ~ ( ~
(2.9)
+ i~)
The associated QM system has the form (h ~ h)
~
h~
i h C t = - -~m ~2qq -}- U r ; U h = ~
i~h -- hOq; Ct -- Uhr
-~- Wh ; Wh -- W(v/-h~);
(2.10)
Uh -- e -(it/h)Hh
(Hh being a self adjoint extension of the symmetric operator Hh). Let z - h-1/2~ and U(p, q) ,.~ D(z) as in ( B ) with U* = U r ~ D t (z) = D ( - z ) ~ V ( - p , - q ) and set h - 1 / 2 l a > U(h-1/2a)lO > with V ( h - 1 / 2 a ) ~ D(z) and z = ( ~ + i T r ) / v / ~ . Then consider (cf. [159,452]) E=<
7r
h-U2a (q-~)
(P---~)1
--
h-1/2a > =
UU* I p - - ~ )
71"
(2.11)
U[O>
where U ~ U ( h - U 2 a ) . However from (2.5)-(2.6)
U*(z) (q v+~ip ) U(z) = u*~u = ~ +
+ ip
__• ; u*pu
+
~ + i~
=~
(2.12)
71"
- p + -~
Consequently E = < OlOi~[O > and it is shown in [452] that as h ~ 0
< h-~/2ai(~h - ~)(Ph - ~)lh-~/2a > ~ 0
(2.13)
=~< h-1/2a[QP[h-1/2a >---+ ~Tr
SUMMARY 1.1. Thus in particular (cf. (2.54) and ( A N ) ) the scaled variables (~ qh ~ ~ and P ~ ifih ~ 7r where ~, 7r correspond to classical values of ~,/~.
We are trying to accomodate here a number of notations from [159, 452, 591,592, 796, 987] (cf. also [392,^593, 594] for difference schemes). Another notation used is (C) (~ ~ qh and P ~ i~h with P - -ih(O/OQ). In [159] we compared P, Q coordinates to scaling ideas in dispersionless or quasiclassical soliton theory and wrote Q ~ eq with Oq ~ eOQ while e 2 ~ h; this produced correct mathematical structures but the e 2 seemed misleading at times (and certain sections are improved with some reorganization). Thus in dispersionless KP for example (cf. [148, 149, 143, 153, 158, 160, 902]) one thinks of q -+ eq - Q and hence ,~ --iOq --o -ieOQ. In [159] we took (D) a = [(Q/e) + eOQ]/X/'-2 = (Q + i [ ' ) / e v ~ - a(e) where P = -ie2OQ. One notes that a(e) ~ ah for e2 = h and this is consistent with qh = v/-hq = eq and iPh = ix/~p = v/hOq = hOQ while a(e) = at, = a. W h a t is slightly misleading is the fact t h a t in [902] for example one thinks of h as the scaling p a r a m e t e r
CHAPTER 1. Q UANTIZATION AND INTEGRABILITY
26
instead of e = v/-h; this obscures the nature of the quantum relations based on formulas such as (2.13). Thus in relation to classicality it is qh -" v/-hq and not q which corresponds in QM to the classical coordinate ~. There have been many papers on various aspects of semiclassical expansions in h or ~ and we do will not attempt to cover all this (see e.g. [434, 435, 436, 548, 549, 723, 873, 987] for developments involving coherent states and cf. also [315,345, 1007, 1008, 1009]). We are mainly concerned here with comparisons to scaling in dispersionless soliton structures. In a more QM spirit one has position and momentum representations via (cf. [842]) QIQ' >= Q'IQ' > and < Q'IQ >= 5(Q'-Q) with 1 = fdQ [Q > < Q[ and I(~ > f dO [Q > < Qla > (here f ~ f _ ~ ) . One writes (E)
<~ Qla > = Ca(O); = f
dQ < Qla > = f dQ~z(Q)r
)
(2.14)
< QIA] Q' > Ca(Q') . For the momentum Also in general (F) - f f dQdQ ' Cz(Q) representation P ~ ihh with ]5 = -ihOQ one has (G) PIP' > = P'lP' > with < PIP' > 5 ( P - P " ) and 1 = f d R [P > < Pl while Is > = f d R [P > < Pla > and
< P]a>= Ca(P); < QIP]a > = -ihOQ < Qla >; ^ ' > = -ihOQ5 (Q - Q'); < QIP > = < Q[P[Q
1
(2.15)
e (i/h)PQ
Note that this gives free of charge a derivation of the Fourier transform formulas Ca(Q) = < QI(~ > = ~ 1 Ca(P) = < P l a > = ~
/ dRCa (p)e(i/h)p Q;
1/ dQr
(2.16)
-(~/h)PQ
In this context vacuum vectors 10 > such that hi0 > = 0 (~ ah]O > = 0) can be represented in a peaked state or Schrhdinger form via the coordinate Q. Thus from < QlalO > - 0 we have (Q + hOQ) < Q [0 > = 0 or < Q[0 > = (hTr)-l/4e-(1/2h)Q2; < QI n > =
(Q - hi)Q)n e-(1/2h)Q2 (hTr)l/4v/n!(2h)ne-(1/2h)Q2 = ( ( h T r ) l / 4 ~
(2.17)
Hn
(--~h )
These will be referred to as oscillator eigenfunctions or the Schrhdinger representation (H~ means Hermite polynomial). If one starts out with Bose operators in a Fock space there is a priori no reference to physics. Then we could define ( n ) ~ = (a + at)/v/2 and ~ = (a - at)/v/2 as operators related to physical objects q and p while Q ,,~ x / ~ a n d / 5 = x/~i5 lead to classical objects and 7r as above. Another approach to classicalization is described in [1007, 1008, 1009]. Thus first one wants to avoid the often ambiguous h ~ 0 approach (see however [315, 345]). Secondly one should begin with quantum mechanics and reverse or retrace the geometrical quantization program where a symplectic geometry and classical phase space (hence classical Hamiltonian dynamics) are posited in advance. This leads to the approach of [987] where the geometric quantization theme is reversed and one obtains a large N limit of a quantum structure as classical mechanics. Note however the retracing technique of [987] can
1.2.
VERTEX OPERATORS AND COHERENT STATES
27
apparently produce a number of corresponding classical systems when rang(~) > 1 (where .~ is the associated Lie g r o u p - see below). The path integral formalism of [548] gives still another direct connection. In any event coherent states seem to play a fundamental role in the quantum-classical correspondence and since the quantum picture appears more basic (cf. [345]) one is led to the path of classicalization from quantum to classical and various forms of semiquantal mechanics (instead of semiclassical). The approach of [1007, 1008, 1009] starts from the geometrical structure of a quantum system determined by a basic Lie algebra .~ (with corresponding Lie group G) and a Hamiltonian H(Ti) where Ti E ~ (note that the mathematical image of a quantum system is an operator algebra ~ in a Hilbert space T/and one can restrict to an irreducible representation (irrp) of G to describe dynamical properties). Let Ir > be the lowest weight bound state of an irrp of G and let H be its maximal stability subgroup. Roughly sketching from [1007, 1008, 1009] (with further detail to follow) one constructs coherent states via transforms ~Ir > where ~ C G/H. The coherent state space so generated will l~e topologically isomorphic to G/H (see below for details) and one has an explicit symplectic form
w = ihO2~(z' Y.)OziO2j d zi A dhj; )~(z,2) = det(I 4- ztz) +=-
(2.18)
(i, j = 1 , . . . , M) where K: is the Bergman kernel, the zi are dimensionless local coordinates of G / H , find ~ = h | k is a standard Cartan decomposition (h = Lie(G) and [h, hi C with [h,k] C k and [k,k] C h); note 5 ~ z* and in some contexts z corresponds to a matrix. Further one has 7: in (2.18) according to the compactness or not of G / H (so the associated irrep space is finite or infinite dimensional) and = is called a quenching index defined via hiIr > = +~1r > (or 0) for hi E H (thus E comes from the map G ~ G / H induced by the fixed point of Ir >). Given this structure one defines canonical coordinates (I) ( 1 / ~ ) ( q + ip) = z/v/1 4- ztz making G / H a quantum phase space. The details appear in the papers listed in [1007] and also with variations in [1008] (cf. also [63, 845]). Now the quantum phase space can be either compact or noncompact depending on the finiteness of g . Starting from the Heisenberg equations of motion (J) ili(dA/dt) - [A, H] one says that a quantum system with M independent degrees of freedom (2M dimensional phase space) is integrable if and only if there exists M quantum constants of motion related to the eigenvalues of M commuting nonfully degenerate (NFD) observables Ai (i = 1 , - . . , M). Here an operator A is fully degenerate if Alr > = cI~Pi > for all ~p~ E ~ . For somewhat more detail now (cf. [1007]) a group element g =
C
D
acting in G / H
is a homomorphism gf~ ~
z ~ -- Cz+B az+B where z ~ and z ~ f~ are complex h x k matrices (h = dim(it) and k = dim([~)). The metric of G / H is
021ogtC(z, ~) gij
-
-
OZi C~2j
; K~ = ~ f ~ ( z ) h ( z ) )~
where f~ is an orthogonal basis of a closed linear subspace s
c
(2.19)
c L 2 ( G / H ) via
M dzid2j f ~ ( z ) f v ( z ) ~ - l ( z , 2)d# = 5~v; d# = g[det(gij)] I I 7r /H 1
(2.2o)
Here/~ is isomorphic to the 2M dimensional space G / H and N is a normalization factor making f d# - 1. There is a closed nondegenerate 2-form w on G / H of the form (K) w +
CHAPTER 1. Q UANTIZATION AND INTEGRABILITY
28
ih ~ gijdzi A dhj
(equivalent to (2.18)) and the corresponding Poisson bracket is
1 where
gijgjk = 5ik. For 1
~ ( q
[ Og Og
Of Og 1 02i Ozi J
(2.21)
semisimple G canonical coordinates can be introduced via
"if-ip) "-- Y ( z ) ; Y ( z ) "--
{
z
(I-4-zt"z)I/2 (l__~Z"-~l/2
compact G noncompact G
(2.22)
where X ~ = is a function of some inherent physical parameters such as spin, particle number, etc., and (2.21) becomes
M 1
w = y~. dpi A dqi; { f, g} =
~l[OfOg Oqi Opi
OfOg] Opi c3qi
(2.23)
(the development (2.19)-(2.23) is handled somewhat better in [1008] via a slightly different representation of z at an early stage- cf. below). In any event there is an inherent quantum phase space constructed from the quantum dynamical degrees of freedom (QDDF) (or the N F D operators) and their corresponding symmetry group G. We want to provide still more detail here from [1007] since this is a very attractive approach. Thus given a quantum system with dynamical group G and Hamiltonian H - H (T i) for i -- 1 , . . - , n along with "transition operators" Aj (which we take as NFD observables 1 < j < M) one will have (L) [~, Tj] = Y~ CikjTk. The vacuum Hilbert space 7-I can be decomposed into a direct sum of unitary irrp 7-/h in the form 7-I - OAYA~-/A (here A ~ highest weight of a representation and YA is the degeneracy). The study of dynamical properties can be focused on one particular irrp 7-/h (called 7-/in the preceeding paragraphs). Let now {X~ } be a subset of generators of ~ such that any state [~ > has the form (M) [~ > = F(X~)[0 > for all [~ >E ~-~h where F is a polynomial. Here [0 > is chosen so that a minimum subset of generates 7-/h. Such a set of X/f is referred to as a set of elementary excitation operators. It is shown in [1007] that the number of X/t is the same as the number of QDDF and one defines (N) /~ ~ {X/t, Xi; i = 1 , . . . , M = MA} (linear span) with G/H generated by
a = ~p [~ n~x~ - ~x~]
(2.24)
Then one will have the quantum phase space (QPS) indicated earlier giving rise to a geometrical structure for the quantum system. The next question involves describing QM on the QPS constructed. This can be done if there exists an explicit map {~,7/h} ~ {g(G/H),s where g E G such that A ~ .A(p, q) with I~I' > ~ f(q + ip). For a quantum system with QPS G/H as above this mapping is uniquely realized by coherent states (cf. [384, 548, 796]). One has (O) gl0 > = Fthl0 > = Ih, f~ > exp[ir for g E G, h E H, and Ft E G/H (we distinguish here h and h). Note hi0 > = 10 > exp[ir and the coherent states ]A, f~ > correspond to G/H. Thus IA, f~ > -
ill0 > = e E~(n'x~-~'x') l0 > -
= K~I/2(z, z)e EM ziXti IO >-- K~-I/2(z, ~lA, z >
(2.25)
1.2. VERTEX OPERATORS AND COHERENT STATES
=< 0
29
I 0 >=
(2.26)
= < h, zlA, z > = l < 0]A, fl > 12 = ~-~fAtz)f2(z) where, for r/corresponding to the h x k block matrix of ~M(rliXi -- OiXti) in the faithfull matrix representation,
z=
r/ Tan(rl~rl)l/2 (~tn)~/=
compact G
Tanh(~t~)l/2
r/- (sty)l/2
(2.27)
noncompact G
From this one also obtains (2.19)-(2.23). Note that IA, z > is simply an unnormalized form of }A, f~ > and (P) fA(z) = < A, AIA, z > where IA, k > is a basis of ~ a with fa/H IA, z > d#H < A, zl = I for dpH = lg-l(z, 2)d# (cf. (2.20)). Note also (Q) fa/g IA, ~ > d# < A, 121 = I (here apparently < A, fll ~ < 12, A[). Now the phase space representation of a wave function can be uniquely defined as (R) f(q + ip) - f(z) = < ~IA, z > where z and q + ip are related by (2.22). The uniqueness of f(z) implies that K: is a reproducing kernel in the sense that
f(z) = fa /H tC(z, 2')f(z')d#g(z', ~')
(2.28)
(f' g) =
(2.29)
while
/U f(z)g(z)dpH(Z, 5) < oc
The closed subspace s generated by such f corresponds to T/A. For an operator A one has the coherent state diagonal element (Q representation or covariant symbol) (S) A(p, q) = A(z, 2) = < A, f~[AIA, ~t > which uniquely describes A. One has
(.4f)(z) =/a/M
z,
(2.30)
Here A(z, 5; z', 2') is an analytic continuation of A(z, 2) to (G/H) • (G/H) with the stipulation (W) A(z, 2.; z', 5') = < A, ~[A[A, ~' >. Further
A1A2 ~ (A1A2)(z, 5) = f
3G/H
Al(Z,5;z'2')A2(z',~';z,2)d#
(2.31)
As for algebra one must determine whether the P bracket of the phase space representatives of two arbitrary operators satisfy the same algebraic structure. For this one notes first that for operators ~ with T / = < A, FtlT/I[A , gt > one has (U) ih[Ti, 7"j] = ~ C~7"k and this will be used in what follows. It should also be noted that a density operator can be defined via p = [~ > < ~ with
p(z) = < A, ftlplA, fl > = [ < A, f~]~
> [2 __
If(z)[ 2
(2.32)
Thus there is a complete kinematical correspondence between quantum and classical frameworks, namely A ( ~ ) --, A(p, q) and [~ > ~ p(p, q). The dynamics is not quite so straightforward. One can describe a dynamical system whose motion is confined to G / H and is determined by the equations
dA(q, p) = {.A(q, p) 7/(q, p)} dt
(2.33)
CHAPTER1.
30
QUANTIZATION AND INTEGRABILITY
or equivalently
(ti = ~p_________~) T-l( q , [9i = Opi
0 7-l( q , p) Oqi
(2.34)
The correspondence principle implied here is to find suitable conditions so that the QM Heisenberg equation can be written as in (2.34). Then if (for AH = UAU-~; U = exp(itH/h)) one has 1 i-h < A ' ~ I [ A H ' B H ] I A ' ~ > - {A,B} (2.35) it follows that the phase space representation of (V) dAH/dt = (1/ih)[Ag, BH] can be given directly by (2.33) and is therefore equivalent to (2.34). Then (2.34) will be referred to as a classical analogy or semiquantal dynamics of the quantum system. Three situations are described in [1007]. 9 (I) Exact quantum solutions. According to (U) no additional requirement is needed for the generators T/ to satisfy (2.35). Hence if g is a linear function of the Ti (V) can be reduced to (2.33) - (2.34) with 7-/= 7-tL (see [1007] for details). 9 (II) Mean field solutions. In this situation whether (2.35) holds or not the solutions of (2.33) provide a general result for the Q mean field dynamics of (V). Thus write U ( t " - t') = e x p [ ( 1 / i h ) g ( t " - t')] with
bl(p",q";p',q';t"-t')=<
A,~'lexp[1H(t"-t')]lA,~'>
(2.36)
Using (Q) this can be written as a path integral
~pP",q"
1.~= ',q' E)#(p,q)e(i/h)S(P(t)'q(t)); ~)p= i.
H dp~)(t),q(t)]; t'
_e II
8 = It, dr[< A, f~ligOtlA, a > -7-l(p(t), q(t))]
(2.37)
The variational principle 6S = 0 yields then (2.34) for 7-l(p(t), q(t)) = < A, aIHIA, a >. Thus the mean field approximation or mean field dynamics is a special case of the classical analogy or semiquantal dynamics. Note that any surface terms arising in the action due to the requirement of invariance of dp will disappear in the variation 5S = 0 and not affect the result (2.34). 9 ( I i I ) Classical limits. If (W) < A,f~IABIA, f~ > = A(p, q)B(p, q) holds and can be applied to H = H(T~) for example, then the mean field dynamics is reduced to (2.34) for (X) 7-/ = 7-lc(p,q) = U(Ti(p(t),q(t)) (so called classical limit). For application of ( W ) to (X) one needs additional requirements. For example if some physical quantity X(B) -* 0 (not the same as h - . 0) then ( W ) can be applied to (X). This is the situation of [987] for example and one refers to [1010] for more on this.
~-~i
Note in the linear case ~ c = 7-/= and more generally 7-/c is an approximation to 7-/ For integrable systems the topological structure of trajectories determined by 7-/ and 7-/c are similar but for nonintegrable systems the global structure may be completely different. The q u a n t u m dynamics is closer to 7-/so this is a good place to study quantum chaos. Note the classical limit above is different from the Ehrenfest theorem (cf. [727, 748]. We now look briefly at [987] as developed in [159]. Let (Y) gh be the Lie algebra generated by el = (i/h)P, e2 = (i/h)Q., and e3 = i/h with corresponding Weyl-Heisenberg (WH)
1.2. V E R T E X O P E R A T O R S A N D C O H E R E N T S T A T E S
31
group (Z) Gh = G = {Uh = exp[(i/h)(a+(1/2)Tr~)]U(Tc,~) where U(~r, ~) ~ exp[(i/h)(~rQ~/5)] ~ D(z) for z = ( 1 / x f ~ ) ( ~ + i~r). One generates coherent states ]u > h = Iu > = Uh[O >h where ]0 >h corresponds to a peaked vacuum for Q (cf. (2.17)). This leads to coadjoint orbits ( A A ) F = {~ = Ad*@; u ~ u(~r, ~, a) E Gh; @ E {7" f i x e d } and for lira < OI(h/i)s >=< @,A > where A ~ (~r,~, a) E gh and/~ ~ (~vel,-~e2,ae3) one has (I u > = U~]0 >) h
< ul--:s
i
>=
h
-
<
OIUh~AUhlO>--->< @,Adu-~(A)
> = < Adu@,,~ >
(2.38)
The variables (Tr, ~) provide coordinates on F and there is a standard symplectic structure. Such operators .A (e.g. h/k) with limits < u].4iu > / < ulu > generate so called classical operators via (AB) f~ = f d)~ f()~)exp[h3,] and covariant symbols (AC) Ah(u) = < uif~iu >h leading to functions a(~) on F. Expectation values of ( h / i ) s distinguish classically inequivalent states and coherent states thus collapse to F via an equivalence relation. Therefore F provides a classical geometrical framework for the Fock space based on (a,a t) via = (1/v/2)(a + a t) and ii5 = (1/v/-2)(a - a t) with (~ = v/-h~ a n d / 5 = v/-~/5. In more detail consider ^
^
Uh(p, q, a) = eia/heipQ/he -iqP/h = e(i/h)(a+(1/2)Pq)U(p, q)
(2.39)
(by Baker-Campbell-Hausdorff = (BCH) this equals Uh of (Z) for q = { and p = 7r - cf. also (B)). One writes now A = ~)~iei E {Th and ~ = ~ i e i E gh and a coherent state Iu > = UhI 0 > with u ~ u(p,q,a) gives rise to a point 4 - Adu@ E F (here 0 > will be a peaked vacuum as in (2.17)). The tangent space Tr C ~ is generated by ad*~(4) where ad~(4),~ > - - < 4, [A,~] > and T~(F) can be identified with equivalence classes [A] (A E gh) via ( A D ) [A] = {A' E gh; ad*~4 = ad*y~}. Let He be the isotropy group of 4 so (AE) He = {u E G = Gh; Adu4 = 4} which produces F ~ G/Hr Explicitly, for the WH group coadjoint orbits correspond to planes 43 = c (c ~ 0) since
' =
Z
^ '
;iei; ~3 = ;3; r = ~1 + P @ ; ~2 = ~2 + q~3
(2.40)
for u ~ u(p,q,a) (cf. [987]). Relabeling ~1 ~ P and @ ~ q one has natural coordinates (p, q) on F and a symplectic structure with P bracket can be exhibited. In particular {f, g} = @[fpgq - fqgp] and the action is S(p(t), q(t)) = f dt[431p(t - Hcl(p, q)] which corresponds to the classical action. Now associated to A = (p, q, a) one has an operator ( A F ) / k = ( i / h ) ( p Q , - q P , a) and h/~ is then clearly a classical operator with ( A G ) < @,A > = lim(1/i) < 0Ih/~10 > existing. Consequently ( A B ) determines classical operators A and covariant symbols as in ( A C ) where Iu >h again denotes a peaked vacuum. Classically equivalent states are those Iu > mapped onto a given 4 E F via (2.38) (i.e. the expectation values of h/~ distinguish classically inequivalent states). For such classical operators A it is shown in [987] that ^
Ah(u) ~ a(~); (AB)h(U) ~ a(~)b(~); i [A,B]h(U) ---, {a(~), b(~)} 1.2.2
Heuristics
for QM and vertex
(2.41)
operators
We give here some heuristic relations between QM and vertex operators but refer to Chapter 4 for different approach and a more extensive development. Let us assume as given a QM situation with q, p, qh "' Q, Ph "" P, a, and a t as in (2.2)-(2.3), (2.5), and (C). We saw
32
CHAPTER
1.
Q UANTIZATION
AND INTEGRABILITY
in ( D ) t h a t this is consistent with a scaling q --, eq = Q provided e2 = h. ( A H ) at(e) = ( Q - i.[:')/ev/2 and [(~,/5] = ie2 leading to
Q_ _
a(e) + a t (e).
Similarly
i P = eOQ = a(e) - a t(e)
(2.42)
Now in dealing with vertex o p e r a t o r s in soliton t h e o r y one encounters o p e r a t o r s involving t e r m s ( A I ) e x p ( x ) ~ - )~-~0) for 0 ,-, 0x which are also related to string t h e o r y for e x a m p l e (cf. [147, 159, 172]). Suppose we identify a ~-- 0x and a t ~ x which will be consistent with Bose o p e r a t o r properties and scaling via x --, ex = X and 0x ---* eOx. T h e n from (2.42) (with e2 = h) one would have (recall a(e) ,,, a and a t (e) ~,- a t) Q, a + at 2 + e20x i P a-- = = ; -- =
at
=
e20x - 2
(2.43)
leading to
2 = r - iP v~ ; J o x -
Q, + iP
(2.44)
F u r t h e r for A e S 1 ( A I ) involves (cf. ( D ) and ( A H ) ) x A - A-10 = Aa t - ~a = A a t ( e ) - Xa(e) =
(2.45)
]-~ (Q+~P)~v~=-J-~v~[~a~)-~aP]= ~[t~)-r
=a\(O,-~P~~v~
for i5 = v/2.~,~ and ~ = v~NA. Now explicitly the main vertex o p e r a t o r s in soliton t h e o r y have the s t a n d a r d form (~'(x, A) = ~ Xn)~ r' and c5 - (On~n) for n >_ 2 with 0 ~ 0x) X(A) = eX~+~'(x'~)e-~-l~
(we treat the
Xn
X*(A) = e-X~-C(x'~)e ~-1~
(2.46)
as parameters momentarily) and X(A r
= A - CX,(r
= A - CX(A)X,(r
__ ex()~-r162
=
(2.47)
-1)
For )~ E S 1 a n d ~ = A- I , using the B C H formula (AJ) exp(A)exp(B)
= e x p { ( 1 / 2 ) [ A , B ] } e x p ( A + B ) = exp([A, B ] ) e x p ( B ) e x p ( A )
one o b t a i n s X()~) = eX;~e~'(x';~)e - ~ ~ e -~'(~'~) =
(2.48)
= e~'(z,;ge-~'(o,~)e-(1/2)[x,O]e x~-~o = el/2e~'(x,~)e-~'(0,~)D(A)
where (AK)
b(~) = e )~at-y~a - e )~at(E)-~a(e)
Recall now (2.4)-(2.5) where D ( z ) = e zat-2ah -- e zat-2"a - e (i/e2)(p~-qp)
(2.49)
1.2.
VERTEX
OPERATORS
AND COHERENT
33
STATES
(here ah "-' a(e.), Q = qh = x/-hq = eq, etc.). On the other hand for/)(A) we have Aaf(e) ha(e) as in (2.45) so [9(,~) = e )'at-~'a = e (i/e)[~O'-4[:'] (2.50) which suggests a correspondence (AL) z = (1/ex/~)(q + ip) ,'., Ale = (1/e)[Re)~ + i~.k] = (1/e)(O + i15) or ~ ,,~ v~q and i5 ~ v~p. This means that (~,i5) corresponds exactly to the classical object (q, p) giving rise to (Q, P). Summarizing the discussion above we have S U M M A R Y 1.2. One can envision our procedure above as follows. Pick a QM background with (q, p) and an associated Fock space with Boson operators a, a t as in (2.2) and (2.13). Let X(A) be a typical vertex operator described via D(A) (and parameters Xn for n > 2). Then for A E S 1, /9(A) can be quantum mechanically realized by identifying the spectral variable A with classical objects (q,p) via (~,i5)= v/2(q, p). In a similar manner,
for ( E S 1 X * ( ~ ) = el/2e-~'(x'~)e((O'r f)(--~)
(2.51)
leading to X(A; ~) = A - ~)((A, r
)( = e~'(x'A)e-~'(&i)e-~'(x'<)e~'(&~)
(2.52)
One could continue this procedure and pass any finite number of Bose operators an ": On and afn "-' Xn to the right; however for now let us keep the Xn as parameters. Also multisoliton tau functions of the form (X(A,~) 2 = 0) ( A M ) VN = l-IN(1 + a j X ( ) ~ j , Q ) ) . 1 could be envisioned. We recall now from ((2.1)-(2.3) and (2.38)-(2.41)) that, with e2 _ h, 1
z = - ~ e ( q + ip); U(p, q) = e (i/h)(pO'-q/') ,.., D ( z )
-
e zat-2a
-- eZath-2ah;
(2.53)
oz >; < h_l/20el.~lh_1/20 ~ > = < O i U , ( h _ l / 2 0 o f i U ( h _ l / 2 a ) l 0 >; [z > = D(z)]O > = -10 7r
U*~U = ~ + -~h ; U*pU = ~ + ---~; D * ( z ) a h D ( z ) = ah + z
Consider first z = (1/ev/-2)(~ + irr) and one obtains e.g.
z 3<
> ; < 0, ,0 z [ Q - ~]z > = e < 0[~]0 > 4 0
(2.54)
Similarly ( A N ) < zlib- 7rlz > = e < 01i5]0 >--+ 0. Look at < ~]Aa f - ~a[z > for general z as in (2.54) and write now ~ = A/e so (cf. (2.49), (2.50), and ( A N ) ) i ,kaf(e) - ~a(e) =/(ibQe - ~p); ~at(e) - ~a(e) = 7~(p Q - q/3)
(2.55)
Now from (2.54) and ( A N )
s
< z l Q - ~)lz > = ip < olplo >; -
s
< zip
~lz > = - i ~ < olplo >
(2.56)
C H A P T E R 1. Q U A N T I Z A T I O N A N D I N T E G R A B I L I T Y
34
and we can assume < 01~[0 > = < 01~10 > - o. Consequently from (2.55) < zlAat(e) - Xa(e) - / ( i 5 ~ -
~)lz >=
i
= - < zlpQ - Ft[::'- p~ - qrlz > = 0 =~< zlAa t (e) -
(2.57) i
Xa(e)lz > = -(~
- ~)
Here (~,i5) comes from A and (~,~) from z. This suggests that operators like D(A) J0e(A) = exp(Aa t (e) - Aa(e)) should have expectation values expressed in terms of (~, ~, ~,/5). In particular it was heuristically conjectured in [159] that < zlbr
(~.58)
> ~ c~ (~/~)(~-~)
for a suitable c and for some suitable meaning of ~ (see below). For confirmation one could perhaps exploit a theorem in [987] saying that for classical operators .4 with A~ = < zlf~lz >h < ~IAI~' > < ~'IBI~ >
(AB)h = J d~(I < ulu' > 12) < ulu I >
< u'lu > + o(1) = Ah(u)Bh(u) + o(1)
(2.59)
but it seemed simpler to work from Theorem 2.1 in [452] specialized to t = 0. Thus given an arbitrary problem (2.9) one has e ~ 0 limits in the strong operator topology
U(h-1/2o~)tei[PQ-qP]U(h-1/2o~) ~ ei(P~-~ ~) U(h-1/2o~)teip(q-h-1/2~)-q(p-h-1/2~r)]V(h-1/2oz)
...o e i(~o-~$)
(2.60) (2.61)
From (2.61) a possible c(/5, ~) emerges as (AO) c(i5, ~) = < Olexp[i(~-(7~)]lO > = < Olexp[AatAa]10 > = e -1/2 (via BCH since al0 > = 0 = < 01at ). Further (2.58) emerges if one inserts I = Iz > < z I and writes
< ale(i/~)[(~Q-OP)-(~-~)]lz > = < zle(i/~)(~Q-4P)lz > < zle-(~/~)(~-~)lz > = = < zle(i/e)(PQ-(tP)lz > e -(i/e)(~-qTr) ----~< 0[e i(i50-~i5) 0 > - - e -1/2
(2.62)
In this spirit the conjecture (2.58) seems well founded modulo a precise meaning for ~. R E M A R K 1.3. The restriction A E S 1 with p2 + q2 = 1 is not pleasing. One could in principle work with general A and exp[Aat(e) - A-la(e)] but the nice geometry of coherent state theory would seem to be lost or seriously compromised. We note also that based on (2.58) one might expect to deal with a class of quasiclassical or semiquantal operators defined by the properties
ftsQ = f f d~d~t](~,Ft)[9~(A);
(2.63) 4)@d4
This enters then the realm of coherent state transforms, oscillatory integrals, Weyl-WignerMoyal theory, etc. (cf. [84, 85, 350, 438, 694] and Chapter 4). Further development in [159] includes some elaboration of the multisoliton situation of ( A M ) . One inserts e following [902] to obtain (AP) ~-~r ,,- 1-IIN[1+ (aj/e)X((T/E), s ~j)]. 1 where T ,,, (tn, n > 2). Subsequently one could carry all the On operators to the right to work on the Fock space vacuum 1 leading to terms _~
(T)~j, ~j) )~j - ~j J0e()~j)J0e(~j); [)e()k)=e (1/e)Ax-eOxX-- e (i/e)(/bQ-~/5) Aj
(2.64)
1.2.
VERTEX OPERATORS AND COHERENT STATES
35
which can be further expanded. One is thus given various /)(A) = exp(/ka t - A a ) and the insertion of e could be thought of as a way of introducing peaked states and coadjoint orbit variables to provide a geometrical background for dispersionless KP theory. One can imagine also a smoothing role for e in e.g. fluid dynamics analogous to the control of quantum fluctuations expressed via Q ~ v/-hq = eq ---. ~. R E M A R K 1.4. We want to indicate next a possible connection of the Maslov canonical operator (cf. [694]) with semiclassical soliton theory. Thus for )~ ~ 1/h large one writes (we discuss one dimension only) 1/2
c~
f_ e-iXp~u(x)dx;
(2.65)
(X)
1/2/_.o eiXp)'v(p)dp (X)
Then consider )~-pseudodifferential (psd) symbols of the type L(x,p,(i)~) -1) E CCC(Rz x Rp x R~-) obeying suitable estimates which we ignore here. To relate these to psd operators ~ _ ~ an(x)O n one would have to specify 0 -1 as an integral operator in some sense and then standard theories of psdo, Fourier integral operators, hyperfunctions, etc. would be applicable (we make no attempt here to provide details or hypotheses). We set D ,.~ (-iOn) and write L(x2, D)u = ~ p l z L ( x , p)JYx___,pU(X); L(lx , D)?.t - ~ p-1 ~ x ~ x ~ p L ( x , p)u(x)
(2.66)
These are typical psd operators based on the symbol L(x, p). The theory can be developed 1
in considerable generality based on A-psd operators of the form L(x2,A-1 D, (iA) -1) and L(1,)~ -1 /~, (i)Q -1) along with Lagrangian manifolds, bicharacteristics, Hamilton-Jacobi (H J) equations, etc. We only illustrate a simple case here. Namely, take e.g. i@t = -(h2/2m)~2xx + V(x)~2 with ~(x, 0) = ~o(x)exp[(i/h)So(x)]. Here for Dt = -iOt one has for A-1 ~ h, the relation L ( t , x , E , p ) = E + (1/2m)p 2 + V corresponding to L ( t , x , hDt, hDx). Look for a solution to Lu = 0 in the form u = exp(iAS(x)) ~ Cj(iA)-J. The leading terms give
L(t, s, St, Sx)r
1
= 0 =~ St + ~_ S 2 + Y ( x ) = 0 ~irt
(2.67)
This is the HJ equation whose solution is a classical action. To solve the corresponding Cauchy problem (data given on a curve x = x~ t = t~ one has recourse to classical methods of characteristic strips, etc. Then the resulting bicharacteristic system has the form dx _ OL p dt OL dp OL -V' (2.68) dt - 019 = m ; d T = OE = 1 ; d--~= Ox = ;
dE dT
OL = 0; xl~=0 = x~ Ot
;l~=0 = ;0(y)
and, since t = T is permissable, t[~=o = 0 with E[~=0 = E ~ = constant. Note E ~ + (1/2m)p2(y)+V(x~ = 0 is required, and along a bicharacteristic, dS - pdx (~ (dp/dt) - L x = - V ' ) . We ignore caustics so J(t, y) = d e t ( ~ ) ~ 0 locally and W 1 - - pdx -~- Edt is to be a closed form on the initial manifold A0 (x = x~ t =t~ Under a displacement along phase trajectories A0 ~ At and, with suitable hypotheses, locally, A 2 = UAt is a so-called Lagrange submanifold, which is embedded in R 4 where coordinates (t, x, E, p) =
36
C H A P T E R 1. Q U A N T I Z A T I O N A N D I N T E G R A B I L I T Y
~'(T, y) can be used. The quantum object L(x, A-1D) corresponds to the classical object A 2. The next term in the asymptotic expansion of u involves the determination of r and this will satisfy a certain transport equation (cf. [694]). It will turn out then that, given a diffeomorphic projection of A 2 on R2(x, t), the operator (2 ~ (x, t), A-1 ~ h) (KA~r
= ~~xp
can be defined on A 2 and [L(t,x, hDt, h D x ) K A 2 r
) = KA2[L(t,x,E,p)r ~ + ihRlr163
+ O(h 2)
(2.70)
where n l ~ Or - ( 1 / 2 ) [ ( 0 2 L / O t O E ) + (02L/OxOp)] arises from the transport equation. In shorthand notation one can write this equation (2.70) as (AQ) s ~ K ( L + ihR1) or K-Is ~ L which has a dressing flavor (here K ~ KA2 is a version of the Maslov canonical operator). Note s ~ A 2 --~ K is a passage from quantum --* classical --~ semiclassical objects. In fact K in (2.69) looks like a semiclassical wave function w ~ exp[(1/e)S] as in dispersionless KP for example (cf. [148, 149, 153, 158, 160] and Chapter 4) so (2.70) is Un+l O-n simply a refined version of Low --* Aw with A P + ~ Un+l p - n (for L ~ 0 + ~ - see below and Chapter 4). In any case there seems to be some motivation for the further study of the Maslov technique in dealing with quasiclassical soliton mathematics (cf. also [902]). 1.2.3
Connections
t o (X, r
duality
We refer here to the idea of (Z, r duality introduced in [317] and developed in various directions in [153, 169] (el. also [2, 937]). The approach in [153, 169] exploits a connection of the Schrhdinger equation to a dispersionless KdV (dKdV) situation which goes as follows. Consider a Schrhdinger equation
Ur
h2
= --Fmr
+ V(X)~E = E~E
(2.71)
where X is the QM space variable with r ~ OCE/OX. Write h / ~ - E now and note that (2.77) has a possible origin from a dKP type situation cx - Z with (AR)L2+ 02 - v(x, ti) with T2 = et2 and T2 = - i ~ T 2 so Or2 = COT2 = --ihOr2. Then if one can write v(x, ti) -- V ( X , Ti) + O(e) (see below) the KP theory gives (r ~ KP wave function) d e " - y ( x , T)r ~ ~ O72 =
0-~
(2.72)
Given r independent of T2 (as will be the situation for KdV) one could write ~ ~ ~E -exp(ET2/ih)r with ( A S ) ( e 2 0 2 - Y ) r = --ECE = 7-lr = ECE which is (2.77). For the approximation of potentials one assumes e.g. v(x, ti) ~ v ( X / e , Ti/e) + O(e). This is standard in dispersionless KP (dKP) and certainly realizable by quotients of homogeneous polynomials for example. In fact it is hardly a restriction since given e.g. F ( X ) = ~ anX n consider ](x, ti) = ao + ~ ( x n / 1 - I ~ +1 ti). Then
--e
= ao +
i_i~+ l Ti
= ~(X,T~)
(2.73)
1.2. VERTEX OPERATORS AND COHERENT STATES
37
and one can choose the Ti recursively so that 1/T1 = al, 1/TIT2 = a 2 , - ' - , leading to F(X) = F(X, Ti). In any event we have now another situation where a QM problem is related to a dispersionless soliton equation, namely dKdV. Now however e = l i / ~ instead of e2 = h = 27rh as in Sections 1.2.1 and 1.2.2. A priori there is no reason why KdV should be connected to QM except for the accidental occurance of a second order differential operator 02 - v common to both theories. Thus one can think here of imposing a related dKdV structure on the QM problem (in the spirit of a WKB theory) and there is a surprising usefulness of various dKdV constructions in dealing with the very quantum mechanical (X, r duality theory of [317]. This is perhaps connected to the fact that integrability ideas are very relevant and appropriate in discussing QM problems as indicated at length in other places in this book. We go now to the (X, r duality of [317] and developed in [153]. Philosophy is omitted at first and we simply list some of the equations. Thus, since E is real, ~E and @E -- ~D both satisfy (2.77) and one introduces a prepotential .T via cO = O.T/OCE. The Wronskian in (2.77) is taken to be W = ~ ' ~ } - r 2~/ih = 2/ie and one has (r - ~ ( X ) and X = Z ( r with Xr = OX/O~2 = 1/r
Y = r162 ~= =
1
1[
2
r 1 6 +2 ~-~; 0r = ~ ~ - ~ x ~
(r always means r but we omit the subscript occasionally for brevity). 0 . T / 0 ( r 2) = r 1 6 2 with 0r = 2 r 1 6 2 2) and evidently 0 r 1 6 2 = - ( ~ / 2 r one has a Legendre transform pair X
= r
O2=
(2.74) Setting r -
(2.75)
One obtains also (ib){r = 2 . T - (2X/ie) (.TO = ~); -(1/ie)Xo = ~2; .TOO = 00/0~. Further from X r 1 6 2= 1 one has X r 1 6 2 1+6 2X~r = 0 which implies
..T'r : E - V 4
(~:r162
E - V ( 2Xr ~ 3 4 ~ i~ /
(2.76)
In fact (2.76) corresponds exactly to the Gelfand-Dickey resolvant equation (cf. also Remark 2.9)
e2.T'" -}- 4(E - V) (.T" - I ) - 2V' (.T - X ) - 0 (since E - I r
(2.77)
= 2 . T - (2X/ie)- cf. [147, 153]).
We refer now to [158, 160, 422, 555, 902] for dispersionless KP (= dKP) and consider here r - exp[(1/e)S(Z, T, A)]. Thus P ~ S ' - Sx with p2 = Y - E and E - -t-A2 real will involve us in a KdV situation as indicated below. Some routine calculation yields (recall Xr = 1 / r and r (P/e)r
r
1 e-(2i/e)~3S; 1Xr -2 e - - i e (2/e)S; Xr -- Re -S/r
(2.78)
1 X r 1 6 2 E p-------w-e - V -s/c; .Tr = r = eglc ;
. T r 1 6 2 e_(2i/e)~S
2__. -2S/e ip e
(2.79)
CHAPTER1.
38 1r
QUANTIZATION AND I N T E G R A B I L I T Y
= e(2/~)~s; -c S = -~ 1 log lr 12 - -~1og 1 ( 2r ) ; P = Sx = P -
2 yawi "----r
(2.80)
Summarizing one has ~gr=
X
---; e
~9~=
1 1 [~s_ Ir 2 -~ = ~e
1 2~P
(2.81)
In the present situation the variables ( A T ) I r 2 - exp[(2/~)~s] and 2r = exp[-(2i/e)<3S] can play the roles of independent variables (cf. (2.80) and we see that (&)]~bl2.~P = - 1 while r162 = (1/2)1~bl2. Now note that for L - 0 + ~C~uiO-i L2+ = 02 + 2Ul; and Ul - 02log(T) where T is the famous tau function. This implies v = -2021og('r) here, from which Y = - 2 F x x for r = exp[(1/e2)F + O(1/e)] in the dispersionless theory (cf. [153]). Then writing out the Gelfand-Dickey resolvant equation (2.77) yields e25_,,, + ($. , - ~ 1 ) ( 8 F " + 4 E ) + 4 F ' " ( ~-_ -X )
=0
(2.82)
which provides a relation between F and $'. We will see below how to embellish all this with a new modification of the dKP and dKdV theory. Thus heuristically (cf. [153]): Under the hypotheses indicated of first order WKB type approximation, the equation (2.82) yields a relation between the prepotential ~ of (X, r duality defined via (2.74) and the free energy F of dispersionless theory. Although in the presence of a background hyperelliptic Riemann surface for a finite zone KdV theory, the dispersionless F corresponds to the Seiberg-Witten (SW) prepotential F(a, T) (cf. Chapter 5), it is hard to imagine any SW involvement here. The details of proof are in [153] and we recall here only a few facts. Dispersionless KP (dKP) is built up from ex = X = T1, eti = Ti (i >_ 2), and L = 0 + E ~ Un+l O-n --~ L~ = cO+ ~,r Un+l(e,T)(eO) -n where 0 H O/OX now and Un+l(e,T) --~ Un(T) + O(e). One sets (recall L r = ~ )
~b = exp ( I s ( T , 1)+ 0 ( 1 ) ) ; r = exp (-~F(T) + O ( 1 ) )
(2.83)
and the KP hierarchy OnL = [Bn, L], Bn = L~_, and L~b = X~b with the vertex operator equation (VOE) defined by the rule (AU) r = T ( T - (1/nln))exp[~C~ Tns lead to
(On = O/OTn, P = Sx) oo
oo
1 = P + ~ Un+Ip-n; P = ~ - ~ WiA-1; 1
1
i)nS = Bn(P) :=>OnP = OBn(P)
(2.84)
where c9 ~.. Ox + (OP/OX)Op. Further for Bn = ~r~ bnmOm one has Bn - ~'~ bump m. We list a few additional formulas which are easily obtained (cf. [153]); thus, writing {A, B} = OpAOA - OAOpB one has oo
0n)k-- {Bn, ~}; S = E
1
oo Tn,~ n + E
Sn+I - -(OnF/n); OmSn+1
:
1
SJ +l)~-j;
(Fmn/n);
1.2. VERTEX OPERATORS AND COHERENT STATES
oo ~n -- )~n ..~ E OnSj+l/~-j;
OSn+l
'~
1
39
OOnF
(2.85)
n
We omit the R i e m a n n surface and Seiberg-Witten aspects here (cf. C h a p t e r 5).
[148, 149, 153] and
We go now to [160, 422, 555] where there is a Hamilton-Jacobi (HJ) theory for KP with
P = Sx. In particular one can take (P, X, Tn) (for any fixed n _> 2) as Hamiltonian variables satisfying
dPn dX Pn -- dTn = OBn; f~n = dTn = --OpBn
(2.86)
Next from (2.79) one has ~ = - Z / e and Ir = exp[(2/e)~S]. In order to have I~12 < 1 as a fundamental variable in the (X, r theory some control over ~ S is needed. One notes that d K d V involves A = ~ik = P(1 + ~ UmP -2m) with A2 = _ k 2 real for k real (cf. [153] for details). In order to satisfy ]r = - 1 we will want A = - i k so t h a t Q = ~ P is negative. The Um are real so P = iQ corresponds to k real and (-ik)2+n+l = B2n+l will be purely imaginary (only "times" t2n+l arise in KdV). Further 0pB2n+l will be real so the equations (2.86) make sense. In addition ~ S = ~ [ ~ T 2 n + l ( - i k ) 2n+l + ~ Sj+l(-ik) -J] = 0 for k real since Sj+I = - O j F / j with 02nF = 0. Now this would imply I~12 - 1, which is absurd, unless we introduce an expansion S = ~ eJSJ (i.e. take into account the O(1) terms in r = exp[(1/e)S + O(1)]). Such O(1) terms arise quite naturally from the VOE via logr = (S/e)+O(1) = log'r[e, T n - ( e / n ) ~ n ) ] - l o g T + ~ Tn)~n/~ with log'r = (F/~2)+O(1/~) and Sn~ = -(OnF/n). Thus e.g.
FT~--~ -- - e ~ l
(~)
~ ) - F(Tn) =
(2.87)
+ -~ e2 E \(Fmn) )~-m-n+O(
This leads to first terms of the form S - S O + eS 1 with
sll
( nm/
;Sx~P+.~-~.-~e ( Y l m n )
(2.88)
Eventually we will want also F - ~ ej F j as well. W i t h this development one finds as a first approximation
Ir
= r 2~S1 :
exp ~ E ~ mn ]
where only terms Y~m+l,2n+ 0 1~-2(m+n)-2 arise and these will be real for k real. Thus S o and p0 _ S ~ __ iQ are imaginary while S 1 and p1 = S~( are real. Before embarking on a full expansion we note from (2.81) that~ in a first order expansion ~ ( p 0 + ep1) = Q which implies P = iQ = - i / 2 ~ . ~ and X/e = -~.~. Hence a possibly fundamental symplectic form in the (X, r theory arises from
dX
A dR ~
i
2(~7) 2
d(~9~ A d(~9~)
(2.90)
In [153] by way of motivation one examines various partial expansions for S and F but we go here to the general case. Thus take a finite zone KdV situation and go to the Schr6dinger equation e2r " + 2 F " r = - E r directly with S = ~ ej Sj and F = ~ ~JFj. Then
40
C H A P T E R 1. Q U A N T I Z A T I O N A N D I N T E G R A B I L I T Y
write t5 = S x = E eJPj and for r = exp[(1/e)S ~ + E ~ eJ-lSJ] one obtains c o x - PO with e2r = e P x ~ +/52~ leading to
0
0
Consequently we have equations (ibm) Pg+2F[~' - - E ; 2PoPI+P~+2F~' - 0; P~+2PoP2+ P~ + 2F~'= 0;... and in general for k >__1 e~k
PiP2k-i + P2k-1 + 2F;~ = 0;
+ 2~
E2k+l
0
[k
(2.92)
]
t! 2 ~ PiP2k+l-i + P~k + 2F2k+l = 0
0
Expanding (2.87) now yields F ( Tn
[
= Ee k -e k=O
and for (AV) S ~ ~ 1 E
0
n~e n ) - F ( T n ) =
\n)~n
1
+~E
(2.93)
l
mn
Tn)~n + y~c~ s ~-~c~z Sjkl/~-j with S O ~ ~ ek -
A-n + 2 ~
1
OO
Tn)~n + }-~c S j % l / ~ - j + O(e2)
\ nm
(3~3
-" --l e e k s 0
E
S jk+ 1 )~-j
1
--
(2.94)
1
There will be further terms -(e2/3!) E(Fkmnp/mnp))~ - m - n - p + . . . in [ ] but the extra terms present no problem conceptually. This leads to the equations S)~ - - F ~ S~ = - F ~ , and sk
j + l ~-~
Fk
1
~
t"~
(Fkl ) m,(j-m)
~yt~ ~ ~ - )
"}- " " " -~-
+ ( - 1 ) k+l ~ F~ ml,.--.,mk__+l (k + 1)! ~__~mZ~/=j YI1k + l m i
(2.95)
One recalls now the derivation of the dispersionless Hirota equations in [158, 168] via the differential Fay identity developed as in [902]. Thus we will expand the formula 1 e-~ E
m,n--1
~-m~-~(-~0)~m(-~0)F=
(2.96)
=log [ 1 + - 1 ~ # -~s - ) ~~- s ~t-~9)axY 1 s
where Xn(t) is the Schur polynomial
1
(Ejkj
[tlkl t~2
XnIt) = ~ [~i k~ '
= k)
]
1
1
; ~ = {01, ~0~, 5O3,...)
(2.97)
1.2. VERTEX OPERATORS AND COHERENT STATES
41
in powers of e (cf. here [153, 169] and remarks below for more details on this). The result should provide all compatibility relations between the coefficients F~b...q at once and we will see how this goes below. Actually one can arrive at many different formulations of dispersionless Hirota equations for F ~ (and of enhanced equations for the F k) by working with any of the dKdV hierarchy equations, upon stipulating the canonical forms for S ~ P0 (and S j, Pj). In using the KdV background via L2r =/~2r we are in fact essentially defining consistency relations between the F~kb...qand thus the relations obtained from (2.92) should at least be a subset of those obtained in (2.96) (although perhaps expressed differently). We should include all possible F~ab...qk of course and here F2nl,2m -- 0 is entirely natural (with F~ "-- 0 mandatory since we will preserve the dKdV origins in the F ~ terms). Now in order to maintain S 2n imaginary with S 2n+1 real for real k we could adopt a strategy based on KP1. Thus introduce imaginary T2n = iT2n and think of F 2j+l - i/~2j+l as imaginary such that l~2j+l *2n+l is imaginary and .~2j+12n is real. This can be achieved e.g. via power series
F 2j+1 ~ i E aaT~ 1T2~2"'" with as real. In this context V ~ V = - 2 E ~ J F f l could be of course be complex which would interfere with the introduction of (X, ~b) duality; however we will see that the consistency calculations indicate F 2j+l = 0. Now look at the compatibility conditions arising from (2.92) (we assume first F 2j+l = 0 along with F2~m = 0). The first terms are (P2k = iP2k)
_Q2 + 2FOx + E = O; 2QPI + Q' = O; P{ + P21 - 2Q152 + 2F~cx = 0;
(2.98)
b~ + 2QP3 + 2PIP2 = o; P~ - / 5 { _ 2Q/54 + 2PiP3 + 2F4x = 0; ... and look at the d S d V situation (e --, 0) in (2.96). Writing/5 ~ t5o = dPo/dA one obtains
(cf. [15s])
log P = log 1 +
1
F jX-J -1
= Z x-j 2
Z
m+n=j m n
(2.99)
whereas from (2.95) (P1 = OS 1) $1
j+l=
1~-~ F~ m(j-m)
p1
=
1 cc 2
Z
m+n=j
F~ mn
(2.100)
Now the first question arises about compatibility in 2QP1 + Q' = 0 or equivalently 2PoP1 + Pg = 0 from (2.98). One can write ()~ ~ -ik)
iQ=P0=a
El01, )~
1-
j
( 2 m - 3 ) ( 2 m - 5 ) . . . I(F~ 2
(2.101)
TM
m[)~ 2m-1
Now P1 = -P~/2Po and P0 = iQ = A[1- (2F~ 1/2 which implies that P~ = -(F~ and hence P1 = F~ From (2.99) and the definition of/~ in (2.100) we see that (1/2)0 logPo =/5~/2/5~ = P1 and hence the compatibility of the (2.100) definition of P1 with 2POP1 + P~ = 0 requires ( A W ) / ~ / 2 P 0 = F~ To see that this is true we write from P~ = -F~ the equation/5~ = (FO11/P~)po so in fact (AW) holds. Actually we have accomplished more, namely we seem to obtain the dispersionless equations of [158, 168] by using the coefficient formulas for P0 and P1 in terms of F~ and the relation 2PoP1 + P~ = 0.
CHAPTER 1. Q UANTIZATION AND I N T E G R A B I L I T Y
42
This is then apparently an example of a derivation of the dispersionless Hirota equations starting from a hierarchy equation and stipulating the canonical form of S and P in terms of F ~ F ~ In the present situation we can produce such equations for a whole hierarchy F j at once, using F = ~_, eJFJ and the machinery of [158, 168, 902] involving the differential Fay identity (with suitable insertion of e). One can continue this process and the next stage from (2.98) (to be confirmed via (2.96)) involves the creation of P2 via 2PoP2 + P~ + P{ + 2F~1 = 0 and the compatibility equation P~ +2(PoP3 + P1P2) = 0 involving P3. We must have F31 = 0 here (which seems to imply in fact that F 3 = 0) since exp[(2/e)~S]~P = - 1 implies that oo
(2(3
2 E ~2jP2j +1 0
co
-~- [ ( E (~2j p~j) / 0
( E s p2j )1
(2.102)
-- 0
0
leading to 2PoP1 + P~ = 0, 2(PoP3 + P1P2) + P~ - 0,.... Thus our odd compatibility equations from (2.98) are exactly the conditions required for (X, r duality (note that (2.102) yields the second formula in (2.92) with *11 Ib-'2j+l = 0), and below for still another point of view. Consider now (2.96) and let us try to write this in a form analogous to (2.99). Thus recall from [158] that one used oo 1
1-
Yl~
=
n
~1 # -n - A-n F~
#-A
P0(a)
_ Po(#)-
n
#-A
(2.103)
Po(A) --+ Po(A)
as # ---, A. In the present situation we recall also that S = S O+ ~_c~ s ' sO __ ~--~TnAn 't" E ~ Sj+IA-J, 0 k -j . The differential Fay identity arises from the Hirota and sk = ~ Sj+IA bilinear identity, using the vertex operator equation (VOE) (AU). Let us write now co A - ~(A, s
= _!
E
E
(2.104)
Xs(-s176 1
so that the right side of (2.96) involves 1 oo # - A + - E (p-s _ A-s) Xs(-eO)F1 = -Z(#, e ) - E(A, e)
(2.105)
1
Then (2.96) becomes for # ~ A 1 5--5E A-m-nx.n(--s163
-- log-'~(A, e)
(2.106)
1
where 1 o0
E(A, e ) = 1 + - y ~ . ( - s E
1~r +c
)A-s-1
([
(2.107)
-- &-Jr-
~]
+
1
Xs(--s
1
o0 E
Yl~ +
)
1
In any event from (2.106) we have 1
~
1
.~I
(2.108)
1.2. VERTEX OPERATORS AND COHERENT STATES
Z~-~-nx~"Yl
•
1
( it 1-
-
43
~-~-I~Yl
e
)
-
(3O
1 E 8)~-s-ixsY11 e 1
(cf. [158, 160] for more on Schur polynomials), and repeat (2.91) in the form eP'+/52+2/~li = - E = A2 with t5 = E ~ eJPj, so t h a t
e00~P + 2PO~P = 2A =~ Po~6o = 2A; J fg;_ 1 "3t- 2 E
PifgJ - i
=
(2.109)
0
0
J FJab...q equations obtained from (2.108) and (2.109) are equivalent.
We suggest now t h a t the
More explicitly we want to show t h a t E of (2.104) corresponds t o / 5 and secondly t h a t (2.108) is equivalent to (2.91) - (2.92) and (2.109). We will sketch a proof. First one notes the fit at the first stages since O = (1/e 2) E / ~ - m - n x m ) ( , n ~ ' l begins with E ~ A-m-n(F~ ~ 25 and 1 - ( l / e ) ~ sA-s-lxsF1 begins with 1 + ~ 1 A-S-iF~ ~ /60. Hence the first term in (2.108) in an e expansion is 2Pllbo =/5~ (known to be true) while/5~ + 2(Po/51 + p1/bo) _ 0 corresponds then to 4PIPo + 2Po/bl = 0 with Po = A[1 - (2F~ 1/2 and P1 = F~ Thus /51 = -2Pl_f='o/Po = -F~ 3 which agrees with /51 = O~(F~ Let us examine the next terms now. We will assume F 2j+l - 0 with F~2j = 0 for convenience (and eventually by necessity). First let us recast the desiderata in simpler terms.
exp(E~ tnAn) = E ~ xj(t)M with X0 = 1, it follows that
In this direction, since cr
cxz
cr
exp(y~ 2tnAn) = ~ X~(2t)A ~ = ~ Ag ~ Xm(t)Xe-m(t) 1
0
0
g-I
g
=~ Xe(2t) =
(2.110)
0
~-~Xm(t)Xe-m(t) = 2 x g ( t ) + ~ Xm(t)Xe-m(t) 0
1
Hence in (2.96) we can write the left side as 1 ~
62 E
-m-n
- _.
1
1 cr
__
2 1
(2.111)
1
oo
= j ~ ~,-~k~(-2~,~) - 2z~(-~,~)]P 2
T h e n (2.108) becomes
]
2
x[11 .
sA_SXs(_e ~. . .
. 1
z
1 ~ s
1
8A-s-lxs(_s
(2.112)
CHAPTER 1. Q UANTIZATION AND I N T E G R A B I L I T Y
44
There is still a lot of calculation so consider an argument based on [158] and (2.105)- (2.106) due to J.H. Chang. First to prove /5 = E observe that via e x p [ - ~ ( e O n / n ) ~ n ) ] F ( T ) F ( T n - (e/n,~n)) one has
S = ~ Tn)~n + 1
1[ e_
(
F
Tn-
( e )) -~
-F(Tn
)]
=
(2.113)
= E Tn)~n + _ E )~-j xj ( - c o ) 1 1
leading to (AX) 15 __ A + ~1 E F )~-Jxj ( - e 0 ) F 1 which is exactly ~ from (2.104). In passing Chang also noted that from (2.106) one can extend a formula of [158] to obtain oo
= ~ x~(Z1 = 0, Z 2 , ' " , Z~))~-~;
(2.114)
0
2j= E ~(-~)~m(-~)j m+n=j
This corresponds to (2.108) in an integrated form and we then ask if this gives the same equations as (2.92) - (2.93) and (2.109). For this we recall that the differential Fay identity is equivalent to the Hirota bilinear identity in characterizing the KP (or KdV) hierarchy equations, so equations arising from L2r = A2r must be compatible with equations generated in (2.109). Whether all such equations arise via (2.92) - (2.93), (2.109) is actually immaterial for the connection to (X, ~) duality, etc. but in any case numerous explicit calculations also show agreement at the early stages and provide expressions for the F2mn in terms of F121. F~ and derivatives theoreof. Finally observe that (2.114) provides coefficient formulas via E o
x~(Z1 = O, Z 2 , " " ", Zl))~-~ = ~ = 1 - - ~-~(t~ - 1)A-~Xg_I/~I s 2 1
-
-
-- X•(21 -- 0, Z 2 , . . . , Ze) (2.115) e for g > 2, and this can be broken down into e series to produce coefficient formulas. Summarizing one can say that the procedure indicated is formally consistent and allows determination of the Pn and F 2n from Q (and the F12P). The ensuing compatibility equations for the Fab... 2n q extend the dispersionless Hirota equations of [158, 168] and are included in those derived from the differential Fay identity via (2.108), (2.112), or (2.115). The adaption to (X, r duality via 1r = - 1 (and hence to the third order differential equation (2.82) and to the derivations of enhanced dKdV in [153]) is automatic in this context (cf. Remark 23+1 2.9) and corresponds to the conditions F{I = 0 (which seems to also imply F 2j+1 = 0). ==~ - - - ( •
-- 1 ) X e - 1 ( - e / ) ) F 1
R E M A R K 1.5. Let us call attention here also (following [728]) to a standard WKB approach for r + ( 2 m / h 2 ) ( E - Y)r = 0 (this is (2.71). One puts r - exp(iW/h) with W = s + (h/i)log A (where s and A are even functions of h) to obtain 2 A/I (s') 2 - 2 m ( E - Y) = h ---~; 2A's' + A s " = 0
(2.116)
From the second equation A = c(sP)-1/2 and the first equation becomes (#)2 = 2 m ( E - V ) +
h2
3
(#-]
2
s,,, 7
- 7J
(2.117)
1.3. R E M A R K S ON THE OLAVO T H E O R Y
45
This equation is now equivalent to the Schrhdinger equation and the last term (Schwartzian derivative) is said to be the quantum correction in [346] (this is of course D. Bohm's quantum potential). Note exp[(i/n)(s + (h/i)log A)] = Aexp(is/h) so one is extracting an amplitude factor A. In our notation we have r = exp[S/e] with S = Z ~ E2j+Is2j+I + ~-~ 62js2j where S 2j+1 is real and S 2j is imaginary (for k real, A = - i k ) so that ~ S Sodd and i.,~S = Seven. Consequently from S/e ,,~ iW/li we have (1/e)(~S + i~S) ~ (is~h) + log A so s ~ ~,~S and log A ,,~ (1/e)Sodd. Writing 15 = ~ , = ~ , + i ~ S ' we have s ' ~ ~ ~ / 5 (or is',,~ ~ D e v e n ) and A ' / A ,,~ (1/e)[='odd. Hence A = c(,~/5) -1/2 is automatic or r = A 2 = c2/.,~[9 in agreement with 1r = - 1 for c = i (the i arises from ,~/5 < 0 as in the text). This indicates that the condition 1r = - 1 is better viewed as a standard consequence of using the WKB formulation (hence no constraint) rather that as an adaption to (X, r duality. The quantum correction term in (2.117), expressed by the Schwartzian - ( I i 2 / 2 ) S C H ( s ') is interesting in itself since the Schwartzian appears in many contexts involving KdV and/or the Schrhdinger equation (cf. [315]). Here we can write this as =
(~'~ 2~/~.~P)
S C H ( s ' ) = .~/5
2
-~]
(2.118)
One is reminded of the third order equation (2.77) but (2.117) seems conceptually different (cf. cca). Actually from [346] s' satisfies (2.117) and 1Is' satisfies the GD resonant equation leading to (2.116).
1.3
REMARKS
ON THE OLAVO THEORY
Notation such as ( B A ) will be used. As indicated in [153] there is an apparent connection between (X, r duality and the theory of Olavo [781] which derives quantum mechanics (QM) via the density matrix and statistical mechanics using an infinitesimal Wigner-Moyal (WM) transformation ((3.1) below). We extract here from [169] some ideas leading to the Schrhdinger equation (SE) following [781] and indicate some possible directions for development in connection with (X, r duality. The physics appears to be plausible enough and in any case represents a refreshing attempt to deal with epistemology (cf. [192, 475, 815]); the mathematics seems perfectly correct in the sense normally acceptable in physics. In any case, given our instinctive reluctance to deal with interpretive questions in QM we feel the present sketch is appropriate and hopefully it is faithfully rendered. The first derivation of (SE) (Article 1 in [781]) goes as follows. There are three axioms: 9 (A) Newtonian mechanics is valid for all particles which constitute the systems in the "ensemble" 9 (B)
For an isolated system the joint probability density function is conserved, i.e.
(d/dt)F(x, p, t) = 0 (we relace X by x in this section) 9 (C) The Wigner-Moyal infinitesimal transformation defined by (3.1) below is adequate for the description of any non-relativistic quantum system (note the emphasis here on the infinitesimal aspect which seems to be a new idea)
(
,x
p x + -~,x-
,x) /
-~,t
=
F(x,p,t)exp ~ - - ~
dp
(3.1)
Then from (B) one has dF/dt --- OF/Ot + (dx/dt)(OF/Ox)+ (dp/dt)(OF/Op) and using dx/dt = p/m with dp/dt = f = -OV/Ox~ upon multiplying the resulting equation by
C H A P T E R 1. Q U A N T I Z A T I O N A N D I N T E G R A B I L I T Y
46
exp(iphx/h) and integrating there results (BA) - O p / O t + (ili/m)O2p/OxO(hx) - ( i / h ) h V p 0 where one writes (OV/Ox)hx = 5V(x) = V ( x + ( 5 x / 2 ) ) - V ( x - (5x/2)) and uses the fact that [F(x,p,t)exp(iphx/h)]~_oo = 0. Changing variables via y - x + (5x/2) and y' = x - (5x/2) ( B A ) can be rewritten as
~
oy2
o(y,) 2 - [v(y) - v(y')]
p(y,y',t) = -i~Np(y,
,t)
(3.2)
which is called SchrSdinger's first equation for the density function. Now one assumes (D) p(y, y', t) = r
t)r
t); (E) r
t) = A(y, t)exp (is(y, t)/h)
(3.3)
and expands p in (D) in terms of 5x to get two equations (F) Ox/Ot + (O/Ox)[x(s'/m)] = 0;
(3.4)
(G) (O/Ox)[(1/2m)(s') 2 + V + O s / O t - (h2/2mA)(O2A/Ox2)] = 0 where we have changed notation a bit in using X = A2 = lira p[x + ( h x / 2 ) , x - (5x/2)] as 5x --~ O. Here X is the standard probability density in configuration space. Now (G) can be rewritten as b f ( B B ) (1/2m)(Os/Ox) 2 + V + (Os/Ot) - (h2/2mA)(O2A/Ox ~) - c = 0 which is (SE) (h2/2m)O2r 2 - V(x)~b - - i h O r (if ~b - Aexp(is/~) is substituted in (SE) one obtains (G)). In summary: If one can write (D) then ~b given by (E) satisfies (SE) along with the equation of continuity (F). With suitable definitions of operators etc. (SE) has the operator form/2ir = ih(O/Ot)~2. Note that one has now [~, 15] = ih and it can be shown that in fact (BC) A x A p >__ h/2 (Heisenberg uncertainty) as a consequence of writing the density function in the product form (D). In this spirit, instead of representing a fundamental property of nature the uncertainty principle simply represents a limitation of the description based on (SE) (i.e. quantum mechanics as developed above is only applicable to problems where the density function can be decomposed as in (D)). Further one remarks that the dispersion relations do not impose any constraint upon the behavior of nature but only upon our capacity to describe nature by means of quantum theory so defined. If e.g. A q A p < h/2 in some situation then this quantum theory does not apply or will not give good results. Since the hypotheses (D) leads to possibly negative values of F (cf. (3.5) below) a different construction was developed in Article 16 of [781] to produce a (SE) with (D) replaced by (BD) F ( x , p , t ) = r t)r p, t) where r is a probability amplitude, on phase space. Some problems indicated in [192] are dismissed via the distinction that operators i1,~ x and i5/~ -ihO/O(hx) apply when working on the density function. The uncertainty principle (BC) is not developed in Article 16 but the same kind of reasoning would assert that the QM determined by the (SE) derived there (see below) holds when F - r162 and not necessarily in other situations. Returning to (BB) one can consider this as a Hamilton Jacobi (HJ) equation with an effective potential Veyy - Y(x) - (h2/2mA)O2A/Ox 2 (the last term being D. Bohm's quantum potential - cf. [475]). Thus formally with initial condition p = s x one can write [9 = -OxV~yy(x) = - O x [ Y - (h2/2mA)A"]. The integration of this with p = sx will give a series of trajectories equivalent to the force lines associated to Veyy. The resolution method goes as follows: First (SE) must be solved in order to obtain the "probability amplitudes" referring to the ensemble. Then the effective potential, which will act as a statistical field for the ensemble is constructed. This potential should not be considered as a real potential
1.3. R E M A R K S ON THE OLAVO THEORY
47
but a ficticious one which acts as a field in reproducing through trajectories the statistical results of the original equation B B ) . Now following Article 16 of [781] one can resolve the problem of producing positive definite phase space densities. The possibly negative values arising from a formal inversion x
5x ~-) exp (_iPhX - - ~ ) d(hx)
(3.5)
can be attributed to the stipulation (D) and the theory is modified as follows. One assumes ( B D ) above so F is positive definite, and defines p = ZQ as before by (3.1). Also one defines also a characteristic amplitude ~(x', x, t) : f eipx'/hr p, t)dp so that (BE) ZQ(x, 5x, t) = f ~*(x',x,t)~(x' + 5x, x, t)dx' which is a convolution. The characteristic amplitude can be negative without violating any physical principles and we note that ( B E ) can also be written as (BF) ZQ(x, hx, t) = f ~*(x',x,t)ei~X#/h~(x',x,t)dx' with/5' = -ihO/Ox'. Putting this ZQ into ( B A ) one obtains an equation for ~(x',x,t) and writing ~(x',x,t) = ~b(x',t)r leads to
Ze(x, hx, t) = r
t)r
t) f r
t)exp (hx[O/Ox']) r
t)dx'
(3.6)
From such formulas one obtains a Schrhdinger equation ( B G ) (~2/2m)r t)+V(x)r t) = Er with i5 = -ihO/Ox where E ~ E = f f[(p2/2m) + V(x)]F(x,p,t)dxdp (cf. (2.71)). Further r t)r t) = f eiPX'/hr p, t)dp which implies (note factors of 2~ are occasionally missing) r p, t) = r t)r t); r t) = f e-iPX'/hr t)dx' leading to F(x, p, t) = ]r162 2 which is a natural probability density here (see [192, 781] for more discussion). Note that the operator constructions here are somewhat different from those of Article 1 for example. A thermodynamic derivation of (SE) is given in Article 15 of [781] (cf. also Article 11). The main idea is to derive the classical (SE) of quantum mechanics (QM) from classical ideas and in particular to show the thermodynamic character of the quantum formalism. Thus one begins with the Liouville equation in the form (F = F(x,p,t))
OtF + P m OxF- OxVapF = dF d--7 = 0; / / Fdp = p(x, t); /_~ pFdp = p(x, t)p(x, t) O0
(3.7)
O0
where p(x, t) is the so called macroscopic momentum. This second equation is a definition which can also be phrased as f pFdp = p < p > with < p > = p(x, t). Then Otp(x, t ) + (1/rn)Oz~J(x, t)p(x, t)] = 0. Note x,p, t are independent variables and f _ ~ Fpdp = F[~_~ = 0 for "reasonable" distributions. Also f_~pFzdp = Oz f _ ~ p F d p = Oz,(x, t)p(x, t)]. Next multiply the Liouville equation by p and integrate using f_~p2F(x,p, t)dp = M2(x, t) to get 1 0 [M2 - p2(x, t)p] + p Otp(x, t) + Ox \
m
- -
X
2m
+ OxV - 0
(3.8)
The first term involves M 2 - p2(x,t)p = f ~ 9 2 - p2(x,t)]Fdp = f ~ - p(x,t)]2Fdp since f p(x, t)Fpdp = p2(x, t). Thus one writes (hp)2p(x, t) = f~9 - p(x, t)]2Fdp = M2 - p2(x, t)p and looks for a functional expression for (hp) 2. In this direction, given ~t(x, t) ~ system accessible states for x in the range [x, x + 5x] and S = k log~(x, t) the entropy, the equal a priori probability postulate implies p(x,t) c< ~(x,t) = exp[(1/k)S(x,t)] (cf. [815]). Let Seq(X, t) correspond to the thermodynamic equilibrium configuration entropy (hx = 0) where S is a maximum with (OSeq/OX)]~z=o = 0 and let Peq be the corresponding density. Some
C H A P T E R 1. Q U A N T I Z A T I O N A N D I N T E G R A B I L I T Y
48
argument leads to p(x, Sx, t) = peqexp[-v(hx) 2] (where V = (1/2k)[O2Seq/OX2[) and then the mean quadratic dispersion (hx) 2 associated with such fluctuations 8p is given by (Sx) 2 = f (hx)2exp[-v(hx)2]d(hx) = 1 f exp[-v(hx)2]d(Sx) 27
(3.9)
A priori there is no relation between such fluctuations and those related to momenta but one can impose the restriction that in the thermodynamic equilibrium situation one should have ( B H ) (hp)2(hx) 2 = h2/4. This is not clear but it implies (for p = exp(S/k)) (BI) p(hp) 2 = -(h2p/4)O21ogp. Writing now p = A2; p(x, t) = Os/Ox = s' one puts (BI) into (3.8) to get as
A2(x,t)O~
1 (as) 2
N + ~-~ N
+ V(x)-
h2
2mA 02A = 0
(3.10)
Then as indicated above, this equation plus pt + (1/m)Ox~gp] - 0 is equivalent to the (SE) - ( h 2 / 2 m ) r + V ( x ) r = ihCt; ~b = Aexp(is(x, t)/h). One notes that the condition p(x, t) = Ozs is in fact some sort of restriction - i.e. p is a gradient of some function s. Thus three axioms are involved, namely: 9 (H) Newtonian particle mechanics is valid for every individual system composing an ensemble 9 (I) The Liouville equation is valid for the description of the ensemble behavior 9 (J) In a thermodynamic equilibrium situation one should have the restriction ( B H ) . In addition, the restriction represented by p = A 2, p = s I must be applicable at this equilibrium situation. One can also make a direct comparison of formulas here with those of Article 1 of [781] and it is shown that (BI) can be derived from first principles without using (D) or ( B H ) (perhaps lending credence to (BH)). Finally the first two postulates (H) and (I) above are the same as the postulates (A) and (B) in the Article 1 of [781] so. regarding the emergence of the ($E), the hypothesis (J) above, or simply (BI), is formally equivalent to assuming the validity of the WM transformation together with the hypothesis that p ~ ZQ in (3.1) can be written as ZQ(x, hx, t) = r (which is (D)). Alternatively we can also say that (J), or (BI), is equivalent to F = r162 as in ( B D ) , since one arrives at the (SE) in both cases. In [153], based on Articles 1-15 of [781] we suggested a formula 1 fi'=~r
(
X+~
5X)
~
(
X-
~_)
X 1 ( 8X + i--~=2P X + - - ~ - , X -
5X2)
X
+--
(3.11)
which can be expressed following Article 16 of [781] as (BJ) fi" = (1/2)ZQ(x, 8x, t) + (X/ie) with ZQ ,.~ f F(x,p,t)exp(iphx/h)dp. We recall that in [317] the equation ]~12 = 29r (2X/ie) of (X, r duality is interperted as describing the space variable as a macroscopic thermodynamic quantity with the microscopic information encoded in the prepotential. Then QM can be reformulated in terms of (X, ~) duality with the (SE) replaced by the third order equation (2.82) for example. Here h can be considered as the scale of the statistical system (cf. [115, 317]). These comments from [317] seem completely adaptable to a connection such as (3.11) with the theory of [781]. Thus suppose we define fi" as in (3.11) and use ( D ) - (E) so that X = A2 = lim p[X + ( h X / 2 ) , X - (8X/2)] as 5X ~ O, leading to (SE). Then fi" ~ 9r = (1/2)X + (X/ie) with the mixing equation ( B A ) in the background.
1.4. T R A J E C T O R Y R E P R E S E N T A T I O N S
49
Further if the t dependence is restricted to t - t2 of the form e x p ( - i E t / h ) for suitable t then the (SE) has the form (2.71). Modifying this now in the light of Article 16 of [782], one can set .T"- (1/2)ZQ + (X/ie) and again ZQ ---, 1r t) as 6x --~ O. Regarding thermodynamic behavior now, the idea is to let the systems ( S Y ) composing an ensemble interact with a neighborhood (O) called the heat bath. The interaction is considered sufficiently feeble so as to allow one to write a Hamiltonian H(q, p) for S Y not depending on the degrees of freedom of (O). The system O is necessary only as a means of imposing its temperature T upon S Y . Now in a state of equilibrium there is a canonical probability distribution F(q,p) - C e x p ( - 2 ~ H ( q , p ) ) where 2/3 = 1 / K B T with KB being the Boltzmann constant, T the absolute temperature, and C some normalization constant. The Hamiltonian may be written H(q,p) = ~ g ( p 2 / 2 m n ) + V ( q l , " " ,qN) where N K B T represents the energy of the reservoir (O). Using the Wigner-Moyal transformation we get a formula which yields (BK) p = Clexp[-23Y(q)] .exp[- ~(mn/43h2)(Sqn)2]. Some analysis (using (D)) leads to
r
t) = v ~ 3 e x p [ - ~ Y ( q ) ] . exp[-iEt/h];
(3.12)
p = C 3 e x p { - 2 Z [ V ( q ) + (1/8)Z(Sqn)2(O2V/Oq2n)]} and taking C1 = C3 = 0 one has (BL) Pea = e x p ( - 2 3 Y ( q ) ) with p = Peq(q + (8q/2)) = Peq(q - (6q/2)) (Taylor expansion of V). Then - ~N(Ii2/2mn)(O2r + V r = E r with E - V(q ~ + N K B T where q0 represents the mechanical equilibrium point. One can now establish (motivated by (D)) a connection between the microscopic entities of the quantum formalism and the macroscopic description given by thermodynamics (we remark that this connection should also prevail locally in various situations without recourse to (D)). Thus define the free energy FG(q) = Y(q) such that FG = - K B T l o g ( r 1 6 2 Writing entropy as 8 = KBlog(r162 we have FG = - T 8 and it must be emphasized here that one is dealing with a "local" entropy of 8(q) as spelled out earlier via ~t(x, t), (x ~ q); it is this locality which suggests an adaption to (X, ~) duality. Thus if we now express ~ as exp(S/e~) as in [153] one has X = A2 = Ir = exp(2~S/c) which implies log A 2 = 2 ~ S / ~ ,~ S / K B or 8 ~ 2 K B ~ S / c . A priori there seems to be no reason not to think of K B ~ S / E as an entropy term, given that Ir refers to a statistical system, and consequently (recall from [153, 317] that - ( X / i e ) = ~ 2 ( 0 9 ~ / 0 ( ~ 2 ) - ~ or equivalently - 9 ~ = r 1 6 2 - (X/ie) for r = 07/0(r
1
1
--'-~exp
1~
~-r
(3.13)
The first equation of (3.13) then would represent a macroscopic formula for 9~, relative to the statistical system involving r but at this point we suggest the analogies here only in a most heuristic spirit.
1.4
TRAJECTORY
REPRESENTATIONS
Notations such as ( C A ) and ( D A ) are used. The idea of a deterministic formulation for QM has a long history, going back at least to deBroglie and D. Bohm, and we will not try to review this background (see e.g. [475, 457] for a review and [107, 108, 109, 110, 111. 112, 113, 114, 457] for some of Bohm's work). The Bohm theory is extensive and profound but seems to have been considerably improved and probably superceded technically by the more recent work of Bertoldi, Faraggi, and Matone (cf. [90, 315, 316, 700] - see also some
CHAPTER1.
50
QUANTIZATION AND I N T E G R A B I L I T Y
related work involving time and classical limits in [155, 345, 346, 347, 348] and some general discussion also appears in books such as [62, 205]). We will disregard philosophical matters such as the implicate order of Bohm etc. and deal only with equations here. 1.4.1
The Faraggi-Matone
theory
In [315, 316] Faraggi and Matone (FM) launched a new era in the study of QM from a deterministic (trajectory) point of view (cf. also [90]). This seems destined to be the logical successor to the Bohm theory which it transcends in several important aspects. It does not require any mysticism regarding measurement and is compatible with the Schrhdinger representation via wave functions ~. We refer to [90, 315, 316] for a full exposition and only report here on some highlights connected with actually using the theory. The heart of the matter is an equivalence principle according to which all potentials are equivalent under coordinate transformations. This principle alone leads to a quantum stationary HamiltonJacobi equation (QSHJE) with Planck constant h as covariantizing parameter. Thus one is reminded of the Einstein equivalence principle in relativity and the level of penentration seems similar. We will be very proletarian here, even brutal, in describing the equations involved for l-D, and refer to [90, 315, 316] for philosophy, elegance, and other gospel. Thus first the classical HJ equation can be obtained from (CA) q = OH/Op and i5 -OH/Oq with transformations q ~ Q(q,p,t) and p ~ P(q,p,t) satisfying ( C B ) ( ~ OTI/OP a n d / 5 = - O ~ / O Q along with P(7- H - P ( ~ - ~ + (dF/dt) for some generating function F depending on t and any of the pairs (q, Q), (q, P), (p, Q), or (p, P) considered as independent variables. Choosing (q, Q) yields immediately p = OF/Oq, P = -OF/OQ, and H(q, p, t) + (OF/Ot) = ~(Q, P, t). Finally the HJ equation emerges by requiring 7-/= 0 so (~ = / b = 0 and P, Q can play the role of integration constants. Then the classical HJ equation with F = S cz is (CC) H(q, OSCl/Oq, t) +OSd/Ot = 0. For stationary states of energy E one writes Sd(q, Q, t) = Sod(q, Q) - Et with OSd/Ot = - E and the classical stationary HJ equation (CSHJE) is (CD) g(q, OSod/Oq) - E = 0. We will write S~L = W d and for Hamiltonians g ( q , p ) = (p2/2m) + V(q) ( C D ) becomes (CE) (1/2m)(OWd/Oq) 2 + W(q) = 0 where • V ( q ) - E. Now one wants to deal with coordinate transformations q --~ ~ alone, treating q and p as dependent via p = OqW (see below). Even though this seems to give q a privileged role there will be a dual role for p developed later. The theme is to develop a quantum stationary HJ equation (QSHJE) of the form 1 (_~_qW)2
2m
h2
+ V(q) - E + -~m {W, q} - 0
(4.1)
where {W, q} = ( W ' " / W ' ) - (3/2)(W"/W') 2 is the Schwartzian derivative. This includes situations where ( C F ) i h C t = - ( h 2 / 2 m ) r '' + Y r with r = ~2(q)exp(-iEt/5) and e.g. (CG) r = R(q)exp(iI?V/h); however the form ( C G ) with R, I/V 9 R is inappropriate for many situations and (2.1) is entirely general. For example a general solution of the Schrhdinger equation (CH) -(52/2m)~2"+Vr = E r is (CI) r = (I?V')-l/2[Aexp(-il~/~)+ Bexp(iI?V/h)]. In the case ( C G ) one obtains the "classical" Bohm type results (W ~ S0) 1 h2R ll 2m (I2d')2 + Y - E - 2m----R= 0; (R21~')' = 0
(4.2)
and ( C J ) (~ = - h 2 R " / 2 m R is the quantum potential of the Bohm theory. Note this can be put in the form Q = (h2/4m){I~, q} via R2I]d' = c and the Schwartzian form is always
1.4. TRAJECTORYREPRESENTATIONS
51
applicable and ( C K ) Q = (h2/4m){W, q} is better called the q u a n t u m potential. The equivalence principle now involves finding, for an arbitrary system with reduced action So - W, a coordinate transformation q ~ q0 such that W~ ~ - W(q) corresponds to the system with W - V - E = 0 (in fact q0 will have the form q0 _ [Aexp(2iSo(q)/h) + B]/[Cexp(2iSo(q)/h) + D] and S O ~ (h/2i)log[(aq ~ + b)/(cq ~ + d)] so t h a t {exp[(2i/h)S~ q0} = 0 = {q0, q0}_ cf. ( C L L ) below). After considerable discussion (all quite essential in establishing a sound theory) one arrives at the formula (CL))/V(q) -(h2/4m){exp(2iW/h), q} which in fact makes the Q S H J E (2.1) into an identity
2m
+ W(q) + Q(q) = 0
(4.3)
where Q is given by ( C K ) and W by ( C L ) . Thus to find W, given V and E and hence ld] = Y - E one could solve ( C L ) or (4.1) for W, the latter being perhaps easier. Further given the formulas /)
Ox (f')
1/2
o~ 1 /) (f,)-1/2 = 0 -- Ox f' Ox (f')
02
Of ~- ~
1
1/2(
(3 1 (3
f')
-1/2f ;
+ _ { f ' x } = (f')l/20x f, Ox(f' )
(4.4)
1/2
one sees t h a t r = ( f , ) - U 2 and r = (f,)-l/2f are two linearly independent solutions of D/r = ( 0 2 + U)r = 0 and (necessarily) 2U(x) = { ( r 1 6 2 (in general by a simple calculation). In addition r = (f')-l/2(Af + B) and r - (f')-l/2(Cf + D) are solutions of D / r = 0 and one is led to the (well known) conclusion that ( C L L ) { 7 ( f ) , x } - {f, x} for Mbbius transformations 7 ( f ) = (Af + B ) / ( C f + D) (in fact {f, x} - {g,x} if and only if g = 7 ( f ) ) . Let us note also a canonical equation ~-fis2 + L/(s)
qv/-~ = 0 = [/)~ + / r
(4.5)
where ( C M ) L / ( s ) = (1/2){(qx/~/v/-~),x } = (1/2){q,s} and s = So(q) = W(q). There is moreover a dual reduced action ( C N ) To(p) = q(OSo/Oq)- So with So = p(OTo/Op) - To via ( C O ) p = OSo/Oq and q = OTo/Op. Further for t = To(p) one has a dual canonical equation ( C P ) [02 + P(t)]Pv/~ = 0 = [02 + P ( t ) ] v ~ where P(t) = (1/2){(px/~/v~),t } - (1/2){p,t}. Next one can show t h a t (2.1), with W = V - E given by ( C L ) , is an identity and this basic Q S H J E is a general QM expression, reducing to C S H J E when tt ---, 0 (i.e. So ~ S~t and Q --~ 0) so t h a t classical mechanics appears as a special case of QM. It is i m p o r t a n t to emphasize here that Q :~ Q since generally Q %+ 0 as tt --~ 0. The correct relation is for
2m
Oq
+ Q(q) = ~
Oq
+ Q(q)
(4.6)
and when ~ c< r if one uses a wave function ~ = Rexp(+il~/h) then S0 = constant (not p e r m i t t e d in the theory of [90, 315, 316]) with Q = Q+(1/2m)(W') 2. Actually when ~p ~ if one picks ~ = Rexp(-iW/l~) and cD = ~ with exp(2ii?V/h) = r162 then Q = Q. The classical S~Z is never constant unless 142 = 0 so in particular as h --. 0 one has l im So ~ S~t in cases where r c< r if the representation r -- Rexp(:hilfd/h) is used. If the representation ( C I ) is used with R = (l~') -1/2 then a reality condition ~ c< r translates onto conditions
C H A P T E R 1. Q U A N T I Z A T I O N A N D I N T E G R A B I L I T Y
52
on A and B rather than 1~ so a consistent classical limit for I/V is possible. Looking now at (4.3) with I/V given by ( C L ) , and using (4.4) with comments thereafter, one sees that the W derived will have the form e x p ( 2 i W / h ) = r 1 6 2 or .~(r162 where c O and r are linearly independent solutions of the Schrhdinger equation ( C H ) and ~/is a Mhbius transformation. Thus one can write
exp and for ft = r 1 6 2 _ r 1 6 2
~
= Cr D + Dr
(4.7)
it follows immediately that
i P = -2 A C ( r
5(AD- BC)ft 2 + ( A D + B C ) r 1 6 2 + BD~2 2
(4.8)
and we set ~ = ( A D - BC)f~. Writing then c~ = - 2 i A C / ~ and/~ = - 2 i ( A D + B C ) / ~ with ~/= - 2 i B D / ~ (2.8) becomes ( C R ) p = hick(Co) 2 + ~ r 1 6 2 1 6 2 a n d / 3 2 - 4 c ~ / = - 4 / f t 2. A further useful form of (2.8) is achieved by setting t~l = 1/c~f~ with t~2 = ~/2c~ and rewriting ( C R ) in the form (CS) p = -t-h~l[(@ D -i- ~2~))2 + ~2@2]-1 __ • _ it, l l - 2 Note next that since (4.3) or (4.1) is a third order differential equation it requires three integration constants to determine So = W. Also from (CS) we have seen that p - OqSo is determined by t~i and ~2 (or by t~ = t~l + itS2 and [) since Ft is determined by ~, @D a n d can be chosen to be an arbitrary fixed value. The additional constant appearing in So is obtained by integrating ( C S ) in the form
exp[2iSo(5)/h] = e ic~w + i[ w-ie
(4.9)
where w = r 1 6 2 E R and 5 ,-~ (&,t~) with & an integration constant. We mention here also that a fundamental feature of the equivalence principle is the result that the QSHJE is defined if and only if w = r (and therefore the trivializing map q --~ q0) is a local homeomorphism of 1~ U { ~ } into itself. Note also that a ~: 0 in ( C R ) and hence ~1 ~ 0 SO SO in (4.9) cannot be constant (and this is essential in defining the QSHJE via {So, q} etc.) . One can also write now ( C T ) p = hf~(t~ + ~.-)/21~D - i~r and setting ( C U ) r = v / 2 e x p ( - i & / 2 ) ( r D - i~r + ~11/2 it follows that ( C V ) exp[(2i/l~)So(5)] = (~/r and p = elr for e = +1 fixing the direction of motion (here r162- Cq~' = -2ie/l~). Note this means p is always r e a l - there are no forbidden regions. We consider next the free particle where 142 = - E (V(q) = 0). One can take ~)D __ a S i n ( k q ) and r = bCos(kq) for kh = v / 2 m E with Wronskian f~ = - a b k leading to
4-h(~E --F ~E)abk PE = 21aSin(kq) _ ieEbCos(kq)l 2
(4.10)
(cf. ( C T ) ) . We can also require that l i m e D = q and l i m r = 1 as E ---, 0 in order to get the correct limiting state I/Y~ This can be achieved via e.g. a = k -1 and b - 1 so f~ = - 1 and (2.10) becomes
PE = 21k_lSin(kq) _ ffECos(kq)l 2
(4.11)
:t:h(g0 + [0) limh---,oPE = +x,/2mE; limE~OPE = P0 = 21@ _ itS012
(4.12)
One must assure now that
1.4.
TRAJECTORY
REPRESENTATIONS
53
Some e x p e r i e n t a l calculations with (4.11)-(4.12) lead one to consider ( C W ) t~E = k - i f ( E , l~)+ AE where f is real and dimensionless. T h e n from (4.11) we o b t a i n two forms (recall
+[I~/2mE + m g ( a . + ~E)/~]
PE = lexp(ikq ) _ ( f _ 1 + kAE)Cos(kq)l 2; PE =
(4.13)
: k [ h 2 ( 2 m E ) - U 2 f + h(AE + AE)] [q _ il~(2mE)_l/2 f _ iAEI 2
where in the second expression one thinks of E ~ 0 ~ k --, 0 so k - l S i n ( k q ) ~ q and Cos(kq) ~ 1 while in the first expression we will let 5 ~ 0. These formulas suggest f ~ 1 as h ~ 0 and E - U 2 f ~ 0 as E --, 0 but further analysis of )~E is still needed. In p a r t i c u l a r we will need A E / h ---, 0 as h ~ 0 (so kAE = ( A E / h ) v / 2 m E ~ 0) in which case with f ~ 1 one would o b t a i n PE --'* =kv/2~nE. To achieve this as well as the other d e s i d e r a t a one considers two f u n d a m e n t a l lengths, C o m p t o n and Planck, given respectively by ( C X ) Ac = 5 / m c and Ap = v / h G / c 3. T h e n two dimensionless quantities d e p e n d i n g on E are ( C Y ) xc - kAc v / 2 E / m c 2 a n d Xp = kAp = v / 2 m E G / l ~ c 3. Since xc does not d e p e n d on h it will be excluded and a n a t u r a l choice of f will t h e n be ( C Z ) f = e x p [ - a ( X p l ) ] where oL(Xp 1) -- ~--~k>l ~ k with a l > 0 for only one t e r m (so e.g. E - 1 / 2 f ~ E - 1 / 2 e x p [ - a l x p l ] --~ 0) and other suitable choices of ak as needed. T h e n ~E • k - l e x p [ a ( X p l ) ] + AE is indicated and some further discussion leads to (3(Xp) = ~ k > l 3kXkp with/31 > 0 and other suitable choices of 3k as needed) eE -- k - l e x p [ - o ~ ( x p l ) ] -~ exp[-~(Xp)]~o (4.14) where g0 = Q(Ac, Ap, Ae) for Ae = e 2 / m c 2 with e ~ electric charge and ( D A ) t~0 ~ h ~ as h ~ 0 for - 1 < 5 < 1 (note Ae/Ac = a = e2/l~c is the fine s t r u c t u r e c o n s t a n t ) . T h e n (4.11) can be w r i t t e n as
2k-1]~exp[-o~(xpl )] -~- hexp[-~(Xp)] (eo -~- ~o ) pE =
+
2[k-lSin(kq)
- i(k,-lexp[-o~(xpl)] -~- exp[-~(Xp)]~o)Cos(kq)[ 2
(4 15)
and one sees t h a t as E ~ 0 we have k - l h e x p [ - a ( X p l ) ] ~ ( h 2 / v / 2 m E ) e x p [ - o L ( x p l ) ] c E - 1 / 2 e x p ( ~ E -1/2) ---, 0 so lim~E = limAE = t~0 and ( D B ) PE ---* P0 -- =t=h(~0 q- ~0)/21q ~ ig012 as desired. For h --, Owe have f = e x p [ - a ( X p l ) ] ~ 1 and e.g. A E / h = exp[--/3(Xp)]/5 exp[--~lh -1/2] ~ O. T h e choice t~0 "-~ h ~ as in ( D A ) arises as follows. One wants lira Po = 0 as h ~ 0 where P0 is given in (4.12) or ( D B ) . One can set ~t~0 = 0 (since it c o r r e s p o n d s to a shift in q0) and distinguish the cases q0 ~_ 0 and q0 = 0. Since ~t~0 =/= 0 the d e n o m i n a t o r in ( D B ) is nonvanishing so consider ~t~0 ~ h ~ as h ~ 0. T h e n P0 ~ +h~t~0/[q ~ - i~t~0[ 2 so P0 ~ • ~+1 for q0 ~ 0 and P0 ~ =kh 1-~ for q0 = 0. In either case P0 ~ 0 as required when - 1 < 5 < 1 (for classical limits see also [345, 346]). T h e r e is much more in [90, 315, 316]. In p a r t i c u l a r the general t h e o r y based on the equivalence principle encompasses t u n n e l i n g a n d q u a n t i z a t i o n of s p e c t r a and n u m e r o u s examples are worked out in detail. T h e higher dimensional and relativistic versions are developed in [90] (cf. also [937]). Let us emphasize now the t r e a t m e n t of t i m e in this theory. T h u s the t h i r d order e q u a t i o n (4.1) or (4.3) leads to microstates as in Floyd, and one avoids a flaw in the B o h m t h e o r y which is based on the erroneous a s s u m p t i o n t h a t particle velocity q is the same as p / m = OqSo = S~o where So = W is H a m i l t o n ' s characteristic function or reduced action (S = S o - Et). T h e correct version here is p = OqW = mOrq where T - TO = m fqo(dX/OxW) represents a time concept developed by Floyd (cf. [347, 348]) in studies of t r a j e c t o r y r e p r e s e n t a t i o n s and microstates. In p a r t i c u l a r one can work with t ~ OEW to write t -- to = OE fqo W I d x and
54
CHAPTER1.
QUANTIZATION AND INTEGRABILITY
arrive at mq = m(dt/dq) -1 = m/W~E = W ' / ( 1 - OEQ) where Q is the quantum potential Q = ( h 2 / 4 m ) { W , q}. One notes also in [315] that a formula p = mQ(t can be obtained via the use of a quantum mass mQ = m(1 - OEQ). The quantum potential Q is regarded here as the particles reaction to an external potential V and no pilot-wave philosophy is needed. Given even alone this important variation in treating time from the traditional Bohm theory (and more generally, given the full theory of FM) many philosophical discussions (some quite recent) regarding trajectory representations should be drastically modified. We will mostly avoid philosophy here however and simply make a few comments about matters related to this use of time. We will consider stationary states but allow E to vary continuously (so discrete eigenvalues En are not indicated although some arguments could be adjusted to include them). It was apparently first observed by Floyd in [347] that Bohm's assumption p = W I = m0 (for particle velocity c)) is not generally valid and the correct version is m ( 1 - i)EQ)(~ : W ' =- mQ(t : W ' ~ mi)rq = W'; mQ=m(1-OEQ);
T-- TO = m
~
(4.16)
q dx W'
0
Then one has, using ( 1 / 2 m ) ( W ' ) 2 + V + Q - E
= 0 and Floyd's effective or modified potential
g=v+Q, t-
t o -- OE
fqq 0
W'dx =
( m ) l/2 fq q ( 1 - OEQ)dx -~ viE U 0
(4.17)
--
and dT/dt = 1 / ( 1 - OEQ). Thus t is explicitly a function of E and we want to expand upon this aspect of the theory following [155]. It is important to note that general solutions of the Schrhdinger equation ( C H ) above should be taken in the form (CI) ~b = ( W ' ) - l [ A e x p ( - i W / h ) + B e x p ( i W / h ) ] and p ~ OqW - - W ' is the generic form for p corresponding to momentum in a QM Samiltonian ( 1 / 2 m ) p 2 + Y ~ ( 1 / 2 m ) ( - i h O q ) 2 + V. Thus p = W ~ corresponds to p ~ -ihOq and this is the quantum mechanical meaning for p; it will not correspond in general to mechanical momentum m0 for particle motion. First one sees that Hamilton-Jacobi (H J) procedures involve t ~ OEW and we will modify an argument in [330] in order to give further insight into the relation t = t(E). We think of a general stationary state situation with S ~ W - E t = W(q, E ) - E t so that OS/Ot = - E and t = t(E). Setting S = - S with OS/Ot = E we can write then (DC) W = E t - S = t S t - S in Legendre form. Now given W = W ( E , q), with q fixed, one has OEW = t + E t w - S t t w = t so ( D D ) S = E W E - W gives the dual Legendre relation. Consequently the constructions in [315] for example automatically entail the Legendre transformation relations ( D C ) and ( D D ) involving ,5' = - S and W. Now one comes to the energy-time uncertainty "principle" which should be mentioned because of situations involving energy dependent time for example (cf. [21,130, 333, 849] for various approaches- we make no attempt to be complete or exhaustive here). First, in a perhaps simple minded spirit, let us recall that microstates are compatible with the Schrhdinger representation by wave functions r and hence one will automatically have a connection of the trajectory representation with Hilbert space ideas of observables and probability (more on this below). In the Hilbert space context the uncertainty A q A p _> h/2 is a trivial consequence of operator inequalities and we take it to mean nothing more nor less (Ap for example represents a standard deviation < r < i5 >)21r > where i5 ~ -ihOq). In this spirit nothing need be said about measurement and we will not broach the subject in any way except to say that sometimes for a trajectory we will think e.g. of physical increments
1.4.
TRAJECTORY REPRESENTATIONS
55
5q ~ q - qo and 5p ~ p - po. Thus we will try to maintain a distinction between 5q and Aq for example and we do not require that 5q be measured, only that it be a natural mathematical concept. After all Aq above is also only a natural mathematical concept without any a priori connection to measurement. The idea of attaching physical meaning to Aq via measurement seems no stranger than thinking of 5q as a meaningful possibly measurable physical quantity. As for energy-time uncertainty we remark first that if one departs by e from a correspondence between observables and self-adjoint operators then an e approximate inequality (DE) A E A t >_ h/2 can be proved in a Hilbert space context (see e.g. [333] for a detailed discussion). There are also various crude physical derivations based on 5q ~ (p/re)St where p is the physical momentum and subsequently, for 5E ~ (p/m)Sp when e.g. E ~ ( p 2 / 2 ) + V(q), one often writes ( D F ) 5 p S q ~ ( p / m ) ( m / p ) S E S t = 5ESt >_ h/2 based on a q - p uncertainty with 5q ~ Aq etc. (displacement version). This would be fine if p = rnq but we have seen that p has an unambiguous quantum mechanical meaning as in Section 2 and the s involving 5q = (p/m)St is generally not valid. A more convincing argument can be made via use of Ehrenfest type relations (/2/~ E) d dt
1 h]d ] = g < i [ ~ ' ~] >; 5ESQ >_ ~ ~-~ < Q >
(4.18)
and an argument that the time 5t for a change 5Q = 5 < Q > should be 5t - 5Q/Id < Q > / d t l , leading to ( D G ) 5ESt _> h/2 (without intervention of p, where however one has d < q > ~dr ~ ( 1 / . ~ ) < p
>).
Since the beginning step 5q ~ (p/rn)St is generally wrong in the crude argument above let us adjust this following (4.16) to be 5q ~ (p/rn)5~- ~ (p/rnQ)St where p = W' is the conjugate momentum (QM momentum). Then with 5E ~ (p/rn)Sp one will arrive at 5pSq ~ 5EST ~ 5 E S t / ( ( 1 - QE) and consequently a correct displacement (or perhaps trajectory) version of ( D E ) should be h 5EST >_ -~ -- 5ESt >_ ( 1 - OEQ)-~
(4.19)
Since in the trajectory picture we are dealing with t = t(E) via t = OEW (with E a continuous variable here) one will have 5 t - WEESE so (4.19) requires a curious condition (DI) (SE) 2 >_ [ ( 1 - QE)h/2WEE]. Thus apparently for any energy change compatible with the microstate picture (DI) must hold. This would seem to preclude any positive infinitesimal 5E unless ( 1 - Q E ) / W E E <__0 (with restrictions on any negative 5E). One can perhaps envision here microstates as developed by Floyd generated at energy E0 with initial conditions Wo(Eo), W~(Eo), and W~(Eo) (or (q0, 00,//0)(E0)) and then imagine E to be changed while keeping the initial conditions constant. This would affect the time relations on the trajectories and lead to a situation where the inequality (DI) could have meaning, but for general situations t = t(E) (4.19) seems untenable, and hence we exclude energytime uncertainty for completely determined microstates (see below however). Further we cannot suggest its applicability in the operator form (D J) A E A T > h/2 since that would clash with ( D E ) which has a more or less substantial foundation. Thus we argue that while ( D E ) may be acceptable its displacement version (D J) is not, except perhaps in the averaged form ( D G ) . As for computation in (4.19) for example one notes that the equations in [347, 348] for example have to be put in "canonical" form as in [315, 316] and, in computing WE, one should only differentiate terms which under a transformation E ~ E ~ :/: E do not correspond
CHAPTER 1. Q UANTIZATION AND I N T E G R A B I L I T Y
56
to a Mhbius transformation of exp(2iW/h) (i.e. one only differentiates terms in which 14; is changed under a transformation E ~ E ~ -~ E). Regarding 1 - QE one can use the relation W'W~E = m ( 1 - Q E ) for computation. As for uncertainty however my interpretation of some remarks of Floyd suggests the following approach. First I would claim that uncertainty type inequalities are incompatible with functional relations between the quantities (e.g. p = OqW = p(q) or t = t(E) via W). Thus if W is completely known there is generally no room for uncertainty since e.g. 5t ~ WEEhE or with adjustment of constants 5t = t - to = WE completely specifies St. Note that one of the themes in [315, 316] involves replacing canonical transformations between independent variables (p, q) with coordinate transformations q ~ with p = Wq(q) depending on q. Now we recall that the QSHJE is third order and one needs three initial conditions (q0, q0, q0) or (W0, Wg, W~~) for example to determine a solution and fix the microstate trajectories. However the Copenhagen representation uses an insufficient set of initial conditions for microstates (and literally precludes microstate knowledge). The substitute for exact microstate knowledge is then perhaps an uncertainty principle. It would be interesting to see if the two pictures interact and one could perhaps think of uncertainty relations involving ~t and ~E as in (4.19) for example as constraints for the microstate initial conditions. However the microstates are always compatible with the Schrhdinger equation for any initial conditions and hence lead to the same operator conclusions in Hilbert space (such as ( D E ) for example). In any event one can continue to use the standard quantum mechanics, knowing that a deliberate sacrifice of information has been made in not specifying the background microstates (i.e. quantum mechanics in Hilbert space is imprecise by construction, leading naturally to a probabilistic theory etc.). We refrain from any further attempts at "philosophy" here.
1.5 1.5.1
MISCELLANEOUS Variations
on Weyl-Wigner
Notations such as ( P A ) will be used. First consider the Husimi transform following [33] (cf. Sections 1.1, 1.3, 3.8, and Chapter 4 for various star products and related ideas). The approach of [33] is to show that the Mizrahi series of an operator product is absolutely convergent for a large class of operators (cf. [458, 638, 741]). By constrast the series for the Weyl transform is often only asymptotic or undefined. We only sketch constructions here and begin with the Wigner function 1 ./
W(x,p) = - ~
y
y
dye ipy < x - ~lt~lx + ~ >
(5.1)
where f~ is the density matrix and h - 1. One smears this with a Gaussian following Husimi to get the Husimi or Q function
Q(x, p) = -~1/ dx'dp'exp[-(x - x') 2 - (p - p')2]W(x', p')
(5.2)
where one assumes x and p have been made dimensionless by suitable scaling. This is known to be the probability distribution describing the outcome of a joint measurement of position and m o m e n t u m in a number of cases and more generally it is the probability density function describing the outcome of any retrodictively optimal joint measurement process. It can be argued (cf. [34]) that Q may be regarded as the canonical quantum mechanical phase space probability distribution playing the same role in relation to joint measurements of x and p as does I < xlr > 12 for single measurements of x. One can derive an analogue of the classical
1.5. MISCELLANEOUS
57
Liouville equation as follows. Let Aw be the Weyl transform of an operator J defined via (PA) A w ( x , p ) = f dyexp(ipy) < x - (y/2)lJlx + (y/2) >. Then the Husimi transform or covariant symbol A s is given by
AH(x, p) = -~1f dx'dp'exp[-(x - x') 2 - (p - p')2]Aw(x', p')
(5.3)
From [741] one has a formula for the Husimi transform of the product of two operators; namely ( P B ) (A/~)H = AHexp(~+-~_)BH where 0+ = 2-1/2(0x 9= iOp). Using this one can derive a generalization of the Liouville equation in the form ( P C ) (O/Ot)Q = {HH, Q}H where HH is the Husimi transform of the Hamiltonian H and {HH, Q}H is the generalized Poisson bracket ( P D ) {HH, Q}H = ~(2/n!)-~(O~HHOnQ). One shows in [33] that there is a large class of operators for which the series in ( P B ) and ( P D ) are absolutely convergent. This can be best appreciated by comparing ( P B ) - ( P D ) with the corresponding formulas in the Wigner-Weyl formalism, namely ( A B ) w = Awe (1/2)( 0 ~~0 p- ~0 p~0 ~)Bw; OW Ot = {Hw, W}w; oc
(_l)n
r~-:-_ --o_
(5.4)
~ --o_ \ 2n+l
{Hw, W } w = ~ (2n + 1)'22nHW [ Oz O p - O p Ox) W o where Hw is the Weyl transform of the Hamiltonian and {Hw, W } w denotes the Moyal bracket. It turns out that the two sets of equations ( P B ) - ( P D ) and (5.4) have quite different convergence properties (cf. [33] for more details). Another set of results in [33] involves expectation values. First one has the standard formula involving the Wigner function ( P E ) Tr(~J)= f dxdpAw(x,p)W(x,p) for the expectation value of an operator A. In certain situations one can also express the expectation value in terms of the Husimi function via ( P F ) Tr(t~J) = f dxdpd~(x,p)Q(x,p) where ( P G ) A/~ = exp(--O+O_)Au with (9+ as before. Since A p can be highly singular there can be problems but one shows in [33] that in many situations one can use instead the series expansion of ( P G ) and interchange orders of summation and integration to obtain oO
Tr(~J) --~-~ o
(-1) n
n!
/ dxdp(O~-On--Ag(x'p))Q(x'p)
(5.5)
which will be often absolutely convergent even when A/~ fails to exist as a tempered distribution. Again we refer to [33] for details. 1.5.2
Hydrodynamics
and Fisher information
We sketch here from [813] (cf. also [8141) and refer to [359] for Fisher information in general. First from [359] let p(ylO) be a probability density function in data variables y and a parameter 0 with ~(y) an unbiased estimator such that (SA) < 0 ( y ) - 0 > = f dy[O(y)- 0]p(yl0)= 0. Differentiate this in 0 to get
(5.6)
/ dy(~ - O)-~ Op - / d y p = O
and note that Oep = p(Olog(O)/O0) so (SB) f dy(O - O)(Olog(O)/OO)p = 1. Write then from
(SB)
[(o-
:.
CHAPTER 1. Q UANTIZATION AND INTEGRABILITY
58
Use now the Schwartz inequality to obtain
/ d y (Ol~
) 2p / d y ( O - 0 ) 2 p >_ 1
(5.8)
The Fisher information is defined as (SC) I = I(O) = f dy(Olog(p)/OO)2p and the second factor is the mean squared error (SD) e 2 = f dy(O- 0)2p leading to e2I >_ 1. In physics this can be used for real probability amplitudes qn(xn) with Pn = q2n leading to (SE) I 4 f dx ~ Vqn" Vqn which is a typical Lagrangian element. This can be extended to (complex) field theory in the form (SF) I = 4N f dx ~N/2 VCn. VCn for example. Now go to [813] and let p(yi) be a probability density (PD) with P(yi + Ay ~) the density resulting from a small change in the yi. Expand p(yi + Ayi) in a Taylor series and calculate the cross entropy J up to the first non-vanishing term
j[p(yi + Ayi) . p(yi)] = / p ( y i + Ayi)logP(Yp(y~) i+ Ay i) d~y ~ Ill
-~
10P(yi)OP(yi)dny]AyjAyk_ijkAyjAy
p (yi ) OyJ
Oyk
(5.9)
k
The Ijk are the elements of the Fisher information matrix. This is not the most general situation but is correct when P(xiiO i) is a PD depending on a set of n parameters 0i having the form P(x~lO~) = P(x ~ + 0~). If P is defined over an n-dimensional manifold M with inverse metric gik one writes 1 OPi c3y OPk dny I = gik Iik= 2gik f pl Oy
(5.1 O)
The case of interest here is for M an n + 1 dimensional configuration space with coordinates (t, xl, --. , x n for a non-relativistic particle of mass m. Then one would have gik _ diag(O, 1/m,..., 1/m). Now in the Hamilton-Jacobi (HJ) formulation the equation of motion is
OS 1 ~ OS OS 0---{ + -2g" Ox---~Ox~ ~- Y = 0 (5.11) where gi, V = diag(1/m,..., 1/m). The velocity field u t' is then (SG) u" - g"(OS/cgx').
When the exact coordinates are unknown one usually describes a system by means of a PD P(t, x ~) which must satisfy f pdnz = 1 and
OPot + ~0 /~Pg ~~OS )_ = 0
(5.12)
Equations (5.11) and (5.12) together with (SG) completely determine the motion of the classical ensemble and can be derived from the Lagrangian
LCL =
/ { O S I v O S O S } P - ~ + -2g~ Ox~ Ox~ + V dtdnx
(5.13)
with fixed end point variation 6P = 6S = 0 at the boundaries with respect to S and P. Quantization of the classical ensemble is achieved by adding to LCL a term proportional to the information I defined via (5.10). This leads to the Lagrangian for the Schrhdinger equation
LQM = LCL + hi =
f P {0S 1 ~,[0SOS ~ OPOP] } - ~ + -2g~ ~ Ox" -~ p20x~ Ox" + V dtd'x
(5.14)
1.5. MISCELLANEOUS
59
Fixed end pont variation with respect to S leads again to (5.12) while fixed end point variation with respect to P leads to
0--{ -~- -2g•
~
Ox u F )~ fi2 0 x , Ox ~'
P Ox, Ox u
+V =0
(5.15)
Equations (5.12) and (5.15) are identical to the Schrhdinger equation provided the wave function r x u) is written via (SH) r = v/-fiexp(iS/h) and A = (5/2) 2. The theory should also be compatible with the FM development in Section 1.4.1. Note that the classical limit of the Schrhdinger theory is not the HJ equation for a classical particle but the equations (5.11) and (5.12) which describe a classical ensemble. It is shown in [813] that the Fisher information I increases when P is varied while S is kept fixed. Therefore the solution derived is the one that minimizes the Fisher information for a given S. In general terms the information theoretical ~ontent of the theory lies in the prescription to minimize the Fisher information associated with the PD describing the position of particles while the physical content of the theory is contained in the assumption that one can describe the motion of particles in terms of a hydrodynamical model. It can be argued that the cross entropy
J(Q)'P)=
Q(Yi)l~
p(yi)
dnY
(5.16)
where P, Q are two PD's measures the amount of information needed to change a prior PD P into a posterior PD Q. The cross entropy (or its negative) is known under various names: Kullback-Leibler information, directed divergence, discrimination information, Renyi's information gain, expected weight of evidence, entropy, and entropy distance, for example. Maximization of the relative entropy = negative of the cross entropy is the basis of the maximum entropy principle, which asserts that of all the PD that are consistent with the new information, the one which has the maximum relative entropy is the one that provides the most unbiased representation of our knowledge of the state of the system. To understand the use of the minimum Fisher information principle in the context of QM it is crucial to take into consideration that one is selecting those PD p(yi) for which a perturbation that leads to p(yi + Ayi) will result in the smallest increase of the cross entropy for a given S(yi). In other words the method of choosing p(yi) is based on the idea that a solution should be stable under perturbations in the very precise sense that the amount of additional information needed to describe the change in the solution should be as small as possible. Thus the principle is to choose the PD describing the quantum system on the basis of the stability of those solutions where the measure of the stability is given by the amount of information needed to change p(yi) into P(yi + Aye). We refer to [813] for more discussion and philosophy. Now one wants to show that the K~ihler structure of QM results from combining the symplectic structure of the hydrodynamical model with the Fisher information metric. First look at the symplectic structure of the hydrodynamical formulation. Introduce as basic variables the hydrodynamical fields (P, S) and consider (all terms have arguments x ~)
-1
0
5'S
dnx
(5.17)
where the matrix will be denoted by ft and 5, 5t denote two generic systems of increments for phase space variables. The Poisson brackets (PB) for two functions Fi(P, S) take the
CHAPTER1.
60
form { F l ( p ' ' ) F2(p~')}: i
QUANTIZATIONANDINTEGRABILITY
(
(5.18)
and the equations of motion are
OP 57i OS Ot = {P' tt} = 5S ; Ot = {S' 7-I} =
57-i 5p
(5.19)
with Hamiltonian
7l-
p
_~g U
~_x_xx U Ox U +
(~)2 10P
OP + V } dnx p 2 0 x ~ Ox u
(5.20)
To introduce the Fisher information metric let 0t' be a set of real continuous parameters and consider the parametric family of positive distributions defined by P(x"lO~ ) = P(x ~ + 0~) where the PD's P are solutions of the SchrSdinger equation (at time t = 0). Then there is a natural metric over the space of parameters 0 u with a concept of distance defined via
l[f
ds2 ( O" ) = -2
1
P ( xU lOu)
OP(xulO") OP(x"lO") dnx I ~OP(~Oa OOP O0~
(5.21)
Using 6P = (OP/OOU)50 u one can write (5.21) as l I f p(xulOu-------1 ~ ~p(xttlOp)~p(xttlOtt)dnx] ds2(O ") = -~
(5.22)
Now use (5.22) to define a metric over the space of solutions of the SchrSdinger equation (i.e. P(x"lOU ) with 0" = 0 by setting
1[/
ds2(hP, 5'P) = -~
1
P(xU) 6P(xU)6'P(xU)d3x] = f g(P)SP(xU)5'P(x~) dax
(5.23)
where (SI) P(x u) = P(xulOU = 0)with 6P(x u) = 5P(xulO u = 0) and g(P) = 1/2P(xU). Now to extend the metric g(P) to a metric gab over the whole space of (P, S) compatible with the symplectic structure one introduces a complex structure Jg and imposes the conditions (SJ) grab = gacJD with jagabJb =gcd and j ~ j b = _sa. A set of (grab, gab, Jg) satisfying (S J ) defines a Kiihler structure. Now set
ab=(0- 1
0
;gab-
.
)
(5.24)
where gab is real and symmetric. Then the solutions gab and J~ to (SJ) depend on an arbitrary real function A and are of the form
gab =
A
A
(hg(P))-l(1 + A 2)
)
; J~ --
-hg(P)
The simplest choice is A = 0 leading to a flat K/ihler metric given by (5.24) with
gab =
0
(hg(P)) -1
; J~ =
-lig (p)
,
-A
0
~ab "" ~
in (5.17) or
(5.26)
1.5.
61
MISCELLANEOUS
The coordinate transformation is (SK) r = x/-Pexp(iS/h) and r and in terms of the new variables (5.26) takes the form
~ab-~
= x / - P e x p ( - i S / h ) (Madelung)
(0 ~)(0~)(~0) -ih
0
;gab =
h
0
; J~ =-"
0
i
,527,
Thus the Madelung transformation is remarkable in leading to the simple Hamiltonian
=
--ffg"~oz. Ox" ~- vr162 dnx
(5.2S)
and the equations of motion become linear. Finally one introduces a Hilbert space structure using gab, ~ab t o define the Dirac product via (arguments are x ~)
>= ~
((r162
[g + ia]
r
=~{,~0.,[(~ ~)~(o ~0~)l(:.)}~nx~~.~nx In this way the natural Hilbert space structure of QM arises from combining the symplectic structure of the hydrodynamical model with the Fisher information metric. Further virtues of this formulation are discussed in [813].
62
CHAPTER1.
QUANTIZATION AND INTEGRABILITY
Chapter 2
GEOMETRY
AND EMBEDDING
This chapter covers a number of topics involving the embedding of curves and surfaces in higher dimensional spaces, the effects of extrinsic curvature and torsion, and the emergence of integrable systems connected with quantum physics. Some of the material was written at an earlier time for lecture notes and we list here a more updated collection of references. Thus first there is a collection of fascinating papers by Matsutani and collaborators which we will discuss later in more detail (cf. [702, 703, 704, 705, 706, 707, 708, 709, 710. 711`. 712, 713, 714, 715, 716, 717, 718]. Next we mention a long and important series of papers by Konopelchenko and collaborators (including the author) which enter into most of the chapter (cf. [157, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576] along with related work of Taimanov (cf. [895, 896, 897, 898, 899]). Then there is a long series of interesting papers by Parthasarathy and Viswanathan and collaborators, some of which are treated in detail (cf. [791, 792, 793, 794, 795, 949, 950, 951, 952, 953, 954, 955, 956, 957]). Additional references for the chapter (somewhat updated) are [52, 132, 99`. 100, 106. 242. 255, 332, 390, 391,417, 418, 430, 431, 432, 446, 469, 483, 470, 453, 518, 541, 544, 620`. 621, 625, 633, 640, 641,666, 687, 721,739, 758, 762, 763, 800, 826, 865, 866, 867, 868, 869, 886, 887, 893, 928, 963, 967, 974, 975, 988].
2.1 2.1.1
CURVES AND SURFACES Background
Notations such as (EA) will be used. We sketch first some results for curves and surfaces. Historically from [446,640] let us have a 3-D curve R(s, t) with arc length s, and unit tangent, normal, and binormal vectors T, N, and B respectively. The Frenet-Serret equations are Ts = aN; Bs - - a N ; where form
a and T correspond
(1.1)
?is = " r B - a T
to curvature and torsion. One
can also write this in a complex
(1.2)
( N + iB)s + i T ( N + i B ) = - a T
Assume e.g. T --* TO as s --, --oo or s --, 0 and write To))
J - - OO
(1.3)
J - - OO
The variable ~ was introduced in this context by Hasimoto in [446] and it plays an important role in the theory. Its meaning was explicated in [633, 762, 763] in different ways (cf. also 63
C H A P T E R 2.
64
GEOMETRY AND EMBEDDING
[6871). In terms of J~ and r one has (* ~ complex conjugate) -. 1 J~s + iTOA/" = - - e l ; fs = ~(r
+ CJ~*)
(1.4)
where iV'. J~* = 2, J~. f = J~*. T = J~. J~ = 0 and one can use J~, J~*, and f to describe the curve. Further the time dynamics will have the form ~ t = hAT" +/3.h~* + 7 f ; ~ = XJ~ + #J~* + u T where one obtains from (163Z) a + a* = 0, /3 = u = 0, and "7 = - 2 , so
= i~
1
+ ~f; ~ = -~(~*~
+ ~*)
(1.5)
where R is real. Equating cross derivatives 02t = 028 gives i Ct + ~ + i(~0~ - h e ) = 0; n~ = ~ ( ~ r
- ~*~)
(1.6)
Thus one has unknowns r R, "7 in (1.6), r = r 7), and there is some flexibility in determining curves via the choice of functional relations between r R, and 7. Thus e.g. (A) if n(s) is prescribed with say TO = 0 then r = n(s)exp(ia), a(s, t) - fs__cr 7ds'. For 7 real one gets R = at and Rs = 7 ( t ) n ( s ) S i n ( a ) . Choosing new coordinates ds' - n(s)ds, dt' - 7(t)dt one arrives at the Sine-Gordon (SG) equation as,t, = Sin(a). (B) Various other substitutions and assumptions lead to the nonlinear Schrodinger equation (NLS)
iCt -4-1r162 + 2 r
= 0; n = Ir = - 2T2; 3' = --2T0tb -- 2ir
(1.7)
or to the modified KdV (mKdV) equation 3
r + ~1r162
+ r
= 0; R 1
- r162
1
i
~01r = - ~3 + ~ ( r
- ~ ) - i~0r
_ ~);
+ r
(1.8)
Next one examines the relations of the Frenet-Serret equations to the linear equations of soliton theory for the wave function and various formulas are obtained. Consider the papers [762, 763] which pick up the m a t t e r some years later and relate the theory also to [390, 391]. Thus keep the notation (1.4) and let R(a, t) be the coordinate vector on a curve with d s / d a = v ~ ( a , t). The curve dynamics can be taken as (a fixed)
Rt = U N + V B + W T
(1.9)
One assumes local motion first in the sense that (U, V, W) depend only on local values of (~, T) and their derivatives in s. Now one notes that the sign of ~ is different here than in [390] and we consider first some 2-D curves (Y - ~- - 0) with the requirement 02~.t = 0 .2 (note O( )~Oh = v/-~O()/Os). Then (1.1) and (1.9) yield ,
D t T = (Us + ~ W ) N ; D r Y = - ( U s + ~ W ) T ; [ 7 - Dtg = 2g(Ws - nU); D t n -
t,C~
(1.10)
(Uss + n2U + nsW)
(note s can depend on t so Dt "~ d/dt and Ot differ). One checks easily the compatability of these equations with (1.1) and (1.9). Now from (1.10) we obtain
= Dts=
Dtx/~da =
-~g-~gda =
(1.11)
2.1.
65
CURVES AND SURFACES
j~00~g~1(Ws
- nU)da =
j~08(Ws
- nU)ds' = W ( s , t) -
jr08 n U d s '
provided W(0, t) = 0. Further since k = Otn + iOsn one has Otn = Us~ + n2U + n~
(1.12)
n U d s ~ = ~2U
where f~ is an integro-differential operator in s. Thus n and hence the 2-D motion of the curve is determined by specifying U(s, t) and integrating. W ( s , t) then helps to determine how points parametrized by a move along the curve but it does not affect the shape of the curve. Now for integrability consider a standard Ablowitz, Kaup, Newell, Segur (AKNS) system (Vl) = ( i ~ q ( s , t ) ) ( V l ) O~ v2 r(s, t) - i~ v2
(1.13)
One determines integrable evolution equations for q, r by specifying Otg and requiring compatibility O~tv2 -. = 02tsg. Now look at (1.4) where T and N have two components satisfying (1.14)
Ostj = nnj; Osnj = --ntj
where j = 1,2. These are equivalent to (1.13) at ~ = 0 with q ~ n and r ~ - q (NLS situation) and one recalls that mKdV and NLS belong to the same hierarchy (called mKdV hierarchy). It follows that any equation from this hierarchy (r = - q ) is compatible. For example if one chooses U = - O s n then 3
Nt -t" -~t~2ns + t~sss --0
(1.15)
(setting n(0) = 0). This is the focusing version of mKdV. One can also obtain nonlocal models such as Ost = Sin(O), SG equation, as follows. Pick U = ft-2ns so that (1.12) becomes t~s = ~ n t = t~tss + t~2~t + t~s n~tds I (1.16) Integrate in s to get then n-
nts + n
fO
(1.17)
nntds'
(one could use 0 or - ~ as a normalization point). Define then O(s,t) = f~ nds' and define F ( s , t by at -- Sin(O) + F so that Ost -- at -- Sin(O) + F. From (1.17) this implies Fs + Os f~ ds'OsF - 0 (note 0s - n and here Cos(O(O)) = 1 is used). Consequently Fo +
(1.18)
/0~FdO' = 0
with solution F = ACos(O) + B S i n ( O ) . Hence Ost = C S i n ( O + 00) which can be rescaled to Sine-Gordon. This approach allows one to relax various restrictions imposed in [390, 391] for example and extends results of [640]. For three dimensional motions one obtains equations generalizing (1.10) in the form D t T = (Us - T V -t- n W ) N + (Vs -t- T U ) B ; D t N = - ( V s - r V -t- n W ) T +
t~5
+
+ 5(u
-
+
=
+
+ -
(1.19)
C H A P T E R 2.
66 T
- [ l o s ( V ~ + ~u) + -(u~ Since D t f ( t , s) = ft + ( W -
- ~v +
.w)]f;
AND EMBEDDING
0 = 2g(W~ - ~u)
f~ aUds')fs this leads to
at = Uss + (a 2 - ~-2)U + as
~t = 0~[~0~(y~
GEOMETRY
nUds' - 2~-Vs - TsV "
jr0
(1.20)
+ ~u) + - ( u s - ~ v ) + ~ j/0s ~Ud~'] + ~ U + ~V~ a
One shows next that the Frenet-Serret equations (1.1) are equivalent to (1.13) at ~ = 0 for r = -q*. Hence equations from the mKdV hierarchy with r = -q* represent possible motions of smooth curves in 3-D. In particular (cf. (1.14)) we consider
Ostj -- anj; cOsnj -- --atj + Tbj; Osbj - -~-nj
(1.21)
Now a general Hasimoto transform enters (with reference to Darboux). Thus set
r
t)~(~, t)d f~ ~(~',~)~' = ~r
(1.22)
(fs ~ f~ or f_scr , or whatever is meaningful) and write Afj = (nj + ibj)e(s, t) (cf. (1.3)). Then (cf. (1.4)) OsAfj = - r and Ostj = l(r + r Write further
Wl --
/ 1/s Afjexp(-~1 s 1CAfj* -- t j ds'; w2 = (1 - tj)exp(-~
1~Afj* - tj ds')
(1.23)
Then there results OsWl
-
1
~r
1
OsW2-----~2
,
Wl
(1.24)
which corresponds to (1.13) at ~ = 0 for q - - r and r = -q*. One emphasizes here that the Frenet-Serret formulas imply (1.16) only at ~ = 0 (cf. [640] where this is not clarified). Now the equations (1.20) become
Ct = [a2 + I~12 + i~
[iO~ + i1r
+ r
/s ds'~r
is
ds'Tr
+ Cs -- ir
/s
is
ds'r
(1.25)
ds'~2s,](Vc)
which can be specialized to cover many situations. In particular for U = 0, V = a 1
Z
"
(1.26)
(focusing NLS) and for U = - a s , V = --nT we get 3 Z which is the complex m K d V equation.
"
(1.27)
2.1.
CURVES AND SURFACES
2.1.2
The
67
role of constraints
We sketch now [255]. Consider a curve or string on an N-dimensional sphere S N ( R ) of radius R which moves on S N (R) subject to no stretching with the d y n a m i c s i n d e p e n d e n t of R (explicitly). Take first S2(R) and let ~ ~ geodesic curvature with g the unit n o r m a l vector to the curve ( t a n g e n t to the sphere) and t" the unit t a n g e n t vector. T h e Frenet e q u a t i o n s are t h e n
Os
t'
=
(0 -A 0
0 -~
n 0
t" ff~
(1.28)
where A = l / R , ~" = A R is a unit radial vector and s ~ distance along the curve ~ arc length p a r a m e t e r . Since R = [/~1 is constant one knows ~'t = ~ = V t ' + U g and no s t r e t c h i n g implies 0s2t = 02s ((s,t) can serve as local coordinates on the sphere). T h e F r e n e t frame is o r t h o n o r m a l so its time evolution is d e t e r m i n e d by an a n t i s y m m e t r i c m a t r i x
Ot
t
(0
=
-AV -AU
0 -A
A 0
~ ff~
(1.29)
A can be d e t e r m i n e d via 0~2t = 0t2s as A = Us + ~ V and this implies (1.30)
nt = (Us -4- tcV)s -4-/~2U; Vs = t~U
which can be w r i t t e n in t e r m s of a m K d V recursion o p e r a t o r 7~ = 0 2 + n2 + ~sC3sl~ as nt -- 7EU + )~2U. Now the d y n a m i c s depends on R so to remove the explicit d e p e n d e n c e one can choose the n o r m a l velocity U as U = r where ~ = _A2r in which case nt = 0. This corresponds to rigid m o t i o n and specifies r in t e r m s of square eigenfunctions of the spectral p r o b l e m (1.28) or its equivalent
0~ r
=2
-K:-i,x
(1.31)
r
u n d e r the i s o m o r p h i s m su(2) "~ so(3). For S3(R) one has
Os
~'
0
A
0
0
~'
~"
-~
0
0 -t~
~ 0
0 T
~
o
o
-~
0
~
=
~"
(1.32)
where b ~ b i n o r m a l and T ~ torsion. T h e kinematics is described by r't - g - V t + U g + W b and 0~2 = 02ts as before with
Ot
~" /' g,
=
0 -AV -AU
AV 0 -A
AU A 0
AW B C
-~w
-B
-C
0
~" t' ~
(1.33)
Some calculation leads to
A = Us + ~ V -
~-W; B = Ws + TU; C - -I (TA + Bs + A2W) t~
(1.34)
68
C H A P T E R 2.
(1 0)
(I)l --
e~ =
(
1 O Os(~)
GEOMETRY AND .EMBEDDING
(1.35) (1.36)
;
--OsT--T(Os)
2 -[- GqsgC~slg -- T2
o~(1)[~o~ +
o~-] + ,~- o~(;)[o~1_ ~_~] + ~o~
~o;1,~) +
Again R dependence can be removed by rigid motions via
and also via suitable Laurent expansion of U and W in terms of A2. One can find such coefficients recursively via rewriting (1.35) as
0110t
--~ t~OslTt
(1.38)
Finally we note that if one introduces a new basis for the normal plane (1.39)
nl - C o s ( a ) g - Sin(a)b; gi = S i n ( a ) g + Cos(a)b
(i is not a running index). Then r = ~(r r E R2), and
C~s
?~1
--
~
ot
~ ~
=
+ ".~((/))?~i "~ (/) -- ~((/))-[- i.~(r
-~
0
~(r
~(r
~"
0
--~(~))
0
0
?~1
0
-~(r
0
0
~
-~v -~(r -~.~(r
o
y~ = ~(~5); r = ~ 2 r
;
0
~
~1
~(4)
-Y
0
~
e2r
(1.40)
~(6)
-.~(4)
~ = u n + w ~ ~ r = (u + i w ) ~ ;
(i.e. r E C
y~ = ~(r
$ - i(r + v~);
(1.41)
Qr = ~[r + r 1 6 2
Here Q is a recursion operator for NLS where (1.40) in 2 x 2 form involves matrices
2.1.3
Surface evolution
We go now to the important papers [563, 564, 565, 566, 576, 578] in order to indicate a general program for surface evolution (cf. also [641, 721] for various points of view and see [157, 791, 793, 949, 952] for connections to conformal immersions and gravity). The main idea is to start with a linear PDE L(01, 02)r = 0 in two independent variables u 1, u 2 with matrix valued coefficients (r is a square matrix). A formal adjoint operator L* is obtained
2.1.
CURVES AND SURFACES
69
via < r 1 6 2 > = f f d u l d u 2 T r ( r 1 6 2 and one has an adjoint equation L*(01,02)~* - 0. It follows that r 1 6 2 - r162 = c91P1 - 02P2 (1.42) where the Pi are bilinear combinations of r and r Thus for solutions r r of L r = 0 and L*r = 0 one has 01P~ k = 02P~ k. This implies that there exists w ik such that
p~k = O2wik; p~k = Olwik and the quantities X i = ~/ikJwkJ (~/ikj
are
(1.43)
constant) given by quadratures
X i = ~ikj f r ( p k J d u l + pkJdu2 )
(1.44)
do not depend on the curve F. Now consider quantities of the type X i (i = 1, 2, 3) as tentative local coordinates of a surface in R 3 induced by L. For example any three linearly independent solutions r of Lr = 0 would induce a tentative surface (for fixed ~/ikj). Assume further that the coefficients of L depend on t and satisfy a t dependent equation
M(Ot, 01, 02)r = O
(1.45)
for some linear operator M. Then compatibility of (1.45) with Lr = 0 provides a nonlinear PDE for the coefficients of L and we also have an evolving family of surfaces - provided of course that the coordinate functions fit together properly to define a surface. Thus in the background one should recall equations like Gauss-Weingarten and GaussCodazzi and in this direction we mention a few basic facts from elementary differential
geometry
(cf. [242, 974]). Thus the first and second fundamental
forms are ((~,/~ = I, 2)
~1 = ds 2 = g~du~du~; ~2 = d ~ d u ~ d u ~ ;
(1.46)
ga~ = O~XiO~Xi; da/3 = O~X2 i. N i; N i = [det(g)]-leikJo1xko2X j (e ikj totally antisymmetric, e 123 -- 1). Embedding of a surface into R 3 can be described via given gaf~ and daf~ (modulo compatability) by the Gauss-Weingarten equations 02X i
Ou~Ouf ~
_ F,. ~ O X i
,~~
ON i OX i - da~N i = 0; ~ + da,yg "~f~OuZ = 0
(1.47)
(i -- 1, 2, 3; ~, ~, ~/-- 1, 2). Here
/.ttJ
gAP(--gttu, p + gvp,t~ + go,,,)
(1.48)
These equations (1.47), determined by. gaz and daz, will have a unique solution, for suitable initial data, provided compatibility conditions (Gauss-Codazzi equations) are satisfied. Such compatibility will then determine a surface locally and they will be nonlinear equations in gaz, daz, and F~v. Thus an analogy exists here between soliton theory in that certain nonlinear equations arise as integrability conditions for the linear equations (1.47). In fact as an example the linear system used to solve the Sine-Gordon equations (with an extra eigenvalue parameter) is a rewriting of the equations of surface theory (see below and cf. [641]). Such an approach gives a natural geometric background for the mathematics associated with inverse scattering etc. and one can ask of course whether all interesting soliton situations have such a geometrical foundation.
C H A P T E R 2.
70
GEOMETRY AND EMBEDDING
We recall also certain special situations (cf. [563, 576]). Thus we will be particularly interested in the situation of minimal lines corresponding to curves of zero length. T h e n ds 2 has the form ds 2 = A(z, 5)dzd2 or ~1 ~ 2912duldu 2, since for real surfaces minimal lines are complex. The Gauss-Codazzi equations for a surface referred to its minimal lines can be reduced to the Liouville equation 02r 2 = Ke r where r - log(g12). Let us look at this more closely following [563, 576]. Thus a classical result of Weierstrass-Enneper for surfaces with mean curvature K m = 0 goes as follows. Let r and r be arbitrary holomorphic functions and define
i(~/)2 _~_r
OzWl =
f i(r
+r
OzW2 = r
X 2=
__ r
f
=
OzW3 -- --2p~2; Z 1 -- ~ W l --
-
X -
=
(1.49)
/ 2 r
T h e n the X i define a minimal surface with z = c and 5 = ~ as minimal lines. Note r and are determined via 02r = 0, 0~r = 0. This is generalized in [563, 576] to a situation 0
cgz
r
p
r
0
(1.50)
with p real and r a 2 x 2 matrix. For c T = transpose r one sees that ~* satisfies the 0 1 there is a same equation as cT so r = c T can be stipulated. Further for a2 = (-10) ~ )" For X i real and
constraint a2r - g,- so a solution of (1.50) has the form ~ = (~r gaz - 0 (a - / 3 ) one obtains (cf. [563, 576]) ~)zXl = i(r
+
cOzX2 = r
~12); 02X 1 _ _i(r - ~ ; OzX3 - - 2 r
g12--OzXiO~X i -
2(r
+r
+ ~22);
OzX2
-(~2
__ ~2 _ ~2;
(1.51)
0zX 3 = - 2 r
2 - 2det2~; d 1 2 - 2pdetr
Further for real p(z, 5) we can write
x =
+ ix
2i fF(r
= 2i f ( i dz 1 -
1- r
Z 1 - iX
X 3-- - 2 jfF(~)2~ldzl + r
where F is an arbitrary curve in C ending at z. Then ~1 - 4det2r and mean curvatures are
K = -det-2r162
(1.52)
=
Km = 2pdet-l~
1) and the Gaussian
(1.53)
and consequently the total curvature is
X- ~
K~
2i/8r / d z A d h ( l o g detr
gdz A d5 -
(1.54)
=2if0--~ s dz(log detr
and hence X is determined by the asymptotics of r and r To examine this write (1.53) as (*) r = Pr r = -Pr and l e t p ~ 0 as Izl ~ oc. Then r ~ a(5) and ~2 b(z) as Iz[ ~ oc where a and b are arbitrary functions. For solutions of (*) defined by Ir --~ Izl n, r ~ 0 a s [z I ~ c~ one obtains X - - 2 n . Minimal surfaces ~ p 0 and
2.1.
CURVES AND SURFACES
71
~/) = ~2~/)2, r -- ~22~ 1 yields the Weierstrass-Enneper situation.
For surfaces of constant
mean curvature one obtains
r
- ~1K ~ ( r 1 6 2- + ~2r162
= 0; r
+ 1Km(~/)l@ 1 -Jr- r
- 0
(1.55)
As for time evolution with u I ~ z, u 2 ~ ~ the simplest nontrivial example is
M(Ot, Oz 0~) = 0t + 03 + 3 '
(~
Pz Ot + 3 ~ w
Oe +
p~
-2 - 2 p ~
Wz]
(1.56)
which corresponds to a nonlinear integral equation for p 3 3 Pt + Pzzz + Peee + 3pzW + 3pe~ + -~PWe + -~pWz -- 0; we = (p2)z
(1.57)
This equation is the first higher equation in the Davey-Stewartson (DS) hierarchy for p, q with q = - p and it can be connected via a (degenerate) Miura type transformation with the Veselov-Novikov NVN-II equation, so one refers to (1.57) as the modified VN (mVN) equation (cf. [106, 558, 563, 576]). In particular, following [563, 576] one can say that the integrable dynamics of surfaces referred to their minimal lines is induced by the mVN hierarchy. The initial value problem for the surface ga~(t = 0) and d ~ ( t = 0) passes to ga~(t) and daz(t) (note that minimal surfaces p = 0 are invariant under such dynamics). For the I-D limit one can impose on p, r the following constraints
(Oz - O~)p = 0; (0~ - C9z)r = 2i~r
(1.58)
(~ real). Then r satisfies the same constraints and consequently the X k are constrained via
(0~ - Oz)X k = 4 i s
k
(1.59)
(k = 1,2, 3). Define now real isometric coordinates a, s via z = -1~ ( s - ia) to obtain p = p(s, t), r = exp()~a)X(s, t) and X k - exp(2,~a)ffk(s, t) (k = 1, 2, 3). It follows that K - 0 and Km = 2 p e x p ( - 2 , ~ a ) - These equations describe a cone type surface generated by the curve with coordinates X ( s , t ) - i.e. the surface is effectively reduced to a curve with curvature p(s, t). The linear problem is reduced to a l-D, AKNS type problem for X with spectral p a r a m e t e r )~, i.e.
and equation (1.57) is converted into the m K d V equation Pt + 2psss + 12p2ps = 0. Similarly the higher m V N equations pass to higher order m K d V equations. In this direction note further t h a t (1.59)implies ( 0 ~ - Oz)X k . ( 0 ~ - Oz)X k = --16)~2XkX k. Via OzXkOzX k = O~XkOxX k = 0 and (1.55) one obtains ( 2 ~ 2 ) X k X k - det2~2. But X k = ezp(2)~o)~2 k and r = exp()~o)x implies then (2x2)2k2 k = det2x. But for (1.58) det)c = constant (say 1) which entails then 2 k 2 k = (2)9 -~. Thus the curve with coordinates 2k(s,t) lies on a sphere of radius 1/2)~. For )~ = 0 one obtains integrable motions of plane curves as in [390, 391, 633, 763, 762]. Note also t h a t (1.44) implies that tangent vectors to the surface will be expressed in terms of biliriear combinations of r and ~*; this will imply t h a t the normal velocity U in (1.29) is specified in terms of square eigenfunctions as indicated (cf.
(1.34)).
C H A P T E R 2.
72 2.1.4
On the embedding
GEOMETRY
AND EMBEDDING
of strings
En passant let us indicate some geometrical background for strings following [868]. We begin with a bosonic action
1 / M v/ggabcOaZ#~
(1.61)
(a, b = 1, 2; # = 1, ..., n). This action is invariant under diff(M) and under Weyl transformations of the metric. The world sheet of this (closed) string is regarded as a two dimensional oriented manifold M embedded in R n (i.e. X : M --+ R n) with local fields X ' ( ~ ) , local charts (Ua, ~a), M = UUa, and ~ : Ua --+ R 2 with ~a~b 1 analytic. We use elementary differential geometry ideas to find structure equations describing the world sheet. Thus at ~ E M introduce an orthonormal frame (~', el, ..., en) and think of a principal frame bundle 9~+ with base M and structure group S O ( n ) x S O ( n - 2) _~ fibre (here take el, e2 to be tangent, and ea (~ = 3, ..., n to be normal, to M at ~ . Let 0t~ be a basis of T~ ( R n) at ~' E M, dual to the et, (< 0 t~, ev > = 5~). Think now o f R n with a constant (flat) metric and consider the connecv The notation used in [868] is somewhat curious here tion I-form w~v defined by d% = wue~,. but is explicated in [869]. One writes here also dx = (dxl, ...,dxn) = E 0~% ~ E O~ | e~ (then dx(ev) = F_, 5~e~ = e.). Thus det~ is a vector valued 1-form so for X a vector field det~(X ) is a vector field, det,(X ) = w ~ ( X ) e v (note w~(e),) ~ F ~ ) . Then d2x = d2et~ = 0 which implies the Cartan structure equations (corresponding here to the M a u r e r - C a r t a n equations) V dwt~V = w p A wp; dOtL = 0 v A w~ (1.62) Now let (U,~ i) be a chart on a neighborhood of ~' E M with 0j = c9/0~j (j = 1,2) a natural basis on U. Then ei = a i Oj with ~ a 2 x 2 nonsingular matrix and 0 i where (a*)} ,,~ ((aT)-1)} (recall here f " U --~ R n gives rise to f , 9 T~ ~ T x ( R n) and f* 9 T ~ ( R n) ~ T ~ ( U ) - i n the present situation f, ~ a} and f* ~ (aT)}). equations become then (i, j = 1, 2; a = 3, ..., n; it, v = 1, ..., n)
The C a r t a n
dOi = Oj A wji ; O = 0j A wj~ ; dw~ = w ~ A w ~
(1.63)
Next one can write the first fundamental form on M via (cf. (1.46)) ~1 = d~'. ~'= (0iei) 2 = 5ijd~Pd~k(a*)p(a*)Jk - gpkd~Pd~ k
(1.64)
(since the e , are orthonormal) and in isothermal coordinates (a*) j has the form
In order to describe how M is curved in II2 one uses the second fundamental form (cf. (1.46))
a2 - ~
Ha'ca;
Ha = - d e c . d~'= -5ijFJp(a*)~d~Pd~ k - hapkd~Pd~ k
(1.66)
This indicates the classical background. Now put 0 i = d~Jpbj into (1.63) to obtain immediately w~ A d~J = 0 and (-hired p + PWm) A d~ m - 0. By a theorem of C a r t a n (linear algebra) this implies ,4'=
r3~d~ " ';
log(p) + o~-
rL3d~~
(1.67)
2.2. SURFACES IN R 3 AND CONFORMAL IMMERSION
73
v i Since w,v = - w v~ one has F~I = r21 = -Ollog(p); F222 = r~2 = -0210g(p); and w,- = Fred ~ . One can combine this now with (1.67) to get
R ,vj k = 0 ~ r ~ - o~r;k + r .p~ r o vj - r .pj r o kv = 0; r ~i j = 0 (i # j)
rm
- r ~ i (# # v); haij
=( hll0 h22~)
(1.68)
; hall = - F ~ I "p; h~22 = -F2~2 9p
P u t t i n g this into R~j k = 0 yields (ka = det(haij)/det(gij))
--(02 "4- 022)log(p) = E k'a" =
r~ ~ rS(r5 +
1
F~I); ~ ~
1
p2; 2 E
01 (F22) 2 -~
02(Fll) 2 = F~I ~
(1.69)
F~I(F~2 + F~I)
Here the Gaussian curvature is K = -021og(p)/p 2 in the gauge gij = p25ij so K =- ~-~ka is the intrinsic curvature of M. Equations (1.69) can be written now in a more condensed form as O1A = 2(A + Ke2r162 02B = 2(B + Ke2r162
r = los(p); A = ~--~(F2~2)2; B -
~-~.(F~I) 2
(1.70)
For K = constant this has solutions A = 2KCexp(2r + clexp(2r B = 2KCexp(2r + c2exp(2r and the first equation in (1.70) is the Liouville equation <>r + 2Kexp(2r = 0 (<> ~ 0 2 ~ Laplace-Beltrami operator). Using complex coordinates z = ~1 + i~2, 2 = ~1 - i~2 the Liouville equation becomes "1
+ 2 K e 2r = 0
(1.71)
with general solution r
1
f'(z)g'(2)
2) = -~log( [1 + ~ ] ( z - - ~ ~ ) ] 2
)
(1.72)
(note ~1 ~ e2r Thus the Liouville equation arises from geometric properties of the embedding of the world sheet of our closed string and it can also arise via action principles as indicated later. Hence, very roughly, solutions arising from geometrical embedding ideas can be thought of as generating the conformal algebra of the Liouville field theory.
2.2
SURFACES
IN R 3 AND
CONFORMAL
IMMERSION
Notations such as ( E A ) are used. We consider a surface in the three dimensional Euclidean space R 3 and will denote the local coordiaates of the surface by u 1, u 2. The surface can be defined by the formulas (see e.g. [242, 292, 352])
X i = zi(u 1, U2), i = 1, 2, 3
(2.1)
where X i (i = 1,2, 3) are the coordinates of the variable point of the surface and the functions xi(u 1, u 2) (i = 1, 2) are scalar functions. The basic characteristics of the surface are given by the first (~1) and second (~2) fundamental forms
1 = d82 = ga~du aduz; ~t2 = dazdu adu/3
(2.2)
74
CHAPTER
2.
GEOMETRY
AND EMBEDDING
where 9af~ and d ~ are symmetric tensors and a, /3 take the values 1, 2. Here and below it is assumed that summation over repeated indices is performed. The quantities g~z and d~z are calculated by the formulas OK i 9 ~ = Ou a
9
OX i
02X i Ou~ ; da~ = Ou~Ou~
9N
i (a,/3
-
1
2)
(2.3)
where N i are the components of the normal vector OX k OX m Ou I Ou 2
1
N i = (det 9) - ~ s
(i --
1,2,3)
(2.4)
and s is a totally antisymmetric tensor with s _ 1. The metric gaz completely determines the intrinsic properties of the surface. The Gaussian curvature K of the surface is given by the Gauss formula K = R1212(det g ) - i where RaZ~ is the Riemann tensor. Extrinsic properties of surfaces are described by the Gaussian curvature K and the mean curvature 2h = gaZdaz. Embedding of the surface into R 3 is described both by ga~ and daz and it is governed by the Gauss-Codazzi equations 02X i
_
Ou~OuZ
0xi a~ ~ - d~f~Ni - 0
F ~
ON i OX i -Ou - - d + d ~ g ~ Z OuZ = 0 (i = 1, 2, 3; ~,/3 = 1, 2)
(2.5) (2.6)
where F~f~ are the Christofel symbols.
Among the global characteristics of surfaces we mention the integral curvature (see e.g. [242, 292, 352])
= ~
1
K(d~tg)~d2~
(2.7)
where K is the Gaussian curvature and the integration in (2.7) is performed over the surface. For compact oriented surfaces (EA) X = 2 ( 1 - g) where g is the genus of the surface and we will generally assume that surfaces are compact and oriented unless otherwise specified. Families of parametric curves on the surface form a system of curvilinear local coordinates on the surface. It is often very convenient to use special types of parametric curves on surfaces as coordinates. We will consider in particular minimal lines (curves of zero length). In this case gll - g22 - 0, i.e. (EB) ~tl = 2g12duldu 2. For real surfaces minimal lines are complex and gtl = 2)~(z,5)dzd5 where bar means the complex conjugation and ~ is a real function. The Gaussian curvature in this case is reduced to K = (1/g12)(O 2 log(g12)/Ou10u2). We recall also that the idea of the method of inducing surfaces following Section 2.1.3 (cf. (1.42)-(1.45)). The method based on (1.49)-(1.52) is essentially an extension of ideas of Weierstrass and Enneper for construction of minimal surfaces (Kin = 0) and we write the Gaussian and mean curvatures now as before with K m - 2h so (EC) h = p d e t - l r The hierarchy of integrable PDE associated with the linear problem (LP) (1.50) arises as compatibility conditions of (1.50) with LP's of the form Ct + A n r = 0; An = ~ - ~ ( q j ( u , t ~q2j+l jVul -k- rj(u, t ~'q2j+l~ JVu2 ). All members of this mVN hierarchy commute with each other and are integrable by the inverse scattering method. Thus the integrable dynamics of surfaces referred to their minimal lines is induced by the mVN hierarchy via (1.52). For such dynamics one is able to solve the initial value problem for the surface, namely ( g ~ ( z , 2, t - 0), d~z(z, 5, t - 0)) ~-~ (g~z(z, 2, t), d~z(z, 2, t)), using the corresponding results for the equations from the mVN hierarchy. This integrable dynamics
2.2.
SURFACES IN R 3 AND CONFORMAL
75
IMMERSION
of surfaces inherits all properties of the mVN hierarchy. Note t h a t the minimal surfaces (p = 0) are invariant under such dynamics. We go next to [791, 793, 949, 952, 953, 957] involving conformal immersions and will sketch some of the results. Consider an oriented 2-D surface immersed in R n, realized as a conformal immersion of a Riemann surface S, i.e. X 9 S ---, R n. This means the induced metric on S can be written in the form gll = g22; g12 = g21 = 0. Pick complex local coordinates z = {1 + i{2 and 2 = { 1 _ i{2 so gze = gez ~ 0 and gzz - gee - O. The Grassmannian G2,n of oriented 2-planes in R n can be represented by the complex quadric Qn-2 in C P n-1 defined by ~ n w 2 = 0 , Wk E C w h e r e w k (k = 1,.. ., n) are homogeneous coordinates in C P n-~. Writing Wk = ak + ibk with A~ = {ak}, /~ = {bk}, Qn-2 involves [[AII = IIBII with A . B = 0. T h e n A, B form the basis for an oriented 2-plane in R n and an SO(2) rotation of vectors gives rise to the same point {exp(iO)wk} in C P n-1 In the conformal gauge above the tangent plane to S spanned by ( 0 ~ I X ' , O~2X p) corresponds to the point (0~1X ]'t -31-i O f 2 X ' ) ..~ O~X" ~ Qn-2. The (conjugate) Gauss map is defined now by G ( z ) = [OzX]. Thus S ---, G2,n ~ Qn-2 and one looks for a function ~b(z, 2) (to be determined) such that O z X " = r162 where Ct~ ~ Qn-2 satisfies r 1 6 2 - 0 (r S ~ Qn-2). Note t h a t a map S ---, q n - 2 corresponds locally to r 2) = (r ..., Cn) ~ c n / o satisfying ~ ? r = 0 (here S ---, G2n ~ (~1 ~2) ~_ (z,2) --* (O~IX t~, Of 2 Z ' ) while G~ n ~- Qn-,2 via (Of~Z t~, Of2X ~) "" (OzX~)i. The line element in S is ds 2 = A2[dz[ 2 where A~'= 2[ OzXI '2 = 2[r162 2 ([1r 2 = r162 and the mean curvature vector field of S is H " = ( 2 / A 2 ) X ~ (note r is tangent to S and H t~ is normal). To see this one uses the Gauss-Codazzi equations in the form (cf. (2.5))
0~0~x = r ~~e a ~ x + H;ey~;~
a~g~ = - H ~ J ~ a ~ x
+ (Nj 9a~N~)Nj
where F ~ ~ affine connection determined by the induced metric 2) are the components of the second fundamental form along the Ni. T h e n one notes t h a t in the conformal gauge only FZz and ~l .~ L ~i a ~.Lr v~. . Assuming r exists it can be determined as follows (cf. terms of ~b and r and write
(28)
gaz and H ~i (i - 1, ..., n n - 2 independent normals F~e are nonzero and H also [439]) Express H p in
(2.9) Here I1r - r and V ~ is the normal component of 0~r ~. Since H tL and I1r ~re re~l, V~ can be written V t' = e x p ( i a ) R t~ for R t' real with a the argument of ~b for r = p e x p ( - i ( x ) . The first two equations in (2.9) are the integrability conditions on the Gauss map (not every G2,n field r forms a tangent plane to a given surface). Now V ~ is a linear combination of n - 2 unit normals to S and V t~ = e x p ( i a ) R ~ so there are n-3 conditions here plus a remaining condition determined by ~rlz = - ( l o g p ) z ~ and az~ = -~?z. Now we concentrate on R 3 although, many results for R n appear in [791] for example (cf. also [578]). Thus G2,3 ~ Q1 ~ C P 1 "~ S 2 and the Gauss map can be expressed by a single complex (Z~hler) function f (z, 2) via r = (1 - f2, i(1 + f2), 2f); r162 = O; [[r 2 = 2 ( 1 + [f[2)2; or via N =
1 + Ill ~
(f + f,-i(f
(2.10) - f ) , [f[2 _ 1)
(N ,- normal Gauss map) and the integrability condition .~]z = az~ is given by ~[(fz2/fz) - ( 2 f f z / ( 1 + [fl2))] = 0
(2.11)
76
C H A P T E R 2.
GEOMETRY
AND EMBEDDING
We obtain then Y ~ = -2f~N~;
h = H ~ N ~ or H ~ = h g ~ ; ~b = -]z/h(1 + If12) 2
(h is the mean curvature scalar). It follows then that (logr this one computes the Euler characteristic (x(g) = 2 ( 1 - g))
If l ~
(2.12)
= - 2 f f ~ / ( 1 + [f[2). From
ifzl i i-f-~ ~- -~dz A d2
(2.13)
Note here that (2.13) is expressed via globally defined objects whereas (1.54) requires e.g. det~b ~ 0 or oc. We will see that for hv/-~ = 1 surfaces det~b = (1/2p) so, assuming p -r at interior points and that p has bounded derivatives pz, p2, pz~ in the interior, one can only use (1.54) when p -~- 0 at interior points. Since this could preclude some interesting situations we will use (2.13) for calculation and refer to this as X throughout. The Polyakov action (or action induced by external curvature) corresponds to the Willmore functional and is
-
2f
sp=g02- -
2f
IIHII2v d2 =g02- -
4/
[[Vl[2 -~dz i A d 2 = --
i1r
[fz[2
(l+lfl
9 A d2 )2 ~-dz
2
(2.14)
and the Nambu-Goto action is ( S N c = a f x/~d2~) SNG
" - (7
/ [r162 .
We will be concerned mainly
with
A d2 = 2a / h2 (z, 2)l[f~]2+ ( If [~)~ 2idzAd2
/
(2.15)
Sp.
A special role is played by surfaces where hv/~ = c. Thus we will introduce a local orthonormal moving frame el, e2, and /Y ~ (~3 where el,e2 are tangent to S and /Y is normal. One can choose e.g. el
1 1+
=
1 1 i f2 1 + ]fl 2 ( 1 - ~(f2 + f2), ~( _ fi2), f + f); s -
(2.16)
i 1 1 (~(f2 _ ]2), 1 + ~(/2 + ] 2 ) , - i ( f - f)); N = 1 + IX]2 ( f + f ' - i ( f - f ) , If 2 _ 1)
The structural equations (2.8) take the form Oz
~2 e3
= Az
~2 e3
;
(2.17)
where Az is a matrix involving f, f , fz, fz. There will also be an analogous identical equation involving 0/02.. Thus Oz~i = (Az)ijej, O~ei = (A2)ijej. Then Az, A~ are components of a vector/~ in conformal gauge which transforms as a 2-D SO(3, C) gauge field. Under a local gauge transformation ~T ~ g ~ T S ~ S I, and A ~ A ~ where in an obvious notation (dropping the arrows) A~= = gA+g -1 - (O+g)g -1 (g E SO(3, C)). Using the SO(2) degree of freedom involved in choosing the ~i to rotate away a component A12 of the tangential connection via g0(r one arrives at
A'z=
0
0
~(:~g
+ hv~) 0
;
(2.18)
2.2. SURFACES IN R 3 AND CONFORMAL IMMERSION
77
l+lfl 2
a~, =
i[(109r
21'.f~ ] + l+l/,12j
0 ai ( H ~
+ h)
- h)
- ~i ( H ~ - h) 0
)
Here one has used Hzz = H~zNU = - 2 f z r where h = - f z / r + Ill2) 2 along with v ~ = 1r162 = 21r + If12) 2. This transformation resembles [887, 886] but works at a deeper level since r is involved (cf. [793]). Further argument with currents and a gauge fixing leaves Tzz unfixed and the condition h v ~ = 1 (or any constant) then singles out a certain class of surfaces (cf. also [887, 886] where light cone gauge is used). In the conformal gauge x/~ = exp(~) where ~ is the Liouville mode. In particular the Polyakov action Sp, or extrinsic geometrical action (2.14), can be considered as a gauge fixed form (in conformal gauge) of the action
i J x/~h~
SP = 9--~
O~fO~f
(1 + I/[2) 2 dz A d5
(2.19)
(this is the same as (2.14) plus terms modulo Euler characteristic- cf. [157]). The EM tensor for (2.19) is Tzz = -OzfOz]/(1 + Ill2) 2 so using Hzz = - 2 f z r h = - ] z / r + ]f12) 2, v ~ = 21r + If12) 2, we get Tzz = (Hzz/x/~)(hx/g) = Hzz/x/~ when hv/-~ = 1. For constant hx/~, (A~)12 in (2.18) becomes iO~log(h) = - i ~ , (x/'-g = exp(~)) yielding the transformation for the induced metric in Polyakov's 2-D gravity so H ~ ~ induced metric for surfaces of constant hv/~ while Tzz = Hzz/x/~ ~ EM tensor. Finally one notes that hx/~ = constant surfaces are characterized by Cf~ = constant (cf. (EI)). Now [949], which begins with a summary of the R 3 situation just discussed, provides further information. Thus first in summary, if one uses H ~ and H z z / ~ as independent dynamical degress of freedom (independent of the X u variables) then the integrability condition O~A~z - O z A ~ + [A~z,A~] = 0 can be rewritten with ~ or directly as an equation of motion 03H~ = ( 0 ~ - H2~Oz - 2(OzH~,~))(Hzz/v/-g) (2.20) Some useful formulas involving Hzz and Hzz/x/-g for hv/-~ = 1 can now be obtained as follows. Thus from OzX tt = r162 CU = (1 - f2, i(1 + f 2 ) , 2 f ) , r = - O z / / h ( 1 + If12) 2, and hv/~ = 1 one finds ( E D ) r = _ 1 while from the Gauss-Codazzi equations (2.8) H ~ = - 2 ~ 0 ~ f . This plus ( E D ) yields (EE) H ~ = O~f/Oz]. Further for by/-~ = 1 the integrability condition (2.11) can be simplified via ( E D ) and (logr = -r/to f 2 z / f z - 2 / f 2 / ( 1 + Ifl 2) = 0
(2.21)
This implies
Tzz= Hzz/v
=
2(
Dzf
(2.22)
Note here that in (2.21) H ~ has the form of a Beltrami coefficient # = O f / O f and Tzz is the corresponding Schwartz derivative. Thus an equation (2.20) becomes 03# = OTzz #~zz2(O#)Tzz. Now one notes also that the independent dynamical fields H ~ and Hzz/x/~ can be parametrized in terms of independent Gaussian maps as H ~ = O~f2/Ozf2 and Hzz/v/~ = Dzfl (the fi determine the image of the X u in G2,3). Then in in [949] an effective action depending on fl, f2 is determined and the equation of motion (2.20) is used to constrain these fields. First one derives an action invariant under Virasoro symmetries (since hv/~ = 1 surfaces have Virasoro symmetry following earlier remarks - cf. here also [801]). The gauge invariant action Feff depends on Az and A~ (we omit t h e ' now) via parametrizations Az = u-lOzu and A~ = v-lO~v. Here u, v are independent elements of
CHAPTER 2.
78
GEOMETRY AND EMBEDDING
the gauge group and this will correspond to taking Hee and Now write (cf. [801])
r~ii : r_(Az)+ r+(A~) where k = n / =
H z z / ~ as independent of X ~.
~Tr f AzA~dzAdZ
(2.23)
the number of fermions and
/ ~abc(~a U)u- l(~bU)u- l(~cu)u- ld2~dt k T +777~ ~ ] Then F+(A~) is given by a similar expression with u --+ v and the sign changed in the last integral (cf. also [801, 802]. Now take A + = Hzz/v@ Az - h v / - g - 1, and A ~ - 0 to get (cf. (2.18) - factors of i/2 are being dropped in integration)
r_(Az) = s_(fl) = -~k f ~O2f l [03zf l~ _
2(/)z2fl
~zfl)2]dz A d5
(2.25)
This corresponds to geometrical action (cf. [21, 37, 38, 165]). Calculation of F+(A~) from (2.18) is not so easy but in light cone gauge an explicit determination is possible, leading to
k .i 02f20zi)z.f2 ( ~
02f20e.f2
Ozf2 Ozf2Ozf2)dz A d2
s+(/2)=-~_~
-
(2.26)
This is exactly the form of the light cone action in 2-D intrinsic gravity theory. Finally the total action on h v/~ = 1 surfaces is k / i)2f1 [63z3fl
87r
i ~zf2 O'zS. OzO.S. ( Oz-------~2-
( 69z2f1)2]dz A d5
(Ozf2) 2
/o.s. ~zf2Dzfldz
~
(2.27)
--
A d2
This is the extrinsic geometric gravitational WZNW action on h v ~ = 1 surfaces in light cone gauge. It combines in a gauge invariant way the geometric and light cone action studied in 2-D intrinsic gravity. The equation of motion for (2.27) is
0~f2 .0~f2, ,,,~ ,Oe.f2 03Z(~zf2) - O~Dzfl - (~zf2Ji)zDzfl - ZOZ~~zf2)Dzfl -- 0
(2.28)
obtained by varying fl and f2 independently, and one can see that this is equivalent to (2.20) which can be regarded as relating H ~ and Hzz/x/'-g. It is automatically satisfied when one takes both Tzz and H ~ as determined by extrinsic geometry via X t~. Now one wants an effective action in terms of H ~ and H z z / ~ through their parametrizations in terms of the fi such that these fields are independent of X ~. First one checks that (2.27) is invariant under Virasoro transformations. Next one shows that Feff(fl, f2) = Feff(fl o f, f2o f) and chooses f = f21(z, 2) where f2(f21(z, 2),5)= z so reff(fl,f2)= F - ( f l o f21) - F+(f2o f l 1) (the last by interchanging fl, f2). This leads to r + ( f 2 o f l 1) = r _ ( f ~ o f ~ l ) = r + ( f 2 ) + r _ ( f l ) -
k /o~f2,_, ~zf2L~zfldzA d5
~
(2.29)
2.2. S U R F A C E S IN R 3 A N D C O N F O R M A L I M M E R S I O N
79
Thus in particular the properties of Fell can be understood from either F+(f2 o f{-1) light cone action of intrinsic 2-D gravity (with f ~ f2 o f{-1) or from r _ ( f l o f~-~) ~ geometric action arising from quantization of the Virasoro group by coadjoint orbits. The last (coupling) term corresponds exactly to the extrinsic Polyakov action Sp modulo X (cf. [157]). In fact the coupling term in Feff is needed in order to make it invariant under Virasoro transformations of F1, F2 (recall H~. = #(F2) and DzF1 = Tzz = Hzz/v/~). Quantization in the 5 sector is developed after modification of the conformal weight of F1 o F2--1 (where one is thinking of the geometric action realization). Fe/f is the conformally invariant extension of Sp where Tzz and H ~ are the dynamical fields. There is also a hidden Virasoro symmetry on hv/-~ = 1 surfaces where H ~ and Tzz transform as a metric and an energy momentum tensor respectively under Virasoro action. The Gauss map is important in establishing the existence of the Virasoro symmetry in h v ~ = 1 surfaces. 2.2.1
Comments
on geometry
and gravity
We make here a few further comments about the Liouville equation, Liouville action, etc. The Liouville equation classically involves e.g. 0z2~r = - 8 9 1 6 2 or (for p = exp(2r 02~log(p) = - K p where K ~,, Gaussian curvature. On the other hand classical conformal unquantized Liouville action involves e.g. (7 ~ h) 1
f V/~(lgab~a~b~ Jr-(~R(g) + ~--' ecp) z 2
SL - 47r~2 a
=
(2.30)
4-~1jv~(-~ [ l~b 0~r162 1 CR(~)+~~)2~2
as in [386] (cf. also [27] for other notations). Note that the second formula follows from the first via r ---. Vr The equations of motion from (2.30) are evidently (EF) 7A~b = n([l) + ( # / 2 ) e x p ( 7 r and from [386] R(exp(2o)[7) = exp(-2o)[R([7) - 2Aa)]. Hence for [7 ~ g = ezp(2a)[7 and 2a = 7r one has 0 = R ( g ) + (#/2) or R(g) = R(exp(7r = -(#/2). Thus the Liouville field r or ~/r is thrown into the metric and one looks for a metric with constant Ricci curvature - ( # / 2 ) . Thus Liouville theory can be thought of as a theory of metrics and and equation such as ( E F ) is sometimes called a Liouville equation. Now we know that the Liouville equation with g = [Texp(7r provides constant curvature Rg = - # / 2 (given a background metric ~). One has equations of motion of the form (~/ = 1) ~zz "~ A~ = R(t~) + (#/2)exp ~ as above. However we must not confuse this with the siutation of [791, 793, 949, 956] where one should emphasize in particular that the Polyakov action of (2.14) or (2.19) is a special action introduced for QCD to cope with quantum fluctuations. It becomes the kinetic energy term of a Grassmannian sigma model (cf. [956, 1004] where the Nambu-Goto action or area term also becomes an action with local coupling l/h2). It is not the same as the Polyakov action of Liouville gravity, which is equivalent to the Nambu-Goto action there, but rather a string theoretic term in QCD (as well as a crucial geometric ingredient for W gravity). This is related to the idea that a geometric realization of W gravity as extended 2-D gravity involves, in R 3, surfaces of constant mean curvature density (hx/~ = 1) in which Tzz ~ (Hzz/v/-g). The corresponding W algebra in this case is the Virasoro algebra. This is accomplished in a conformal gauge for the induced metric (~ H22). The mathematics however, involving the K~hler function f of (2.10), then leads to formulas similar to those of Liouville-Beltrami intrinsic gravity a la [430, 432, 699, 801, 802, 865, 869, 963] for example (e.g. formulas such as (2.20), (2.25), (2.26), etc.). In particular the Polyakov action Sp or Sp leads to the basic EM tensor
80
C H A P T E R 2.
GEOMETRY AND EMBEDDING
Tzz and metric H~e above which can be used as basic variables (via Kiihler functions f) in formulating an effective action F e / / a s in (2.27). Further, following [793], one has to be careful to distinguish conformal gauge and light cone gauge (cf. here [886, 887] where light cone gauge is used). Also we must recall that in [793], the condition h v ~ = 1 is a gauge fixing, and some formulas hold more generally before such a fixing. For example the Gauss-Codazzi equations imply Hee = -2~Oef and in general one has also (cf. equations after (2.19)- this is organized below).
x / ~ - 21r
1 + 1f12)2;
H~z =
-2Ar
h =
L
~(1 + If[2) 2
(2.31)
On the other hand after gauge fixing, hv/-~ = 1, one has (cf. also below) r
1
(2.32)
= --~; Tzz = Hzz/x/~
The formula Tzz = Hzz/v/-g arises after gauge fixing but is not itself a fixing (cf. [793, 949]). One notes also that hx/~ = 1 is the conformal analogue of the condition h = 1 of [886, 887] where light cone gauge is used with ~ = 1/4 (in conformal gauge ~ exp(~) where ~ is the Liouville mode or field). Similarly in [886, 887] one uses Tzz "~ Hzz. The role of Hz~ as induced metric corresponds then (for hx/~ = 1) to p = O f ~ o f being the induced metric. Equations such as (2.20) take the form then 03# = [ 0 - # 0 -
HZZ
2(O#)]Tzz (Tzz = ~ )
(2.33)
and as in (2.27) for hv/-~ = 1 we have Tzz = D z f . Such formulas also arise in [430, 432, 865. 886, 963] and we will look at this below. We will want to compare now various formulas for various actions involving Beltrami coefficients (divergence terms are frequently added without changing the theory). 2.2.2
Formulas
and relations
It is clear that there is a strong interaction between the material just sketched on induced surfaces and conformal immersions; we will establish some precise connections here. This will provide some new relations between integrable systems and gravity theory (cf. [157]). First consider (cf. (1.51)) (EG) OzX u = (i(r162 r162 2, -2r Evidently OzXU.OzX ~ - 0 with [10zX#[[ 2 -- Oz x t t " Oz-X # = 2 ( [ r 2 + [r 2 = 2det2~ (2.34) (r will be used for the matrix involving r r and ~ will be used in OzX u = ~r We note that the Weierstrass-Enneper (WE) ideas motivated work of Kenmotsu [541] which underlies some work on the Gauss map (cf. [469]) so there are natural background connections here (some of this is spelled out later). Now let r coordinatize the map S ~ QI and be represented by (2.10) for some complex function f. We can determine f and ~ in terms of r ~b2 via i(r
+ ~}2) = r
- f2); ~12 _ r
= ir
+ f2); - - 2 r
--
2r
(2.35)
Consequently G W E inducing (1.50), (1.52) and the Kenmotsu representation are equivalent and one has (EH) f = i~1/r r = ir 2.
2.2. SURFACES IN R 3 A N D C O N F O R M A L I M M E R S I O N
81
To prove this note from the formulas
for real h one gets r
fz
2ff~
= - --------~; 1 + Ifl h = - r
(logr = eft; r
(2.36)
+ If12) 2
= Czf. The Kenmotsu theorem gives the condition h[fz~
2ff~f~
1 + Ifl 2] = hzf2
(2.37)
for existence of a surfaces with a given Gauss map and mean curvature h and (2.36) with its conclusion correspond to this (cf. [469], second reference). Writing now r
-- - - ] ~ - - i ~ ;
r
-~ V/-~--ir
P =
(2.38)
+ Ifl 2)
one shows that equations (2.36) and its conclusion are equivalent to the system L ~ - 0 of (1.50), or r = Pr r = -Pr Evidently ( E H ) holds and the Jacobian of the transformation (f, r ~ (r r is equal to 2. We will now develop some relations between the r P, ~, and f. Situations arising from the constraint hx/~ = 1 will be distinguished from the general case when possible, but the derivations are often run together for brevity. The situations of most interest here involve hx/~ = 1 and we will therefore concentrate on this. First in general, from (1.53) and ( E C ) , one has h = pdet-l(b and ds 2 = A2dzd2 where A2 = 2110zX~l12 (we choose this definition for A and will change symbols for other A). Hence from (2.34)
A2 = 4det2~, detr = A/2; h = 2p/A
(2.39)
Note also the agreement of K in [577, 791]. Now recall (after (2.19)), h = - ] z / r + If12) 2 and x/~ = 21r 1 + If12) 2 so hv/-~ = -2fz1r162 = - 2 f z r and since (fz) = f~, one has (EI) hx/~ = 1 - r = - 1 / 2 . Also for hx/~ = 1 from (2.22) Tzz = H z z / v / - g - D z f and the integrability condition (2.11) takes the form (2.21). This leads to (r = pr ~ ~lZ =
1 = i~22(i~1/~22)~ = _~22 ( ~2
r
-- - p ( l r
2
2 + 1r
~ det(b = Ir
+ 1r
~2
)
(2.40)
= 1/2p (hv~ = 1)
P u t t i n g this in det~b = A/2 gives A = 1/p and h = 2p 2 = 2/A 2 while the function K is determined via g = -det-2(b(logdet~)z~ = -4p2(logdet~)z~ (note by/-~ = c is of interest h e r e - not h = c). We also write ( f = - i r 0~]= -(i/r162162 ~1~2z))
gzz
fz _ r162 - r fz p(1r e + 1r
~_~ 2 ( ~ 2 r
- r
-- 2 ~ 2 0 2 ( ~-1
r ) (hv/-~ -
1)
(2.41)
Also from (2.22), noting that Ozf = -ip(Ir 2 + 1r = - i / 2 ~ 2, which implies 0z2 ] (i/2)2(~2z/r 3 = - i p r 1 6 2 3 and 03] = -ipz~bl/~b 3 - i p ~ l z / r 3 - 3ip2~b2/~ 4, one obtains
2
Tzz = ~222(pzr162 + P r 1 6 2 Consequently for hx/~ = 1 we have (2.40) - (2.42).
2 -~-O~(p~;1) ( h x / ~ - 1) ~p2
(2.42)
CHAPTER 2.
82
G E O M E T R Y AND EMBEDDING
R E M A R K 2.1. We indicate here some calculations designed in particular to confirm various results in [793]. Thus for h v ~ = 1 we have first (recall (Fz) = F~) (2.43)
~_ e~ __ [ r 1 6 2
R~c~ll n~xt that h v ~
:
A2 1 + if[2)2 = 2det2~ = _~_; A - - ; h = 2p 2 P
2[r
= 1 ~ Cf~ = - 1 / 2
~nd from ~
= 21r
+ If12) 2 on~ g~ts h -
2/A 2 = 2p 2 (also h = - f 2 / [ r + 1f12)2] - cf. (2.36)). From Hzz - - 2 f z r = - ( 2 f z / f z ) f z ~ and (2.41), namely H22 = f2/fz, we see that H22 = (Hzz). Note that in general one expects only (Hzz) = ((Hz)z) = (Hz)2 = / ~ 2 . Further from [577] K = -4p2(logdet~)z~ = -4p2(log(1/2p))z~ (1/2p = [r [r Evidently (logh)2- -~2. Further ~ exp(~)= 1/2p 2 implies 2p 2 = exp(-~) = h and ~ = -log(2p 2) with ~z~ = -2(logp)z~ while K = -4p2[log(1/2p)]z~ implies g = 2exp(-~)(logp)z~ so K = 2exp(-~)(-~z~/2) = -(z~exp(-~) and hence ~z2 = -Kexp(~) or ~z~ = - K / h = - K v @ Note that in [798] one writes ~z~ = g e x p ( - ~ ) which is equivalent to (-~)z2 = - K e x p ( - ~ ) or ~z2 = -Kexp(~). Also we have for hv/-~ = 1 the equations r = - ( 1 / 2 ) and this with h = - ( f z / ~ ( 1 + [f]2)2) implies h = (20fOf/(1 + [f[2)2) while in general H ~ = - 2 r plus hv/-~ = 1 implies g ~ - O f / O f (cf. (2.21).
We want to exhibit next the restrictions (if any) on p, ~bl, r which are imposed by the requirements (2.15), hx/-d = 1 and Tzz = Hzz/v/-g (note (2.42) is the calculation Tzz - D;.f and Hzz = - 2 r = -2ir162 -- 2(~1zr - ~l~2z)). One obtains first then Tzz = Hzz/V~ = 4p2(r162 r which must equal (2/'~2)0(pr by (2.42). Hence we have the following conditions on p, r r 2P 2 ( ~ l z ~/)2 -- ~1~/)2z) ---- G1 0 ( p ~ l )
Ir
+ 1r
1
~- ~pp; r
----Pr
r
;
(2.44)
--" --Pr
(the latter equations being equivalent to ~12 = p~2 and ~2z = - p ~ l ) plus (2.20) (which will turn out not to be a restriction). Recall also
of
# ~" H z z ~- 0-7 - - - 2 ~ / ) f 2 - -
2(~2~/J12- ~1~22)
(2.45)
which leads to Tzz = 2p2# which is quite pleasant and equation (2.20) has the form 0 3 # [ 0 - # 0 - (20#)]Tzz. One can now show with a little calculation that (2.44) and (2.45) are compatible and we have the result: Given the basic evolving surface equations ~lz - p~2 and ~2~ = - p r with p real one achieves a fit with comformal immersions via ( E H ) . The condition hv/~ - 1 implies then that det~ = [~)1[2 + [~2[ 2 - - (1/2p) (and h - 2p 2) and these conditions imply the first equation of (2.44) which says that Hzz/v/~ - Tzz - Dzf. This all implies Tzz = 2p2# (Tzz = Hzz/v/g, # = fi2/fz) and the only additional condition then on p, r r is that (2.20) hold in the form 03# = [ 0 - # 0 - (20#)](2p2#). However this equation is always true when Tzz = D z f with # = f2/fz a Beltrami coefficient (as is the case here). This is stated e.g. in [430, 963] and verified in [157] (it is also implicit in [165]). This means that (2.20) is automatically true and hence there are no a priori restrictions on r p imposed by the fit above, beyond the condition det~ = 1/2p. The Liouville equation
2.2. SURFACES IN R 3 A N D C O N F O R M A L I M M E R S I O N
83
{z2 = - K e x p ( ~ ) also holds automatically here as do the equations (cf. [793]) O# + O~ - 0 and 0# + 0~ = 0. All that remains for proof are the last two equations which arise in [793] when ~
-
exp(~) and the second fundamental form (Haz) are used as independent variables. We check these as follows. Since 2p 2 = exp(-~) one has - ~ = log(2) + 2log(p) so the requirement involves 21og(p)z = O# and 21og(p)~ = 0#. Then from # = 2 ( ~ 2 ~ 1 ~ - @1~2~) we get for example Pz - 2(~2zelZ + ~2r
- el~)2z2
- r
(2.46)
= 2[-p~1~212 -~- ~2 (p2~22 -[- p~)22) -- P~)2~222 '] ~)1 (P2~l -[- P~12)] = 2{p2([r
2 -]-[r
2) -- P~1r
-- P~2~2~ -}- P[~2(--P~I) -}- r
Now the last [ ] is zero and from (2.44) we have ~1~1~ + ~22~2 - -(P~/2P 2) which implies #z = 21og(p)2. The equation #~ = 21og(p)z is then automatic. We consider next the various actions in terms of the r
i~/r
A = ip(lr
2i/
SP = g-~
2 + 1r162
Thus from (2.14) (f
-
h v ~ = 1)
IAI 2 (1 + If[2) 2dz
i/
2i/
dzAd5
A d2 = g--~o p2dz A d2 - 2g---~o ( ell2 + 1r
2
(2.47)
while from (2.25) the geometrical action with fl = f becomes (cf. calculations in (2.42)) S_=~
47r
0 ~ [ 0~ - 2 ( 0 z f
[(Oz(P~)l)/r
--
(2.48)
ADS--
'
P2(~)l/~)2)2]dz A dY_,
(one notes that calculation with f is appropriate since #, T are defined via fl, f2 ~" f). From (2.25) and (2.48) we can write now
= -021ogr This leads to
- 2(01og~2) 2
(2.49)
k /[021og~2 + 2(Olog~2)2]d z A d2 47r
(2.50)
We consider also the Nambu-Goto action of (2.15), which we write as S N G = Or
/ v/gd2~ = T =
(1r
ia
[r162
A dy, = ia
+ 1r
A d5 = -~-
/
[@[2(1+ [f[2)2dz A d5
ia / dz
A
p2
d2
(2.5:)
Further in general we look at S+
--
k
-
izzi~]dz A d~
(2 521
CHAPTER 2.
GEOMETRY AND EMBEDDING
(]~/]z) so #z = ( ] z ~ / ] z ) -
(f~fzz/]2z) while #z = 2(log(p))~ as
84 a n d recall h o w e v e r t h a t # = well. A l s o
]zz ~ =
_~/~(~
= _~ (~
k/
Consequently one has
s+ = ~
(2.54)
(log(~2)~(log(p))~
Finally we compute also x(g) via (2.13) to get 27rx(g ) = f Rv/-gd2~ = i /p2[1 -Ipl2ldz A d5
(2.5~)
Thus we can state that for hx/~ = 1 the quantities Sp, S_, SNG, S+, and X are given via (2.47), (2.50), (2.51), (2.54), and (2.55). We remark in passing that the genus of our immersed surface corresponds to the degree of the mapping S ~ C P 1 and the total curvature is X = 2 - 2g For immersions into R 3 this is the only topological invariant whereas for R 4 one obtains the Whitney self-intersection number, which has an interpretation in QCD (cf. [956]). See here also the discussion in [558] (second book), pp. 169 and 181, in connection with charge and the Ishimori equation (cf. [157]). Consider now the extrinsic Polyakov and Nambu-Goto actions (cf. (2.14) - (2.47) and (2.15)- (2.51)) which we rewrite here as ( h v ~ = 1) SP :
2i .i g'-~O-- p2dz A d2;
ia .l dz A d5
S N G = --4 __
p2
(2.56)
Now go to the modified Veselov-Novikov (mVN) equations based on (1.56) to obtain for
M~2=O
3
o,r
r
+r
+r
+ 30r
+ ~w~r
+ 6(--~-2 z~22z + 3WOlz = 0
r
+ r
+ r
+ 3Wr
+ 3~r
3 r + ~Wzr + 3(-~-2 )~1~ = 0
where w~ = -[(~21zr
(2.57)
From the mVN equation (1.57) one has also
Consequently we obtain (assume a closed surface or zero boundary terms)
dSp 2i ] 0(P2)dz A d2 = 0 dt - g2 Ot o
(2.59)
Thus Sp is invariant under the mVN deformations which means there is an infinite family of suraces with the same Sp. In particular this would apply to minimal Sp surfaces which in the corresponding quantum problem would correspond to zero modes. Further one knows via [564, 575, 900,988] that the integrals of motion are common for the whole mVN hierarchy (where the n th time variable would correspond e.g. to Mn ~ Ot + 0z2n+1 -~- 02n+1 + - " "). In the one dimensional limit this hierarchy is reduced to the mKdV hierarchy. In any event we can state that for compact oriented surfaces Sp is invariant under the whole mVN hierarchy of deformations (hv/~ = 1).
2.2. S U R F A C E S I N R 3 A N D C O N F O R M A L I M M E R S I O N
85
We note however that separately (2.58) does not yield zero for O t S N G or ORS-. From the point of view of inducing surfaces one continues to ask what is the role of the condition hx/~ = 1 and this has the following features. Thus consider ~]z - Pr ~2~ = - p r under the constraint [r 2 + [r 2 = 1/2p, which leads to r
1
r
- ~ ( ) z Ir ~. +
1
= 0; r
r
+ ~ ( ) z Ir ~ + Ir
- 0
(260)
This system has several simple properties. First it is Lagrangian wih Lagrangian -- ~-)l~2z nt- ~1~D22 -- ~22~12 -- ~2~Dlz Jr- log(]r
i + 1~212)
(2.61)
(confirmation is immediate). Introducing coordinates z - (x + iy)/2 one has the system
1
r
r
1
(2.62)
~)lx -- i~)ly -- ~ ( , ~21[ , [ -~-[['~3202 ) "- O; ~2x -+- i~)2y q- ~ ( [~)112 q-[ ['~)2'2 ) -- 0
where x plays the role of time. This system has 4 real integrals of motion, namely c+
P --
/
= [
J
+
dy(~21y~2 - ~1r
+
+
7-[ -
c_
=
+
J
/ dy{i(~21y~2 -~ r
-
(2.63)
-
1
q- ~log(l~ll 2 Jr
1~12)}
Again confirmation is straightforward (note Pz = - ( 1 / 2 ) f dyOylog(lr 2 + 1r and 7-/x f dy.O). Next we see that the system can be represented in the Hamiltonian form (E J) ~lx {r 7-/}; r = {r 7-/} where the Poisson brackets are given via
5/ ~r 5g {f' g} = f dY[5r
5f 59 5r 5r
(f ~ 9 ) ]
(2.64)
The corresponding symplectic form is ~ = dr A dr + d~l A dr The equations (E J ) are easily checked and we omit calculations. One can also say that the interaction part of the Hamiltonian 7-I is (EK) ~-~int = l l o g d e t ~ which has a pleasant appearance. Thus for hx/~ = 1 we have (2.60) - ( E K ) . Thus (2.60) is a Hamiltonian-Lagrangian system inducing surfaces with the property hv/~ - 1. Let us next consider particular classes of surfaces with hx/~ = 1 which are generated by the Weierstrass-Enneper formulas in the case Pz - Pz (one dimensional limit corresponding to curves). In this case (z - (1/2)(x + iy)) referring to [563] we can write ( E L ) ~ 1 = r(x)eAY; r = s(x)e Ay where r,s are complex valued functions and A = i# is imaginary. The system (2.62) becomes now
1
s
1
r
r~ + ~r - ~( irl2 + i~1--------~)= O; ~ - ~ + ~( ir12 + 1~12) = 0
(2.65)
We write r = r] + ir2, s = 81 -~-i82 then to obtain (E = rl2 + r 2 + s 2 + s~) 81 rl~+#rl-~=O;
82 r2~+#r2-~=O;
rl Sl ~ - # S l + ~ = O ; z.,-,.
s 2 ~ - # s 2 + ~ =Z.-. O
r2
(2.66)
CHAPTER 2.
86
GEOMETRY AND EMBEDDING
It is easily checked that this system has the following two integrals of motion --#(r181 -[- r282) q- llog(.~); M - r182 - r2sl
(2.67)
Further the system (2.66) is Hamiltonian with (EM) ri~ = {ri, ~}; si~ = {si, 7-l} where the Poisson brackets arise from (2.64) in the form {f'g} =
f
51 5g dY[Srx 581
5f 5g 582 5r2
(f ~ g)]
(2.68)
One checks that ~ and M are in involution ( { ~ , M } = 0) and thus the system (2.66) is integrable in the Liouville sense with two degrees of freedom. The induced WeierstrassEnneper surfaces (developable surfaces generated by the curves) then have the form X 1+
iX 2 = 2ie -2i~y/[(rl - ir2) 2 -
(81
-
is2)2]dx';
X 3 - - 2 / ( r i B 1 + r2s2)dx'- 2My
(2.69)
and we refer to [564] for more on this. Consequently for hv/-~ = 1 and Pz = P~ with ~b~ - ~bz = 2 i ~ , ~ real, we obtain ( E L ) - (2.69). There is much more in [157] in connection with Liouville-Beltrami gravity and calculations based on [430, 431, 432, 865, 866, 867, 869, 963]. This leads to further results concerning Sp, S+, and S_ etc., some of which have a partially heruistic nature until suitably embellished with further information about projective connections (cf. [17, 1011]). In most cases one already knows however from [469, 791,793, 949, 956, 952, 991] that the integrands involved are in fact well defined global densities. Various other miscellaneous results are also proved. Further results on Willmore functionals, Weierstrass-Enneper embedding, elastica, strings, etc. appear in [569, 570, 571, 572, 575, 706, 709, 711, 713, 714, 715, 716] and will be discussed in Section 2.4.
2.3
QUANTUM MECHANICS ON EMBEDDED OBJECTS
Notations such as (FA), ( G A ) , and ( H A ) are used. We go now to the Matsutani papers listed earlier ([702]-[718]) where a main theme is the study of QM on embedded curves and surfaces (cf. also [206, 739] and Section 2.1.1). There are of course well known constraint techniques in QM (cf. [451]) but it is quite fascinating to look at a hands on approach which examines geometrical effects of embedding. We will pick up the story in [707, 710, 709] and work in 2 or 3 ambient space dimensions. One recalls first that the methods of soliton mathematics for KdV for example involve a sort of fictitious QM involving inverse scattering ideas, Lax operators, etc. However the physical meaning of the mathematics is obscure since there is no inherent probability density or other intrinsic QM feature and the time develoment of the wave function is not really quantum mechanical (cf. however Chapter 4). Now recall that the dynamics of a thin elastic rod has been studied in the framework of soliton mathematics (cf. Section 2.1.1 and in particular [390, 495, 557, 640, 446, 926, 927, 960, 961]) leading e.g. to the mKdV equation. Submanifold quantum physics also arises in condensed matter physics and it is known that if a submanifold has an extrinsic curvature it sometimes generates an attractive potential in the quantum equation (SchrSdinger equation); one refers here to [206, 504] for example. In [702] in R 2 one studied the Dirac equation along a thin elastic rod in the spirit of submanifold quantum physics and found that the Dirac
2.3.
QUANTUM
MECHANICS
ON EMBEDDED
OBJECTS
87
operator can be regarded as the Lax operator of the mKdV equation. Since the dynamics of the rod is governed by mKdV this implies that the fictitious linear differential equation in the soliton physics can be regarded as a real QM wave equation on the soliton as its base space (not entirely clear here but meaning should emerge in the discussion). The difference between the time development in the real physics and the soliton physics is interpreted by the Born-Oppenheimer approximation, i.e. by the geometrical phase (cf. [863]). Thus the fermion in the thin rod can be expressed in the language of QM in the soliton physics. As for R 3 one arrives at the NLS equation and a corresponding collection of analogies. There is also an anomaly here connected to the index theorem (cf. [709]). There will be a number of apparently reasonable physical arguments and assumptions and we will simply accept them naively (without claim of full digestion). Such approximations are not always clear and one could also imagine other physical arguments reaching the same goals. However the overall achievement of putting together soliton mathematics and QM for realistic physics in one theory seems so attractive that the threads of argument deserve a sketch, however incomplete or tenuous, and we proceed in this spirit, following Matsutani. 2.3.1
Thin elastic rod
One assumes that the physical states are identical at space • so the space is regarded as a 2-D torus T 2 - S 1 • S 1. Let K be the position vector along the curve C c T 2 with arc length s ~ ql. Crossing points will be permitted but not cusps or wild topologies. Take as orthonormal basis the unit tangent vector nl and unit normal vector n2 with Frenet-Serret equations (FA) O s F - gl, 0s?~l -- k -- k?~2, and Osg2 - - k n l where k(s) is the curvature vector, k ~ curvature, and Os = O/Os. For gi written as ( F B ) ( C o s O , SinO) - ?~1 a n d ~2 - ( S i n O , - C o s O ) the curvature is k - OsO. The Lagrangian density and action for the elastic rod are 1 ~-, = -~p(Ot~ 2 -- 1 Aft2; S = / dt f0L dss
Ot~
(3.1)
where L is the length of the rod, p is mass density, and A is an elastic constant. In terms of an infinitesimal value e(s, t) the variation of CE from C is (FC) ff~(s~) - ~'(s) + (Osc)g2. Consequently (FD) OsFc = (1 - ek)gl + (0He)g2 and ds~ - dKcdK~ -- (1 - 2ek)ds 2 + 0(~2). The Jacobian is OsS~ - 1 - c k and the change of curvature is then -~ 02_, ke ---- ~se2re -- 7~1 (--(0ss
-+- ?~2[k -~- (k 2 -4- C~s2)E]-4- O(~ 2)
(3.2)
Hence the kinetic term becomes (FE) k2ds~ = [k 2 + (2kO2s + k3)e]ds + O(e2). Due to ( F B ) one has Otgl = (OtO)g2 and in order to simplify the problem one considers the case in which the shape of the rod does not change (this corresponds to a one soliton solution). Then g l ( s , t ) = g l ( s - ut, O) or K = Ugl. Suppose that the local length of the rod does not change as well so i)sOtK-- Otgl. Then one has the relations (FF) ku - OtO, i)su - O, and OtF~ = (u - eOtO)gl + Oteg2 with kinetic term (Or~'~)2ds~ = (u 2 - 3uei)tO)ds + O(e2). The variation of s then gives the equation of motion of the rod as 3 08 .030 A/00\ -~pu--~ + d ~ s 3 + -~ ~
3
--0
Scaling the time appropriately and differentiating in s (3.3) becomes for v (1/2)Os 1 3
Ot + ~0 s + Osss = O; vt + 6V2Vs + Vsss - 0
(3.3) (1/2)k = (3.4)
C H A P T E R 2.
88
GEOMETRY AND EMBEDDING
The calculation is for a one soliton solution but the equations (3.4) extend to a general situation with multisoliton solutions; this agrees with [390, 495, 557]. This sometimes arises with a different time scaling in the form (cf. [461,710])
4vt + 6v2vs + vsss = O; v = Oslog (~__ ) ; D2sT+T_ = 0; (D 3 = Dt)T+T-- = 0; D n ( f g ) -
f(~s-
(3.5) -~s)g
Now consider the Lax equation for mKdV. One introduces a fictitious quantum system with wave function r (fictitious means that r does not represent any physical particles or other quantity). One of the Lax operators for mKdV involves
L = a20s
+ olv; L r = A r
(3.6)
where A expresses the infinite degrees of freedom in mKdV. The partner of L is the time development operator B where
i r 1 6 2= B e ; B = -4i02 - 3iOs[v 2 - i ( O s v ) a 3 ] - 3i[v 2 - i(OsV)a3]Os; O t L - i[L, B]
(3.7)
Note the factor of i is artificially introduced in B in order to give a QM type development in iOtr = B e although this time development is a priori different from say the Dirac equation. 2.3.2
D i r a c field o n t h e r o d
One constructs now the fermionic QFT on the rod C c T 2, considering a ( I + I ) - D dimensional curved space time embedded in (2+l)-D space time. This will be different from standard fermionic string theory ~ la [411] since the theory here involves the extrinsic curvature effect on the submanifold (cf. [93, 364]). Thus assume the system is compact with positive definite metric diaghij - (1, 1, 1) and a periodic connection in time so one has T 3 -- S 1 • S 1 • S 1. One assumes that the dynamics of the rod is much slower that that of the fermion and the adiabatic approximation is employed, i.e. Oov ~ 0 where 00 = O/Oq ~ with q0 the fermion time (thus for the fermion the base space does not appear to move). This implies that the kinetics of the base space and that of the fermion are separated and expressed via two times q0 and t. Note OtA = 0 means that the base space (rod) can adiabatically change its shape without changing the eigenstate of the particles. The relation of time scales is somewhat confusing and we hope the discussion will sort this out" in fact the discussion in [710] is somewhat cleaner here and we go to that. It is pointed out there in particular that the original development of the fermion-boson correspondence in soliton mathematics due to Date-Jimbo-Miwa-Sato-Kashiwara et. al. (cf. [147, 210]) is mathematical. There is no physical (or geometrical) meaning to the fermionic field while the soliton is a physical object. On the other hand Ishibashi, Matsuo, Ooguri, Saito, and Sogo applied these ideas to string theory (cf. [491,831, 884]) where the fermionic field of the theory can be regarded as the fermion over the string (but the soliton equation becomes an auxiliary object). In the Matsutani theory on the other hand both the soliton and fermion can be considered as physical or geometrical objects. Now one considers only the space geometry since the time qO direction is essentially trivial and writes ~ = ~'+ ?~2q2 for a nearby point to C. There is a Jacobian element e!t~ ----- O''xi~ where (FH) e~i _ [1 - k(ql)q25p,1]nip (no " = 5~. " The field operator is sum i n # ) . Let 0 < p < 2 a n d q 0 = x 0 with %0 = 5~0 and e~) = (91, ~2)T and one obtains the Dirac equation ( F I ) 7 i 0 i = V~0~ (cf. [707] for more detail). The original Lagrangian is given by ~ ( 2 + l ) D
--
ieff27~'Ot,~ - e~Vq2
(3.8)
2.3. Q U A N T U M M E C H A N I C S ON E M B E D D E D O B J E C T S
89
where the (2+ 1)D 3' matrix is Vt~ = 3'ieit~ with e = det(ei~ = (1 - kq 2) and V is a confinement potential of the form V = (q2)2/25 with 5 ~ 0. Since the potential behaves like a mass one can avoid the Klein paradox on the confinement (more detail on this is given in [135, 206, 504, 707] and we will try to discuss it further below). Since the measure on the curved system is dax = e. d3q one writes ~ = ( 1 - kq2)1/2~. Due to the confinement potential V, q2 vanishes and the Lagrangian density becomes (F J ) Z ; = i~(v~ + 7101 + where r ql) = r ql,q2 = 0). Then the Dirac operator in this sysem is given as ( F K ) z~ = i(7~ Let now a, ~ , . . . run over 0, 1 and write ( F K ) a s ~ = 7~:D~ where :Do = 00 and :D1 -" 01 ntAfter the confinement z~ is not self adjoint and the penentration of the energy spectrum to the complex plane generates a gauge anomaly (see below);:_ The left derivative part is ( F K K ) - i ~ = i ( - 0 c~7~ + v72) - - i v Ol+':'-:Dc~ where ~ 1 = 0 1 - iv7 ~ One uses now the representation of (1+1)D 7 matrices in terms of Pauli matrices via (FL) 7 ~ - a 3 with @ = a 2 and 72 - -i7~ = -(71. It will be seen below that these Dirac operators agree with the Lax operator of mKdV. In terms of the confinement prescription one obtains the (1+1)D partition function
v',/2)~)
iv70.
(3.9) In the action integral the integral region along q0 is [ t - (5q~ t + (5q~ for suitable 5q0 (time scales are discussed in [707, 710]). Thus the partition function Z depends on the elastica time t via v(t) and
Z[~?, ~, v, t] = det(~)exp ( f d2q~-Tp-lzl)
(3.10)
where ~ - 1 is the Green's function. Now one shows the relation between this system and the Lax equation of the mKdV equation; also the adiabatic approximation is "justified". Since for the fermionic field the time q0 direction is flat one can write
~(qO, ql) : ~
1
~-~e-iEq~ E
r
Similarly the measure becomes (FLL) D e = leading to
Z: / liE D~JEDr
1
ql) : ~ ~-~eiEq~ ~/oq~ E
YIqo,qld r
~ q l ) (:x:
(3.11)
l-IE,qld~E(q 1) -- I-IED~zE
(-- ./dEdqlf_..[~JE,~E] --[-~E~)E--[-~2E?]E) ~/,class
(3.12)
For the partition function one can define the classical field ( F M ) ~E (ql) _ 5lo9(Z)/~flE ~/.class - w,~,class(ql) for and setting r/c< 5(q 1 - q~) one has (FN) H r ass = i ( a l 0 1 - u _2~ujq; E = t'r ql :/= q~. This is the same as the Lax eigenequation for mKdV. Although the partition function (3.12) does not include information on t development one can argue for the t dependence of the wave function ~o/,dass (thus considering a justification of the adiabatic E approximation). Generally E depends upon t due to the adiabatic condition and in order to retain the adiabatic approximation OtEht should be smaller that 5E = IEn - En-ll. However 5q~ and the length L of the rod are sufficiently large so one uses the decoupling conditions (FO) OrE = 0 and [Or, 00] = 0. In terms of the "gauge" freedom one replaces these conditions with weaker ones, namely ( F P ) [ O r - B3, H] = 0 and [Or - B3, E] = 0
CHAPTER 2.
90
GEOMETRY AND EMBEDDING
where B3 is generated by 01 and v as before (see also below). Due to ( F P ) W ,,/,class is the E simultaneous eigenstate of the operator (FQ) (Or - - D = 0. In general ( F P ) gives us J - ) 3 ~ol.dass )WE the equation of v in t while the system obeys the mKdV equation so ( F P ) can be identified with the Lax equation of mKdV. The definition of B3 can be given canonically via (note
~o = o.3)
=
0
LKdV+
; LKdV+ -- --0~ -- v 2 -t- iv
(3.13)
In terms of PSDO one can write
L1/2
iv O_ 1
KdV+ -- Ol -i- - ~
1
+'";
B2n+ l -
( B2n+l
0 ) + _ rL(2n+1)121 B+n+l ; B2n+l t KdV+ J+
0
(3.14)
(cf. Section 1.1.1). Going back to (3.9) one derives now the gauge transformations for this system (cf. [60, 363, 707]). Since the curvature v represents the symmetry of the submanifold one can regard the curvature as an external gauge field. In fact the shape of the elastic rod breaks the parity of the (2+1)D space time locally. Thus perturb v around itself in the space direction via ( F R ) v ~ v' = v(q 1) - 0 1 a ( q 1) with a change in Lagrangian (FS) s s iOxa(b~/2r The extra term is cancelled by the gauge transformation ( F T ) ~b ---, ~' = exp(i'yoa)r with r ---, ~ ' = ~exp(iToaT) where T is the operator of parity transformation T 9q0 ___,_q0 and one stipulates that it acts only on the differential operator T 900 ~ - 0 0 (cf. [707]). In relativistic theory ( F T ) does not have physical meaning but in nonrelativistic theory it does since time t is a parameter. Further since v depends adiabatically on t one has neglected 6q~ Now for the anomaly one follows [60, 363, 707]. On the quantum level Z changes as Z[v'] = S Df~DOexp ( - i d2qs
= f D~Dr ~r ~ee~
#2, v ' ] ) = Z1--
(3.15)
(- J d2qf_,[~,r v]) - z2
Due to the natural relation between the left and right derivatives of the Dirac operator ( F K ) and ( F K K ) one can expand the fermionic fields via (FU) r = ~ n anon and ~) - ~ m bnXtn where On and Xtn are defined via ~ r = AnOn and Xtni~ = --AnXtn with normalization f Xtn(q)Om(q)d2q = 6n,m and fermionic measure DODr - Hm damdbm. Then the transformation ( F T ) becomes (FV) r ~ m a~mOm(q) = ~ m e~W~~ 9One looks now at the fermionic Jacobian in the transformations
am = E
'
n
i
d2qXtm(q)eia(q)'Y~
= E Cm,nan
(3.16)
n
Since one is dealing with Grassmann variables the Jacobian 5r162 ( F W ) 1-[ da~ = [det(gm,n)] -1 rim dam where
[d~t(C~,n)]-i = {d~t [5~,n + i f
d2qa(ql)x~(q)?oOn(q )
is expressed via
=
= exp [ / 2qo/ql/A/q/}
(3.17)
2.3. QUANTUM MECHANICS ON EMBEDDED OBJECTS
91
However A(q) is not well defined so one uses heat kernel regularization (cf. [383] and Section 4). The heat kernel is defined as
K(q, r, T) = ~ e-(ilD(q))2rCm(q)xtm(r )
(3.18)
in
(cf. here (3.52)) where 0 r ~ = _ ( ~ ) 2 ~ with T the Schwinger proper time. One can redefine A as ( F Y ) A(q) = limr-+olimr-+qTrToK(q, r, "I-). Although _~2 _ - 0 ~ -012 + i~/~ - v 2 is not Hermitian, 01v and v 2 are bounded for an elastica, so the spectrum A2m exists in a cone around the positive real axis in the complex spectral plane. Furthermore the stipulation r -+ q in ( F Y ) implies that one is taking a high energy limit and A(q) is determined only by the circumstances surrounding cx~ in the momentum plane. There the fields behave like free fields. In any event for such an operator, for small T, we can write 1
K(q, r, ~-) - -47VTe-(q-r)2
/4T oc
E en(q, r)~-n o
(3.19)
where (FZ) el = -i~/~
has r
+ v 2 and, picking up a factor in the trace over the spin space, one = -i(1/27001v. One notes that T has no effect on this procedure so similar terms
arise from the bn leading to _
5~' 5~2'
exp
J
Note that the Jacobian is determined by only the zero mode of the eigenequations ( F U ) , namely An = 0. Next one derives the boson-fermion correspondence following [707]. Thus the so-called Ward-Takahashi identity for (3.15) is (GA) (6/6.)(z1_-- 0 giving directly an anomaly (GB) (1/5q ~ E E < (0I(~E')/2@E) > - - --(i/Tr)OlV. Using the Pauli matrices (FL) one obtains (GC) 01 < J+ - J - > = (i/Tr)OlV. Here given the periodicity of the fermionic fields and the fact that 5q~ is sufficiently large, one defines 1 J + = T r f dEr162
j_ = 7r 1 f dEg~,E+gJE-
(3.21)
Integrating ( G C ) over qX one gets (GD) < J + - J_ > = ( i / ~ ) v = ( i / 2 ~ ) 0 1 , . Another integration leads to an index theorem (cf. [707]). Furthermore, in a similar manner, one obtains (GE) 01 < J+ + J - > = v2/Tr and finally one has the boson-fermion correspondence 1
1
< 01j+ :>= ~(v 2 + iOxv); < 01j- >-- -~(v 2 - iOlv)
(3.22)
where j+ = 7rJ+. The right hand sides of (3.22) are the KdV solitons due to the Miura map. Next one can give a partial physical meaning for the Hirota bilinear identity (this needs expansion and we only sketch the argument here following Matsutani). Choose c~(q1) via(GF) ~(ql') = _(iTr/hqO)(ql'-q~)O+(q~ _ q l ' ) where 0 is the Heaviside function O(x) = 1 for x > 0 and O(x) = 0 for x < 0. The action and Jacobian (3.20) become
/ One writes now ( G G ) exp(ir
= < exp(f ql d q l ( j + - j _ ) > and defines the tau function
( G G G ) T+(q 1) = < ezp(f ql dql'j+) > (cf. [461, 480]). Then consider the situation where
CHAPTER 2.
92
GEOMETRY AND EMBEDDING
@E+ (resp. @E-) does not interact with CE- (resp. @E+) (reflectionless situation) where one obtains ( G H ) iv = 01(T+/T-) and D2T+T_ = 0 which recovers the anomalous relations
(3.22).
Next one expresses the system formally in terms of free fermionic fields. Adopting the gauge transformation a(ql) in the form (GI) c~(q1) = (1/2)r 1) the action is S = f d2qs where s = s - 0]. Then
~r162 _
i
.x.
(q~)] ;
=
_
where g0 = ( ~ 2 ~ / ~ ' ~ 2 ' ) is related to the Lie algebra in the infinite Grassmannian. Now one goes to the argument of Kuratsuji on geometric phases (cf. [488]) and considers the effect of the time t on the partition function. As with (3.11) one writes the field operators via ( G N ) @E(q1) = f dACE,A6(ql,A) and @E(q 1) = f are defined via i'y2(~/101
-F
~/2v)r
A)
=
AC(ql,A);
with normalization f dqlx(q 1, A')72r
dA~)E,AX(q~,a)wh~r~ the
x(ql,A)i~/2(~/l+-o1 + ~f2v)
=
a x ( q l , A)
functions (3.25)
1, A) = 5 ( A ' - A). The partition function (3.9) is
Z[v, t] = Z[~ - 0, 7] = 0, v, t] - . / H
d~E,A H dCE,A •
E,A
(3.26)
E,A
The coherent state I > is defined via
CE,AI >-- CE,AI >, I >= I I IE, A >= H (1 - CE,A~tE,AI0 > E,A
E,A
(3.27)
and in the fermion Fock space the unit operator is written as
f YI d~E,AdCE,Ae-fdEdA(bE'h~2r
><1-- 1
(3.28)
E,A
Consider now the effect of t on the functional integral. Assume that the time scale goes beyond the fermionic time q0 unit and reaches the elastica time t scale. To see the effect introduce a Green's function
G(t,E,A;t',E',A') = < I~tE,A(t)i71~E,,A,(t')l > = < E,A, tlE',A',t' >
(3.29)
The initial and final states are in the mass shell states and the diagonal part of the Lagrangian corresponds to E = =hA and E ' = :hA'. Due to the adiabatic condition and ( F O ) one has (GL) < E, A, tiE' , A', t' > = 5A,A'bE,E' < E, A, tie , A, t' > (cf. [488]). One divides the time interval by N with t = ty, t I -- to, and tn - tn-1 We. The total Fock space does not deform with time t because of the adiabatic condition. Hence any functional of the fermionic field in time t development can be represented by the vectors in the fermionic space at a certain time t'. However the basis IA, t > rotates in the space and the geometric phase a p p e a r s with the rotation indicated by the Green's function (3.29). To see the rotation one can insert thecomplete set via N
G(E, A, tl; E, A , to) -- limN-,cr / I I
1
"4J'(k) ,r ~WE,A~WE,A •
(3.30)
2.3. QUANTUM MECHANICS ON EMBEDDED OBJECTS
• ~xp
93
~E,A~ ~,AI H < E, A, tklE, A, tk_ ~ > /
1
According to [488] one can write now
,7(k) 2.(k-l) < E,A, tklE, A, tk-1 > = exP(WE,AV YJE,A ); ,/,(k)
,/,(k-1)
~ ( k ) WE,A -- WE,A
WE,A
(3.31)
= ~E,A~/2OtCE A
~t
Consequently (3.29) becomes
(f
G(E, A, t; E, A, t') = exp -
dt"(~E,A~2Ot,,~E,A
)
(3.32)
and therefore the partition function including t development is given by
Z[v, t, t'] = f
H
d~E,A II
E,A,t/~q ~
(3.33)
deE,A•
E,A,t/~q ~
One notes that the notation f dt f dE is not rigorous but can be expressed more precisely. N o w one shows very heuristicallyhow to pass this to the free fermion fieldtheory related to soliton mathematics. Recall that the boson-fermion correspondence is determined by the zero mode of (FU) or the diagonal part in (3.33). Due to the adiabatic approximation (GL) implies that in the time t develoment the diagonal part plays the most important role. Thus in the partition function (3.33) one employs the semiclassical approximation (i.e. the probability of the diagonal part is higher than that of the off-diagonal parts) and one considers only the diagonal part A = • in the partition function. Define then ( G M ) @A+ = CE,A+IE=A and CA- -- CE,A_IE=-A. Then use the gauge transformation (GI) and replace (3.25) by
r
A) ~
; x(q 1, A) ~
)
(3.34)
CA will denote the free field now. Now use the gauge transformation ( G F ) again and consider the time t development of the tau function (defined by ( G F ) without t dependence). Thus estimate the current in terms of the free field (3.34) via
j~0ql dql'j• - ~ql fj dAr177
l/0q dqI'/ dE / d A I/ dA ue
,.
•
-
(3.35)
~',7:~A,• ~E,A'~E,A"
Since one is thinking of the solitary solution of mKdV it is assumed that (@~+, @A-) is decoupled from (@~_, CA+). For the time development of t in the partition function (3.33) one assumes now @thia2Ot@A = 2@tAia2r (3.36)
7r
Note 0t in (3.36) acts on @A which does not directly depend on v(q 1). Now in forming the Jacobian of the Grassmannian field ( G E ) A(q) is determined in terms of the information
CHAPTER 2. GEOMETRY AND EMBEDDING
94
around oo in momentum space. Now after taking the free field expression A is defined on the Riemann sphere CP 1. One complexities the momentum space A and considers the residue at oo. Thus write
CA = ECnAn; n
etA -- E C t n A - n " /cr dA An = 60,n n ' 27riA
(3.37)
From (3.33), (3.35), and (3.36) one can now write the tau function ( G G G ) as
T+(ql, t) = / H d~AdCAgOexp [--i E(qlr162177 A
n
+ 4tr162
(3.38)
In the fermionic functional measure rIA indicates the product over the representative elements of the complexification A. The elements should not depend on the. parametrization and the space {A} should be taken modulo coordinate transformations. Further, due to energy conservation, g, which is a function of r162 has an exclusion law and A 2 must be (A') 2. These conditions are apparently characteristic for KdV as a subset of KP (some of this is unclear but believable and it seems worth reproducing since it indicates a thread of reasoning with interesting consequences). Thus as with (3.38) one redefines the tau function as
w+(t'g)=/Hdr162177
-~-~nr162177 n
(3.39)
9 ) and (GN) 7/+ = ~ m > l t2m+l En ~:F,n@-t-,n+2m+l 9 where g+ = goexp ( ~n n~mp,n~+,n (socalled Hamiltonians). Here the 7-/ term should be the Legendre transformation from the {~} system to the {t} system and then {t} might be considered as the external field in the theory (there may be some connection here with the (X, r duality of Section 1.2.3). Now introduce a new functional average via (JM ,-., Jimbo-Miwa)
< O >JM: / H dCtndCnOexP( - En nr162
(3.40)
The anticommutation relations are (GO) [era,n, ^* Cm,m]+ -6n,m6m,+ and the tau function is denoted via ( G P ) T+ --< exp[7-l+(t)]g >JM where there are obvious connections to Section 1.1 and Chapter 4. Further one has wave functions ~A+ (t) -
r177
A'5 < r162
7-+
A~ < Cm,-lexp[~+(t)]r ~_•
=
>JM
;
(3.41)
>JM
where 6 = I for the + field and 6 : 0 for the - field. T h e discussion should be considerably amplified and is included as it is because of its seeming ingenuity.
In terms of the operator f o r m m s m (GQ) fA:~(dA/2~i)r + ~)r
A
(cf. [5O6]) one c~n derive re1~tions of the form ~) : 0 which c~n ~iso be expressed vi~
dAe~(s'A)AT+(t-s+e(A-1);g)T_(t+s-e(a-1)'g)-0
=cr 27ri
(3.42)
where, in standard notation (cf. Section 1.1), ~(t,A) = ~cr h2n+ll~2n+l; e(A -1) - ( A , 3 A1 3 , . . . ) 1
(3.43)
2.3.
QUANTUM MECHANICS
ON EMBEDDED OBJECTS
95
T h e n (3.42) yields the Hirota bilinear equations for mKdV, namely ( G R ) D~-+~-_ - 0 and (Dal + 4D3)T+T-- = 0. Thus the physical partition function leads to the Jimbo-Miwa constructions for mKdV. In the second paper of [710] one discusses the physical (geometrical) meaning of the Goldstein-Petrich scheme ([390, 391]) in terms of a generalization of a Noether scheme to the submanifold system. T h e n the arc length is complexified and the Hirota bilinear equations for m K d V emerge naturally. There are a number of approximations involved however and since quantization is not featured prominently we will not discuss this here. 2.3.3
The anomaly
in R a
We will follow [709] here and refer to [705] for the Dirac field on rod in R n. Thus one considers a one dimensional non-relativistic closed thin rod C in R 3 with a position vector ~,(ql) describing its centre axis. The length L of C is to be sufficiently large as indicated later. Take an orthonormal system along C of the form (~1, g2, g3) with ?~1 a s the tangent vector ?~1 -- 01r' where 01 ~ c3/Oq 1. The Frenet-Serret relations are
01
g2 g3
=
(0 k -k 0
0 --T
7 0
(3.44)
g2 g3
Here k = 01~) is the curvature and ~- = 0 1 0 is the torsion; they are to be functions of ql alone. One rotates the orthonormal frame fixing d l = gl to obtain
(Ol
(~2
ga
--
(0
--t~l
0
0
(~2
-~2
0
0
ga
(3.45)
where al = kCosO and a2 = kSinO with 0 = fql 7dql" For convenience one defines a complex curvature as ( H A ) Vc = vl + iv2 = (1/2)(al + i~2) - (1/2)kexp(iO). If the rod is an elastica without elastic torsion it obeys (cf. [446, 926]) ( H B ) iOtVc-(1/2)O21vc+21vc12vc = O. Note here t h a t elastic torsion can exist even if there is no Frenet-Serret torsion ~- (e.g. via twisting the rod). Now for the Dirac field one writes 2 = ~'+ d2q 2 + d3q 3 where x i ~ Cartesian coordinates and qU ~ curved coordinates along C. One has a tetrad in the vicinity of the rod ~ .i - O,x i with ( H e ) ~ = (1 - hSt~,i)a ~ where 0 u ~ O/Oq t~ and h(q t~) = •2(ql)q 2 + aa(ql)q 3. Next extend R 3 to (3+1)D spacetime and use the Euclidean fermionic field theory as before. Let # run from 0 ~ 3 and the imaginary time of the fermionic system is q0 _ x 0 with ~o _ 5o _
and ~ = 5~. The original Lagrangian density is ( H D ) s = iCp(ViOi _ m o - V)~P for the s t a n d a r d 4 x 4 7 matrices (constant), Oi = O/Ox i, mo is the bare mass, and V is a confinement potential of the form V = [(q2)2 + (q3)2]/26 with 5 ~ 0. The potential is not coupled with 7 ~ and behaves like a mass so t h a t one avoids the Klein paradox. REMARK 2.2. We include the following physical remarks from [709] without comment: It is known t h a t the Dirac particle cannot be confined in terms of the potential coupled with 7 ~ in a region smaller that the C o m p t o n wavelength because the particle is exchanged with the antiparticle at the barrier owing to the distortion of the Dirac sea. However, since one effectively changes its mass now, the Compton wavelength can be extremely small and. due to the mass-like potential, the particle cannot interact directly with the antiparticle.
C H A P T E R 2.
96
G E O M E T R Y AND EMBEDDING
In terms of the curved system the Lagrangian density can be written as /~(3+l)D __ ~([[i(TttOtt _ mo _ g ) ~
-Jr-O(q 2, q3)
(3.46)
where 7 t~ = 7 i ~ and ~ = det(~) = 1 - h. Because of the measure (1 - h)d3q in the action integral the space derivative iOu cannot be Hermitian and to avoid this difficulty one redefines the field as ~(3+l)D = (1_h)1/2~. Now look at the squeezing limit 6 ~ 0 so (q2 q3) vanishes and h --~ 1. The density along the rod becomes (HE) /2[~, ~, Vl, v2] = ~ i ( ~ + m ) ~ where r q:) = r ' rn is a renormalized mass with the ground energy of V, and ( H F ) ~ = 7000 + 7107 -~- C1/2)71tCl -t- (1/2)72tc2 9 After taking the confinement limit the matrix 7 ~ becomes independent of qU. In order to simplify the argument now one takes the massless limit and writes
r =~
1
V~q~
~ eiEq~162 E
(3.47)
Here 5q ~ is the integration region of the q0 variable and can be taken sufficiently large. The Dirac equation becomes (cf. [705]) ( H G ) ECE = i(7~ + 7~ + 7073v2)r where v{~ = ha/2. Now choose the 7 matrices to be
(0 i
0
;7
--
(0
a3
0
;
0"1
0
;
~2
0
(3.48)
For (r = (r T r T one has then ( H H ) Er : i[a301 + (1/2)alkexp(iOcr3)]~E+. Equation ( H H ) has the same form as the Lax eigenvalue equation for NLS (cf. [446]) and therefore if we deal with an elastic rod the quantum system is isomorphic to the QM of the soliton (cf. [705]). In terms of the confinement prescription one obtains the (1+1) dimensional partition function (HI) A[vl, v2] = f d~DCexp ( - f d2qs r Vl, v2]). Consider now the gauge transformation arising from the fact that the shape of the rod makes parity break down. If one perturbs the curvature v's around themselves for space directions via ( H J ) v ~ ~ v ~ - v ~ ( q : ) - 0:a~(q 1) then the Lagrangian changes as ~ [ r , r- Vl, V~]-- /:[~, ~), Vl, V2] -- i01OZl~72~d- iOla2~'y3~d
(3.49)
The extra terms are cancelled by the gauge transformation ( H K ) ~ ~ ~' = exp[i(~/:~/2al + 7173{~2)]~ and ~ ~ ~ ' = ~exp[i(7172al + 7173a2)T] if O~1 -- (Vl/V2)O~2 (here 7 - : D O - D o acting only on the differential operator). This transformation T has no meaning in relativistic physics but in nonrelativistic submanifold physics the embedding breaks the parity locally (cf. [709] for more detail). Now rewrite a l = v i a and (~2 = v2a and following the procedures of Section 2.3.2 one has
(3.50)
:i
.~r
~xp
(i -
e~qC[~, r ~1, ~21
)
= z2
One expands again as in ( F U ) - ( F V ) , (3.215) where now ( F V ) has the form
d(q) = ~ a~mCm(q) = E eia('yl~2v:+'Y:~/3V2)am~m; ?71
7Yt
(3.51)
2.3. QUANTUM MECHANICS ON EMBEDDED OBJECTS
, =~ am
/
d2qx~(q)eia(@~2Vl+@@v2)r
97
= ~ Cmnan
n
n
One obtains again ( F W ) and the first and last terms in (3.17) can be identified as before (the intermediate steps look slightly different). Again A(q) is to be regularized by heat kernel methods and one writes (cf. here (3.18) which is somewhat vaguely written)
K(q,r, T) = ~ e-~r
(3.52)
m
(here T is a time as in Section 2.3.2). Again one has asymptotics as in (3.19) and (HL) .A(q) = limr__,O,r__+qTr(~/l~/2vl + @~/2v2)K(q, r, 7) (cf. (FY)). The coefficient el is now ( H M ) el = -i(@~/201Vl + @@01v2) + v~ + v22 and the trace over the spin space gives ( H N ) A(q) = -(i/Tr)(VlOlVl + v201v2). Again T has no effect on the procedure and one obtains (3.20) exactly. Now we go to the boson-fermion correspondence and the anomaly on the submanifold. The Ward-Takahashi identity in (3.50) (cf. ( G A ) ) gives an anomaly 1 2i 5q---~~-'~[< VlOI(~)E"/2~)E) > -t- < V2OI(~)E'~3@E) > ] - - - - - - ( V l O l V l E 7r
+ V201V2)
(3.53)
Recalling ( H A ) one obtains ( n o ) (1/bq ~ ~E[O'I < (~E~/2exp(~/2@O)r > ] - --(i/7r)01 vc where 0[ does not act on 0. If one defines F 2 = 72exp(~/2@O) it seems to be constant in ql. Noting the periodicity of the fermionic fields one defines then
' / dE~b*E_alei~3~162 J+- ~i / q l dqlOl
i f qld q l O ~f a~YJE+a e
J- = -~
(3.54)
~2E-
Choosing an appropriate initial point q~ one integrates ( H O ) over ql to obtain ( H P ) < J+ - J_ > = (1/2~r)010. The right side is regarded as the absolute value of the NLS soliton if the rod is an elastica. If one integrates it again there arises a global property of the system, namely (HQ) f dq 1 < J + - J _ > = (1/27r)(r r Since the rod is closed the right side is an integer and one notes that ( n P ) and ( H Q ) do not depend on q~ because of periodicity. Define now the projection H : C ---+C' via II : O(q1) --~ 0; then the right side of ( H Q ) is the sum of the signed intersection numbers of the curve C ~. and one chooses 0 at ql as the origin (i.e. O(q1) = 0). Consider now the geometrical meaning of the right side of ( H Q ) , namely # = [ r r This # seems to be associated with the winding number around a circle S 1, i.e. with the fundamental group 7rl ($1), and this is realized geometrically in terms of a closed loop. There follows a discussion of knots, links, Reidemeister moves, etc. (cf. [532]) leading to the conclusion that # = w(C) is a geometrical object and the index theorem is stated as follows: SUMMARY
2.3.
(i) For a closed space curve C one can draw a new curve C' with
/ ~ - w(C) - w(C') determinable as a geometrical index related to 7r1($1). (ii) Restricting the Dirac operator to a tubular neighborhood of C and the time to the region C x R 1 one obtains ~ - ~ 7 9 ~ as
7/) =
0 -iOo -+-8301 + alve ia30 ) iOo + 8301 + alve ia3~ 0
(3.55)
CHAPTER 2.
98
G E O M E T R Y AND EMBEDDING
The analytic index related to ~ is defined as ~, = f dq 1 < J+ - J_ > where J+ are given by (3.54). (iii) The indices p and u agree. The anomaly ( H P ) is given as the local version of this theorem and the theorem can be extended to R n as in [705]. In an appendix a corresponding index theorem is derived for a rod in R 2.
2.4
WILLMORE
SURFACES,
STRINGS,
AND
DIRAC
Notations such as (IA), ( J A ) , ( K A ) , and (LA) will be used. We go now to [569,570, 571, 572, 575, 706, 709, 711,713, 714, 715, 716]. 2.4.1
One loop effects
Let us begin with [560] which establishes some background related to Section 2.2 (cf. also [157]). The idea is to use a generalized Weierstrass representation introduced in [563] (cf. also [157, 713, 714]). This is related to Section 2.2 and there is some repetition; we give many details since the paper is very instructive in showing relations between QM and geometry. Thus any surface in R 3 can be generated via the generalized Weierstrass representation of [563] and the Nambu-Goto and Polyakov actions have a simple form. This allows one to calculate exactly the one loop correction to the background for the full Polyakov action. The propagators of fields are found and their infrared behavior is analyzed. Quantum corrections to the classical Nambu-Goto and spontaneous curvature actions are evaluated perturbatively. Then first the generalized Weierstrass (GW) representation for a surface conformally immersed in R 3 via )((z, 5) 9C ~ R 3 is given by formulas
s
§ _
where F is a contour in C and the functions r r satisfy (IA) /)r - pC and /)r = - p ~ where p(z, 2) is real valued. This defines a conformal immersion of a surface into R 3 with the induced metric (IS) ds 2 = (1r 2 + 1r 2 dzdS. The Gaussian and mean curvatures are
K-
(1r = + ir
z~'
(1r = + Ir ~)
Any surface can be represented in this form and at p = 0 one has a minimal surface (H - 0). The GW representation is equivalent to another Weierstrass representation used in Section 2.2 which involves the Gauss map. The equivalence is established via (IC) f = i(~p/O) and r/ = (1//2)r 2 but this produces a nonlinear system of somewhat greater difficulty to manipulate. Another advantage of the linear GW representation is that the extrinsic Polyakov action Sp = f H2[dS] (where [dS] is the area element) has the simple form (ID) S p ~- 4 f p2[d2z] where [dz 2] = (i/2)dzd2. The Nambu-Goto action S N G - - Olo f[dS] becomes (IE) SNC = c~0 f(lr 2 + 1r The representation (4.1) with (IA) makes it possible to define an infinite class of integrable deformations of surfaces generated by the modified Veselov-Novikov (mNV) hierarchy (eft [587] and Section 2.2) and a characteristic feature of these deformations is that they preserve the extrinsic Polyakov action (cf. [157, 564]). This circumstance has been used in [713, 714] to quantize the Willmore surface and one can regard the GW rePresentation as a parametrization of a surface in R 3 in terms
of p, r r One loop corrections for the Polyakov action have already been studied in [801,802, 956,
2.4. W I L L M O R E SURFACES, STRINGS, AND DIRAC
99
957]. However nonlinear constraints associated with the Gauss map did not allow calculation of one loop corrections for the full Polyakov action. This can be accomplished however with the GW representation. Thus start with the classical action
s = ~0 S (ir + ir
+ r S,'["z]
(4.,)
where s0 and J~ are the tension and extrinsic coupling constant respectively. The firststep is to take into account (IA) which relates the primary fields @, r p. Introducing complex Lagrange multiplier fields and requiring the action to be real one obtains a constraint term
s. = f[d~z][~(or
- pC) + ~(0r + pC) +
cc]
(4.4)
where cc denotes complex conjugate. This is added to the action in (4.3). Once constraints are introduced into the generating functional of Green's functions Z = f[DII]exp(-S) then the correct definition of measure [DII] requires the evaluation of the Faddeev-Popov determinant. In the present case the fields are constrained by the Dirac equation (IA), i.e.
,0)(:)=0
To evaluate det(L) one follows the heat kernel procedure (cf. [293, 713, 714]). The FaddeevPopov term is defined via (A = LtL = LL +)
SFp =-log[det'(L)] = [ d s ~'(slA) = Tr' [(A) -~] =
(s]L) 1
8--0
/0
-
G
(slA)
s=-O
;
t~_lTr,[exp(_tA)]
Here the Riemann function ~ is constructed from eigenstates of the operator A and the prime means that the contribution of zero modes is omitted. The heat kernel of A is defined as in [293]
Kt(z, z'lA ) = [exp(-tA)](z, z'); OtKt(z, z'lA ) + AKt(z, z'lA ) = 0
(4.7)
with limt~o+Kt(z,z'lA ) = 6(z,z'). The small t expansion of A is (IF) Kt(z,z'lA)lz=z, ~(1/87rt) ~ kn(z)t n where the k~(z) are matrix valued functions, a s a result one gets
~'(OIA) = ~1 fJ [d2zlTr'[kl(z)];
[ d ~'(slA) ]1 s=o - 3'~'(OIA)
(4.8)
where -y is the Euler constant. Since for the operator A there results TrY[k1] = - 2 p 2 (cf. [293]) one gets (IG) SFp = -(7/47r)fp2[d2z]. Thus the effect of the FP determinant is reduced to a redefinition of the extrinsic coupling constant fl~ which we denote by fl, assumed to satisfy fl > 0. Therefore the total action becomes (IH) ST = SNC + Sc + Sext where SNa and Sc are defined by (4.4) and (IE) with Sext = fl f p2[d2z]. First look at the pure extrinsic action ST = Sc + Sext (i.e. s0 = 0). The correspnding action in one loop approximation can be derived by employing the background field method. Thus split all fields into slow and fast components (p = P0 + Pl, etc.) and keep only terms quadratic in fast variables to get the following one loop action
f[d2z][Al(Or J
p0r
- p i e 0 ) if- (71 ( 8 r
if- p0~)l q-
pl~0)ff-
CHAPTER 2.
100
-[-pl(2/JlO" 0 --r
t-/~
GEOMETRY AND EMBEDDING f
Jp~[d2z]
(4.9)
In Fourier space one has ~(2) = f[d2k]~(2)(k) where S(2)(k) is written out in [572]. Here one symmetrizes and uses the formula
f(z, 5) - 1 /[d2k]f(k)ex p
[ _ 2 ( k 2 + ~z I
(4.10)
where the integration is performed over the domain/~ < [k[ < A. Exact propagators can be calculated using standard methods and the two point functions in the UV regime are written out (see [572] for details). Then, using subscripts R and I for real and imaginary parts and suppressing the subscript 1 for simplicity, one obtains at the one loop level NG
--
+2(r
+ r
+ r
+ r162
2]
(4.11)
The contribution to the classical Lagrangian from the intrinsic geometry term treated perr(2) >" Using the propagators following from the turbatively is ( I J ) A ( ~ ) = f[ d2z] < ~NC calculation of correlators one obtains
(l 01 + where (IK) DR = Z(IkIe-4p~)(Ikl2-4~p~) with ~ = [1+2(/3~//3)]. Therefore the counterterm to the classical Nambu-Goto action turns out to be Ar"-'NG- 3(--~~_ a0 1)I(A,/~,Pg,~, 3) f
[d2z] (1~ol2 + 1r
2;
(4.13)
I(A,A,P~,~,a) =- log [ (lkl2 -- 45P~)~+al lkl:A .
(ikl 2 _ 4p2)l+a
Ikl=A
This gives rise to the renormalized Nambu-Goto action (IL) S x a = & f[d2z](lr 2 + where & = c~0[(1 + (7r/3)-1(~- 1)-1I(A,/~,p02, ~, 3)1 is the renormalized string tension.
Ir =)
Another term discussed in [5721 is the spontaneous curvature defined as in [213] as ( I N ) SH = (~0/2)f[dSlH. This is also given a renormalized form (cf. [572] for details). Thus in summary the complete Nambu-Goto-Polyakov action has been treated and propagators for primary fields have been calculated in the case of a pure extrinsic action. In general they have the structure (IP) Np/Dp where the Np are functions of slow fields and momentum while Dp is given in (IK). The FP determinant contributes to ( I P ) by means of/3 and there ig a simple geometrical interpretation of the results since via (4.2) the P0 field acts as a link between extrinsic and intrinsic geometry of the string world sheets. The appearance of singularities which depend on background geometry as in ( I P ) seems to be new but the singular behavior is naturally removed once an infrared cut-off A is introduced; the cut-off must be sufficiently large with respect to 2p0. Further discussion is given in [572]. 2.4.2
CMC
surfaces
and
Dirac
We go to [709] now where the compatibility between geometrical and physical ideas is developed; one uses the thickness of the surface to investigate Dirac fields and compares
2.4. W I L L M O R E SURFACES, STRINGS, AND DIRAC
101
with the geometrical structure. One recalls here e.g. that the mean curvature of a minimal surface based on a soap film vanishes due to the surface tension and the surface tension comes from the existence of a thickness for the surface (here CMC means constant mean curvature). One reviews some notation first for a surface S embedded in R 3. Since the imaginary time and Euclidean QM are useful in the path integral method one uses only the Euclidean Dirac field. A confinement potential mcon/is used to constrain the particle to be on S and it allows one to consider only the geometry in the vicinity of S or in a tubular neighborhood T of S. One expresses a metric Gt~v in T in terms of the variables of S via coordinates (ql, q2, q3) describing S after one degree of freedom is suppressed. In particular let (ql, q2) indicate the position attached to S and let (X 1, X 2, X 3) be Cartesian coordinates. One has a moving frame (Jacobian element) ( J A ) E / = O~X I where /)~ = 0/0q t~ and Gu~, = 5IjE~EJ; also let E~ ~ (EI) -1. When a position on S is represented via the affine vector ~(ql, q2) and the normal unit vector of S is denoted by e3 one can uniquely write )( = (X1,X2, X 3) in the vicinity of S via ( J B ) ) ( ( q U ) = 2(qa) + q3e 3. Only tubular neighborhoods where such a coordinate representation is valid are considered. Define the moving frame along S as ( J C ) e / = Oax I with inverse matrix e 7 . Divide the derivative along S into the horizontal part ( J D ) Vab = Oab'-- < 0ab, e3 > e3 and a vertical part. Here < , > denotes the canonical inner product in R 3. The 2-D Christoffel symbol 7~a attached to S is ( J E ) Vaez = 7~ae 7 and the second fundamental form is ( J F ) 7~c~ = < e3, cgaez >. On the other hand from the relation < e3, oqae3 >~-~ 0 the Weingarten map -VZ,3e~c~ is defined via ( J G ) ~/~,3ea a = V~e3 with 7 a~,3 - - < ea, /)r >. Because of 0a < e 7, e3 > 0 it follows that 7 Z,3 a is associated with the second fundamental form through the relation ( J H ) V~a = -~/3,agT/~ 7 where gaz = 5Ijeaer I J is the surface metric. It should be noted that for the dilation (q~, q2) ~ / k ( q ~ , q2) the Weingarten map does not change. Now one can I + q 3 V3,aez. ~ I The metric in T express E~ (= Ox I/Oq ~) in T in terms of e Ia via (JI) EaI - ec~ is explicitly written as ,
G3,~ - G~,3 = 0; G33 = 1; G = det(Gu~) - ~g - ~det(gu, ) w h e r e ~1/2 = (1 +
(4.14)
Tr2 (')'3,~) 7 q3 + det2( Va 3,o)(q 3) 2) . Here Tr2 and det2 are the 2-D trace
and determinant over c~ and ~ respectively. These values are invariant for coordinate transformations if S is fixed and they are known as the mean and Gaussian curvatures, 7 namely (J J) H = -(1/2)Tr2(v~,~) and K = det2( V3,Z)" Accordingly the Christoffel symbols associated with the coordinate system of Y are given as ( J K ) F ~ p - (1/2)GUT(G,,~,p + Gp,r,u - Gu,p,r) and in terms of them the covariant derivative Vu in T is naturally defined a s (JL) V u B ~ = O . B ~ for a covariant vector Bu.
F~.~B~
Now consider the Dirac field ~ = (~1, ~2) in Y and confine it to S by taking a limit (cf. [135]). Start with the original Lagrangi.an given via
s
- ~(x)i(FIoi _ mcon/(q3))q2(x)d3x; mcon/(q3) = ?#~ + co2(q3)2
(4.15)
where ~ = ~ t F I while co is large with # >> v/~. Since the confinement potential depends only on q3 the thin surface has the same thickness overall. Further singularities in the curved coordinate system will have no effect since 1/x/~ is a unit of the thickness with #0 >> v/~ >> maxs, a,~7~,Z. Further mcon/ is not coupled with F ~ and the Klein paradox is avoided (cf. [702, 707]). The normal component of the Dirac field is factorized and each energy level splits and can be regarded as an effective mass indexed by a
C H A P T E R 2.
102
GEOMETRY AND EMBEDDING
non-negative integer. The Lagrangian can then be expressed by the sum of Lagrangians corresponding to each level. The ground state of the normal direction exists as a mode of the lowest level and is localized in the region ( - 1 / x / ~ , 1/v/-~). By paying attention only to the ground state the Dirac particle is approximately confined to the thin tubular neighborhood To ~ ( - 1 / v ~ , 1/x/~) • S. After taking the squeezed limit one can realize the quasi-two-dimensional subspace in R 3 and, by integrating the Dirac field along the normal direction, express the system using the 2-D parameters (ql, q2); this subspace will then be interpreted as S itself (cf. similar arguments in previous subsections and the development below). For the coordinate transformation ( J A ) the Dirac operator becomes ( J M ) i F I O i iFt'O t, and the spinor representation of the coordinate transformation is given as ( J N ) ~(q) e x p ( - E I J ~ i j ) q 2 ( x ) where E IJ is the spin matrix E IJ - (1/2)[FI, F J] and E I J ~ I j is a solution of ( J O ) O u ( E I J f ~ I j ) - ftu for ftu = (1/2)EIJE'[(VuEj,~,). Hence the Lagrangian density in (4.15) can be written as
s
= @(q)i(r"Dt~ - mcon/(q3))q2(q)V~d3q
(4.16)
where F t~ - F IE~ and ( J P ) D;, = 0 t, + ftu is the spin connection. After straightforward calculation (cf. [135]) the spin connection becomes (JQ) Da = 0a + ~ a with 03 - 03. Now since the measure on the curved system is d3x - x/~d3q and -i/)3 is neither Hermitian nor a m o m e n t u m operator, the field is redefined as r = ~1/2~ and (4.16) becomes
f,,d3x = ~(q)i(FUD, - mconf(q3))r
(4.17)
H - Gq 3 /)alog(~); D3 =/)3 + 1 - 2Hq 3 + G(q3) 2
Da = Da -
In the deformed Hilbert space spanned by @, -i03 is the momentum operator and represents the translation along the normal direction. Now due to the confinement potential m c o n / t h e Dirac field for the normal direction is factorized as mentioned before. There is a lowest energy mode (n = 0) such that r
q2, q3) ~ V/6(q3)r
q2);
(4.18)
(F3/)3 _ mconi(q3))~5(q3)~(ql, q2) = mo~5(q3)~(ql, q2) The Lagrangian density is decomposed into modes designated by n and it can be assumed that each mode is an independent field. By restricting the function space of the Dirac field to that with n = 0 mode the Lagrangian density on S is defined as
s s
= (/d3qgx/~).=od'q;
(4.19)
v/-~d2q = i@(V1D1 + 3,292 + HV3 + rno)#,x/~d2q
where ( J R ) e / = E/Iqa=0, v"(q 1, q2) = r.(ql, q2, qa = 0), and wa(q 1, q2) = aa(qZ, q2 q3 = 0) with 7:)a = cOs + wa. Since a confined space is expressed via (qZ, q2) and can be regarded as a 2-D space one identifies the confined space with S itself and then (4.19) is interpreted as a Dirac operator in a surface S embedded in R 3. Hence the inner space can be expressed via (JS) wa = (1/2)EiJe#i(Vaej#) where the indices i , j run from 1 to 2. One notes that this Dirac operator is not Hermitian in general (nor is that of a space curve immersed in R3).
2.4.
W I L L M O R E SURFACES, S T R I N G S , A N D D I R A C
103
It is natural because the extra term appearing as an immersed effect in the Schrhdinger equation behaves as the negative potential if one confines a Schrhdinger particle in a lower dimensional object; roughly speaking the square root of the negative potential appears as a pure imaginary extra field in the Dirac equation (cf. [135, 504, 702, 803]). Note also t h a t if there is a crossing in a surface the crossing can be moved in another direction by embedding S in a higher dimensional space, as long as it has a negligible effect on the curvature. Such an operation will not affect the above computations if the thickness of the surface is appropriate. In general m0 does not vanish but in order to concentrate on properties of the Dirac operator itself one neglects m0 from now on. The behavior of the high energy of the surface direction and the lowest energy of the normal direction will be investigated using the independence of the normal modes. Consider a surface with the conformally fiat metric (cf. [803]) immersed in R3; thus ( J T ) g ~ # d q ~ d q # = p6(~,~dq~dq #. This seems strong but it is a very natural condition. Then the Christoffel symbol is ( J U ) ~ / ~ = (/2)p-Z(6~O,~p + 6 ~ O # p - 6#,~O~p). We have a natural Euclidean inner space denoted by the parameters ya, yb (a, b = 1, 2) and the moving frame is written as ( J V ) epa _ ~f~l/2~a va and the ~ / m a t r i x is connected to the fiat a a via ~a = e ao ~ a . Thus the spin connection becomes ( J W ) wa -" - - ( 1 / 4 ) f l - l o ' a b ( c g a f l ~ a , b Obfl~a,a) where 0 "ab = (1/2)[O "a, o'b]. The Dirac operator can then be expressed as "/~
+ (1/2)p-3/2(Oc, p)]
~- o'a~[fl-1/2Oo~
(4.20)
As in (4.17) one redefines the Dirac field in the surface S via ( J X ) r = pl/2~ so the Dirac operator (4.20) becomes simpler, namely ( J X X ) 3,c'Dar = p-laa6$oc~r Noting that the metric for direction q3 is G33 = 1 and now that the q3 direction is regarded as an inner direction, one should write ,yc,=3 = o.a=3 where a 3 = - i a l o 2 . Since ~ ' y l ~ v ~ d 2 q is the charge density one expects ( J Y ) r = r = ctalpl/2" Then the Lagrangian density (4.19) is reduced to s176 = iCp-1/2(o'ah~oa + pl/2Ho'3)r (4.21) If the complex parametrization z = ql + iq2 is employed with ( J Z ) /) = (1/2)(0ql - iOq2) and c9 = (1/2)(0ql + iOq2) then the a a g r a n g i a n density (4.21) becomes
s (~) v/-~d 2q = [r162
+ r162
- lpl/2H(r162
- r162
(4.22)
where r is not the complex conjugate of r177and 2id2q = dhAdz = d2z. Hence the equations of the Dirac field in the complex surface S immersed in R 3 are
0r
= p(z, ~)r
0r
- p(z,
~)r
~r
= -p(z, ~)r
~r
= -p(z,
where the external field p is defined as p = (1/2)pU2H. leads to
(4.23)
~)r Under the on shell condition this
/4 4/ where q~+ is the complex conjugate of r177 It is thus somewhat surprising but eminently satisfactory t h a t (4.23) is in fact the same as the equations from [563, 566] discussed in Sections 2.1, 2.2, and 2.4.1. The dilation factor is obtained as in [563] as ( K B ) t) 1/2 =
104
CHAPTER
2.
GEOMETRY
AND EMBEDDING
2(r162 - r 1 6 2 = 2(r162 + r 1 6 2 and the Gauss curvature is K = -2[OOlog(p)/p]. The Euler number can be calculated via (cf. [607])
1/
X = ~
1/
K p d z A d5 = -~-~
0p-1/2012(r162
- r162
A d2
(4.25)
For a CMC surface (H = constant) the Dirac field is coupled with the dilaton (cf. [803]) = log(z) and s176
q = [r162
+ r162
- leo/2H(r162
- r162
(4.26)
This dilaton is governed by the Sinh-Gordon equation which is related to the Liouville equation. When H - 0 the equations of motion (4.23) become the Weierstrass-Enneper formulas and the Lagrangian (4.26) becomes that of ordinary classical conformal field theory. 2.4.3
Immersion
anomaly
In [713] one deals with the quantized Dirac field and investigates the gauge freedom which does not change the Willmore functional. In other words one looks for a symmetry in the classical level and computes its anomalous relation at the quantum level. A new regularization is used which can be regarded as a local version of a generalization of the Hurwitz zeta function. In particular one finds a relation between the expectation value of the action of the Dirac field and the Willmore functional. Thus consider a compact surface S immersed in R 3 via ( K C ) & 9 E ~ S c R 3 where S and E are 2-D conformal manifolds and S is parametrized by ( q l q2). There may be some repetition of notation but we include it for coherence. Thus let position on S be represented using the affine vector Z(ql,q2) _ ( x l , x 2 x 3) and let e3 be the unit normal vector to S. One also uses a coordinate representation Z - x 1 + ix 2 E C with x 3 E R. The surface has a conformally fiat metric ( K D ) g ~ d q ~ d q ~ = p b ~ d q ~ d q ~ and one writes z = ql + iq2 with (KE) 0 = (1/2)(Oql-iOq2), 0 = (1/2)(Oql +iOq2), and d2q = dqldq 2 = (1/2)d2z - (i/2)dzd2. If a given function f on S is real analytic one writes f ( q ) and if regarded as complex analytic one writes f ( z ) . The moving frame is (cf. ( J C ) ) ( K F ) e as = Oax s with e zI = Ox I where 0a - O/Oq ~ The inverse matrices are denoted by e~ and e} and the metric is (KG) (1/4)p = < ez, e~ > a b 5abeze ~. The second fundamental form is (cf. ( J F ) ) ( K H ) 7 ~ = < e3, O~e 3 > and using the relation < e3,0ae3 > = 0 the Weingarten map -7~,3ea is defined by (cf. ( J G ) ) (KI) 7 ~a, 3 = < e a , 0ze3 > " Via 0a < e~ , e 3 > = 0 o n e has (K J) 7 ~ = --73,ag'y3 ~ ~ = -- 7 3,c~P where g~f~ is the surface metric. 0 h e introduces again (cf. ( J J ) ) ( U U ) S - - ( 1 / 2 ) T r 2 ( 7 3 ~ ) and K = d e t 2 ( ,ya 3 , ~ ) and by the Gauss theorem egregium (KL) K = - ( 2 / p ) O 0 log(p). Further one has ( K M ) S = (2/p) < 00Z, e a > = (4/ip2)EIJKOOxSOxJOxK. Using the independence of the choice of local coordinates one introduces now a change of coordinates diagonalizing the Weingarten map, namely ( K N ) T r m ~ a l~ 3 ,,~a , J 5 rr~ - d i a g ( k l , k 2 ) leading to ( K O ) K - klk2 and H = (1/2)(kl + k2). The surface will be regarded as a thin elastic surface. Its local free energy density is given as an invariant for the local coordinate transformation but the difference of the local surface densities between inside and outside surfaces is proportional to the extrinsic curvature for a local deformation, due to the thickhess of the surface. By the linear response of the elastic body theory and independence of the coordinate transformation the free energy is therefore written as 1 f = B o l l 2 + B I K = -;Bo(k 2 + k ~ ) + (Bo + B1)klk2
(4.27)
2.4. WILLMORE SURFACES, STRINGS, AND DIRAC
105
where the B's ,,~ elastic constants. But via Gauss-Bonnet ( K P ) f pd2q K = f pd2qklk2 = 27rX where X is the Euler characteristic exhibiting a topological property of the surface. Hence the second term in (4.27) is not dynamical if one fixes the topology of the surface. Hence the free energy becomes (KQ) W = Bo f pd2q H 9 which is the Willmore functional or Polyakov extrinsic action. Fixing B0 = 1 for convenience and introducing p = (1/2)v/-fiH = (1/2)gl/4H one obtains ( K R ) W = 4 f d2qp 2. From [709] (cf. Section 2.4.2) one knows that the Dirac field confined to the surface S obeys the equations 1 Of 1 = Pf2; Of 2 = -Pfl; P = -~v/-fiH (4.28) As in [707, 709] one also obtains the quantized Dirac field over S by using a confinement potential so one starts here with the quantized Dirac field of S with partition function (KS) Z[~, r p, H] = f D(bDCexp(-SDirac[~, r p, H ] ) w h e r e (SD ~ SDirac etc.)
SD[~, r p,H] =/pd2qs
s
= ir162 ~=
7aT)a + 7 3 H ; ~ = ~+crlpl/2;
(4.29)
1 t o.ab (Oaphc~b -- Ob~aa); ~/c~ -- CaO c~ a ; o.ab = [(7a, ab]/2 ~)c~ = 0(~ + 02(~; 02~ = --~p-Here one has used the conformal gauge freedom eaa __ pl/26a where a = 1 , 2 . Then ~P = (7aDa + 73H) = o ' a p - 1 / 2 ~
=2
[Oc~ Jr- ~11p - (Oc~p)] -Jr-a3H =
(4.30)
H/2 p-l~(pl/2)) fl_l/20(fll/2) -H/2
Noting that the r are integral variables in the path integral the kinetic term of the Dirac operator is Hermitian since after some calculation one finds
/ I \
0 p-lOpl/2 p-1~pl/20 -- --i =
=
-i/d2z
pd2z i
d2z (r
) ~b > = <
>=i/pd2z~MO
0 jol/2~) 2 -4- ~ s o l / 2 ~ 1 )
(~pl/2~fll/2~)
P-lc~pl/2
r162
0
-
--
2 -4- ~)lpx/2*-~pl/2~21)
-
~2 o'1pl/2~ ~< iM~'l~ >
As in [709] now one redefines the Dirac field in S via f =
pl/2~
l:/v/gd2q=iftal(aa~aOa+Pl/2Ha3)fd2q=if(
(4.31)
with
pO -pO ) fd2z
(4.32)
Then the equations (4.27) ( K T ) 0 f l = Pf2 and 0f2 = -P fl are obtained as the on-shell motion for (4.32). One notes that the Willmore functional as expressed via p and p consists of multiples of H and Vrfi. Hence by fixing p there still remains freedom to choose p; fixing p means deforming p without changing the Willmore functional. Corresponding to this the Lagrangian of the Dirac field (4.29) has a similar gauge freedom which does not change So.
CHAPTER 2.
106
GEOMETRY AND EMBEDDING
The scaling r --, f utilizes this gauge freedom. Now in QFT even though the Lagrangian may be invariant the partition function is not invariant generally due to the Jacobian of the functional measure. One now constructs an infinitesimal gauge transformation which does not change the action of the Dirac field. Thus introduce the dilatation parameter (I) = (1~2)log(p) (sometimes called the dilaton) and rewrite the Dirac operator in (4.30) as ~=2(
pp-l/2 p-i~pi/2 ) p-lOpi/2 _pp-i/2
(433)
One deals with variation of the dilaton preserving p so ( K W ) @ ~ @+a, p ---, pexp(2o~), and p ~ p. For the infinitesimal variation of the dilaton the action of the fermionic field changes its value and ( g x ) SD --~ S 5 -- SD + i f pd2qa(p-iS~aCgz~aapi/2r + ~7~r However this change can be classically cancelled by the gauge transformation ~ ~ ~ = exp(-a)r and ---, ~ ' = ~. In other words one has an identity (KZ) SD[r ~, p', p] --~ SD[(Y, ~', p', p] = SD[r r p, p]. Now one evaluates the variations ( K W ) and ( K Y ) in the framework of QM (cf. [363] and previous sections). _
Z(PI' HI)= .l D~)DCe-SD[CP'r
-: Zl - /D~fDr
-SD[(p''r
= / D(~Dr 5r162exp(-SD[~, ~, p, p]) = Z2 5r 5r Noting that the r are Grassmann variables the Jacobian is (5r162162162 complete sets as in (FU), (FV), ( F W ) , and (3.51) in the form
~r
f pd2qXtm(q)r
)~r
(pXtn)i~ = /~nP)(.tn;
= 5ran;
= ~ almCm = m
(4.34) Now one uses
(4.35)
e-aamCm
Similarly one writes (LB)a~m = En f pd2qx~exP(-a)r - E m Cm,nan and the change of measure is (LC) Hm datm = [det(Cm,n)]-1Hm dam. More explicitly (LD) [det(Cm,n]-l= exp[f pd2qaA(q)] and .4 needs regularization where here A(q) = Ern xtm(q)r 9One uses a modified type of regularization partially proposed in [28] which is a local version of the regularization (cf. [293]). Thus one introduces a finite positive parameter (LE) p2 > -minn(~A2n) > 0 and define /Cr
r, sip ) = ~-~(A2m+
p2)-SCm(q)Xtm(q)(r);
m
]CHK(q, r, T[#) = ~ e-(~+"2)rCm(q)Xtm(r )
(4.36)
m
By the Mellin transform (LF) ~r q, s[#) = (1/F(s)) f ~ dT 7s-IKHK(q, q, 7-lp) and tracing K:r q, sip) over q one obtains a generalized Hurwitz ~ function (LG) ~(s, #) = Em[1/(A~+ p2)s]. Then redefine ,4 via ( L H ) A ( q ) = lims_~olirnr_~qTr~i(q, r, slp). For small 7 one has the asymptotic form (LI)]~HK(q,r, TIp) ~ (1/4~T)exp[--(q- r)2/nT] E ~ e~(q,r)w ~. Accordingly one calculates /Cr
q, ql#) -- ~
1(/0 1 + e
dT"rS-l~-'nK(q'q' wl#) =
2.4. W I L L M O R E SURFACES, STRINGS, AND DIRAC
1(/0
r(s)
ill
dT T s-1 ~ T 1
dT Ts-1]~HK(q, q, 81~ )
E enTn "~ n
(1
r(s + 1 ) ~
n~
107
En+s-1
s ~ n-- 1
)
=
+ sG(s))
(4.37)
(recall r ( s + 1) = s t ( s ) - we have corrected a typo in (4.37)). Since ]CHK(q,q, slp ) c< exp(-AT) as T ~ c~ (with A > 0) the second term is a certain entire analytic function over the s plane denoted by G(s). Since P(1) = 1 there results (LJ) fit(q) ,,~ el/47r. On the other hand from [383] since
_p~2
--4p1/2(Opp-1/2) --400 + 2p-l(Op)O+ (Kp -
( - 4 0 0 + 2p-l(Op)O + ( K p - 4p 2) = _4p1/2(Opp-1/2
4p2)
)
(4.38)
it follows that el = 4p2p -1 - #2 _jr 2pl/2~as~O~pp-1/2 _ 5 K _ 6
(4.39)
= 4p2p-1 _ #2 - 56 K + 2Pl/2aas~Ozpp-1/2 Noting that the trace over the spin index generates a factor of 2 one gets 1 (
A(q)= ~
5
4p2p - 1 - # 2 - - ~ K
)
1 (100c~r
= ~
k3p
-
#2+H2)
;
exp [f pd2qa(q)A(q)]
(4.40)
For the boson-fermion correspondence recall (LK) (5/Sa(q))(Z1- Z2)[a(q)=0 = 0 which gives an anomaly 1
2
_~5 K _
2_~H 2
(4.41)
The right side of (4.41) is closely related to the conformal anomaly in string theory and the Liouville action. Further information comes from integrating both sides in (4.41) to get <
1
0
(4.42)
The first term is locally expressed as a.total derivative j = < 5aa~o'apl/2r > dq a so let the surface be divided as (LL) E = E+ tJ E_ with S 1 "~ E+ N E_ and E+ ~ E_ ~ R 2 where ~_ means homeomorphism and j+ are defined over E+. Then the integration of the first term becomes (LM) fEd 9 j = foE+ *J+ + foE_ *J- - foE+ (*J+ - *J-) - B2u where u is an integer and B2 a constant parameter. Thus it can be regarded as the generator of the fundamental group of E+ N E_ while the Euler characteristic X expresses the global topology of the surface. If the current is conserved then u vanishes. Furthermore the third term means the area of the surface (LN) A = fE pd2q 9 Using all this one obtains a global expression from (4.41) of the form (LO) < SD >-- (1/2~)(#2A - W ) + ( 1 / 3 ) X - B2u. Even though the current is not conserved B2u can be regarded to be a topological quantity.
CHAPTER 2.
108
2.4.4
GEOMETRY AND EMBEDDING
Quantized extrinsic string
We go now to [714] and consider the quantization of a Willmore surface and relations to MVN equations (cf. Section 2.2). Again there will be some repetition of notation for coherence. Thus let S be a compace conformal surface immersed in R 3 _~ C • R with (LQ) Z(z, 2) = X 1 + iX 2 and E ~ z ~ ( Z , X 3) E S . Here E can be represented as E = C / F where F is a Fuchsian group so E becomes a complex analytic object. The volume element and infinitesimal length of S are dvol = (i/2)pdzAd2 = (i/2)pd2z with ds 2 = pdzd2. The author designates the equations r
OX 3= - ~ + ~ _ ;
= iv/-~; r
( )0 ( -;V)
~r
v
0
+ _
(4.43)
1
=0; v=~gv/- ~p
as the K K W E relations (Konopelchenko-Kenmotsu-Weierstrass-Enneper). Here V is a real valued function of (z, 2). More general forms for a non-conformal surface are given in [135]. The energy integral or Polyakov extrinsic action is (LR) E = f pd2zH 2 = 4 f d2zV 2 and the Willmore surface is realized as its stationary point. The partition function is given as
Zreg : / DXexp(-Z f pd2zH 2)
Yol(Af f)Yol(Conf)
(4.44)
(cf. [711]) where Yol(Aff) and Yol(Vonf) mean the volume of the group of the affine transformations and the conformal reparametrization group of E --. E respectively. One looks for the deformation flow of the surface which preserves the Willmore functional or Polyakov extrinsic action and this is the modified Veselov-Novikov (MVN) equation as indicated in Section 2.2. The MVN equations are given as 3
Vt = Vt+ + Vt-; Vt+ = 02V + 30VU + -~VOU;
(4.45)
Vt- -- 03V+3OVU-~- 3VOU; OU-- OV2; O U - O v 2 Along the curve z = 2 the MVN equations reduce to the mKdV equation. Since the FrenetSerret relations can be regarded as the inverse scattering system of the mKdV equation, the K K W E equations can also be regarded as the inverse scattering system of the MVN equation. Thus (LT) (Ot+ - B + ) ~ + ~ , A +] = 0 recovers (4.45) for
A+_(03
0
-3(OV)O+ 3VU ) (03+U0+30(]/2 03 + 3uO + 3(0U)/2 ; A- . . + VU . . 30VO
B+= 3 B- = 3
(
0 -(OV)O- (02V) - UV
(OV)O-VY) 0
( -(OV)3o+ V(] (~v)~+(~2v)-vS) 0
The variation of the Dirac field is (LU) Otr = Ot+r + Ot-r with variation of t obeying the MVN equations we recall that (LV) 40t f the integrand can be expressed by the boundary quantities Vt2=0
;
O) 0 ' (4.46)
Ot+r = A+~b. For the d2zV 2 = OrE = 0 since
(VO2V_ ~1( 0 V ) 2 + 3V2U) + ~ (V~2 V _ _~(~V)2+ I ~V2FJ)
(4.47)
2.4. W I L L M O R E SURFACES, STRINGS, AND DIRAC
109
Next one checks whether the conformal structure of S is preserved under MVN by an argument of Taimanov. First note that the metric is represented by the Dirac field as (LW) p = (1r + I@~)2 (cf. the first equations in (4.43)). Thus if these equations in (4.43) are covariant or preserved for the MVN flow the conformal structure is maintained. Thus consider OtZ = Or+Z + Or-Z and OtX 3. By straightforward computations one gets
Ot+ Z = 2i iz(~) d(f + -~-g=h); Ot X 3
-- --
3 f+ = -~Ur ; g+ = r162
h2 = r 1 6 2
(4.48)
1 )2 ; - -~(0~_
1 )2 ; g - = @+c~2@+ _ ~(c5r
f-=73 0 r hi = ~ + 0 2 r
/z(~) d(hl + h2);
+ r + ~ - + r162
- 0r
+ 3U@+r
- 0r162
+ 3Ur
One writes also U (+) = U and U(-) = U with df = Ofdz + Ofdh. Let now the infinitesimal flow obeying the MVN equations (4.45) modulo e2 be denoted via (LX) (ZE, X 3) - (Z, X 3 ) + eOt(Z, X3)+O(e2). Then (4.48) means that the infinitesimal variation is given as the integral of a closed form defined over E and can be regarded as a single function on E. Since (Z, X 3) is also a periodic function on E it follows that (ZE, X 3) is globally defined over E as a function on E. On the other hand (4.48) guarantees that [0, ch]Z~ = 0, which means that we can locally define the independent coordinates z and 5 for the X~ surface. Thus one can locally find a complex coordinate system on any open set in E and these will fit together globally. Hence the MVN flow preserves the conformal structure of S. One notes that the MVN system involves an initial value problem, namely, for any shape of compact conformal surface the MVN flow conserves the energy functional and conformal structure. Hence the trajectory of the deformation parameter yields states (presumably @+(t)) of the same energy, and the associated volume is the degeneracy of the energy level E (this is partially clarified in what follows but one should read all of [714] here). The measure of the functional d# can be decomposed as (LY) dfit = ~ E dfitE and, denoting the moduli of Zreg restricted to E by ~-E, one writes
Z... SdDe -'E: ~-'~e-'Ei~..dDE= ~-~e-'EvoI(2E) E E E =
(4.49)
Now the Willmore surface S has no natural measure of length from the point of view of the value of the Willmore functional. However along the curve z = 2 the MVN flow obeys the mKdV equation which conserves the local length of the curve. Thus one can adjust a conformal reparametrization of z so that the curve is in a fiat plane (which suppresses conformal freedom of the surface). On the MVN flow the length L of the curve is also a conserved quantity and is well defined. Due to compactness of the survace L must be finite and one can redefine the partition function via (LZ) Zreg = Z r e g l ( l e n g t h o f z--2)=L Then define the decomposed measure and the space of trajectories by d # E , L -- d f t E I L and "~'E,L -- '~'LIL giving ~
Z~g = Z E
dD',L E,L
=Z
(4.50)
E
One refers here to the first part of [714] involving the mKdV situation to justify all this and we temporarily omit details. In any event the space of trajectories of the MVN equation
110
C H A P T E R 2.
G E O M E T R Y A N D EMBEDDING
can perhaps be classified by the genus g of the mKdV equation (and the MVN moduli via the moduli of the Jacobi variety associated to the corresponding mKdV equation); then an argument is sketched involving density of states etc. suggesting that (4.50) can be reexpressed as
=
~-E,L~176
e
where ~tg(E) is the volume of the subset of the moduli of the Jacobi variety of the MVN equation in which the Willmore surfaces live with given boundary conditions and have energy in the interval (E, E + dE).
2.5
CONFORMAL
MAPS
AND
CURVES
Notations (C1) are used. In this section we sketch some results from [590, 729, 730, 973] on conformal maps, interface dynamics, tau functions, analytic curves, and Whitham-Toda equations; there are obvious relations to some of the material in this chapter and to Whitham theory in Chapter 5. In particular in [590, 973] one shows that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2-D Toda hierarchy singled out by conditions known as string equations. The same hierarchy locally solves the 2-D inverse potential problem of reconstruction of a domain out of a set of its harmonic moments. This is the same solution which is known to describe 2-D gravity coupled to c = 1 matter. One also introduces a concept of tau function for analytic curves. Thus one can characterize a closed analytic curve by a set of harmonic moments of e.g. the exterior domain and its area; here (C1) Ck = -- f fext z-kdxdy for k > 1 with Co = f Lnt dxdy. Assume z - 0 lies in the interior and c~ is outside and a suitable regularization at k = 1, 2 is assumed. Given this set of data one wants to find (1) The harmonic moments of the interior (C2) C-k - f lint zkdxdy for k > 1 and to reconstruct the form of the domain and (2) An invertible conformal map of the exterior of the unit circle onto the exterior of the domain. Explicitly the area and the set of moments of the exterior (and their complex conjugates) are identified with the times of a 2-D dToda hierarchy via (C3) t - to = Co/~r, tk -- Ck/Trk, and tk = Ck/rk. These parameters should be treated as independent variables and one proves (C4) OC_k/Otj = OC_j/Otk with OC_k/Otj = OC-j/Otk. It follows that there is a function T(t, {tk}, {t-k}) such that ( C C 4 ) v k - C - k / r = Ologw/Otk and vk -- C'_k/Tr = OlogT/i)tk for k __ 1. In its turn the conformal map is determined by the dispersionless limit of the Lax-Sato equations for the Toda hierarchy. Let z(w) be a univalent function providing an invertible conformal map of the exterior of the unit circle Iwl > 1 to the exterior of the domain and write (C5) z(w) = rw + E ~ ujw-J. The potentials of the conformal map obey an infinite set of differential equations with respect to the times (harmonic moments) which are the evolution Lax-Sato equations. One identifies z(w) with the Lax function (as in dispersionless theory) and the inverse map w(z) has the form
(CCh) logw = l o g z - [(1/2)(02/0t 2) + Ek>l(Z-k/k)(O2/OtOtk)] log7. We sketch (somewhat brutally) some of the details. Thus the equation of the curve can be written as (C6) 2, = S(z) where S is defined also outside of the curve and is analytic in some domain containing the curve. S(z) is called the Schwartz function of the curve and its singularities encode the information about the curve. Further (C7) S(S(z)) = z and one can write (C8) S = S + + S - with S+(z) = (1/~r) E ~ Ck zk-1 and S - ( z ) - (1/Tr) E ~ c C_kz -k-1 (since (C9) Ck = (1/2i)~c z-kS(z) dz for - o c < k < oc). Thus one has ( C C 9 ) S(z) -
2.5. C O N F O R M A L M A P S A N D CURVES
111
~_c~ kt, kzk-1 _jr_(t/z) -Jr-~_,~ VkZ -k-1 and S(z) = 2 ~ ktk zk-1 -Jr-(t/z) -Jr-~_,c~~;kz-k-1 The unitarity condition (C7) is resolved via ( e l 0 ) z(w) = S(5(w-1)) and 2(w -1) - S(z(w)) while the conformal maps are given via ( e l l ) z(w) = rw + ~ ujw-J and 2(w -1) rw -1 + ~ , ~ UjWj. Next one finds a set of differential equations determining the "evolution" of the potentials and moments of the interior in terms of times ~ moments of the interior. Thus one defines a Poisson bracket via ( C 1 2 ) { f , g } = w ( f w g t - wgwft) and there results (C13) {z(w, t ) , 2 ( w - l , t ) } = 1 which corresponds to a string equation. One introduces a generating function ft(z, t) with dft = Sdz + log(w)dt so that (C14) S = f~z and log(w) = ftt leading to (C15) ft = ~ tkz k + tlog(z) - (1/2)vo(t) -- ~ ( v k ( t ) / k ) z -k where Otvo = 2log(r). Considerable argument and calculation leads then to (C16) df~ = Sdz+log(w)dt+~(Hkdtk-Hkdtk) where Hj = (fttj)lz and Hj = -(ft~j)lz (z is fixed). Further (C17) cgtr = Ot~Hjlz with Hj = zJ - (1/2)Otjvo - ~c~(Otjvk)(z-k/k) (similarly for Hk) leading to Lax-Sato type equations (C19) Oz(w)/Otj - { H j , z ( w ) } and 02(w-1)/Otj = {Hj,2,(w-1)} along with zero curvature equations i)tjHi -i)t~Hj + {Hi, Hj} - O. Such Lax-Sato type equations plus string equations give the complete set of differential equations for the potentials uj of the conformal map as functions of t, tk,tk; they uniquely determine small deformations of the map and eventually the curve. The tau function is determined via ( C C 4 ) for Vk, ~k plus v0 = i)logT"/Ot. Note from ( C 1 0 ) and (C13) one has {z(w),S(z(w))} = 1 and in (C19) OS(z(w))/Otk -- {Hk, S(z(w))}. From (C16) (with cOkS = OzHk etc.) one adjoins e.g. O~kS(z ) -- -OzH(z) and (C19) is enlarged via Recall from Section 1.1.3 that 2-D Toda is defined via (C20) L = r(to)exp[h(O/• + uk(to)exp[-kh(O/Oto)] and L = exp[-hOo]r(to)+ ~ exp[kl~Oo]ftk(to) acting on a space of functions of to (plus tk, tk as parameters - here 00 ~ O/Oto). Note uk and gk are not necessarily conjugate but will in fact be so. The Lax-Sato equations are (C21) 5cgkL [Hk, L] and hOwL = [n, Hk] (similarly for L - note 0~ ~ 0/0tk). One has (C22) Hk (Lk)+ + (1/2)(Lk)o and/~k = (Lk) - + (1/2)(Lk)0; the first Toda hierarchy equation is then (C23) (92ir = exp[r + h) - r - exp[r - r - h)] where r 2 - exp[r + l~) r One can write (f(tk)"~ f(t0, t l , " "))
~
~b(z, tk ) = Thl (tk )zt~
Y'~k>0tkZkeh }--~k>0(z-k/k)Ok Th (tk )
(5.1)
and among many possible solutions of the hierarchy the one of interest is selected via the string equation (C24) [L, L] = h (corresponding to the normal N matrix model). One goes to dispersionless limits as in Chapters 1 or 5 with ~b ~ exp(f~/h) and Th ~ exp(logT/l~ '2) to get for D ( z ) = Ek>l(Z-k/k)Ok a n d / ) ( 2 ) = Ek>l(2-k/k)O~
l o g ( W ( Z ! -__ W ~( ~ ) ) -log
(1-
12 = ---~OOlogT + D(z)D(~)logT;
(5.2)
1 ) = D(z)D(~)logT
w(z)ff~(~)
leading to (C25) (z-~)exp[D(z)D(~)logT] = zexp[-OoD(z)logT]-@xp[-OoD(~)logT] (dispersionless Fay identity) which generates an infinite set of relations between the second derivatives of logr (see Section 5.1 and [158, 902]). These can also be displayed as (cf. [590]) (C26) 1 -exp[-D(z)D(~)log7] = (1/z~)exp[Oo(Oo + D(z) + D(~))logT]. In particular one can write down the lovely formula for the Schwartzian derivative T(z)
-
2
=
Zi nz
OzO Z~
-
z-~
112
CHAPTER 2.
GEOMETRY AND EMBEDDING
leading to (C27) T(z) = 6z -2 ~k,n>_l z-k-nOkOnl~ In any event the infinite set of Poisson commuting flows from Section 2.4 (expressed via the Hk, [-Ik) form a W h i t h a m integrable hierarchy. In [729, 730] one shows how this theory can be aplied to interface dynamics and the Laplace growth problem (e.g. Hele-Shaw flow). We omit the details.
Chapter 3
CLASSICAL A N D Q U A N T U M INTEGRABILITY 3.1
BACKGROUND
We will try to give an overview involving quantum groups, Poisson-Lie (PL) structures, etc. Notation such as ( 5 A ) , . . . , etc. appears in various Sections and will be indicated at the beginning of the Section; notations such as ( h A ) , (A), (XI), (.), (~), etc. are introduced subsectionally. For basic material in this direction, from which we will extract see e.g. [6, 20, 39, 50, 127, 173, 187, 303, 305, 464, 584, 587, 588, 589, 603, 630, 648, 663, 670, 671, 672, 817, 819, 821,858, 859, 860, 861,862, 876, 917, 918, 919, 924, 935, 1003]. For basic Lie theory, quantum groups, and symplectic topics we refer to [552], [187, 361,529, 672,919], and [720] respectively. Now from [187] the general background is classical mechanics with a phase space M ~ R 2 and A = COO(M) with a Poisson bracket {f, g} = fpgq - fqgp and dynamical equations (d/dt)f(m(t)) = { S c , f}(m(t)), where rn(t) is the "state" of the system at time t and Hc is the classical Hamiltonian function on M. In particular {p, q} = 1. In quantum mechanics (QM) M ~ rays in a complex Hilbert space V (there are many good books on Hilbert space and we refer to [143, 147, 171] for basic information on functional analysis) with A ~ Op(V) ~ operators on V (not necessarily bounded). The time evolution of an operator O is expressed via dO/dt = [HQ, O] where HQ is the quantum Uamiltonian operator. For example, in motion along the real line V = L2(R), P = -ihOq, Q = q (as multiplication), and [P, Q] = - i h (we use h for h = h/2~ or h ~ - i h sometimes). For quantization one would like a map Q: f E A ~ Q(I) E Op(V) such that Q({/, g}) = (i/h)[Q(f), Q(g)] but it is well known that such a map does not exist. As an alternative various noncommutative products f *h g on A were envisioned by Moyal and others such that 1
limh-~O-~ (f *h g - - g *h f ) -
{f,g}
(1.1)
We recall from Gelfand that any compact topological space M is determined by its commutative C* algebra of complex continuous functions and consequently one may ask for a space Mh dual in this spirit to Ah(M) = A(M) with the product f *h g. Recall also that any affine variety over C is determined up to isomorphism by its algebra of regular functions. This leads to the environment of noncommutative geometry as in [193] and one thinks of quantum spaces Mh associated to deformations Ah of A(M). In this spirit consider a (Lie) group with multiplication # : G • G ~ G which induces a homomorphism #* = A : A(G) ~ A(G • G) (note A(G • G) ~_ A(G)~A(G) for a suitable
113
114
C H A P T E R 3. C L A S S I C A L A N D Q U A N T U M I N T E G R A B I L I T Y
completion - cf. [171]). Similarly the inverse map i : G ~ G (g ~ g - l ) induces a map i* = S : A(G) ~ A(G) via S ( I ) ( g ) = f ( g - 1 ) (antipode) and e(f) = f ( e ) for e E G identity defines a map e : A ~ C (counit). There are then certain compatibility properties ( o ( f | g) = g N f - flip homomorphism) (id | A) o A = (A | id) o A; A ( f g ) = A(f)A(g); S ( f g ) = S ( f ) S ( g ) ;
(1.2)
A o S -- a o (S @ S) o A; (id | e) o A = (e | id) o A - id and #((id | S) o A) -- # ( ( S | id) o A) _-- e. 1 which provide A(G) with a structure of Hopf algebra. Thus one anticipates a category of quantum groups Gh dual in some sense to a category of (not necessarily commutative) Hopf algebras and consequently one looks for deformations Ah(G) of A(G) in the category of Hopf algebras. In fact the existence of a deformation Ah(G) automatically endows G with the structure of a Poisson-Lie group (see below) so one can try to extend such possible (first order) Poisson-Lie structures on G to full deformations. This is the approach of [187] and integrable systems arise via natural connections to Poisson-Lie structures. We note also that the universal enveloping algebra U(~) for ~ - Lie algebra of G, is also a Hopf algebra, essentially the dual of A(G) in the category of Hopf algebras (details below). Thus one can also look for suitable deformations Uh (~) in order to study quantum groups Gh. As background we mention also (cf. [14]) that some of the motivation for Poisson-Lie (PL) structures was provided by the dressing transformation group for classical integrable systems; this is the symmetry group of the system in the Lax pair formulation and it carries a natural multiplicative Poisson (P) structure. A quantized integrable system will have a quantum group symmetry so PL groups generate interest as the classical counterparts of quantum groups. Associated to any PL group G is a PL group G* and a Lie bialgebra (~, ~*) with ~* dual to ~ as a vector space. A multiplicative P structure and a Lie bialgebra are both defined in terms of a 1-cocycle which in many cases (e.g. semisimple (ss) groups) is constructed from a solution of a Yang-Baxter (YB) type equation as an r-matrix (r E G • G). In particular when a PL group is linked with a quasitriangular r-matrix (i.e. a nonantisymmetric solution of the classical YB equation (CYBE)) then one can construct an isomorphism between U(~) and U(~*) in such a way as to determine a unique factorization property of G (factorizable G and (~, ~*)). An important class of factorizable groups is provided generically by the double group (e.g. SL(2, C) is the double of SU(2) and any element of SL(2, C) factors into a product of an SU(2) element and an SU(2)* element). This idea of double group provides a natural setting for the study of a dual P pair (cf. [263, 663, 858]). The P bracket for. the double describes actions of G and G* on each other and represents an abstract description of dressing transformations. Let us remark that the idea of dressing comes from Zakharov-Shabat (cf. [1000, 1001, 1002]) and in classical situations it is related to scattering problems and to the idea of transmutation as developed by the author in [143, 144, 145, 146, 147]. 3.1.1
Classical and quantum
systems
We digress here to give some brief background remarks about classical inverse scattering, Lax equations, dressing and Hamiltonian methods, etc. The hierarchy picture is developed in Section 5.1 and in Chapter 4 so we will not repeat that; rather we will concentrate on the Hamiltonian approach of [305] (cf. also [221]). As background let us first recall the dressing idea of Zakharov-Shabat which goes as follows (cf. [147] for extensive discussion and [376] for "ur-ideas"). One begins with a bare operator M0 (e.g. M0 -- i)2 on (-cx~, cx~))
3.1. B A C K G R O U N D
115
and "dresses" it to a more complicated operator M (e.g. M = 0 2 - u(x)) via operators (expressed via kernels here) K+ such that M(1 + K+) = (1 + K+)Mo (the "ur-examples'" here are KdV and KP). Thus one has a "transmutation" of M0 --~ M via M B = BMo with an intertwining operator B ~ 1 + K+ and this already involves a whole industry concerning inverse scattering, transmutation of operators, etc. (see e.g. [143, 144, 145, 146, 147, 181, 645, 681]). Upon noting that intertwining ideas arise naturally in group theory it is not surprising to find extensions of transmutation or dressing occuring in theories of quantum integrable systems (QIS). The analysis of kernels K+ was often facilitated in the author's work by introducing (sometimes in a virtually ad hoc manner) a spectral variable and constructing K+ via spectral pairings of generalized eigenfunctions (~p0,~P)(x, ~) where M e = Ar and M0r - Ar The analysis of kernels involved then a pleasing mixture of hard analysis, asymptotics, and functional analysis, whose group theoretical nature emerges most naturally in studying the nonlinear Schrhdinger equation (NLSE) phrased as a 2 • 2 system. Here one can combine analysis with algebra in a potent mixture and this approach is developed for example in [305] and sketched briefly below (cf. also [3, 4, 140, 151,144, 145, 146, 147]). Subsequently the algebraic features have dominated, especially those involving PL structures, because of the eventual aim of quantization via e.g. { , } ~ ( i / h ) [ , ] etc. We note again that there is a natural confusion here since with infinite dimensional systems such as KdV, NLS, etc. one is automatically using partial differential operators (PDO) whereas the quantization of finite dimensional Hamiltonian systems via p ~ -ihOq also introduces PDO. However the quantization of infinite dimensional problems in PDO involves also operator valued spectral data, normal ordering, Fock spaces, etc. (cf. [147, 209, 584]), remarks below, and Chapter 4. In this connection we recall that KdV equations on (-oc, co) in a Hamiltonian spirit can take the Magri-Lenart form (no corrections at -4-oo)
OnU = O 5Hn+l = $ - -5Hn 5u
(1.3)
5u
where g = i)3 + 2(u/) +/)u) and ut = (1/4)u'" + (3/2)uu' (here t ~ tl). Thus for Hi = (1/8) f u2dx and/-/2 = (1/8) f(2u 3 - (ux)2)dx one has
ut = {u, H2}1 = {u, H1}2;
{f' g}~ =
~(x) 0~ ~(x); {f' g}2 =
(1.4)
~(x) E~~(x)
(cf. [165, 221]). Thus the P brackets involve differential operators but one notes that this approach is different from that based on double Lie algebras, etc. (cf. [861]). Go now to [305] and consider briefly the NLSE on (-c~, c~) in a matrix form (e - + l cf. also [147])
U = Uo--[- ~U1; V ~- Vo-~- )~V1--[- )~2V2 ; Uo - x//~ ( 0q 0q ) ' 9 1 (1
u~=~/
0
0 ) -
1
1
=~3;Y0=i~lq12~3-i4/
(
0 -qx
(1.5)
qx) 0
with V1 = -Uo and V2 = -V1. One considers U, V as connection coefficients in a trivial bundle R 2 x C 2 over R 2 with (q = q(x,t))
Fx = U(x, t, )~)F; Ft = V(x, t, )~)F
(1.6)
116
CHAPTER 3. CLASSICAL AND QUANTUM INTEGRABILITY
where F is a two component vector or a 2 x 2 matrix. The compatibility condition F.~t - Ftx leads to the zero curvature equation
Eft - Vz + [U, V] = 0 - iqt = -qxx + 2e[q]2q (NLSE)
(1.7)
Gauge transformations F ~ G(x, t, A)F are associated with maps (.) U --, GzG -1 +GUG -~ and V ---, GtG -1 +GVG -1 leaving (1.7) invariant. For a curve 7 in R 2 from (x0, to) ---, (x, t) parallel transport along 7 is expressed via (P)f~7 =ex~p (f-y Udx + Vdt) (path ordered e x p o n e n t i a l - cf. [54] for a good discussion). Vanishing of curvature means that f~7 depends only on the end points and f~7 = I for a closed curve. Evidently f~1+~2 = ~"~')'2~')'1 and under a gauge transformation (.) one has f ~ ~ G(x, t)~TG-l(xo, to) With this notation one can write ( - . ) F(x,t) = fLyF(xo, to) which evidently satisfies (1.6). Consider next the parallel transport (. 9 .) T(x, y, A) =ex~p fy U(z, A)dz (with t fixed and suppressed). T h e n
OxT(x, y, A) = U(x, A)T(x, y, A); T(x, z, A)T(z, y, A) - T(x, y, A); OyT(x,y,~) = - T ( x , y , ~ ) U ( y , ~ ) ;
T(x,y, ~) =
(1.8)
T-~(y,x,~)
and det(T) = 1 since TrU(x,A) = 0 (note OzDet(T) = Tr(V)det(T) from (1.8)). In the quasiperiodic situation a(x + 2L, t) = exp(iO)a(x, t) with (U, Y)(x + 2L, t, A) - Q-1UQ for q = exp(iOa3/2) the matrix TL(A, to) =ex~p f_L U(x, to, A)dx is a monodromy matrix. We stay here on ( - c o , oc) and write E(x - y, A) - exp[(A/2i)(x - y)a3] for the solution of Ez = UE with E = I for x = y. Then T• A) = limy__.i~T(x, y, A)E(y, A) exists for A real and one can write
T_(x,A) = E(x,A) +
r_(x,z)E(z,A)dz;
(1.9)
O0
T§
= E(x,a) +
r+(x,z)E(z,a)az
for suitable kernels F• Let T+ = (T~T~:) for column vectors T i and then in fact T_1 and T~ can be analytically extended to ~)~ > 0. Setting T(~) = limE(-x,)~)T(x, y, )~)E(y, )~)
asx--*e<~andy-"-ec~
( abeT)) ~ ) = TwhereT-(z + ( z ' ~ ) T ( ''x ) g ~
with a(A) = det(Tl(x,A)T~(x,A)) = detS+(x,A) and b(A) = det(rl(x,A)Tl_(xA)). Thus a(A) extends analytically for ~A > 0 and for simplicity we assume here a(A) ~= 0 for ~A > 0 (no sol• The scattering matrix has the form S(~)=(
l/a -b/a
eb/a ) 1/a
(1.10)
and one sets S_(x, A) = (TI(x,A)T2__(x,A)) which is analytic for .~A < 0 with S_(x,A) S+(x, A)S(A). Now one defines G_(x,A) = S_(x,A)E-I(x,A); G+(x,A) = a(A)E(x,A)S+I(x,A)
(1.11)
with G(x, A) = E(x, A)G(A) so that G• satisfies the Riemann-Hilbert (RH) type of problem
(&)G+(x,A)G_(x,A)=G(x,A)whereG(A)=
( - 1b
eb) 1 is a priori defined only on R
and G• are to extend analytically to the upper (resp. lower) half plane with G ,-~ I + o(1) as IAI ~ oo. One can deal with this by Wiener-Hopf (WH) techniques or by deriving
3.1. BACKGRO UND
117
Gelfand-Levitan-Marchenko (GLM) equations using (1.9). The latter GLM equations have the form r + ( x , y) + A(x + y) + F+(x, s)A(s + y)ds = 0 (y > x); (1.12) r _ ( x , y) +/~(x + y) + (X)
r _ ( x , s)A(s + y)ds = 0 (y < x)
where
A(~)=
0 c ~0( x ) ) ; ~ ( x ) = ~ 1 ~ ~ abexp(i)~x/2)d)~; ~(x)
(
~-
0
)
;~(x) = ~
/
(113)
~ ~-~xp(-i~x/2)d~
Note that we have assdmed no discrete spectrum (i.e. a(A) ~= 0 for .~A > 0); with discrete spectrum the terms w and @ acquire additional finite sums of terms exp(ii.kj/2) where a()~j) -- O. Standard inverse scattering theory determines the time evolution of spectral data a, b from (tb) OtT(t, )~) = (iA2/2)[a3,T()~)] which specifies at = 0 and bt = -iik2b (the equations follow from Tx = UT and Ut = V x - [U, V] and x asymptotics). Note direct scattering gives (a, b)(t = 0). Then the potential q (or U) is determined from the GLM kernels directly (upon solving GLM). Alternatively the WH technique gives G+ and setting q0 = 0 there results
U = A d * G + I ( ~ . 3 ) = A-~d*G_ ( ~ / 3 )
(1.14)
where Ad gU - gtg-1 + gUg-1. This is standard inverse scattering and we will not belabor the matter here (cf. [3, 4, 143, 145, 146, 147, 181,305, 645, 681] for many results). We remark also that the technique of r matrices is developed in [305] for classical integrable systems. For NLS the basic P brackets are
S G S F ) 55~ ( t 5G {F, G} = i / ( h F-~q -~q dx (we assume functional variational calculus is k n o w n - cf. introduces a notation
8B {A e, B} = i f ( 5~q | 5(t
(1.15)
[143, 147, 221; 377]) and one
8A 8B) 5(t | -~q dx
(1.16)
where A,B ~ 2 x 2 matrices. Thus (AQB)jk,mn : AjmBkn and {A ~, B}jk,mn : {Ajk, Bkn}. For transition matrices T(x, y, A) one will have e.g.
{T(x, y, ~) ~, T(x, y, ~) } = [~(~ - ,), T(x, y, ~) 0 T(x, y, ,)] for a suitable r matrix (indicated below for this situation).
(1.17)
One notes that {q(x), q(y)} =
iS(x- y) so e.g. (&&) {U(x,)~) ,e U(y,#)} = [ r ( A - #),U((x,) 0 | 1 + 1 | U(y,p)]8(x - y) which is called ultralocality (here r(A) = -(el)Oh where a(~ | r/) = r] | ~ and one notes that U contains x derivatives of q). Note
a=
1000 0 0 1 0 01 00 0001
(1.18)
118
C H A P T E R 3. C L A S S I C A L A N D Q U A N T U M I N T E G R A B I L I T Y
There is also a general discussion of classical integrable models via loop groups and r matrix ideas in [305] but we omit this here (cf. also [808]). Let us next indicate a quantum version of NLS following [147, 209] (cf. here [966] for remarks on quantum integrability). Thus write now ~ ~ q and think of the NLSE ~s = -~ + 2~1r162 We take e > 0 momentarily (no solitons) and T for the transition matrix. The Hamiltonian formulation here involves [-I = f dx((~tx~x + e ~ t ~ t ~ ) with [@(x, t), ~t (y, t)] = h b ( x - y) where @ is the operator corresponding to @ and normal ordering will be used which puts ~ (annihilation) operators to the right of ~t (creation) operators. The equations of motion in Hamiltonian form are then @t - (1/ih) [@, /:/] with i@t = -@zz + 2e@t@r One thinks of a Fock space with @(x, y)10 > = 0 and an n particle state with momenta k l , . . . , kg has the form (setting h = 1 momentarily)
iCt
Iki"'kN r
>= ~
1 / uH d x i r
(1.19)
>;
1
k) = < 0l@(Xl)...@(XN)lkl...
kN >
Here r is the N particle wave function. In this context N
N
fIlkl...kN>= /l~1 dxi[-~l02 -j-2EES(xi-xj)]r i<j
...@t(xN)lO>
(1.20)
a n d , ] k l . . , kN > is an eigenstate of/:/corresponds to r being an eigenfunction of the many body Schrbdinger equation with delta function interaction, i.e. [- ~ N 0~ + 2e ~ i < j 5(xi -xj)]r = Ar with A = ~ g k 2. Solutions to this equation are expressed via the Bethe Ansatz r
k) = ~ exp(i ~ P
kpjxj) H
kpm - kpj -
m>j
(1.21)
kpm -- kpj
in the region Xl < ... < XN (here P ,,~ (pj) corresponds to permutations of 1 , . . - , N). We recall the form of U, V etc. in (1.5) and note that the auxiliary linear problem (1.6) takes a normal ordered form (for e < 0 here- F ~ r and A ~ 2~)
$. = :
05
_
(11) 0
0
0"_
0 0 ) (1.22) 1
0
Now recall r(A) = - ( e / A ) a from (11,8) which is replaced by r(~) = (e/2~)q in [209] (this seems to introduce a spurious minus sign but it apparently arises here since e < 0 is under consideration). Next from (&&) one obtains with some calculation (upon setting (tblb) R ( ~ ~') = I - ihr(~ - ~')) R(~ - ~')[U(x, ~) | 1 + 1 @ U(x, ~') + he(a_ | a+)] = fr(x, ~) | 1 + 1 | {](x, ~') + he(a+ | a_]R(4 - ~')
(1.23)
(note R is still a c number matrix). This represents a quantized version of (&&) to which it tends as h ~ 0 via (1/ih)[, ] ~ { , }. Next write
T(x, y, ~) =" T(x,
~fx
y, ~) "=:exp
dzU(z, ~) "
(1.24)
3.2. R M A T R I C E S A N D PL S T R U C T U R E S (note e.g. r - i.e. r
r
119
is a functional of the dynamical variables and is defined to be normal ordered Differential equations based on (1.8) will now have the form, e.g.
o~(x, y, ~) =: r2(x, ~)~(x, y, ~)
-i~3~+
:=
leading to
~'(x,y,() = I +
dz " O ( z , < ) T ( z , y , ( ) := I +
dz " ~'(x,z,<)O(z,<) "
(1.26)
Commutator brackets of the form [r
2P(x, y, ~)] = i ~ ~ a + T ( x ,
y, ~)', [r
2n(x, y, ~)] = - i h ~ a - T ( x ' 2
y' ~)
(1.27)
etc. also arise. Next one defines (&&&) 2Pl = T | I and T2 = I | T and uses R as in (lb~) to obtain e.g. R ( ~ - ~ / ) r 1 (x, y, ~ ) r 2 ( x , y, ~') = r 2 ( x , y, ~ l ) r l (x, y, ~ ) R ( ~ - ~t)
(1.28)
(quantum analogue of (1.17) to which it goes when h -~ 0). There are many other formulas involving e.g. operator valued spectral quantities 5(~), 5t(~) etc., which we omit here. Further from (1.28) one obtains the Yang Baxter (YB) equation R12(r - ~')R13(r -
- r
= R23(~' - ~r162
- ~ " ) R 1 2 ( ~ - ~')
(1.29)
which reduces to the classical YB equation (~" = 0 for simplicity) [r12(~ -- ~/), r13(~)] nt- [r12(~ -- ~I), r23(~/) _it- [r13(~), T23(~/)] = 0
(1.30)
when h ---+ 0. One should compare with the CYBE Br = 0 (cf. (2.12)) which does not contain a spectral parameter.
3.2
R MATRICES
AND
PL STRUCTURES
Notations such as (5A), (11A), and (12A) are used. We have seen r and R matrices already in the NLS example of Section 3.1.1 and will be more systematic now. Originally we simply extracted from [304, 507, 585, 589, 671, 819, 821, 861, 877, 917, 918, 924] to provide sample spin chains, transfer matrices, Bethe Ansatz equations, etc., leading up to the presentation of [972] describing the role of the Hirota bilinear difference equation in both classical and quantum integrable systems. This seemed to be a reasonable way to introduce matters but it left too much unsaid to be instructive and said too much at the same time. What we would like to do now is to give a more concise historical development based on [919] (to which we refer for many details) and propel the discussion toward a rather more general perspective which will at least bring us into contact with [57, 82, 79, 127, 300, 322, 324, 355, 356, 357, 397, 464, 526, 527, 649] (as well as [603, 972]). Thus we go now to the superb lectures [919] and provide extractions to serve as a springboard. First we recall that the classical inverse scattering method (CISM) involves continuous situations {U(x, s o, U(y, #)} = [ r ( s
#),U(x,)~) | I + I | U(x,p)]5(x - y);
(2.1)
120
C H A P T E R 3.
CLASSICAL AND QUANTUM INTEGRABILITY
[r12()~ -- ]_t), r13(/~) -~- r23(#)] -~- [r13()~), r23(/-t)] : 0 (the latter is the classical Yang-Baxter equation (CYBE)). Also a unitarity condition of the form ( l l A ) r12(/~) + r21(-A) =- 0 usually holds. In passing from the continuum to the lattice (2.1) is deformed into ~.z / X n + l
{Ln(A) ~ Lm(#)} = [ r ( A - #), Ln(A)| Ln(#)]~Smn; Ln(A) ,~exp
dX n
U(x,A)dx
(2.2)
(eft (1.17) with a classical r matrix as in (2.1)). The quantum inverse scattering method (QISM) uses a (quantum) R matrix (as in (~tb)) with R ( A - p)(Ln(A) | 1)(1 | Ln(#)) = (I | ]-,.(#))(]-,n(A) | I ) R ( A - . ) ;
R 1 2 ( ) ~ - #)R13()~)R23(~)
--
R23(#)R13()~)R12()~-#)
(2.3)
(the latter is the quantum YB equation (QYBE)). Now the lectures [919] are on quantum groups and in this spirit we will sketch first some group and algebra ideas, then PL structures with Yang-Baxter (YB) equations (CYBE) and modified YB equations (MYBE or mCYBE), and then quantization ideas with examples. Referring to (1.2) we recall that an algebra A over C (associative with unit) requires (A) a multiplication m " A | A ~ A with m ( a | b) = a . b and (a. b ) . c - a . (b. c) (B) an inclusion i " C ~ A ' i ( a ) = a . 1. Thus A ~ ( A , m , i ) and for A* - H o m ( A , C ) (linear functionals on A) one has a counit A 9 A* ~ A* | A* via A(g)(a | b) = g(a. b) where g E A*. Associativity for m implies (id @ A ) . A = (A o id) o A (as in (1.2)) and i induces the evaluation map e" A* ~ C via e(g) = g(1) satisfying (id | e) o A = (c | id) o A = id (dualizing the axioms for a unit). A vector space L over C is a coalgebra if there exists A and e satisfying the above axioms, so A - A* has a natural coalgebra structure. Further if an algebra A is also a coalgebra satisfying the additional axioms of (1.2), namely A ( a . b) = A ( a ) . A(b) and e(a. b) = e(a)e(b) then A is called a bialgebra, A ~ (A, m, i, A. e_). Note if A is a bialgebra so is A* with multiplication (12A) rn*(gl | = (gl | and A*(a | b) -- g(a. b) while 1" ~ e and the counit is the evaluation map. A coalgebra is cocommutative if (12B) A = a o A and e.g. if A = C ( X ) for X = Mn(C) (n x n matrices) then A is not cocommutative. Finally a bialgebra is a Hopf algebra if there exists an antipode S with (12C) S(a . b) = S(b)S(a) and m o (S | id) o A = m o (id | S) o A - i o (cf. (1.2)). Thus a Hopf algebra involves (A, rn, i,A, e, S) ~ A and in this situation A* also has a natural Hopf algebra structure with S*(g)(a) = g(S(a)). E X A M P L E 3.1. Let X = Mn(C) and A = P o l ( X ) where the polynomials are generated by linear functions tij defined via tij(g) = gij for g = ((gij)) where 1 < i , j < n. Then A = C[tij] and A ( t i j ) = tik @ tkj provides a coproduct. One writes this as A(T) = T for T = ((t~j)) w h e r e , corresponds to summation over repeated indices (this is equivalent to the formula for A(tij) above). It follows also that e(T) = I. One can also consider the algebra ,4 freely generated by n 2 variables tij with no relations. This algebra A has a natural bialgebra structure with A and e as above, extended to ,4 via the connection axiom (12D) A ( a . b ) = A ( a ) . A(b) with c(a. b ) = c(a). c(b). However A is neither commutative nor cocommutative. One remarks that if A is a commutative bialgebra then A = C ( X ) for X a (smooth or topological) space and the coalgebra structure plus the connection axiom implies that X is a semigroup (but maybe not a group). For commutative Hopf algebras however the underlying X has a group structure. In particular for G a Lie group A - C ( G ) is a commutative Hopf algebra which is cocommutative if and only if G is Abelian. Another
3.2. R M A T R I C E S A N D PL S T R U C T U R E S
121
important Hopf algebra is the universal enveloping algebra U(~) for ~ a Lie algebra. Here for X E ~ A(X)=X@I+I@X; A(1)=1| (2.4) e(X) = 0 ; e ( 1 ) = 1; S ( X ) = - X and these maps are extended to U(~) by the connection axiom (12D) and the antihomomorphism property of S. U(~ is cocommutative but is commutative if and only if .0 is Abelian. Roughly any cocommutative Hopf algebra has this form and in particular U(~) _~ G-[~c(G) which is the space of Schwartz distributions on G with support at the identity e. Finally one can think of a quantum group as a noncommutative and noncocommutative Hopf algebra A although in fact it is the putative "spectrum" of A which would be the underlying geometric object. Next one considers,Poisson (P) structures on a manifold M via the P bracket { , } : A @ A ~ A for A = C ~ (M) satisfying
{ f , g } = - { g , f}; {f, gh} = g { f , h } + h{f,g};
{r {r x}} + {x, {r r
+ {r {x, r
=0
(2.5)
Then M is called a P manifold and it can be considered a phase space with A the algebra of classical observables. One has (1 < i, j < n)
{r
r
=
0r 0r
n~j(x)Ox~Oxj;
Ofljk
Of]ij
~
= -~J~;
(2.6)
O?]ki
~"-ggx~ + ~-~x~ + ~J~-g[x~ = o Note that if ~* is the vector space dual to a Lie algebra ~ then it has a P structure via (12E) rhj(u) = ckijuk where [ X i , X j ] - ckjXk and u = ~ u i x i for x i e .q* a dual basis to Xj E ~. Suppose now that a L i e g r o u p G is a P manifold v i a a P structure { , } on A = C ~ ( G ) . Then G • G is also a P manifold with observable algebra A @ A ~_ C ~ ( G • G) and P bracket (12F) {r @r r | "-- r @{r @2}-~"{~)1, r162 The coproduct A is defined via A ( f ) ( g l , g2) = f(glg2) and one says G is a PL group if it is a P manifold satisfying (12G) A({r r = {A(r A(r 2 (i.e. m : G • G ~ G is a P map). It will be instructive to follow [919] here by introducing the notation (12H) (Agf)(g') =
f(gg') and (pgf)(g') = f(g'g) so that
{r ~)}(glg2) --- { ) ~ g l r
(2.7)
t- { p g 2 r
Recall also (12I) (Ox f )(g) = (d/dt) f (gexp(tZ))lt=o and (O~xf )(g) = (d/dt) f (exp(tX)g)lt= o. Thus X ~ Ox is a representation of ~ by first order differential operators on G with k Ok; similarly 0 'i ~ -- 0~c9~ = --CijO k k. , Since OxOg-0,/01,( = O[x,y] and (12F) OiOj-OjOi = cij ' the vector fields c9/ give a trivialization of the tangent bundle TG any P bracket on G has the form (12J) {r r = ~?iJ(g)Oir162 with ~ij = _r/ji where r/defines a map G ~ A2~ via rl(g) = rlij (g)Xi @ Xj. Note here r//j ~ rhj of (2.6) defined for a manifold M but there is more structure now. In particular the structure (12J) provides a PL group structure if and only if ~7(glg2) = r/(g2) + Ad~ 1. r/(gl); (2.8)
~
= ~"o~ ~ + ~o~
~j + ~/~oe~ ~ + c ~ J ~ ~ + 4 f ~
~ + ~f~/~
- 0
C H A P T E R 3. C L A S S I C A L A N D Q U A N T U M I N T E G R A B I L I T Y
122
T h e first condition means th a t rl is a one cocycle for A d a action (i.e. A d ( g ) . ( X | = Ad(g). X| and to prove it one can use (2.7) with AgOX = OXAg and OxPg = pgOAd(g-1).X. REMARK 3.2. Here it is useful to digress briefly to recall the cohomological picture at the infinitesimal level. Thus (cf. [589]) X . Y ~ a d x Y = [X, Y] for X, Y e t) with X . (Y | Z) = X . Y | Z + Y | X . Z and ad2x = a d x | 1 + 1 | adx. T h e n for b - bijXi ~ X j one has ad2x b - biJ ([x, xi] | X j + Xi | [X, Xj]) and
(2.9) T h e space of skew symmetric k-linear maps on ~ with values in a vector space V is called the space of k-cochains ~ - , V and the c o b o u n d a r y operator is k
~u(Xo, X 1 , ' " , X k ) = ~ - ~ ( - 1 ) i x i 9( u ( X o , . . . , X k , ' "
,Xk))+
(2.10)
0
+ ~-~(-1)'+Ju([X,,Xj],Xo,
"'" ,f(i, "" ",f(j,'" . , X k )
i<j
and and and ~so Y =
as usual 5(5u) - 0 since we are dealing with representations. Note for k - 0 : u C V 5u = x . u (small and big x are used interchangeably here) while for k = 1 : u : ~ ~ V for x, y e ~ with v e Y one has ( 5 8 ) 5v(x, y) = x . v(y) - y . v(x) - v([x, y]). We note ~ ( ~ ) ( x , ~) = x . ~u(~) - ~ . ~ ( x ) - ~ ( [ x , ~l) = x . (~. ~) - ~ . ( x . ~) - Ix, ~ ] . ~ ~nd for ~0 with adjoint representation 5(5u) = [x, [y, u]] - [y, [x, u]] - [[x, y], u] = 0 by Jacobi.
REMARK 3.3. For a change of perspective here, following [589], a Lie bialgebra can be simply defined as a Lie algebra ~0 with a linear m a p ? : .0 ---* ~0 | t) such t h a t (i) 3'* : t~* | g* ~ g* determines a Lie bracket on ~* and (ii) ? is a 1-cocycle of ~0 with values in | ~ (with ~ acting via the adjoint representation ad2). Thus condition (ii) asks one to look at x --, ad 2" ~ --, End([? | ~) (i.e. Y = ~ | with ad2z(y | z) = [x, y] | z + y | [x, z]. T h e n consider ?(x) ~ b E ~| and 57 = 0 corresponding to 5?(x, y) = x . ? ( y ) - y . ? ( x ) - ? ( [ x , y]) = 0 and this a m o u n t s to ad~?(y) - ad2y?(X) - ?([x, y]) - 0 (2.11) We note also t h a t x --, ad~ 9~ --, ~* (coadjoint representation of ~ in ~0") with < ~, adxy > - < a d ~ , y > leads to ad~u = [~, rl], with < a d ~ , x > = - < U, ad*~x > and ~ ---, ad*~" .(l* --* ~0 is the coadjoint representation of ~* in ~0. The double of ~ (a~ = ~0 ~ g) is defined as ~ | t) with the unique Lie algebra structure such th a t ~ and ~* are Lie algebras (as indicated) and the n a t u r a l scalar product on ~ | ~ is invariant (the natural scalar product is defined via (5 C ) (x[y) = 0, (~[rl) = 0, and (x[~) = < ~,x > for x , y C ~ and ~,r / C ~*). Next one goes to r-matrices and the YB equations in somewhat more detail and we summarize as follows. Since ? : ~ ---, ~ | ~ determining a Lie bialgebra is necessarily a cocycle with 57 - 0 one asks w h a t further conditions on a c o b o u n d a r y ? = 5r are needed to determine a Lie bialgebra. These are evidently (A) 5r must take values in A2~ (skew s y m m e t r y of the bracket [, ],) and ( B ) T h e Jacobi identity must hold. Note ? = 5r implies 57 = 0 and ? ( Z ) = [ Z | I| r] from which the Jacobi identity follows from (hD) [X | 1 | 1 + 1 | X | 1 + 1 | 1 | X, Br] = 0 where Br =
[r12,r13]-t- [7'12,r23]-1- [r13, r23] E g(~g(~)g;
rabZa•
l(~Z
b ~..
r13; etc.
(2.12)
3.2.
R MATRICES
AND PL STRUCTURES
123
Here ( h D ) is sometimes called the modified YB equation (MYBE) and Br = 0 the classical YB equation (CYBE) - cf. (2.1) for rij = lim~__.c~rij(A). In particular we see t h a t a PL group is a Lie group equipped with the additional P structure and it is this structure which induces on 0 a corresponding additional structure, namely it makes ~* into a Lie algebra. Thus bialgebras arise and a Lie algebra structure on ~0" in fact amounts to considering a second Lie algebra structure on ~ determined by an r matrix. The modified Yang-Baxter equation (MYBE) is then a sufficient condition for r to define a second Lie bracket on ~. Now return to [919] for a more detailed account as follows. Consider r/ : G ~ /~20 independent of g in the form r/(g) = r = riJXi | X j with e.g. r12 = riJXi | X j | 1 etc. and C Y B E in the form Br = 0 from (2.12). Then ~ijk = 0 in (2.8) will hold and ( 1 2 K ) {r ~2}teSt = riJOir with r ij -- - r ji is a P bracket if and only if r E A2~ satisfies C Y B E (note the PL property (2.8A) may not hold). Similarly {r ~2}right = riJO'r is a P bracket if and only if r E A2~ satisfies CYBE and hence any linear combination a { , }teSt +t3{ , }right is a P bracket under these circumstances with ~7(g) = a r + / 3 A d ( g - 1 ) . r and~iJk = 0. Further for c~ = - 1 a n d / 3 = 1 it follows that r/(g) = A d ( g - 1 ) . r - r is a c o b o u n d a r y and hence a cocycle. Indeed r/(9192 ) = A d ( g l g 2 ) - 1 . r - r = Ad(921) 9( A d ( g ~ l ) . r - r ) + + A d ( g 2 1 ) . r - r = Ad(g21) 9r/(gl) + r/(g2)
(2.13)
Thus for r E A2O satisfying C Y B E the pair (G, { , }) with {r r - r i J ( O ~ r 1 6 2 Oir is a PL group. A variation on this (cf ( h D ) , (2.12), ( h E ) ) posits r E A2t) with r/(g) A d ( g - 1 ) r - r (so { , } defined by r/ satisfies the PL property (2.8A). T h e n the Jacobi identity (2.8B) holds if and only if ~ ~ Br E A3~ is A d c invariant which is equivalent to ( h D ) corresponding to MYBE. REMARK 3.4. Let C E $2(~) be a central element, i.e. [C,X] = 0 for all X E .~) (Casimir element). Since [A(C), A(X)] = [A(C), X | 1 + 1 | X] = 0 the element (12L) t = (1/2)(A(C)-C| 1- 1 | E {7| satisfies [ t , X | 1 + 1 | = 0 or equivalently t is A d c invariant. If we write C = c i J x i x j with C ij = C ji then t = c i J x i | X j and one can set ~ Br = c~[t13, t23] E A3~ since the ad o invariant subspace of A3~ is one dimensional. This will then lead to the M Y B E and a PB { , } r i g h t - - { , }teSt for c~ = --1 (achievable by scaling when c~ =fi 0); for c~ = 0 the C Y B E arises. One notes that the difference between C Y B E and M Y B E is simply r E A2t) instead of r E ~ | ~. Namely if r E A2~ satisfies M Y B E then = r + (1/2)t E ~ | ~ satisfies CYBE, but ~: r A2~. Let now ~ be a Lie algebra with r = riJZi | Z j E A2g an r m a t r i x satisfying CYBE. T h e n r is called reduced if the n • n matrix ((riJ)) is nondegenerate. It can be shown easily (cf. [919]) t h a t any such r can be reduced in the sense t h a t there exists a Lie algebra g0 c t) and r E A2~0 with a nondegenerate matrix. Further r E A2~ is a reduced solution of C Y B E if and only iff the corresponding nondegenerate m a p w : A2~ ---, C is a 2-cocycle, i.e.
~(x, [v, z]) + ~o(z, [x, v]) + ~o(Y, [z, x]) = o
(2.14)
where w ( X , Y ) = rijuiv j for X = u i X i and Y = v i X i . Thus the Lie algebraic meaning of C Y B E is t h a t its reduced solutions correspond to nondegenerate 2-cocycles. For simple Lie algebras H2(~) = 0 so any 2-cocycle is a coboundary and there exists 0 E ~* such that ( 1 2 M ) w ( X , Y ) = O([X, Y]) (here 0 ~ uix i where x i is a dual basis in ~*). However it follows (cf. [919]) t h a t ((wij)) is degenerate so there are no reduced solutions of C Y B E for simple Lie algebras. Fortunately M Y B E can be applied to simple Lie algebras as follows. One says that
C H A P T E R 3. CLASSICAL AND QUANTUM I N T E G R A B I L I T Y
124
an element r e End([?) is a classical r-matrix if (12N) [X, Y]r = (1/2)([rX, Y] + [X, rY]) satisfies the Jacobi identity, i.e. equips ~ with a second Lie algebra structure. Now write B ( X , Y) = [X, r Y ] - 2r([X, Y]r) and one can rewrite the Jacobi identity in the form
[X, B(Y, Z)] + [Z, B(X, Y)] + []I, B(Z, X)] = 0
(2.15)
This can be satisfied in two ways: (A) B ( X , Y) = 0 or (B) B(X, Y) = (~[X, Y]. One shows that case (A) implies CYBE so there is nothing new. Assuming r is skew-symmetric now the reduced r must live in a smaller algebra than ~ and for case (B) with ~ simple one obtains ( 1 2 0 ) [ r X , r Y ] - r([rX, Y] + [X, r Y ] ) = - [ X , Y ] (setting (~ = - 1 so that r = id can be a solution). Introducing ((r})) and r ij = r}g ej where rXj - r}Xi (and u j = giku k for Cartan-Killing form gij with ((gij)) = ((gij))- 1) one can write r = r ij Xi | Xj and there results
[~12, ~la + ~2a] + [~la, ~2a] = -[tla, t2al; t = #Jx~ | xj
(2.16)
We recall that t is Ado invariant and we have obtained MYBE. A simple way of finding solutions of MYBE now can be given as follows. Let ~ - ~+ | ~_ with P+ the corresponding projections. Then r = P + - P _ is a classical r-matrix which lies in A2O if ~+ are isotropic with respect to some nondegenerate invariant bilinear form on ~. In particular if ~ = •+ | h | fi_ is the Cartan decomposition of a simple Lie algebra with P+ the projections onto fi+ then r = P+ - P_ is a classical r-matrix with r E A2~. If A+ is the set of positive roots of ~ and hi, Xa, X - a is the corresponding orthonormal basis then ~
r =
~ ( X - a | Xa - Xa | X - a ) E A2~ aEA+
(2.17)
satisfies MYBE.
3.3
QUANTIZATION
AND
QUANTUM
GROUPS
Notations such as (SA), (12A), and ( 1 3 A ) are used. Let now M be a phase space, i.e. a P manifold with P bracket { , } and set A = C ~ (M). By a quantization of A one means a deformation Ah of the the commutative algebra A which coincides as a vector space with A or perhaps with A[[h]] ~ formal power series in h but has a new product *h " Ah | Ah ~ Ah of the form r *h r = r r + h{r r + . . . satisfying r *h (r *h X) = (r *h r *h X. In addition one wants r *h 1 = 1 *h r = r and (1.1) which connects the quantum and classical pictures. Since physically one expects h ~ - i h one imposes also the condition r *h r -- ~ *h r which ensures that real valued classical observables go into self-adjoint quantum observables. This approach goes back to H. Weyl and there is an extensive literature (cf. also Chapter 2). One of the most general theorems states that if the phase space M is a symplectic manifold and if H 3 ( M ) = 0 then there exists a quantization of A = C ~ ( M ) . However it is a pure existence theorem without any effective construction of the deformation product. The sample construction of H. Weyl in this direction goes as follows. m
m
E X A M P L E 3.5. Let M - R 2 with coordinates p,q and P bracket determined by {r r = (Or162 - (Or162 Let P, Q be the QM operators with commutation relations [P, Q] = (h/i)I = hi and set U(u) = e x p ( - i u P ) with V(v) = e x p ( - i v Q ) (one parameter groups). Then one has the Weyl commutation relations U(u)V(v) = exp(-huv)V(v)U(u). Now for any f E A N S' write
](u, v) -- -~r
1 L2 ](u, v)e-ipu-iqvdudv
2 f(p' q)eipu+iqvdpdq; f(p' q) = ~
(3.1)
3.3. Q U A N T I Z A T I O N AND Q U A N T U M GROUPS
125
One can extend the Fourier transform to A C T)' in various ways (cf. [171] for references) but the integral formulas are more intrinsic to S ! . Consider the correspondence f ---, Af where A / i s an operator in the same Hilbert space as P, Q defined via
A/= ~1 fR ]exp[(h/2)uv]U(u)V(v)dudv
(3.2)
Using the formula Tr V(v)U(u) - (2~/ih)6(u)5(v)one arrives at the formula (12P) ](u, v) i h T r d / V ( - v ) U ( - u ) e x p [ - ( h / 2 ) u v ] . Therefore the associative product d e . d e induces a new product *h on their symbols (~ functions r r Using Weyl commutation and the Fourier inversion formula one obtains
(r
r
~) =
~1
/ R ~ 5 ( U l -~- U2 -- U ) ~ ( V l -~- V2 -- V)"
9e-(hl2)(ulv2-u2vl)r
(3 3)
vl)~(u2, v2)duldu2dvldv2
One writes now (12Q) ff = (O/Op) | (O/Oq) - (O/Oq) | (i)/Op) so that ff(r | ~) - Cp ~ Cq - Cq @ r Then writing m 9A | A --, A for the usual multiplication one writes in [919] 9h ~" m o exp[(h/2):Z-] for Weyl quantization (cf. also [448] for this notation). Star products are treated in more detail in Chapters 1 and 4 and Section 5.8 and we simply use here the notation (ha oh) (r *h r q) = e (h/2) 0~ 1 0-~2 0~10p 2 r , ql)r , q2)lp~=p,q~=q (3.4) (cf. also [43, 44, 68, 88, 116, 117, 118, 137, 141, 217, 219, 229, 295, 308, 320, 321, 340, 343, 433, 448, 520, 530, 536, 579, 649, 650, 652, 653, 663, 664, 785, 820, 891, 910, 947, 968, 984, 985, 986] for more on this). We go now to quantization for PL groups associated with CYBE and MYBE and discuss the QYBE following [919]. We only sketch the main points here and refer to [919] for details .. and references. Thus let G be a PL group with P bracket (12Q) {r r = r~3(0~r 0ir where r = riJ(zi | Xj) E A2t) satisfies CYBE in the form (12R) [r12, r13 + r23] + [r13, r23] = 0. We consider the quantization problem for A = C a ( G ) , which is in general underdetermined and quantization is not unique. However in this case one can make it essentially unique if the PL property is taken into account via (12S) A ( r *h ~) -- A(r *h A ( r where A(f)(gl,g2) = f(glg2). Since r satisfies CYBE one knows from above that both the left and right invariant terms in the P bracket are also P brackets so one can quantize these separately and then combine. Therefore consider - { r r - -riJOir ' and look for an associative product 9 such that r r - r r (h/2){r r + " " with r = 1,r and)~gr162 = A g ( r 1 6 2 One c o n s i d e r s , in the form ( 1 2 T ) , = m o where m is the usual multiplication a n d / ~ = 1 + ~ h n[~n for linear differential operators /~n. Left invariance of 9 is equivalent to (Agl | Ag2) o/~n = /~n o (Agl | Ag2)" Let ~ be the representation of the universal enveloping algebra U(~) by differential operators via ~v~(Zi) = 0i so that /~n = ( ~ | ~ ) ( F n ) w h e r e Fn C U(~)| V(~). In particular F1 = - ( 1 / 2 ) r and generally /~ = ( ~ | ~ ) ( F ) for F e U(~)[[h]] | U(~)[[h]]. Now set X = ( X 1 , . . . , Xn) and wr.ite any element a E U(~) in terms of noncommuting variables Xi with X i X j - X j X i = c~jXk. Introduce a second set of variables Y = (Y1,""", Yn) commuting with X and satisfying YiYj - YjYi = c~jYk. Identifying Xi ~ X~ | 1 and Y~ ~ 1 | Y~ one can write any A ~ U ( ~ ) | U(~) in the form A(X, Y). The Hopf algebra structure of U(.~) then involves e(a) = a(0) with A ( a ) ( X , Y ) = a(X + Y) = a(Y + X) - A(a)(Y, X) and S(a)(Z) = a ( - Z ) where - indicates the antihomomorphism of U(~) defined on generators
126
CHAPTER 3. CLASSICAL AND QUANTUM I N T E G R A B I L I T Y
by Xi ~ - X i . Then if F(X, Y) - ~_,cc~,z(h)XaY ~ (in standard multiindex notation) one has (12U) ( r r ~ cc~z(h)Dar162 and we have the requirements
F ( X + Y, Z)F(X, Y) = F(X, Y + Z)F(Y, Z); F(X, O) = F(O, Y) = 1
(3.5)
(cf. [919] for proof). Consequently 9 = m o / ~ defined as indicated determines a left invariant quantization of the P bracket - { , }Zelt. A similar argument can be applied to (12V) , ' = m o/~' for the P bracket { , }right via /~' = (Trp | 7rp)(F-1) where 7rp is an antirepresentation 7ro(Xi_) = 0~. The procedure then leads to the same equation (3.5). Consequently f: = F o F' will yield the desired quantization *h = m o fi" for ( 1 2 Q ) . The details are in [919] and finally one can invoke a theorem of Drinfeld (cf. [263,264] which says that for r 9 A2~ satisfying CYBE there exists F satisfying (3.5) (the proof" is also sketched in [919]). One can write now
R(x, Y) = v-~(y, X)F(X, Y) 9 U(~)[[h]] o U(~)[[h]]
(3.6)
and this will satisfy the quantum Yang-Baxter equation (QYBE)
R(X, Y ) R ( X , Z)R(Y, Z) = R(Y, Z)R(X, Z)R(X, Y); R(X, Y)R(Y, X) - 1
(3.7)
and R = 1 - hr + O(h2). It is interesting that QYBE which is cubic in R reduces to the main equation for F which is quadratic. Finally the algebra Ah ~ (A, , h , A , i, e) actually has a Hopf algebra structure (i.e. there is an antipode S). Going now to MYBE let r 9 A2~ satisfy MBYE in the form (2.16) with [ t , X | 1 7 4 0 as in (12L) and let G be the corresponding PL group with P bracket ( 1 2 Q ) . Guided by the development for CYBE one looks for *h in the form *h = m o/N o/~ = m o/~ o / ~ where /~ = (Tr~ | Try(F) a n d / N = (Trp | 7rp)(F-i). One cannot assume that F should satisfy (3.5) again so this equation is modified to be
F ( X + Y, Z)F(X, Y) = a(X, Y, Z)F(X, Y + Z)F(Y, Z)
(3.8)
where a 9 [U(~)]| is Ada invariant via [ a , X | 1 | 1 + 1 | X | 1 + 1 | 1 | X] = 0. It is then proved in [919] that under these stipulations on a and F the product *h above is associative. Additional conditions on a then arise in the form (A) a = 1 + h2a2 + . . . where Alt a2 = 4[ti3, t23] with Alt ~ alternation defined via
Alt a2 = a2(X, Y, Z) - a2(Y, X, Z) + a2(Y, Z, X ) -
-~2(z, Y, x ) + ~2(z,x, Y) - ~2(x, z, Y)
(3.9)
Further a should satisfy
~(x, Y, z ) ~ ( x , Y + z, u)~(z, z, u) = ~ ( x + Y, z, u)~(x, Y, z + u)
(3.10)
(pentagon equation in 2-D conformal field theory (CFT). Finally one wants the relation ( 1 2 W ) a(X, II, Z) = a-l(Z, Y, X) and
eht(X+Y'Z) = a - I ( z , x , Y ) e h t ( X ' Z ) a ( X , Z , Y ) .eht(Y'Z)a-I(X,Y,Z)
(3.11)
Then a theorem of Drinfeld states that there exists an AdG invariant a E [U(~)]| satisfying these requirements and the proof reveals deep connections with differential equations in C F T for correlation functions. Finally for a as indicated and r as above satisfying MYBE with F satisfying (3.8) and F = 1 - ( h / 2 ) r + O(h 2) one has an R "matrix"
3.3.
Q U A N T I Z A T I O N A N D Q U A N T U M GROUPS
127
(12X) R ( X , Y) = F - I ( Y , X ) e x p [ h t ( X , Y ) ] F ( X , Y) satisfying QYBE. Consequently Ah is well defined and will in fact be a Hopf algebra. It is instructive to look at matrix examples based on a representation p : ~ ~ Mn(C). Let T = ((tij)) be the corresponding coordinates and set F = (p| p)(F). Then (12Y) T1 *h T2 = F - 1 R @ T/~ where e.g. T1 ~ T | 1 | 1 and e.g. t~-'12,3~,~ (p| p | p ) ( F ( X + Y, Z)); for P defined as P(vl | v2) = v2 | Vl one can write (12Z) T2 *h T1 = ( P - F P ) - I T | T P F P and consequently for/~ = ( p / ~ p ) - l / ~ one has/~T1 *h T2 = T2 *h T1/~. More generally one can use in these commutation relations any m a t r i x / ~ of the form ( p / ~ p ) - I . I n v . F where the n 2 • n 9 matrix I n v is invariant via [Inv, T | T] = 0 and setting I n v = exp(ht) one obtains (13A) /~ = ( P [ ~ P ) - l e x p ( h t ) F leading to (13B) /~T1 *h T2 = T2 *h T1/~. This clarifies some of the connection between QISM and QYBE; the quantization procedure for PL groups is in some sense the conceptual background for the algebraic approach. E X A M P L E 3.6. Consider the following example from [918, 919] (cf. Section 6.1.4 for more examples). We set Mq(2) = SLq(2) with generators a, b, c, d satisfying
ab = qba; ac = qca; bc = cb; bd = qdb; cd = qdc; ; a d - da = (q - q-1)bc
(3.12)
where q e C*. Then (cf. (1.2)) A(a)=a|174
A(b)=aQb+b|174
A(c)=c|174 with e(a) - e(d) = e ( 1 ) -
A(d)=c|174
(3.13)
1 and e ( b ) = e ( c ) - 0. S e t t i n g T = ( a c
db / E Mq one has
\
/
A(T) = T @ T (where 9,,~ matrix multiplication and @ ~ Mq | Mq) with e(T) = I (cf. [917] for more on this). This leads to RqT1T2 = T2TIRq for T1 = T | I and T2 = I | T while q
Rq=
0 0 0
0 1 q_q-1 0
0
0
0 0 1 0 0 q
(3.14)
Then for R = Rq one has R12R13R23 = R23R13R12 and Detq(T) = a d - qbc E center(Mq). It follows that ( M q / D e t q T = 1) is a Hopf algebra with
S(T)=
(
d -qc
-q-lb ) a ; T . S(T) = S ( T ) . T -
I
(3.15)
We enhance this with some formulas involving RqT1 *h T2 = T2 *h TIRq and A(tij +h tke) = A(tij) *h A(tkt) where A(tij) = tik | tkj. Thus a,ha=a2;
b,hb=b2;
(2)1/2 ~
b *h a =
b*hc=C*hb-C,hd:
(
C,hC--C2; d , h d = d 2 ;
ab;
( l+q_2 2 )1j2ac;
2bc . b *hd = qWq-1,
2 )1/2 cd;
1+q-2
a *h c :
d ,h
a,hb--
( l+q_ 2 2)1/2 bd; ( 2 ) 1 / 2 cd; c= ~
and finally d *h a = ad - [(q - q-1)/(q + q-1)bc.
( 2 ) 1+q-2
1/2
ab;
c, a=( )lJ2ac d*hb=
(2)1J2 ~ bd;
a *hd - ad + q-q-1 q+q- 1 bc
(3.16)
C H A P T E R 3. C L A S S I C A L A N D Q U A N T U M I N T E G R A B I L I T Y
128 3.3.1
Quantum
matrix
algebras
We continue to follow [919] (omitting most proofs). Thus to begin let A - C < t;ij > be the bialgebra of noncommutative polynomials in n 2 variables tij with R the n 2 • n 2 m a t r i x which is the nondegenerate solution of QYBE, i.e. R12R13R23 = R23R13R12 (with det(R) =/= 0). Set T = ((tij)) E Mn(A) with T1 = T | 1 etc. belonging to Mn2(A ). Let IR be the two sided ideal in A generated by the matrix elements of the RT1T2 - T2T1R (i.e. by elements Rik,pstp~ts,m- Rps,~mtkstip for 1 < i, k,~, m < n). Then AR -- A / I R is called the algebra of functions on the q u a n t u m matrix algebra of rank n associated with R. Thus in AR the generators tij satisfy the relations RT1T2 = T2TIR. Note for R - I the tij c o m m u t e so AI = C[tij]. One shows t h a t AR has a bialgebra structure inherited from A, i.e. there is a coproduct A(T) - T | T and a counit ~(T) - I (cf. E x a m p l e 5.1). Now in the commutative case there is an action of the matrix algebra Mn on the vector space C given by (g,x) ~ gx where (gx)~ = g~kxk. This induces a C-algebra h o m o m o r p h i s m 5" c [ x l , . . . , x ~ ] ~ c[tq] | C[x~,...,Xn] where c[t~3] ~ Pdy(M~) and C [ x l , . . . ,Xn] ~ Poly(Cn). This map 5 is called a coaction and is defined on the generators by the formula 5(xi) = tik | xk while satisfying (13C) (id | 5) o 5 = (A | id) o ~. Based on this for a given bialgebra A, a C-algebra V is called an A-comodule if there is a coaction m a p 5 9 V ~ A | V satisfying ( 1 3 C ) . For example C < X l , ' " , X n > is a comodule for the bialgebra C < tij > with the same coaction as in the commutative case, namely ~(x) = r ~ x.
Let now P be the p e r m u t a t i o n n 2 • n 2 matrix in C | a n d / ~ - P R . For any polynomial f E C[t] let I/,R be the two sided ideal in C < X l , . . . , Xn > spanned by the components of f ( R ) x | x (i.e. by the elements f(R)ij,k~XkX~). The algebra C~, R - C < X l , ' " , X n > / I I , R is called the algebra of functions on the q u a n t u m n-dimensional vector space associated with R and f. This means simply that the generators xi in Cf, R satisfy f ( R ) x | x - O. C~, R is also called a q u a n t u m vector space and the map 5 9 C~, R ~ A | C~, R defined via 5(x) = T ~ x provides an AR comodule structure. Consider now some examples beginning with the An-1 or {] = sl(n) situation. corresponding R m a t r i x Rq has the form
The
n
Rq = q E eii| eii+ E eli | ejj+ 1 ir
(3.17)
+(q _ q - l ) E eij | eji; q C C / { 0 } i>j where eij is the matrix with 1 in the i, j position and 0 elsewhere. One notes t h a t Rq - PRq satisfies the Hecke condition ( 1 3 D ) / ~ 2 = (q_q-1)/~q + I. Writing now Aq = ARq one shows t h a t the element detqT = y~ (-q)e(S)tls~ . . "tns, (3.1s) sESn (g(S) is the length of the element s E Sn or the minimal number of permutations in s) belongs to the center of Aq and is called the q u a n t u m determinant. It is group like in satisfying (13E) A(detqT) = detqT| Finally the quotient algebra of Aq by the relation detqT 1 is called the algebra of functions on the q u a n t u m group S L ( n ) and denoted by SLq(n) (also called q u a n t u m group). It is a Hopf algebra with coproduct A(T) = T ~ T and counit e(T) = I. The antipode S arised from the q u a n t u m cofactor matrix (cf. (3.15)) and has
3.3.
QUANTIZATION AND QUANTUM GROUPS
129
the property S2(T) = D T D -1 where D = diag(1, q2, ... ,q2(n-1)) 6 Mn(C). To introduce the quantum vector space associated with Rq for Example 5.6 we note first that/~q = PRq satisfies the quadratic equation (13F) (/~2 - q I ) ( R q + q - l I ) = 0 so that in the definition of C}, R one can consider only polynomials f ( t ) of first order. Of particular interest are linear functions f s ( t ) = t - q and fn(t) = t + q-1. In the first case C~, R is a quantum analogue of the symmetric algebra and writing down the defining relations Rqx | x = qx | x one arrives at the definition of C~. This is C [ x l , . . . ,Xn] with relations xixj = qxjxi (q-polynomials) and is called the algebra of functions on the quantum vector space (or simply the quantum vector space). In the case n = 2 there is only one relation xy = qyx and one has the formula
(x +
; (N)
= Z 0
q
q
= (qN--1)...(qN-k+l--1) ( @ - 1 ) . . . ( q - 1)
(3.19)
For the case f ( t ) = fA(t) = t + q-1 the algebra C}, R can be considered as a quantum analogue of the exterior algebra. Thus via the defining relations Rqx | x = - q - i x | x one has (for q2 r --1) the algebra (ACn)q with generators xi satisfying x i2 = 0 and XiXj : - q - 1xjxi; this is called the q exterior algebra of the quantum vector space Cq. E X A M P L E 3.7. For the case q E R the algebra SLq(n) with the 9 anti-involution T* = U S ( T ) T U -1 for U - diag(el,. . ",s and ei2 = 1 is called the quantum group S U q ( e l , . . . , e n ) (superscript T means transpose). For U = I one gets the compact form of the quantum group SLq(n) defined via T* = S(T) T. As an example consider n = 2 so
b,) (d qc)
T * = ( a'c, d*
=
-q-lb
a
,320,
Thus d = a* and c = - q - l b * with
T=
(a b) -q-lb*
a*
; detqT = aa* + bb* = I
(3.21)
Consequently one has (13G) ab = qba, bb* = b*b;ab* = qb*a, and aa* = a*a + (q-2 _ 1)bb* (using the determinant result one can replace the last relation by a*a + q-2bb* = 1). This describes then the quantum group SUq(2). Consider now the limit q = exp(h) with h ~ 0 (for the S L ( n ) situation). One has Rq = I - (r - P ) h + O(h 2) where - r is the canonical r matrix corresponding to the Cartan decomposition of sl(n). Assuming that Aq turns into the commutative bialgebra Poly(M~) (i.e. f E Aq --~ fd E A) and defining the P structure on A by r = C d ' ~ d + ( h / 2 ) { r ~;d}+ O(h 2) one obtains from T1T2 = RqlT2T1Rq the result {T e T} = [r,T | T]. This means that the Hopf algebra SLq(n) can be considered as a quantization of the PL group S L ( n ) . Note however that the explicit form of *h is missing in the algebraic approach. Many other examples are discussed in [919] (cf. also [508]).
3.3.2
Quantized enveloping algebras
Let G be a Lie group with Lie algebra ~ and universal enveloping algebra U(~). We have noted already that U(~) _~ C e ~ ( G ) . We have defined for simple Lie groups of classical type the quantization of C ~ (G) to obtain the Hopf algebra Gq. Hence it seems natural to define the quantum analogue of U(~) as a suitable subalgebra in Gq = Hom(Gq, C). Consider
C H A P T E R 3. C L A S S I C A L A N D Q U A N T U M I N T E G R A B I L I T Y
130
first AR defined in Section 5.3.1 via the Q Y B E with defining relations RTIT2 = T2TIR. T h e n (see the beginning of Section 5.2) A~ = H o m ( A R , C) has a bialgebra structure with multiplication (61.62, a) - (61.62)(a) = (61 | 62)(A(a)) and coproduct A*(6)(a | b) = 6(ab) (also 1" - e and the counit is given by the evaluation map). One takes now UR to be the subalgebra of A~ generated by the elements 6~ e A~; (i,j = 1 . . . , N) where ( 1 3 H ) L+(T1 . . . T k ) = R ~ . . . R [ . Here L + = ((6~)) ( N • Ti = I | 1 7 4 1 7 4 1 7 4 E MNk(AR); the matrices R~ E MNk+I(C ) act nontrivially on factors numbered 0 and i and coincide there with R + where R + = P R P and R - = R -1 UR is called the algebra of regular functionals on AR. One shows that this definition is consistent with the defining relations in AR and it follows that (13I) L + ( R T 1 T 2 - T2T1R) = 0. Thus in fact in order t h a t the definition ( 1 3 H ) make sense it is necessary and sufficient t h a t R satisfy QYBE. The algebra UR has a bialgebra structure with coproduct A(L i ) = L + ~ L + and counit e(L +) = I. Further UR acts naturally on AR and with any 6 E UR one can associate operators De" AR ~ AR and D~ " d R ---* AR defined via (13J) D e ( a ) - (id | t~)(A(a)) and D~(a) - (6 | I ) ( A ( a ) ) . These play the role of q u a n t u m left and right invariant operators in the sense t h a t (id | De) o A = A o De and (D~e | id) o id) o A - A o D~e. Thus UR can be considered as a (coarse) q u a n t u m universal enveloping algebra associated with R. In the meaningful examples the doubling of generators for UR is explicable and UR is a genuine q u a n t u m universal enveloping algebra. EXAMPLE there results
3.8. For G = SL(2) one has R = Rq given by (3.14) with q = exp(h) and
0
e_hH/2
; L- =
_(q _ q - 1 ) X +
chH/2
(3.22)
X + and H are generators of the Lie algebra and it follows from the constructions here that ( 1 3 K ) [ H , X +] = • + with [ X + , X -] = ( e x p ( h H ) - e x p ( - h H ) ) / ( e x p ( h ) - e x p ( - h ) ) = S i n h ( h H ) / S i n h ( h ) . Further A ( H ) = H | 1 + 1 | H; A ( X + = X + | e -hH/2 -p e hH/2 | X+;
S(H) =-H;
S ( X +) = - q T 1 Z • 1 7 7
(3.23)
hH/2
Therefore UR is a quantization of U(s/(2)). This construction was generalized by Drinfeld and Jimbo to arbitrary simple Lie algebras (cf. [919] to which we also refer for further material).
3.4
ALGEBRAIC
BETHE ANSATZ
Notations such as ( 8 A ) are used. Following [304] we consider the
XXX1/2
model
defined
in a Hilbert space 7-/N = 1-IN | where each space hn is two dimensional hn ~ C 2 and the spin is s = 1/2. The spin variables Sna act on hn as Pauli matrices a a divided by 2 where
(01) 1
0
with a 2 -
(0 /
(10)
and 0 -3 --" for the s = 1/2 representation i 0 0 -1 (see below and cf. also Section 5.3.3). There are several important observables such as total spin and the total Hamiltonian where S a = ~ n Sna and H = ~a,n[SnaS~+l - ( 1 / 4 ) ] with [H, S a] - 0 (the latter equation reflecting the s/(2) s y m m e t r y of the model) while S~+ g = ~ SnO~ ~ . Here a = 1, 2, 3 and [S Cm, Sn~] = iIheo~Z'~S~nhmn with S + _ _ S 1 i iS 2 (~o~,~ is completely
3.4. A L G E B R A I C B E T H E A N S A T Z
131
antisymmetric with e123 = 1). Here the Sna define a Lie algebra sl(2) with finite dimensional representation realized in C 2s+1 where s = 0, 1/2, 1 , . . . and Sn~ = (h/2)r s for s = 1/2. This is the X X X model and a more general Hamiltonian of the form (SA) H = E s , n JsS~S~+I with parameters j s corresponds to the X Y Z model. The first problem is to investigate the spectrum of H and one is eventually interested in the situation N ~ o~ where analytic methods are needed. One uses a Lax operator and an auxiliary space V ~ C 2 where
L n s = )~In | Is + i ~-~ S~ | ' s
" s "" (
A + iSsn iS~ iS + A-iS~
)
(4.1)
where In and S~ act in hn while Is and a s are unit and Pauli matrices in V. The 2 • 2 matrix form acts in V with entries being operators in the hn. Another form uses the permutation operator in C 2 • C 2, namely, P = (1/2)(I | I + E a s | a s ) satisfying P(a | b) = b | a to write (8C) Ln,s = [)~- (i/2)]In,s + iPn,s (note hn ~ V ~ C2). Consider now two examples of Lax operators Ln,sl (A) and Ln,s~()~) with the same quantum spaces hn and auxiliary spaces Vi. The product make sense in a triple tensor product hn | V1 | V2 and there is an operator Rsl,s2 ()~- #) in V1 | V2 such that R S l , S 2 ( )~ -- # ) L n , s l
()OLn,s2(tt) = Ln,s2(p)Ln,sl ()ORs~,s2()~ - #)
(4.2)
where (8D) Rsl,S2(A) = AIsl,S2 + iPs~,s2. To check (4.2) one can use the form ( 8 D ) and the commutation relation for permutations Pn,siPn,s2 = Ps~,s2Pn,sl = Pn,s2Ps2,Sl with Psl,S~ = Ps2,sl. One calls (4.2) the fundamental commutation relation (FCR). The Lax operator Ln,s()O has a natural geometric interpretation as a connection along the spin chain, defining the transport between sites n and n + 1 via the Lax equation (BE) r = LnCn where Ca = ( r Cn2)T with entries in 7-/. The ordered product over all sites between n2 and nl is (8F) Tnn~,s and the defines the transport from nl to n2 + 1; the full product Tn,s = Ly, s " " Ll,s()0 is a monodromy around a circle and this operator is given as a 2 x 2 matrix in V as
TN's=
( AN(A) CN(A)
BN(A) ) DN(,,k)
(4.3)
with entries being operators in the full quantum space 7-( (example below). As in the classical cituation the map from local dynamical variables Sna to monodromy TN, s(A) is a tool for solving the dynamical problem. One will see that TN, s acts as a generating object for the main observables such as spin and Hamiltonians as well as for the spectrum rising operators. For this one shows first the easily proved relation
RSl,S2 (,~- #)Tsl (/~)Ts2 (#) : G2 (#)Tsl (/\)RSl,S2 ( )~ - #)
(4.4)
The monodromy TN, s is a polynomial in A of the form TN, s()~ ) ~- A N -~- i)~ N - 1 E ( S s
s ~ 0 "s) -~" " "
(4.5)
so that the total spin S a appears via the coefficient of the next to highest degree. For the Hamiltonian note first that the (FCR) (4.4) shows that the family of operators (8H) F(A) = Tr T(A) -- A(A) + D(A) is commuting (i.e. [F(A), F(#)] = 0) and its expansion has the form (8I) F(A) = 2)~N + E N-2 Qk)~k with N - 1 commuting operators Qk. One can show that in fact H belongs to this family. To see this note that A = i/2 is special since
CHAPTER
132
3.
CLASSICAL AND QUANTUM
INTEGRABILITY
(8J) L n , a ( i / 2 ) = iPn,a and for any ~ one has (d/d)~)Ln,a(A) = In,a. Hence one can control the expansion of F(A) near ~ = i/2. Thus TN, a ( i / 2 ) = iN P N , a P N - I , a "'" Pl,a which is easily transformed into P 1 , 2 P 2 , 3 " ' " P N - 1 , N P N , a 9 Now the trace over the auxiliary space is T r a P N , a = IN so (SK) U - i - N T r a T N ( i / 2 ) = P1,2P2,3"'" PN-1,N is a shift o p e r a t o r in T/ (since Pnl,n~Xn~Pnl,n~ = X n l ) ; here the X n or Xna are dynamical variables with X n ~ I | I | . . . X . . . | I with X in the n - th position. Hence (8L) X n U - U X n - 1 and U is u n i t a r y via U*U - UU* - I since P* - P and p2 _ I. Thus U - 1 X n U - X n - 1 and we can introduce m o m e n t u m P as a shift along one site with (SM) e x p ( i P ) - U. T h e n e x p a n d F(A) in the vicinity of )~ = i / 2 to get
•
]
Ta(A) ~=i/2
=iN-I~PN,"'"P~""'n, n
~
'
(4.6)
where/Sn,a means t h a t t e r m is absent. Now we transform this after taking the trace over V into
d
~=i/2
~-- i N - 1 E
pI'2
" " Pn-I'n+I
"
(4.7)
"
and one can cancel most of the p e r m u t a t i o n s here, multiplying by u - l ; as a result one gets
d
d~ F~ (:~)F~ (:~)- ~ ~=i/2
_
d log Fa()~)]
d)~
~=i/2
_ 1_
i En Pn,n+l
(4.8)
Using ( 8 B ) we can rewrite the expression for H at the beginning of this section as
1~ H = -~
id P n , n + l - - ~N ==~H = -~-~log F()~) I
N
(4.9)
~--i/2
n
T h u s H belongs to the family of N - 1 commuting operators generated by the trace of the m o n o d r o m y F()~). One component of spin, say S 3, completes this family to N c o m m u t i n g o p e r a t o r s and establishes the integrability of the classical counterpart of our model, which can be considered as a system with N degrees of freedom. T h e r e is a procedure to diagonalize the whole family global F C R (4.4) and some simple properties of local Lax alizes the simplest q u a n t u m mechanical t r e a t m e n t of the n = ~b*r based on c o m m u t a t i o n relations [~b,r = I and Ow = 0. One writes the relevant set of F C R as
of operators F()~) based on the operators. In a sense this generharmonic oscillator Hamiltonian existence of a state w such t h a t
[B()~), B(p)] = 0;
(4.10)
A()~)B(#) = f ( , ~ - # ) B ( # ) A ( A ) + g ( , ~ - #)B(A)A(#); D()~)B(#) = h()~ - # ) B ( p ) A ( ) ~ ) + k()~ - # ) B ( ) ~ ) D ( # ) where ( 8 N ) f(A) = [ ( A - i ) / ) q and g(~) = read these relations from the one line F C R in V | V. Take 4 x 4 matrices in a n a t u r a l and e4 = e_ | e_ in C 2 x C 2 where e+ =
p=
1 0 0 0
o 0 1 0
o, 1 0 0
o 0 0 1
i / ~ with h(A) = [(A + i)/A] and k(A) = -i/)~. To (4.4) one uses the explicit representation of F C R basis el = e+ | e+, e2 = e+ | e_, e3 = e_ | e+, (1 0) T and e_ = (0 1) T so t h a t
; R(~)=
a(~) 0 0 0
o o o b(A) c(A) 0 c(~) b(~) 0 0 0 a(A)
(4.11)
3.4. ALGEBRAIC BETHE ANSATZ
/
133
where a = A + i, b = A, and c = i. Further
T ~ (~) =
A(A) 0 C(A) 0
0 A(A) 0 C(A)
B(A) 0 D(A) 0
0 / B(A) 0 D(A)
; Ta~ =
I A(#) B(#) 0 0 I C(#) D(p) 0 0 0 0 A(#) B(#) 0 0 C(#) D(p)
(4 12)
"
Thus one has
A(A)A(#) A(A)B(#) B(A)A(#) B(A)B(#) A(A)C(#) A(A)D(#) B(A)C(#) B(A)D(#) C(A)A(#) C(A)B(#) D(A)A(#) D(A)B(#) C(A)C(p) C(A)D(#)D(A)C(p) D(A)D(p)
Tal(A)Ta~(#)=
(4.13)
Now to get (4.10) one uses the (1, 3) relation in FCR (4.4), namely (8D) a(A-p)B(A)A(p) c(A- #)B(#)A(A) + b(A - #)A(#)B(A) and interchanging A ,-, # one obtains
A(A)B(#) = a(#A) ~c(#B ( A A) )A(#) b(# - A)B(#)A(A)b(~~)
-
(4.14)
Other relations are obtained in a similar manner. Now exchange relations such as (4.10) are a kind of substitute for the relations (SP) r (n + 1)r and r = ( n - 1)r for an harmonic oscillator with Hamiltonian n = ~p*~b. The analogue of a highest weight vector w with Cw = 0 will now be played by a reference state f~ such that (SQ) C(A)f~ -- 0. To find this state one observes that in each hn there is a vector Wn such that the Lax operator Ln,o~(A) becomes triangular in the auxiliary space when applied to Wn, i.e.
Lna()~)wn = ( A + 0(i/2) A - *(i/2) ) ' and this vector is given by
Wn =
e+. Now for f~ = Kin |
T(A)~= (aN(A) 0
Wn
(4.15)
one gets
* ) 5N(A ) Ft
(4.16)
where a(A) = A + (i/2) and 5(A) = A - (i/2). In other words one has C(A)Ft = 0, A(A)~ = aN(A)f~ and D(A)f~ = 5g(A)f~. Thus ft is an eigenstate of A(A) and D(A) simultaneously and hence of F = A + D. Other eigenvectors can be sought in the form (8R) ~({A}) = B(A1) ... B(Ak)Ft (k is the number of eigenvalues). The condition that q~ be an eigenvector of F leads to a set of algebraic relations on the parameters Ai. To derive these equations one uses the exchange relations (4.10) to get k"
A(A)B(A1) ... B(Ak)f~ = rI f(A _ )~m)aN(A)B(A1) .. . B(Ak)f~+ 1 k
+ ~ Mk(A, {A})B(A1)-../~(Am)""" B(Ak)B(A)f~ 1
(4.17)
The first term in the right hand side has a nice form and is obtained using only the first term in the right side of (4.10B). All other terms are combinations of 2 k - 1 terms and the coefficients Mk can be quite involved. However M1 is simple enough and one obtains
134
C H A P T E R 3. C L A S S I C A L A N D Q U A N T U M I N T E G R A B I L I T Y
(8S) MI(A, {A}) = g ( A - A1) YI1k f(A1 - Am)CtN(A1). One notes also that due to the commutativity of B(A) all other Mj(A, {A}) are obtained from M1 by a simple substitution A1 ~ Aj so
k
Mj(A, {A}) = g(A - Aj) r I f(Aj - Am)a N(Aj) 1
(4.18)
Analogously for D(A) one obtains k D(A)B(A1)""" B ( A k ) a = H h(A - Am)~N(A)B(A1) . .. B ( A k ) a + 1 k at- E Nk(A, { A } ) B ( A 1 ) ' ' ' / ) ( A m ) ' " B ( A k ) B ( A ) ~ 1
(4.19)
where Nj = k ( A - Aj)Ilk h(Aj - Am)SN(Aj). Next one notes that g ( A - Ao) - - k ( A Aj) and this allows one to cancel unwanted terms in (4.17) and (4.19) for the application of A + D to q~. There results (8T) [A(A)+ D(A)]~({A}) = A(A, {A})q~({A}) where A(A, {A}) = ~N(A) 1~1k f ( A - A j ) + s N ( A ) 1-Iklh ( A - A j ) . If the set of {A} satisfy the equations (8U) II k f(Aj - Am)oLN(Aj) = li k h(Aj - Am)SN(Aj) for j = 1,... ,k then using (8N) and the formulas for c~ and 5 after (4.16) one can rewrite (8U) in the form
(AJ+(i/2)) Aj - ( i / 2 )
N k - I I Aj - Am + i mr Aj - Am - i
(4 20)
and this is a main result (Bethe Ansatz equation = (BAE)) which can be used to investigate the N + cc limit (cf. also Section 3.9). Equation (4.20) shows that the superficial poles in the eigenvalue A(A, {A}) actually cancel so that A is a polynomial in A of order N as it should be. We will see that solutions with nonequal Aj are enough to give all of the spectrum. The algebraic derivation here is completely different from the original approach of Bethe but in any case the approach here is called the algebraic Bethe Ansatz (ABA) and R)({A}) is called the Bethe vector. To find explicit values for the eigenvalues of spin, etc. consider taking the limit p --+ oc in (4.4); using (4.5) one gets ITch(A), 2ac~ + S c~] = 0
(4.21)
which expresses the s/(2) invariance of the monodromy in the combined space 7-I | V. From this one has in particular [S 3, B] = - B and [S +, B] = A - D and for the reference state ft one has S+ft = 0 with S3f~ = (N/2)ft showing that it is the highest weight vector for spin S c~. Further S3O({A}) = [(N/2) - k]O({A}) and one can show that S+~({A}) = 0 since S+(I)({A}) = ~ B(A1)... B(Aj_I[A(Aj) - D(;~y)IB(;~y+I) . .. B(Ak)a J
= ~ Ok({A})B(A1)... [3(Am)'" B(Ak)ft
(4.22)
?'n
and one can show that all Ok = 0 when BAE holds. Thus O({A}) are all highest weight vectors and this means that k cannot be too large because the S 3 eigenvalue of the highest weight vector is nonnegative; more exactly one must have k ___ (N/2). Note when N is even the spin for all states is an integer while for N odd the spins are half integer. Now consider
3.4. ALGEBRAIC BETHE ANSATZ
135
the shift operator. For A = i/2 the second term (and many of its A derivatives) in A vanish and this eigenvalue becomes multiplicative. In particular
U(I)({A})= igF(i/2)O({A})= ~ Aj + (i/2) 9 ~ j -(i/e) r
(4.23)
Taking the logarithm one sees that the eigenvalues of the momentum P are additive and (SV) P~({A}) = ~-]p(Aj)(I)({A}) where p(A) = (1/i)log{[A + (i/2)]/[A- (i/2)]}. The additivity property holds also for the energy and differentiating log(A) in A and putting A = i/2 one gets (SW) Hr = E j E(Aj)r where e(A) = -(1/2){1/[A 2 + (1/4)]}. One can use here a quasiparticle interpretation for the spectrum of observables on Bethe vectors. Each quasiparticle is created by the operator B(A); it diminishes the S 3 eigenvalue by 1 and has momentum p(A) with energy E(A) = (1/2)(d/dA)p(A). In this interpretation A is called a rapidity of a quasiparticle. Note e(p) = Cos(p) - 1. Also note that the reference state cannot be taken as a ground state (state of lowest energy) since all eigenvalues of H are negative. Consider now the thermodynamic limit N ~ c~ (ferromagnetic situation). From (4.20) one sees that for real Aj both sides in BAE are functions with values on the circle and the left hand side is wildly oscillating when N is large. Taking logarithms one gets (SX) Np(Ay) = 2rQj + Ekl r - Am) where the integers Qj, 0 _< Qj <_ N - 1, define the branch of the logarithm and r is a fixed branch of log[(A + i)/(A - i)]. For large N and Q and fixed k the second term in (8X) on the right is negligible and one obtains the usual quasicontinuous expression pj = 27rQj/N. In the ferromagnetic case when the Hamiltonian is - H the energy of this particle is e(p) = 1 - Cos(p). The correction term in (8X) expresses the scattering of particles and a comparison with the usual quantum mechanical treatment of a particle in a box shows that r - Am) plays the role of a phase shift of particles; thus the function (BY) S(A-#) = [(A-p+i)/(A-#-i)] is a corresponding S matrix element. Another analogy is with the classical inverse scattering method where Z(A) = B(A)A-I(A) is more important that B(A). The factor S enters the exchange algebra Z(A)Z(#) = Z(p)Z(A)S(A- #) valid in the limit N ~ cc since the second term in the right side of (4.10B) effectively vanishes; Z(A) can be interpreted as a creation operator of a normalized particle state. The BAE allows also for complex solutions (corresponding to bound states). The first nontrivial case is k = 2 and from the BAE
(Al-~-(i/2)) N A1 -
we see that
(i/2)
A1-A 2 +i -- A1 - A 2 - i ;
(A2+ A2-
(i/2)) N (i/2)
A2-AI+i =
A2 -
A1 -
i
(4.24)
( Al + (i/2) ) N ( A2 + (i/2) ) g A1 - ( i / 2 ) .
A2 - ( i / 2 )
= 1
(4.25)
so that p(A1)+ p(A2) is real. Further for ~A1 ~: 0 the left side in (4.24) grows (or decreases) exponentially when N ~ c~ and to compensate it in the right side we must have ~(A1-A2) i (or - i ) . Since A1 and A2 can be interchanged we can say that for N --, ~ they acquire the form A1 = AU2+(i/2 ) and A2 = A~/2-(i/2) where A1/2 is real. Further calculations are given for momentum and energy in [304]. For k > 2 the complex solutions can be described in an analogous manner. The roots are combined in complexes of type M where M = 0, 1//2, 1,-.. defining the partition k = ~-]~MUM(2M + 1) with 1/M the number of complexes of type M. The set of integers {UM} defines a configuration of Bethe roots with each complex containing roots of the form AM,m : AM + im ( - M <_m < M). Here AM is real and m is integer or
136
C H A P T E R 3. CLASSICAL A N D Q U A N T U M I N T E G R A B I L I T Y
half integer together with M. We omit further details for the moment (cf. [304] and Section 3.9).
3.5
SEPARATION
OF V A R I A B L E S
Notations such as (6A), ( T n ) , and (9A) are used. In [877] it is argued that separation of variables (SOV) understood generally enough could be the most universal tool to solve integrable models of classical and quantum mechanics. It is shown that the standard construction of the action-angle variables from the poles of the BA function can be interpreted as a variant of SOV and moreover for many particular models it has a direct quantum counterpart. The list of models discussed includes X X X and X Y Z magnets, Gaudin models, the nonlinear Schrhdinger equation, etc. Briefly SOV can be characterized as a reduction of a multidimensional problem to a set of one-dimensional ones. For the classical integrable systems subject to the inverse scattering method (ISM) the standard construction of the action-angle variables using the poles of the BA function is in fact equivalent to a SOV. Further in many cases it is possible to find the precise quantum analogue of this construction. Start with the classical case and consider a Hamiltonian system with a finite number D of degrees of freedom and integrable in the Liouville sense. Thus one is give a 2D dimensional symplectic manifold (phase space) and D independent Hamiltonians Hj commuting with respect to the Poisson bracket, i.e. (6A) {Hi, Hi} = 0 for j , k = 1 , - - - , D . A system of canonical variables (iff,~) with {Xj, Xk} -- {Pi,Pj} -- 0 and {pj,xk} -- ~jk will be called separated if there exist D relations of the form (6C) ~ j ( x j , p j , H 1 , . . . , HD) -- 0 for j -- 1 , . . . , D . Note that fixing the values Hj = hj one obtains from (6C) an explicit factorization of the Liouville tori into the one dimensional ovals given by the equations (6D) a2j(xj,pj, h l , . . . , h D ) = 0. The action function S(h,~) is defined as the generating function S of the canonical transformation from (g, ~) to the action-angle variables (f, r i.e. -.
-~
-.
h = h(I); pj =
SS(h(I), :~) OS Oxj ; Cj = OIj
(5.1)
such that the HJ equation
H i ( OS ( f~(:~ ~) ' :~) , :~ = h j ( I-')
(5.2)
holds for each Hj considered as a function of (iff,~). The relations (6D) allow one immediately to split the complete solution to (5.2) into the sum of terms (6E) S(h, x l , . . . ,XD) -SI(h, x l ) + " .+SD(h, xD) satisfying each the ODE's (6F) Cj(xj, (OSj/Oxj), h i , " ' , hD) -- O. This justifies apparently the term separation of variables. Now for the quantum case the condition (6A) is replaced by ( 6 G ) [ H i , Ilk] = 0 and (6C) retains the same form with the only difference that we have to fix some operator ordering of the noncommutative operators xj, pj, Ha. One assumes in [877] that the operators are ordered by the rule that x preceeds p and p preceeds H. It is convenient to work in the x representation and to pick the quantum states as functions ~(~) from some Hilbert space ~-/. The canonical operators xj and pj ,,~ -ih(O/Oxj) satisfy then (6I) [xj, xk] - 0 - ~j, Pk] with ~gj, Xk] = --ihhjk. Let ~ be a common eigenfunction (6J) H j r - hjr then ~ satisfies (6K) r h)r : 0 and this suggests the factorization (6L) r - li D ~j(xj) where (6M) C j ( x j , - i h ( O / O x j ) , f~)r = 0. Semiclassically r - exp[(i/5)S(:~)] with Cj(xj) = exp[(i/l~)Sj(xj)] and this provides the correspondence between (6E) and (6L).
3.5.
S E P A R A T I O N OF V A R I A B L E S
137
Now for the magic recipe for SOV one goes to the classical case and recalls that commutative Hamiltonians can be obtained as the spectral invariants of a matrix Lax operator L(u); this means that the Hj can be expressed in terms of the coefficients tn(u) of the characteristic polynomial N W ( z , u) = det(z - L(u)) = ~ ( - - 1 ) n t n ( u ) z N - n ; o
tn(U) = T r A n L(u)
(5.3)
where to(u) = 1 and t y ( u ) = d e t L ( u ) . The characteristic equation (6N) W ( z , u ) = 0 defines the eigenvalue z(u) as a function on the corresponding N-sheeted Riemann surface and the BA function ~(u) is defined via ( 6 0 ) L ( u ) ~ ( u ) = z ( u ) ~ ( u ) . Since an eigenvector is defined up to a scalar factor, to exclude ambiguity in the definition of ~ one must fix a normalization via (6P) ~ N an(U)~n(U) ---- 1 where an may depend on the dynamical variables (the ~tn are presumably the components of gt). Then Ft(u) becomes a meromorphic function on the Riemann surface (6N). The poles xj of the BA function play an important role now (cf. [268, 604] and in particular the time evolution of the xj for any Hamiltonian flow based on an Hn can be expressed explicitly in terms of the Riemann theta functions corresponding to (6N). Further for many models the xj Poisson commute and together with the eigenvalues zj - z ( x j ) of L ( x j ) , or some functions pj of the zj, provide a set of separated canonical variables for the Hn. One of the reasons why the poles of ~t(u) should provide a SOV is easy to understand; since zj = z ( x j ) is an eigenvalue of L ( x j ) the pair (zj, xj) should lie on the spectral curve (6N), i.e. (6Q) W ( z j , x j ) = 0. Further if zj is a function of pj the equation (6Q) fits exactly the form (6C) since the coefficients tn(xj) contain nothing except xj and Ilk. However the relations (6C) alone are not enough to produce SOV. It is necessary that in addition the number of poles xj be exactly the number of degrees of freedom D and the variables ( p ( z j ) , x j ) be canonical as in (6B). These properties are by no means obvious and the lattter can be difficult to verify. To perform the calculation it is necessary to transform the above definitions of (z j,, xj) into a more convenient form. Thus let ~J = Resu=x~Ft(u). From ( 6 0 ) and ( 6 P ) there follows N L ( x j ) ~ j = zj~J; ~ an(Xj)gtJn = 0 1
(5.4)
Let (~(u) denote the 1-row matrix ( a l , . . . , aN). The existence of a nonzero solution ~J to (5.4) is equivalent to the condition
Rank
c~(xj) ) = NL ( x j ) -- z j I
(5.5)
1
which in turn can be expressed generically as the vanishing of any of two minors of order N, e.g. al a2 a3 999 aN L21 L22 - z L23 "'" L2N 9
.
=0;
.
LN1
LN2
LN3
"'"
LNN-- Z
LN1 al
LN2 a2
LN3 a3
9"" L N N 9"" aN
z
LN1
LN2
LN3
9"" L N N -
z
=0
(5.6)
CHAPTER 3. CLASSICAL AND QUANTUM INTEGRABILITY
138
The pairs (zj,xj) are then obtained as the roots of the system (5.6) (note Lij - Lij(x) here); this allows one to count them and to calculate the Poisson brackets. For the NLS (nonlinear Schrbdinger) model and X Y Z model one needs the formulas for the N = 2 case so these are treated as an example. Thus (5.6) becomes (~I
L21
Ct2 L22 - z
=0;
Otl
Ol2
Lll - z
L12
=o
(5.7)
Elimininating z leads to (6R) B(xj) = 0 where B = ct2L12 - CtlO~2(Lll - L22) - c~22L21 and the eigenvalue zj is then obtained from Zj =
(
Lll
a l L12
-
ct2
)
(
L22
u=xj
Ctl
L21
)
u=xj
(5.8)
Generally there is no guarantee that one obtains from (5.6) the canonical Poisson brackets for some pj(zi) although it is true for a fairly large class of integrable models. The magic recipe then is to take the poles of the properly normalized BA function and the corresponding eigenvalues of the Lax operator to obtain a SOV. The choice of normalization is critical and is unsolved in some cases (cf. [97, 276]). 3.5.1
r-matrix
formalism
There are techniques which simplify calculations allow one to verify (sometimes) SOV for whole families of L operators instead of handling them individually. According to a remarkable theorm of [51] the commutativity of the spectral invariants tn(u), i.e. {trn, tn} = 0, is equivalent to the existence of a matrix rl2(U, v) of order N 2 • N 2 such that the P brackets between the components of L are represented in the commutator form 1
2
1
2
{L (u),L (v)} = [rl2(U,v),L ( u ) ] - [r21(v,u),L (v)] 1
(5.9)
2
where e.g. L - L| L - IQL, and rl2(U, v) = Pr12P where 7)(xQy) = y| for x, y E C N. Generally r12 is a function of dynamical variables and very little is known apart from a few particular examples. The best studied one is that of purely numerical r-matrices satisfying CYBE [ r l 2 ( U l , U2), r l 3 ( U l , U3) -t- ~'23(U2, u3)] -- [ r l 3 ( U l , u3), r32(u3, u2)] = 0 (5.10) which ensures the Jacobi identity for (5.9). Especially important is the case of unitary rmatrices satisfying in addition (6T) r12(Ul, u2) = -r21(u2, ul) and depending on Ul - u2; for such situations (5.9) takes the form 1
2
1
2
{L (u),L (v)} = [ r l 2 ( U - v),L (u)+ L (v)]
(5.11)
and (5.10) becomes (6U) [r12(u), rl3(U + v)] + [rl2(U), r23(v)] + [r13(u + v), r23(v)] = 0. To a unitary numerical r-matrix one can associate not only the P algebra (5.11) whose right side is linear in L but also the algebra 1
2
1
2
{L (u), L (v)} = [ r l 2 ( U - v), L (u) L (v)]
(5.12)
with a quadratic right side. Formally (5.12) can be put into the form (5.9) with the dynamic 2
2
r - m a t r i x ?~12(u, v)= (1/2)[r12(u-v) L (v)+ L (v)?~12(u-v)] and ?~21(u, v) = -(1/2)[r12(u-
3.5. S E P A R A T I O N OF V A R I A B L E S 1
139
1
v) L (u)+ L ( u ) r 1 2 ( u - v)]. However the special structure of ?=12 allows one to consider the formula (5.12) rather as a modification of (5.11). Evidently (5.11) can be obtained from (5.12) if one Substitutes L = l + e L + O ( c 2) with r ~ Er and e --~ 0 (cf. also [881] for another quadratic algebra). There exists a profound algebraic theory of unitary numerical r-matrices (cf. [76]) which allows one to classify them in families labelled by Lie algebras. A particularly important example for any semisimple (ss) Lie algebra ~ is (cf. (2.17)) r12(u) = ~ ~ a Ia | Ia where p is constant and Ia E ~ is an orthonormal basis with respect to the Killing form. Taking various finite dimensional representations of .~ for the Ia leads to the family of r-matrices related to ~. In particular for ~ = gl(N) and the fundamental vector representation one has (6V) f12 = (p/u)7' and it is only for this example that a general SOV construction is obtained. E X A M P L E 3.9. Following [877] we look here at the G L ( N ) model where any choice of normalization as in ( 6 P ) yields SOV and take for simplicity (6Z) ai - 0 (1 < i < N - 1 with O~y --- 1 so ~ g ( u ) = 1. The corresponding separated coordinates xj are defined as the poles of t2(u) and the momenta (6I) are pj = - p - l z j for the linear P brackets (5.11) and pj = - p - l l o g ( z j ) for the quadratic P brackets (5.12). For the linear case see [?] and for the quadratic situation cf. [375, 852]. For the normalization indicated in (6Z) the equations (5.6) can be simplified a bit (cf. [877]) but in fact it is possible to eliminate zj from them to produce one equation (TA) B(xj) - 0 where B is a certain polynomial of degree Y ( N 1)/2 in the components of L(u) The corresponding eigenvalue is then (TB) zj = A(xj) for a certain function A expressed rationally in components of L(u). For example when g = 2, B(u) = L21(u) and A(u) = Lll(U) so for u - xj the matrix L(u) becomes triangular
L(xj)=- ( zjO L12(xJ)
(5.13)
For N = 3 one obtains B ( u ) - - L31(u)
Lll L31
L12 L32
(u) + L32(u)
L21 (u) L31
L22(u) L32
(u)
(5.14)
There are two possible ways to choose A(u), namely
Al(u) =
Lll L12 L31 L32 L32(u)
(~) ; A2(u) = -
L21 L22 L31 L32 L31(u)
(u) (5.15)
These are equivalent modulo B(xj) = 6 r A l ( x j ) = A2(xj) = 0 with (TC) zj - A l ( x j ) A2(xj) (cf. [852] for general formulas - note that due to a different choice of normalization the formulas above differ from those in [375, 852,878, 879]). The advantage of using A and B as above is that the B Poisson commute {B(u), B(v)} - 0 which implies the commutativity of the xj. However the correct P brackets between xj and zj is not enough to establish SOV; one needs the correct number of variables xj and in some degenerate cases this number could be less that D (in which case additional variables can be added). In [877] one remarks that so far (1995) no generalization is known of such results to the Lie algebras other than the An series (problem of normalization).
C H A P T E R 3. CLASSICAL A N D Q U A N T U M I N T E G R A B I L I T Y
140
3.5.2
Quantization
In the quantum case the components Lmn of the N x N matrix L(u) become quantum operators and the P brackets should be replaced by commutation relations satisfying the correspondence principle [, ] ~ - i h { , }. A nice feature of the P algebras (5.11) and (5.12) is that it is well known how to quantize them. For the linear case (5.11) one has 1
2
[L (u), ~ (v)] = [T~12(u- v), L (u)+ L (v)]; T~(u)= - i h r ( u )
(5.16)
while in the quadratic case (5.12) is replaced by 1
2
2
1
R 1 2 ( u - v) n (u) n (v) =L (v) n ( u ) R 1 2 ( u - v)
(5.17)
(Yangian of gl(N)) where R satisfies the QYBE R12(u)/~13(u -3t- v ) R 2 3 ( v )
-- R 2 3 ( v ) R 1 3 ( u
-~- v)R12u
(5.18)
where (6W) R(u) = 1 + ihr(u) + O(h2). In the G L ( N ) case (6V) the quantum T~ and R matrices are (6X) 7~(u)= -~l(P/u) and R(u) = 1 + rI(P/u) where 77 = ibp. The quantum integrals of motion tn(U) for the Yangian are obtained by appropriate deformation of (5.3). namely
tn(u) = Tr L(u) A L(u + ~) A . . . A L(u + (n - 1)r/); [tm(u), tn(u)] = 0
(5.19)
The quantity t y ( u ) (quantum determinant of L(u)) produces central elements of the Yangian via the equation (6Y) [ty(u), L(u)] = 0 (ty(u) is numerical on the irreducible representations). E X A M P L E 3.10. The quantum analogue for Example 3.9 with N - 2 or N - 3 is given as follows (cf. [878, 879] where the normalizations are slightly different). By virtue of (5.17) the operator family B(u) turns out to be commutative, i.e. (TE) [B(u), B(v)] -- 0 which allows one to define the operators Xn as the commuting operator roots of the equation (TA) (cf. [878]). To give a sense to the formula (TB) in the quantum case it is necessary to fix the operator ordering and we assume that x variables in (TB) are ordered to the left, e.g. ( T F ) d ( x j ) = ~ m x ? d m for A ( u ) = ~ m u m A m 9 Then using the commutation relations (5.17) it is possible to verify that
[x~, xj] = [z~, zj] = 0; zjx~ = (x~ + ~ j ~ ) z j
(5.20)
which suggests the realization of zj as shift operators in an appropriate Hilbert space 7-I of functions on the common spectrum of the xj, e.g. (TG) zj~2(2) = ~(xj)~2("" ,xj + ~,...). The choice of ~(x) is dictated by the model (see below). Note that there is a certain liberty in choosing ~ due to the canonical transformation (TH) z~ - zjtl(xj) + t2(xj) - 0 which generalizes the classical characteristic equation (5.3) and fits the form (6H) required for the quantum SOV. Denote by ~-n(u) the eigenvalues of the commuting tn (u) and one obtains for the corresponding separated equation (6M) the finite difference equation ~ ( X j nt- T ] ) ~ ( X j ) ~ 2 j ( X j nL 2?7)
- TI(Xj)r
-
-
~ ( X j ) T I ( X j nt- 7 ] ) ~ j ( X j nt- ?7) -~- T 2 ( X j ) ~ j ( X j )
= A_(xj)r
- ~) + A + ( x j ) r
+ ~) - 0
-- 0 --
(5.21)
where (7I) A+(u) = ~(u) and A_(u) = t 2 ( u - r I ) / ~ ( u - rl). In the GL(3) case the quantum B(u) and A(u) are obtained as deformations of the classical formulas (5.14) and (5.15) in the form B(u)
= L31 ( u -
rI)[L32(U)Lll ( u -
?7) - L31 ( u ) L 1 2 ( u
- ~7)]+
3.5.
SEPARATION
141
OF V A R I A B L E S
+ L 3 2 ( u - rl)[L32(u)L21 (u - rl) - L31 (u)L22(u - r])]
(5.22)
A l (u) = n321(u)[n32(u)n11(u - ~) - L31(u)L12(u - r])];
A2(u) = - L 3 1 ( u ) [ L 3 2 ( u ) L 2 1 ( u -
L 3 1 ( u ) L 2 2 ( u - ~)]
~)-
(5.23)
where zj - A I ( X j ) - A 2 ( x j ) . The rest is similar to the GL(2) case. The quantum characteristic equation is (TJ) z 3 - z 2 t ( x j ) + z j t 2 ( x j ) - t3(xj) - 0 and the separated equation is now a third order finite difference equation (el. [879]). In the case of the 7~ matrix of the form (6X) it has been proved in [324] that for any Lie algebra ~ the quantum commuting operators do exist and are deformations of the spectral invariants of the classical L matrix; the proof does not produce any effective formulas however. Nevertheless the simplest integrals of motion ar easy to construct. Note that t l ( u ) = T r L ( u ) is a trivial central element of the ensuing algebra so it can be safely put to zero and this corresponds to dealing with s l ( N ) instead of g l ( N ) . The first nontrivial invariant is quadratic in L(u) and coincides with the classical expression (7K) t2(u) = ( 1 / 2 ) T r L2(u) with [t2(u),t2(v)] = 0 (for the general simple Lie algebra one should use the corresponding Killing form). This quadratic invariant is enough to serve for the s/(2) case. The definitions for A and B remain the same (cf. (TA) and (TB)) while the commutation relations (5.20) and the quantum characteristic equation (TH) are replaced respectively by (TL) [Xj, Xm] : [Zj,Zm] -- O, [Zj, Xm] -- Zj~jm and z~. - t2(xj) = 0 (~ = 1 for simplicity). Then use the realization (TM) zj = O/Oxj + ~ ( x j ) to obtain the separated spectral problem in the form (cf. [880]) (TN) r + 2~r + (~2 + ~,)r = T2~. The above scheme of quantum SOV is general enough to serve for the variety of quantum integrable models obtained by taking concrete representations of the Yangian (5.17) or the algebra (5.16). The main problem of adjusting the general scheme to a concrete model is to find the spectrum of the commuting operators B ( u ) and to describe the functional space in which the one dimensional spectral problem (5.21) or (TN) can be solved. 3.5.3
XXX spin chain
We look here at the X X X magnetic spin chain from the perspective of Section 5.2. The general finite dimensional irreducible representation of the Yangian Y[s/(2)] is realized in the tensor product of D finite dimensional irreducible representations of the Lie algebra s/(2). Thus iS 3 , Sin] = -}-S~n6mn; [STm, Sn] -- 2S36mn (m, n = 1,..-, D);
1
(sX) + - (sgs
+ s
s:) = e (en + 1);
and one has a factorized form (9B) L x x x ( u )
(1
122 (U -- ~2)/~l (U -- ~1) where
= Ks
c~(~) = ~I + ~&; & =
}
(5.24)
s~
-s~
(5.25)
and K is a constant numerical matrix (T is often used here instead of L). The representation given by (9B) has dimension l~lD(2en + 1) and is parametrized by a 2 x 2 matrix K, the number D of degrees of freedom, spins ~n and shifts 5n. The Casimir operator (i.e. quantum determinant (5.19)) t2(u) takes the value (9C) t2(u) = det K and the resulting quantum integrable model is called the inhomogeneous X X X magnet. The parameters of the representation being in generic position the representation (9B)
142
C H A P T E R 3. CLASSICAL A N D Q U A N T U M I N T E G R A B I L I T Y
of the Yangian turns out to be irreducible and the spectrum of the operators xj defined from (7A) for B(u) given after (TB) turns out to be the finite set (9D) spec[xj] = Aj = {)~f, )~j +~, " ", )~+-rl, A+} where A~= = 6j• For the separated finite difference equations (5.21) to be well defined on the finite set Aj the coefficients A+ must satisfy the boundary conditions (9E) A+(hn • gnrl) = 0. The most convenient choice of A+(u) is
D A+(u) = ~+ H ( u - 6n ~/nrl); ~+~- - det(K)
1
(5.26)
The s/(2) Gaudin model is the degenerate case of X X X obtained in the limit 77 --~ 0. The corresponding L operator is produced by renormalizing the L operator (9B) via Lxxx(u)
I].(U-- ~n)
= 1 + rlLaaud(U ) + O(rl 2)
(5.27)
The L operator
D LGaud = ~ + ~
1
8n U - 5n
(5.28)
satisfies the linear commutation relations with the T~ matrix (6X) for r / - 1. The spectral invariant t2(u) in (TK) produces the commuting Hamiltonians which are quadratic spin operators 1 D Hn D gn(gn + 1)
t2(u)--" -~Tr ](:2 _f_E1 u - ~------~-f- V ( u - ~ni ~;
(5.29)
D TrSmSn Hn -- Tr)~Sn + E 6n - h m
rnr
The normalization (5.26) in (5.21 corresponds to the normalization
1
~
(5.30)
In the limit rl ---* 0 the 2t~j + 1 points of the spectrum Aj of the operator Xj merge into one point xj = 5j of multiplicity 2t~j + 1. The space of functions on Aj is understood, respectively, as the ring of polynomials in xj modulo the ideal (xj - 6j)2~J+1 - 0. The spectrum of the Hamiltonians (5.29) is given then by the values of Hn for which the differential equation (TN) with ~(u) given by (5.30) in each of the points x = 6jl D has a regular solution Cj(x) = 1 + E F ( x - 6j)sr If one realizes the spin operators Sna in (5.24) as differential operators ( 9 F ) S 3 = y n O / O y n - ~n with S + = Yn and S n = ynO 2/Oy n2 - 2~nO/Oyn then the equation B ( x j ) = L21(xj) = 0 defines a purely coordinate change of variables {Yn} ~ {xj}. The separated coordinates xj can be described as generalized ellipsoidal coordinates and in fact all the models allowing SOV in generalized ellipsoidal coordinates can be considered as degenerate cases of the Gaudin model (cf. [622, 517]). The particular simplicity of the Gaudin model makes it attractive and the fact that the eigenvalue problem for the Gaudin Hamiltonians (5.29) coincides with the KZB equations on the critical level can be exploited in various ways (el. [322, 818]). The X X X model, as well as the s/(2) Gaudin model, allows comparison of SOV with the algebraic Bethe Ansatz (cf. Section 3.2 and [356, 877, 878, 880]). The
3.6.
143
HIROTA EQUATIONS
SOV and ABA both lead to the same equation (5.21) ~ ( 7 N ) (with different interpretations). In the SOV method the equation is solved on the finite set A = t2jAj whereas in ABA r is suppposed to be a polynomial whose zeros Vm parametrize the Bethe vector (9G) ~b~ = [IM L21 (Vm)lO >. The power of SOV is revealed most clearly in cases when the representation of the Yangian does not possess the highest weight vector 10 > such that L1210 > = 0 and hence ABA cannot be applied (for more on all this see [877])
3.6
HIROTA EQUATIONS
Notations such as (5A) are used. We have been pursuing a shortcut to general perspective by sketching from various sources in the preceeding sections and displaying some of the natural and important ingredients. We go now in the same spirit to [972] from which we extract as follows. We want to emphasize the fundamental role of Hirota equations in the theory of integrable systems (see e.g. [147, 162, 167, 459, 558, 607, 740, 757, 992, 993, 994, 995] and Section 1.1.5 for details). A particular equation of great interest is the bilinear difference equation with three discrete variables g, m, n o~w(n,g + 1, m ) w ( n , g , m + 1 ) + / 3 T ( n , g , m ) ' r ( n , g + 1, m + 1)+ +q/T(n + 1,g + 1, m ) T ( n -
1 , g , m + 1) = 0
(6.1)
where a + / 3 + "7 = 0 (T(n, g, m ) can be identified with the tau function of a continuous hierarchy- and this will be discussed later). In the present context different continuous limits with different boundary conditions in (6.1) reproduces continuous soliton equations such as KP, Toda, etc. On the other hand the same equation (6.1) (with a particular boundary condition) arises in QIS theory as a fusion relation for the transfer matrix (corresponding to the trace of the quantum monodromy m a t r i x - cf. [551,614]). Thus classical integrability emerges in quantum systems not as a limiting process but rather as an inherent and exact feature. It seems that while solving an integrable quantum system one can in fact deal with classical integrable models in discretized form. In particular the Bethe Ansatz equations and the eigenvalues of the scattering matrix and transfer matrix may be obtained by solving classical Hirota equations. Thus let Ri,A (u) be the R matrix acting in the tensor product of Hilbert spaces V/<9 VA. Then the transfer matrix TA is defined via TA(UIyi)
- - R N , A ( u -- Y N ) " " " R 2 , A ( U
-- y 2 ) R 1 , A (
u --
Yl);
TA(U)
= Yr TA(ulyi)
(6.2)
The TA commute for all u and different auxiliary spaces, i.e. one has (5M) [Ta(u), TA,(U')] = 0. Thus they can be diagonalized simultaneously (i.e. the eigenstates do not depend on A and u). The family of eigenvalues of the transfer matrix is the first question to be addressed for an integrable system and the spectrum of a quantum problem can be expressed in terms of eigenvalues of the transfer matrix. Moreover the transfer matrix may be used to obtain the scattering S matrix. Indeed, at u = 0 the fundamental R matrix becomes a permutation R ( u = O) = P so, choosing A to be the same as the representation of say the first particle, and setting u = Yl, the transfer matrix becomes the S matrix of scattering the first particle with rapidity Yl by the other N - 1 particles, i.e. SA(uIY2, . . . , YN) = RN, A(U -- YN) " " " R2,A(U -- Y2)
(6.3)
The transfer matrix corresponding to a given rpresentation in the auxiliary space can be constructed out of transfer matrices for some elementary space by means of the fusion
CHAPTER 3. CLASSICAL AND QUANTUM I N T E G R A B I L I T Y
144
procedure; this is based on the fact that at certain values of the spectral parameter u the R matrix becomes essentially a projector onto an irreducible representation space (more on this later). A combination of the fusion procedure and the YB equation results in numerous functional relations (fusion rules) for the transfer matrix and they were recently combined into a universal bilinear form (cf. [551,614]). The bilinear functional relations have the most simple closed form for the models of the Ak-1 series and representations corresponding to rectangular Young diagrams (cf. [147, 365]). Thus let Ta(u) be the transfer matrix for the rectangular Young diagram of length a and height s. They obey the relation
Ta(u + 1 ) T a ( u - 1) - Ta+i(u)Ta_i(u ) = T s a + l ( u ) T s a - l ( u )
(6.4)
Since the Tsa(u) commute at different u, a, s the same equation holds for eigenvalues of the transfer matrices so we can treat Tsa(u) in (6.4) as numerical valued functions. The bilinear fusion relations for models related to other Dynkin graphs were suggested in [589]. It is remarkable that the fusion relations (6.4) appear to be identical to the Hirota equation (6.1). Indeed one can eliminate the constants a, ~, "y by the transformation
r(n,g, m) = [1 + (7/(~)] gm rn(g, m);
(6.5)
leading to
rn(e + 1, m)rn(s rn + 1) - rn(e, rn)rn(e + 1, m + 1 ) - - T n - k l (~. Jr-
1, m)rn-1 (g, m + 1) = 0
(6.6)
Then one can change variable from light-cone coordinates n, g, m to the direct variables a=n;
s=g+m;
u=g-m-n;
7n(t~,m)-=T~+m(g-m-n)
(6.7)
At least at a formal level this transformation provides the equivalence between ~6.1), (6.4). (6.5), and (6.6). Relations between Hirota equations and fusion relations goes much further. One knows that a general class of Hirota equations are related to the geometry of Grassmann manifolds and can be constructed out of general Young tableaux (aft [147, 777]). If e.g. the Young diagrams Y = [a~2, (al +a2) sl] consist of two rectangular blocks (one with al lines of length Sl + s2 and the second with a2 lines of length s l) then there are higher Hirota equations involving e.g. symbols '-Fa1'a2-1 (see [972] for some formulas). In any event in [972] one -~ 81~82 treats specifically the relation (6.4) not as an identity but as a fundamental equation which (together with particular boundary and analyticity conditions) completely detrmines all the eigenvalues of the transfer matrix. The solution of the Hirota equation then appears in the form of the BAE. This approach should then make it possible to use some specific tools of classical integrability such as finite gap integration techniques. The origin of Tas(u) as an eigenvalue of the transfer matrix (6.2) imposes specific boundary conditions (5N) Tsa = 0 for a :J: 0, 1 , . . . , k and what is more important, certain analyticity properties. As a general consequence of the YB equations (more below) the transfer matrices may be always normalized to be elliptic polynomials in the spectral parameter (i.e. finite products of Weierstrass a-functions). Such (elliptic) solutions are well known in the classical soliton theory (cf. [15, 605]). To illustrate elliptic solutions consider ([606]) the elliptic solution of the KP equation ( 5 0 ) u(x, y, t) = 2 E N P(x - xi(y, t)) where all the
3.6. HIROTA EQUATIONS
145
dynamics in t and y is hidden in the behavior of the poles xi(y, t). The poles obey the elliptic Calogero-Moser (CM) system of equations x" = - E
- xj)
(6.8)
J
and such elliptic solutions can be lifted up to the discrete case (cf. [619, 770]). Then the poles xi obey a fully discretized version of the CM model which can be recognized as the BAE. In [972] one considers now the discrete version of the classical Liouville equation. Its elliptic solutions give the eigenvalues of the transfer matrix and the S matrix of the QIS associated with the A1 algebra (for extension to the Ak-1 algebra see e.g. [603]). The Hirota equation for the A1 situation may be obtained from the more general (6.4) by using the boundary conditions (5P) Tsa = 0 for a -~- 0,1, 2. Then T ~ obey the discrete Laplace equation while Tls(u) = Ts(u) obeys the discrete Liouville equation (referred to as a Hirota equation)
Ts(u + 1)Ts(u- 1) - Ts+l(U)Ts-l(U) = T~ The values of T~
and T2(u) can be easily determined as T~
T2(u) : 7(u + s)v(u + s + 4) 7(u- s)',/(uN
r
= II
s
+ 4);
(u)
(6.9)
= 1 and (cf. [612])
7(u + 2) 7(u) = r
qk-1
a(~?(u--yk+i))
H
(6.10)
k=l i---qk+l
Here a(x) is the Weierstrass a function and qk is a spin of the kth quantum space. The roots Yk and degree N and parameters of the Weierstrass function are considered parameters of the quantum model. Below one concentrates on the simplest case qk = 1. Apparently different solutions of (6.9) have a number of common zeros and Ts(u) for general parameters Yk are polynomials of degree s ~ qk. These zeros can be gauged out, so all Ts(u) (different eigenvalues at different s and u) remain polynomials of the same degree N. The maximal gauge transformations is N
Ts(u) ~ Ts(u)
r 2
qk-s
+ s - 2p) l--[ r I a ( u ( u - s + qk + 2 - 2p))
(6.11)
k = l p--1
where the last producd holds for qk > s. This gauge transformation simplifies the right side of (6.9) so Ts(u + 1)Ts(u- 1) - Ts+l(U)Ts-l(U) = r + s ) r s - 2) (6.12) Here all the Ts are polynomials of the same degree with no common zeros. A general solution of the discrete Liouville equation (6.9) is parametrized by two arbitrary functions of one variable. Not all of them correspond to eigenstates of the quantum transfer matrix but only elliptic solutions do and elliptic polynomials may be characterized by the roots z~ via N
Ts(u) = Asexp(#sU) H aO?(u- z~))
(6.13)
where As and #s do not depend on u. Similarly to the continuum case the roots obey some dynamics in a discrete time variable s which is determined by certain Bethe Ansatz like
C H A P T E R 3. CLASSICAL A ND Q U A N T U M I N T E G R A B I L I T Y
146
equations (cf. [972] for details). The equation (6.9) is a discrete version of the Liouville equation written in terms of the ~- function. It can be recast to a more universal form in terms a discrete Liouville field Y : ( " ) -- Y~(") = r
which hides the function r
Ts+l(U)Ts-l(U) + ~)r ~ - 2)
(6.14)
in the right side of (6.12) which becomes
Y s ( u - 1)Ys(u + 1)
= [Ys+l(U)-Jr-
1][Ys-l(U) + 1]
(6.15)
with a boundary condition Yo(u) = 0 (the same functional equation but with different analytic properties appears in the thermodynamic Bethe Ansatz - see below). In the continuum limit one puts Ys(u) = 5-2exp[-r Then for 5 --~ 0 one gets
02r
t)] with u = (~-lx and s = 5-1t.
02ur = 2exp(r
(6.16)
(Sine Gordon (SG) equation). For illustrative purposes one can compare this with the discrete SG equation which requires quasi-periodic boundary conditions (see here [459]) (hQ) Tsa+l(u) = exp(o~))~2aTsa-l(u- 2) where c~, A are parameters. Putting this into (6.4) yields T l ( u + 1)Tsl ( u - 1 ) - Tls+l(u)Tls_l(U ) = exp(o~))~2T~ ( U ) T ~ 2); (6.17)
T~
+ 1)T~
1 ) - T~176
) = exp(-a)r2(u)rJ(u-
2)
Introduce now two fields pS,U and Cs,u on the square (s, u) lattice via
T~
= exp (pS,U + CS,U) ; Tls (u + 1) = A1/2exp (pS,U _ CS,U)
(6.18)
Putting these in (6.17) and eliminating pS,U one obtains the discrete SG equation
Sinh (r
+r
= ASinh (r
- r
- r u-l) =
(6.19)
+ cs-l,u + Cs,u+l + CS,U-1 + o~)
The constant a can be removed by redefinition of CS,U. Another useful form of the discrete SG equation appears in the variables Xa(u ) _
-.
T a ( u + 1)Tsa(u- 1) Tsa+l(u)Ta_l(u ) = - 1 - y a ( u )
(6.20)
Under the condition (hQ) one has (hR) s a + I ( u ) = x a - l ( u - 2 ) a n d / ~ 2 x a + I ( u + I ) X a ( u ) = 1 so there is only one independent function (hS) zls(u) - Xs(U) = - e x p ( - o ~ ) e x p ( - 2 r s ' u 2r s'u-2) and the discrete SG equation becomes then
xs+l(u)xs_l(?.t) _
(~ + Xs(?.t + 1))(A + X s ( U - 1)) (1 + )~xs(u + 1))(1 + AXs(U- 1))
(6.21)
In the limit )~ ~ 0 this turns into the discrete Liouville equation (6.15) for Ys(u) = - 1
~-lx~(u).
-
The relation between the Hirota equation and Pliicker relations of the coordinates in Grassmann manifolds suggests numerous determinant representations of the solutions (cf.
3.7.
147
SOV AND HITCHIN SYSTEMS
[71, 147]). The most familiar one allows one to express Tsa(u) through T~(u) or Tls(u). For instance the determinant formula giving the evolution in a is Tsa(u) = det (T~ +i-j (u + i + j - s - 1)); i, j = 1 , . . - , s; T~(u) = 1
(6.22)
Other expressions are also given in [972]. Regarding the Bethe Ansatz one can write (aft (6.12) and (6.13)) (hU) Q(u) = exp(u~u)1-I M a ( ~ ( u - uj)) so uj are zeros of Q(u). Then evaluating the A1 determinant forms
T~
= r
T:(~) = r
+ ~) =
~ - 2) =
Ts(u) =
R(u + s) Q(u + s) R(~ + 2 + ~) Q(~ + 2 + ~) /~(u-s-1) /~(u-s+l)
Q(u + s + l)
0(~- ~)
(6.23)
(~(u-s-1) Q(u-s+l)
R ( u + s + l)
R(~- ~)
(note ( h i ) Q(u) = Q ( u - 1) and R(u) = R ( u - 1)). Then evaluating (6.23) at u - uj and u = uj - 2 there results (hV) r = Q(uj + 2)R(uj); r - 2) = - Q ( u j - 2 ) R ( u j ) from which r Q(~j + 2) = (6.24) r - 2) e ( ~ j - 2) and this is the BAE
~p(-4v.)
r
r
_ 2) = -
~
~ ( ~ ( - j - -k + 2)) ~ ( ~ ( - j - ~k - 2))
(6.25)
There is much more in [972] which we omit here. The conclusions involve the deep connection of quantum integrable models and classical discrete soliton equations. The fusion rules for quantum transfer matrices are identical to the Hirota difference equation with a certain boundary condition and analyticity requirement. Eigenvalues of the transfer matrix are represented as T functions and positions of zeros of the solution are determined by B AE. Thus information usually obtained via YBE may also be found from classical discrete time equations without quantization. For other work in these directions see also [72, 584].
3.7
SOV
AND
HITCHIN
SYSTEMS
Notations such as ( 1 0 A ) are used. We follow here [397] and recall the magic recipe of Sklyanin in Section 3.3 which says that the poles of a properly normalized BA function and the corresponding eigenvalues of the associated Lax operator determine a SOV. Thus one looks at L ( z ) r A) = A(z)r A) and the separation vairables should be the poles zi of and Ai = A(zi). The geometry behind this can be expressed in terms of Hitchin systems, their deformations, and many body problems considered as degenerate systems (cf. Section 5.2.3 and references there plus [74, 260, 296, 482, 754, 755, 767]). For a complex surface X we write X [h] for the Hilbert scheme of points on X of length h (cf. [498, 755]). Thus we recall the Hitchin setup as in Section 5.2.3. One starts with the compact algebraic curve E of genus higher than one and a topologically trivial vector bundle V over it. Let G = S L N ( C ) and ~ = Lie(G). The Hitchin system is an integrable system on the moduli space Af of
148
CHAPTER 3. CLASSICAL AND QUANTUM I N T E G R A B I L I T Y
stable Higgs bundles (strictly on the moduli space of semi-stable bundles). A point of Af is the gauge equivalence class of a pair (.~, r (i.e. an operator ~A : ~ + A and a holomorphic section r of ad(V)QK). The holomorphic structure on V is defined via ~A and the symplectic structure on Af descends from the two form f Tr 5r A 5.~. The integrals of motion are the Hitchin Hamiltonians ( 1 0 A ) T r r k E H ~ k) "~ C (2k-1)(g-1) for k > 1. Their total number is (10S) ~ N - l ( 2 k - - 1 ) ( g - - 1 ) = ( N 2 - 1 ) ( g - 1 ) - (1/2)dimcH (note this is a slightly different point of view than in Section 3.2.3). Thus Af can be represented as a fibration over B ~ g - 1 H 0 ( E , K k) with the fiber over a generic point b E B being an Abelian variety Eb. This variety is identified by Hitchin with the quotient of the Jacobian Jb of the spectral curve Cb, identified as the divisor of zeros of (10C) R(z, A) - Det[r - A] E H~ K N) in T*E. The curve Cb has genus N 2 ( g - 1) + 1 which is g higher than the dimension of Eb and in fact Eb = Jb/Jac(E). The Jacobian of E is embedded into the Jacobian of C as follows. The spectral curve curve covers E so the holomorphic 1-differentials on E are pulled back to Cb, giving the desired embedding. An open dense subset of A/" is isomorphic to T*.A4 ,,~ T*T~g,N, the cotangent bundle to the moduli space of holomorphic stable G bundles on E. The space iV" is a non-compact integrable system and one can compactify it by replacing T*E by a K 3 surface. This is a natural deformation of the original system in the sense that an infinitesimal neighborhood of E imbedded into K 3 is isomorphic to T*E (since c1(K3) = 0). Instead of studying the moduli space of gauge fields on E together with the Higgs fields r one studies the moduli space of torsion free sheaves supported on E (cf. [260]). One can also think of this model in terms of D branes wrapped on E but we omit discussion of this here (cf. [397, 933, 979]). In particular in the compactified case the curve C ,,~ Cb is imbedded into K 3 and is endowed with a line bundle s which is simply the eigen-bundle of r and this determines a point on Jb. Assume deg(s = h and take the generic section which has h zeros P l , " ' , P h . Note that conversely given a set S of h points in K 3 there is generically a unique curve C in a given homology class/3 E H2(K3, Z) of genus h with a line bundle 1: on it such that the curve passes through these points and the divisor of/2 coincides with S. This identifies the open dense subset of the moduli space of pairs (Cb,s with that in the symmetric power Symh(K3) itself. The symplectic form on the moduli space is therefore the direct sum of h copies of the symplectic forms on K3. Thus summarizing, the phase space of the integrable system looks locally like T*A~ where M is the moduli space of rank N stable vector bundles over E and E is holomorphically imbedded in K3. It can also be identified with the moduli space of pairs (C b, ~) where the homology class of Cb is N times that of E and the topology of s is fixed. This identification provides the action-angle coordinates on the phase space. Namely, the angle coordinates are (cf. [463]) the linear coordinates on the Jacobian of Cb while the action variables are the periods of d - 1 K along the A cycles on Cb. The last identification of the phase space with Symh(K3) provides the SOV. In more detail now let ~ be a compact smooth genus g algebraic curve and p be the projection p 9T*~ --~ ~. Let V be a complex VB over ~ of rank N and degree k and consider the moduli space .MN, k of semi-stable holomorphic structures E on V. It can be identified with the quotient of the open subset of the space of 0A operators acting on the sections of V by the action of the gauge group OA --* g-lOAg. The complex dimension of .AdN, k is given by the Riemann-Roch formula (10D) dimc.MN, k -- N2(g- I) + 1 - h. Explicitly the points in T*AZtN,k are the equivalence classes of pairs (E, ~) where E is the holomorphic bundle on 5] and 9 is a section of End(E) | Kc. The map ~ is given by the formula (10E) ~(E, ~) {Tr(~), Tr(~2), ... ,Tr(~N)}. Let ?-/denote the h dimensional vector space C h in the form (10F) T/ - oNH~ Hitchin shows that the partial compactification of T*.MN, k is an
3.8.
DEFORMATION
149
QUANTIZATION
algebraically integrable system, i.e. there exists a holomorphic map (lOG) 7r : T*.A4N,k whose fibers are Abelian varieties Lagrangian with respect to the canonical symplectic structure on T*.A4N, k. The generic fiber is compact and geometric SOV in the Hitchin system is expressed as follows: There is a birational map (10H) r : T*.MN,k --~ (T*~) [hI which is a symplectomorphism of open dense subsets (cf. [397, 482]). Here the open dense set in X [hI coincides with ( X h - A)/,Sh where A denotes the union of all diagonals and Sh is the symmetric group. The theorem implies that on this dense set one can introduce coordinates (zi,s where zi E ~, )~i E T ~ such that the symplectic form in these coordinates has the separated form (10I) f~ = y]h ~is A ~zi. The coordinates (zi, s are defined up to permutations (cf. [397] for proof).
3.8
DEFORMATION QUANTIZATION
Notations such as (14A) and ( M A ) are used. We will try to give here a somewhat up to date rendering of basic mathematical material on deformation quantization following [43, 44, 68, 88, 116, 117, 118, 137, 141, 176, 177, 178, 215, 217, 219, 229, 295, 308, 318, 320, 321, 340, 343, 413, 433, 448, 520, 530, 536, 579, 649, 650, 652, 653, 663, 664, 726, 785, 820, 891,910, 947, 968, 984, 985, 986] (cf. also Chapter 1 for a more earthy approach and Chapter 6 for more details in certain situations). Existence of deformation quantization on symplectic manifolds was established in e.g. [219, 320, 321,785] while classification is determined in e.g. [88, 217, 320, 321,769, 985] (other references above also develop these themes with various embellishments). Connections to geometric quantization are indicated in [448] for example (cf. also [425,809,929, 935, 982]). We will follow here [88, 321,968, 985] at first. Thus from [985] and go directly to the Fedosov quantization. Generally a star product or deformation quantization is a family of associative products *h (we continue to use h which in the present context is h)in Cm(M([[h]] ofthe form (14A) f * h g = f g + h C l ( f , g ) + . . . + h k C k ( f , g ) + ... satisfying (A) C l ( f , g ) - C l ( g , f ) = - i { f , g } with (B) Ck(1, f ) = C k ( f , 1) = 0 (k > 1) and (C) C k ( f , g ) is a bidifferential operator, while (D) C k ( f , g ) = (--1)kCk(g, f). On a symplectic vector space V there exists a standard Moyal-Weyl 9 product (cf. Example 5.5) I,hg=Z
m ( ~- ) k o
1 i~:1
rikJk
Okf
Okg
Oyil . . . Oyik ogyJl ... OyJk
(8.1)
where y l , . . . , y 2 n are linear coordinates on V and 7rij - - { y i y j } Now let (M,~) be a symplectic manifold of dimension 2n so that each tangent space T x M has a linear symplectic structure which can be quantized via (8.1). One defines a formal Weyl algebra Wx associated to T x M to be an associative algebra over C with unit consisting of formal power series in h whose coefficients are formal polynomials in T x M . Thus the elements have the form ( 1 4 B ) a ( y , Ii) = ~ h k a k , a y a where ( y l , . . . ,y2n) is a linear coordinate system on T x M and f *h g is given as in (8.1). Let W = tJ~Wx (Weyl bundle or quantized tangent bundle) whose sections FW form an associative algebra under fiberwise multiplication; thus FW gives rise to a deformation quantization on T M . Let Z ( W ) c F W be the space of sections not containing y variables so Z ( W ) ~ C~(M)[[h]]. Given an assignment of degree via deg(y i) = 1 and deg(h) = 2 there is a natural filtration CC~(M) C F(W1) C - - - C F(Wi) C F(Wi+I C - - . C F(W) with respect to the total degree (note that the terms in (14B) have degree 2k + lal). One also considers differential forms (14C) a(x, u, Ii, dx) = ~ hkak,il...ip,jl...&y il ... yipdxJl A . . . A dx& where the coefficient ak.i,j is a covariant tensor, symmetric in the i and antisymmetric in the j; the space of such
150
CHAPTER
3. C L A S S I C A L A N D Q U A N T U M I N T E G R A B I L I T Y
sections is denoted by F W | Aq. One defines now (a E F W | A*)
Oa 6a = y ~ d x i A ~ ; Oy i
6-1(a)= y~
1 p + q yiI (o-Q-~)a
(8.2)
where I(O/Ox~)a is the contraction of O/cgx ~ with a (interior multiplication). Here p + q > 0 and 6-1a = 0 when p + q = 0 while a E F W | A q is homogeneous of degree p in y. One also defines (14D) 5*a = F_,ykI(i)/i)xk)a with 5-1apq = ( l / p + q)6*apq for p + q > 0 so t h a t for aEFW| a = ~ * a + ~*~a + a00; ~2 = (~,)2 = 0
(8.3)
Let now V be a torsion free symplectic connection on M and 0 : F W ---, F W | A 1 the induced covariant derivative. We follow now [985] where the notation and formulation differ occasionally from [321]. Recall that (~ la [321]) given a Riemannian manifold M with natural bases Oi and dx i in T M and T ' M , a connection in T M corresponds to a covariant derivative 0 satisfying c9(fu) = df . u + f O u for f E C ~ ( M ) and u E C ~ ( T M ) (this is called an affine connection). Torsion is defined via S = 00 E C ~ ( A 2 | T M ) being 0 where 0 is a fundamental l-form section of A | T M such that I ( X ) O = X for X E T M . Thus e.g. 0 ,'., dx i | Oi will do with S = F{ A dx i | Oi = rikdX J k A dxi | cgi and S~k j __ ( 1 / 2 ) ( F ~ " i - F/k j )" then S is torsion free if e.g. S~j = 0 or FJi = F~k. Generally a torsion free agfine connection on a symplectic manifold (M, w), which preserves a;, is called a symplectic connection (this corresponds to dw = 0 or OjWik + cgiwkj + Oka;ji = 0 where a; ~ (1/2)wijdx i A dxJ). Now consider a connection of W of the form (14E) D = - 5 + O + ( i / h ) [ " / , .] where-y E F W | 1. T h e n for a, b E F W
D(a ,h b) = a ,h Db + Da ,l~ b; D2a = -
f~, a ;
(8.4)
i f~ = ~ - R + & / - 0-y - ~,y2 where R = (1/4)Rijk~yiyJdx k A dx ~ and Rijk~ = WimR'jmk~ is the curvature tensor of 0. R E M A R K 3.11. D is called Abelian if f~ is a scalar 2-form, i.e. f~ E f~2 (M) [[h]] (note ~ 2 ( M ) ,-~ A2T*M). For such a connection the Bianchi identity implies df~ = Df~ = 0 so f't E Z2(M)[[h]] and f~ is called the Weyl curvature. D is called a Fedosov connection if in addition -y E FW3| 1 and a theorem of Fedosov (cf. [321,968, 985]) then says that for any torsion free symplectic connection V ,,~ c9 and any perturbation f~ = w + ha~l + . . . E Z2(M)[[h]] of w in Z2(M)[[h]] there exists a unique -y E FW3 @ A 1 such that D given by ( 1 4 E ) is an Abelian connection with Weyl curvature ~t satisfying 5-13, = 0 (Fedosov connection). This then indicates that a Fedosov connection D is uniquely determined by a torsion free symplectic connection V ,-~ 0 and a Weyl curvature f~ = ~ hiwi E Z2(M)[[h]]; hence one writes D ~ (0, f~). Now for a Fedosov connection D the space of all parallel sections WD C F W a u t o m a t ically becomes an associative algebra and one writes a : WD ~ C~162 defined by a(a) = aly=o. Note a E WD ,'., D a = 0 so a is a flat section. In fact Fedosov proves (cf. [321, 968, 985]) that for any ao(x,h) E C~(M)[[h]] there is a unique a E WD such t h a t a(a) = ao so a is a vector space isomorphism WD ~ C~162 Note now t h a t if V ,-~ 0 is flat and f~ = w then D = - 5 + 0 with the solution a = a - l ( a o ) having the form (14F) a = ~ ( 1 / k ! ) ( O i l ...Oikao)y il . . . y i k _ Taylor series of expxa o at the origin. Thus the correspondence C~(M)[[h]] --, WD is the pullback via the C ~162 jet at the origin of the
3.8. D E F O R M A T I O N Q U A N T I Z A T I O N
151
usual exponential map. Hence for a general Fedosov connectioI~ D one may consider this correspondence as a quantization corresponding to the exponential map. More precisely one defines a quantum exponential map as an h linear map p : Ca(M)[[h]] ~ F W such that (A) p(Ca(M)[[h]]) is a subalgebra of F W (B) p(a)ly=O = a (C) p(a) = a + 5-1da mod W2 (D) p(a) is a formal power series in y and h with coefficients which are derivatives of a. Thus a quantum exponential map determines a 9 product. Further for any Fedosov connection D the correspondence Ca(M)[[h]] ~ WD is a quantum exponential map which establishes the existence of such maps for any symplectic manifold. In fact one shows in [985] that quantum exponential maps are equivalent to Fedosov connections (cf. also [769, 968]). The characteristic class of a star product algebra .A (always isomorphic here as an algebra over C[[h]] to Ca(M)[[h]])is defined as (14G) cl(A) - (1/h)~ 9 H2(M)[h-I,h]] and via [320, 321] this does not depend on the choice of Fedosov connection. Here C a ( M ) [ h -1, hi] corresponds to formal Laurent series in h and W + designates W extended via C[[h]] C[h -1, hi] so W + ~ WD[h -1, hi]. Now (following [968]) let *h be a star product on a symplectic manifold M and .A = Ca(M)[[h]] its star product algebra. Write A - CeC(M)[h -1, h]] which will have an induced associative algebra structure. Let Ck(fL, fL) be the space of k-linear maps A | | A ~ A which are multidifferential operators when being restricted to C a ( M ) N " " @ C a ( M ) . Each element of Ck(.,[t, A) can be identified with a multilinear map OO
c = ~ hece 9 C a ( M ) |
| C a ( M ) ~ C a ( M ) [ h -1, h]]
-N
where each cg is a multidifferential operator on M. Define the coboundary operator via (14H) b. Ck(A,A) ~ ck+~(A,A) ~s (i/n)~ where b is the usual Hochschild coboundary operator kD1
(bc)(u0,""", ttk) ~- tt0
*h
C(ttl,""",
,
ttk) + E ( - - 1 ) i + I c ( u o ,
0
"'" , tti *h
+(--1)k+lc(u0,""", ttk-1)*h ttk
Ui+I,""", ttk)+ (8.5)
for u o , " ' , U k E A and c E Ck(A,A). The Hochschild cohomology H*(.A,.A) of the star product algebra .A is defined as the cohomology of this complex. One proves that for a general symplectic manifold M (14I) H k ( A , A ) "~ Hk(M)[h-l,h]]. Further for WD ~ A there is an isomorphism ~ " H2(WD, WD) ---, H2(m)[h -1, h]] such that if T" WD~ ~ WD2 is an isomorphism of star products with induced isomorphism T, 9 H2(WD~, WD1) H2(WD2, WD2) then ~2 o T, = id o ~1. Another important concept which we mention only briefly here is the idea of the derivative of a star product. Thus define (14J) c(f,g) (d/dh)(f *h 9). Then for the corresponding star product algebra A = Ca(M)[[h]] it follows that ~[c] = - i h 2 ( d / d h ) ( c l ( A ) . 3.8.1
Path
integrals
We refer here to [179, 238, 579, 580, 864] in particular and begin with [864] (cf. also [68]) where one goes back to the classical situation of Example 3.5 with
(f * g)(q, P) = f exp =Y~rV-~sv.a 1 (~)
o
""
~ Op
Op Oq
r + s ( - l ) sOr+sf~ or+sg
OqrOps OprOqs
g=
(8.6)
C H A P T E R 3. C L A S S I C A L A N D Q U A N T U M I N T E G R A B I L I T Y
152
(cf. also (8.1)). It is then shown that the Feynman path integral is the Fourier transform over the momentum variable of exp*(f)(q,p) = 1 + f ( q , p ) + (1/2!)(f 9f)(q,p) + . . . , namely
dq(T)dp(T)exp [~ fot(po- H)dT =
-
(8.7)
iHt)(q"+q'
p
where the integral on the left is over all paths q(T) with q(0) = q' and q(t) = q". The proof is based on a lemma for functions F(~, rl) which can be Taylor expanded in ~ on R and which go to 0, along with all partial derivatives in rl, as r / ~ +oc (F is said to belong to the space S). Set then
(8.8)
.~ij(qi, qj) = /_x) ~dP h exp [hp(qi - qj) J F (qi + qj , p ) 2
Then if ~'21 is based on F(~,~) and 9r32 on a(~,r/) the function (14K)~T'31(q3,ql ) -f dq2.T'32(q3, q2)gv21(q2, ql) belongs to S and
~hexp
-7"31 =
-~P(q3 - ql) ( a 9F)
q3 +2 ql 'P
(8.9)
Now let (14L) Fi+l,i(q,p) = e x p [ - ( i / h ) ( t i + l - ti)H(q,p)] where to = 0, t l , - . ",tN = t is a partition of (0, t) and H is the Hamiltonian. Let 9~i+l,i(qi+l, qi) =
= / d p i +27rhl e x p { - h
(8.10)
/ qi+l + qi i r [Pi+l(qi+l - qi) - (ti+ 1 - ti)H ~ ,2 Pi+I)] }
Then repeating (8.9) over and over one obtains (using the associativity of ,)
/ dpNdqN-l...dqldpl { i ~-~ exp -~ [PN(qtt-- qN-1) +""-~- Pl(ql -- qt) -- (tN -- tN-1) • x H =
/"
~exp
2
1,PN ) . . . . .
(tl -- to)H
]
( ql +2 q' ,Pl )1}=
p ( q " - q') (FN 'N-1 * " " * F21 * Flo)
(
q'' + 2 q' 'P
)
(8.11)
Now as the partition ti is refined the left side tends to the Feynman path integral over all paths (p(r), q(r)) with q(0) = q' and q(t) = q". On the right the difference between e x p * [ - ( i / h ) ( t i + l - t i ) H ] ( q , p) and the ordinary exponential is of order ( t i + l - t i ) 2 so formally
Nlim - , ~ (FN 'N-1 * " " * F10)
q2
'p
= exp*
-- t H
2
'p
(8.12)
(using the associativity of the star product). We go next to [238] where a field theoretic version is developed (there are many books on Q F T and we cite [447] for background). One deals with star products of the form f ,~ g = E r > o u r C ~ ( f , g ) (with classical observables N corresponding to functions on M - then N(u) denotes formal series in u as indicated). In the quantum theory of course ! u = ih/2. Two star products *u and *u are called c-equivalent if there exists a formal
3.8. D E F O R M A T I O N Q U A N T I Z A T I O N
153 !
series Tv ---- id + Er>l ~rTr such that ( 1 4 M ) T ~ ( / , ~ , g) - (T~,f),~, (T,g) where the T~ are differential operators vanishing on constants. This notion of c-equivalence is related to operator ordering in quantum theory in the sense that defining a new star product by c-equivalence corresponds to a change of ordering in quantum theory. The star exponential is defined via ) n (. g ) n exp* ( t-H~ ) = y 0 1 ~. ( t-~ (8.13) In [238] one develops the star quantization of the free scalar field and the star product is the equivalent of normal ordering in QFT. First one recalls some notions from [83]. Thus consider the complex Hilbert space ~ of L 2 functions on R d (d > 1); this may be considered a real Hilbert space if to each { E ~ one associates the couple ({*, {) where {* is the complex conjugate. Functionals on 7{ are denoted by (I)(~*, ~) and one can define a Gaussian measure # on ~ via { 1 } d~*(k)d~(k) (8.14)
d.(C,~)=exp --~ f dpC(p)~(p) H
2~in
This measure has the properties (14N) f d#(~*,~)= 1 and
/
m
d#(~*,~)~*(kl)" "'~*(km)~(pl)" "'~(Pn) = 5mn hm E
H 5(ki --Pa(i))
(8.15)
aESm 1
where Sm is the permutation group on rn elements. Let now jr be the space of functionals of the form
9(C,5)= ~
/dkl...dk~dp~...dp~K~(k,p)5*(k~)...~*(k~)~(p~)...5(p~)
(8.16)
m,n>O
where the distributions Kmn are such that the series is convergent. One constructs now a star product on a subspace of jr and the holomorphic representation is used where the free field Hamiltonian takes the form ( 1 4 0 ) H(a*,a) = f dkwka*(k)a(k) (here wk = v/k 2 + m 2 with m 2 > 0. Clearly H is not defined on the whole space 7-/. One can show that the P bracket in (a*, a) is i times the P bracket expressed in "canonical" variables (r 7r). Set now a(k)
(n) ai(k)=
--
i- 1 i= 2
a*(k)
(B) < 5i~...irF, Sj~...j~G > -
/ d k l . . . d l k r sail(kl).:_-Sair(kr)Sa~l(kl~....Sa~r(kr)
; ( C ) A ij -=
(8.17)
(o )1o
Then the P bracket can be put in the form (14P) i{F, G} = A ij < 5iF~ 5jG > which allows one to formally define the Moyal bracket *M on jr via F *M G = E
r>O
s
r! Ailjl " " "AirJr < ~il'"irF, 5Jl'"JrG >
(8.18)
which can be written out explicitly as (h)r
F *M G = r~>o -~
1 ~ ( ~ ) f ( (-1)s
"~" s=O
~rc
5rF 5a*(kx) ...Sa*(ks)Sa(ks+x). "Sa(kr)
)
5a(kl)...Sa(ks)Sa*(ks+l)...Sa*(kr) dkl...dkr
(8.19)
CHAPTER 3. CLASSICAL AND QUANTUM INTEGRABILITY
154
This Moyal product corresponds to the Weyl ordering (completely symmetric) and it is known that this ordering is not appropriate in field theory for the quantization of the free Hamiltonian H above. In the star product approach this is expressed through the nonexistence of the star exponential of H; in fact one finds already that H *M H diverges. In Q F T normal ordering removes this (trivial) divergence and then one proceeds to quantization. In the star product quantization this problem is solved by looking for a new star product *N such the star exponential of H is well defined. One shows in [238] that in fact such a star product (corresponding to normal ordering) exists and is formally equivalent to the Moyal product. To define *N as a c-equivalent star product to *M one uses the following (formally equivalent) integral representation to *M
(F *M G)(a*, a) = =
dp(~*,~)dp(rl*,r/)F
a* + ~ , a
+ ~
(8.20) G a* + ~ , a -
Then using the c-equivalence operator (14Q) T = exp{-(h/2)fdk(52/ha(k)ha*(k))} defines F *Y G = T-I(TF *M TG). After some calculation there results
(F *Y G)(a*, a) = / d#(~*, ~)F(a*, a + ~)G(a * + ~*, a) This yields a Taylor series by expanding F(a*, a + ~)G(a* (a*, a); thus F *N G = ~r>o(ili/2)rcg(F, G) where (2)r
CN = _-_
(~rF
1 /
_~
one
(8.21)
§ ~*, a) in a Taylor series about (~rG
dkl--.dk~a(kl).:_-~a(kr) ea*(kl)---)~a*(k~)
(S.22)
Here ili/2 has been chosen for deformation parameter in order to have *N a deformation of the normal P bracket. Further *N is related to normal ordering by N ( F , N G ) = N(P)Af(C) where Af is the map F(a*,a) --, Af(F) = normal ordering of F. One shows next that the star exponential (8.13) makes sense for *N (note H is not defined on all 7-/ so one works on the subspace S of rapidly decreasing Schwartz functions). First e.g. H *N H = HF + h f dkwka*(k)(hF/ha*(k)) and one restricts here F to the subspace 59 C 9c where 59=
FEJc"
when (a*, a) E S. It follows that
5rF(a,,a )
5a*(kl)...Sa*(kr)
ESx'"xS=Sr
}
(8.23)
(,NH) n C 0 for all n _> 1 and one can compute 1
exP*N (ti-~) = exp {-~ / dk(e-ia~kt -1)a*(k)a(k) }
(8.24)
Note that while H is only defined e.g. on S the star exponential is defined on all of 7-/. Now for the Feynman path integral, in the holomorphic representation the kernel of the evolution operator is given by a path integral over the paths t ---, (a~, at) and it turns out that the star exponential for *g is nothing else than the normal symbol of the evolution operator. More explicitly one recalls first (cf. [68]) that in the holomorphic representation the commutation relations [a(k), a+(p)] --- h h ( k - p ) are represented in the space of analytic functionals F(a*) via (14R) (a(k)F)(a*) = 5F(a*)/ha*(k) and (a+(k)F)(a *) = a*(k)F(a*). There is then a scalar product < FIG > = f d#(~*,~)F*(~)G(~*) under which the operators
3.8. DEFORMATION QUANTIZATION
155
a and a + are self adjoint. With each operator A is associated its kernel /C(A) defined by ( 1 4 S ) ( A F ) ( a * ) = f d#(~*,~)~(A)(a*,~)F(~*). It is remarked that /C(AB)(a*,a) = fd#(~*,~)lC(A)(a*,~)lC(B)(~*,a). Further if A is normal ordered with kernel /C(A) one can associate with it the functional $ ( A ) obtained by the substitution (a +, a) ~ (a*, a); S ( A ) is called the normal symbol of A and is related t o / C ( A ) via (14T) /C(A)(a*,a) = exp[(1/li) f dka*(k)a(k)]S(A)(a*, a). In this setting one then shows that
/ 1 - I d~s 27rili
x exp
-~
ds =exp
exp {1_~/ dp[a'(p)~t(p) + a(p)~(p)] } x
dp
2
{1/ } (tH) (a*,a) -~ dpa*(p)a(p) exp*N ~
(8.25)
where ~s(p) = (d/d~)~,(p) and the integration is over paths s ~ (~s, ~s) restricted to boundary conditions ~;(P)l,=t = a*(p) and ~,(P)l,=0 = a(p). We refer to [238] for the proof (for Hamiltonians H = H0 + V in normal ordered form) and note that (8.25) is stating that exp *H N(tg/ili)(a*, a) is the normal symbol of the evolution operator. A rather more current point of view for path integrals is developed in [179] based on the Kontsevich approach to deformation quantization (cf. [378, 530, 579, 580, 958]) showing that classification of deformation quantizations is equivalent to the formal classification of P structures (cf. also [420]). Kontsevich's formula for deformation quantization of the algebra of functions on a P manifold involves an expansion reminiscent of a 2-D QFT on a disc D with boundary. This is a simple bosonic topological QFT with a field X : D --, M taking values in the P manifold M and a 1-form 77 on D taking values in the pullback X*(T*M). The formula for the star product is
(f 9g)(x) - fx(~)= f(X(1))g(X(O))
(8.26)
where 0, 1, oc are three distinct points on OD. The integral is over all X : D ---, M and rl E F(X*(T*M)| subject to boundary conditions X(oc) = x, ~(u)(~) = 0 for u E OD and ~ tangent to OD. The Kontsevich construction is nicely reviewed in [179] and we refer to that and [579] for details. The QFT model goes as follows. We have a sigma model with two bosonic fields X and ~; X is given by d functions Xi(u) for u E D = {u: lul < 1} and rl(u) = rli,t~(u)du~. The action is 1 i"3(X(u))rli(u) A rlj(u) S[X, rl] = D rli A dXi(u) + -~c~
(8.27)
The boundary conditions on r/require for u E cOD that rh(u) = 0 on tangent vectors to OD. The semiclassical expansion of (8.26) is understood as an expansion around the classical solution X(u) = x and ~(u) = 0. The quantization is somewhat subtle due to the presence of a gauge symmetry which only closes "on shell" (i.e. modulo the equations of motion) so BRST quantization fails and one needs the BV quantization (cf. [451] for BRST and [66,850] for BV). Then the action is invariant under the following infinitesimal gauge transformations with infinitesimal parameter fl which is a section of X*(T*M) vanishing on OD
5zX i = a ij (X)C/j; 5zr/i = -d/~i - OicJk (X)rlj/3k
(8.28)
CHAPTER 3. CLASSICAL AND QUANTUM INTEGRABILITY
156
Here the P structure on M is given via {f, g} = E d a ij (X)OifOjg where a is skew symmetric and satisfies (14U) O~ffOeO~jk + oJeOeo~ki + akeOeaiJ -- 0 (Jacobi). In the symplectic case where a comes from a symplectic form w one can integrate formally over rl in (8.26) to obtain (in the spirit of Feynman)
f . g(x) = ~(•
p [~ /2 d-lw] d7
(8.29)
The integral over trajectories 7 : R ~ M is to be understood as an expansion around the classical solution 7(t) = x which is a constant in time since the Hamiltonian vanishes. We refer to [179] for details and do not pursue the constructions here. It is interesting to speculate here on the connection between Kontsevich deformation and renormalization in view of the recent papers [131, 194, 195, 597].
3.8.2
Nambu
mechanics
There is at present some active interest in Nambu mechanics and its Zariski quantization in connection with M theory (cf. [22, 40, 49, 69, 188, 201,239, 240, 335,341,342,405, 368,423, 477, 485, 731,751,790, 843, 847, 889, 920, 921,936, 939]). It is proposed in [731] for example that the problem of a covariant formulation of M theory can be solved in this context. We extract here first from [423] which connects the Nambu-Poisson (NP) theory to dKP and dToda in the twistor formulation (cf. [22, 188, 406, 423, 477, 902, 909, 907, 912, 920, 921]). Here we only give a sketch of Nambu mechanics in relation to integrable systems. Thus one looks at the volume preserving differmorphic integrable hierarchy of 3-flows which will be related to the Nambu mechanics originating in [765]. In the beginning Nambu mechanics was designed to formulate a statistical mechanics on R 3 emphasizing that the only feature of Hamiltonian mechanics to be preserved was the Liouville theorem. Nambu considered the equations ( M A ) (df'/dt) = V u ( ~ A V v ( ~ for f' = (x, y, z). The Liouville theorem follows from the identity ( M B ) V . ( V u ( ~ A Vv(r-)) = 0 and the equations of motion can be put in the form
df dt
--
=
O(f , u, v) O(x,y,z)
(8.30)
where the Jacobian on the right can be interpreted as a generalized Poisson bracket. The basic principles of Nambu mechanics have been formulated in [920, 921]. Now it turns out that various classes of integrable systems (e.g. self dual Einstein equations, dKP, etc.) can be written in the form ( M C ) dfl (n) = 0 with f~(n) A t2 (n) -- 0. In [423] one unifies such systems via Gindikin's pencil or bundle of forms and the Riemann-Hilbert (RH) problem (cf. [385, 406, 902, 912]). Let M be a smooth finite dimensional manifold and C~(M) the algebra of C a real valued functions on M. M is called a NP manifold if there exists an R-multilinear map ( M D ) { , . . . , }" [C~176 | ---, C~(M) called a Nambu bracket of order n such that for all fl, f2,'" ", f2n-1 E C c~ ( M ) o n e has {fl,'", {flf2, f3,""",
fn} = (--1)e(a){fa(1), " " , fa(n)};
(8.31)
fn+l} -- fl{f2, f3,""", fn+l} + {fl, f3, " 99 fn+l }f2
as well as
{ { f l , ' ' ' , f n - l , fn},fn+l,''',f2n-1} -t- {fn, { f l , ' ' ' , f n - l , fn+l},fn+2,''',f2n-1}~-
(8.32)
3.8. D E F O R M A T I O N Q U A N T I Z A T I O N -'F" " " Jr- { f n , " " " , f 2 n - 2 , { f l , " " ", f n - l ,
157 f2n-1}}
= {fl,''',fn-l,{fn,''',f2n-1}}
where a E Sn "~ the symmetric on n elements and e(a) is its parity. These are the standard skew symmetry and derivation properties found for the ordinary (n = 2) Poisson bracket plus a generalization of the Jacobi identity called the fundamental identity. When n = 3 this fundamental identity becomes
{{fl, f2, f3}, f4, f5}+
(8.33)
+{f3, {fl, f2, f4}, f5} + {f3, f4, {fl, f2, f5}} -- {fx, f2, {f3, f4, f5}} It is also known that Nambu dynamics on a NP phase space involves n - 1 so called Nambu Hamiltonians Hi E C ~ ( M ) and is governed by the equations of motion ( M E ) (df/dt) = {f, H 1 , - - . , Ha-l}. A solution to these equations produces an evolution operator Ut which by virtue of the fundamental identity preserves the Nambu bracket structure on C ~ ( M ) . Now f E C~176 is called an integral for the system if ( M F ) {f, H x , . . . , H a - l } = 0 and like the Poisson bivector the Nambu bracket is geometrically realized by a polyvector e F ( A n T M ) (i.e. a section of A n T M ) such that ( M G ) {fl, "" ", fn} = ~](dfl,"" ,dfn) which in local coordinates is given by = ~i~...i~ (x)
"1
A--. A
(8.34)
OXin
One knows from [920, 921] that the fundamental identity (8.32) is equivalent to the following algebraic and differential constraints on the Nambu tensor r/i~...i~ : Sij + P(S)ij = 0 for all multiindices i = ( i 1 , ' " , in) and j = ( j l , " " ,jn) from the set ( 1 , . . . , N) where Sij = T]il...in ?~jl...jn Jr- ?~jnil i3...in ?]jl ...jn_ l i2 Jr-'''-Jr-]-?Tj~i2""in-lil l]jl""jn-lin -- ~jni2""in ~jl""jn-lil
(8.35)
and P is the permutation operator which interchanges the indices il and jl of the 2n tensor S; also N
E
~=1 N
= E
e=l
( ~i2"''inOI]jl'''jn OX e
7]Jlj2""jn-l~
+ l]jnei3...i n
(8.36)
OT]jn i2""in -OXi --
Ol]Jl"'jn-li2 O?]Jl'"jn--lin) OX e "nt- " " " -Jr-?]jni2""in-ll "b-~
It was conjectured in [912] that Sij = 0 is equivalent to the condition that the n tensor ~? is decomposable and this has been proved in [22, 686] for example. Thus any decomposable element in A n v where V is an N dimensional vector space over R endows V with the structure of a NP manifold. As an example one considers the motion of a rigid body with a torque about the major axis introduces in [96]. Euler's equations are then
fin1 = alm2m3; fin2 = a2mlm3; ?:r~3-- a3mmm2 q- u
(s.aT)
where u = - k m l m 2 is the feedback, al = ( 1 / 1 1 ) - (1//3), a2 = ( 1 / / 3 ) - (1/I1), and a3 = ( 1 / I 1 ) - (1//2) with say I1 < / 2 < / 3 . These equations can be recast into generalized Nambu Hamiltonian equations (MI) ( d m i / d t ) - {H1,/-/2, mi} with two Hamiltonians
H i = 1(a2m2 - alrn22); H2 = ~l ( a 3 - kalm ~
_ m2 )
(s.3s)
CHAPTER 3. CLASSICAL AND QUANTUM I N T E G R A B I L I T Y
158
When a l = a2 = a3 = 1 and u = 0 these equations reduce to a famous Euler equation or Nahm's equation ( M J ) (dTi/dt) = Cijk[Tj, Tk] where the Ti are SU(2) generators. Now following mehtods of Takasaki-Takebe, based on self dual vacuum Einstein equations and hyperkghler geometry, one considers L, M, N(A, p, q) with the volume preserving integrable hierarchy equations
ON o n = {Bln, B2n L}; O M = {Bin B2n M}; ~ n Otn ' Otn ' '
{Bin B2n, N} '
(8.39)
and { L , M , N } = 1 (volume preserving condition) where Bin = (nn)n>O and B2n (Mn)n>_O. One can compare with the case of the area preserving hierarchy in Section 1.1 (sdiff(2) K P hierarchy) given via ( M E ) (Os {Bn,s with (OA4/Ot~)= {B~, A/I} and { s = 1 where /: = A + E~Un+l(t)A -n and B b - (f-,n)n>_O with ( M L ) M E ~ ntn s + x + ~,~ vis -i-1 (tl = x). In any event since the flows are commuting one finds easily that the Lax equations for L, M, N are equivalent to
OBln B2n -~- Bin, Otm ' Otm OBlm Otn
_
Blm ,
Otn
+{/2/1, B2m}-
(8.40)
+ { Blm H2 } = 0 '
where ( M M ) //1 = {Bin, B2n, Blm } and I212 = {Bin, B2n, B2m }. In the case of the ordinary Poisson geometry and the sdiff(2) hierarchy this is simply a zero curvature equation. Now let ~ be a 3-form (X)
CO
= E dBln A dB2n A dtn = dA A dp A dq + ~ dBln A dB2n A dtn 1 2
(8.41)
It is clear that gt is closed and in fact one proves that the volume preserving hierarchy is equivalent to the equation ( M N ) ~ = dL A dM A dN. To see this expand both sides of (8.41) as linear combinations of dA A dp A dq, d)~ A dp A dtn, dA A dp A dtn, and dp A dq A dtn. Equating coefficients gives first (MO) {L, M , N } = 1 and subsequently
O(Bln, B2n)
0(~,p)
=
O(L,M,N)
0(~,p,t~) ;
(8.42)
O(Bxn, B2n) O(L, M, N) O(Bln, B2n) O(L, M, N) O()~, q) = O()~,q, tn) ; O(p, q) = O(p, q, tn) Multiplying these equations by (OL/Oq), (OL/Op), and (OL/Os respectively one finds after some addition and subtraction (using ( M O ) ) the equation ( M P ) (OL/Otn) = {Bin, B2n. L}. Similar calculations yield the other equations in (8.39). Now one can write down the fundamental relations ( M Q ) ~ = dL A d m A dN as d(MdL A dN + ~ BlndB2n A dt~) - O. This implies the existence of a 1-form Q such that ( M R ) dQ = Md(LdN) + Z ~ Blnd(B2ndtn). This is an analogue of a Krichever potential and one can write
OQ I 9 M = O(LdN) B2~,tn /ixed'
OQ Bin = O(B2ndtn) L,N,B2m,tm(re#n) /ized
(8.43)
3.8. D E F O R M A T I O N Q U A N T I Z A T I O N
159
To set the stage now one recalls how area preserving diffeomorphisms appear in the self dual gravity equations. Begin with a complexified metric of the form
ds 2 = det
ell el2 ) = elle 22 - el2e 21 e21 e22
(8 44)
where eij are independent 1-forms. Ricci flatness is then the closedness (MS) dft ke = 0 of the exterior 2-forms f/ke = (1/2)Jije~k A eje where (J) is the normalized symplectic form (i.e. it is a 2 x 2 matrix whose entries are 0, 1 , - 1 , 0 respectively. Then the above system of 2-forms can be rewritten as ( M T ) t2(~) = (1/2)Jij(e il + ei2)~) A (ejl + eJ2)~) satisfying ( M U ) f/()~) A ~t(A) = 0 and dt2(A) = 0. This suggests the introduction of a pair of Darboux coordinates ( M V ) f/(A) = d R A d Q which are sections of the twistor fibration 7r: T ~ C P 1 where T is the Penrose twistor space. Basically each fiber is endowed with a symplectic form and as the base point moves these forms must deform with area preservation. Two pairs of Darboux coordinates are related by f(A, P(A), Q(A)) = P'; g(A, P(A), Q(A)) = Q'
(8.45)
where {f, g} = 1. Locally f and g (after twisting with A) yield patching functions. A Ricci flat Kghler metric is locally encoded in this data and this arrangement is nothing but the RH problem in the area preserving diffeomorphism case. One goes now to the Gindikin bundle of forms. One knows that anti-self dual vacuum equations govern the behavior of complex 4-metrics of signature (+, +, +, +) whose Ricci curvature is zero and whose Weyl curvature is self dual. These two curvatures are independent of coordinate changes so in one particular choice of coordinates the metric becomes automatically K~hler and can be expressed in terms of a single scalar function ~ (the K~hler potential). Then the curvature conditions lead to the first Plebanski heavenly equation
02~ 02~
02~ 02~
Ox02 OyO~l
OxO~lOy02
= 1
(8.46)
with anti-self dual Ricci flat metric ( M W ) g(gt) = (02~/OxiO2J)dxid2 j where 2 ~ - 2, y and x j = x, y. This system is completely integrable and writing
= dx A dy + )~2d2 A d~]+
(8.47)
+)~(~x~dx A d2 + ~x~dx A d~] + t2y~dy A d2 + t~y~dy A d~]) one has ( M X ) dr2 = 0 and ~t A t2 = 0. As an example consider dKP in the form Os = {Bn, s with Bn = (s and s = A + E ~ Un+l (t)A -n. Then { , } is a P bracket in 2-D phase space with respect to ()~, x) and one writes (MZ) ~t - dA A dx + E ~ dBn A dtn and A t2 = 0 is equivalent to the zero curvature condition (NA) (OBn/Otm) - (OBm/Otn) + {Bn, Bm} = 0. The corresponding version for multidimensional integrable systems related to volume preserving diffeomorphisms is then that the 3-form oo
~(3) = dA A dp A dq + ~
dBln A dB2n A dtn
(8.48)
2
satisy (NB) d~ (3) - 0 and ~(3) A ~(3) = 0. The Gindikin method of pencils of two forms is the most effective way to study such systems (cf. [385]). Thus consider the following system
C H A P T E R 3. C L A S S I C A L A N D Q U A N T U M I N T E G R A B I L I T Y
160
of first order equations depending on a parameter T = (71,7"2)" el(T)
=
e11T1k + . . . + elkT"k;
=
=
(8.49) +...
+
where eij are 1-forms. Let ~k(T) be the bundle of 2-forms ( N C ) Qk(7-) = el(T ) /~ e2(T ) _~_ 9" .+e2~-l(T)Ae2e(7-) satisfying ( N D ) (gtk) TM = 0 with (wk) t # 0 and d~ k = 0. This bundle encoldes the integrability of the original system. For t~ = 1, k = 2 one recovers the Ricci fiat metric g = e11e 2 2 - et2e 21. This leads one to consider a higher dimensional analogue of Ginkikin's pencil via (7- = (7-1,T2, W3) E C 3)
r r r
ellT1 + e12T2 + e1373 =__ e21T1 nt- e2272 + e2373 -- e31T1 + e32T2 + e3373 -
-
(sbo)
One thinks now of three forms ~t (3) satisfying ( N E ) d~ (3) = 0 and ~t (3) A ~(3) = 0 describing the volume preserving multidimensional systems, the corresponding metric here being ( N F ) g = e11e22e33-elle32e 23 +e12e31e23-e12e21e33 +e13e21e 32 +el3e 31e 22. For connections to RH problems see [406, 902, 912]. For the volume preserving case consider two solution sets L, = f l ( L , M , N ) ; ]f~ = f 2 ( L , M , N ) ; IV = f 3 ( L , M , N ) (8.51) where fi()~,P,q) are arbitrary holomorphic functions defined in a neighborhood of A = oc. Assuming ( N G ) { fl, f2, f3 } = 1 one has a kind of RH problem and s d i f f (3) symmetries acting on (fl, f2, f3) can be lifted to (L, M , N ) and (L, /1~/, /V) via RH factorization.
3.9 3.9.1
MISCELLANEOUS Geometric
quantization
and Moyal
It would be remiss not to say something more about geometric quantization (see e.g. [119, 281, 406, 448, 809, 883, 888, 982] for general ideas and for connections to noncommutative geometry cf. [142, 194, 195, 597, 627, 941, 942, 983]). We sketch here some material from [406] which connects geometric quantization (GQ) to Moyal deformations (cf. also Section 6.1.4 and Section 6.3). One follows ideas on symplectic groupoids developed in e.g. [969]. Thus if M is a manifold with symplectic form w let M be the symplectic manifold (M,-w). A groupoid is a set with a partially defined associative multiplication and a symplectic groupoid is a pair of manifolds (G, G0) where G has a symplectic form ~ and a partially ordered multiplication with domain G2 C G • G together with two submersions (~ 9 G --. Go and ~ 9 G -~ Go and an involution x -~ x* of G such that (A) The graph AA = {(x,y, xy)" (x,y) C G2} of the multiplication is a Lagrangian submanifold of • G • G; (B) The set of units G0 may be identified with a Lagrangian submanifold of G (identified with Go); (C) For any x E G one has c~(x)x - x - x~(x) and (~(x) - xx* with ~(x) - x*x - moreover (x,y) E G2 if and only if ~(x) = (~(y); (D) The graph I - {(x,x*) 9 x E G} of the involution is a Lagrangian submanifold of G • G; and (E) Whenever (x, y) and (y, z) belong to G2 then (xy, z) and (x, yz) lie in G2 and (xy)z - x(yz) (note Go ~ objects in G - cf. Section 6.1.4). We recall that Lagrangian submanifolds are characterized by the vanishing of the symplectic form (cf. [720]). As a consequence of these postulates one finds that c~(x*) - ~(x), c~(x)* = a(x) = c~(x) 2, and ~(x)* - ~(x) - / 3 ( x ) 2.
3.9. M I S C E L L A N E O U S
161
Also x x * x = a ( x ) x = x, a ( a ( x ) ) = a ( x ) and /3(/3(x)) = fl(x). Further if ( x , y ) E G2 then ( R A ) a ( x y ) = x y y * x * = x a ( y ) x * = x ~ ( x ) x * = xx* = a ( x ) and/3(xy) = a ( y * x * ) = ~(y*) = ~(y).
Consider M = R 2n with co a nondegenerate alternating bilinear form. Writing &(u) : v ~ co(u, v) gives a skewsymmetric invertible map & : R 2n ~ (R 2n)* and one writes
ft(x, y), (z, w)) = co(x, z) - co(y, w) = &(x - y) z +
-&(z-w)Ix+Y]2
(9.1)
On the other hand R 2n x R2n can be identified with the cotangent bundle T * ( R 2n) and if (u, r (v, X) are elements of R 2n x (R2n) * regarded as local coordinates of covectors in T * ( R 2n) the cotangent symplectic structure reduces to the alternating bilinear form ( R B ) E((u, r (v, X)) = X ( u ) - r Thus R 2n • 1~2n can be identified with T * ( R 2n) as symplectic manifolds by the linear isomorphism (RC) (I): (x, y) ~ ((1/2)(x + y ) , & ( x - y)) for which (I)*E = ft. Prequantization for a 2n-dimensional symplectic manifold (M, co) proceeds by finding a real linear map S ~ ] from the Poisson algebra of smooth functions on M to an algebra of operators on the Hilbert space L2(M) for which i = I and {f,g} = (i/h)[f,~0}. The right recipe is ] = f - i h V x s where X I is the Hamiltonian vector field of f and V x = X - ( i / l i ) O ( X ) where 0 is a symplectic potential, i.e. a one form for which dO = co. When co is not exact local potentials must be patched together so that V becomes a linear connection on a Hermitian complex line bundle L --~ M whose curvatur form is - ( i / h ) w . The elements of the prequantization Hilbert space are sections s E FL of this bundle. Geometric quantization then involves finding a positive polarization of (M, co), i.e. a subbundle F of the complexified tangent bundle T * M c which is maximally isotropic for co with F n / ~ of constant rank; one should also have integrability in the sense that both F and F n F are closed under the Lie bracket and positivity in that -ico(I7", Y) _> 0 when Y is a section of F. A polarized section is any s E FL for which V g s = 0 for Y E F F . The quantizable observables are those g E C ~ ( M ) for which ad(Xg) preserves r E . Then one checks that ~0 preserves the space FFL of polarized sections. The remaining problem is to endow FFL (or some modification thereof) with a suitable inner product so that the quantizable observables be represented as operators in a Hilbert space. This is done by using the idea of a half-form pairing. One follows here [812] (cf. also Section 6.3) and takes the canonical line bundle of F as K F = AnF ~ where F ~ C T * M C denotes covectors which vanish on F. For example if M is a Kghler manifold with local holomorphic coordinates ( Z l , " - , Zn) and F is spanned by 0 / 0 2 1 , . . - , O/02.n then K F is spanned by dzl A ... A dzn and F n / ~ = 0. A contrasting example for which F is a real polarization, i.e. F = F, is obtained by taking local Darboux Coordinates ( q l , " ' , q n , P l , " " ,Pn) for M with F spanned by 0 / 0 p l , . . . , O/Opn whereupon KF!is spanned by dql A . . . A dqn. Now suppose we have two positive polarizations F and P; it turns out that K F and K P are isomorphic as line bundles over M and that B2F | K P is a trivial bundle. There is an obvious map from this bundle to A2nT*M c (replace tensor by exterior product) which is an isomorphism if and only if F N P = 0. The Liouville volume )~ = (-1)n(n-1)/2w^n/n! trivializes the latter bundle. Hence one has a pairing< c~,/3 > of e r K F and ~ E r K P defined via (RE) i n < a,/3 > = ~ A/3 provided F n P = 0. In particular if P N F = 0 then < , > is an inner product on F K F. Matters are less straightforward if F n P 7~ 0. Thus set /~ N F = D c where D is an isotropic sub-bundle of T M . If D • is the symplectic orthogonal of D then D • becomes a symplectic vector bundle with an induced symplectic form wu of which F / D and P / D
162
C H A P T E R 3. C L A S S I C A L A N D Q U A N T U M I N T E G R A B I L I T Y
are nonoverlapping maximal isotropic sub-bundles. One obtains then as before a pairing of
K F/D and K P/D. One can try to pull this back to a pariring of K g and K P by suppressing the common real directions in D. Thus suppose that the foliation of M induced by D has a s m o o t h space of leaves M / D , t h a t D is spanned locally by O / O y l , . . . , O/Oyk, and t h a t ( x l , . . . , xk) are conjugate local coordinates to ( y l , ' " , Yk). If c~ = adxl A . . . A dxk A dzl A 9.. A dZn-k E F K F and/3 = bdxl A . . . A dxk A dwl A . . . A dwn-k E F K P where the coefficient functions do not depend on the yj, then one can define c) = ad2l A . . . A dzn-k E F K F/D and/3 = bd@l A . . . A d(Vn-k E F K P/D, where the tildes denote corresponding coordinates on M / D , and try to set < c~,/3 > = < (},/3 >. However this step is coordinate dependent and (after incorporating a correction factor of A2) the change of variables formula shows that the result is a 2-density on the leaf space M / D . Since we could integrate a 1-density over M / D to get a scalar valued inner product one abandons K F in favor of the line bundle QF of half-forms on M, which is defined by the requirement that QF | = K f . For c~ E F K F we write x/~ = # E FQ F if # | # = c~. Then (~F | QF carries a pairing, whose values are l-densities on M / D , determined up to a sign by the requirement < v/d, ~ > 2 = < c~,/3 >. One can circumvent the problem of existence of QF, which requires metaplectic structures by using techniques of [824] requiring M p c structures which always exist (this is akin to passing from spin to spin c structures on Riemannian manifolds). Finally one replaces the prequantization bundle L by L | g with F F ( L | QF) its polarized sections (those killed by V y for Y E F F ) . The pairing of two sections s | E F F ( L N Q F) and t| v/-/~ E F p ( L | P) is given by
< s|
v/-d, t e v/-~ >= f (s,t) JM /D
(9.2)
where ( , ) is the Hermitian metric on L. W h e n F = P the geometric quantization space is obtained by completing FF(L @ QF) with respect to this inner product. Now one describes relations between pairings and the Weyl correspondence. On the symplectic manifold G = R 2n take coordinates (x',x") = (X~l,...,xn, X~, .,Xn) so aJ dx' A dx" = ~ k dx~ A dx~ and for convenience take n = 1. Then w is a bilinear symplectic form on R 2 with co((x,x'),(z,z')) - x ' z " - x " z ' (cf. [720]). The symplectic groupoid G = R 2 x I~ 2 has coordinates (x/, x"; y~, y") with which its symplectic form may be written as ( R F ) a = 7r~co- 7r~c0 = dx' A d x " - d y ' A dy" so (x', y', x " , - y " ) are Darboux coordinates for G. On T * R 2 one uses Darboux coordinates (ql,q2,Pl,P2) with E = dql Adpl +dq2 Adp2. T h e n q~ in ( R C ) is given by /
x ~ + y~
ql = ~
; q2 = x' - y'; Pl - x" - y"; i02 =
x" + y .... 2
9 9
t/
(9.3)
Consider two real polarizations of G, namely ( R G ) F = span[(O/Ox"), (O/Oy")] and P = span[(O/Opl), (O/Oq2)]. Then from (9.6) one has ( R H ) o/opl = (1/2)[(0/0x")- (O/Oy")] with O/Op2 - (O/Ox") + (O/Oy") so one can rewrite F - span[(O/Opl), (O/Op2)]. Therefore /~MP = D C where D is spanned by O/Opl. With abuse of notation one can regard (ql, q2, p2) as local coordinates for the leaf space G / D and the pairing FQ F x rQ ~ -~ D I ( G / D ) is determined by (RI) < v/dx'A dy ~, v/dql A dp2 >= dqldq2dp2. The polarized sections in FFL are of the form fso where f E C ~ ( G ) and so is a nonvanishing section of L satisfying V x s o = --(i/l~)OF(X)so and (so, so) = 1. The symplectic potential O f for (G, f~) may be taken to vanish on F and thus ( R J ) OF -- - x " d x ~ + y"dy ~ = - p l d q l -p2dq2. In this case fso E FFL if and only if X f = 0 for X E F, i.e. f = f(x', y'). Similarly if to is a section of L satisfying V z t o = - ( i / h ) O p ( X ) t o and (to, to) - 1 with ( R K ) OR - - p l d q l + q2dp2 being the symplectic potential which vanishes on P, then a typical element of F p L is of the
3.9. MISCELLANEOUS
163
form gto with g = g(ql,p2). Clearly to = r for a nonvanishing r C COO(G); indeed from Vxto = (XO0)s0 + r one obtains (RL) (dr162 = ( i / h ) ( O F - OR) = -(i/h)d(q2p2) and so r = Cexp(-q2p2/h) for some positive C. Since (x0, to) = r we can now compute the half form pairing of a = f(x', y')so | v/dx ' A dy' and fl = g(ql, q2)t0 | v/dql A dp2 as (9.4)
< a,/3 > = C f f(x', y')g(ql,p2)e-iq~P2/hdqldq2dp2 -
= C
/ f(x',y')g
2
,p2 e~P~(Y'-X')/hdp2dx'dy ' =< f, Tg
where
Tg(x', y') - C
>L2(R 2
/(x'+y') g
2
' ~ eii(Y'-X')/hd~
(9.5)
is the kernel of the operator on L2(R) whose Weyl symbol is g. Unitarity of T is achieved by taking C = (27rh) -1. In other words the pairing of the nontransverse polarizations F and P of the symplectic groupoid R 2 • R2 yields the well known correspondence between kernels of Hilbert-Schmidt operators and the Weyl symbols of these operators. Thus the groupoid forms a bridge between conventional QM and the phase space formalism. The importance of symplectic groupoids in general is that the partial multiplication in G induces an associative product of polarized sections so that the geometric quantization Hilbert space becomes in fact a Hilbert algebra. By suitably modifying its topology one can obtain a C* algebra in the spirit of noncommutative geometry. On the other hand the basic idea of Moyal quantization is that by working with functions on phase space rather than wave functions one may describe both states and observables of QM systems in classical terms; thus phase space functions are to be equipped with a noncommutative product which gives the quantum formalism directly without invoking a Hilbert space a priori. We will see now how the Moyal product of phase space functions is inherited from the groupoid structure of R 2 • R2 equipped with the polarization P of ( R G ) . Thus for any groupoid G one defines a convolution product (for Az a suitable measure on the set {(x, y) E G2; xy = z})
(f 9g)(z) = fx
y=z
f(x)g(y)d)~z(X, y); (f * g)(x, y). = I'M f(x, t)g(t, y)d)~(t)
(9.6)
where G = M • ~ / a n d )~ = )~x,y is a multiple of the Liouville volume on M. When G has a real polarization with a regular leaf space the polarized sections are represented locally by functions covariantly constant along the leaves; in general their convolution products will however fail to be covariantly constant. To obtain a new polarized section one must average over the leaves by integration; by projection one then recovers a twisted product of functions on the leaf space. In the case of G = R 2 x 1~2 the diagonal A = { (x', x"; x', x") } E G; (x !, x/~) E R 2} is a Lagrangian submanifold of G which is transverse to the leaves ql -Cl; q2 = c2 of the polarization P; thus a polarized section is determined by its values on A and one may identify A with the leaf space G/P. Now look at (9.3) as a linear change of variables and rewrite the groupoid product ( R M ) (x', x", y', y") = (x', x", t', t")-(t', t", y', y") in a more suitable form. Thus substitute
1 q=~(x'+
y,
);
ql
1 =~(~'+t');
q,,
1 -~(t'+y');p-~
l(x,,
+ y") ;
p, = 21(x,, + t,,) ; p,, = 21(t,, + y,,) ; ~ . x" . y";. ~' . X" . t"; . ~"
r/=y~-x~; ~7t = t t-x~; r / " = y ~ - t ~
t"
(9.7) y,""
C H A P T E R 3. CLASSICAL A N D Q U A N T U M I N T E G R A B I L I T Y
164
Now ( R M ) takes the form (RO) (q, p, ~, r/) = (q', p', ~', r/'). (q", p", ~", r/") determined by the four relations (RP) q = (1/2)(q' + q " ) - (1/4) (r/' - r/") with p = (1/2)(p' +p") + (1/4)(~'-~c'') and ~ = 2(p' - p"); r/ = 2(q" - q'). Then a(q,p,~,rl) = (q - (1/2)r/,p + (1/2)~) with fl(q,p,~,rl) = (q + (1/2)rl, p - ( 1 / 2 ) ~ c) so the product ( R O ) i s subject to the compatibility conditions ( R Q ) q ' + (1/2)r/' = q " - ( 1 / 2 ) r / " and p ' - ( 1 / 2 ) ~ ' = p" + (1/2)~". We can intepret the coordinate changes (9.7) as follows" The parameters (q,p) label points of the leaf space G/P (since A is the submanifold ~ = r / = 0) while (~, r/) are parameters along the leaves. Since (x',x",y',y") = ( q - (1/2)~,p + ( 1 / 2 ) r / , p - (1/2)~) each leaf carries a natural volume form 2-4dr/A d~. The pointwise product of two functions on G representing sections in Fp(L | QP) is (RR) (2rh)-2g(q',p')exp[-ip'rf/h]h(q",p")exp[-ip"rl"/h] which is of the form f (q, p, q', p', q", p")exp[-iprl/h] with
f(q,p,q',p',q",p") = (27rh)-2g(q',p')h(q",p")e -(i/h)(p'n'+p%''-pv =
(9.8)
= (27rh)-2g(q ~,p~)h(q", p~)e -(2i/h)(pq'-qp'+p'q''-q'p''+p''q-q''p since (RQ) and (RQ) imply (RS) r / = 2 ( q " - q') with r / ' = 2 ( q " - q) and r / " = 2 ( q - q'). The twisted product (g x h)(q,p) is thus an integral of (9.8) over (A) The parameter region (t',t") E R 2 determined by (RQ) which underlies the prequantized convolution product and (B) The leaf of P through the point (q,p) E A which is parametrized by ( q - (1/2)r/,p + (1/2)~). Since
1 l q,,) p,,) dt' A dt" A (2-4dr/A d~) = -4d(q + A d(p' + A d(q" - q') A d(p' - p") = = dq I A dq" A dp I A dp"
(9.9)
one arrives at (g • h)(q, p) = (27rh) - 2 / R a g(q', p')h(q", p") • e - (2i/h)(pq'-qp'+p'q''-q'p''+p''q-q''p)
x
(9.10)
dq~dq,dp ~dp"
which is the Moyal product of the symbols g and h. Thus the geometric quantization data (G, ~, P) incorporates the essentials of Moyal quantization in the linear case.
3.10
SUMMARY
REMARKS
The material in this chapter is complicated and we will try to summarize briefly what has been covered. Thus 9 Section 3.1 involves general comments about quantization via deformation of A C ~ ( M ) which for C~162 leads to PL groups and Hopf algebras. 9 Section 3.1.1 gives examples for the NLS in Hamiltonian form of classical integrability via inverse scattering and a quantum version is also sketched. R and r matrices are introduced in a pragmatic spirit. 9 Section 3.2 develops PL structures, r matrices, and YB equations. 9 Section 3.3 indicates the Weyl quantization (developed later in Section 3.8) but mainly sketches quantization for PL groups associated with CYBE and MYBE. 9 Section 3.3.1 is on quantized matrix algebras.
3.10. S U M M A R Y R E M A R K S
165
9 Section 3.4 on the algebraic Bethe Ansatz looks at the XXX1/2 model. The trace F()~) of the monodromy operator generates N - 1 commuting operators (including the Hamiltonian H) and one has a quantum integrable system (QIS). The Bethe Ansatz equation (BAE) of (4.20) is crucial for a detailed study. In the text we have only shown calculational aspects; one looks for eigenvectors in a particular form and wants to show completeness of BA (which means that all eigenvectors have this special form). For G = SL(2) it is shown in [356] that the existence of an eigenvector of the Gaudin operators with given eigenvalues (cf. Section 3.5.3) implies that the corresponding projective connection generates a trivial monodromy representation of p 1 / m a r k e d points. In fact this is precisely equivalent to the B AE and enables one to prove completeness of BA in the SL(2) Gaudin model. In [356] Frenkel deals with affine algebras, Langlands duality, and BA (cf. also [322, 323, 324]). For rational curves X and G = SL(2) the geometric Langlands correspondence amounts to a separation of variables (SOV) in the Gaudin system (as an alternative to BA). The Langlands correspondence is developed in [356] (which is readable) and we do not pursue the details here. It provides a natural framework in which to view SOV and BA. 9 Section 3.5 discusses classical and quantum SOV and gives some examples. As just mentioned above one should view BA and SOV in the Langlands context for a deeper unified picture. Hopefully someone will write a book about this.
9 Section 3.6 gives just a sketch of topics related to Hirota equations (again a book on this subject is needed). The equation (6.1) unifies all known (classical) continuous soliton equations and it also arise in QIS theory as a fusion relation for the transfer matrix (,,~ trace of the quantum monodromy matrix). Further in [603, 607, 994] one encounters the BAE, which is usually considered as a tool in quantum integrability, arising as a result of solving classical nonlinear discrete-time integrable equations. This suggests intriguing connections between integrable QFT and classical soliton equations in discrete time. We mention also [736] where generalizations of the tau function and integrable hierarchies are interpreted in a group theoretic framework with an immediate quantization procedure. This is all very rich and one expects the tau function idea (suitably extended) to embody enough combinatorial information for unification of QIS and classical integrable systems (CIS). 9 Section 3.7 gives a good illustration of the naturalness of Hitchin systems and classical integrability via a connection to (classical) SOV. 9 Section 3.8 on deformation further developments.
quantization
is more or less self explanatory
and presages
9 Section 3.8.1 is very limited but should be self explanatory; obviously more new results on this can be expected. 9 For connections of quantum groups to physics one is well advised to study [672] where many fascinating ideas are developed. In particular the theme of Hopf algebras as a setting for the unification of QM and gravity is expounded. Plank scale physics is discussed in the quantum group context as well as quantum random walks and combinatorics. There is also a nice discussion of q-deformation as natural for QM; the transition from differential to q-difference operators as a controlled discretization process is indicated. The papers indicated in [672] came out just as this book was being finished. They go much further in terms of quantum groups, noncommutative
166
CHAPTER 3. CLASSICAL AND QUANTUM I N T E G R A B I L I T Y
geometry, and the foundations of quantum physics; in addition they provide fascinating reading and marvelous speculation. 9 Another road to the future is developed in [825] which deals with the left spectrum of an associative ring and the first natural steps toward a theory of noncommutative affine, quasi-affine, and projective schemes. The technology becomes very category and scheme theoretic. Some background to this appears in e.g. [680] where noncommutative algebraic geometry is broached in contrast to noncommutative differential geometry and topology. The question of localization is crucial and one essentially replaces the idea of a space by a category of sheaves.
Chapter 4
GEOMETRY
DISCRETE MOYAL 4.1
AND
INTRODUCTION
The main themes here are discrete geometry and its connection to integrability and QM (references are given as we go along) and a series of papers by S. Saito and collaborators concerning strings, Moyal, Hirota, integrability, discrete geometry, M theory, etc. (cf. [536, 537, 538, 539, 540, 766, 828, 829, 830, 831,832, 833, 834, 835, 836, 837, 838, 839, 840, 962]). The constructions in this section will amount to a collection of formulas and motivation. Notations such as (15A), (17A), (19A), (20A), (21A), and (23A) will be used. 4.1.1
Phase space discretization
We go now to [536, 537, 538, 539, 828, 829] for an interesting version of Moyal (M) quantization via discretization of the classical phase space (cf. also [307, 697]). This leads to a difference operator approach to M quantization with applications to integrable systems. First one shows that discretization of the phase space leads to a natural definition of a difference analogue of vector fields and the M bracket follows as a difference version of the Lie bracket in place of the P bracket. Thus (we use the notation of [539] here which refines [536]) define
CA0 e-~o lsinh(ikO) (0 Ox); 2)~ )~ 1 V~ = -~sinh(.k ~ aiOi) (Oi "~ O/Ox i) w
_
_
(1.1)
The aj correspond to indices j of local coordinates xJ and one defines ( 1 5 K ) X D = f ddv~(~,d)V~ (note V~ is not written as a vector and f dd ,.~ f l-I daj - the lattice vectors d play the role of indices j for local coordinates xJ). A particular space discretization corresponds to v~(~,d) = 5 ( d - d o ) . A difference 1-form is now defined via ( 1 5 L ) ~ D = f ddw~(2, d)A ~ where < A 6, V~ > = 5 ( b - d). Thus
< ~D,xD >--/ddfdb<
w~(2, b)A6, v~(~, d)V~ > - f ddw~(~,d)v~(i,d)
(1.2)
The operator A ~ can be realized via
e-;~ 5 : 2/~E e-A(2n+l)a'O
,~csch[A(d. 0)] = e ~ 5 --
0
167
(1.3)
CHAPTER 4. DISCRETE GEOMETRY AND MOYAL
168
(so < A g, Va > = 5 ( b - d ) and again we emphasize that A ~ = A ( E aiOi) is not a vector). Then with integrals f dd ~ f 1-Idaj (see also below) and omitting vector signs occasionally
iV~(A a A A z) = 5(7 - a ) A z - 5(-y - / 3 ) A a ; s
= A . ixD + ixD 9A
(1.4)
Note AA = 0 since [Va, VZ] = 0 and (15M) (/k ax A/~a2)(~Tbl , ~7b2 ) : det One considers now the phase space ~ = (p, x) and in place of (-.) Xlg = (fpOx - fxOp)g one writes (15N) X ~ = f dalda2v~[f](x,p, al,a2)Vg where
v~[f](x,p, al,a2) =
~
dbldb2exp[-i)~(alb2-a2bl)]f(x + )~bl,p+ )~b2)
(1.5)
This is a better notation than in [536, 537] where factors of ~ are treated differently and alb2 -a2bl is written as ~ • b (cf. however the notation in (1.26) for example coming from [930]). We note that (_1/)~)(~ • b) is the area in the units ~ of the parallelogram in the phase space formed by g • b. The symplectic structure in ( . . ) is then retained via interchange of g and b. From (15N) and (1.5) one obtains now
X ? g = -~Sin['~(OxlOp~ i - cOplOx~)]f(pl,xl)g(P2, X2)l(p,x) = {f,g}M
(1.6)
(in [539] there is a misplaced i) and thus X ? can be regarded as a difference operator representation of the M bracket. Note in [537] one defines Va = (1/,X)Sin(~ E aiOi) which causes some problems in comparing formulas). To obtain (1.6) we note that
X~g=
-i)~
(2rr)2 / / d a l d a 2 /
f dbldb2x
(1.7)
xSin[~(alb2 - a2bl)]f(x + )~bl,p + )~b2)exp[)~(alOx + a2Op)]g(x,p) since ( 1 5 N ) can be written as X ? = (27r)2
dalda2
dbldb2Sin[)~(alb2-a2bl)]f(x + ~bl,p+ )~b2)e"x(a~O~+a2Op) (1.8)
Note that the philosophy involving ai running over R = (-oc, oc) is crucial to the calculations. Then for a --, - a one has Va ~ --Va from (1.1) and f_~ exp(-i~a)Vada = f_~ ezp(i)~a)Vada (with obvious notation). Hence one can write 2 f ezp(-i)~ab)Vada = -2i f sin()~ab)Vada so -
X}9 =
(27r)2
da
oo
dbSin[~(alb2-a2bl)]f(x+)~bl,p+)~b2)Sinh[)~(alO~+a2Op)](1.9) AaOx
oc
Also f~ Sin(,Xab)Sinh(~aOx)da = (1/2)[f_~cr daSin(ikab)e - f-oc daSin('Xab) e-~a~ with -fo_c~ daSin(,Xab)e_)~aOx = fo_~ daSin(,Xab)e~aOx. Consequently (1.7) and (1.8) are verified. We obtain then by Jacobi (cf. below) ( 1 5 0 ) [X}9 X D] = X D For (1.6) now write ' {/,9}M" from (1.8) =
(2rr)2
da
dbSin[A(alb2 -a2bl)]f(x + A b l , p + Ab2)g(x + Aal,p+ Aa2) (1.10)
4.1. INTRODUCTION
169
Now consider
/da/dbei)~(alb2-a2bl)f(x + Abl,p + Ab2)g(x + Aal,p + Aa2)
(1.11)
and set e.g. x + Aal = al and p + Aa2 = a2 so (1.11) becomes --/~21 f / dadbei[b2(al_x)_bx (a2-p)] f (x + Abl, p + Ab2)g(o~l , o~2)
1/(f(x
-'- ~-'~
+ i~00~2, p -- i~(~1)
=
/
)
ei[b2(~176
g((~, c~2)d~ =
(1.12)
(f(x + iAOa2,p - iA0al)5(al - x,c~2 -p))g(Otl, ol2)do~ ~-~ =
f ( x - iAOp,p+ iAOx)g(x,p) ~
g 9f
Similarly putting - A in the exponent in (1.11) leads to (23H)(27r/A)2f(x § iAOp,p-
iAOx)g(x,p) which corresponds to (27r/A)2f 9g; note here (cf. (1.19) below) f * g = exp[t~(OxlOp2 -- Ox2Opl)]f(xl,Pl)g(x2,P2)l(x,p)
(1.13)
--~
= ~ - ~ ( - 1 ) r ~ r+s Or+sf Or+Sg
0
r!s!
OxrOpsOprOx s = f e ~ ( O x O P - O P O ~ ) g - - g ( x - ~ D P ' X - ~ O x ) f - -
= / ( x + ~Op, x - ,~o~)g = ~ 7. 0
(-~)~
Then the Moyal bracket is often defined via (cf. ~ ~
{f,g}M=l{fsin[~(OxOp = ~ (-1)s~2s 0 (2s + 1)!
[~176176176
g]
0
[343, s91])
~ ~
1
OpOx)]g}=-~(f .g--g.
f)--
(1.14)
[rgJ:92s+~-j f][0;2s+ -Jc~pg] o (-1)J 2s J+ 1 Lvxvp
(note also Sinh(iz) = iSin(z) and Sinh(z) = -iSin(iz)). Hence (1/2i)(-iA/(27r)2)[(1.11) (23H)] gives -1 X f g -- -~-~(g * f - f * g) -- { f , g } M (1.15) according to (1.14) (cf. also (2.69)). Now for ( 1 5 0 ) consider
X I Xg =
(27r)2
da
db
de'
db'Sin[A(alb2 - a2bl)]Sin[A(dlb~2 - ar2b~l)]• (1.16)
• f(x + Abl,p + Ab2)g(x + Ab~ + Aal,p+ Abe2+ )~a2)eA(a'~Ox+a~20p In [539] one suggests using here the addition relation 2Sin(A)Sin(B) = C o s ( A - B) Cos(A+ B) so that one would have exponentials (23I) exp[:l:iA(alb2-a2bl • (a~b~- a~b~))] but direct calculation is complicated. In fact there is a better approach based on the Jacobi property ( { , } - { , }M)
{{f,g},h} + {{h,f},g} + {{g,h},f} = 0
(1.17)
CHAPTER 4. DISCRETE GEOMETRY AND MOYAL
170
(cf. [59, 65, 312, 313]). We know X~g = {f,g}M from (1.15) so consider
[Xff, XD]h - Xff{g,h} - x D { f , h } ----{f, {g, h}} - {g, {f, h}} = {{f,g},h} = X~,g}h
(1.18)
as desired. We note in passing also (cf. (1.13)-(1.14))
or+sg g(xWt~Op,p--t~Ox)f = E O-~-~ps(--1)s(nOP)r(gox)sf = g ' f = f(x--~Op,p+~ax)g (1.19) In any event (1.6) shows that X~)g can be regarded as a difference operator representation of the M bracket; moreover, given ( 1 5 0 ) , the X}9 form an infinite dimensional Lie algebra. Further one can define a 2-form
f~= ~---~//dalda2//dbldb2ei)~(alb2-a2bl)/kaA/k b
(1.20)
satisfying (15P) ixyf~ = A f which is analogous to ixsco = dco for a symplectic form a~. Consequently (1.20) is interpreted as a difference version of the symplectic form. R E M A R K 4.1. In [310, 311,344] for example the same algebraic structure as in ( 1 5 0 ) is studied with shift operator generators =
l__l_f ( p -
i O ,x +
=
1
fmneimp+inxeA(mOx_nOp)
(1.21)
for f(p,x) = ~ fmnexp(imp + inx). The algebra of ( 1 5 0 ) is a subalgebra consisting of antisymmetric combinations of generators (cf. (1.8)) and ( 1 5 0 ) reproduces some of the results of [68, 789, 891,910, 913]. For quantum mechanics (QM) now the "ad hoc" replacement of P brackets by M brackets for quantization (instead of the natural Lie brackets which don't work) is substituted by a more geometrical direct discretization of phase space wherein the natural difference analogue of Lie brackets leads directly to the M bracket. In [539] one identifies now ~ ~ 5/2 and defines XAQ = hX~ leading to
X Q = ihOp; xxQ = 2ihXOp; XpQ =-ihOx; X~ =-2ihpOx; ...
(1.22)
The P bracket of physical quantities is then replaced by the commutator ( 1 5 0 ) and one obtains for the time evolution of a physical variable A(x) the equation (15Q) -il~(d/dt)XQA = [XAQ,XH Q] (both A and g can contain h). This is then equivalent to ( 1 5 R ) ( d / d t ) d {A,H}M and via the Weyl correspondence this is equivalent to the Heisenberg equation (cf. Section 4.4). We recall that if Fw is the Wigner distribution function with f Fwdx = 1 then < .4 >= f dxFwA where A ~ .4. In order to express this in the difference operator framework we recall that the association of an operator X D with a c-number follows (1.2), namely, choosing w,x in (15L) as
W)~ = /~2/ f dbldb2ei'x(albz-azbl) f(x nt-/~bl, p + )~b2)
(1.23)
the inner product in (1.2) with X D becomes f dxf(x)g(x). N o w a sum corresponds to an observable A and the Wigner distribution should then be expressed in terms of some difference 1-form which is stipulated to be
hff/ PEw = -~
dalda2 / /
h h b2) / k a dbldb2eih(alb2-a2bl)/2Fw( x + -~bl,p+ -~
(1.24)
4.1. INTRODUCTION
171
The dimensions here are dim[all = dim[bl] = (MLT-1) -1 and dim[a2] = dim[b2] = L -1. The inner product between XAQ and PEw becomes then (15S) < PFw,X~ > = f dxdpFw(x,p)A(x,p) and the right side is nothing but the expectation value of .4. In the case of a pure state ]r > one gets (15W) < PF~,X~ > = < r >. From this we consider the time dependence of < PFw, XAQ > which can be twofold. In the Heisenberg picture one has < PFw,XQA > t = < PFw,XQA(t) > while in the Schrhdinger picture the last term will be < PF~(t),XQA >, and these two expressions must be physically equivalent. Hence the solution of (15Q) becomes
XQA(t) = exp[-iXQi_It/h]XQAexp[iXQHt/h ]
(1.25)
which corresponds to a solution of (15R) of the form (15U) A(t) = exp[iXQHt/h]A. Now in the Heisenberg picture d
Q
(1.26)
- i h - ~ < PFw,XQA(t) > = < PEw, [XQA(t),XQH] > = < PFw,X{A(t),H}M >
and one can show that the right hand side equals < P{H,Fw(t)}M,XQA > upon defining (15V) Fw(t) = exp(-ihxQt/h)Fw. Consequently (15W) (d/dt)PFw(t) = P{H, Fw(t)}M which determines the time evolution of the quantum state in the Schrhdinger picture and is equivalent to (15X) (d/dt)Fw(t) = {H, Fw(t)}M (see also remarks in Section 4.3.1).
4.1.2
D i s c r e t i z a t i o n and K P
Following [537] (cf. also [828]) now consider a complex phase space (z, ~) (in place of (p, x)) and functions (17A) A(z,~) = E Em,~ezam~zmC n (cf. also [310, 311, 538, 540]). In this case X D becomes (note here (17AA) v~[A] ~ (A2/nTr2) f dbexp[-iA(a x b)]A(x + ib) and Va ~'J (1/)~)Sin()~a. 0))
1
X D = -~ ~ amnzm4nsin m,n
{ A(
o
o))
nolog(z) - molog(~)
= E amnzm~nVmn m,n
(1.27)
The factor V m n plays the role of •a in the last section with say a ~ (m, n). If we define the "components" xDn of X D by zrn/'nv~ mn then these constitute the basis of a Lie algebra
[Xmlnl , z ~ ] = D
D
1
Sin {)~(nlm2 - n2ml)}XDml+m2,nl+n2
(1.28)
The XDmn satisfy ( 1 7 B ) [ X ~ , X g] = - X { l , g } , and X ~ g = --{f,g}M (cf. ( 1 5 0 ) - an / i s removed in defining X}9 - cf. (1.6)). Here
1
{f , g}M = --~Sin
{(
o
o
)~ Olog(zl) Olog(~2)
0
0 )}
Olog(~l ) Olog(z2)
x f(zl, ~l)g(z2, ~2)[(z,~)
x (1.29)
For a Wigner function one takes now (cf. remarks after (1.39))
FKp(z, ~) = / dx E ~2(qt/2z)~*(q-e/2z)~-e f.EZ
(1.30)
CHAPTER 4. DISCRETE GEOMETRY AND MOYAL
172
where the parameter A in (1.27) ... -ilog(q) (Iql < 1 - or sometimes later )t = log(q) for convenience or adaption to other notation) and ~ (resp. r is the BA function (resp. adjoint BA function). Here one thinks of a Wigner function
Fw(p,x) = ~
dye x+ ~y
x - -~y e -ipy
(1.31)
(cf. Chapter 1, (1.1) and Section 4.4); the y integration corresponds to the summation over t~. The integration over x does not exist in (1.31) but the BA function has x dependence via
~(z, t) = 1 + ~ wjO -j exp
trz r
= W
1
ezp(~)
(1.32)
(with x = tl and L~ = z~) so in (1.30) the x variable is simply smeared out (presumably to isolate the time variables and the phase space variables are (z,r - note in d K P the phase space variables are (x,p) with p ~-, 0x). Next a 1-form f~g~p is defined as follows. First note that (1.30) can be formally expanded as FKp = ~ m ~ f ~ z - ~ r -~ (since e.g. r ~ ~ ~n#-n). Then introduce the dual basis A mn of Vmn and define
(1.33)
aFKP(Z' r -- E fmnz-m~-nAmn mn The expectation value is then
d(FKg(z,r162 <~F~,xJ ~>=- I ~dz i 2~ir via < A mn, ~Tn,m, > - ~mm,~nn,. In particular if we use the
(1.34)
XDn one gets
finn =< f~Fgp,XmD >---- f dx i 2~iz zmr162
(1.35)
We recall now the auxilliary flows generated by using the Orlov-Schulman operator M in conjunction with L (cf. Section 1.1.1) and see [7, 10, 58, 163, 222,223, 424, 742,786,787. 788] for further information). Also considerable enhancement is given in Section 4.2.4 after (2.70); the formulas below are extracted heuristically from [537] with some corrections. Thus following in part the notation of [537], L r = zr Ore = Lr+r r = Wexp(~) for ~ = ~ trz r, and Ozr = M e where (17C) M = W ( ~ rtrOr-1)W -1. One writes (17D) z k ~ r ~ MeLkr for k E Z and g E Z>0 (note from [222] (233) LkMer = ~ z k r and MeLkr = zkJzr but recall [L,M] = 1); then one defines vector fields (17E) Oker = -(Menk)-~2. There is a Lie algebra isomorphism zk~ -~ Oki with
[Ok~,Ok,e,] = D { ( ~ ) (:)--(kj')(~)}jIOk+k'-j,~+~,-j Since the BA functions ~, ~* can be written in terms of vertex operators r r
t) = t) =
v(z)
r
_
-
(1.36)
If, V* as
T(t)
where t • (1/[z]) ,,~ tl • ( l / z ) , t2 • (1/2z2), .. ., one obtains the action of function via
Om+,,,Olog(T) =
(1.37)
_-
i ~dz(zm+,~+(z)) ~O,(z)
Ok~ on the tau (1.38)
4.1. INTRODUCTION
173
(cf. Section 4.2, (2.76) for more details). Now the right side of (1.35) becomes the expression (17F) f dxDmn (--(O/Ox)log(T)). To see this one writes (A = log(q) here but A = -ilog(q) is also used for calculation at times, depending on context- cf. (1.30), (2.84), (2.95)- (2.96), etc.).
--/dx/~dz
cc : _:m,.z
@mqn[Zo.+(m/2)]@(z))@*(z) =
(n)~)j
,. /,x /
(1.39)
dz
0 ~ (n)~)J = f dx [qnm/2Ecje j! Om+e,e (---~xlog(T)) = ./dxDmn (--~z0 log(T)) e<_j (note
qnzO'r
r (qn/2z)mr
=
enzOzl~162
-
enl~176
-- ~ ( l o g ( z )
+ log(qn))
--
= @(qnz) and the integrand in (1.35) can be written as z ~ qn/2z" (dz/27riz)x while exp[nlog(q)zOz] = F_,(nlog(q))J(zOz)J/j!). Thus Dmn is defined
as (we have changed the indexing from [537])
Dmn = qmn/2 ~c~ cje (n)~)J j! ~)m+e,e e<_j
(1.40)
where j, t~ > 0 and ce0 = c0e = 5~,0. Now since 021og(7) = u = u2 is the basic solution of KP (cf. [163]) one can say that f~Fgp "induces" the KP solution via (17F). On the other hand ~)mn comes from XDn and is represented via the Oke. The Oke are connected to W symmetry (cf. [222]) and the above suggests that the discretization which preserves integrability is related to W symmetry (since W symmetries are believed to possess the universal properties of various integrable systems). In any case defining (17G) DA -- ~ amnI)mn one can write the expectation value as
>/,.,.
Now observe that
o
Ot~
= _ f dxD
( O og( ))=-/dxDAJr
(1.41)
(1.42)
and Jr = CgrOlog(r) is in fact a first integral of KP (i.e. Hr = f dxJr is a Hamiltonian - recall here from [1631 that the Sn+l = O-10nU = OnOlog(7) are conserved densities). Since 7:)A generates independent symmetries from the ordinary KP flows Or this equation means that the time evolution of the expectation values naturally yields the Hamiltonian structure of the KP hierarchy. For example one has (17H) (O/Ots) f dxT:)AJ~ = 0 and the integral is simply a Hamiltonian for KP. This should not be surprising since a form such as r162 generates the conserved gradients as in [163, 221] (cf also Section 1.1.1). Turning back to the definition of Fgp in (1.30) we see that FKp essentially has this form. On the other hand we can introduce the vector field associated with A to write the right side of (1.42) as (17I) - f dx~DAJr = Z)j. Hr ~vhere the action arises from ~)mn as in (1.35) and hence refers back to Xmn structure as in (1.28). In any event via (1.29) one can say that D j . Hr provides the Moyal bracket structure in KP, i.e. we can write (1.42) as (17J) (O/Otr)f4 = {-4, Hr}(M gP). One is struck here by the lack of visible phase space variables in ,2, or Hr to the notation {A, Hr} gP seems ill defined. However this Moyal bracket
CHAPTER 4. DISCRETE GEOMETRY AND MOYAL
174
refers to (1.29) with entries defined via A = E amnZm~ n as in (17A). Thus in ( 1 7 J ) one seems to imagine {ft,, Hr} KP defined via phase space variables zm( n suitably inserted in and Hr (conceivable via the constructions) and then ignored after calculation (note (z, ~) are actually not needed here- see Section 4.3.1 for clarification and expansion). In any event T~A generates the KP flow Or and it is observed to be remarkable that the integrability of K P (T function, Hamiltonian structure, etc.) arises from the particular forms of FKp and ft,. We note also that the notation ( 1 7 J ) is consistent with the Heisenberg relation ( 1 5 R ) and should therefore be "morally" correct; in fact this consistency is the basis of proving ( 1 7 J ) . From these facts one asserts in [537] that the Moyal structure implies the origin of KP integrability (we will address all this again in [172] more systematically and cf. Section 4.3.1 for more details). To pursue this theme from (1.40) one can associate XDmn with a holomorphic vector field. For the holomorphic function f(z) = ~ fmz m+l the operator which generates an infinitesimal conformal transformation is defined via
L/ = f(z)C)z= ~
mEZ
,. m+1.--, :mZ Oz = ~ fmLm
(1.43)
This can be understood as a vector field in the sense that (17K) [LI, Lg] = LL:g-Lgf. The component Lm is the basis vector for the Virasoro algebra and by analogy with the case of a Hamiltonian vector field one writes a difference analogue of (1.43) via
L~ = ~
~
q-n(m+2)/2 fmzmqn[ZOz+(m/2)] = ~ q-n(m+2)/2 fmLmn
(1.44)
mEZ nEZ>0
In this situation one has a Lie algebra structure ( 1 7 L ) [ L ~ , L gD] = LL~g_LD: D and this (1.44) is seen as a discretization of the holomorphic vector field corresponding to the Moyal deformation. Schematically one has
X:g (Poisson) ~ L : g - Lgf
x?. (Moyal)
Lf g- L,D f
(1.45)
In [538] it was shown that such a type of q-diference operator realization of the Wl+oo algebra can be understood as a specific realization of the discretization of the conformal symmetry. We will provide a sketch of [538, 539] and more on [537] in Section 4.2 after (2.84). There are also relations here to the polarization process in geometric quantization
(cf. [982]).
The above diagram for the holomorphic vector fields indicates the existence of some new bracket corresponding to the Moyal bracket. This is then discussed for the KdV equation u t - 6 u u ' - u ' " = 0 as an example (cf. also [357]). Define the functional ( 1 7 M ) s = f dxxm+lu. It is known (cf. [381]) that the following bracket characterizes the Virasoro algebra / ,
J dxxm+l(uo nt- Ou)x n+l = (m - n)~-,m+n
(1.46)
(the central term based on 03 is omitted here). This bracket has the properties of the Poisson bracket and provides a part of the second Hamiltonian structure for KdV. Then D in (1.43), namely one proposes a functional corresponding to the difference analogue of Lmn (17N) s n = f dxq-nxmu with the definition
D mDl n I }q {f-'mn,
/ dxq-nx m (qn'XOuqnXO _ q-n'XOuq-nXO) q-n'xm'
(1.47)
4.1. INTRODUCTION
175
With a partial integration f dxf(x)(qX~ yields
= f dx(q-XOf(x))g(x) the right side of (1.47)
(qnm'-n'm q-nm'+n'm~rD --
/*"m+m',n+n'
(1.48)
After substitution of -ilog(q) for A and some change of normalization (1.47) becomes the same algebra as in (1.28). In this sense the term qn'XOuqnXO--q-n'XOuq-nXO can be interpreted as a deformation (discretization) of the term uO + Ou.
4.1.3
D i s c r e t e surfaces and K P
The point of departure in [828] is the Hirota bilinear difference equation (HBDE - cf. Section 3.6 and 1.1.5) whose importance has already been discussed in connection with integrable systems. This single equation describes the totality of soliton equations in the KP hierarchy for example and has connections to QIS as well. It can also be given an interpretation in terms of the discrete geometry of two dimensional surfaces (cf. [98, 249, 250, 251, 252. 253, 254]). There the HBDE turns out to be a recurrence formula satisfied by projective invariants of the Laplace sequence of the discrete analogue of conjugate nets. In [828] one asserts that this corresponds in fact to the constructions in the difference analogue of differential geometry in Sections 4.1.1 and Section 4.1.2. Thus we recall the HBDE in the form C~fn(g + 1, m)fn(g, m + 1) + ~fn(g, m)fn(g + 1, m + 1)+
+'Yfn+l(e + 1, m)fn-l(e,m + 1) = 0
(1.49)
Here g, m, n E Z and a,/3, 7 E C are parameters with constraint ~ +/3 + 7 = 0. In [250] one considers in Euclidean 3-dimensional space a quadrilateral < Y(g, m), Y(g + 1, m), Y(g, m + 1), Y(g + 1, m + 1) > (elementary quadrilateral). If the coordinates satisfy the discrete Laplace equation Y(e + 1, m + 1) - Z(g, m) = A(g, m)[Y(g + 1, m) - Y(g, re)I+
+B(g, m)[Y(g, m + 1) - Y(/~, m)]
(1.50)
then they form a planar quadrilateral (PQL). A collection of such PQL is called a discrete conjugate net (DCN). The Laplace transform of the net Y = Y0 defines two other conjugate nets Y1 and Y-1 by connecting the intersection points of the directions of the opposite edges of the four elementary quadrilaterals which intersect at Yo(g, m). This is illustrated by pictures in [249, 250] for example and will be discussed in more detail below. The same process can be repeated to the images Y:i:l generating a sequence of conjugate nets Yn, n = 0, +n. At every step of this process there exists an invariant under the projective transformations, called a cross ratio /n(g,m), which is formed via the four points {Y~(g, m - 1 ) , Yn(g, m), Y~+I(g, m), Y~-I(g - 1, m)}. Since this cross ratio is determined by the coefficients An, Bn of the Laplace map (1.50) the sequence of maps defines a recurrence formula among the cross ratios which is in fact equivalent to HBDE (after some normalization). The details will be indicated later. Now we refer to Sections 4.1.1 and 4.1.2 for the discrete notation of differential geometry and the corresponding Wigner-Moyal theory. The meaning of the discretization of phase spaces is not based on discretizations of x and p separately as would be the case for discrete surfaces based on (x, p) but in fact they are simply different views of the same thing. We recall now two viewpoints for KP. One is based on (1.49) with three discrete variables g, m, n; the other involves KP flow parameters tn and many soliton equations. The variables are
CHAPTER 4. DISCRETE GEOMETRY AND MOYAL
176
related to each other via the Miwa transformation ( 1 7 0 ) tn = ( l / n ) E ~ kjz] where the zj are arbitrary parameters and t~ = p+q+r-(3/2) with m = - p - (1/2) and n = p + q - 1; here (p, q, r) are an arbitrary three of the kj. The geometry of the discrete surface is described by means of the shift operators exp(O~) and exp(Om) while the third shift operator exp(On) generates the sequence of Laplace transforms. The question is whether or not they correspond to some difference vector fields in the form (17P) X D = f dalda2vA (x, p, al, a2)Val,a2 when they are translated into the language of the tn. Here one is thinking of d b l d b 2 e - A ( a l b 2 - a 2 b l ) A ( x + Abl, p + Ab2)
VA =
(1.51)
as in (1.5). After the change of variables ( 1 7 0 ) one has l
(1.52)
The right side can be interpreted as simultaneous shifts to all directions with the amount of shift in the tn direction equal to (1/n)z~. Since g,m,n are linear combinations of k~s the shifts to these directions are produced by shifts to the tm directions of amounts (1/m)(z~ + z~ + z~n), -(1/m)z~, and (1/m)(z~ + z~) respectively. Hence it is convenient to introduce a shift vector field (17Q) X~ = f dSA({, d)exp[A En an(O/Otn)]. One can show that (17R) [XSA,XSB] = XS_{A,B}s where (a ~ ~ a etc.)
•
( t A , ~a + b ) B ( t B , ~a- - b )l
(1.53)
tA--tB=t
(this is not clear but see (2.162)-(2.163) for explanation). covariant shift operator defined via
One also introduces a gauge
1
]
(1.54)
u(t) is defined as ( 1 7 S ) u ( t ) = U(k)U-l(k ') for k' = {kl, k2,...,kj + 1, kj+l,-.-} exp(Ok~)f(k) = f(kl,...,kj + 1,'")exp(Okj)). The correspondence of XkS[U ] to XAs of (17Q) is achieved by choosing (17T) A(t,a) = u(t)l-I 5Jan -(z~/nA)] (so XAs = u(t)exp(E(1/n)z20~) = u(t)exp(Ok~)). These new operators will enable one to translate various terms in KP theory in the'tn language into the language of the kj. In particular one where (note
will have the following situations (some typos are corrected). ..
9 First consider the case (17U) U(k) = with Iql < 1 and n = 1,2,.... Then
s
l-ltr
(
Xk~[U]=I- I 1 - q z~ ~#j zj
and let
tn = ( l / n ) E kjqnz2
eO/OkJ =
= (1--q)kJexp [~1 qntnzjn] exp [ ~ l n z ~ ~ n 0 ]
(1.55)
4.1. INTRODUCTION
177
(
Note here via Miwa
logH l-q-e#j zj
( 1 )kjZ~
=Ek~-'~;
e#j
= E Z? nqn tn -n
n
1( ) n \qz~"
=
-- E [tnqnzjn l - kj E - n
The last expression in (1.55), after dividing by ( 1 - q)kr and taking the q --+ 1 limit, is the standard KP vertex operator. 9 Next one can generalize the vertex operator by doubling the space of in; one defines (17V) U(k) = 1-Ie#j[1 - ( e s 1 - - ( z j / z l ) ] -k~kj (note q is removed) and
tn
_
1
nll ~
kjz~,
n = 1,2,...,
to = ~
kjlog(zj)
(1.57)
In terms of the tn, tn one has then (note the cancellation of log[1 - (~j/zj)] terms) S Xkj[U ] = exp - ~-~(tnh~-n - tnZjn ) •
(1.58)
0 •
[log(zj)O o
C~ l(z;Otn+hjno~n)J ---eiX+(zJ,2J)eiX-(zJ,hJ
)
o+ E n 1 where the last expression is to remind one of the relation to string coordinates X+ (cf. [831] for example). This produces a symplectic structure such that the (tn, tn) form a symplectic pair of an infinite dimensional phase space and one will have Moyal brackets as in ( 1 5 0 ) rather than ( 1 7 R ) . 9 B~icklund transformations are generated by operators of the form ( 1 7 W ) B(zj, ze) = X kjs [U]Xk~[U]; they form an algebra following ( 1 5 0 ) . 9 HBDE can be derived as an integrability condition of the following linear problem (cf. [830]) X[[U]IO > = c+X[+n[V]IO >; XSm[V]IO > = c-X~-n[U]I~ > (1.59) It was shown in [830, 834] (cf. Sections 4.2.3 and 4.2.5) that under the gauge condition Y(t~, m, n) = U(t~, m, n - 1) the gauge potential U(t~, m, n) satisfies HBDE with = -c+c_a. The gauge potential U in the covariant shift operator has an obvious geometrical meaning as a connection and HBDE is nothing but a geometrical constraint imposed on the connection. In [756] it was shown that this set of equations (1.59) reduce to a Lax type equation for the Toda lattice in an appropriate continuous limit. 9 The BA function of the KP hierarchy can be given explicitly in terms of the solution T of HBDE; in the present formulation it can be interpreted as a wave function of the form (17X) I~(zj) > = Xk~[T-1Uo]IO > where U0 is of the form (17U). Note then by (1.55)
X~ [T-1Uo] ~ T-1UoOk~Uo1T ~
(1.60)
(1 -- q)k5 e ~ qntnZ;ne~(1/n)z~OnT T
so dividing by (1 - q)k~ and letting q ~ 1 gives (1 _ q)-ks Xkjs [7-1U0] which tends to the expression e x p ( ~ t n z f f n ) T ( t n clear.
~-[Zj])/T ~,~ r
1) but the 10 > notation is not
C H A P T E R 4. DISCRETE G E O M E T R Y AND M O Y A L
178
Thus according to [828] one has shown (modulo many details still - see Sections 4.2.4 and 4.2.5 for enhancement) that various objects in the KP theory can be expressed differently via discrete surface coordinates k~ or soliton coordinates tn. In all examples they are translated from one to the other by the operator XkS [U] which itself has two faces, a gauge covariant shift operator or a difference analogue of a vector field. If U is an arbitrary function of the tn or kj it generates a large symmetry algebra via (17R). From the viewpoint of geometry this algebra should describe some topological properties of the discrete surface, as the kj are restricted to p, q, r, and an extension to higher dimensional hypersurfaces should be straightforward. One knows that the coordinates tn and kj are linearly dependent via the Miwa transformation but it is not known what is the meaning of the kj in the space of soliton coordinates and what is the role played by the tn on the discrete surface. 4.1.4
Multidimensional
quadrilateral
lattices
We go now to the fascinating series of papers [98, 102, 103, 190, 249, 250, 251, 252, 253, 254, 560, 654]. We will emphasize here the viewpoint of [252, 253] where one shows that the Q F T formulation of the tau function theory has a geometric interpretation within the classical transformation theory of conjugate nets. In particular the Hirota bilinear identity for the multicomponent KP hierarchy becomes, through a generalized Miwa map, a bilinear identity for the multidimensional quadrilateral lattice (QL) equations. Thus the Japanese school of Sato et. al. introduced the tau function in the framework of QFT for free fermions in 2-D space time (cf. [147, 210] for a survey of this). They revealed thereby an unexpected link with the isomonodromic deformation theory of linear differential equations. Later it was shown that the same QFT formulation applies equally well for describing the KP hierarchy and its multicomponent extension in terms of b - c systems of ghost fields (cf. here [147, 224]). This description was also related to certain aspects of string theory and in particular the Grassmanian model of the tau function theory is strongly related with the operator formalism in string theory and CFT (again cf. [147] for a sketch of this). On the other hand the theory of conjugate nets is a classical subject in differential geometry. We recall (cf. [560]) that a net is a system of curves on a surface involving two 1-parameter families of curves on the surface such that through each point of the region considered passes just one curve of each family, the tangents of the two curves at each point being distinct. Two curves are called conjugate if their directions at each point are orthogonal with respect to the second fundamental form of the surface. It turns out that the basic operations associated with the multicomponent KP hierarchy (MKPH) have a distinguished geometric interpretation in terms of conjugate nets, their transformations, and their discretizations. One begins now with b - c systems of quantum fields satisfying cj(z')}
= e
je(z - z');
=
cj(z')}
= 0
(1.61)
where bi(z) and ci(z) for i = 1 , . . . , N are free charge d fermion fields defined on the unit circle S 1. The Clifford algebra generated by the b - c system admits a representation in terms of bosonic variables where the fields act on the Fock space H of complex valued functions (19A) T = ~ e T(g, t)A e where g ~ ( e l , . . . , eN) with t ~ ( t l , " - , tN) C C N'~ having components ti = (ti,l,ti,2," ") E C ~ while A = ( A I , . . . , AN) with Ae = ffl 1-.. AeN. The representation of the b - c generators is written in two forms in [253] and we will use the second based on vertex operators (19B) Vi+(z)f(t) = f ( t + [1/z]ei) where [1/z] = (1/2, 1/222, 1/323, .- .) and (el) N are canonical generators for C N. Thus the b - c action is given via
bi(z)r(~,t) = (-1)EJ>~eJzei-lexp[~(z,t)]r(~-ei, t - [1/z]ei);
(1.62)
4.1. I N T R O D U C T I O N
179
ci(z)r(e, t) = (-1)E~ >'e~z-eJexp[-~(z, ti)]7-(g + ei, t + [1/z]ei) The Fock space decomposes into a direct sum of charge sectors (19C) H = | where Ha = {7- 9 H; QT- = qT-} where the total charge operator Q = E N Q i is the sum of N commuting partial charges Q~ = Ai(0/0A~) for i = 1 , . . . , N; they correspond to the N different flavors of fermions of the model. The N component KP hierarchy can be formulated as the following bilinear identity (cf. also [86, 162, 512]) N
13(7- | 7-) = 0 (7- 9 H0); 13 = Js --dz ~ bi(z) | ci(z) 1 z
1
(1.63)
In terms of the components 7-(t?,t) we have N
~--~(- 1)EJ>~ ~J+~ exp[~z, ~ ' tm - t~m)]Zem--e'--2X
/sdz 1
1
•
(1.64)
t-- [1/zlem)T(g'--em, t' + [1/z]em) = 0
for any t, t ~ and t?, t?~ such that ~1 -}-"'"-}-/N -- 1 = t?~ + . . . + t~v + 1 = 0. It can be shown that (19D) [r~ | = It; | = 0 (i.e. for any solution 7- of (1.63) the functions FiTand F~'T also satisfy (1.63)). Here one defines
r~(z) = ~xp[~(z, t~)]y((z);
r~(z) = ~xp[-~(z, t~)]y~ + (z)
(1.65)
and ~(z, ti) = E ~ znti,n with the V/+ as in (19B). In what follows one fixes t? and writes (19E) T(t) = T(i,t) with 7.ij(t) = SijT(t) = T(g + e i - ej,t). One notes that the vectors c~ij - ei - e j are the roots of the AN-1 root system so that any linear combination with integer coefficients is a point in the corresponding root lattice. The shift operators Sij along the root lattice vectors c~ij correspond to Schlesinger transformations in monodromy theory (cf. [210, 640]) and satisfy (19F) S i j 0 S j i - - id; Sij o Sjk = Sik, and S i j 0 S k i - - S k j . Now the N • N matrix BA function r and its adjoint r tau function as
r r
can be defined in terms of the
[1/z]ej) exp[~(z, tj)]; ' t) = cijz ~ j - 1 T i j ( t - T(t) (z, t) = Eijz~'r
(1.66)
ti)] Tij(t + [1/z]ei) ~(t)
where e i j = s g n ( i - j) for i :/: j, ti ~ tin, and eii = 1. One can write now (19G) ~(z, t) = X(z, t)r t) and r t) = r t)-lx*(z, t) where
Co(z, t) = diag{exp[~(z, t l ) ] , ' ' ' , exp[~(z, tN)]} and X, X* are the "bare" BA functions with asymptotic expansions (19H) X ~ 1 + / ~ Z -1 + O(Z -2) and X* ~ 1 - ~z -1 + O(z -2) as z ~ ce with matrix/~ given by
Z.(t) =
Olog(T(t)) (i = 1 , . . . , N ) ; Ou~
Tij(t) (i ~ j; i , j = 1 "'" N ) 9~j(t) = ~j ~(t) ' '
(1.67)
180
CHAPTER
4. D I S C R E T E
GEOMETRY
AND MOYAL
where Uk = tk,1 (k = 1 , - . . , N). Thus by setting t~ ~ t~ + ei and 6' ~ t~ - ej in (1.64) one obtains the Hirota formula ( 1 9 H H ) fs1 d z r t)r t') = 0 in a standard form. The symmetry operators Fi and F~ induce a corresponding action, say Gi and G~, on BA functions via Gi(p)r
c~(p)r
= [Vi-(p)~2(z,t)]
t) = [y~ + ( p ) r
t)]
-
;
(1.68)
-
where Pi is the matrix with elements (Pi)jk : ~ij~ik 9 The notations involving vertex operators here and in Section 1.1.1 are different but no confusion should arise. Notice that in order for V i + ( p ) r to be convergent one must have IPl > Izl 9 The form ( 1 9 n i l ) of the Hirota identity is most suitable for formulating the Grassmannian approach to KP as in [854] (cf. also [147]) which in turn is convenient for deriving the linear system of equations for the BA function. To this end let W be the set of N • N matrices r such that (19I) fs1 d z r 1 6 2 t') - 0 for all t' in the definition domain of r Under appropriate conditions W belongs to an infinite dimensional Grassmann manifold as in [854] and from ( 1 9 H ) it follows that W is a left M y ( C ) module (ring of N • N complex matrices). Let Eij be the standard basis in M y and note in particular that E i i - Pi. As a consequence of the asymptotics for r above one has for any t W =- e n > _ o M N ( C ) " V n ( t ) ;
Vn(Z,t) =
r
0
(1.69)
Note that Vn(Z) ~ (z n + O ( z n - 1 ) ) r as Z ~ OC. Thus the linear system for M K P H results from the decompositions of the time derivatives of r in terms of the Vn, n = 0,-... In particular by decomposing Pi(Or (i ~ k) one obtains (19J) ( O r - /3ik~k where r = ( r 1 6 2 Similarly for r one gets (19K) (Or -- ~ / 3 k j where r = (r r T. The compatibility of ( 1 9 J ) or ( 1 9 K ) leads to the Darboux equations
0~j = ~k~kj
(1.70)
Ouk
for i, j, k different. As for the bilinear identity (1.64) the evaluation of residues at oc provides the Hirota representation of MKPH; in particular 02r
q OUiOUj
Or Or
OU i OUj
OTij OT T~U k -- TiJ-~U k
TijTji = 0 (i ~ j);
(1.71)
eijeikekjTikTkj = 0
(the latter for i, j, k different is the Hirota form of the Darboux equations). By setting now ---, t~ + ei and l' ~ t~ + ek -- e~ -- ej in (1.64) one obtains s163
-Jr-s163
-- s163
-- 0
(1.72)
for i, j, k,g different. This relation can be found in [512] and is a Fay trisecant formula for theta functions on Riemann surfaces (cf. also [732]). Now the Darboux equations (1.70) for the so-called rotation coefficients ~ij characterize N dimensional submanifolds of R M (for M >_ N), parametrized by conjugate coordinate systems (multiconjugate nets) and are the compatibility conditions of the linear system ( 1 9 L ) ( O X j / O u i ) = ~ j i X i for i , j = 1 , . . . , N with i ~= j; the Xi are Mdimensional vectors tangent to the coordinate lines. The so-called Lam6 coefficients Hi
4.1. I N T R O D UCTION
181
satisfy (19M) (OHj/Oui) = flijHi for i 7~ j in terms of which the points x of the net are found by integrating the equation (19N) (OY/Ou~) = XiHi (i = 1 , . . . ,N); note there is no summation over repeated indices here. Thus given the BA function r one can construct conjugate nets with the flij appearing in (1.67) and with tangent vectors Xi being rows of the matrix ( 1 9 0 ) X(t) = fsl dz~(z, t)f(z) for some distribution matrix f(z) E MNxM(C). Given the adjoint BA function r the Lam6 coefficients are provided by the entries of the row matrix (19P) H(t) = fsl dzg(z)qd*(z, t) (i = 1 , . . . , N) for some distribution row matrix g(z) E C N. This leads to the statement that the solutions of the N-component KP hierarchy describe N-dimensional conjugate nets with coordinates ui = ti,1 (i = 1 , . . . , N) while the remaining times ti,k for k > 1 describe integrable iso-conjugate deformations of the nets (i.e. integrable non-stretching evolutions of the conjugate nets). Now to motivate the idea of quadrilateral lattices as the discrete generalization of conjugate nets we go momentarily to [249]. In the continuous case of conjugate coordinates on a surface look at the tangent planes to the surface at two near points along a coordinate line. These planes intersect in a straight line which in the limit as the points approach one another is the tangent direction to the second coordinate line. On a discrete level a surface is defined as a mapping from Z 2 ---, R M and if one considers two neighboring "tangent planes" along the first coordinate line, defined by triples < x, Tlx, T2x > and < Tlx, T~x, T1T2x > where Ti is the corresponding shift operator, then their intersection line coincides with the direction of the second coordinate line if and only if the elementary quadrilateral {x, TlX, T2x, T1T2x} is planar. Hence the natural discrete analogue of 2-D conjugate net is the 2-D QL defined via: A 2-dimensional QL is a mapping x 9 Z 2 - - , R M for M _> 2 such that all the elementary quadrilaterals {x, Tlx, T2x, T1T2x} are planar. A multidimensional conjugate net is characterized by the property that any surface made by varying two parameters forms a 2-D conjugate net with respect to these two coordinates. Hence one defines an N-dimensional quadrilateral lattice (MQL) as a mapping x - Z N ~ R M (M >_ N) such that all elementary quadrilaterals with vertices {x, Tix, Tjx, TiTjx} are planar. It is better to describe this projectively with x" Z y ~ p M and homogeneous coordinates x - [x] E p M Now returning to [253] one considers basic transformations of conjugate nets (we refer to [253] and the other papers in the set [98, 102, 103, 190, 249, 250, 251,252,254, 560, 654] for pictures, diagrams, and further discussion). 9 The Laplace transform s (i 7~ j) of a conjugate net is defined by (19Q) s x - (Hj/flij)X~. The corresponding transformation for the rotation coefficients flij involves
r.~j (fl~j) = ~ j
1
fl~jflj~ flkj
log(fl~) OuiOuj
;
(1.73)
flik
& ; fl~e where all the indices i, j, k, ~ are different. The Laplace transformations also satisfy (19R) l:ij o s = f-.ji o s = id w i t h / : i j o s = 12ik and ~-'ij o l~ki --" ~-'kj" One recalls that the Laplace transformation/2ij of a 2-D conjugate net provides the geometric meaning of the 2-D Todd system (cf. [208]) and in fact interpreting/2iy as translation
C H A P T E R 4. D I S C R E T E G E O M E T R Y A N D M O Y A L
182
in the discrete variable n one can write from (1.73) and ( 1 9 R )
021og(/3ij(n)) OuiOuj
-
-
-
~ij(n) f l i j ( n - 1)
D
~ij(n + 1) /3ij(n)
(1.74)
9 The L6vy transformation f-.i(x) of a conjugate net is a net conjugate to the i th tangent congruence (cf. [292]- this means the lines < x , s > are tangent to the i th coordinate lines at x); thus the position points of the new net are given by (19S) s = x - (ft[(,H]/~i)Xi where the transformation data (i i = 1,-.. , N are solutions of the linear system ( 1 9 J ) and ft[~, H] satisfies (19T) (Oft/i)uk) = (kHk for k = 1 , . . . , N. The corresponding transformations for the tangent vectors Xi are
l:i(Xi) =
OXi 10(i. (k Xi Oui + ~-~ui "4i; Ei(Xk) -- Xk - (i
(1.75)
(the latter for k =fi i - ft and f~ below are called potentials). 9 The adjoint L~vy transformation involves the lines < x, s > being tangent to the i th coordinate lines of the new net. The position points are given by (19U) E~(x) x - ( f ~ [ X , ~*]/~*)Hi where ~ is a solution to ( 1 9 K ) and ft[X, ~*] satisfies the equation (19V) (Of~/Ouk) = Xkr The corresponding tangent vectors Xi transform via
(1.76)
F_,*(X{) = _ n [ X , ~*~]", E,~(Xk) = Xk - ~ki ~ [ X ' ~*~]
r
r
9 A fundamental transformation ~ ( x ) has the property that the lines < x,.T'(x) > intersect both nets along the coordinate lines. It can be viewed as the composition of L~vy and adjoint L~vy transformations, i.e..T'i = s o f_.~. One notes that 5ci - 9cj and the transformation formula are ft[r H]
9ri(x) - x- f~[X, (*]ft[(, r ; where ft[r r
.T'{(Xj) = X j -
a[X, r
ft[r r
Cj
(1.77)
satisfies ( 1 9 W ) (Oft/OUk) = ( k q (no sum).
These transformations map solutions of the MKPH into new solutions and are sometimes called Darboux transformations. It is a common belief that Darboux type transformations of integrable P D E generate their natural integrable discrete versions (cf. [646]). Further, if the original P D E have a geometric meaning the Darboux type transformations provide the natural discretization of the corresponding geometric notions (cf. [251]). For example if one considers a conjugate net x and two fundamental transformations Jci (i = 1, 2) then the points {X,~'l(X),.~'2(X),~C'l(.~2(X))} a r e coplanar (cf. [208]). In any event it turns out that a lattice x : Z N --. R M with n ~ x(n) ( N < M ) whose elementary quadrilaterals are planar (i.e. a QL) is the correct discrete analogue of a conjugate net and the planarity condition can be expressed by the equation (cf. (19L)) (19X) A j X i = ( T j Q i j ) X j with compatibility conditions (19Y) AkQi j = (TkQik)Qkj the discrete analogue of (1.70). The points x of the lattice can be found by means of discrete integration of (19Y) A i x = (T/H{)Xi where A i H j = QijTiHi (Ti is the translation operator in the discrete variable ni determined by (19Z) T i f ( n l , . . . , n i , . . . , n g ) = f ( n l , . . . , n i + 1 , . . . , n N ) and Ai = T1 - 1 is the partial difference operator). Now one notes that the algebraic relations ( 1 9 R ) involving Laplace transformations are
4.1. I N T R O D U C T I O N
183
the same as those for the shift transformations &j indicated in (19F). In fact one can prove that the root lattice shift Sij in the direction aij is the composition of ~ij with a trivial scaling symmetry of the Darboux equations. To see this look at the Laplace transformation in the light of tau functions and, using (1.67), one starts from the first equation in (1.73) to obtain ~ij~ij = --s where (1.71) has been applied in the form r 2 ( 0 2 1 O g ( T ) / O u i O u j ) = rijrji. Applying Sij one g e t s Ti2j(O210g(Tij)/OuiOuj) -- (Sijr)r so that (20A) s = -Sij3ij. For the next three equations in (1.73) one has trivially
l:ij~ji = --Sij~ji;
E, ij3ki = ekis
(1.78)
~ i j ~ j k -- s
The next two equations in (1.73) come easily from (1.71) once the shifts are applied properly, namely
OTij OTik' Tik-~u i -- Tij-~u i -- ekjs163 From
OTij OTkj = Tkj--~U i -- Tij OUj Ekj{ki~ijTSkjTij
(1.79)
this it follows that
~-,ij~ik = eki~kjs and (1.72) leads to (20B) s
s
(1.80)
= s
= ekiEkj~giCgjSij3kg. This may all be summarized as ak Sij3ke -- s
ak = s
ag
(1.81)
One notes that the Darboux equations have the scaling symmetry 3ij --+ (aj/ai)/3ij coming from the freedom in the choice of the Lam@ coefficients Hi ~ aiHi where ai = hi(U). Thus a tau function of MKPH describes not only the integrable deformations of a single conjugate net but also all its Laplace transforms. One considers next the action of basic vertex operators at the level of BA functions; it turns out that Gi(p) can be identified with the classical L~vy transformation. Indeed it is shown in [253] that given tangent vectors Xj associated with ~ ( z , t ) as in ( 1 9 0 ) one has (20C) s = p&JVi-(p)(X j) where t:i stands for the L@vy transformation with data (20D) ~j(t) = (dn~bji/dzn)(p, t) for j = 1 , . . . , M and n _> 0 is the order of the first nonzero z-derivative of the i th column of r t) at z = p. The proof is more or less straightforward but nontrivial. Similarly in this context one shows that (20D) s - (1/p~J)Vi+(Xj) where/2~ stands for the adjoint L~vy transformation with data ~ ( t ) = (dm~/J*./dzm)(p,t) with m being the order of the first nonvanishing z-derivative of the i tn row of ~*(z,t) at via
s162162
-
Pi ;
& ("r
=
(~)Pi
[c~(p)r
( --)v
"~
(1.82)
z
where (Pi)jk "- r this is not quite clear and we refer to Section 4.5 for some clarification. These correspondences, derived from the bilinear identity for BA functions, are direct consequences of Fay identities for the tau function which are specified in [253]. Consider next the bilinear identity ( 1 9 H H ) and for each complex number p one can introduce new functions depending on N additional discrete variables n E Z N via 9 (z, t, n) = r
t - n[1/p]); q;*(z, t, n) = r
- n[1/p])
(1.83)
C H A P T E R 4. D I S C R E T E G E O M E T R Y A N D M O Y A L
184
where t - nil~p] = (tl - nl[1/p],..., tg -- ng[1/p]). Then ( 1 9 H H ) becomes a continuous discrete bilinear equation of the form
I s 1 dzql(z, t, n)9*(z, t', n') = 0
(1.84)
From ( 1 9 G ) one has the equations (20E) 9(z, t, n) = E(z, t, n)90(z, t, n) and 9*(z, t, n) g0(z, t, n ) - l E * ( z , t, n) where
go(z, t, n) = Co(z, t)diag
1- P
...
1- P
;
(1.85)
E(z, t, n) - X(Z, t - n[1/p]) ~ 1 - pQz -1 + O(z-2); E*(z) = x*(z, t - n[1/p]) ~ 1 + pQz -1 + O(z -2) as z ---, oo, where Qij(t, n) = - ( 1 / p ) f l i j ( t - n[1/p]). If we fix attention on the n dependence the asymptotic module structure is now (20F) W = | vn(n) with Vn(Z, n) =
( ~ N Ak)n- ~(z, n). The linear systems for 9 follow from the decomposition of the discrete derivatives of 9 in terms of the Vn. A similar analysis holds for 9" and it follows that Q, 9 , 9" satisfy \
/
Ak'~i = (TkQik)q2k (i # k); A k g ; = Qkj(Tkr
(j # k)
(1.86)
The compatibility of (1.86) gives the discrete Darboux equations ( 1 9 Y ) while Xi(n) and Hi(n) can be obtained by the analogues of ( 1 9 0 ) and ( 1 9 P ) . Hence we have a QL in the discrete variable n. In the present Miwa-like scheme the translation Ti in ni corresponds to the vertex operator V~-(p) which by ( 2 0 C ) corresponds in turn to a L6vy transformation" hence x(n) describes a QL. This approach gives through the Miwa transformation a tan function formulation of the QL and a Q F T representation of them in terms of b - c systems. The tan function expression of the QL equation ( 1 9 Y ) is
(TiT)(TjT) -- T(TiTjT) -- (TiTij)(TjTji) = 0 (i # j); T(TkTij)
-- ( T k T ) T i j -- e i j e i k e . k j ( T k T i k ) T k j
-~ 0
(1.87)
(i,j,k different)
(recall Tij = SijT from (19E)). We want to clarify somewhat the passage to discrete variables based on (1.83) by extracting from [252] where the treatment of vertex operators is slightly different. Thus go back to (1.66) and the formula (1.69). Define tangent vectors as in ( 1 9 0 ) ( i . e . ) ~ i fsl r t ) f ( z ) d z - i ~ row) and Hi as in ( 1 9 P ) (i.e. Hi(t) = ~ N1 fS~ 9J(Z)q2;i( z, t)dz). One defines now (21E) Yi(p, q)f(t) = f ( t + ([iv] - [q])ei) where (ei)N1 is a basis of C g and shows that V i determines a fundamental transformation of the conjugate net (cf. (1.77)). The Hirota bilinear identity ( 1 9 H H ) is used extensively here. First note that for p, q E C with [P[, Iql > 1
'
q
z--p
Thus Vi(p, q)r extends to Izl _> 1 meromorphically with a simple pole at z = p where the residue satisfies (21F) Resp[Vi(p, q)r - Pi) = 0. Then if t ~ t + ( ~ ] - [q])ei and t ~ ~ t in ( 1 9 H H ) the only nonvanishing contribution comes from residues at z - oc and z = p leading to Resp[Vi(p, q)r162 (p) = (1.89)
4.1. INTRODUCTION
185
[Vi(p,q)fl] [ 1 - ( 1 - P ) P i
: -p (1-P)Pi4-
I - [1-(1-P)Pi]~
Given the expansion (19H) and multiplying (1.89) by Pj on the right and left (1 < j _< N) there results Vi(p,q)~ii=/3ii+q ( 1 - P ) + qResp[Vi(p, q)~i]~*i(P); (1.90) q
,
qvi(p, q)flij = flij 4- pResp[Vi(P, q)r162 (i r j); P PVi(p, q)~ji =/3ji 4- Resp[Vi(p, q)~2ji]~2i*i(P)(i r j); q Vi(p, q)fljk = fljk 4- Resp[Vi(p, q)~ji]~zi*k(P) where i, j, k are different i~ the last expression. Similar identities for the BA function ~ can be obtained by considering the function F = Pj LResp[V~(p ' q)r
Vi(p, q)r
(1.91)
The only possible singularity for ]a] > 1 is at z = p and this is removable via (21F). Hence F is analytic for ]z] > 1 and from (1.88) one has (21G) Yi(p, q)r ,.o {1 - [ 1 -(p/q)]Pi + O(z -1 }r as z --+ oc. Hence the first terms of the Laurent expansion of F on S 1 are
[ Resp[V~(p'q)r - 1 ] { 1 - ( 1 F = Pj REesp[Vi(p,q)r ]
-p)
q Pi + O(z-
~}
~o(z)
(1.92)
Therefore by the uniqueness of the BA function in the Grasmannian element W of (1.69) there results Pj
{ Resp[Vi(p, q)~2]
R--~Sp[~-/-(~,q-y~]- 1
} vi
(p, q)r
= Pj
{ pResp(P, q)[Vi(p, q)~2] qResp[Vi(p, q)~dii]
1}r
(1.93)
In terms of rows this yields
Resp[Vi(p' q)r Vi(p, q)r = Cj 4- Resp[Vi(p, q)r
(Vi(p, q)qdi - P r q
(1.94)
(for i r j. This implies
r
p
r
(j =/=i)
(1.95)
Next one shows that the operator 9c defined via
Jci(flij) = Pq V i (p,q)13ij;
(1.96)
T'i(Zji) 9 = PVi(p, q)flji; .T'i(Zjk) -- Vi(p, q)/3jk is a fundamental transformation (cf. (1.77) and the discussion thereabout). In particular via (1.90) and (1.95) (i) (i), ~ @ k ) = Z j k - v3 ~k ~h~r~ (1.97) f~(i)
C H A P T E R 4. D I S C R E T E G E O M E T R Y A N D M O Y A L
186
v
5i)
p r q Resp[Vi(p, q)r
~ (i), , _ = ~2ji(q); % _ ~bik(p); ~(i) _
Evidently @i) and @)* are solutions of (19J) and (19K). Then to show that ~ i is fundamental with data as indicated in (1.97) one must show that f~(i) is a potential, i.e. (21H) (Of~(i)/Ouj) = vSi)vj(i), for i -J: j; for i = j one starts with the equation (21I) f~(i) _ {r q)r for j g: i, takes the i th derivative, and uses the third equation in (1.90). As for action on the tangent vectors Xi one notes that
.yi(xi) = pq v ~(p, q)Xi; ~ ( x j )
= v ~(p, q)Xj (j 7~ i)
(1.98)
which can be written as
C i ; C i = v~(v, q)X~ v•i)
7(xj) =xj-~-~
~x~
(1.99)
Resp[Vi(p, q)~2ii]
with Ci a so-called Combescure vector satisfying (21J) (OCi/Ouj) = XivJ i)*. For i # j the calculation involves taking the derivative and using (19J), ( 1 9 0 ) , (1.90), and (1.95); for i = j one uses a different formula for C i (via (1.95)), namely ( 2 1 K ) C i = [Vi(p, q)Xj Xj]/{Resp[Vi(p,q)~2ji]}, plus (19J), ( 1 9 0 ) , and (1.90). Now consider QL's via Miwa transformations
9 (z,t,n) = r
+ n ( [ p ] - [q])); ~ * ( z , t , n ) = r
+ n ( [ p ] - [q]))
(1.100)
where n = nlel + . . . + nNeN E Z N. Thus the discrete variables are introduced by a composition of fundamental transformations acting on tangent vectors of the original net. Hence via [251] the corresponding lattice is a QL with suitable renormalized tangent vectors given by the Combescure vectors {C~}N. However finding the rotation coefficients Q~j (cf. here (1.84) and (1.85)) requires consideration of the iterations 5"~ o ~J = bcq and in that direction one writes V q (p, q)r t) = (1.101)
[viJ(p,q)x(z,t)]r
) [1-(1-P)
z z-p
(Pi + Pj)]
with a simple pole at z = p. In fact this is (1.88) after replacing P~ by P~ + Pj. Thus by substituting t ~ t + ( [ p ] - [qJ)(ei + ej) and t' ~ t in ( 1 9 H H ) the only nonvanishing contributions to the integral come from z = oc and z = p yielding
Resp[V ij (p, q)r162 (P) = -p(1
(1.102)
- p q ) ( P i + P j ) + { V q ( p , q ) f l } - ( 1 - q )[1( P i + P j ) ] P
- [1-(1
- p q)
(P~+ PJ)]fl
One uses now the convenient notation
B =
13ji 3jj
; bk =
CJ~ Cz
;r
fljk
Cjk
;
=
;&=(~k~kj)
(1.103)
4.1. I N T R O D U C T I O N
187
which allows one to express the components of (1.102) in the form
viJ (P' q)B = B + q ( 1 - P + pResp[ViJ(p,q)rb]cb*(p); q
(1.104)
q-V ij (p, q)bk = bk + q Resp[V ij (P, q)O]C~(P) (i, j, k different) P P P-V ij (P, q)l)k = t)k + Resp[ Vii (P, q)r q
(i, J, k different);
Vij (P, q)Zkt = 3ke + Resp[ Vii (P, q)r162 - *
(i , j,k , e different)
For the BA function one has from (1.101) that (21L) Resp[Y ij (p, q)~] = Resp[V ij (p, q)~] (Pi-~ Pj) which implies
Resp{Pk[Resp(ViJip, q)qd)pij[(Resp(ViJ(p,q)42))-1 ] - 1]vij(p,q)~} - 0
(1.105)
where pij : M2x2 ~ M y x Y with Pij[(mab)a,b=i,j] = (mabhakhb~)a,b=i,j with k,/~ = 1,-.. , N is the canonical embedding of 2 x 2 matrices into N x N matrices in the cross of the i th and jth columns and rows (one is renaming a, b). The same arguments as before lead to
Pk {Resp[ Vii (P, q)~)]]Pij[(Resp( Vii (P, q ) O ) ) - l ] -- Pk [P Resp(ViJ(P, q)r
_
1]vij (p, q)~p =
q)~))-x] - limb
(1.106)
In terms of the rows of ~ this reads as
V ij (P, q)r
= qdk + nesp[ Vii (P, q)r
•
(1.107)
• [Resp(Vij (P' q)(b)]-l ( vViJ"(p, (P' q)r _- ~qr CJ Finally from (1.101) it follows that YiJ(p,q)qd(q)(Pi + Pj) = O, which together with (1.107) implies ~k(q)~(q) -1 -- P Resp(V ij (p, q)~k) [Resp(V ij (p, q)(I))] - 1 (1.108) q It turns out then (cf. [251]) that q .TiJ (bk) = pViJ (p, q)bk; .T'iJ(bk) = PqViJ(p,q)bk (k 7s i,j);
7 j (/3kt) = V ij (P, q)/3kt (i, j, k, e different)
(1.109)
defines an iterated (or vectorial) fundamental transformation. Vectorial fundamental transformation of the tangent vectors take the form ( 2 1 M ) 3 c i J ( x k ) = (q/p)V~J(p,q)Xk for k = i,j and 3ciJ(Xk) = viJ(p,q)Xk for k 7~ i,j. Note here by (1.107) one has
3ciJ(Xk) = Xk -- r
( )Vij(p' (p, V q)Xj q)xi i j - ~q ~qXj xi
(1.110)
Some further calculation also yields
ViJ(p,q)Xj - ~qXj
-
Cj
(1.111)
CHAPTER 4. DISCRETE GEOMETRY AND MOYAL
188 so that
.~J(xk) = Xk- r
-1
dJ
(1.112)
Finally since the fundamental transformation of the Combescure vectors satisfies
.T'iJ(c i) = Ci
f~}i j d J (i 7~ j)
(1 113)
~jj
by performing the discretization (1.100) $-J becomes the shift Tj in the nj variable. In this manner one obtains the QL equations (21N) AjC i = (TjQij)C j where
TjQij
-
[ResP(YiJ(P'q)r ~esp-(~i](p ' q)r
[Resp(ViJ(P'q)~iJ)]~JJ(q)
-
[R--~Sp(-V--~(~,q)~ii)]~jj(q)
(1 114)
are the shifted rotation coefficients of the lattice. 4.1.5
0 methods _
In [254] it is shown how c0 methods can be used in studying MQL and multidimensional circular lattices (MCL). The cOtechnique of deriving Darboux equations Oi/3jk - ~ji/3~k for example goes back to Manakov and Zakharov and is expounded at length in [102, 103, 154; 1006]. We extract first some general material on c~ methods from [154] and then sketch some further developments from [102, 254]. This approach also brings out again the role of the Hirota bilinear identity and tau functions. Thus first a background situation here goes back to [162, 558] where the Hirota bilinear identity was derived from the D-bar framework. This connection involves algebraic techniques from Sato theory on one side and analytic techniques from D-bar on the other. Connections based on Hirota as in [162, 558], or more generally in [102], form a bridge or marriage between the two types of technique and touch upon the intrinsic meaning of the whole business. Now for the background derivation of [162, 558] we consider the (matrix) formula _
&r
~, ~) = f f~ r
~', ~')R0(~', ~', ~, ~) d~' A ~';
(1~15)
&~(x', .,,, ~) = - f f~ Ro (,~, .~, .,,', ~')~(x', ,v, ~'),~),' A ,~' Multiply by ~(x', ,~, ~) on the right in the first equation and by r second to obtain
J J~ O[~2(x,)~,~)(b(x',)~,~)]d)~A d ~ = - f o ~
A, ~) on the left in the
~2(x,)~,~)(~(x',)~,~)d)~-O
(1.116)
which is the Hirota bilinear identity when Oft ,.~ C is a small circle around oc. We go next to [103, 102, 104, 1006] and from [102] take (iblb) r #, g) - g-l(p)X()~, #, g)g()~) with r / = ( A - ~ ) - 1 ( X ( / ~ , # ) ~ ?7 as )~ ~ p) and g ~ exp(~Kixi) where Ki(A) are commuting meromorphic matrix functions. It is assumed now that there is some region G c C where R(A, #) = 0 in G with respect to A and # (we will take this to mean that R = 0 whenever A or # are in G). Also G contains all zeros and poles of the g(A) and G contains a neighborhood of oc. For Ki(A) a polynomial in A this involves only {ce} c G whereas for g = ()~- a) -1 it requires {a, oe} c G. Some examples are also used where g ,.,, exp(~_, xiA -i)
4.1. INTRODUCTION
189
with G a unit disc. For now we think of G as some region containing oc and 77 = ( A - p ) - i which leads to (gl ~" g(A, x) and g2 ~-, g(A, x')) cO,xX(A,#) = 27ri~(A - #) + / c d2z"X(v'P)gl(v)R(z"'A)gl(A)-l; (~xX*(A, #) = 27ri~(A- # ) - / c
(1.117)
d2zJg2(A)a(A'v')g2(z")-lx*(z"'#)
One can also take R(x, A',A) = g(x,A')Ro(A',A)g-I(x, A) and in (1.117) we should think of R as an R0 term and write (1.117) as
Ox [g1(p)~b(A, P, gl)g11(A)] = 27ri~5(A- #)-+-
+ / ~- [g~(.)r
(1.118)
gl)gl 1(-)] gl (.)R0(.. ~)g;l(~)
Then/~(t,, A) = gl(~)R0(v, A)g~-I(A) plays the role of R in [162] and (1.118) becomes for gl analytic 0x~b(A, #, gl) - 2rrigll(#)6(A - #)gl(A) + / d2ur
gl)Ro(z.',A)
(1.119)
(actually/~ ~ /~1 here). One can also stipulate an equation (.) OiR(~, >,m) = Ki(,~)RRKi(#). Some calculations now give (&&) X*(#, ~, g) = -X(k, >, g) and, generally and aside from the formulas (1.117), we know that the functions X, X*, and the gi are analytic in C/G so by Cauchy's theorem (~, # E G)
O = -- ~G X(z"'A'gl)gl(z")g21(z")X*(z"'#)d~ -- s
X(tJ'A)gl(~')g21(z")X(#'~)dz"
(1.120)
Note here that from (1.117) one knows X(A, #) ~ (A-#) -1 for A --+ # and X*(A, #) ~-" (A-#) -1 as well. Then from (1.120) and a residue calculation one obtains for gl - g2 another proof of (&&), requiring only that the gi be analytic in C/G. Finally we can write (1.120) in terms of ~b via (&&), namely
0- f r Jo G
gl)~b(I.t, z4g2)dz.,
(1.121)
which is a more familiar form of Hirota bilinear identity (but now generalized considerably). One can derive the Darboux-Zakharov-Manakov (DZM) equations immediately from the Hirota bilinear identity (1.120) as in [102]. Thus write g(z~) = exp[Ki(v,)xi] with Ki = Ai(l,,)~i)-1 SO that g1(z~)g21(tJ) = exp[Ai(~, - )~i)-l(xi - xi)] ' ' (gl ~ g ( . , ~ , x~), g2 ~ g ( - , ~ , x~)). / Look at (1.120) and differentiate in xi (with x 'i fixed); then let x i ---+xi to obtain
0=
G OiX(u'#'gl)]X('~'u'gl) + X ( u ' P ' g l ) u - )~i ;g(~'~''91) &'
(1.122)
Computing residues yields (g ~ gl) -O~x(a,.,g)
+ x(a~,.,~)A~x(X,a~,~)
+
Ai #-
hi
x(a,.,~)
Using the relation ( ~ b & ) r 1 6 3 = g-~(#)X(s equivalent to (20FF) 0ir p, g) = r #,g)Air
- x(a,.,~)a
Ai
a~ = o
(~.l~a)
this is immediately seen to be
)~i,g). To derive the DZM system
C H A P T E R 4. DISCRETE G E O M E T R Y AND M O Y A L
190
take for G a set of three identical disconnected unit discs Di with centers at A = 0. The functions Ki(A) have the form Ki(A) = Ai/A for A E Di and Ki(A) = 0 for A ~ Di (the Ai are commuting matrices and each Ki is defined in its own copy of C in [103, 102] - alternatively take e.g. three distinct discs in C and translate variables accordingly). Evaluating ( 2 0 F F ) for independent variables A, # C {0i, 0j, Ok} one obtains 0ir
#) = r162
Oi), 0ir
Oig,(Oj, #) = r
#)Air
0j) = ~b(0i, Oj)Ai~(A, 0i);
0j); Oig,(Oj, Ok) = r
(1.124)
Oi)Ai~2(Oi, Ok)
Now one can integrate equations containing A, # over OG with weight functions p(A), r so there results (no sum over repeated indices)
Oi(b = ]ifi; Oifj =/3jifi; Oifj = ]i3ij; Oi3jk = 3ji/3ik
(1.125)
r = f ~(#)g,(A, p)p(A)dAd#; /3ij = (Aj)I/2~2(Oj, Oi)A~/2",
(1.126)
where
=
f
The system of equations (1.125) implies that
=
f
OiOjfk -- [(Oj~)](-1]Oi]k -Jr-[(Oi]j)]?l]Oj]k;
(1.127)
O~Oj~O= [(Oj~)f(1]O~O + [(O~)~-l]oj~ OiOjfk -- (Oifk)(f(lojfi) + (Ojfk)(fj-loifj);
(1.128)
The first system in (1.127) is the matrix DZM equation with the first system in (1.128) as its dual partner. At this stage the development is purely abstract; no reference to Egorov geometry or T F T is involved. In this note, regardless of origin, we will refer to
Oifj =/3jifi (i r j); Oi/3jk = ~ji~ik (i 7(=j -r k); /3ij =/3ji;
(1.129)
as a (reduced) DZM system. In addition one will want a condition (20G) O/3ij = Ofj =
0 (0 = Y:~Ok) discussed in [154]. One recalls also that
Ok~ij = /3ik~kj (i ~ j r k); O~ij =- Oil3ij + Oj/3ji + ~
mr
~im~mj = 0 (i r j)
(1.130)
are referred to as Lam~ equations. They correspond to vanishing conditions Rij,ik -- 0 and Rij,ij - 0 respectively for the curvature tensor of the associated Egorov metric. Compatibility conditions for the equations (1.128) give the equations for rotation coefficients /3ij as in (1.125). One has the freedom to choose the weight function fi keeping the rotation coefficients invariant, and this is described by the Combescure symmetry transformation (20H) (f~) ~ - 1 cOifj ~/ = ][-loi~. Similarly the dual DZM system admits (Oifj)(f~) , , -1 -_ (Oifj)f[ -1. The function (I) is considered as a wave function for two linear problems (with different potentials) corresponding to the DZM and dual DZM systems. A general Combescure transformation changes solutions for both the original system and its dual (i.e. both p and t5 change). We note also that, according to [925], the theory of Combescure transformations coincides with the theory of integrable diagonal systems of hydrodynamic type.
4.1. I N T R O D U C T I O N
191
It is natural now to ask whether some general WDVV equations (cf. Section 5.6.3 for example) arise directly from DZM as formulated here, without explicit reference to Egorov geometry etc. and this is discussed in [154]. We emphasize however that the role of Egorov geometry and its many important connections to integrable systems, TFT, etc. is fundamental here but one can in fact use the Egorov geometry to isolate some algebraic features, after which the geometry disappears. In this direction consider a scalar situation where fj ~ r and assume/~ij -- ~ji. Then Oifj =/3jifi and Ojfi =/3ijfj which implies Oif ] - cOjf2 This corresponds to the existence of a function G such that ]'/2 = OiG = Gi. Then look at (1.128) where (no sums)
fkO~ojSk =
Af~o~Ao~f~+ Af3o3Ao~f~=> f~
(1.131)
f]
:=>"fki)iOj fk =
GkiGij GkjGji 4Gi ~ 4Gj
Also from 2fkOjfk = Gkj one gets 20ifkOjfk + 2fkOiOjfk = Gkji. Similarly 2fiOjfi = Gij implies 20kfiOjfi+2fiOkOjfi = Gijk and 2fjOifj = Gji implies Gjik = 20kfjOifj+2fjOkOifj. Hence, using (1.131), we get
2f~Gijk -- 4f~OifkOjfk + 4f2OiOjfk = GikGjk + f~ GkiGij Gi + GkjGji Gj I This implies
2Gijk =
G~kGj~ t GkiGij ~ GkjGji Gk Gi Gj
(1.132)
(1.133)
(such a formula also appears in [677] with origins in the Egorov geometry).
Similarly,
]2 = 0iG and (20I)Oi(~ = ]ifi =:~ fifii)jOi (~ = 89(GiGij + GiGij) while from (1.127)(1.128) one has OjOi(~ = fiOj~ + fjOiL = ~Ojfi + LOifj (1.134) Consequently
1 Further
1
(1.135)
/3ij -- ~ji -- coif j - Oj f i ::~ f i f j /3ij -- 1
f~ -
f~
1
-~a~j; L]jg~j = -~ ~j
(1.13~)
Note also from (1.125) O i ~ j k = ~ j i ~ i k , O k ~ i j -- ~ i k ~ k j , and Oj/~ik -- /~ij/~jk, while from (1.136) we have (20J) (Okfi)fj~ij + fi(Okfj)/~ij + fifjOk~ij -- 89 This leads to
-1G~jk -- { fkfjgk~9~j + fkf~9~jgjk + f~fjg~kg~j 2
(1.137)
fkfjOi/~kj + fkfiOj/3ik + fifjOk/3ij
Such relations seem interesting in themselves. Now one can reverse the arguments and, starting with G satisfying (1.133), define/~ij :
(1/2)[Gij/(GiGj) 1/2] = ~ji. Then immediately Gijk Gij Ok~ij = 2(GiGj)I/2 - 4(GiGj)3/2 [GikGj § GiGjk] 1
[GikGjk
= 4(a~Oj)l/~ [
a~
GkiGij ~
~
GkjGji] +-
aj
-
4(s
Gij
[Gik
(1.138)
Gjk
[-bT + W
C H A P T E R 4. D I S C R E T E G E O M E T R Y
192
GikGjk = 4Gk(GiGj)I/2 Also for fi = ~
AND MOYAL
=/~ik/~kj
one has
Gij _ 1 1/2~i = /~ijfj Ojfi = 2(Gi)1/2 - 2G~/--------52(GiGj) 3
(1 139)
which is the reduced DZM system (1.129). This shows that (1.133) characterizes reduced DZM. Thus, if we stipulate N indices, a solution of reduced DZM yields a function G satisfying (1.133), (1.136), (20J), and (1.137) for example. Conversely given G satisfying (1.133) one can define /3ij = (1/2)[Gij/(GiGj) 1/2] such that (1.138) and (1.139) hold for fi = (Gj) 1/2, and this corresponds to reduced DZM. Connectins of reduced DZM to WDVV are developed in [154] and we do not dwell on this here. Now a passage from the c5 formulation to discrete and q-difference DZM equations is indicated in [102] for example but we will follow [254] for convenience since the connections to MQL and MCL are spelled out there. The notation here involves planarity of elementary quadrilaterals via (19X) with compatibility (19Y) where Ti is given via (19Z) (cf. also [102]). In the limit of a small lattice parameter e one can write c2 Ti=l+e0i+~-0/2+...;
_2,~(2) Qij - e/3ij + e pij + " "
(1.140)
and Xi = Xi + . . - (we will distinguish now vector X and scalar X explicitly. The natural discretization of the further orthogonality condition (20K) Xi" Xj - 5ij requires that the elementary quadrilaterals of the MQL be inscribed in circles (producing MCL or discrete orthogonal lattices). The corresponding perpendicularity constraint is (20L) X i . T ~ X j + X j 9T j X i = 0 which in the continuous limit goes into the orthogonality condition (20K). Now one considers the matrix nonlocal 0 problem 0r 0h (cf. here (1.115) etc. the dependence of the One assumes that the that R depends on N
= fcg,(p)R(#,A)d#Ad)~
(A,# e C)
(1.141)
but we follow here the notation of [254] for simplicity- in particular M • M matrices r and R on ~ and # is omitted except when needed). c~ problem is uniquely solvable with solution ~b(A) ---, I as A ~ oc and < M discrete variables n via
TiR(p,)~,n) = ( I -
Pi + # P i ) R ( # , ) ~ , n ) ( I -
Pi + ~Pi) -1 (i = 1 , . . . , g )
(1.142)
where Pi ~ (Pi)jk = 5~jS~k with 1 < i < N and 1 < j , k < M . Equation (1.142) admits for example the solution N R(#,)~,n) - r I ( I 1
N Pk + #Pk)nkRo(#,A) I - [ ( I - Pk + )~Pk) -nk 1
Standard methods in c~ philosophy say then that the solution r ization as indicated, solves the discrete linear equations
Lij()~)g,(s Lijr163 = P j T i r 1 6 3
(1.143)
of (1.141), with normal-
- 0;
Pi + )~Pi) - (TiQji)r163 - Pj~2(s
(1.144)
4.1. INTRODUCTION
193
where r = I + A-1Q -+- A-2Q (2) + O(A -3) as A --+ oc and Qji = PjQPi. Now in order to completely exploit (1.144) one mtiltiplies from the right by Pk and Pi where k ~: i to obtain AiCjk(s = (TiQji)g'ik()~) (j, k # i); (ATi- 1)r
(1.145)
= (TiQji)g'ii()~) (j ~: i)
where Cjk()~) = Pjg'(s Evaluating (1.145) at ~ = cc one gets the integrable nonlinear equations associated with the compatible linear system (1.144) and solved via the c~ problem (1.141). The ~-1 terms in the expansions of (1.145) give
AiQjk = (TiQji)Qik (j, k # i); TiQ~ ) = Qji + (TiQji)Qii (j # i)
(1.146)
while the ~-2 term in (1.145) gives (20M) "-',"jkA~(2) --(TiQj~j~Q(2)~k (j,k ~ i). Since the equations (19Y) which characterize the N-dimensional QL in R M are obtained from (1.146) by choosing j, k = 1,..., N and j :/: k, one can say that (19Y) is solved by the nonlocal c~ problem (1.141). It is also convenient to evaluate (1.145) at two other values of ~, namely the distinguished point ~ = 1 and a generic point ~0 E C. At ~ = 1 one gets (20N) /kir ) -- (TiQji)~ik(1) for j =/= i and comparison with (19X) leads to identification of the M-dimensional vector Xi, (i = 1,... ,N, with the i th row of the matrix r The remaining M-N rows of ~(1) can be identified with the additional elements Xj (j - N + 1,..., M) of the natural basis of R M at points of the MQL. Therefore (20N) can be written as (200) A i X j -- (TiQji)Xi for i ~ j; these are the extended version of the linear system (19X). Finally in the neighborhood of a generic point ~0 E C where (20P) r - r + ( ~ - ~0 ~/~!-~)(~0) + O((~ - ~0) ~) one obtains AiCjk(s = (TiQji)r (j, k # i); (1.147) (s
1)r163
(TiQji)g'ii(s
(j # i)
and A../,(1) q/,(1) (A0) (j, k ~: i); z,~jk (A0)= (TiQji),~ik
(AoTi - 1)r ) (s
+ (TiQji)r
= -~r163
(1.148)
()~o) (j # i)
Generally the 0 method provides complex solutions of MQL equations (19X) and (19Y) and in order to obtain real solutions the datum R must satisfy the reality constraint (20Q) R(#, #, )~, X) = - R ( # , #, X, ~). In this event the solution 9(~, X) of (1.141) has the reality properties r ~) = r ~); Qij = Qij (1.149) Explicit solutions of (1.141) are often obtained via separable kernels of the form (cf. [102, 558]) (20R) R(#, ~) = (i/2) ~ K Uk(#)Vk()~). With this choice (1.141) leads to _
~()~) = I + ~ which reduces to
M - )~
~(#)R(p,)~')d# A d#
(1.150)
K
~J + E ~k < VkO~l uj >--< Uj > (j = 1 , . . . , K) 1 for unknown matrices ~j = < Cuj > where < f > = ~i / c d~ A d~f(~); (0~-1/)(~)= ~ 1 J'cdA'Ad~' ~ ' - ~ f(~')
(1.151)
(1.152)
CHAPTER 4. DISCRETE GEOMETRY AND MOYAL
194 Then r
and Q are expressed in terms of these solutions via K 1 K = I + Z ~ k ( O f ~k)(~); Q = - Z ~ k < vk > 71" 1 1
r
(1.153)
In particular if K = 1 one has = < u > ( I + < vOflu >)-1; Q § r
< u > (I+ < vc~flu >)-1 < v >;
I + < u > ( I + < v0~-lu >)-1(0~-1v)(/~)
(1.154)
Real solutions of the MQL equations (1.146) and ( 2 0 0 ) are obtained by imposing the reality condition (20Q); this implies that for arbitrary matrix functions fk, gk one will have
R = -~
Uk(#, #)Vk(A, ~) + ~ [fk(#, #)gk(A, ~) + fk(#, #)gk(~, A)] 1
(1.155)
where Uk(A, ~) = uk(~, )~) and Vk(A, ~) = vk(~, ,k) with K = Ks + 2Kp. One considers also integrable reductions of the MQL equations via a linear constraint of the form (208) R T ( p - I , A -1) = 1#I4~2[F(,X)]-IR(,X,#)F(#). This will be acceptable if and only if F is a diagonal matrix with (20T) ,~F(A) - +)~-IF(A-1) which is conveniently parametrized in the form AF(A) = A(A) -t- A(A -1) in terms of an arbitrary diagonal matrix A. Such constraints (20T) imply a nonlocal quadratic constraint on ~ of the form
; ~ ~2(A)F(A)~2T(A-ldA + / c ~(A)[O~F(A)]~T(A-1)dA A d~ - 0
(1.156)
where Coo is a circle with center at the origin and arbitrarily large radius and the integration is counter-clockwise. To utilize all this one chooses A in suitable classes for which numerous examples are given in [254]. In particular for MCL one uses A(~) = a(,X)I with a = 1 / ( ~ - 1) and f+(A) = (1/A)[a(A) :h a(A-1)] leading to (20U) r -- (1/2)[~(0) + ~T(0)]. One also spells out the MCL conditions in terms of MQL but we will omit the details here.
4.2
HIROTA, S T R I N G S , A N D D I S C R E T E S U R F A C E S
Notations such as (21A), (22A), (23A), (24A), (25A), and (26A) are used. We will try now to unify the treatments of [536, 537, 539, 828] and those of [102, 249, 250] for 2-D situations (cf. also [582, 830, 831, 835, 836, 870]). In particular we want to clarify the notations in Section 4.1 and examine connections between Moyal and KP suggested in [828] (cf. (1.52) - (1.59) and the discussion thereabout). We also want to spell out more explicitly the connections indicated in [828] and discussed briefly in Section 4.1.1. Another theme worth noting here is the coincidence of the discrete geometry studied in Section 4.1 with the "classical" string model (cf. [147, 830, 831,834, 835, 836, 870, 884]); they share in common the HBDE. The muse of D-brane technology in recent work on strings and M theory (involving K theory, coherent sheaves, etc.) ignores sometimes the role that integrability has played (and continues to play) in the development of string theoretic ideas. However in any case there are often impulses to believe that physics, and other natural "structures", correspond to (or perhaps are) mathematics. So much for platonism here.
4.2. HIROTA, STRINGS, A N D D I S C R E T E SURFACES 4.2.1
Some stringy
195
connections
One goes back here to strings as seen in the 1987-1988 era (a point of view still of interest I think). A number of mathematical connections to integrable systems were indicated in [531, 828, 831, 834, 835, 836, 837, 838, 884] for example (cf. also [147] for some survey material and [210] for fundamental background material- the free fermion picture of [253]. sketched in Section 4.3.3, is important here and this goes back to [210] for example). We extract now from [835]. Thus one recalls that HBDE includes all of the soliton equations belonging to the KP hierarchy in various continuum limits and the same equation is also satisfied by string correlation functions (cf. [831, 884]) as well as by transfer matrices of solvable lattice models in statistical physics (cf. [524, 603, 651] and see Section 3.6). Note that from a mathematical point of view HBDE can be identified with the Fay identity arising in the study of algebraic curves. Further the deformation of integrable systems to discrete form preserves integrability when the deformation satisfies certain geometrical constraints. The purpose of [835] is to show that the higher dimensional extension of discrete conjugate nets can be naturally interpreted by means of string theory. We summarize briefly some results of Section 4.1.4 in the repetition of formulas but this should contribute to clarity). Z D the D-dimensional QL x is a map from the lattice q to elementary quadrilateral is planar (note x .-~ x and q ~ q). Laplace equation
notation of [835] (there is some Thus for q - (ql, q 2 , " ' , qD) E R M, M ~ D, such that every It can be characterized by the
A ~ A v x - T~((A~,Ht~)H~l)At~x- T~,((A~Hv)H;1)A~,x - 0
(2.1)
where Trois the shift operator qg -o q~ + 1 and Ag = T ~ - 1. The Lam6 coefficients H~ satisfy At~AvH p - T t ~ ( ( A v H , ) H ; I ) A , Hp - T v ( ( A , H , ) H ; 1 ) A , H p - 0 (2.2) If one defines tangent vectors Xv via ( 2 1 0 ) Avx = (TvHv)X, f o r , = 1 , . . . ,D then (2.1) and (2.2) turn out to be AuXv = (T~Qvu)Xu; AuH,,, -- (T~Hu)Qw ,, (p 7/= z,,)
(2.3)
The consistency conditions for these equations yields for the rotation coefficients Qt~- the relation (21P) ApQt, v = (TpQ~p)Qpv for # =/= ~ =/= p -y= #. These equations (2.3) and ( 2 1 P ) are discrete analogues for the conjugate net equations
0H.
09~
0X~ = fl~,X~,; = fl~,~H~,; = flppflp, Ou~ Ou~ 0%
(2.4)
Now a D-dimensional congruence ~(q) is a map from the lattice to lines in R M such that every two neighboring lines t~ and Tit~ are coplanar. If all points of the QL x(q) belong to the rectilinear congruence t~(q) then x and t~ are called conjugate. The focal lattice y~(t~) is a lattice constructed out of the intersection points of the lines t~(q) and ~(q + %) and one can show that focal lattices of congruences conjugate to QL are QL. The map from x to the focal lattice y~(t 0 is called a Laplace transformation and is given by (21Q) s = y ~ ( ~ ( x ) ) = x - (Ht~/Quv)X u. In [249, 250, 251] it is shown that HBDE arises as an equation satisfied by invariants under this transformation. On the other hand the map of x to the QL conjugate to the #th tangent congruence of x is called a Darboux transformation and is given by (21R) s = x- (r162 x where r is a solution of (2.1).
CHAPTER 4. DISCRETE GEOMETRY AND MOYAL
196
Now for strings, the HBDE characterizes an analytical property of the string correlation function. Thus following [831, 884] (cf. also [147]) let x E C be a complex proper time of a string. Then the positive and negative oscillation parts of the string coordinate X ~ (in the target space), defined via X t~ - X~_ + X_~ (1 _< # _< D), where X+(z) -- iplog(z) + E~(1/v/-d)a~z -n and X_(z) = x+E~(1/v/-d)a~z n, satisfy (21S) [X~(z),X'_(z')] = 5"~log(z z ~) (requiring k~. ku = 0 - see below and note k~ ~ k~et~). Interaction of a ground state particle with the string takes place through the vertex operator (see Section 4.4.5 for more details)
where k E R D is the momentum of the interacting particle. Since the string is quantized the vertex operators satisfy (kk' is unclear- it seems to mean E~D=I k~k~ with k k ' ~ k . k' and et~ .e~ = 5~v)
V(k,z)V(k',z') = ( z - z')kk' " v ( k , z ) V ( k ' , z ') := (-1)kk'v(k',z')V(k,z)
(2.6)
where :: denotes normal ordering. For a given configuration of strings (say IG >) the correlation function will be (21T) F a ( k l , . . . , k g ) = < 01Y(kl, Z l ) " - V(kg, zg)lG > where 10 > is the vacuum state annihilated by the X_~. When IG > itself is the vacuum the amplitude becomes (cf. [837]) F0(kl,.'' ,kg)=<
01Y(kl, Z l ) ' ' ' Y(kN, zg)lO > = I](zj -- zi) kjk~
(2.7)
i>j
(kjki = Y]~ kjtt ki# = k j . ki). It was shown in [531, 831, 838, 884] that if one defines f(kl,'",kN)
F a ( k l , ' ' " ,kN) = Foo(-~I: ,kN)
(2.8)
(cf. also Section 4.2.5) then it satisfies the HBDE (for fixed # in k~)
Za(Zb- zc)f(ka + e~,kb, kc)f(ka, kb + et,, kc + e~)+
(2.9)
-~-Zb(Zc- - za)f(ka, kb + e~, kc)f(ka + %, kb, kc + e~)+ +Zc(Za - zb)f(ka, kb, kc -~-ett)f(ka -t- ett, kb +ett, kc) = 0 Thus discrete geometry and string models share HBDE and this might imply that the discrete geometry is realized by the string model; in fact this is claimed to be the case in [835]. The main part of the argument comes from [253] where the KP hierarchy was described in terms of discrete geometry (cf. Section 4.1.3). On the other hand the correspondence between KP and the string model was established in [831, 884]. Therefore the remaining problem is to find a translation from one language to the other. One shows that the string model variables can be directly associated to the discrete conjugate net. To see this one recalls that the KP hierarchy (following [210]) is a set of infinitely many bilinear differential equations of Hirota type (cf. also [147]). All of their solutions are given by a single tau function T(tl, t2,'" "). The D-component KP hierarchy (as in [210]) involves D-component vectors tn (we use here a more correct vector notation) and in addition one needs charge variables q taking values on the D-dimensional lattice space. Writing the MKP tau function as T(q, t l , ' " ") one defines (for %v - s g n ( v - #), %~ - 1)
W t ' u ( z ) = e t ' u z l - ~ " ~ ( T ( q + e t ' - - e u ' t lT(q, - - z etl, u ' t 29-9")- ( z 2 / 2 ) e u ' " ' ) )
eEnz-ntn~
(2.10)
4.2. HIROTA, STRINGS, A N D D I S C R E T E SURFACES
197
(where # , v = 1 , . . . ,D). Note this is simply a change z --. 1/z in the last two terms of (1.66) with a corresponding correction in the first term (cf. also (3.31). This W~. is known to satisfy (21U) (OWt~u/Otnp) = ~t~n(nP)w'~uwhere (B(np))t~v is a matrix differential operator with #, v, p = 1 , . . . , D (cf. [210]). The notation of [211] is (W ~ matrix of wave function components et~u- cf. (1.66))
0
W(t) = e~(t, O)W(t); 0 ,..,
(2.11) c~-'-1
where c~ = 1 , . . . , m
~ D here, n = 1 , 2 , . . . ,
W is an m • m matrix, and t = (tn~) with
B~ = EaO n + ' . . and Ea = 5tLahva for #, u = 1 , . . . , m so B (no) above depends only on O/Ot~ for c~ = 1 , ' " , D 9 Further, writing .=.(t,k)= Y~n=I ~ ~ =m1 tnEak n, one sets W(t, k) ITV(t,k)exp(.=.(t,k)) with ITd = 1 + Wl(t)k -1 + .... The tan function in this situation is (cf. also [147, 210])
dlog(T) =-Resk=ccTrlTV(t,k) -1
1 cO ) IM(t,k) dS A dk O k - ~cr ~m kn+l 0tn~
(2.12)
n=l a=l
The approach of [253] interprets the variables t~ in t l as the coordinates u of the conjugate net, whereas the remaining variables tn (n >__2) describe iso-conjugate deformations of the nets. T h e n Wt~v is associated with the components of the tangent vector (X~)v and Bp. is associated with the rotation matrix ~ u ; thus (21V) tl ~-~ u with W ~ ~-, ( X ~ ) . and B~u ~-~ ~t~u. In terms of the M K P tan function this means (cf. (1.67))
f l ~ = %~
T(q + et~ -- e u ' t l ' " " ") T(q, t l , ' " ")
(2.13)
Moreover an operation of a KP vertex operator to W can be interpreted as a Levy transformation of the tangent vector X via (this is not clear - cf. however ( 2 1 X ) below)
s (Xv ) ~-, z-5,~, exp [ f i
-1
l zn 0 ]
j
(2.14)
where W v is the v th row vector of W (thus Xu ~ Wu with components (Xu). - Wu.). This corresponds to a vertex operator of the form -O/Ok ~ = -O/Ott~ - ~-~(zn//n)(O/OtPn) (cf. (1.75), (1.82), etc. for the Levy t r a n s f o r m a t i o n - the relation to (2.14) is not entirely clear but see below for more detail). Next one shows the realization of the conjugate nets by means of string theory. The key is the Miwa transformation given by (N ~ N external particles) N
to = ~ 1
1
N
kj; 'tn = - ~ kjz~ (n = 1, 2,..-) n 1
(2.15)
One introduces to here as the center of mass m o m e n t u m of external particles. It was shown in [831, 884] that the M K P tan function is the same as the ratio of correlations functions in (2.8), i.e. ( 2 1 W ) T(t0, t l , ' . . ) = [ F c ( k l , " ' , k N ) / F o ( k l , " ' , k N ) ] (the F's are correlation functions for various string configurations- F0 ~ vacuum). Here to is interpreted as the fundamental lattice q of the conjugate net. The other coordinates are related via (21X) (O/Okjt~) = (O/Oto~)+ Y~C~(z~/n)(O/Otnt~) for j = 1 , . . . , N (cf. (2.15)). Denote now by Tj, the operation shifting kj to kj + e , , i.e. (21Y) Tj, = exp[O/Okj~]. Then one can
C H A P T E R 4. D I S C R E T E G E O M E T R Y
198
AND MOYAL
prove that, in the original notation of [835], (zj ,.,, kj but ke in [835] is ambiguous- it seems to mean (ke)t, - cf. (2.7) and recall kk' ~-, k . k')
UolT,.U0 = I I ( z , -
z,) ~ = z~~
-
z;ntn.
(2.16)
g
We observe here that
N
t~ = 1 ~
=
k~ "=
(:,)n
1
z~
oo
k~ z~ =~ ~
n i=1
=-
k~log
n
z~ n t~n
( :,):
(2.17)
1
1-z~
-
i=1
k~log Zj
i=1
while log 1-I(zj - zi) ky - EN=I k~log (zj - ze). Thus
exp
-
z~nt~n
= exp
log =
= z, z ~ ;
zj - ze Zj
=
(2.18)
I I ( z , - z~) ~; = z ~ I I ( z , - z~) ~; g
t~
so zj ~ e x p ( - ~ z~-nt~n) = 1-Ig(zj- zg) ky as desired in (2.16). One notes that the total m o m e n t u m to of the correlation function F0 does not change because it has no background (not clear). Let T~ be the shift of q to q + e~ as before. Combining all this the tangent vector corresponding to (2.10) can be rewritten as ( 2 1 Z ) ( X , ) j , = %~F~-IT~Tj,-1FG (it" seems that (2.16) for # = • provides the exponential factor in (2.10) note also the bracket in (2.10) will have the form (Fo/FG)T~Tj-,I(FG/Fo)). Similarly the rotation matrix becomes (22A) Q~u = %uFGITpT~-IFG (cf. Section 4.5 for more on this). The operator of Darboux transformation is proportional to Tj~ 1 as one sees from (2.14), (21X), and (21Y); however it is more simply defined via (22B) s = F0-1Tj-~IF0 (Darboux transformations historically refer to Levy or Laplace transformations, among others, and we recall here (1.81)). Here it is assumed that Tj~ 1 acts on all functions to the right and there is a correspondence here to the vertex operator (2.5) of the string theory. From these relations one can draw a picture of the lattice conjugate net which is realized by the string models. One considers the space of momenta of N external particles ( k l , . . . , k g ) . Also consider the momenta of the background so that the total momenta q of the external particles become independent variables qi. Suppose the background space is compactified into tori in all D directions. Then its momenta, and hence q involves only integer values. Consider the lattice space of q E Z D as the fundamental lattice of the conjugate net which is embedded in the space a DN of the momenta of the N external particles. Based on this setting one can interpret the tangent vector X~ of (21Z) via the gauge covariant shift operator acting along the #-direction of the q space. Similarly the rotation operator Q~, of (22A) has the meaning of a gauge covariant rotation operator in this space. The Darboux transformation operator of ( 2 2 B ) is somewhat different. It acts on the space of momenta of external particles {kj } E a ON, again in a gauge covariant form. We will write out a few more details later. 4.2.2
Discrete
surfaces and Hirota
Let us spell out the QL situation for maps x " Z 2 ~ p M with shift operators Ti (i - 1, 2) and TIT2x E < X, Tlx, T2x > (cf. here in particular [249, 250] as well as general remarks in
4.2. HIROTA, STRINGS, AND DISCRETE SURFACES
199
Sections 4.3 and 4.4.1 - we put in vector notation now for increased clarity). The Laplace equation is then AIA2X = (TIA12)A2x + (TIA21)A2x (2.19) where the A12, A21 " Z 2 ~ R define the position of TIT2X with respect to x, TlX, and T2x (recall/Ni = T i - 1). The Laplace transformation is defined via (22C) s = x(1/Aji)/N~x and one has
Aj~ = TjAji(TiAij + 1 ) - 1;
s s
= T71(Tis
(2.20)
(Aji + l)) -
Assuming the transformed lattices are non-degenerate we have as before ( 2 2 C C ) ~'ij 0 ~ j i = s o s = id and one can define a sequence of QL via (22D) x g = s with s = s From (2.20) the above Laplace sequence leads to /k2Ag21
A~I
TIAgl2- ''12 Ag+ 1 (Ag+l + 1) (TIA~2 + 1)t~-12
/N1A~2
Ai2
(2.21)
T2A~9. - A~2-~1 (T2A~21 + 1 ) ( A ~ 1 + 1)
We note also from [249] A~+I+ 1
Ae21
A~]-I + 1
A~2
(2.22)
T1A~12 + 1 = I"2A~1; T2Ae21 + 1 = T1A12
Now the basic numerical invariant of projective transformations is the cross ration given via (22E) Cr(a, b; c, d) = [(c - a)(d - b)/(c - b)(d - a)] and one defines the function Kij as the cross ratio of x, s Tix, and TjEij(x). Elementary calculations give first
Ti Aij -+-1 TjAji
Aji A/x; Tjs Aji
Tix- s
= 1+
TjE.ij (x) - s (x) =
- x = ---/kix;
1 + TiAij ) TjAji /Nix
1
Aji
(2.23)
Consequently
Kij = Cr(x,s From (2.20) there results then
~ij(gij)
Tjs
s
= Aji(TiAij + 1) - TjAji (1 + T~A~j)(1 + Aj~)
= Kij with
= T~_1 (K ij(TiTjK ij) (TiKij + 1)(TjKij + 1) T~Kj~ + 1 \ (T~K~j)(TjK~j)
1)
(2.24)
(2.25)
These equations can be rewritten in terms of K = K12 as
7'2
-K--I-+--1 T1 \ - ~ -_(_~[
= (T1K e) (T2 K e)
(2.26)
CHAPTER 4. DISCRETE G E O M E T R Y AND MOYAL
200
which is the gauge invariant form of the Hirota equation. Note here K ~ refers to K~2 determined via ( 2 2 D ) and (2.24). This should be compared to the standard gauge invariant form of HBDE from [993], namely (1 + Y(Xl, X2, X3 + 1))(1 + Y(Xl,X2, X3 - 1)) Y ( x l , x 2 -+-1, x3)Y(xl,x2 - 1, x3) - (1 + Y - I ( x l + 1, x2, x3))(1 + Y - I ( x l - 1, x2, x3)) ;
Y(xl,x2, x3) = T(Xl,X2, X3 + 1)7(Xl,X2, X 3 - 1) T(Xl + 1, x2, X3)T(Xl -- 1, X2, X3)
(2.27)
and this will be discussed later. Now one reformulates the Laplace equation (2.19) as a first order system via suitably scaled tangent vectors Xi for i = 1,2 in the form (22F) Aix = (T~H~)Xi in such a manner that (22G) A j X i = (TjQij)Xj. The Lam~ coefficients Hi satisfy (22H) AiHj = (TiHi)Qij (i =/=j) with Aij = (AjHi/Hi). Further
s
s
QijAj
~
(2.28)
and the tangent vectors of the new lattice are given by
AiQij f~iy(Xj) - - A i X i nt- Qiy Xi; f-.ij(Xj) - - ~ i j Xi
(2.29)
The following equations also hold 1
t:~j(Qj~) = Q~;
(9,.3o)
( QijTiTjQij ) f-.ij(Qij) = Tj-1 TiQij TjQij (1 - ( T i Q j i ) ( T j Q i j ) ) leading to
IQ~ = TI I[\ T2Q~ ) A2 ~q~ Q~-I
T2Q~+ I q~
(2.31)
To give geometric meaning to the tan function one now introduces the vectors :~i pointing in the negative directions, namely ( 2 2 I ) s - (T(-1Hi)Xi or Aix -- Hi(TiXi) where /~i = T/--1 - 1 is the backward difference operator. The scaling factors /ti are chosen so that the /~i variation of :~j is proportional to :~i only. One then defines backward rotation coefficients ~ij via (22J) s = (Ti-l(~ij)Xi or A i X j = (TiXi)(~ij which implies (22K) AjfIi - (TjQij)Hj. Since the forward and backward rotation coefficients describe the same lattice x they must be dependent and, defining Pi " Z 2 ~ R as the proportionality factors between Xi and Ti:Xi (both vectors being proportional to Aix), one has (22L) Xi -pi(Ti:Xi) and TiHi = -(1/pi)Hi (i = 1,2). Consequently one can can show that
pjTjQij = piTiQji; Tjpi = 1 -(TiQji)(TjQij) Pj
(2.32)
Since the right side of the last equation is symmetric with respect to interchange of i and j there exists a potential T 9 X 2 --~ n such that (22M) pi = (TiT~T) (note Tj(TiT/w) = (TjTi'r/TjT) ) and
(TiTjT)T (T~)(Tj~) = 1 -(T~Q~)((TjQ~j)
(2.33)
4.2. HIROTA, STRINGS, A N D D I S C R E T E SURFACES
201
Further from (2.30) and (2.31) one has
Q ij Ti Tj Q ij rTi Tjr = TjQijTiQijTirTjr
1 - (Tis163
(2.34)
which, due to (2.33) allows the identification s = rQij (note rij is defined in [249] as rQij and thus Tij ~ s - cf. also (19E) for another form of Tij based o n Sij and recall the relations between Sij and s indicated in (2.25) and thereabout). In any event, writing (2.33) in terms of the t a t function and its Laplace transforms gives the original Hirota form of the discrete Toda lattice, namely (cf. equation 3 in (2.37))
TeT1T2 Te = (TITe)(T2 Tg) -(T17-e-1)(T2 Te+l ) 4.2.3
(2.35)
More on HBDE
In this sectin we want to sort out the various forms of HBDE (see e.g. [459, 460, 830, 834, 836, 993]). First let us recall that the Fay identity for tau functions is (see e.g. [7]) E ( s 0 -- Sl)(S2 - S3)T(t + [SO]+ [Sl])'r(t + [82] + [83]) = 0
(2.36)
cp
where cp ~ cyclic permutations of 1, 2, 3. This is the same as (1.139) and we refer to Section 5.1.4 for the differential Fay identity (1.64). For HBDE we have the equivalent forms (1.127). (1.128), and (1.129), namely
a f ( A + 1, #, u ) f ( A - 1, p, r,) +/~f(A, p + 1, u)f(A, #
-
1, u) + 7f(A, p, u
-
1)f(A, p, u + 1) = O;
a(b - c)r(p + 1, q, r)T(p, q + 1, r + 1) + b(c - a)r(p, q + 1, r)T(p + 1, q, r + 1)+ +c(a - b)r(p, q, r + 1)T(p + 1, q + 1, r ) - - O;
(2.37)
ag(n, e + 1, m)g(n, g, m + 1) +/3g(n, e, m)g(n, g + 1, m + 1)+ +Tg(n + 1, g + 1, m ) g ( n - 1, g, m + 1) = 0 To connect (1.129) and (1.127) we can write (cf. [834]) t~ = (1/2)(A + tt + u) and m = (1/2)(p--A--L,) with n = u while setting f(A, #, u) ~., g(g+(1/2), m + ( 1 / 2 ) , n). As for deriving the HBDE the derivation of Miwa [740] involved looking at a KP t a t function in a particular operator form and then proceeding to derive (1.128) in Miwa variables (cf. [147, 210, 211, 212, 525] for such operator calculations). It is instructive to sketch here another proof following [836] (cf. also [147]) which employs relations to string theory. Thus one deals with free fermion fields satisfying (22N) {era, ~bn} = 5ran and {r ~bn} = {~b~, ~bn } - 0 where {f, g} ~ f g + gf. The vacuum state of the fermion system is defined by ( 2 2 0 ) ~bn[0 > = < 0[r n = 0 f o r n >_ 0 and < 0[r = ~bn[0 > = 0 f o r n < 0. Let V (resp. V*) be the linear space of ~bn (resp. ~bn) and consider operators g E Clifford group satisfying (22P) g~bn = E Cmgamn and ~bng = E anmg~bm (dmn E C). Then t a t functions can be defined via (22Q) T(t, g) = < O[[exp(H(t))]g[O > where H(t) = E ~ tn EmeZ gam~bm+n (note H(t)]0 > = 0 but < 0[H(t) -~ 0). To make connections with t a t functions and string vertex operators (as in Section 4.1) one writes ~b(z)= En~Z ~bnzn and r = En~Z ~bnz-n-1 with
x = ~1 an=
dz~(z)r VmVn+m = ~m~z*
~-~
p=
2~i
dz~(z)r
1 j dz~(z)V(z)z-n;
2~X/~
(2.38)
CHAPTER 4. DISCRETE GEOMETRY AND MOYAL
202
at. = V/~i . ~ ~ r
= 27rv/-n1~l dzr
n
-
The integrals a r e around z : 0 and with suitable c a r e (cf. [147]) one obtains (228) [am, at] 5mn with Ix, p] = i. Then the string coordinates can be written as Co 1
X(z) = x + iplog(z) + ~ - ~ = (anz n + at z-n) = = 2--~ 1 / dwr
log ( w - p z )1 Z
-
-
pW
(2.39)
p--*l-0
This leads to the particle emission vertex operator (cf. (2.5) - (2.6))
V(k, z) =" e ikX(z) "= eikz+(z) eikX-(z) ; Co 1
(2.40)
Co -~n
Z+(z) --iplog(z) + ~l -~atnz-n; X-(z) - x + ~I
anzn
leading to the tau function nature of expressions such as (2.7). Now put in Miwa variables via (22T) tn -- ( l / n ) ~--~x~pjz~ w i t h g(p) = g(t) + itoz and H(t) = i ~ tnanv/-n, where in fact the zj ~ Koba-Nielsen variables and the pj ~ momenta of strings in the ground state (cf. [147, 836]). Set then
dz < oleH(P)r
I = ~
o
><
oleH(P')(~(z)gl o >
(2.41)
(~ around z = 0 with g as in ( 2 2 P ) ) . Using ( 2 2 P ) and integration leads to I = Z
< 01~H(P)V~gl ~ > < 01~H(P')r
~
>--
nEZ
~
-
< 01~u(~)gr
>< 01~U(p')gr
>-- 0
(2.42)
nEZ
(via ( 2 2 0 ) ) . This proof has the same form as the standard proof of ( 1 9 H H ) by free fermion operators (cf. [147] for example) but now S(t) is replaced by H(p). Now a little argument shows that co
< oleu(P)r
= 1-[(z- zy) p~ < OleiZj~JX-(zJ)+iX-(z);
0 Co
(2.43)
< 01~H(P)~(z) = 1-I(z- zj)-PJ < 01~E~ pjx-(zj)-~x-(z) 0
and one chooses (22U) p~) = P o P u t t i n g this in (2.41) gives
0= ~ • < 01e
1
1, p; = pj + 1 for j = 1,2,3 with pj = 0 otherwise.
/dz (Z-
i ~ p~x-(z~)+ix-
z-zo Zl)(Z-
Z 2 ) ( Z - Z3) X
(2.44)
i ~ j p~x-(z~)-ix(;)gl0 > < 01e (z)gl0 >
Since X - is expanded in positive powers of z it is analytic at z - 0 so contributions to the integral occur only at Zl, z2, z3. Evaluation gives z0 - zl)
(Zl--Z2)(Zl--Z3)
r(p0,pl + 1,p2,P3)T(po -- 1,pl,P2 + 1,p3 + 1)+
(2.45)
4.2. HIROTA, STRINGS, AND DISCRETE SURFACES + +
zo
Z2
(z2-zl)(z2-z3) T(pO,Pl,P2 + 1,p3)T(p0z0
-
-
203
1,pl + 1,p2,p3 + 1)+
Z3)
T(pO,Pl,P2,P3 + 1)T(p0 -- 1,pl + 1,p2 + 1,p3) = 0 (Z3--Zl)(Z3--Z2) which is exactly (1.127) where (22V) f(A,#, y) = f(P2 +P3 + 1,p3 + p l + 1,pl + p 2 + 1 ) = T(PO,Pl, P2,P3) with a = zl(z2 - z3), 13 = z2(z3 - zl) and 7 = z3(zl - z2) while z0 = 0 (the Po term could be ignored ouside of the string context). -
-
We mention still another aspect covariant difference operators (22W) while D+ = exp(O~) - 1 so D+f(6, m, with D_ f (6, m, n) = f (6, m + 1, n) -
of HBDE, following [830, 834]. One considers gauge V+ = U+D• 1 where U+(6, m, n) are integer valued n) = f(6 + 1, m, n) - f(6, m, n) and D_ = exp(Om) - 1 f(6, m, n). Define also
E+ -- c+U_exp((Os .-~ 0n)U-1; E_ - c_U_t_exp((~m - (~n)U~ 1
(2.46)
(c+ arbitrary) and consider V+fn(6, m) = E+fn(6, m); V-fn(6, m) = E-fn(6, m)
(2.47)
(f(6, m, n) ~ fn(6, m)). These equations are invariant under local gauge transformations fn(6, m) --~ V(6, m, n)fn(6, m); U+(6, m, n) ~ V(6, m, n)U+ (6, m, n)
(2.48)
for V(6, m, n) arbitrary. It follows that ( 2 2 X ) [ V _ , E + ] = [V_,E+] = 0 and applying V• to (2.47) gives V-V+fn(6, m) = E+E_fn(6, m); V+V_f~(6, m) = E_E+f~(6, m)
(2.49)
The equations (2.47) are compatible if the equations in (2.49) give the same result and this is assured if (22Y) [V+, V_] = [E_,E+]. In this event (2.49) is equivalent to a single linear equation (22Z)[V+,V_]fn(6, m ) = [E+,E_]fn(6, m). Thus ( 2 2 Y ) d e t e r m i n e s the behavior of the gauge fields Ui whereas (22Z) is to be solved for fn(6, m); the combination is a necessary condition for (2.47) if IV+, V_] # 0. Note if [~7+, V_] = 0 (corresponding to U+ = U_) then (22Y) is trivial and the equivalence between (2.47) and ( 2 2 Y ) - ( 2 2 Z ) is lost. Equations (22Y)-(22Z) are written out explicitly in [830] and if one writes (23A) U+(6, m, n) = gn(6, m) with U_(6, m, n) = gn-l(6, m) then (2.47) becomes the duality equations of [834], namely V+fn(6, m) = c+ gn(6 gn-l(6, + 1,m) m) fn+l (6 -~- 1, m);
(2.50)
V - f n ( 6 , m) -- c_ gn(6, m) gn_l(6, m) fn_l(6, m + 1) (note these are form invariant under interchange of the roles of fn and gn - proved in [830]). The compatibility conditions (22Y) will have a bilinear form in the gn(6, m) and it will be satisfied if (1.129) holds for gn(6, m) = g(6, m, n) (third equation in (2.37)). Moreover/3/a could depend on (6, m) as well so thi~ is really an extension of HBDE. The corresponding linear equation for fn(6, m) will be Z
gn(6, m)gn-l(6, m) gn(6, m) 1) fn(6 -+- 1, m + 1 ) + fn(6+ 1 , m ) + 1 , m)gn-l(6, m + gn(6 + 1, m) -4
gn-l(6, m)
gn-l(6, m + 1)
fn(6, m + 1) - fn(6, m) = 0
(2.51)
(corresponding to a linear B/icklund transformation). The whole matter is then studied at some length in [830]; in particular Lax pairs are constructed and auto-B/icklund transformations developed for (the extended) HBDE.
CHAPTER 4. DISCRETE GEOMETRY AND MOYAL
204 4.2.4
Relations to Moyal (expansion)
In this section we add a few more equations and comments to Sections 4.1.1, 4.1.2, 4.1.3 and 4.2.1. We recall first some calculations from [789] as follows (cf. also [31, 59, 65, 84; 85, 123, 218, 278, 350, 379, 407, 419, 421, 479, 545, 804, 853, 940, 947, 959] and other calculations below); the calculations and viewpoint are instructive and will be useful in dealing with [537] etc. Thus one develops a Weyl symbol calculus as follows. Work with
dw = dp A dx and ~
(x ' p), setting {f,g} = V f . JVg for J = ( - 01 \
01
/ so { f , g } /
(fx, fp)" (gp,-gx) = fxgp- fpgx and recall i5 ~ -iI~Ox. The Wigner-Weyl theory (symmetric ordering) can be phrased as follows. A phase space function A({) is quantized (roughly) by replacing 4 - (x,p) with r (5,i5) via (du = duldu2, etc.)
A(~) = / a ( ~ ) ~ ~ ; (note e ivr ~ a = 5 ( v - u) ~ eiVr (2.52). In general (~ ~ (5,15))
~i = / a ( ~ ) ~ ~
~ A(~)
(2.52)
The Weyl symbol of A is aw(.~) or Aw = A(;) as in
1/ du / d4Aw(4)e iu'(~-~)
= (27r)2
(2.53)
Writing Aw(4)e -iur - [Aw(iVu)e -iur in (2.53), after integration there arises the interesting formula (23B) A = Aw[(1/i)Vu]exp(iu. r note here (1/27r) / Aw(iVu)exp(-iu . ~)d~ = (1/2r)Aw(iVu) f exp(-iu . ~)d~ - Aw(iVu)5(u) (23B) clarifies the ordering as the symmetric product; for example
One can also characterize operators via formulas like (cf. also [350])
< xIAlY >= ~I
/ dpeiP(X-Y)/hAw ( x + ) y2 ' p
(2.55)
which follows from (2.53)via < xl(-. ")IY > upon using < xlexp[i(ul , u2)'~]ly > = exp[iul((x+ y ) / 2 ] 5 ( x - y + hu2]; we recall that [i5,5] = - i h and when [A,/~] commutes with ft. and /~ then ezp(A)exp(B) = exp[(1/2)(.~ + B)]exp((1/2)[.~, B]) (Baker-Campbell-Hausdorff BCH formula). In the opposite direction one has
Aw(x,p)
/ d y e -ip'v/h < x + (1/2)vlAIx- (1/2)v >
(2.56)
which follows from (2.55) by changing variables and inverting the Fourier transform. Generally for a symbol calculus one needs the following (see below - z plays the same role as 4. i.e. z ,',., (z,p))
(AB)w(z) - (Trh)-2 / / dCdr
+r
+r
(2i/h)r162
(2.57) (2.5S)
(~B)w = ~xp -~V~. JVz, Aw(z)Bw(z') Zr
4.2. HIROTA, STRINGS, AND DISCRETE SURFACES
205
Here (2.58) is a consequence of (2.57) and provides a small h expansion of (AB)w while (2.57) can be viewed as providing a 9 product (23C) Aw * Bw =- (AB)w (definition). It follows that (23D) ih{Aw, BW}M = Aw * Bw - Bw * Aw = [A,/)]w (the last equation also being a definition). To verify (2.57)-(2.58) one shows first, from the definitions, that ^
,.
^
^
;B = f du f dwa(u)b(w)e-(in/2)u'Jwei(u+w)'r
(2.59)
where B ~ b as in^(2.52) and BCH has been used (ihJ arises from commutation relations). Now (2.59) for AB is essentially parallel to (2.52) for A so we have the analogous formula as in (2.52) ( ; / ) ) w ( ~ ) = f du / dwa(u)b(w)e -(ih/2)uJwei(u+w)'r (2.60) Now to get (2.57) put in the Fourier representations of a and b to obtain first (-4/))w(z) =
f d~ f" d~'Aw(~)Bw(~') f f dUdWeui(z-~)eW i (Z-~)(2'7r)4
e-(itV2)u'Jw
(2.61)
from which (2.57) follows upon shifting the ~ and ~ integrations by z leading to a uintegral equal to 6(~ + (hJw/2)), after which some rearranging does the job. To get (2.58) directly from (2.61) one rewrites the exponential factors in the integrand as {exp[(ih/2)Vz. JVz,]exp[i(u.z + w. z')]}[z,=z and pulls the operators out (cf. (23B) et suite). Some of the calculations above are not entirely clear and later we give more detail regarding the Weyl-Wigner-Moyal (WWM) calculus following [31, 59, 65, 84, 218, 278, 350, 379, 407, 419, 421, 479, 545, 804, 853, 940, 947, 959]. In [84] for example there are some useful explicit calculations. It seems appropriate here to insert a few formulas to complement the calculations in Section 4.1.1 (cf. also (4.43)-(4.48) in Section 4.4.2). Thus looking ahead to (4.43)-(4.47) and the analysis in (1.7)-(I.12) we can write some Fourier formulas as follows. First from the density of Sx | Sp C S(x,p) one can write
F(Ab) = ~
dc
F(a)da
(2.62)
Consequently in (4.48) one can write also AVbf
f ei)~(a•
U(~)~[f]);
(2.63)
~,~[f](a) = A$'~-~a(Vbf); X}~ = A / $ ' - l ( V b f ) ( A a ) V a d a We would also like to include a formalization of an ingredient in (1.12), namely
f (x + iA o~2 , p - iA o-~l) f dbei[b2(~l-x)-bl(~2-P)] =
(2.64)
= / f(x + Abl,p + Ab2)eib2('~-z)-b2('~2-P)]db = (2:r)2f(x +
iAO~2,p-iAO~)6(c~l- x,o~2- p)
Now one sees that the variables a and b (but not A) in (2.62)-(2.63) are Fourier variables so one is creating a copy of the phase space P ~ {(x,p)} via 7:'4 ,.~ {(al,a2)} and letting
CHAPTER 4. DISCRETE GEOMETRY AND MOYAL
206
ai run over (-oc, oc). Hence the difference operator X ~ on 7) is parametrized via 7)d. To see how this creates an analogue of fxOp- fpOx consider (B46) Va ~ a - 0 as A --~ 0 and V a f --~ alfx + a2fp. Then
jc'-l(Vb/) ~ lim-~ 2 e-iA(bxa)(blfx
(2.65)
+ b2fp)db =
A2
1 A / e-i~la2+i~2al(~1~ -'[-~2fp)dd~= = l i m ~-rS~2 f dbe-i~(bla2-b2al)(blfz + b2fp) = lim 4zr2 e-i(la2+i(2a~d~= limA4(fxOa2 - fpOal)5(a2, al)
=/im47r2 A(fzOa2 - fpOal) Consequently
X ? --* i f da[(fxOa2 - fpOal)~(a2, al)](alOx + a2Op) = i(fpOx
-
-
(2.66)
fxOp)
as desired. Now going back to (1.6)-(1.12) it seems more appropriate to write simply X?=
2-~ ( ~ ) 2 / d a / d b e _ i A ( a x b ) f ( ( x , p ) + A b )
[eAa.0.6_ha.O]
-
(2.67)
87r2 with ((x,p) 4- Aa = (0~1,012) and da = da/A 2)
X~(+)g = 1
8~-2A
da
dbe-iX(axb) f((x,p) + Ab)g(x + Aa) =
(2.68)
/ dot f dbeTi[b2(C~l-x)-bl(C~2-P)]f(x ~- ~bl p-~- )kb2)g(o~l, 0~2)=
_-- 8~2A1 f da (f(xTiAOc~2,
p+ i a O ~ l ) f d F~[b~-(~-x)-bI(~-p)I) g(~l, ~2) -
1/ da(f(xTiAO~,p+iAO~x)5(al-X,
= 87r2A
a2-p))g(c~,~2) =
1
= 2Af(x 4- iAOp,p T iAOx)g(x,p) Hence we have
X~(+)g=
1
f .g; X ~ ( - ) g = ~ g .
f;
x
g
=
1
(f * g -
g
* f)-{f,
g }M
(2.69)
as in (1.15). Now go to Sections 4.1.2 and 4.1.3 along with [537, 835]. Note first that there are some typos in [537] and a different definition of Va but we will not belabor the matter; the computations seem to be basically correct in [537, 539, 828, 835]. Our calculations give
( 1 5 0 ) and (1.6) if X}9 is defined as in (1.8). In connection with discretization and KP it will be instructive to sketch the development in [538], but only after a rather long excursion to rigorous developments in [7, 10, 222,742, 788]; the combination will considerably enhance Section 4.1.2. Thus write (cf. Section 1.1.2) C()
L - WOW-I; W -
1 + E a j ( x , t ) o - J ; OrW = -Lr_W; 1
(2.70)
4.2. HIROTA, STRINGS, A N D D I S C R E T E SURFACES
207 (3o
Br = ( w o r w - 1 ) + = L~_; L r = zr
r = W e ~ (~ = ~
tnzn); 1
co
Ozr = M e ; M = W ( y ~ rtrOr-1)W-1; [L,M] = 1 1
Now regarding additional symmetries of the form (23J) Oktr = - ( M t L k ) - r we refer to [222] for a careful t r e a t m e n t (cf. also [7, 10, 163, 742]). Due the the relation [L, M] - 1 some care is required but as indicated in [215] it is however permissible to write (23J J) L k M t ~ ~ z k r and M t L k r = z k ~ r To see this think e.g. of L ~ z and M ~ W ( ~ ntnOn-1)W -1 or M ~ Oz and L = 0 + ~ ajO -j. Then L M r = Mz~b = Oz(Zr and ML~b - LOz - Z~Pz (corresponding to [L, M] = 1). More precisely consider e.g.
L M r = WO(y]~ ntnOn-1)e ~ = W(1 + y ~ ntni)n)e ~ = We~-f -
(2.71)
- + - w ( E ntnOn-1)ze ~ = r + zWOze ~ = r -+-ZCz = iDz(Z~) Going back to [538] one defines flows (23K)Om~W - - ( M ~ L m ) - W and there follows [Omt, Or] = 0 with (1.36). The operators spanned by {zmi)~z} form an infinite dimensional Lie algebra w~+cr with flows Om~ and its central extension (cf. [147, 513, 808]) Wl+c~ is related to infinitesimal B/icklund transformations T ~ T + eX(z, ~)~- of the KP hierarchy (cf. also [222]). We also write V(z)
zJtj
=
jz- Oj ;
1
V*(z) =
-
zJtj 1
) ( llZ j
and r = V ( z ) 7 / T with r = V*(Z)T/T. The remaining steps leading to (1.38) and variations in [538] are best developed in [7, 10, 223, 742, 788] to which we go now (cf. also [222, 424, 415, 416, 786]). We will not embark on a survey of additional symmetries here but simply indicate some relevant formulas. It is instructive to follow [788] at first and write W = 1 + ~ wj(t)O -j with w = -2Wl. One inserts a to variable here with tl = x and sets
r = W e r = er
+ ~
A-nwn(to, t)) ; r
= ( W * ) - l e -r = e-C(1 + ~
A-nwn(to, t)) (2.73)
where 4 = ~ Aktk + tolog(A). A lemma of Dickey [221] is used in the following which states that for P = ~ p k O k and Q = ~ qjO -k with R e s o ( ~ anO -n) = al (definition) one has (23L) R e s x ( P e x p ( A x ) ) ( Q e x p ( - $ x ) ) = ResoPQ*; versions of this have been used frequently (cf. [147, 210]). One can write the KP hierarchy as (23M) OnW - 2 R e s o W O n W -1 and in terms of (2.73) this leads to
Omw(to, t) = 2ResxAmr
to, t)r
t0, t)
(2.74)
which in fact corresponds to (23N) 2r162 = 2 + (wz/A 2) + +(Wy/A 3) + (Wt//~ 4) -4-''" (cf. [163]). To treat the additional symmetries based on Akc~r of Ok~ consider now ( 2 3 0 ) F = x + too -1 + ~ k # 1 k t k Ok-1 (in [788], as in [222], the theory embraces negative times t n as well as tn for n > 0 but we concentrate only on tn for n > 0 here). Note F ~ x + ~ ktkO k-1 where tn = 0 for n < 0 so M = W F W -1 is our standard M; however in [788]
C H A P T E R 4. D I S C R E T E G E O M E T R Y A N D M O Y A L
208
M = W r W -1 for general F as in ( 2 3 0 ) . Next one defines ( 2 3 0 0 ) VmnW = [Omn, W] = - ( W F n C O m w - 1 ) _ W and F n is also defined for n < 0 in [788] (but we omit this here). For n > 0 the algebra of vector fields Vmn (with commutator [Vmn, Vm'n'] ~ [xnCOm, Xn'Om'] for nn' ~ 0 and [Vmn, Vm'n'] -- 0 for nn' -- 0) is denoted by W~. Now work with L = W O W -1 and M = W F W -1 (with [L,M] = 1) via L r = Ar and M~b = COxr (note M - I ~ = (~;1~2 = f~ r to, t)dA' for It[-r 0). Then as a theorem one proves easily that COmnW= 2Reso(WFnCOmw-1) = 2ResoMnL m = 2Res~AmCO~r162 *
(2.75)
Recall w = 2COxlogT(to, t) is a well known relation (cf. [147, 163, 162, 558]) so we have in particular COmnCOlog(T) = R e s A A m C O ~ r 1 6 2
=
~
1/
(2.76)
[AmCO~r
(proving (1.38)). This follows immediately from (23L). One can find many lovely formulas involving "additional symmetries" and Orlov-Schulman operators in [7, 8, 9, 10, 11, 163, 172, 210, 221, 222, 223, 424, 742, 786, 788]. In particular to relate this to vertex operators one recalls (2.72) which, in order to accomodate to, we enlarge via V --. X in the form
( (
X(A, to, t) = exp tologA + ~ Aktk 1
)
exp
-COo-
~-~COk ;
) (
x*(~,to, t) = ~xp -tolog~- ~ ~ktk ~xp Oo + 1
-;vOk
)
(2.77)
(d. ~lso [42a, 415, 416, 786, 787]) and one defines (23P) 5 ( A - #) = ( l / A ) ~ _ ~ ( # / A ) n so that ~ f ( A ) 5 ( A - #)dA = f ( # ) (cf. also [210]). We recall a few formulas involving such delta functions 1 t -I (A) [210] 5 ( t ) = ~ t n = t ; (2.78) 1-t 1 - t -1 nEZ 1
(S) [10] 5(A, z)
A
z
1
2_,
1
--
1
zl-(A/z)
--(X3
1
I AI-(z/A)'
f ( z , A)5(A, z) = f(A, A)5(A, z); (# - A)5(A, z) = (# - z)5(A, z) = 1 - (p/z)
p 1 - (z/p)
1 - ( A / z ) + -s 1 -(z/~))
(C) [ 7 8 8 1 5 ( A - z ) = ~
1
(_~)n
1 =~1
1
-
1
1
(z/A) + - z l - ( A / z )
Thus everything fits together and is equivalent. One can write ~- = 7n(t) where n ~ to since to plays the role of a discrete variable. Now by these constructions (23Q) ~(A, to, t) = IX(A, to, t)T(to + 1, t)/T(to, t)] with a similar expression for ~b* and one has the bilinear identity (23R) Res~=~r to, t)r to, t ) = 0. Further by (2.75) we have also (23S) V ~ w = -2Resk=~(V~t~r162 ). It is also worth noting that V~z7 = X ( A ) X * ( # ) 7 is an infinitesimal group transformation of T. One
wants to look at V,xu acting on the vertex operators and the tau function and a
4.2. HIROTA, STRINGS, A N D D I S C R E T E SURFACES
209
general picture can be extracted from [7, 10, 222, 742] (cf. [172] for more on this). Here we only need a few formulas and recall first the differential Fay identity from Chapter 5, (1.64) (we continue with the X notation even when tn = 0 for n <_ 0). (2.79)
r(t)(gT(t + [81]- [82]) -- T(t -~- [81] - [82])(gT(t) : ~- (811 --
S21)[T(t q-[81]-
[82])T(t) -- T(t -+- [Sl])~-(t- [82])
(cf. [7, 10, 742]). Following [7] for example one writes A = s{ 1 and # 821 to obtain from (2.79) ~*()~, t)@(~, t) = e- E tiAi T(t -~- [/~--1]) e E titti T(t -- [#-1]) = (2.80) =
~(t)
~(t)
=~(9(e~t~(u~--,x~)T(t+[)~--l]--[#--l])) #-~ ~(t) = ~
( # - A)J-I(9 jv
1
~- ~ _~
1 ~/~_n_Jw(J)(T)
9
=l__~oX(t,A,P)r(t)_
- -cr ~-
~(t)
(._)~)j-1 j!
1
w(J)(T) (9
T
Consequently one has in particular (upon differentiating (2.80) in p and then setting p - A)
ur162
= (9
~ )~-n-~'W~(r) --00
where one uses (cf. [7, 742] and note exp[-
X*(a, t)X(., t). -
~ _)~
W)~(~-) 7"
E~(1/i)(#/~) ~]= ~/(~ -
(2.81)
~))
e~--~7 ti(tti--Ai)T(t _~_[/~--1]_ [#--1]) :
)~ e ~ l t~(t~-~)e~(~-~-t~-~)(1/i)(~176 ~-#
- # ~=o
- 0
( , - ~)~ k!
-~E
=
~x(t,~,,)~
~-#
(2.82) =
~_,_~w(~)
(~)
(note (2.47) in Chapter 1 uses X(#, A) ~ - X ( A , #)). Now one has directly from (2.82)
Re8 #a(gbx(t, )~, ~)TI,k= # = Re8 E # a - n - b w b ( T ) -- Wa_b+l(T ) b
(2.83)
SO Re8 ~n+i(9~X(T) ~,o W~%1(T ) (not W~n+l as in [538, 539]). Consequently we write
/ ~~i~n+s
~)lz:~ T ~_. W~+I(T ) ~.o On+s T ,.~ ~n+s T
(2.84)
(which defines the t) notation). We go now to [538, 540, 839] in order to spell out the nature of the D-Virasoro subalgebra D-Vir (+'~ (discrete Virasoro) and its relation to Wl+oo as a subset of the Moyal algebra based on { , }M (cf. also [311, 806]). This goes back to a q-deformation of Virasoro (see e.g. [182, 204, 837]) based on
Lmn = z TM
qn(ZOz+Am) _ q-n(zOz+Am) m ~ zm[nzOz + Am]n; A m -- -~ nu 5 qn _ q-n
(2.85)
C H A P T E R 4. D I S C R E T E G E O M E T R Y A N D M O Y A L
210
where [X]n -- [x]/[n] and [n] = [n]q = ( q n _ q - n ) / ( q _ q-l). The notation needs a little embellishment (cf. [187, 191, 529, 581]). Thus in [538, 540, 837] one seems to be using [n] as indicated with [n]! = [hi. [n - 1].-.
1;
=
- hi!
whereas a more standard notation the the mathematics literature would follow [581] with (a; q)k = (1 -- a)(1 -- q a ) . . . (1 -- qk-la);
\to,]
q
-- (q;q)k(q; q)n-k --
(1 - qn) 9". (1 _ qn-k+l) = ( 1 - q)(1 - q 2 ) . . . ( 1 - qk)
(2.87)
Then in the algebra Cq[x, y] generated by x and y with xy = qyx one has e.g. (24B) (x + y)n = ~ r d [r~]qyn-kxk (note xy -- qyx is not symmetric in x, y). In [529] for example one uses the notation _
(n)g = 1 + q + . . . + qn-1 _
qn_
1
(q-
q - 1 ; (n)!q = (n)q... 1 -
1)(q 2
1)..-(qn
-( q _ 1) n
(nk) q = ( k ) ! q(n),q (n-k)!q
_
1)
.
(2.88)
and again ( 2 4 B ) will hold. Evidently ( 2 4 C ) ( q ; q ) n = ( 1 - q ) . . . ( 1 - qn) and formally (qn;q-1)n = ( 1 - q ) " " ( 1 - qn) as well. On the other hand (24D) (q2n_ 1)/(q2 _ 1) = Y~on - 1 q2p and[n]q __ q-n+1 ~ - ~ - 1 q2p _ q - n + l (n)q2. One finds a rich source of information on q-analysis in [581] with many interesting formulas but we only mention here some connections to ordinary differentiation and integration (cf. also Section 4.4.2 and [357, 616]). Thus the q-derivative D q f is defined by
(Dqf)(x) = f ( x ) - f(qx) (x 7~ 0); (Dqf)(O) = f'(0) (1 - q ) x
(2.89)
Thus in particular limqTl(Dqf)(x ) = f'(x) for f E C 1. The q-integral can be defined via
f0 a f(x)dqx = a(1 - q) ~oo f(aqk)q k 0
(2.90)
This can be viewed as an infinite Riemann sum with nonequidistant mesh width and as q T 1 the right side tends to f~ f ( x ) d x . There is apparently no canonical choice for a q-integral Now returning to [538, 540, 837] (2.85) is a q-differential analogue of ZOz + Am so the algebra generated by Lmn is a discrete version of the corresponding Lie algebra. Note first a c~ gm,n,] a' that the shift operators gmn = zmqnzOz+a form a Lie algebra with (24E)[gmn, = (qm'n _ qmn'~pa+a' and there is a link to the Moyal bracket as follows. For a function ]~m+m',n+n ~ f ( z , y) = )~ ~ f m n ( q ) e x p ( m x ) e x p ( - n y ) define (24F) K f = (1/2)f(x+2)~Oy, y-2)~Ox). Then
{Ks, Kg } = KSinh(2:~{S,g}) ; Sinh(2,k{f , g}) -- Sinh
[ (00 00) 2~
Ox S Oyg
Oy I OXg
f ( x s ' ys)g(xg' yg) x,y
(2.91)
4.2. HIROTA, S T R I N G S , A N D D I S C R E T E S U R F A C E S
211
Putting x = log(z), y = log(w), and A = (1~2)log(q) in (24F), K f becomes the important expression (24G) (1/2)A E fmng~n(Z)gi~,m(W) 9 Therefore f~nn ~ a block of generators for K f and since (2.91) corresponds to the Moyal bracket algebra one can say that ( 2 4 E ) forms a part of the Moyal algebra. The goal here is to have the replacement of differential by difference operators in the Virasoro algebra provide a quantization scheme for the x plane. The algebraic connection to quantization is through the Moyal algebra and one would like to connect this with a q-deformation of Virasoro (cf. [358, 774, 871]). Now go to [538] for more details as follows. First let Am = m / 2 in (2.85) and call the resulting operator s thus
s
= zm
qn(ZOz+(m/2)) _ q-n(zOz+(m/2))
(2.92)
qn _ q-n
These operators satisfy the commutation rules n s [ m,
m,n
n!
--" [y.m,n f n + n ' [y.m,n f n - n ' . ~m~,n~,..,m+m t ~ ~ m ~ , _ n ~ m + m ~ ,
[(1/2)(nm' -- n'm)][n + n'] In] [nq
Cm"n' :
(2.93)
qX _ q-X ; [X] - -
q __ a -
1
In the limit of q ~ 1 one has s ~ gm = zm(zOz + (m/2)) and the gm form a Virasoro algebra [gm, gm'] = ( m ' - m ) g m + m ' . Thus we can regard (2.93) as a deformation of the Virasoro algebra and one calls { s a q-discretized Virasoro algebra or D-Virasoro algebra. Actually there is a more general form of D-Virasoro algebra (which we ignore here- cf. [538]) and the algebra (2.92)-(2.93) is identified with D-Vir (+,0) which can also be identified with a subalgebra of the Moyal algebra (cf. [311, 540, 806]). The central extension D-Vir~ +'~ also has a free field representation indicated in [538]. Now think of z in (2.92) as a spectral parameter and expand s in powers of ZOz (cf. (1.39). Thus f n _ Z m qn(zOz+(m/2)) _ q-n(zOz+(m/2))
(2.94)
qn ~ q-n
This can be written out roughly as in (1.39) and we give more detail here. Thus e.g. qnzOz = exp(AnzOz) = ~_,(An)J(zOz)J/j! for A = log(q). We note that (zOz) 2 = zOz + z20~ 3 ... leading to a formula (24H) (zOz) j = ~Jo cjgZgJz with with (zOz) 3 = zOz + 3z202z + z 3 O~z, ~0 - 1. We do not use the cjt of [538, 540] or the same indexing since the exposition there is not clear. Consequently let us write
q+n(zOz+(m/2)) = q+nm/2 ~0 (:kn)~)J ~ 9
(2.95)
0
leading to
cn =
c~ qnm/2 j=0
q-am~2 ~
--(--1)J qn _ q-n
oo
J zm+ dz -- Z( 0
m
(2.96)
l~j
Consequently one considers the flow as (2I) L n = E~j(Sm)ntOm+~,~ and writes (using the form (2.92) for s
Lm(T) =
s
()lz=~T =
an ~
q-rt
(2.97)
C H A P T E R 4. D I S C R E T E G E O M E T R Y A N D M O Y A L
212
while from (2.82), (2.83) and (2.96) (using the notation Ohm~as in (2.84)) one has (24J) Ln~ ~-](Sm)n~Om+~,tT = ~(Sm)n/W~m+l (we will develop this further in [172]). Thus, in summary, the simplest D-Virasoro subalgebra D-Vir (+'~ can be regarded as W1+cr which is considered to be a universal symmetry structure for integrable systems in the sense that such a symmetry often appears in various theories. For example the W-infinity algebras provide a unified viewpoint for various integrable hierarchies originating in KP. On the other hand. some discretization of the variables preserves the integrability of the system, so D-Vir (+'~ is not only a deformation of the Virasoro algebra, but is also a discretization of the spectral parameter which preserves the integrability of the KP hierarchy. Further D-Virasoro is seen to be a subalgebra of the Moyal algebra (cf. [310, 311]) and this suggests that the Moyal structure must contain much information about integrability. For example, in [615] it was shown that quantizations on C~162 generally parametrize realizations of the integrable dynamical system, and the Hamiltonian form on the phase space was established. In particular some soliton equations were discussed for the case of the Moyal formalism. Also we indicated earlier some results concerning the KP hierarchy as the Moyal like deformation of d-KP (cf. [374, 891,892]). These examples support the claim that study of symmetry with Moyal structure is of value in obtaining a general perspective on integrable systems. 4.2.5
Further
enhancement
The material on string connections in Sections 4.2.1 and 4.2.3 is more or less clear but Section 4.1.3 should be enhanced further. In this direction we provide first some further material from [830, 834] to supplement the discussion around (2.46)-(2.51) in Section 4.2.3. Thus lattice gauge theory provides a useful tool for investigating the nonperturbative nature of particle physics at short distances. In [830, 834] one derives a Lagrangian associated with HBDE via the study of a linearized version of this equation obtained by introducing gauge potentials (cf. also Section 4.2.3). It turns out that the corresponding Lagrangian is described by a lattice gauge model with a particular form of links. Under appropriate conditions the system appears symmetric under the exchange of fields defined on the lattice sites and the gauge potentials defined along the links. This "duality" feature enables one to investigate the behavior of solutions. First recall HBDE in the form of the first equation in (2.37). As was discussed in Section 4.2.3 this equation can be derived as a consistency condition for the linear problem (r m, n) ~ fn(g, m) in Section 4.2.3 and there is some readjustment of notation regarding c+ etc.) V+r
m, n) = c•162
m, n);
(2.98)
V+ -- U+(g, m, n)(e a+ - 1)U~__1(g, m, n); E+ - U~=(g,m, n)ea++anU~l(g, m, n) with (24K) O+ = O/Og, O_ = O/Om, and On - O/On while g = (1/2)(A § # § ,), m ( 1 / 2 ) ( # - A - v) and n = v. The c+ are constants related to c~ and V by c+c_ - -V//~. In fact under the gauge condition (24L) U+(g, m, n) = U_(g, m, n § 1) = U(g, m, n) the coupled equation (2.98) is compatible if U(g, m, n) satisfies HBDE. If U satisfies HBDE then r is obtained by solving the second order linear difference equation (24M) V _ V + r m, n) = c+c_E+E_r m,n). The remarkable feature of (2.98) is that it can also be written as (24N) V+q~ = c + E + r where ( 2 4 0 ) V+ = U + ( 1 - e-a• 1 with/~+ = -~f~e-o+~=On~f~=1 and q~ - U with U = r Thus in ( 2 4 N ) the roles of the gauge potential U and the wave function r are exchanged and this property is called duality between U and r It is easy to prove that the compatibility condition of ( 2 4 N ) again requires b', and hence r to satisfy HBDE; thus r satisfies HBDE if U does and vice versa.
4.2. HIROTA, STRINGS, AND DISCRETE SURFACES
213
In [830] one derives the Lagrangian which leads to the equation of motion given by (2.98) and (24N). First one observes that there are four equations linear in r and r Second they are covariant under the local gauge transformation (cf. (2.48)) (24P) U+(t~, m, n) V(t~, m, n)U+(t~, m, n) and r m, n) -~ V(t~, m, n)r m, n). There are two alternatives for the choice of time variable. One is to regard one of the three variables as a discrete time variable (recall HBDE was first derived as a discrete analogue of the 2-D Toda lattice). Another is to introduce the time t and the most general suitable Lagrangian is then ~
4
L = 2i z
Z
_
j c~=l
iA0j) +
4
Z
""jk wk + Lu
(2.99)
j k c~,/3=1
and the equations of motion for the field r are (24Q) 2i(Ot-iAoj)r + 1~Ek E4~=1 '~rc~'aZ vv j k 'Ok" In (2.99) Lu represents the kinetic term of the gauge fields and Lu = 0 is assumed in [830] for simplicity. Then U+ bre not dynamical fields. The Lagrangian is still invariant under local gauge transformations of the type (24P). One can fix U+ either by hand or by solving for them as functions of the r Instead in [830] one fixes U+ such that (24Q) becomes 9 compatible when one requires ( 2 4 R ) C j IX = Cj2X where X = R,L. With this constraint (24Q) becomes (2.98) and (24N) with r _ r = Cj and r _ r _ ~j if the t dependence is ignored. Similarly one derives the same set of equations for Cj t as those for r under the same conditions. Therefore one concludes that the system described by the Lagrangian (2.99) is characterized by HBDE when the gauge potential U satisfies HBDE and there is no t dependence. The system described by the Lagrangian (2.99) is a lattice gauge model in which the fields Cj defined on the lattice site j interact with each other through the gauge potential defined along the links. It is a three dimensional lattice which consists of layers of two dimensional square lattices. Two adjacent layers couple with each other with strength c+ through particular links. If one does not consider the variable t the Lagrangian describes a two dimensional lattice which evolves along the discrete time variable n. Then in [830] various soliton solutions are generated which we ignore here. We gather next a collection of formulas from [146, 210, 211,531,831,836, 837, 838, 884] concerning vertex operators and string coordinates. There are some differences in notation and various points of view which we will try to clarify. First recall the free fermion operator approach via (24S) { e r a , Cn} -- 6ran, { e r a , Cn} ~- { e r a , Cn} ~-- 0 and C n l 0 > = < 01~ n = 0 for n > 0 while < 01r = Cn]0 > = 0 for n < 0. This is the notation of [836, 837] for example while in [210, 211, 831, 884] one uses (24T) r > = < 01~ n = 0 for n < 0 with r > = < 0]r = 0 for n > 0 (in [1471 both notations are used and a somewhat different approach and/or notation is adopted in [512, 514, 515]). Note if one interchanges ~ and r then (24T) becomes (248). In [836, 837] one writes oo Z _ n
oo
zn
Z+(z) = iplog(z)+ ~ --~atn; X_(z) = x + ~ -~an; an =
~
i
(2.1oo)
i
~ z CmCn+m; atn = v/-~z~ CmCm-n; [x,p] = i; [am, atn] = 5ran
(using free fermion operators), while in [884] the notation is Oln = E ~)m+nCm;* Z X(z)
O~-n "-- E ~ * Z =
-
pZog(z) -
~
:rm-nFm[x,P] ,
i ; OZm ,
z n)
C~-n -- ~ O l n
i
1
n
-- m ~ m + n , O
(2.101)
C H A P T E R 4. D I S C R E T E G E O M E T R Y A N D M O Y A L
214
so we see that interchanging ~bn and Cn gives an ~ - ( x / ~ / i ) a n and a-n ~ (x/~/i)a~. Thus we will adopt (24S) and when interchanging formulas from [884] via ~bn ~ ~bn we must also take z ---, 1/z and k ~ - k . Then e.g. from [884] one obtains
V(k, z) = exp
k
-1 na-n
--~exp(ik~l
exp
-k
-n an
z-n
(2.102)
--~
Zn
--~atn) e x p ( i k ~ l
-~an )
which will correspond to the part of Y(k, z) in (2.5) coming from the an, atn terms. Note also under z ~ 1/z, iplog(z) --~ -iplog(z), giving the correct term in (2.101) for X ( z ) . Following [884] now one has also
V ( k , z ) ~ ek((t'z-~)e-k((O'z)
(x)
oo z n
1
1
~ = E tnz-n; ~(O,z)= E
n
(2.103)
and this gives one connections to traditional vertex operator formulas. In particular (cf. (2.82))
X ( z , t ) = e~(t'z)e-~(~
~x(t,
X*(z,t) -- e-~(t'Z)e~(~
~,,) =
=
(2.104)
e~(t,p)_~(t,A) e~(b,~- 1)_~( b,t~- 1 )
A variation on this is needed for (2.6) in the form (recall here k ~ - k and z ~ 1/z)
9(k, z)9(k', z') =
~ - z ) . 9(k, z)9(k', z')-
(2.105)
Note here by the Baker-Campbell-Hausdorff formula eAe B -- e BeAe [A'B] we can write
ekg(t, z- 1) e_k~(5,z ) ek,~(t,(z, )- 1) e_k~(~,z, ) __
(2.106)
__ ek~(t,z -1) ek'~(t,(z')-l) e-k~(O,z) e-k'~(O,z') e-kk'[~(O,z),~(t,(z')-l )]
where-[~(ch, z), ~(t, (z') -1] = - E ~ ( 1 / n ) ( z / z ' ) n = l o g { 1 - ( z / z ' ) } . This yields (2.105) since 9 9simply brings the creation operators to the left. We note in (2.6) for scalar k, k t other factors appears via (21S) so that
at -n - - ~ z ,x + ~
[iplog(z) + ~ -[~'~ tnz -n, ~
an 'n" - - ~ z ] ~ i~o, x]log(z)-
( z,)
z'----~nOn]= log 1 - -n
z
(2.107)
+ log(z) = log(z - z')
confirming (21S). Let us make a few remarks concerning the formula (2.8) (referring to Section 4.4.3 and [531,836, 837, 838]). G in (2.8) refers to internal configurations of diagrams with 1-,N
Fa(kl,'",kN)
1--,N
~ < O] H V ( k j , z j ) l G >=< ~ n
V(kj,zj)G]O >
(2.108)
4.2. HIROTA, STRINGS, AND DISCRETE SURFACES
215
and in the background there are free fermion operators as in (248) with operators g such that gCn = }-~)mgamn and r = ~anmgr (cf. remarks before (2.39)). Recall also r = ~ CnZ n and r = ~ r -n-1 before (2.38). Then one can show directly that for such g one obtains general tau functions via (3O
~(t) --< 01eg(~)gl0 >; H(t) = ~ ~-~ " CtmCm+~ 9 1
where Ct = r
Z
(2.109)
(cf. [147, 838]). Note here oo
(24U) exp[H(t)]r
= e x p ( - ~ tnz-n)~g(z); 1 oo
exp[H(t)~(z)exp[-H(t)] - exp( y~ tnz-n(b(z) 1
Consequently
dz
< lleH(t)r
><-lleH(t')~P(z)g[) > = 0
(2.110)
This can be proved in various ways (cf. [147, 210, 838]). We mention here also a few other relations (see e.g. [210, 831,838]). Thus first to use formulas from [210] we interchange ~n and Cn" Then note that CmCn =" ~mCn " + < Cm, Cn > where < ~m, ~n > - ~mn; this means in the H(t) of (2.109) with n _> 1 we have" CmCm+n = ~ m + ~ and therefore we can write from (2.100) (as after (2.40)) H(t) = ~ tn 2 ~2mr - i ~ tnanv/-~. Let us note also < o I E H ( t ) ~ ( z ) = ef(t,z)-f(O,z-1) < 01ell(t); < oleH(t)~(Z) = e--~(t,z)+~(O,z-1) < 0[e H(t)
(2.111)
Now the correspondence between (2.8) and a tau function is sketched in [838]. This is achieved by the identification
[1//
G = gF = exp ~
dx
B(x, y ) = < Olr
dyB(x,y)(b(x)~b(y) >=
]
;
-1 y--x
(2.112)
which needs further clarification (cf. [26, 147, 291,491, 764, 934, 945, 996]). For the physics the use of Riemann surface methods is really essential but we will not spend time here on background mathematics (cf. Chapter 5 for a dose of Riemann surfaces). It is worthwhile to sketch here some formulas from [491] (cf. also [147]). The notation involving free fermions is slightly different from either [210] or the Saito papers but connections are easily written down (a good exercise- cf. [147] for some of this). For convenience we simply follow the notation in [491] here and refer to free fermion operators Cn, Cn satisfying (24V) q~n[0 > - - Cnl 0 > : 0 for n > 0 with {Ca, Cm} = ~m+n. The index n E Z + (1/2)here (corresponding to spin 1 / 2 ) a n d (24W) C n ( z ) = ~nEZ+(1/2)Cn z-n-(1~2) with r
= ~nEZ+(1/2)r *z-n-(1~2). Elements of the Clifford group G can be written as
g=
--(X)
L
=(:X)
S(z,
(2.113)
C H A P T E R 4. D I S C R E T E G E O M E T R Y A N D M O Y A L
216 A =
E Amnz-n-(1/2)w-m-(1/2) =: r162 m,nEZ+(1/2)
"
and the current operator is (24X) Jz = E n e Z Jn z - n - 1 = A(z, z). The universal Grassmann manifold (UGM) is the space of G orbits of the Fock vacuum [0 > and since g[0 > = g'[0 > for g = g'h with h =: e x p ( ~ amnAmn :, for m, n E Z with n > 0 or m > 0, the G orbit space can be identified with the coset space G i g = {gh}. Each point g[0 > in UGM is referred to as a g vacuum. The Pliicker coordinates (see e.g. [147]) of ordinary Grassmann manifolds are defined by embedding into a higher dimensional projective space. For UGM one introduces the time translation operator H belonging to the Cartan subalgebra of G defined via (24Y) H(t) = Y]n>_otnJn and the tau function for a point g[0 >E U G M is defined by ( 2 4 Z ) r ( t ) = < O[exp[H(t)]g[O >. The Pliicker coordinates are given then as coefficients of T(t) expanded in terms of the character polynomials. In order to characterize the locus of GM in projective space one uses bilinear relations for the Pliicker coordinates and for UGM the situation is essentially the same; to be the tau function of UGM r(t) must satisfy the Hirota equations which are the Pliicker relations for UGM. Now one introduces vertex operators via
V ( z ) = exp
tnz n
exp
Oolog(z) -
_ n
'
and the Hirota equations are again expressed via ( 1 9 H H ) which can naturally be put in a vertex operator form (24ZZ) fz=oo d z ( Y ( z ) T ( x ) ) ( Y * ( z ) r ( y ) ) = 0. For r(t) as in (24Z) the action of vertex operators is equivalent to the insertion of free fermions. Indeed for charged vacua < m I with m = 0, =t=l, =k2,... defined via
< 0[01/203/2"''r < O] < 01r "" r
< m[ =
(m < O) m = 0 (m > 0)
(2.115)
the following identities can be proved (cf. [147, 210]) < mleH(
)r
= V(z) < m-
= V*(z) < m + lie H(t)
lleH(t); <
(2.116)
This can be interpreted as the boson-fermion correspondence in the operator formalism (cf. [147, 210, 514]) and when r(t) is given as in (24Z) one can easily check (24ZZ). On the other hand one can also recover gl0 > from a r(t) which satisfies the Hirota equations. Indeed one uses the generalized Wick theorem
< oleH(t)r
= det
9"'(~(Zn)qS*(Wn)''' (~*(Wl)gl0 > < OleH(t)g[O > < Oleg(
)O(z
(2.117)
)O*(wj)glO >
< oleg(t)glO >
to obtain
1 2 ~ =oodz fw = m d w f ( z ' w ) a ( z ' w ) gl 0 > = T(0)" exp ( (27ri)
/(z,w) = 70-1 V(z) V*
) "[0 >; (2.118)
4.2. HIROTA, STRINGS, AND DISCRETE SURFACES
217
In particular one can write (cf. (2.112) in a different notation)
< olr162
> = V(z)V*(~)~(t)
< OlglO >
~(t)
I
(2.119)
It=0
Thus in a certain sense when T satisfies the Hirota equations it is a generating function of the Pliicker coordinates parametrizing UGM. Let us write out further details in the above spirit, using the notation of [210] (cf. also [147] for such Calculations). Thus the situation of ( 2 4 T ) is adopted where Cn]0 > = < O[psin = 0 f o r n < 0 and < 0[r = Cn[0 > = 0 for n > 0. H e r e g ( t ) = ~ t e ~ n ~ 2 n + e = $ t e ~ 9CnCn+g " since CmCn =" r 1 6 2 " + < r 1 6 2 > and n ~ m + g with g >_ 1 so # 0 for < r 1 6 2 m+g > = 0. Further g(t)lO > = 0 but < oIg(t ) # 0 since < Ol~n+e~n * n + g >_ 0 and n < 0. To,check the relations (2.120) Z
Z
consider He = ~
te ~
CnCn+e ~
Cm zm = ~ te ~ Cn(-~2mCn+e + 5m,n+e)z m =
--" E tg E ~2m~2n~)n+g -+-E tg E zn+gCn -- CH + ~r
(2.121)
Similarly H 2 ~ = H~2H + ~H~ = ~2H2 + 2~2H + ~2~ and ~2exp(~)exp(H) consists of such terms leading to (2.120). We want to show next
< mleH(t)r < mleH(t)r
= kin-ix(k)
< m-
lleH(t);
= k-mX*(k) < m + lie H(t)
(2.122)
where ( 2 5 A ) X(k) = exp(~(t,k))exp(-~(O,k-1)) and X*(k) = exp(-~(t, k))exp(~(O,k-1)) with (25B) < n I = < 0 [ r 1 6 2 for n < 0 w i t h < n[ = < 0 [ r 1 6 2 for n > 0. We note t h a t the cases m - 4-1 in (2.122) are especially i m p o r t a n t since
< l[eH(t)r
> = X(k) < O[eH(t)g[O > = r
(--lleH(t)r
k)~-(t);
(2.123)
> = kX*(k) < OleH(t)g[O > = r
where T is the tan function associated to g. Now in order to u n d e r s t a n d (2.122) we recall from [505] t h a t for H(t) ,.~ ~ tm ~ z CnCm+n = ~ tmam one has an isomorphism (25C) el0 > 4 | < n]exp(H(t))alO >. Here a 9 A = End(F) ~ Clifford algebra. F fermionic Fock space, and one writes V = | where Vn ~ C[x] (usually indexed by k to be Vn "~ knC[x] - cf. [514]). This correspondence ( 2 5 C ) can be lifted to the operator level in writing x(k) =
d(~,~)~-~(~,~-') = E xjkJ; x*(k) = _~(~,~)~(~,~-1) = E x;k-J Z
(2.124)
Z
T h e n define X, .,Y~ via ( f 9 t~)
.f(jf = X j - e f E t~+l; ) ( : f = X*j-g+lf E V~-I
(2.125)
C H A P T E R 4. D I S C R E T E G E O M E T R Y A N D M O Y A L
218
It follows that (25C) is an A module isomorphism via (25D) r ---, ,~j and r --, )(~. This means that the fermionic Fock space F can be realized as a bosonic Fock space V = | Now one could try to prove (2.122) by writing e.g.
< m-
1]r162
>= km-lX(k) < m-
lleH(t)alO >
(2.126)
with e x p ( H ) r = exp(~)r from (2.120) to suggest e ~ r 1 6 2 H .,~ ~--~z X j k j e H . However further analysis seems to require Schur polynomials in the form exp[~(x,k)] = ~ - ~ p n ( x ) k n for example and this does not prove an immediate success (note Schur polynomials are used in [211]) and in fact the main key here is Wick's theorem as in [506] (see e.g. (2.142)). It is also possible to use techniques from [513, 514, 515] to obtain specific bosonization formulas but using them to obtain (2.122) is not entirely trivial. In this spirit we note in passing from [996] that (2.122) is equivalent to
< nlr
u(z) = u n-1 < n -
< nlr
lIeH(x-[u-1]);
H(x) = u - n < n + lleH(x+[u-1])
(2.127)
This can be seen e.g. via (2.120) which implies that < n l e x p ( g ) r = exp(~) < n l C e x p ( H ) so that (2.122) becomes exp(~) < n l C e x p ( H ) = u n - l e x p ( ~ ) e x p ( _ ~ ( ~ , u - l ) < n - l l e x p ( H ) which is (2.127). Similarly one can look at a formula from [610] (cf. also [86, 506, 639]) which works with free fermions such that (25E) Ckl0 > = 0 for k < 0 and r > = 0 for k > 0 (different from (24S) or (24T)). We won't pursue connections but state the formula as
= Qz~~
r
r
~ko(Z-k/k)~k;
= Q-~z-"oeE~o(Z-~/k)"~
(2.128)
where (25F) Q]0 > - r > Qr -- g~k+iQ, and Qr -- r with ak = Y~Z " r J* " iWk and r = ~r k with r = ~ r 9z -k . This still needs embellishment to be useful in proving (2.122) which turns out to be a rather sophisticated formula. Given all this we return to the Japanese school and will give a more suitable background for (2.122) along with a sketch of proof. Thus we return to ( 2 4 T ) and think of F* and F generated by < 01 and 10 > with < Ola.blO > = < ab >. Then (25FF) < 1 > - 1, < ~2i~ j > 0, < r162 > = 0 and < r162 > = 6ij for i , j < 0 (= 0 otherwise) while < r162 > = 5ij for i, j _ 0 (-- 0 otherwise). Wick's theorem is written in terms of a general product W l . . . w ~ of free fermions w E ( O C r (|162 via
< Wl'" "Wr >= 0 (r odd); = ~
6r
(2.129)
sgn(a) < Wa(1)wa(2) > . . . < Wa(r_l)Wq(r) > (r even)
where a runs over permutations such that ~(1) < a ( 2 ) , . . . , a ( r 1) < a(r) and a(1) < a(3) < . . . < a ( r - 1). One can easily check the commutation rules
[E
9
E
!
" - - * ; aiJ" = E aikakj -- E aikakj' = E aijvAv2J
(2.130)
The set t~ = { ~ aijr aij E C} is a Lie algebra isomorphic to the Lie algebra of infinite matrices having a finite number of nonzero entries. As a Lie algebra ~ is generated by ei-- r162
fi = r162*
* - @i@; hi = ~)i-l~)i-1
(2.131)
4.2. H I R O T A , S T R I N G S , A N D D I S C R E T E SURFACES
219
along with r162 One can extend ~ to include linear combinations (25G) X - E aij " g ' i ~ " where " r 1 6 2 := r 1 6 2 < r162 >. Then (2.130) has the same form except that we obtain [, ] = ~ a ~ ' r 1 6 2 1 where c - ~i<0,j>0 a i j a ~ i - ~i>0,j<0 aija~i. One defines
a d ( X ) ( w l , . . . , wr) - ~ W l . . . a d ( X ) ( w i ) ' " w r ; 1 ad(X)(r
(2.132)
= ~ g'iaij; ad(X)(g,~) - - ~ ~;aji
and the action of X on F, F* is given by X . hi0 > - a d ( X ) ( a ) [ O > -+-a. y~. aijr162 i>_O>j
>;
(2.133)
< Ola. X = - < Olad(X)(a)+ < 01 ~ aijg, i g , ; . a j>_O>i One defines A ~ = gl~ = {F_,aij 9r Now write
lI/~ --
i
"; aij = 0 for l i - Jl > N} ~ C . 1 (N can vary). (g = 0)
~I/t --
i ~D*--1 (~ -- 0)
(2.134)
Then < gl = < 0 l ~ and Ig > = ~I,tl0 > give highest weight vectors such that e~lg > - 0 and h~lg > = 5~srg > for all i. One writes Ft = < glAo and Ft = Aolt~ > with < gig > = 1 where a E Ao means a = linear combination of monomials r 1 6 2 1 6 2 1 6 2 with r - s = g = 0 (here g signifies charge). One notes the existence of a central element Ho = ~ : r162 : with ad(Ho)(r = r and ad(Ho)(r = - ~ ' ; generally Hn is defined via (25G) Hn = Y~ " r "E glcr with [Hn, Hm] - n•m+m,o. Then Hn and 1 span a Heisenberg subalgebra h in gl~. Again H(x) = ~cr xnHn and since H(x)lO > = 0 one can write exp(H(x))alO > ~ a(x)lO > where a(x) = e x p ( H ( x ) ) a e x p ( - H ( x ) ) corresponds to the formal time evolution of a. Note as an example
eH(x)r
eH(x)r
= E r 0
-- ff)i "3t- Xl~2i-1 -~ x2 -~ -~x
-= E ~)i+nPn(--X) = ~)i -- X l r 0
~i-2 - + ' " ;
(2.135)
-~- --X2 -4- ~X 2 ~2i+ 2 -+- ' ' '
where the Pn are Schur polynomials (exp({(x,k)) = ~ p n ( x ) k n ) . This leads to the isomorphism ( 2 5 C ) where hi0 > + 9 < glexp(H(x))alO >. The Fock representation of A = Cliff(el, r also has a representation on the right side of ( 2 5 C ) given via X, X* as in (2.124) where Xi, X.~ are functions of (x, 0x) and some of the Xi, X ] are written out explicitly in [506]. The rules (2.125) also apply along with (25D). In fact the realization ( 2 5 D ) of r r in terms of the X, X* in (2.124) is considered in [506] as a consequence of (2.122) which is derived by another approach. In order to get to (2.122) in [506] one begins with the irreducible representations pg :glcr ~ End(Ft) arising in the Ft defined after (2.134). One notes that the Pt are all equivalent via an automorphism it of glcr defined by Q(r = r and e t ( r r Then Pt "~ Po o it and < glalg' > = < g - m l i m ( a ) l g ' - m >. The Pt can in fact be realized via
Pg(: ~)i~); ") = Z i - l , j - 1
(o) x,-~x
+ eijOt(i);
(2.136)
C H A P T E R 4. D I S C R E T E G E O M E T R Y A N D M O Y A L
220
oe(i) = < ~-t~z~*-e > - < ~ ; ~
>=
1
(0_<~_<e-1)
-1
(e <_ i <_ - 1 )
0
otherwise
Here Zij(x, O/Ox) = ~ Zij(x, O/Ox)piq -j are generating functions defined via
q-: Z(p,q) = q - P
exp(
(pn _ qn)xn)ex p
_
1 ( p - n _ q-nOn
1 n
_ 1
(2.137)
One notes here that
po(Hn) = ~ Zi,i+n x,--~ iez
=
-<
(n>o)
0 -nx_,~
(n = O) (n < O)
(2.138)
and in fact
z~j = ~
n,mkO
< r162
> p~(--0)p.(5)-- < ~ ;
0 < ~(~)r
>=
(2.139)
>;
(j >_0)
~j ~oJ-lpn(-x)pi_j_n(X)
(i < 0, j < 0) (i > 0, j < 0)
Explicit formulas are given for some of the Zij and the correspondence (25C) involving X, X* as in (2.124) are exhibited explicitly via Young diagrams (cf. [147, 778]). We omit this since we don't want to draw diagrams here. As for (2.122) we recall (2.120) and although r and r do not belong to A the inner products < g[r and < g[r make sense; (2.122) then follows immediately by construction in the form
< g]r
> = k t-1 < g - lleH(z-[k-1])a]O >;
(2.140)
< g]~*(k)eU(x)a]O > = k - t < g. + l[eH(x+[k-~])a]O > Note this is not a trivial construction since Wick's theorem is involved and we illustrate with an example. Take g = 1 and a = r then use (2.120) and Wick's theorem to obtain < ll~(k)eH(~z)alO > = E = < Ol~)~(k)eHr >= (2.141) = < O[r162162
> = e ~(x'p)-~(z'q) < O['~)r162162
Now from (2.129)we must evaluate < r 1 6 2 > < r From ( 2 5 F F ) we get < ~ ; r > = 1 = < r162 > and
< ~(p)r
>= ~
< CY
> - < 9~9(P) > < g,(k)g,*(q) >.
* _~
= ~ i,j
while < ~b(k)~2*(q) > = q / ( k -
= e((X,p)_((x,q)
q)
~
= q- P
leading to
q(k - p) (p-q)(k-q)
= e((X_[k-~],p)_((x_[k-~],q) < 01~(p)~,(q)l 0 > =
- < oleH(x-[k-1])r162 since (([k-1],p) = ~ ( 1 / n ) ( p / k )
>
>
(2.142)
n = - I o 9 [ 1 - (p/k)] so the adjoined exponential terms are = ( k - p ) / ( k - q ) while < 01~(p)~*(q)10 > = q / ( p - q ) .
ezp(log[1-(p/k)])exp(-log[1-(q/k)])
This procedure explains (2.122) which one sees is really a rather intricate formula.
4.2. HIROTA, STRINGS, AND DISCRETE SURFACES 4.2.6
Matrix
221
models and Moyal
We sketch first a recent paper [697] on a susy extension of the Moyal algebra and its application to the IKKT matrix model (cf. [308, 492]). One works here with the IKKT model which looks like the Green-Schwartz (GS) action of a type IIB string in the Schild gauge; it has manifest Lorentz invariance and N = 2 space time susy. Thus the action is defined via
Au][AU, A~'] + I@FU[Au , ~]) SIKKT = - ~22Tr (I[Au, -~
(2143) .
Here @ is a ten dimensional Majorana-Weyl spinor field with Au and @ being N • N matrices. It is formulated in a manifestly covariant manner which is believed to have some advantage over the light-cone formulation. This action can be related to the GS action by using the semiclassical correspondence in the large N limit 25H) - i [ , ] ~ ( l / N ) { , }p and Tr ~ N f d2ax/r~. In fact (2.143) reduces to the GS action in the Schild gauge
,..,.,,-:,,~
:..144>
In this correspondence the eigenvalues of Au are identified with the space time coordinates X ~ and the N - 2 susy manifests itselfin SschiZd as 6(1)~/) :
1 -~a,~r''el
(a,~ = OoX, O1X, - 0 i X , OoZe);
(2.145)
5(1)X~ = i~1F1@; 5(2)r = e2; 5(2)X~ = 0 The N - 2 susy relations (2.145) are directly translated into the symmetry of SIKKT as 5(1)r
~i[At,, Av]FUVel; 6 ( 1 ) A , - i~lFt,@; 5(2)r = e2; 5(2)A, = 0
(2.146)
By way of algebraic background one writes the basis for the s u ( g ) algebra as (25I) J(ml,m2) = wmlm2/29ml h m2 where
1 0 0 w 9 =
0
0
.
.
0
0
0 0 .
0 0
0 0
w2 0
0
0
.
;h .
.
.
0 0
1 0 0 1
.
.
.
O1 0 0
002 g-1
0
with (25J) gN = h N = -1, hg = wgh, and w = exp(2ri/N). expressed as (m ~ (ml, m2)) [Jm, J n ] = - 2 i S i n [ N ( m x n ) ]
Jm•
9 0 0
(O~_mi, n i ~ _ N - 1 ,
0
0 0
(2.147)
1
0
With this basis su(N) is
m,n:fi0)
(2.148)
The Poisson operator ( 2 5 K ) X f = (Of/Oq)(O/Op)- (Of/Op)(O/Oq) = V f • V satisfies [XI, Xg] = XU,g}p which can be expressed in Fourier components as [Zm, Z . ] = - ( m • ~)Xm+.; Zm = - i ~ - ~ m " m • V
(2.149)
where f ( q ) = ~ m f m e x p ( - i m , q). Thus the commutation in (25K) coincides, up to a constant factor, with (2.149) in the N ~ oc limit and it is this agreement which underlies the correspondence (25H). We know that the Moyal bracket
1[ (00 00)]
{ f , g}M = limq,~q;p,~p-f ~ Oq' Op
Op' O-q
f (q', P')g(P, q)
CHAPTER 4. DISCRETE GEOMETRY AND MOYAL
222
1
= lirnq,~q~Sin [A(V' • V)] f(q')g(q)
(2.150)
is the unique one parameter associative deformation of the Poisson bracket (cf. [343]) and the algebra (2.149) is modified into (of. [311]) (25L) [Kin, Kn] = ~Sin [),(m • n)] Km+n. Thus we see that the Moyal algebra corresponds to su(N) when A = 7v/N and to the Poisson algebra when s ~ 0. a susy extension of the algebra (25L) was discussed in [311] where
{F~, F~n} -- Cos[A(m • n)]Km+n;
[Km,F ' ]
-
(2.151)
~ Sin[A(m • n)]F~+ n
( { , } denotes an anticommutation [, ]+). These relations are realized by
s
i---ei(2Amlp+m22) Ix,/)]
Km = i A m = 2hA
;
=
i/~
(2.152)
as well as by
1 , i eimqexp[_A(m • V)] Km=-$Fm=
(2.153)
In order to generalize this superalgebra to include fields one uses the basis independent differential operator realization K f introduced in [68, 311,344], namely (2.154) However a problem arises in trying to incorporate a fermionic field in a similar way and using (2.151) for the algebra. This problem can be avoided by using an alternative realization of the Moyal bracket for fermionic fields as in [539]. It is a slight modification of (2.154) in the form
9
[(00
Bf = l~mq,__,q,p,_w-~Sin A Oq' Op
00)]
Op' O-q
f(q',
)=
(2.155)
i = limq,__,q~Sin [A(V~ x V)] f(q') One checks that (25M) [BI, B~] =
B~{I,gIM. For the fermionic counterpart one uses
Fr
[/~( 0 0
Oq' 019
Op' O-q
~b(q',p') =
(2.156)
= -limq,~qCos [A(V' • V)] r It is then claimed that (25N) {Fr - A2Bi{r M and [Bf, Fr = Fi{f,r M when the statistics of the fields is considered (the anticommutation arises via statistics); here the Moyal bracket for fermions is defined as
1 [(~176
{~, X}M -- limq,~q,p,~p~Sin )~ Oq 019' 1
= limq,~q~Sin [s
Op Oq'
• V')] r
~2(q,P)X(q', P') = (2.157)
One can see that behind this superalgebra ( 2 5 M ) - ( 2 5 N ) there exists an algebra of generators admitting the su(N) reduction. To see this one writes the operators in Fourier components Bf-- E f m S m ; f ( q ) = ~ jm,~e,~imq; (2.158) m m
4.2. HIROTA, STRINGS, AND DISCRETE SURFACES Fr - E ~)mFm; r m
- E emeimq m
where ( 2 5 0 ) B m = (1/A)exp(-imq)Sinh[A(m x V)] and Fm = V)]. These generators satisfy a closed algebra IBm, Bn]
= -iASin[A(m
223
- -~Sin[A(m
-exp(-imq)Cosh[A(m x
x n)]Bm+n; [Fm, Fn] -
(2.159)
x n)]Bm+n; [Bm, Fn] = - ~i Sin[A(m • n)]Fm+n
In any event one can now express the matrix model Lagrangian and the GS action in a unified form in Moyal formalism via
I F(pFt~[Bx,, Fr S - - 1 T r (1 [Bx,, Bx~][Bx,, Bx~] + -~ g
(2.160)
where [, ] is a commutator of operators (not matrices) and Tr denotes integration over the world sheet parameters and the sum over the complete set of functions on the world sheet. This action is invariant under the N = 2 transformations 1 ~(1)Fr = -~E~uF~Uel; E~u = 8(1)Bx, = ielFt~Fr
OoBx.OIBx~ - 01Bx.OoBx~.;
5(2)Fr = e2;
5(2)Bx,
= 0
(2.161)
Differences between this model and the Fairlie model in [308] are indicated and a Moyal extension of the N = 1 model is also exhibited. We omit here various remarks on philosophy related to branes etc. (cf. also [172, 844] for more on this). 4.2.7
Berezin star product
and path integrals
We go here to the recent paper [840] and sketch some of the results (cf. also [962] and Section 6.3). Basically a new star product is introduced which interpolates between the Berezin and Moyal quantization. A multiple of this produce reduces to a path integral quantization in the continuous time limit. In flat space the action becomes that of free bosonic strings. Relations to the Kontsevich prescription are indicated. A motivational point here is the fact that the path integral interpretation of geometric quantization does not seem to fill the gap between the local nature of quantization and the global nature of correlation functions. One works in the spirit of Berezin quantization (cf. [63, 84, 85, 845]) and the new star product is to reduce to Moyal in a flat space and to the Poisson (P) bracket in the classical limit. Recall that f *B g -- g *B f ~ P bracket but f *M g --* P bracket directly. To begin consider a simple Moyal product
ei(m'x+n'P) *M ei(mx+np) ---- eiA(Ox'Op-OxOp')eim'x'+n'P')ei(mx+np)x'=x,p'=p= -- eiA(mn'-m'n) ei(m'x+n'p)ei(m'x+n'p)ei(mx+np) -= / d~'d~'
(2.162)
d~d?Te(i"~-r)A/' e[ix(r-/)l-P(")]-'A/e(im"'+nr')/e(im+' nr)l27rA
This information is sufficient to derive an integral representation of the Moyal star product of arbitrary functions which can be expanded into Fourier series, namely (Y *M g)(x,p) =
/ d~'dr//d~drlei(r 2~
,)g(~, rl)
(2.163)
CHAPTER 4. DISCRETE GEOMETRY AND MOYAL
224
In order to establish a correspondence between the Moyal product and the Berezin form of quantization one rewrites the above formulas in terms of variables (26A) a = (x + ip)/x/~ and 5 = ( x - ip)/x/~ with b = ( n - ira)Ix~2, b = (n + irn)/x/2 and v = (~ + irl)/x/2, ~ = ( ~ - irl)/x/~. Then (2.163)is equivalent to
dudg e-(v~-vg-ag) /)~ 27ri)~ e-(Uf~-ua-af~)/)'f (u, f~g(v, ~)
dvdf~
(2.164)
Here f(a, 5) means the same function as f(x, p) but written in the new variables. One notes that the Moyal star product can be split into two parts
(f *stg)(a, 5)=e~~176 (f *ar g)
5)g(a',5 ') =
= e-AOaOa'f(a,
/ dvdf~ -(v~-va-a~+aa) / ), ~e f(a,f~)g(v,5)
5)g(a', 5') =
(2.165)
/ dudfteUf~-u~-af~+aa)/'X f (u, 5)g(a, ~) 2riA
Now for the Berezin product (cf. here [85]) one introduces a supercomplete set of state vectors exp[K(a, 5')] where K(a, 5') is defined by the analytic continuation of the K~hler potential K(a, 5') of the manifold in question. A multiplication of covariant symbols is given by
(f *B g)(a, 5) = / d#(v, fOe(~(v'v'a'a)/)~f (a, fOg(v, 5)
(2.166)
where d# is the integration measure and (I) is the Calabi function defined via the equation (26B) (I)(v, 9, a, 5) = -K(v, 9) + K(v, 5) + K(a, 9) - K(a, 5). In a flat space K(a, 5) = aS. The Berezin product does not reduce to the Moyal product in the fiat space but rather to the half star product (2.165A). The new star product of [840] is defined via
(f .g)(a,•)=
f d#(v,f~)
f
ee(V'v'a'a)/a d#(u, ft)e,~(u,f~,a,a)/~f(v,~)g(v, ft )
(2.167)
Note that this reduces to *M in the fiat space case. Now it is straightforward to see that (26C) ( f *B g) *B h = f *B (g *B h) but in the case of the new product the formulas do not show associativity in an obvious manner. If the space is fiat the integrations can be manipulated to show associativity but in general another approach is needed. It turns out that this star product plays the role of time ordered product which is manifestly associative if one considers the path integral representation of the product defined along a closed loop. Thus one defines the forward product
+AN(a, a)--(eab(N+l)--b(N+l)a:r162 (eab(2)-b(2)a, (e a~(1)-b(1)~*eab(~176
(2.168)
By repeated application one obtains
~-.N Abb = /
•
(
N+I H dP( r
NII~I e~(r162
e~(r
1
1) ,ff(J))d#(X (J) , ~(J -1))x
)
(2.169)
N+I
• H e(x(J)b(J)-b(J)f((J)) 1
where 4)(o) = X (~ and r = a. For an infinitely large sequence so that j becomes a continuous variable T one gets in the limit
~-Ab,g(a,a) = / D#(r X ) / ) # ( X , r
(2.170)
4.2. H I R O T A , S T R I N G S , A N D D I S C R E T E S U R F A C E S where (26D) S = (1/i) f~ dT (lr(dr - (dr via 0K(r r OK(X,r =
o~
-
Notice that in (2.170) one puts r product
0~
225
and the canonical variables are defined 0K(r
;~=
r
0K(r
0r
= a and r
-
)()
(2.171)
or
= g. Similarly one defines the backward
--~N(a,~t)--(((eab(N+l)-b(Y+l)a,e a~)(g)- b ( g )-a ) , e
a~(Y-1)--b(N-1)~)
N
= f H d#(r
*
..
-*e
ab (0) -b(O)~
q~(J+l)) x
)-(2.172)
0
X
(nO0 eO(x(J)'~P(Jt-1)'r e~(j+l),.fC(j),r
1-I e 0
(X(j)~)(j)-b(j).,~(j))
= a and r _ x(N+I). In the continuous time limit this is exactly the s a m e where r as the result of the forward product except for the boundary conditions which we must impose as r a whereas we had to put r a in the forward product. The formulas thus obtained provide us with a natural interpretaton of the products arising in f 9 (g 9h) and (f 9g) 9h in terms of path integration. Here S is nothing but a kinetic action of path integral quantization. In the case of fiat space the integration over r and r can be performed in (2.170) and one obtains a familiar path integral representation of a free particle propagation
Ab,~(a, a) = ~ 7:)XT:)fieiS/~efo (Xb-bX)d~-; S = -~ .,, d'r X d---~
(2.173)
One sees that the asymmetry between the forward and backward situations disappears if the path of integration over 7 is closed to a loop and the condition r = r - a is imposed. Knowing this one can provide a star product which manifestly satisfies associativity. First one should discuss the external source functions b (j) and one defines a generating functional by multiplying an arbitrary functional w(b, b) to Ab~(a , ~t) and integrating over the b 's via (26E) F(w, a, 5) = f I)bI:)bw(b, b)~b,~(a,~). As an example consider
(
w(b,{~) = 6 b(T) -- y ~ b(J)6(T -- Tj) under the conditions r yields
0
= r
)(
6 {~(T) -- ~
0
{~(J)6('r -- 7j)
)
(2.174)
= a and ~(N+I) _ $(t) = ~. Substituting into ( 2 6 E )
l
where X j means the value of X(T) at T = Tj. Now suppose the path of integration is closed, i.e. (26F) S = (1/i) f dT (Tr(d~)/dT) - (dr in (2.175). Assume also r = r _ a and ~(0) = ~(g+l) = d. Then the result is clearly symmetric under cyclic permutation of the b(J). It is more convenient to use if we multiply Fourier components ~ of a function fj and integrate over the b (j) to obtain N
N
(fN "~ " " " * f2 "~ fl)(a, a) -- / I ~ db(J)dt)(J) 1-I ]J(b(j), t)(J))F( b(1), " " " , b(N); at
1
1
C H A P T E R 4. D I S C R E T E G E O M E T R Y A N D M O Y A L
226
N
= fa
r
T)#(r Jr
is/A I I f g ( X j , f(j)
,a
(2.176)
1
Defining the integration loops as in [840, 841] the rule of the * product is (26G) fN * * fr+l * fr * " ' * fl = ( f Y * ' " * fr+l) * (fr * ' " * fl) and this guarantees the associativity (26H) f * (g * h) = (f * g) * h. One remarks (cf. Chapter 3) that Kontsevich proposed a constructive method for a star product which works for arbitrary Poisson manifolds (cf. [624]) and it was subsequently shown to be equivalent to a path integral for a nonlinear sigma model (cf. [179]). A similar formula can be obtained from (2.176) by restricting to
N=2. To illustrate the correspondence of this formulation to string models one concentrates on flat space where the fields can be expanded into Fourier series 2 # ( 0 r) -- X # "Jr-E
- ' ~ f(pne-ina;
x#
~- p#O" -- ~1
(2.177)
XPneina
1
The upper index # = 1 , . - . , d specifies the space time components. The fiat space integrations over X can be carried out and the canonical conjugate of X " is nothing but X ". This leads to
E o
Z~ nd'f(~n dw
dX~nx~n dT
)
-- ~
1
/0 ( 2,r
da
OXu(a) of(u(a) 02X~'(a) f(u(a) Oa 0"I- - OTOa
)
(2.178)
with free bosonic string action (26I) S = (1/47r) f~ dT f2~r da(OXr,/Oa)(O.f2•/O7.). Consider now a compact domain D in C whose boundary is given by a = 0. Instead of (w, a) use a complex coordinate z to specify a point of D. Let zj for j - 1 , . . . , N + 1 be N + 1 points on the boundary which are fixed in the order corresponding to 7- = 7 1 , ' . ' , 7- = 7"N+1 and let k ~ , . . . , k~v be constant vectors associated with 71,..., TN. If one substitutes
= c ~(j
1
Zj -- Zj+I
,(
b.(~) - ~
1
k}'~(~ - ~j)
)(
- ~
1
k~5(~ - ~ )
)
(2.179)
into (26E) there results
F ( k l , . ' . ,kN;a,~) --
1
zj - Zj+l
(2.180)
,a
which represents an N-point open string correlation function with external momenta k~. In these formulas one obtains C by fixing three points, say za, zz, and z~ and writing C_ 1 _
dzc~dzzdz.y (zc~ - z z ) ( z z - z.y)(z.y- zc~)
(2.181)
so that the integrations over these zj are not over counted (not quite clear). In particular one choose ZN+I = zo to be one of these. Since one assumes k~v+l = 0 the dependence on X~'(zg+l) = X~'(zo) = a~' is implicit. The three fixed points (0, 1, oc) of the Kontsevich formulation are to be identified with these points.
4.3. A FEW SUMMARY REMARKS
4.3
A FEW
SUMMARY
227
REMARKS
Notations ( 2 5 A ) a r e used. We will now try to summarize some of the relations between Moyal, discretization, KP, Hirota, strings, conjugate nets, etc. developed in the preceeding sections (but omitting matrix models and M theory). Further remarks are added in Section 4.4 to follow.
4.3.1
Equations and ideas
Let us first make a list of equations and constructions appearing in Sections 4.1 and 4.2 and try to make clearer the relations between all these relations. Thus: 9 In Section 4.1.1 we have phase space discretization defined via (1.7)-(1.8), i.e. X} ) =
(2~.)2
da
dbSin[)~(alb2 -a2bl)]f(x +)~bl,p+)~b2)e :~(al~176
(3.1)
where (cf. (1.6))
Z~ =
dav)~[f]Va; Va =
1
Sinh()~ ~ aiOi);
(3.2)
X?g = {f , g}M; [X? ' X D] = X {f,g}M D (cf. (3.5) below for {f, g}M). If Fw ~ Wigner function then there is a difference 1-form (1.24), i.e.
h / da f dbeih(al b2_a2bl)Fw ( x + -~bl, h ~ b2) A a PF~ = -~ p + -~
(3.3)
where < A a, Vb > = 5 ( a - b). Then as in (15S), one has the relation < PEw, XAD > = fdxdpFw(x,p)A(x,p) = < A >. 9 In Section 4.1.2 one considers (1.27), namely for A = ~ amnZm~n 1 X 2 = --f ~
am~z~4nSin
{
( 0 0 )} )~ n olog(z) - m olo9(~) - ~
lYtn
amnZ'~r
(3.4)
lYtIt
where Vmn ~ Va and again (17B) says [ x D , x D] = --X{A,B}M. Here we compare (1.6) and (1.29)via -~ i Sin[A(OzlOp2 _ OplOz2)]f(xl,pl)g(x 2 P2)l(p,z) = {f,g}M; {f,g}M =
lsin{A( O 0 0 0)} = --~ Olog(zx) Olog(~2) - Olog(~l) Olog(z2)
f(zx, ~x)g(z2, ~2)l(z,()
(3.5)
The standard Wigner function Fw of (3.3) is often portrayed in terms of wave functions as in (1.31), i.e. 1/
( h ) r
Fw(p, x) = ~r
dyr z + -~y
x - -~y e -ipy
(3.6)
and in the KP situation one uses (1.30) or
FKp(z, ~) = f dx E r162 ,./
gEZ
~-e
(3.7)
CHAPTER 4. DISCRETE GEOMETRY AND MOYAL
228
where r is the BA function. Writing FKP = Y~ fmnZ-m~ -n one defines for < A rnn, •pq > = 5 ( m - p ) 5 ( n - q ) the 1-form ~Fgp(z,~)= ~-]fmnz-m~ -nAmn a s in (1.33) and
d~FKp(z,~)A(z,~)= ~-~fmnamn < a F ~ , , x D > = - / ~dz f 2~i~
(3.8)
Here the formula (1.35) is important, namely f~
=< ~ F ~ , x ~
>=-
f dx f 2-Vf~ dz z m~" qn / 2 z)r
(3.9)
as well as the formula (1.38), namely
Om+e,e0 log(T) =
~/
(z)
(3.10)
leading to (1.39) based on the relation (A = log(q) for example or A = -ilog(q) depending on context)
qnzOzo(z) = exp[nA(O/Olog(z))]~b(log(z)) = ~b(log(z) + log(qn)) = r
(3.11)
Consequently
fmn = - f dxDmnOlog(T); Dmn = qmn/2 ~ 5jg (nA)J j! Orn+t,g t<j
(3.12)
Thence for DA = E amnDmn one has (1.41)-(1.42), namely (3.13) 0
Otr so that for (17I) - f dxDAJr = D A 9Hr one apparently obtains a Moyal bracket structure in KP via (173) 0r,2. = {.A.,Hr}M (note from Xf)g = --{f,g}M as in (17B) and the structure for X D in (3.4) leads to the comparable {A, B}M in (3.5) while the additional structural choice for FKp leads heuristically to (17J) (see comments after (1.42) and [172]). We return to (1.42) now and the philosophy thereafter in order to check further the expression (17J). Thus first we write down in one place the ingredients (A = -ilog(q) or A = log(q) as needed)
A(z, ~) = ~ amnZm~n = ~ amnXDn; ~
- ---s
~ ~n o log(~)
(3.14)
log(z)
One takes {f, g}M as in (3.5) with
FKP(Z, ~) -- / dx E r162 g a~.
= ~ f~z-~-~
= E fmnZ-m~-n ~
(3.15) (3.16)
4.3. A F E W SUMMARY REMARKS
229
1Sin I m ~ - n Vm~ - - ~ Olog(~)
Ologz)
(3.17)
;
1
Va "~ -~Sin(Aa. 0); a ~ (m,n); ( x , p ) ~ (log(~),log(z))
< ~FKp'X2 >= E fmnamn; fmn
=<
(3.18)
[-~FKp,X2n >
We want to show now ( 1 5 W ) - (15X), namely
d dFw -~Pg~(t) = P{H,F~(t)}M -- dt = {h, Fw(t)}M via D < ~Fw, X{A(t),H}M
>~-<
(3.19)
[']{H,Fw}M X 2 >
One first proves
i dxFw(XDA) = - f dx(XDFw)A
(3.20)
To see this we write (following [537] - L(R)HS ~ left (right) hand side) f
(3.21)
L H S = - 7 dxFw(x){A,H}M =
- _
47r 2~ldxFw(x)jdaibsin[~(axb)]A(x+i~bH(x+i~a)-
_- 4r2 a i d x i d a i d b s i n [ a ( a x b ) ] F w ( x + i a b ) H ( x + i a a ) A ( x )
=
= S dx{g, Fw }MA(x) -- R H S (cf.
(1.6) and (17AA)). Combining this with A(t) = exp(iXDt)A and Fw(t) -
exp(-ixDt)Fw one obtains D >=< < ~Fw, X{exp(iXDHt)A,H}M
(3.22)
~{H,exp(_ixDt)Fw}M X 2 >
This follows from (recall XDA = - { H , A}M here)
L H S = f dxFw{eiXDtA, H}M -J = E--n-~.~(it)n 176 0
dxFw{(XD)nA'H}M= E--n-~. 0 =
.....
OO
E(--1)n+l 0
dx
(3.23)
Fw[(XD)n+IA] =
oo (it)n S dx(XDFw)[(xD)nA] = 0
(it)nn' i
Jdx{H,F (t)} A-
RHS
consequently one obtains ( 1 5 W ) - (15X) (the notational shift in ~7a and the absence of h are immaterial here). Now to deal with l?mn, Jr, Hr etc. arising from (1.39) (1.42) we organize first the ingredients. Thus X ~ --0 ainu and fmn in f~f~p(z,~) comes from (1.35) or (3.12) with ( 1 7 J J ) fmn = - f dxTPmnOlog(m) where 1:)ran is given by (1.40). then 79A = ~ amnl?mn and < f~Fgp,Z D = - - f dXTPAOlog(T)= A ( t ) w i t h Ortt = - f dx:DAJr = Z)~. Hr. We want to show that this corresponds to a Moyal
-
CHAPTER 4. DISCRETE GEOMETRY AND MOYAL
230
bracket {.4, Hr}M based on say (1.29) or (3.5) and (17B). Thus think of tr as a time variable in the Hamiltonian theory and 0r-4 = T)A 9Hr as its Hamiltonian flow in this spirit. This has the structure Or < ~Fgp,X D > = 0r,4 which one then asserts to be necessarily < ~tFKp, X {A,Hr D }M >" The reasoning goes back to the equivalence '
{A,H}M =- Ot < ~Fw,X~(t)
:>=<
~Fw,X{A(t),H}M > (3.24)
in the Heisenberg picture. It is not necessary for us to insert spectral variables z, ~ in Hr etc. in order to arrive at this conclusion. 9 It could be instructive now to look at discrete surfaces as in Section 4.2.2, which will correspond to Toda situations for HBDE (cf. (2.35)), and write out the corresponding KP analogies as in Section 4.1.3, along with their stringy counterparts as in Section 4.2.1. One could avoid some of the heavier notation of multicomponent KP and perhaps see more clearly the correspondence to the phase space discretization just outlined above (for Toda see also [250, 832, 833, 834]). For our purposes however the correspondences specified in Sections 4.1.3 and 4.2.1 (with embellishments as indicated) are sufficient. We will however summarize some of the material. 9 Now for Section 4.1.3 following [828] one recalls the HBDE as in (1.49)
afn(g + 1, m)fn(g, m + 1) + flf,.,(g, m)fn(g + 1, m + 1)+ -}-~fn+l(~ + 1, m)fn-l(~, m ~- 1) = 0
(3.25)
and the Miwa transform ( 2 5 P ) t n = ( l / n ) Y ~ kjz~ related to three arbitrary kj (p,q,r) w i t h f = p + q + r - ( 3 / 2 ) , m =-p-(1/2), andn =p+q-1 in (3.25). Then exp(Og) and exp(Om) discribe the discrete surface geometry with exp(On) generating Laplace transformations. Instead of difference vector fields X D based on vn = (A/27r) 2 f dbexp[-A(alb2-a2bl)]A(x+Abl,p+Ab2) as in (1.5)or (1.51)one thinks of exp(Okj) = exp[E~(1/n)z~On] and a shift vector field defined via the equation (17Q) X~ = fdaA(t,a)exp[AEanOn] for which ( 1 7 R ) [ X AS ' XBs] = X -{A,B}s s where {A, B } s is given by (1.53). The formula (1.53) seems to make no sense whatever but it is really just a funny choice of notation. Thus for X-{A,B}s = -- f dc{A, B}sexp(AcO) (where cO ~ ~ CnOn) we have from (1.53)
- { A , B } s = / d b [-A (t + A (2 - b) , 2 + b) B (t, 2 - b) +
(3.26)
+ A (t, 2 + b ) B ( t + A ( 2 + b ) , 2 - b ) while [XAs, XBS]= / d a A ( t , a ) e A a 8 / d b B ( f , b)e AbO- / d b B ( f , b ) e A b O / d a A ( t , a)eAa~ = f f
] ] dadb[A(t, a ) B ( t + Aa, b) - B(t, b)A(t + Ab, a)]e ~(a+b)0
(3.27)
Hence write a + b = c and set a = (c/2) + b with (c/2) - b = b or b = (c/4). Also shift operators XkSj[U] are introduced as in (1.54) so that X~ ~ u(t)exp[Okj] for
4.3. A FEW SUMMARY REMARKS
231
u(t) = U(k)U-l(k ') and for A(t, a) = u(t)1-I 5[an-(z~/nA)] with tn = ( l / n ) E k j q n z ~ one has (1.55) or
(
= H 1--q ze X k~[U] S eCj zj ] = (1 -
q)k~ezp
eO/Okj-_
qntnz~n exp
-zr~-~nn
Doubling the space of t~ gives vertex operators as in (1.58) and a symplectic structure so Moyal brackets can be used instead of (17'R). H B D E is an integrability condition for s (a.29) xS[Ullr > = c+XS+.[v]lr >; XmS[V]lr > = c-X;,_.[U]lr > and the BA function of g P (sort of) arises via ( 1 7 X ) I q e ( z j ) > = Xksj[T-1U0] 0 > where U0 has the form ( 1 7 U ) ; see here (1.60) for the interpretation
XkjS [T-1Uo]
~,~ T - 1 U o O k j U o 1 T
(3.30)
~,~
(1 - q)kj e ~ qntnz~ne~-]~(1/n)Z~On T T
This section is still unclear as to what is accomplished in [828]; there seem to be several sources of confusion. 9 For Sections 4.1.4 and 4.1.5 we temporarily suspend further comments. 9 Let us try to picture the discrete conjugate net associated to the string model as in Section 4.2.1 (cf. [835]). Thus take e.g. just three k vectors k~ with i = 1, 2, 3 and # = 1,2 for D = 2 say (or simply k~ = ki when D = 1). We could think of the k~ ~ p, q, r in ( 1 7 D ) for example with g = p + q + r - (3/2), m = - p - (1/2), and s = p+q1 (say klu = Pu, k~ = qt~, and k~ = ru). The connection to discrete surfaces as in Section 4.1.3 involves shift operators exp(Oe) and exp(Om) with exp(Os) generating the sequence of Laplace transforms. We use n for indexing exp(Ok~) = exp(~(1/n)z~On) and zj ~ kj with j = 1,2,3. The shifts in directions g,m,s are respectively (1/n)(z~ + Zq + zn), -(1/n)Zp, and (1/n)(Zp + %) in the tn direction. At this point we should look more carefully at W ~ in (2.10) in order to u n d e r s t a n d what is going on. In [211] for example one looks at such matrices (as indicated in (2.11)) and in [162] a 2 • 2 situation was written out explicitly. Variations on notation appear also in [36, 102, 253]. Thus we recall from Section 4.1.4 the equations (1.66)-(1.67) where e.g. (%~ = s g n ( # - u) with Euu = 1 and t~ ~ t,~)
t) =
T~u ( t - [1].eU))e~(Z,t~); T~,(t) = T(g + e p -
e,,t)
(3.31)
and strictly speaking T(t) -- T(g, t) (recall we have changed the notation of [835] in (2.10) to agree with this under z --, l/z). Evidently g ~ q and one has D components (1 or 2 here). We note also that /~uv in (2.13) corresponds exactly to /3ij in (1.67) which we recall as (p, u = 1 , . . . , D = 2)
=
( , # .); Z . . ( t ) =
Olog(T) (u u = tl~) Ou,
(3.32)
C H A P T E R 4. DISCRETE G E O M E T R Y AND M O Y A L
232
Then for D = 2, N = 3 we have k~ (j = 1,2,3 and # = 1,2), tn~ (0 _< n < ec and # = 1, 2), q~ (# = 1, 2), and zj (j = 1, 2, 3). We recall here also from [253] that the Levy transformation is defined via (1.75) as s
=
-
0 X t,
+
1 0~t, X o --S
";
~
=
where O4u/Ouu = /~t~u(u (no sum). Note here that a formula in [253] where it is proved that for the 4u(t) = (dnCu,/dzn)(z, t), where n > 0 is the order the #th column of r (u = 1,... , D - cf. ( 1 9 0 ) )
- -r
(3.33)
(2.14) is apparently referring to Levy transformation with data of the first nonzero derivative of one can write
where # = 1 , . . . , D = 2. This takes some proving which we omit here since the calculations are not unlike those in Section 4.1.4 for the vertex operators of [252]; note the Xu e a M arise as in ( 1 9 0 ) where X(t) = fsl dzr t)f(z) with f(z) e MD• Now replacing z by 1/z (3.34) is the same as (2.14). Note here from O~2,/Ou~ fl~uCu or OCt~e/Ouu = fl~uCve one gets (O/Ouu)Dnr = fl~vDnr corresponding to O~t~/Ouv = 13~u(v. Now from (1.66)-(1.68), (2.10), (2.13), etc., in a strictly 2 x 2 format, Wt~u ~ (X,)u and Bt~u ~ 3,u so for given z ( X t t ) u ,~
s
1-5"~ e e l z-~tX . e- E 7 (z~/n)O~T#u(t)
%u(t) = T(q + et~ -- eu, t); flt~u = %u %.(t) T(t)
(3.35)
Note the fl,~ arises from the asymptotic expansion in ( 1 9 H ) so the association B , . Z , , of [8351 is rather too loose. The Levy transform (3.34) is spelled out in [253] (cf. (20C), (20D), and (1.82)) with VT(z)f(t ) = f ( t - [z-1]e.). Note in (3.34) (X,)~ ~ Wt,~ (#th row vector of W ..~ Xt, ) so V~(z) acts on a quotient of tau functions. It is perhaps worth sketching a proof of (3.34) following [253]. Thus one notes first that
V~-(p)r
= V~-(p)x(z,t)
(z)
1 - pP,
r
=
(3.36)
Pt~ + I - - V ~1 - ( / 3 P , ) + O ( z - 1 ) ] r , P where (Pi)jk = 5ijhik with ~0 "~ diag(exp[~(z, tl)]exp[~(z, t2)]), and X is the bare BA function with expansion X(z) ~ 1 + ~z -1 + O(z -2) as z --~ c~ (cf. (1.65), (1.67), (19E), (19H), etc.). To see this observe that = [-
V I ( p ) r - (V~- (p)x) )diag z
(e'(Z'tl--~9-1]), c'(z't2) ) :
= (VI(P)X)diag ( ( 1 - p ) e ( ( Z ' t l ) , e ( ( Z ' t ~ ) ) =
10r P~
z
(VI(P)X)(1--P1)~0 P
One compares this with Or = [zP, +/~P, + O(z-1)]r considerations and asymptotics there results (cf. [253]) V~ (p)r =
(3.37)
and by Grassmannian
[ 1 ] t- 1 - p(V,--(p) - 1)~P~ ~
(3.38)
4.3. A F E W S U M M A R Y R E M A R K S
233
Further, from the matrix form of (1.70), namely (25Q) Pv(Or ) - Pu3Pu~ for # ~ u (recall (19J) (Or = 3u~,r for row vectors r ~ . ) one gets
0r = Pmu 0r + E p~zp,r = p , =or - + (zp, - p, z p , ) r v#u Ouu Ou u
(33o)
(think of consolidating the rows). Thence (3.38) becomes
v,-(~)r
0r
1
- ~2 - pP,~_.ouu +
1
PuZP.r - P (Vff (p)3Pu)~2
(3.40)
or equivalently
P~Vff (p)r = p ~ r 1
1
~Ys162
0~
Pu Vff (P) r : - - P ~ 7 + f(t)Pur p out~
(. #
~);
(3.41)
1
f(t) = 1 + - ( 1 P
(recall Etw is the matrix with 1 in the (p, u) position- Pt~ = E~u). From (3.36) one deduces that V~-(p)r t)P u = 0 so putting p = z in (3.41) yields
1 (Vff (p)3u~)(t) = 4~(t)
10log4,
dn
~u(t); f(t) = P Ou u (p, t); ( , = d---~r
P
t)
(3.42)
Consequently (3.41) can be written
(Vy(p)r (V~-(p)r
, t) = r
t) = -Pl OCmu o--7(z,
t) - ~r - ~ W u ( z , t);
t) +
10log~ P Ouu
(3.43)
(t)~,(z, t)
Since the rows of ~ and Vu- (p)r provide tangent vectors for conjugate nets we compare (3.43) with (1.75) to obtain s (r
= P6"" Vu- (r
(3.44)
which leads to (3.34) for vectors in R M via ( 1 9 0 ) . Now for the connection to strings one picks three kj as indicated before (3.31) and looks at vertex operators etc. as in (2.5)-(2.9). A tau function can be produced as in ( 2 1 W ) and shift operators Tju arise as in ( 2 1 X ) - ( 2 1 Y ) . Note to = ~ 3 kj ~ t~ = ~ k~ for # = 1,2 and q = (ql, q2) ~ to means ql = t~ and q2 = t02. Consider (Xu)v = Wu~ where Xu is the pth row of W and look at (21Z) (Xu)jv = etwF~ltuTfvlFc. Note the t~ shift is independent of zj and Fo, F0 in (2.8) are apparently considered as unshifted. Thus to obtain elements as in (2.10) using (2.8) one looks at (Fo/Fo)TuTfvl(Fo/Fo) as in [835]. 9 We can now go to the explicit calculations of Section 4.2.2 with the background model just sketched in terms of strings. First the rotation matrix is given as (22A) Qu, = %vF~ITuT~-IFc and we recall from [250] (cf. (2.32)-(2.34) et suite) that TQu, = eUvs (we inserted euv here to enter the multicomponent KP picture- one recalls also Su~ ~ s with TUv = Su~(T ) from [253]). Thus Qu~ ~ eu~('cu~/T) ~ 3u~ from (2.13) and this needs comparison to (22A), which means we should look at
Tu----Y-u Fc1TuT~-IFc %v "r = euv Fo l Tu Tj-1F 0
(3.45)
C H A P T E R 4. D I S C R E T E G E O M E T R Y A N D M O Y A L
234
.y
However for Fo - 1-Ie>j(zj - ze)~ k3 k'[ we get via T~T~ -1 9 q ~ q + ep - e,
FolTt~T~-IFo = H (zj - Ze)k~+kY-k; -k[+2 j<e
(3.46)
Thus the f a c t o r FolT#T~-lFois a constant (independent of t) and can be removed from Qt~u in describing the geometry (at least in certain respects).
4.4
MORE ON PHASE
SPACE DISCRETIZATION
We want to add here another section which greatly expands the scope of the previous sections in this Chapter in the sense of providing still other points of view and possibilities for comparison. First from [203, 204] for Schrhdinger wave functions ~ one defines Wigner functions ( W F ) via
f ( x , p ) = ~1 / d y e *
(x - ~ y ) e x p ( - i y p ) r (x +-~5 y )
(4.1)
Then defining f 9g via (cf. also [68])
(OxOp-
f,g=fexp
f(x,p) , g(x,p) = f
(x
OpOx
g;
(4.2)
i t ~O p , p - --~ i~-g)x g (x,p) + -~
time dependence of WF's is given by (H ~ Hamiltonian)
cOtf(x, p, t) = -~1 (U 9 f ( x , p , t ) -
f ( x , p , t ) , H) = {S, f } M
(4.3)
where { f , g } M ~ Moyal bracket. As h ~ 0 this reduces to Otf = {H, f} = 0 (standard Poisson bracket). A natural generalization of this from [930] (cf. also [278]) involves first a reformulation via
< xl~lx' >= ~(x-
x'); <
xl~lx' >= -i~o~(x-
x'); <
xl~lx' >= p(x,x')
(4.4)
(where fi ~ density matrix. Then with Fourier transforms -4(r/,~c) = ~1
f
dqdpe-i(oq+@)A(q,p); A ( q , p ) =
l
(4.5)
one will write z = (q,p), P. = (~,15), and a = (r],~) and the Weyl transform of A(q,p) is defined via
ftw " A(q,p) ~ ftw(A) = A = 1 f da.~(a)eia ~
(4.6)
with the Wigner transform of .4 being
W" A ~ W(A) = A(q, p) = 2 / d t e -pt/2 < q - t l A l q + t > From the identities
e i('7~+~) = ei'7O/2ei~PeiOq/2;
(4.7)
(4.8)
4.4. M O R E ON P HA S E SPACE D I S C R E T I Z A T I O N
235
< x[ei(nO+~P)]x' > = ein(z-i~)5(x ' - x + 27ri#~) (for # = ih/2) one obtains (A) W(f~w(A)) = A and f~w(W(ft)) = A so W ~ f~w1. Further from (4.6) one has the Weyl ordering
n(;)
aw(qnpm) = E 0
okpm(ln-k
(4.9)
whereas (4.7) implies that for any t5 and A (B) Tr(~f~) = 7-~ 1 f pAdpdq so p ~ f2wl(r and A = f~wl(A). In fact p/27rh is the Wigner function associated with the density matrix r The 9 product is defined now via (note 2# - ih) 1
1
f * 9 = f~wl(f~w(f)f~w(9)); { f , g } M = ~-~p[f,g]M = -~p(f * 9 -- 9 * f)
(4.10)
where [, ]M ~ Moyal bracket. The corresponding phase space formulation of quantum statistical mechanics follows from the above via (B). Thus (z* = 2 and f~wl(A)= Aw(q,p))
9 (i) A = A t
r
9 Tr(fl)= 1 r 9 ~wl(0k)
__
Aw=A~ f Awdqdp= 27rh
qk, f~ol(~k) = pk
9 If A =/~ = 1r > < r
is a pure state then
/ pw(q,p)dp=
/ pw(q,p)dp=
2~1r
27rlr
2
(4.11)
A substantial and natural generalization of (4.6), which includes many useful operator orderings distinct from (4.9), arises from
1/
a " A(q,p) ~ a(A) = ~
dcra(a)fi(a)e i ~
(4.12)
One assumes the weight function 9t to be an entire analytic function with no zeros. If e.g. A(q,p) = qnpm then (4.12 gives an ordering different from (4.9). Setting (C)a~(z) (1/2r) f daexp(iaz)(1/f~(a)) the inverse f~ transformation exists and is given by 1
f~-l . A ~ A(q, p) = -~ f dq'dp'dtw(q - q', p - p')e -(p't/u) < q' - tlA q' + t >
(4.13)
The proof is straightforward and by (4.7) one obtains
a-l(A) = ~1 /
dq'dp'w(q-q',p-p')Aw(q',p')
1
= -~--~a~@Aw
(4.14)
where | ~ convolution (in fact any lir~ear transformation of quantum operators to phase space functions which is phase space translation invariant is of this form). Consequently f | g = (27rf~-g) and (D) a - l ( A ) = A(q,p) =~ .,~(a) = (1/a(~)).~(~) with
A~(n,~)
- 2 i # f dqe -irlq < q + i#~lfilq - ip~ >
(4.15)
This leads then to ft_l(A) = _ i #
f ei[~(q-q')+~P] q~ dq'd~d~ < + ~ ~(n, ~)
i~lAI q~ _ i~5 >
(4.16)
CHAPTER 4. DISCRETE GEOMETRY AND MOYAL
236
We obtain then in analogy to (4.10) a nonabelian associative algebra structure 1
1
f *~ g - ~2-1(gt(f)~t(g)); ~ [ f , g ] - -A-::(f *~ g - g *~ f) ztt ztt
(4.17)
This is called a generalized Moyal product corresponding to the associated generalized Weyl and Wigner transformations. Now (4.17) can be given an explicit form as follows. From (4.12) one has
1 f dada, f(a){7(a,)gt(a)gt(a,)e(i~)e(i~,~) a(f)a(g) = (2~)2
(4.18)
Using (4.8) one obtains (E) exp(ia2)exp(ia'~) = exp[p(a' Aa)]exp[i(a +a')2] where a' Aa = (~', ~') A (~, ~) = ~'~ - r]~'. This leads to
1 (f *a g)(z) = (2~)2
/ dada'](a)g(a')~(a)~(a')et4a'Aa)~-l[ei(a+~')~]
(4.19)
However by (4.12) (F) ~[exp(iaz)] = exp(ia~)~(a) and hence
(f *a g)(z)
1 = (27r)2
~,(~,~)~(~+~,)z dada'/(a)~(a') a(~)a(~') a ( a + a')
(4.20)
Some further manipulation gives also
1
1
2---~[f, g](z) = (2r) 2
/
~(a)gt(a') sinh[p(a' A a)] ](a){l(a,)ei(a+a,)z dada' a(a + a') #
(4.21)
One can then show that from (G) Uf(z) = f da~(a)](a)exp(iaz) there follows U ( f , ~ g ) = U f 9Ug. This shows that the 9 and *a products define isomorphic algebras. In fact if in (4.20) the kernel [~(a)~(a')/~(a+a')]ezp[#(a'Aa)] is replaced by B(a, a'), and associativity is required, then (4.20) is the only possibility; this is related to the uniqueness of the Moyal algebra (cf. [343]). In particular one has now (H) (1~2#) If, g] = U -1 ((1/2#)[Uf, Ug]M) and some further calculation shows that if ~t is # independent then the Moyal bracket is the only *a product whose associated Lie bracket tends to the P bracket as # ~ 0. We note finally the generalization of (B) in the form
~(0)/
Tr(~(f)~(g)) = ~
f *a gdz
(4.22)
We refer also to Sections 1.1, 4.1, and 4.2 for further details and expansion. 4.4.1
R e v i e w of M o y a l - W e y l - W i g n e r
We follow here the classical formulation as in [59, 65, 313, 458, 901]. Most of the formulas appear already in the book with derivations and we only gather together here some basic ideas in one place in order to discuss discretizations. Philosophically the background theory of Moyal is statistical in nature. Thus given a QM system in which observables may be expanded via (qk,Pk) one represents the system by a quasi probability distribution (not always positive). The uncertainty principle will be related to the occurance of negative probability. This approach can be made equivalent to the standard QM treatment. Thus following [458] the classical average (X) < A > d = f f dqdpA(q, P)Pd(q, P) for a phase space distribution is replaced by (Y) < .4 >Q= Tr(.4~) where f~ is a density matrix and in terms
4.4. MORE ON PHASE SPACE DISCRETIZATION
237
of a quasiprobability distribution one has (Z) < A > Q = f f dqdpA(q, p)PQ(q, p). Now one must make explicit the relation A ,-, A and r ~-~ PQ(q,p). The Wigner distribution will be denoted by Pw and with that choice A ~ A is the Weyl correspondence leading to the Moyal theory. We will concentrate on this situation. Thus one defines
iS
Pw(q,P) = --~
dy < q - y]f~]q + y > e2ipy/h
(4.23)
O0
For a pure state r one has
P(q,q') = <
qlPlq' >= ~ Wn~2n(q)r
= r162
(4.24)
y)e 2wy/h
(4.25)
n
1/ dyr
which implies
Pw(q,P) = ~
+ y)r
The following properties for a pure state r are among those of special interest for any admissable P(q, p) (here Pw(q, P))
9 (i) f dpP(q,p) = < q[Plq >; f dqP(q,p) = < p]~]p >; f dq f dpP(q,p) = Tr~ = 1 9 (ii)r ~ r P(q,p-p~) 9 (iii) r
+ a) =~ P(q,p) ~ P(q + a,p); r
~ r
=~ P(q,p) ~ P(-q,-p); r
9 (iv)Ifdqr162 2 = 27chfdqfdpPr162 ::~ f dq f dpPr162 =0
~ eip'q/hr
~ r
~ P(q,p)
~ P(q,p) ~ P(q,-p) f d q f d p P ~ ( q , p ) = )-~, 1 . r _L
The last condition shows that P(q, p) cannot be everywhere positive. One recalls now the correspondence
A(q, P)
f dze ipz/h < q - (z/2)]ft[q + (z/2) > ~ Aw(q, p) J
and the result
f
f dq J dpA(q, p)B(q, p) = (27rh)Tr(A/3)
(4.26)
(4.27)
with the consequence that f f dqdpA(q,p) = 27rhTr(A). Further from (2.1) and (4.27) one has / ,
] dq ] dpA(q, P)Pw(q, p) = Tr(~fI) corresponding to ( X ) - ( Y ) as desired. Set now (AA) r and then (2.3) can be rewritten as
1/ dp'r
Pw(q,P) = - ~
+ p')r
(4.28)
= (27rh) -1 f dqr
- p')e -2iqp'/h
(4.29)
One should note the restriction here to a nonrelativistic theory and the fact that not all functions P(q, p) qualify as quasiprobability functions. For the Moyal-Weyl correspondence one writes (cf. (4.28)) <, ~lA]~ > = f dq f dpPw(q, p)A(q, p) = Tr(Dft)
(4.30)
C H A P T E R 4. D I S C R E T E G E O M E T R Y A N D M O Y A L
238 via the association (h = 1 here)
' A(~, i5) = / da f dTa(a, 7-)ei(~O+'p)
(4.31)
Putting this .A into (4.30) and replacing A by the integral of a in (4.31) leads to an identity (the BCH formula is used also- cf. [458]). One recalls also the idea of characteristic function (BB) C(a, ~') = Tr(~C(a, T)) where C(a, 7-) = exp[(i/li)(a~ + Ti5]. It is then shown in [458] that
Pw(q,P) =
(:)'/ / ~h
da
dTe-(ffh)(aq+~P)C(a,~-)
(4.32)
For the dynamics now we refer to [901] first where one writes A w as in (2.4) with
(AB)w = Awexp
~ [Op
0 q -- 0 q
Op
BW = A w * Bw;
(4.33)
(cf. Section 4.2 and [789]). Then from [901]
Opw Ot = { H w , PW}M
(4.34)
dAw d.4 _ l__[f~,HI =:> = { A w , HW}M dt ih dt
(4.35)
Ot
= 1[:t, ih
Similarly for the Heisenberg equation
Note that the right side in (4.34) is not in general approximable by the classical Liouville equation even for small h. Now the evolution operator U(t, s) mapping r ---+~bt = U(t, s)~2s is the unitary operator solution of ( C C ) i h ( O / O t ) U ( t , s ) = _f-I(t)U(t,s) with initial condition U(s,s) = I. In classical mechanics the classical phase flow gc(~l') maps an initial state z ~ (q,p) e R 2d to gc(TIZ ) -- Zc(T) on the trajectory Zc starting at z and satisfying (DD)(d/dT)zc(~') = JVHc(zc(T), T) where the Poisson matrix J is inverse to the matrix of the symplectic form dqi A dq i (cf. Section 4.2 for more on notation) One writes also g ( t , s, x, y ) ~ < x]U(t, s)ly > in the form (EE) ~t(x) = fRd dyK(t, s, x, Y)~2s(Y). One wants now to systematically inject classical dynamics into the formulation of the quantum evolution problem. First consider the Heisenberg operators T(zo) for z0 E R 2d satisfying (FF) T(zo)*~.T(zo) = ~. + zoI (cf. here also Section 2.1). Now T(zo) can be defined via
T(zo) = exp
-~zo . g-12
= exp
]
(Po " O - qo "P) =
(4.36)
= e-(i/2h)poqoe(i/h)pOOe-(i/h)qoP Note that the special cases T(qo, O) = e x p ( - q o . Vz) and T(0,p0) = exp[(i/h)po. O] lead to (4.36) i n c o m b i n a t i o n w i t h j =
( - 0I
0I )
where I is a d-dimensional unit matrix. We
note that for ~ = T(z0)r ( F F ) implies (GG) < 2 > r r > = < 2 >r +z0. Further all moments about the mean are invariant with respect to translations T(zo) so T shifts the expectation values but does not significantly alter the shape of a wave function. One shows 9 (i) The translation operators are unitary and obey T(z0) -1 = T ( - z o ) = T(zo) t.
4.4. M O R E O N P H A S E S P A C E D I S C R E T I Z A T I O N
239
(ii) The integral kernel of T(zo) is < xlT(zo)lx' > = exp (~Po " (x + x')) 5(x - x' - qo) 9 (iii) If z(-) is a smooth curve in phase space then
= - T ( z ( t ) ) k ( t ) . j - 1 ( l~z +( t )-2)
ih~T(z(t))
(4.37)
To see all this note
[T(zo)r
=- e(i/h)(-(1/2)p~176176 e-q~162
- e(ih)P~176162
- qo)
(4.38)
One checks that (ii) above leads to the same value of [T(zo)@](x). ( F F ) can be proved by straightforward calculation based on (4.38). For (iii) one considers
d eA(r ) = eA(r ) dT
) ~01d)~e),AO.) dAe)~AO. dT-
(4.39)
To apply
based on ( H H ) (d/d)O[exp(-AA)(d/dT-)exp()~A)] = exp(-)~A)(dA/dT-)exp()~A). (4.39) choose A(7-)= (i/li)z(7-). j - 1 2 and compute
i l i d T( z(7-) ) = ih = --T(z(T)) = --T(z(T))
/0
e AO-) = iIie A(r)
/o 1d)~e-)~A(r) -~i z(,, 7-) . J-12e)~AO-)
1
~0
d)~T()~Z(T))ti(T)
9
=
J-~T(~z(r)) -
d)~:~(T), j-l(~, + )~Z(T)) = --T(z(T))k(7.)" j - 1
(4.40)
(
~ + 2z(7.)
)
Now one constructs an alternate description of the quantum dynamics that incorporates the phase space translation operators. Let z(t) be an arbitrary time indexed C 1 curve in phase space. Take @t to be the time dependent state whose motion is governed by /:/(t). The transformed state r = T(z(t))*@t has the companion evolution operator (II) Uz(t, s) = T ( z ( t ) ) t U ( t , s ) T ( z ( s ) ) the the time derivative of (II) determines the equation of motion for the unitary operator Uz. Stated as a Schrbdinger initial value problem the result is (J J) ih!O/Ot)Uz(t, s) = [-Iz(t)Uz(t, s) with Uz(s, s) = I and a transformed Hamiltonian ( K K ) Hz(t) - ~(t) . j - l ( ~ + (1/2)z(t)) + T(z(t))t[-I(t)T(z(t)). Note that even i f / : / i s static [-Iz(t) acquires a time dependence from z(t). [789] deals with many topics of no concern here and we only note the emergence of the Moyal-Weyl-Wigner machinery. Thus by a change of variable z ~ - v (2.4) becomes (4.41) Note that for operator functions of position or momentum alone one has h(q)w(z) = h(q) and f ( ~ ) w ( z ) = f(p). In order to extract from the quantum operator/:/(7.) its underlying classical Hamiltonian Hc one has (LL) Hc(~, 7) -/:/(T)W(~)Ih=0 and as usual the full Weyl symbol for /:/(T) has the decomposition [-I(T)W = He(', T) + h~v(. , T, h) where h~v ~ O(5). Then one has the standard formulas based on (4.41)
J = A w ( ~ V u ) exp(iu. ~)
u---0; [T(zo)tAT(zo)]w(z)=
A w ( z + zo)
For the dynamics we refer to Section 4.1 and cf. [662] for constraints.
(4.42)
C H A P T E R 4. D I S C R E T E G E O M E T R Y A N D M O Y A L
240 4.4.2
Various forms for difference operators
It would be nice to have a more symmetric form connecting X}9 Va, and v)~[g] in Section 4.1. To this end consider (cf. (1.5), (1.7), etc. and see also (2.62)- (2.69)) v~[f](x, a) =
dbe -i)~(axb)e'xb'Sf
~
(4.43)
Hence (using b --~ - b )
dbei~(axb)e~.Sf =
v,~[f](x , - a ) = and as before V - a = - V a
L 1/
X? = ~ 47r2
da
2
dbe_i~X(a•
(4.44)
so
X ? = J dav~x [f] (x, a) Va = Consequently
A
dbe -i~x(axb)
dav),[f](x,-a)V_a=-
da [v~[f](x, a) 2A
-
/
dav),[f](x,-a)Va
v~ [f] (x, - a ) ] Va
-
(4.46)
-
f V a -- 4 7 r 2 f d a / d b e - i ) ~ ( a x b )
VbfVa
This gives perhaps a nicer Fourier theoretic viewpoint. We should check here X ~ . g has the form A3
X ? " g = "~2 / d a / dbe-i~X(axb)X~bfVag
(4.45)
which
(4.47)
This looks nice but leads back to calculations as in (1.10)-(1.12) which are no simpler. The formula (4.46) should perhaps be thought of as providing another representation for X}~, namely X? =
/
"/
da~,x[f] (x, a)Va; ~.x[f](x, a) = ~
dbe-i~(axb)vbf
(4.48)
The object now is to obtain a better conceptual picture of Va, etc. Recall in some sense
X ~ ,.~ (Of/Op)Ox -(Of/Ox)Op (cf. ( . . ) in Section 3). One could think of going back to standard difference operators as in [19, 53, 128, 215, 228, 229, 230, 231,232, 233, 234, 235, 236, 237, 337, 366, 367, 437, 449, 820, 827, 875, 989, 999] but the Va is tuned already to the Moyal framework. For completeness however we recall some standard notation regarding difference and q-differential operators. There are both algebraic and geometric aspects and we will sketch some of the ideas. In one dimension the connection between noncommutative differential calculus and lattice theory becomes already apparent. Thus going to [231] for a more of less coherent exposition one develops a lattice differential calculus as follows. Let .M = Z with eij -r 0 r j = i + 1 and write x = g F_.j jej for g > 0 where ej is defined via e~(j) = 6~j (i.e. the e~ are functions in .,4 = algebra of C valued functions on 3,4). Then eiej = 6ijei and ~ ei = 1 while ( T T ) dei ej + ei dej = ~ijdej with ~ dei = 0. One writes here dei = eji - e i j (as a special case of df = ~ij[f(J) - f(i)]eij with eli = 0). Then
dx -- ~. E idei : ~"E i(eJ i - eiJ) -- ~"E i(ei-1, i - ei,i+1) -- ~ Z ei,i+l
(4.49)
Using now eil.-.ir = eili2ei2i3 "" .e#_~,i~ (r > 1); eil...i,.ejl...j8 -- 5i,~jleil,...ir_ljl...js
(4.50)
4.4.
MORE
241
ON PHASE SPACE DISCRETIZATION
one has [dx, x] - e 2 E j [ e i , i + l , e j ]
ij
- e2E
ei,i+l - e d x
(4.51)
Hence ( U U ) [dx, x] = edx leading to dx x = (x + e)dx and d x f ( x ) = f ( x + e)dx. Further 1 [dx, x] = -~[(a+f)dx, 1 1 df = ( a + f ) d x = -~0+ x] = -~[df, x] = -- 89( d ( f x -
(4.52)
l ( d x f - f dx) = ~ ( f ( x + e) - f ( x ) ) d x x f ) - [f, dx]) - -~
Consequently (with similar calculation for df = dx O_ f ) 1 1 O+f = -~[f(x + e) - f(x)]; S _ f = ~ [ / ( x ) - f ( x - e)]
(4.53)
It looks a little weird but this is the basis for general noncommutative differential calculi. As for integration one would like to have f df = f + c where c is a function such that dc = O. Such c are the periodic functions with period e and every such function can be integrated. Thus xo +n~
[
Jxo-mg.
n- 1
f(x) x = e Z
k---m
f(xo + ke)
(4.54)
with natural limits as m, n ---, oc. Now let .A be the associative and commutative algebra over R freely generated by x ~ where 1 < # < n. The ordinary differential calculus on .4 has the property [dx ~, x"] = 0 and one introduces now a class of noncommutative differential calculi such that ( V V ) [dx/~, x"] Cir~v dx_ k . Then from [ d x " , x v] -- ( d x " ) x u - xUdx" - d ( x " x " - xUx") - x " d x " + (dx~')x" - [dxU, x/~]
(4.55)
one gets ( W W ) C~u - C~'. Similar calculation yields C ~ " C ; p = C ; " C 2 ~ - C " C " = C"C~; ( C " ) [ = C ; "
(4.56)
One notes that ( V V ) is invariant under (suitable) coordinate transformations. Finally set ( X X ) g,U = T r ( C , C u) as a metric (although C~ u is the more fundamental geometrical structure). As an example consider the commutative algebra .4 of functions on Ad = e0Z x el Z and define [dt, t] = eodt; [dx, x] = eldx; [dr, x] = [dx, tl - 0 (4.57) This is a standard lattice differential calculus and one can write for Z = (t, x) dt f ( Z ) = f ( ~ + Q)dt; dx f ( ~ ) = f ( ~ + e l ) d x
(4.58)
where ~ + e0 = (t + e0, x) and Z + el = ( t , x + el). Further 1 1 df = K0 [ f ( e + e0) - f(e)] + K1 [ f ( e + el) - f(e)]
(4.59)
Acting with d on (4.57) gives ( Y Y ) d t d x = - d x d t and dtdt = 0 = d x d x (this familiar anticommutation does not however extend to 1-forms). One can show easily that every
242
C H A P T E R 4. D I S C R E T E G E O M E T R Y
AND MOYAL
closed I-form is exact. Indeed for w = wo(t, x ) d t + wl(t, x ) d x the condition dw = 0 means i)+tWl = O+xWO = 0. For simplicity set t~0 = el = 1 and define t-1
F(t, x) = ~
x-1
Wl (t, j )
wo(k, O) + ~
0
(4.60)
0
(integrate w along a path ~, : N --~ Z 2, first from (0, 0) ~ (t, 0) along the t-lattice direction, and then from (t, 0) ~ (t, x) along the x-lattice direction- this will not depend on the p a t h via Stokes' theorem). Then e.g. x
OxF = F ( t , x + 1 ) - F ( t , x ) = E W l ( t , j ) 0
x-1
- E
wl(t,j) = Wl(t,x)
(4.61)
0
and using dw = 0 one obtains x--1
O+tF = F ( t + 1, x) - F(t, x) = wo(t, O)+ ~
O+tWl(t, j) --
(4.62)
0 x-1
=
o(t, o) +
j) =
o(t, o) +
o(t, x) -
o(t, o) =
o(t, x)
0
Consequently w = dF. Now one goes to the conditions for a 9 operator which is defined generally as follows. Let A be an associative and commutative algebra with unit 1 and (Ft, d) a differential calculus on it (f~(~4) = (~r>0 f~r (.4) where the F/r are .4 bimodules and f~~ = .4; further d : fV --~ FF +1 with d 2 = 0). Assume now that 9 : f~l ~ f~l is an invertible linear m a p such that (ZZ) , ( w f ) = f , w and w , w ' = w' , w . Further one requires ( A A A ) dw = 0 ~ w = **d)~ with X e .4. Also for a e G L ( n , A ) and A = a - I d a one obtains ( B B B ) F = d A + A A = 0 since dz -1 = - a -1 (da)a -1. These definitions are made in such a way that the field equation of a generalized a - model ( C C C ) d , A = 0, and a construction of an infinite set of conservation laws in 2-dimensions, generalizes to a considerably more general framework. For the situation at hand one introduces g,U via ( D D D ) d x " , dx ~ = gPUdtdx. W i t h w = w , dx ~ ( Z Z ) becomes [w,(2)w~(2 + e, - e~) - w ; ( 2 ) w ~ ( 2 + e, - g~)]g"" = 0
(4.63)
' This yields ( E E E ) g "u = c"5 "u with arbitrary functions c"(2~) which for all w , and wtL. must be nonzero in order for 9 to be invertible. This includes the metric ( X X ) . For the generalized Hodge operator we obtain ( F F F ) 9dt = c~ - Q ) d x and , d x = - c 1 ( 2~ - el)tit which extends to f~l via (ZZ). Now choose g"~ = r/"~ (2-dimensional Minkowski metric). T h e n 9 w(2) = w ( 2 - t~0- t~l) which, together with the above result on exactness of closed 1-forms, implies ( A A A ) . Therefore the construction of conservation laws will work in this case (the development of this is in [232] and we just indicate what happens in this example). Thus look at the simplest generalized a - m o d e l where ( G G G ) a = e x p ( - q ( t , x)) and qk(n) = q(neo, kel). Then 1
[Leqk(n)-qk(n+l)
- lj] dt + ~1 [eqk(n)_qk+l(n ) _ 1] dx"
1 [eqk_l(n)_qk(n ) _ 1] dt 9A = ---~ol[eqk(n-1)-qk(n)--l]dx -- -~1
(4.64)
4.4.
MORE
ON PHASE
243
SPACE DISCRETIZATION
and the field equation d , A - 0
takes the form
Replacing .A with the algebra of functions on R • glZ which are s m o o t h in the first a r g u m e n t one can take limits g0 ~ 0 leading to the Toda lattice equations ijk + 1~ [eqk_qk+~ _ eqk_~_qk ] = 0
(4.66)
T h u s there is a n o n c o m m u t a t i v e geometry n a t u r a l l y associated with the T o d a lattice. We want to expand now on the relations between the viewpoints of a differential calculus over an algebra A based on ft(A) and d as above and the perspective of discretization or calculus over a finite (or discrete) set. T h e idea is to find another way to express the material involving Va and A a in the Saito theory. Thus we go to [230] and sketch some material in detail. Thus first for an associative algebra A over C with unit 1 one defines a differential calculus over .4 via a Z graded associative algebra ft(.A) = @r>_o~r(.A)with .A-bimodules ~r(,A) and a C-linear map d 9 f~r(.A) ~ ~ r + l ( , A ) satisfying d 2 - 0 and ( H H H ) d ( w w ' ) = ( d w ) w ' + ( - 1 ) r w d w ' where w E a r and w' E f~. One requires w l w = w l and from 11 = 1 there results d l = 0. F u r t h e r one requires t h a t d generates the ftr for r > 0 in the sense t h a t f V ( , A ) = . A d f t r - l ( , A ) A . T h e n given a differential calculus (ft(.A), d) over an associative algebra ,4 a linear (left A-module) connection is a C-linear m a p V " ~ I ( A ) --, a l ( A ) @A a ( A ) such t h a t ( I I I ) V ( f a ) = df @.4 (~ + fVc~. This extends to a m a p V " ft(A) | f~l(A) ~ f/(A) | ~1(-A) via V(w |
a) = dw |
a + ( - 1 ) r w V c t ; w E f~r(A); Ct C ~ I ( A )
(4.67)
T h e torsion of a linear connection V is the map O" f~l(A) ~ f/2(A) given by ( J J J ) O ( a ) = d a - ~r o V a where 7r is the n a t u r a l projection ftl(A) | f~l(A) ~ f~2(A). It satisfies ( K E g ) O ( f a ) = f O ( a ) and extends to a map O" f t ( A ) | f~l(A) via O(w |
a) = d(wc~) - 7r o V(w |
a) (w C fI(A), a E Ell(A)
where now 7r denotes more generally the projection t2(.4 |
(4.68)
ftl(.A) ~ t2(A). T h e n one has
O(Vc~) = dTr o V(c~) - 7r o V2(a) = d ( d a - O(c~)) + 7r o T~(a)
(4.69)
where the curvature 7~ of V is the map T~ = - V 2 satisfying T~(fc~) - fT~(c~). This leads to the first and second Bianchi identities d o @ + @ o V = 7r o 7~;
(4.70)
(V~2~)(Ct) = V(~2~(Oz)) -- n ( V C t ) = --V3Ot nt- V30~ = 0
Now go back to calculus on a finite set. Let A/I be a set of N elements and .A the algebra of C valued functions on it with basis e i (i = 1 , . . . , N) where e i ( m ) = ~ . T h e n as before eie j = 5iJe j and ~ e i = 1 (cf. [235, 236, 237]); one knows t h a t first order differential calculi on a finite set Ad are in bijective correspondence with digraph structures on Ad. Given a digraph with a set of vertices Ad one associates with an arrow from i to j (denoted as i ---, j ) an algebraic object eij and defines fl I = s p a n c { e i J ; i --, j } . This is t u r n e d into an A-bimodule via ( L L L ) eie ke = 5ikeke and ekee i = 5eieke 9 Introduce then p = ~k,e e ke where the sum is restricted to those k, g for which there is an arrow k ~ g in the digraph. T h e n
C H A P T E R 4. D I S C R E T E G E O M E T R Y
244
AND MOYAL
( M M M ) df = [p, f] defines a C-linear map d : A ~ fll satisfying the Leibnitz rule. If there is an arrow i ---, j in the digraph then ei pe j = e ij and otherwise eipe j - 0. The subspace ~1 = eif~l is generated by the 1-forms e ij corresponding to the arrows originating from i in the digraph. It may be regarded as the cotangent space at i E NI and ~1 = | The complete digraph where all pairs of points in A4 are connected by a pair of antiparallel arrows corresponds to the largest first order differential calculus on A4, called the universal first order differential calculus since every other such calculus can be obtained from it as a quotient with respect to some sub-bimodule. Now one defines a canonical commutative product in f~l via
a.df
= [a,f]; ( f a f ' ) . ( h / 3 h ' )
= fh(a./3)f'h'
(f,f',h,h'
E A, a,/3 E ~1)
(4.71)
More generally this product exists for every first order differential calculus over a commutative algebra (cf. [53]). In the present situation it is given explicitly by ( N N N ) e ~j ,, e kt = 5ikSJte ij. The space of 1-forms f~l is free as a (left or right) A module and a special left .A-module basis is given via
pi = ~
eJi (pei ~ 0); A = ~
j
Aije ij - ~
ij
Aip i (Ai = ~ i
Ajie j)
(4.72)
j
Note that ~ i AiP i = 0 implies, via multiplication with eJ from the left, that Aji -- 0 and thus Ai = O. Concatenation of 1-forms e ij leads as before to r-forms as in (4.49) which can be written as e i~ = ei~ i l p . . , pe i" (4.73) Using (LLL) this space is turned into an A bimodule and the exterior derivative extends to higher orders via de i = pe i - eip; d p - p2 + E eip2ei (4.74) i
In particular this leads to
deiJ = pe i pe j _ e i p2 ej + e i pej p;
(4.75)
deijk __ pei pe j pek _ ei p2 ej pek -k e i pe j p2 ek _ ei pej pek p Starting with the universal first order differential calculus on Ad these formulas generate the universal differential calculus (known also as the universal differential envelope of A). Note a missing arrow from i to some other point j (in the complete digraph) means eipe j = 0 and acting with d on this equation one obtains ( O O O ) i 74 j ::v. ei p 2ej - O. Consider now the set X defined as the dual of fil as a complex vector space. Let {Oij} be the basis of X dual to {e ij} and if < , >0 denotes the duality contraction one has ( P P P ) < eij, Okt >0 = 6~5~. X is turned into an A bimodule by introducing the left and right actions < a, f . X > o = < a f , X >0; < a , X . f > o = < f a , X >o (4.76)
e k = 5kiOji. An element X E X can be uniquely decomposed as ( R R R ) X = ~_,~...,j X(i)JOj~. Now introduce a duality contraction < , > of fil as a right A-module and X as a left A-module by setting (SSS) < e ij, X > = e i < e ij, X >o for all X E X. There results A s a consequence ( q q q )
e k 90j~ = ~ ] 0 j ~ and Oji"
< fa, X. h >= f(a,X
> h; < a , f . X > = < a f , X >
(4.77)
4.4. M O R E ON PHASE SPACE DISCRETIZATION
245
The elements of A' become operators on j( via ( T T T ) X ( f ) = < df, X > and using the Leibnitz rule for d one obtains ( U U U ) X ( f h ) = f Z ( h ) + (h. X ) ( f ) for f, h E A. Further ( V V V ) ( X . f)(g) = X ( g ) f . The duality contraction extends now to the pair of spaces | ~tl and A' | ~ via ( W W W ) < w | a, X | w' > = w < a, X > w' and the space A'i = A'e i = { X . ei; X E A'} may be regarded as the tangent space at i E A/i. It is dual to Ft~ with respect to < , >0. The set {0ji; i --, j } is a basis of A'i dual to the basis {eiJ; i --~ j } of ~t~. Now for linear connections on a finite set let V : ~1 __, ~1 | ~1 be a (left A-module) linear connection. Using ( I I I ) and the properties of p one obtains ( Y Y Y ) U(a) = p | - V a as a left A-homomorphism U : gt 1 ~ ~1 | ~1 (i.e. U(fo~) = fU(o~)). This is called the parallel transport associated with V and in particular one has U(e ij ) -- eiu(e ij ) with an expansion [
u ( ~ ) - ~ u(i) ~k~~ |
~ -- ~ ~ |
k,~
~ U(i)J~e ~
(4.78)
k
with constants U(i)Jkt. Via ( Z Z Z ) e ik ~ eikUiJ = E t U(i)kte jt for fixed i,j the parallel transport defines a linear map ~ ~ ~t~ with associated matrix Uij. One has (A1) U(a) = ~ j eij | [(eia)UiJ]. Given a linear connection on ~1 there is a dual connection V" A" ,u | ~1 such that ( A 2 ) d < a , X > = < Va, X > + < a, V X > (cf. [233]). Using d < a , X > = [p, < a , X >] one can prove that the dual parallel transport defined by (A3) < a, U(X) > = < U(o~),X > acts on A' via (cf. (4.78))
u(x) = x |
p+
vx;
u ( x . f) = u ( x ) f ~ u(oj~) = ~ U(k)~jO~k 0.~ ~k~
(4.79)
The parallel transport (and thus also the connection) extends in an obvious manner to gt | ~1 and A' | gt as graded left (resp. right) ~t-homomorphisms, i.e.
U(w @~t a) = ( - 1 ) r w |
U(a); U ( X |
w) = ( - 1 ) r U ( X ) |
w
(4.80)
where w E gF. The map A'j --, A'i dual to the parallel transport map with matrix U ij defined ( Z Z Z ) is given by
ok~ ~ ~ u(j)fko~j = u ~j(ok~) ~ u ( x ) = Z U~j ( x " ~) |
~J
(4.81)
i,j
One may introduce the curvature as the right gt-homomorphism 7~' 9 A' | ~ --~ X | defined via (A4) ~ ' = V 2. Its dual 7~" ~ @A ~tl ~ ~ | ~1 is then given by 7 ~ - - V 2 in accordance with the previous general definition. One obtains then
T~(eij) = ~ R(i)Jk~meik~ |
Jm
=
(4.82)
u~]}
(4.83)
k,g.,m
k,~,m
where (Ab) U(i)i~ = 5~. One also has a formula
n ( . ) = Z ~jk | i,j,k
{(~.)[u~Jujk _
246
C H A P T E R 4. D I S C R E T E G E O M E T R Y
AND MOYAL
where U ii = idn~. For the torsion one finds that
o(~) = _~d~j + ~ u(i)jk,e ~ke = k,~
~-~(6Jk
-
6~ + U(i)Jke)e ~ke
(4.84)
l,~
Metrics and compatible linear connections on finite sets are discussed in [231, 230] along with geometry for an oriented lattice and Hodge operators for noncommutative algebras. Also many examples are given. We omit discussion of this for the moment. 4.4.3
More on discrete phase spaces
We go here to [19, 366, 367, 437, 827, 999] for still another approach to quantum phase spaces. We begin with [437] to describe the Schwinger cyclic unitary operator basis. Thus one considers a unitary cyclic operator ~r acting on a finite dimensional Hilbert space HD spanned by a set of orthonormal basis vectors (lu >a} for k - 0 , - . . , D - 1 with the cyclic property ~rD __ 1 via ]U >k+D-- Iu >k; kK UlU :>k'= ~k,k'
~rlU > k = Iu > k + l ;
(4.85)
In the {lu >k} basis U is represented by ( S l ) U - ~ D - 1 lu >k+l k< ul. The action of corresponds to a rotation in HD with axis of rotation along the direction in HD given by the vector Iv >e whose direction remains invariant under the action of U; thus
~l~>~=~'~~
D-1
1
v~e)I~>~ (0 < ~ < D _
0
_
-
1)
(4.86)
where v if) - exp(-i"/okg) and "70 = 2 ~ / D . On the other hand the new set {Iv >e}e=0,...,D-1 also forms an orthonormal set for which one can define a second unitary operator V such that ~D _ 1 and (B2) VIv > ~ - I v >t+l with Iv >t+D--I v >~ and Vlu > k - exp(-i~/ok)lu >k where k E Z and 0 < k _< D - 1. The basis vectors {lu >k} and {v >e} define two equivalent and conjugate representations in the sense that (4.86) is complemented by
1 V-lu~k )
u~k) ei,yoek
(e)*
0
The corresponding operators U and V satisfy ~.fml?m2 ~. ei"yomlm2?m2uml; Umlq-n : Uml; ?m2q-n __ 71/l 2
(4.88)
An operator ~ with projection in the lu >k representation given by q2(uk) is given in the Iv >~ representation as ~(ve). It follows then that
9 (~k)- ~
1
D-1
~ k< ~1~ >~ ~(~); k< ~lv >e= r
(4.89)
0
In analogy with the elements of the discrete Wigner-Kirkwood basis one now defines now Sm ---- e.-i"{~
ml~rm2 "- ei"y~
ml
(4.90)
where m ,,~ (m~, m2). One can now represent the transformations in equations (4.86) and (4.87) using the unitary Fourier operator/~ defined as { Iv >k } = /~{ lu >k } and { ]u >k } F - l { I v >k} where/~t =/~-1. Then lu >k F Iv >k F lu >_k F IV >_k F [u >k;
(4.91)
4.4. M O R E O N P H A S E S P A C E D I S C R E T I Z A T I O N
247
Next one defines a transformation R~r/2 in the space of the lattice vectors m such that
R~r/2" (ml, m2) -* ( - m 2 , ml) and one can show that (B3) P~mP -1 = SR./~.~ with/~4 _ 1 and R4/2 = 1. This implies that (4.90) is invariant under simultaneous operations of P and R~/12 and Sm has the properties
Sim = S-m; T r { S m } = Dhm,o; SmSm, - ei~~215
(4.92)
(SmSm,)Sm,, -- Sm(Sm'Sm"); S0 = 1; S m S - m -
1
where m x m ' = mlm~2-m2m~ 1. Using equations (4.90) and (4.92) there results (B4) (Sin) D = SDm = S - D m = (--1) Dmlm21 where Sore commutes with all elements Sin'. The unitary Schwinger operator basis /~m defines a discrete projective representation of the Heisenberg algebra parametrized by the discrete phase ^space vector m in ZD • ZD. Excluding m = 0 ^ and for D prime the elements of the basis U TMV m2 form a complete set of D 2 - 1 unitary traceless matrices providing an irreducible representation for su(D). If D is not a prime then the prime decomposition D = D1 .-. Di permits the study of a physical system with a number of quantum degrees of freedom with each degree of freedom expressed in terms of an independent Schwinger basis with cyclic properties determined by the particular prime factor Dj. One assumes now that D is prime. The eigenspace of Sm is spanned by eigenvectors [m, r >0 in for example the [u >k basis with coefficients erk(m) - k< u[m, r >. From this definition and (4.85) it is clear that the coefficients are periodic, i.e. erk(m) = e~+D(m ). The coefficients and eigenvalues are then determined via
Ar(m)erk(m) =e-i~k(ml'm2)~rk_ml(m); ~k(ml,m2):
~0?Tt2(2k
-- ml)/2
(4.93)
which yields
M-I H ~r(m)e-i~k-nml
~r(m) -- eiTrmlm2e-27rri/D; erk(m) "-
('nzl"m2)
0
1
e~--l--l(T/%)
(4.94)
where M = M ( D , m l) E Z. It is known that the ;~m basis has an explicit deformed algebraic structure defined via operators Dm = (D/27r)Sm. Indeed the commutator
[L)m, On]-- 2ASin ( - ~ m • '7o
n)
Dm+n
(4.95)
describes the Fairlie-Fletcher-Zachos sine algebra (cf. [310, 311]). The generators jm can be represented by the Weyl matrices (cf. (2.147))
g=
1 0 0 --" 0 w 0 "" 0 0 w 2 ... 9
.
0
0
9
. .
0
...
...
0 0 0 0
w D-1
; h=
0 1 0 0 0 1 " : :
""" ...
0 0 0
0 0 0 1 0 0
... ...
0
(4.96)
1
with Jm = cvmlm2/2gmlh m2 satisfying hg = wgh and gD : h D : 1 with wD - 1 and w = exp(iTo). It is then possible to verify that [Jm,-in] - ( 2 i / v o ) S i n [ ( v o / 2 ) m • n]Jm+n.
CHAPTER 4. DISCRETE GEOMETRY AND MOYAL
248
In this format one can consider the discrete Wigner-Kirkwood (WK) operator basis /~(V) acting on the quantum phase space spanned by vectors V = (V1, V2) ~ V. The WK and Schwinger bases are connected by the dual form
s
1
-" - ~ E e-i'r~ m
Srn -
/
dVei~/~163
(4.97)
where m • V = mlV2 - m2V1 (we insert m for m etc. when the vector nature is especially important). The range of the integral over V here is the entire 2-torus (cf. also [19, 366]). The Wigner function We(V) is defined as the projection of A(V) in a physical state I~ > via (Bb) We(V) = < r162 >. Any operator /~(U, V) with II/~11 <_ oc can then be associated with a classical function f(V) via
1 < r162
>=
/
dVf(V)Wr
f ( V ) = Tr[/~/~t(V)]
(4.98)
where dV = dVldV2. The normalization used here is based on f ~ dxexp(ixn) = 2~5n,0 leading in the continuous limit to limD-~cr ~D-1 exp(ixn) -- 2~5(x). Next one checks that (B6) /~/~(V)/~-1 = /~(R~)2V ) where R~)2V = (I12,-Vt). One can check now that all of the standard properties of the Wigner function will hold. In particular We(V) is real, integration over one phase space variable Vi yields the marginal probability distribution of the physical state in the eigenbasis of the other variable Vj, W~(V) is Galilean covariant and covariant under space and/or time inversion, and the following integral properties hold. First f dVWr162 (4.99) = ~1l < r r > [2 which arises via
/
dVWr162
1
-~ ~ m,m'
/
dYe -i'y~247
< ~)lSml~) > < ~)'lSm, l~)' >
(4.100)
and ( B 7 ) ] r > = E k ek[ u >k with 1r > = E k r u >k. The V integral yields D25m_m, and one subsequently uses Smlu > k = exp(-i~/o(2k + ml)m2/2)lu >k+ml. Secondly if ]~ and 2 are two dynamical operators of U and V then (BS) ( 1 / D ) T r [ Y Z ] ^ - f d V y ( V ) z ( V ) where y, z are classical functions on the phase space corresponding to Y, Z. This follows from (4.98) and Tr[Sm] = Dbm,O. One can say all this in a slightly different manner following [827]. Thus write (B9) A = (1/27rh) f dpdqa(p, q)A(p, q) with (note a(p, q) ~ Aw(p, q)- one can interchange q --. p and p -~ - q without loss of physical content- cf. (2.4)) (4.101) A(p, q) = / dve ipv/h
q+ ~1>i q- ~1
Thus a(p, q) is the Weyl transform of A and A(p, q) ~ elements of an operator basis. Hence the Weyl transform of an operator .4 is understood as the coefficient of its decompostion in the operator basis A(p, q). The Weyl transform is also obtained via (B10) a(p, q) = Tr[A(p, q)fi.] and one has
Tr[A(p, q)] = 1; Tr[A(p, q)A(p', q')] = 27rhb(p - p')5(q - q');
(4.102)
4.4. MORE ON PHASE SPACE DISCRETIZATION Tr[A(p, q)A(p', q')A(p", q")]
249
4e (2i/h)[p(q''-q')+p'(q-q'')+p''(q'-q)]
=
It is not difficult to show then that ( B l l ) Tr[fl] = (1/27rh) f~ (1/27rh) f dpdqf(p, q)g(p, q). From (4.102) follows also
dpdqa(p, q) and Tr[[~d] =
1 2 f dp"dq"dp'dq'a(p', q')b(p", q") x (AB)w(p, q) = 7r2h {2i
xexp --h-[(p' - q)
(p,,
- p) -
(q,,
- q)(p' - p)]
}
(4.103)
After some algebraic manipulation one obtains as expected
0
(A[3)w(p, q) = exp ~
0 0)]
Opa Oqb
Oqa Opb
a(p, q)b(p, q)
(4.104)
This leads then to the Moyal bracket {a, b}M in the form ([A,/)])w(P, q) = 2iSin -~
0
Opa Oqb
0 0)]
Oqa Opb
a(p, q)b(p, q)
(4.105)
(note there will be a minus sign difference due to the interchange here q ~ p and p ~ - q ) . For the dynamics one writes (B12)/5(t) - I r >< r and ~b(q, t) = < ql~(t) > leads to
pw(p, q, t)
f :~ dveWV/hr
- (1/2)v, t)~b*(q + (1/2)v, t)
(4.106)
J-- oo
T~[~(t)AI
Consequently as usual (B13) .A(t) with A.(t) - (1/27rh) f dpdqa(p, q)pw(p, q, t). We recall also that the time evolution of a quantum system can be described by the Liouville equation for the density operator, namely (B14) O~(t)/Ot - -(i/h)[H, p(t)] and this can be put into the equivalent form 0
otPW(p,q,t) = {h, pw}M
(4.107)
Now in the same spirit one can look for a discrete phase space associated with a quantum finite dimensional degree of freedom without a classical counterpart. Setting h - 1 now consider (B15) O(m, n) = Tr[Bt(m, n)O] for 0 ~-- ~ 2 0 ( m , n ) B ( m , n) where the operators B(m,n) are elements of any given complete orthonormal operator basis. For example one can use the Schwinger basis (B16) S(m,n) = [UmVn/x/-N]exp[(iTr/N)mn] for m, n = 0, 1 , . . . , N - 1 as above. In order to take advantage of the quantum discrete canonical like symmetry of the phase space one uses now basis elements which implement the T(m,n,j,e) for h = ( 1 / 2 ) ( N - 1) with symmetry, namely (B17) G(m, n) n_ ~-~j,g=-h h
T(m,n,j,g)=
uJ Vg eilrjg/Y C-(2rci /N)(mj+ng) cirrd)(j+h,g+h,g)
N
(4.108)
where (B18) r N) = N I N I N - j I N - g i N for I N = [k/N] the integral part of k with respect to N. In this way the phase is introduced in order to carry out all the mod(N) arithemetic. The exponential of the modular phase exp[iTrr + h, g + h, N)] is easily seen to be 1 when {j,g} both lie in the interval [-h, h]. On the other hand the presence of this phase ensures a symmetry in the 7'(m, n , j , g ) factors so that (B19)J~(m, n, j, g) 7"(m,n,j(modN),g(modU)) for j,g E Z (of. here [366, 367]). The basic properties are (B20) Tr[CJ(m, n)] __ 1 with Tr[C_,t (m, n)Cl(r, s)] . . . .AT,~[N] m,rt,,~[N] n,s and 1
Tr[Gt(m, n)G(u, v)C,(r, s)] = N2
h
~
a,b,c,d=-h
e(i~/h)(bc-ad)ei~o(a'b'c'd'g) X
(4.109)
CHAPTER 4. DISCRETE GEOMETRY AND MOYAL
250
where (B21) (I)(a, b, c, d, N) = - r + c + h, b+ d + h, N) (we refer to [827] for more details). These expressions are the discrete counterparts of equations (4.102). Since the operator basis elements are Hermitian it can be seen that Hermitian operators have real phase space representative functions. Furthermore (B22) (Ol02)w(m,n) = Tr[Gt(m,n)Ol02] or in more detail
1
( 0 1 0 2 ) w ( m , n) = N2
h ~ O1 (u, v)O2(r, s)Tr[G t (m, n)G(u, v)G(r, s)] u,v,r,s=-h
1
(Ol()2)w(m,n) = N4
h
(4.110)
h
E E Ol(U'v)O2(r'8)e(izr/N)(bc-ad)• u,v,r,s=-h a,b,c,d=-h
(4.111)
• eizr~b(a,b,c,d,N) e(27ri/N)[a(m-u)+b(n-v)+c(m-r)+d(n-s)] The trace of an operator follows directly as (B23) Tr[01] = ( l / N ) Em,n=-hh 01(m, n) while the trace of a product can be obtained via Hermitianness of the basis elements and (B20) as 1
T~[OIO21= N2
h
h
Z
~
m,n=-h j,g=-h 1
-- -N
Ol(m..)O2(j.e)T~[dt(m.~)d(j.e)-
h
E O1 (m, n)O2(m, n) m,n=-h
(4.112)
which is the discrete analogue to ( B l l ) . Similar calculations give now 2i [O1, ()2]w(m, n) = N4
h
h
E E Ol (u, v)O2(r, 8)x u,v,r,s=-h a,b,c,d=-h
x ei~(a'b'c'd'N)F(m, n, u, v, r, s, a, b, c, d, N)
(4.113)
~ (bc - ad) ] e (2~i/N)[a(m-u)+b(n-v)+c(m-r)+d(n-s)] F = Sin [-~
(4.114)
where
Equation (4.113) can be compared directly with (4.105). For dynamics in the discrete phase space one still uses a density operator (B24) r >< r whose Weyl representative is called the Wigner function (B25) pw(m, n, t) Tr[Gt(m, n)th(t)]. As in the continuous case one writes ( B 2 6 ) A ( t ) - Tr[~(t)ft] which upon using (4.112) gives (B27) ft.(t) - ( l / N ) ~u,v=-h h a(u, v)pw(u, v, t). Again one has the Ir
Liouville equation (828) (O/Ot)~(t) = -i[/:/, r phase space if one uses (4.113) as (h ~ Hw) 0
2i
-~pw(m, n, t) = N4
which can be mapped into the discrete
h
E h(u, v)pw(r, s, t)x u,v,r,s,a,b,c,d---h
xeiZr~b(a'b'c'd'N)F(m, n, u, v, r, s, a, b, c, d, N)
(4.115)
4.4. M O R E ON PHASE SPACE DISCRETIZATION
251
This can be rewritten as (B29) (O/Ot)pw(m, n, t) = Er,s=-hh 2i
h
s = N4
h
~
~
s
n, r, s, N)pw(r, s, t) where
h(u, v)ei~(a'b'c'd'X)F
(4.116)
u , v = - h a,b,c,d=-h
is now identified as the discrete mapped expresson of the Liouvillian of the system. To clarify all this consider the simple case of a time independent Hamiltonian and write (B30) t)(t) ~[(t, to)~(to)[s to) where (B31) ~[(t, to) = e x p [ - ( i / h ) ( t - to)]. The corresponding discrete phase space mapped expression is (32) pw(u, v, t) = Tr[GT(u, v)/((t, to)~(to)K t (t, to)] which, upon using the standard decomposition (B33)tS(t0) = Er, s G(r,s)pw(r,s, to), can be written in a general form as (B34) pw(u, v, t) = Er,s 7)( u, v, tlr , s, to)Pw(r, s, to) where
p(u, v, tit, s, to) = Tr[Gt(u, v)~[(t, t0)(~(r, s ) / ( t (t, to)]
(4.117)
This last expression is the mapped propagator of the Wigner function in the discrete phase space. Now recalling
~(t) = ~(to) - i(t - to)[H, ~(to) + " "
(4.118)
h
one sees that its mapped expression is f(~[N]5[N]
L
t) - Z
r~8
+
i2(t-t~ 2!h2
~ s
_
v, x, y, N)s
x,y
i(t - to) s
v, r, s N ) +
y, r, s, N) . . . .
}
(4.119)
pw(r, s, to)
This series is a solution to ( B 2 9 ) and by a direct comparison one can identify
P(u, v, t]r, s, to) +
i 2 ( t - to) 2 2!h2
:
~
x,y
(~[N](~[N] _ r~u
s
8~v
i(t -h to) s
v ~ r, s N)+
(4.120)
v, x, y, N)t:(x, y, r, s, N) . . . .
This expression associates, through the repeated action of the time independent Liouvillian, the Wigner function of integers (r, s) at to with a Wigner function of integers (u, v) at time t, while the phase space grid is kept constant in time (cf. [367] for more general situations). We go now to the continuous limit of the discrete space formalism (DPSF). Thus consider the basis elements (B35) (~(j, g) = ~-~m,n=_h~(j,g,m , h n). Since the phase r h, n + h, N) will always be zero in these sums we may just write 1
(~(j,g) = ~
h
umvne-(2~ri/N)(mj+n~)e (i~r/N)mn
~
(4.121)
m~rt:-h
One can also consider the operators (B36) U + exp[-ieQ] and V = exp[ieP] with e2 = 2 ~ / N which becomes an infinitesimal as N ~ c~. Now perform the change of variables (B37) - q = cj, - p = eg, u = em, and v = cn defining the intervals A u ~Am and Av = can with A m = A n = 1. This leads to -eh
1 G(p, q) = E2N Y~ u,v--s
^
^
AuAve-iuQeivPei(qu+pv)eiuv/2
(4.122)
C H A P T E R 4. D I S C R E T E G E O M E T R Y A N D M O Y A L
252
Considering N ~ oc one has Au ~ du and Av ~ dv yielding
1/~/~ 6;(p, q) = ~
cr
^ ^ dudve-~uQeive eiuV/2ei(qu+pv) =
_- __1/_~ f_~ dudveiU(q+(1/2)v_O.)eiV(p+p ) 271"
~
(4.123)
e~
which with the help of the identity (B38)]q > < ql = (1/2~-)f_~ d x e x p [ i x ( q - ~))] (note luj > < ujl = ( I / N ) ~ h = _ h u~U k) gives
J:l S
(~(p, q) =
dv q + l v
(:X)
G(p, q) =
>( I
q + l v eiV(P+P);
(4.124)
dveiVp[q + (1/2)v > < q - (1/2)v[
(X)
Upon recalling (4.101) one identifies (B39) (~(p, q) = A(p, q) and from this the phase space mapping procedure can be rewritten as (840) .4 = ( l / N ) Em,n=-h h A m A n A(m, n)G(m, n) so that by assuming p = em and q = en with e as in (B36) there results (B41) A = (1/27r) E mh, n : - h ApAqA(p, q)(~(p, q). As N ~ oc (B42) A = (1/27r) f-co dpdqA(p, q)A(p, q) which is exactly (B9) up to the omitted factor of h. All of the further Weyl-Wigner results follow directly and in particular one obtains the Moyal bracket as follows. Start from (4.111) and use (4.114) to write 1
(()l(~)2)w(m, n) = N4
h
h
E
E
u,v,r,s--h a,b,c,d=-h
AuAvArAsAaAbAcAdx
(4.125)
XO1 (it, v)O2 (r, 8)e (iTr/N)(bc-ad) e iTro(a'b'c'd'N) X
X e (2iTr/g)[a(m-n)+b(n-v)+c(m-r)+d(n-s)] Makeachangeofvariablesa+c=j,b+d=t~,
1 (Ol()2)w(m, n ) = 4N 4
• AuAvArAsAjAgAxAz01
h E
a-c=x,
2h-lJl
andb-d=ztoget
2h-lg[
2h
E
E E u,v,r,s=-h x=-(2h-lj[) z=-(2h-]gl) j,g=-2h
(u, v)O2(r, 8)e (iTr/2N)(jz-gx) e iTrdp(j+h'l+h'N) X
X e (i~r/y)[j(2m-r-u)+x(r-u)+t(2n-s-v)+z(s-v)]
(4.126)
where summations over indices (x, z) is restricted to run in steps of 2. Realizing that the phase r has a discontinuous nature it is convenient to break the summations over (j, g) into 9 different intervals, according to different phase values (see [827] for details). Only the first of these nine intervals, namely ~-~j,~=-hh will give a non-zero contribution in the continuous limit (in the other terms the phase term is non-zero and gives rise to oscillations in the resulting integrals which cause the integrals to vanish). The contributing term involves zero phase r since (j,g) both lie in [-h, h] and one sets em - p with en = q and generically ey - ~/. Constraints in the (~, 2) summing intervals can be dropped and one gets
1 4 /_~cr d~d~dfd$djdg-clid501 (~, ~)01 (~, ~) • (()IO2)w(P, q) -= 4e8N X e(iz/4)~-2(v-s)]e (ii/4)[[-2(f-ft)] •
(4.127)
4.4. MORE ON PHASE SPACE DISCRETIZATION
253
• e(i/2) [J(2m-f-~)+g(2-- s-~)] Integration over (1/4)(i, ~)yields 5 ( ~ - 2 ( ~ - 5 ) ] (j, ~) gives
4/
(Ol02)w(p,q) = (27r)2
and 5 [ j - 2 ( 9 - ~ ) ] so that integration over
dgd~dfdgOl(g,f~)O2(f,g)x
(4.128)
x e (i/2)[(~-~)(2p-(f~+f))+(e-~) (2q- (~+~)] which can be finally written as
4/
(~)l~)2)w(p,q) = (27r)2
d~df~dfd~Ol(~,f~)O2(~,g)x
(4.129)
• e2i[(s-q)(f~-p)-(v-q)(f-p)] which is exactly (4.103) up to the omitted factor of h. This leads then directly to the Moyal bracket as desired. It is interesting to note here that although one can go from discrete to continuous for the Moyal bracket the opposite direction does not work at all. The expression for the Moyal bracket casts its roots in only one term of the discrete expression for the commutator so discretizing the Moyal bracket can not be expected to lead to the discrete
dynamical bracket.
254
C H A P T E R 4. D I S C R E T E G E O M E T R Y A N D M O Y A L
This Page Intentionally Left Blank
Chapter 5
WHITHAM
THEORY
Whitham theory arises in various contexts and the corresponding Whitham times play roles such as deformation parameters of moduli, slow modulation times, coupling constants, etc. When trying to relate these roles some unifying perspective regarding Whitham theory is needed and we try here to provide some steps in this spirit following hep-th 9905010 (cf. also [149]) with some updating and modification; the format was essentially developed in typed lecture notes and earlier papers (see e.g. [148, 150, 151, 156, 158, 152, 153] and hep-th 9410063, 9511009, 9712110, 9802130, 9804086). The idea is to display and review Whitham theory in many of its various aspects (with some sort of compatible notation) and thus to exhibit the connections between various manifestations of Whitham times. We prefer to leave open the question of why all these topics are related to Whitham systems except to say that they all seem to involve moduli dynamics and/or tangent deformation objects (in some related sense). In particular one will see how Whitham theory, with its WKB aspects, mediates in various (sometimes curious) ways between classical and quantum behavior. Some background work in this direction appears in [32, 87, 94, 95, 124, 125, 126, 146, 148, 150, 151, 152, 153, 155, 156, 160, 158, 168, 256, 257, 269, 270, 272, 273, 282, 283, 284, 287, 338, 339, 362, 382, 393, 394, 395, 396, 397, 398, 399, 400, 401, 403, 427, 428, 472, 473, 474, 497, 499, 542, 543, 555, 599, 600, 601,602, 637, 642, 643, 644, 658, 659, 660, 661, 682, 684, 688, 689, 690, 691, 733, 734, 735, 744, 745, 746, 747, 760, 761, 773, 782, 783, 902, 903, 904, 905, 906, 907, 908, 909, 946, 971, 990]. Symbols such as (.), (.I,), (A), (XlII), etc. may be repeated in different sections but references to them are intrasectional or intrasubsectional.
5.1
BACKGROUND
Perhaps the proper beginning would be Whitham's book [971] or the paper [338] where one deals with modulated wavetrains, adiabatic invariants, etc. followed by multiphase averaging, Hamiltonian systems, and weakly deformed soliton lattices as in [272, 273, 599]. The term Whitham dynamics then became associated with the moduli dynamics of Riemann surfaces and this fit naturally into recent work on e.g. topological field theory (TFT), strings, and SW theory to which some references are already given above. In Section 5.1 we begin with the averaging method following [148, 362,599] plus dispersionless theory as in [158, 160, 168, 555, 902] (which is related to TFT); then we develop relations to isomonodromy and to Hitchin systems following [642, 643, 644, 782, 783, 904, 906, 907, 908, 909] in Section 5.2, while connections to Seiberg-Witten (SW) theory, the renormalization group (RG) and soft supersymmetry (susy) breaking appear in Sections 5.3-5.5. In Section 5.6 we add material 255
CHAPTER 5. W H I T H A M T H E O R Y
256
about WDVV and the Picard-Fuchs equations. The averaging method for soliton equations goes back to [338] and early Russian work (cf. [272, 273] for references) but the most powerful techniques appear first in [599] (with refinements and clarification in [148, 362]). For background on KdV, KP, and RS see e.g. [78, 148, 147, 268, 558, 894]. 5.1.1
Riemann
surfaces and BA functions
We take an arbitrary Riemann surface E of genus g, pick a point Q and a local variable 1/k near Q such that k(Q) = oc, and, for illustration, take q(k) = kx + k2y + kat. Let D = P1 +'" "+Pg be a non-special divisor of degree g and write r for the (unique up to a multiplier by virtue of the Riemann-Roch theorem) Baker-Akhiezer (BA) function characterized by the properties (A) r is meromorphic on E except for Q where ~2(P)exp(-q(k)) is analytic and (*) after normalization r ~ exp(q(k))[1 + ~ 1 ( j/kJ)] near Q. (B) On E/Q, ~ has only a finite number of poles (at the Pi). In fact r can be taken in the form (P E E, P0 -/= Q)
r
O(A(P) + xU + yV + t W + zo) y, t, P) = exp[ f P ( x d a I + yda 2 + tda3)] 9
o(A(P) + z0)
J Po
where df~ 1 = dk + . . - ,
df~2 = d(k 2) + . - . ,
df~3 = d(k 3) + . . . , U j
(1.1)
= fB~ df~l, Vj =
fB- df~2, Wj = fB~ df~3 (J = 1 , . . ' , g), Zo = - A ( D ) - K, and O is the Riemann theta functio~n. The symbol ~ will be used generally to mean "corresponds to" or "is associated with;'; occasionally it also denotes asymptotic behavior; this should be clear from the context. Here the df~j are meromorphic differentials of second kind normalized via fAk df~j -- 0 (Aj, Bj are canonical homology cycles) and we note that xdf~ 1 + ydf~ 2 + tdf~ 3 ~ dq(k) normalized; A ~ Abel-Jacobi map .A(P) = (fPPodozk) where the &ok are normalized holomorphic differentials, k = 1,.. ",g, fAj &ok = 5jk, and K = (Kj) ,.~ Riemann constants (2K = - A ( K z ) where Kr~ is the canonical class of E ~ equivalence class of meromorphic differentials). Thus O(.A(P) + zo) has exactly g zeros (or vanishes identically). The paths of integration are to be the same in computing fPPodfti or A ( P ) and it is shown in [78, 148, 268] that ~ is well defined (i.e. path independent). Then the ~j in (*) can be computed formally and one determines Lax operators L and A such that Oy~2 = L~2 with Ot~2 = A~. Indeed, given the (j write u = -20x(1 with w = 3~10x~l - 302~1 - 30x~2. Then formally, near Q, one has (-Oy + 02 + u)r = O(1/k)exp(q) and (-Or + 03 + (3/2)UOx + w)r = O(1/k)exp(q) (i.e. this choice of u, w makes the coefficients of knexp(q) vanish for n = 0, 1,2, 3). Now define L = 02 + u and A = 0za + (3/2)uOz + w so Oyr = Lr and Otr = A~. This follows from the uniqueness of BA functions with the same essential singularity and pole divisors (Riemann-Roch). Then we have, via compatibility L t - Ay = [A, L], a KP equation (3/4)Uyy = Ox[ut- (1/4)(6uux + uxxx)] and therefore such KP equations are parametrized by nonspecial divisors or equivalently by points in general position on the Jacobian variety J ( E ) . The flow variables x, y, t are put in by hand in (A) via q(k) and then miraculously reappear in the theta function via xU + y V + tW; thus the Riemann surface itself contributes to establish these as linear flow variables on the Jacobian. The pole positions Pi do not vary with x, y, t and (t) u = 202logO(xU + yV + t W + zo) + c exhibits O as a tau function. We recall also that a divisor D* of degree g is dual to D (relative to Q) if D + D* is the null divisor of a meromorphic differential d~ = dk + (~/k2)dk + . . . with a double pole at Q (look at ~ = 1/k to recognize the double pole). Thus D + D* - 2Q ~ K~ so ,4(D*) - A(Q) + K = - [ A ( D ) - A(Q) + K]. One can define then a function ~*(x, y, t, P) = e x p ( - k x - k 2 y - k 3 t ) [ l + ~ / k ) + .. .] based on D* (dual BA function) and a differential d~ with zero divisor D + D*, such that r = r162 is meromorphic, having for poles only a double
5.1. B A C K G R O U N D
257
pole at Q (the zeros of d ~ cancel the poles of r 1 6 2 Thus r ~ ~*(1 + (~/k 2 +...)dk is meromorphic with a second order pole at oc, and no other poles. For L* = L and A* = - A + 2 w - (3/2)ux one has then (Oy + L*)r = 0 and (Or + A*)~* = 0. Note t h a t the prescription above seems to specify for r r
(L~ = xU + y V + tW, z~ = - . 4 ( 0 * ) - K )
~ e-f~o (xdal+yda~+tda3) . O ( A ( P ) - U + z~) O ( A ( P ) + z~)
(1.2)
In any event the message here is t h a t for any Riemann surface E one can produce a BA function r with assigned flow variables x, y, t , . . . and this r gives rise to a (nonlinear) K P equation with solution u linearized on the Jacobian J ( E ) . For averaging with K P (cf. [148, 362, 599]) we can use formulas (eft (1.1) and (1.2)) r : e px+Ey+flt. (~(Ux nt- Y y + W t , P)
r
= e-px-Ey-at . r
- Y y - W t , P)
(1.3) (1.4)
with r r periodic, to isolate the quantities of interest in averaging (here p - p(P), E E ( P ) , t2 = ~ ( P ) , etc.) We think here of a general Riemann surface Eg with holomorphic differentials dwk and quasi-momenta and quasi-energies of the form dp - df~ 1, d E = d~ 2, dr/ = d~3, --. (p = f/~ d~ 1 etc.) where the d~ j = d~j = d(A j + O(A-1)) are meromorphic differentials of the second kind. Following [599, 600] one could normalize now via ~ fA t d~ k = ~ fBj d~k = 0 so that e.g. Uk -- (1/27ri)~Ak dp and Uk+g = -(1/27ri) ~Bk dp (k - 1 , . . . , g) with similar stipulations for Vk ~ ~ dr22, Wk ~ ~ dt~ 3, etc. This leads to real 2g period vectors and evidently one could also normalize via ~Am df~k -- 0 (which we generally adopt in later sections) or ~ ~A,~ dt2k = "~ ~B,~ d~k = 0 (further we set Bjk = ~Bk dwj).
5.1.2
Hyperelliptic averaging
Averaging can be rather mysterious at first and some of the clearest exposition seems to be in [94, 338, 362, 722] for which we go to hyperelliptic curves (cf. also [78, 148, 147]). For hyperelliptic Riemann surfaces one can pick any 2g + 2 points )~j E p1 and there will be a unique hyperelliptic curve Eg with a 2-fold m a p f : Eg --+ p1 having branch locus B - {Aj}. Since any 3 points Ai, Aj, Ak can be sent to 0, 1, oc by an a u t o m o r p h i s m of p1 the general hyperelliptic surface of genus g can be described by (2g + 2) - 3 - 2g - 1 points on p1. Since f is unique up to an automorphism of p1 any hyperelliptic Eg corresponds to only finitely m a n y such collections of 2 g - 1 points so locally there are 2 g - 1 (moduli) parameters. Since the moduli space of algebraic curves has dimension 3 g - 3 one sees that for g > 3 the generic R i e m a n n surface is nonhyperelliptic whereas for g - 2 all R i e m a n n surfaces are hyperelliptic (with 3 moduli). For g = 1 we have tori or elliptic curves with one modulus T and g -- 0 corresponds to p1. In many papers on soliton m a t h e m a t i c s and integrable systems one takes real distinct branch points Aj, 1 < j < 2g + 1, and oc, with ~1 < )~2 < ' ' " < )~2g+l < CO and 2g+l #2 = H ( A - AJ) = P2g+I(A, Aj) 1
(1.5)
as the defining equation for Eg. Evidently one could choose A1 = 0, A2 = 1 in addition so for g = 1 we could use 0 < 1 < u < oc for a familiar parametrization with elliptic integrals, etc. One can take dA/#, A d A / # , . . . , A g - l d A / # as a basis of holomorphic differentials on Eg but
C H A P T E R 5. W H I T H A M
258
THEORY
usually one takes linear combinations of these denoted by dwj, 1 <_ j <_ g, normalized via fA~ dwj = 5ij, with period matrix defined via ~B~ dwj = IIij. The matrix II = (II~j) is symmetric with ~ I I > 0 and it determines the curve. Frequently in situations arising from KdV (Korteweg-deVries equation) for example one regards the intervals [A1, Ag],---, [A2g+l, oc) as spectral bands and intervals (A2,/~3),"', (/~2g, A2g+l) as gaps with the ai cycles around (A2i,)~2~+1) (i = 1 , . . . , g). One will also want to consider another representation of hyperelliptic curves of genus g via 2g+1
#2 = 1-I ( )~ - )~J ) = P2 g+ 2 ( )~, )~j )
(1.6)
0
where oc is now not a branch point and in fact there are two points #+ corresponding to We recall next some of the results and techniques of [338, 722] where one can see explicitly the nature of things. The presentation here follows [148]. First from [722], in a slightly different notation, write qt = 6qqx - qxxz with Lax pair L = - 0 2 + q, B = -4/)x3 + 3(qOx + Oxq), Lt = [B, El, L r = ) ~ , and r = B r Let r and r be two solutions of the Lax pair equations and set 9 = ~r these are the very important "square eigenfunctions" which arise in many ways with interesting and varied meanings (cf. [158, 160, 168, 163, 147]). Evidently ~I, satisfies
[-03 + 2(qOx + Oxq)]~ = 4A0xg; Or9 = -2qxq2 + 2(q + 2A)0x~I'
(1.7)
From (1.7) one finds immediately the conservation law (C): Ot[~]+Oz[6(q-2A)~-202x ~] = O. If one looks for solutions of (1.7) of the form ~ ( x , t , A) - 1 + E ~ [ ~ j ( x , t)]A-J as A --~ oc then one obtains a recursion relation for polynomial densities Oxff~j+l
--
1
[---~03 "Jr-(qOx + Oxq)]~j (j = 1,2,'--); ~0 = 1
(1.8)
Now consider the operator L = - 0 2 + q in L 2 ( - o c , oc) with spectrum consisting of closed intervals separated by exactly N gaps in the spectrum. The 2N + 1 endpoints Ak of these spectral bands are denoted by - o c < A0 < A1 < "'" < )~2N ~ OO and called the simple spectrum of L. They can be viewed as constants of motion for KdV when L has this form. We are dealing here with the hyperelliptic Riemann surface determined via R2(A) = l-IgN ( A - Ak) (oc is a branch point) and one can think of a manifold M of N-phase waves with fixed simple spectrum as an N-torus based on Oj E [0, 27r). Hamiltonians in the KdV hierarchy generate flows on this torus and one writes q = q N ( 0 1 , ' " , ON) (the 0 variables arise if we use t h e t a functions for the integration - cf. also below). Now there is no y variable so let us write Oj = xnj + twj (we will continue to use dwj for normalized holomorphic differentials). For details concerning the Riemann surface we refer to [78, 147, 207, 268, 338, 772] and will summarize here as follows. For any qN as indicated one can find functions p j ( x , t ) via 9 (x,t,)~) - I-IN()~- # j ( x , t ) ) where # j ( x , t ) E [~2j-1, ~2j] and satisfies
Oz#j = - 2 i ( R ( # j ) /
l-I("J - "i)); ir
(1.9)
2N
OtPj = -2i[2(~-'~ Ak - 2 ~-~ .i)]" ( R ( p j ) / 1 - I ( # J - . i ) o ir ir In fact the #j live on the Riemann surface of R(A) in the spectral gaps and as x increases #j travels from/~2j-1 to/~2j on one sheet and then returns to A2j-1 on the other sheet; this
5.1. BACKGROUND
259
path will be called the jth #-cycle (~ Aj). In the present context we will write the theta function used for integration purposes as O(z, 7) = ~meZN exp[Tri(2(m,z) + (m, Tm))] where z E Z N and T denotes the N • N period matrix (T is symmetric with .~T > 0). We take canonical cuts Ai, Bi (i = 1 , . . . , N) and let dwj be holomorphic diffentials normalized via fAj dwk = 5jk (the cycle Aj corresponds to a loop around the cut Aj). Then qN can be represented in the form 2N
qg(x,t) = A + F - 2021ogO(z(x,t);T); A = ~ Aj;
(1.10)
0 N
and z(x, t) = -2i[c g ( x - x0) + 2(Ac y + 2cg-1)t] + d where (cg)i -- ciy arises from the representation dwi = ( ~ g cij)~j-1)[dA/R(A)] (d is a constant whose value is not important here). Then the wave number and frequency vectors can be defined via g = -4izTT - l c N and = --8iTrT-l[Ac N + 2c N-l] with Oj(x, t) = ajx + wjt + ~o (where the ~o represent initial phases). To model the modulated wave now one writes now q = qy(01,'-" ,0Y; )~) where /~j
Aj(X, T) and A ~ (Aj). Then consider the first 2N + 1 polynomial conservation laws arising from (1.7) - (1.8) and C for example (cf. below for KP) and write these as OtTj(q)+OxA'j(q) 0 (explicit formulas are given in [338] roughly as follows). We note that the adjoint linear KdV equation (governing the evolution of conserved densities) is OtTj + 037j - 6qOxTj = 0 (~/j ~ VHj) and (1.8) has the form 07j+1 = ( - ( 1 / 2 ) 0 3 + qO + Oq)Tj. One then rewrites this to show that 6qOx~/j = 0x[6~/j+l - 6q~/j + 302~/j] so that the adjoint equation becomes Otq/j + 0 [ - - 2 0 2 ~ j q-
6q~/j -- 6 ~ j + l ] -- 0
(1.11)
which leads to (1.12) and (1.16) below (after simplification of (1.11)). Then for the averaging step, write Ot = COT, etc., and average over the fast variable x to obtain
o~ < ~(qN) > +ox < xj(qN) > - o
(1.12)
((1.12) makes the first order term in e vanish). The procedure involves averages
< Tj(qN) > = limi--,cr
1 /_ L
(1.13)
L fj(qN)dx
for example (with a similar expression for < 2dj(qN) >) and an argument based on ergodicity is used. Thus if the wave numbers aj are incommensurate the trajectory {qN(x,t); x E (--c~, c~)} will densely cover the torus A/[. Hence we can replace x averages with 1
< Tj(qN) > = (27r)N
"'"
Tj(qN(O~) YI dOj 1
(1.14)
For computational purposes one can change the 0 integrals to # integrals and obtain simpler calculations. By this procedure one obtains a system of 2N + 1 first order partial differential equations for the 2N + 1 points Aj(X,T), or equivalently for the physical characteristics (g(X,T),v~(X,T)) (plus < qN >). One can think of freezing the slow variables in the averaging and it is assumed that (1.12) is the correct first order description of the modulated
CHAPTER 5. W H I T H A M T H E O R Y
260 wave.
The above was rather loose so let us be more precise (cf. [338]). One looks at the KdV Hamiltonians beginning with H = H(q) = limL+oo(1/2L) fLL(q2 + (1/2)q2)dx (suitable for (averaged quasi-periodic situations) 9Then {f, g} = limL__,m(1/2L) f'-L(hf/hq)Ox(hg/hq)dx L Gardner bracket) and qt = {q, H}. The other Hamiltonians are found via
,hHo (:95gm+l = (qO + O q - 103 ) 5Hm (m > O); =1 5q
(1.15)
where ",/j ,,~ V H j ,,~ (hHj/hq) (cf. here [147, 338]). It is a general situation in the study of symmetries and conserved gradients (cf. [163]) that symmetries will satisfy the linearized KdV equation (Ot-6Ozq+O3)Q = 0 and conserved gradients will satisfy the adjoint linearized KdV equation (Or- 6qOz + 03)Q t = 0; the important thing to notice here is that one is linearizing about a solution q of KdV. Thus in our averaging processes the function q, presumed known, is inserted in the integrals. This leads then to
~hHj
Tj(q) = 5q ; Xj(q) = -202
~hHj
-6
~hHj+l 6q
+ 6q
~hHj 5q
(1.16)
with (1.12) holding, where < 02r > = 0 implies 1
< Xj > = limL--,oo-~
L (-65Hj+1 + 6qN
f_ L
6qN
~qN
)dx
(1.17)
In [338] the integrals are then simplified in terms of # integrals and expressed in terms of abelian differentials. This is an important procedure linking the averaging process to the Riemann surface and is summarized in [722] as follows. One defines differentials 1 dA ~1 = ___~[~N __ EN CJz~j-1] R(/~) 1
=
1AN+I
+ 1
(1.18) \
]
+ ZN E, j-1] 1
where the cj, Ej are determined via ~b, h l = 0 -- s h 2 (i --- 1, 2,---, N). Then it can be shown that ~ oo
<xj>
0
(2#)J ; < X > ~ ~0
(2#)J
(1.19)
w i t h ~'~1 "~< ~Y > (d~/~ 2) and < X > (d~/~ 2) ,-~ 12[(d~/~ 4) - ~ 2 ]
where # = ~-2 cx~ (# ,-~ (1/v/~) 1/2) so d# = -2~-3d~ =~ (d~/~ 2) ,-~ - ( ~ / 2 ) d # ~ - ( d p / 2 v ~ ). Since ~1 o(#g/#g+(1/2))d# = O(#-(1/2)d#, ~2 = O(#l/2)d# (with lead term - ( 1 / 2 ) ) we obtain < ~ >,,~< T > = O(1) and < X > = O(1). Thus (1.7), (1.8), (C) generate all conservation laws simultaneously with < Tj > (resp. < ,u >) giving rise to ~1 (resp. ~2). It is then proved that all of the modulational equations are determined via the equation O T ~ I = 120x~2
(1.20)
where the Riemann surface is thought of as depending on X, T through the points Aj(X, T). In particular if the first 2N + 1 averaged conservation laws are satisfied then so are all higher averaged conservation laws. These equations can also be written directly in terms of the Aj
5.1. BACKGROUND
261
as Riemann invariants via 0TAj = SjOx)U for j = 0, 1 , . . . , 2 N where Sj is a computable characteristic speed (cf. (1.29)-(1.30)). Thus we have displayed the prototypical model for the W h i t h a m or modulational equations. Another way of looking at some of this goes as follows. We will consider surfaces defined via R(A) -- r111 ] 2 g + l ( A - Ai) For convenience take the branch points Ai real with A1 < 9" < A2g+l < oc. This corresponds to spectral bands [A1,A2],..., [A2g+I, oc) and gaps (A2, A3),'..,(A2g, A2g+I) with the Ai cycles around the gaps (i.e. ai ~ (A2i, A2i+l), i = 1 , . . . , g). The notation is equivalent to what preceeds with a shift of index. For this kind of situation one usually defines the period matrix via iBjk -- J~Bkwj and sets A
g c j q i q- l d A
k
dwj = 27rhjk (j, k = 1 , . . . , g ) ; dwj = ~
v/R(A)
1
(1.21)
(cf. [78, 148, 147]). The Bi cycles can be drawn from a common vertex (P0 say) passing through the gaps (A2i, A2i+l). One chooses now e.g. p = f dp and ft = f dft in the form
p(A) =
~(A)
f dp(A) = f 2v/R(A); P(h)dh P = Ag + ~
f da(A)-
f
J
J
6Ag+l -Jr O ( A )
x/~
1
g
ajAg-J;
dA; O = ~ b j A g - J ;
bo = - 3
0
(1.22) 2g+1
~
Ai
1
rA2~+l dp(A) -- 3A2i rA2~+l drY(A) = 0; i . . . . 1, with normalizations Jh2i , g" We note that one is thinking here of (*) r = exp[ipx + if~t], theta functions (cf. (1.1) with e.g. p(A) = -i(logr f~(A) = -i(logr Recall that the notation < , >x simply means x-averaging (or ergodic averaging) and (logr ~ < (logr >z7~ 0 here since e.g. (logr is not bounded. Observe that (*) applies to any finite zone quasi-periodic situation. The KdV equation here arises from L r = Ar L = - 0 2 + q, Otr = Ar A = 403 - 6 q O - 3qx and there is no need to put this in a more canonical form since this material is only illustrative. In this context one has also the Kruskal integrals Io,..., I2g which arise via a generating function p(A)-
5
> x = x / ~ nu E (2v/-~)2s+1 0
-i ( (lo9r
= ~
where Is = < Ps >x = Ps (s = 0, 1 , . . - ) w i t h - i ( l o g r larly
-i(logr
= -i
= 4(v~) 3 +
(1.23)
+ ~[Ps/(2v/-A):S+l]. Simi-
(2x/~)2~+1
(1.24)
and one knows OtPs = Ozfts since (lb) [(logr = [(log~b)t]z. The expansions are standard (cf. [148, 158, 160, 168]). Now consider .a "weakly deformed" soliton lattice of the form
O(TIB ) =
1
~
exp(---~ ~ Bjknjnk + i ~ njTj)
--oo
j,k
(1.25)
j
with the A~ (i = 1 , . . . , 2g + 1) (or equivalently the parameters u ~ - I~; i = 0 , . . . , 2g) slowly varying functions of x, t (e.g. u i = ui(X, T), X = ex, T = et, i = 0, 1 , . . . , 2m). Now one wants to obtain a version of (1.20) directly via (a). Thus insert the slow variables in (ilb) and average, using ei)x or e0T in the external derivatives, to obtain
Or(logr
= cOx(logr
(1.26)
C H A P T E R 5. W H I T H A M T H E O R Y
262
or/)Tp(A) = / ) x E ( A ) . Then from (1.22) differentiating in A one gets
/)Tdp =/)xdf~
(1.27)
Now (recall /)tPs = /)xf~s) expanding (1.27) in powers of ( v ~ ) -1 one obtains the slow m o d u l a t i o n equations in the form
/)TUs = OX~s (s = 0 , . . . , 2g)
(1.28)
where f~--~is a function of the u i (0 < i < 2g). This leads to equations 0TAi = vi(A1,... ,A2g+l)/)xAi
(i = 1 , . . . ,2g + 1)
(1.29)
for the branch points Ak as Riemann invariants. The characteristic "velocities" have the form dgt I = 2 6Ag+l + f~(Ai) (1 < i < 2g + 1) vi = T p i : h ~ p(Ai) - -
(1.30)
To see this simply multiply (1.27) by ( A - A~) 3/2 and pass to limits as A ~ A~. 5.1.3
Averaging
with r162
Let us look now at [599] but in the spirit of [148, 362]. We will only sketch this here and refer to [148] for more detail. Thus consider KP in a standard form 3a2Uyy +/)x(4Ut 6uux + uxxx) = 0 via compatibility [/)y- L , / ) t - A] = 0 where n = ~r-l(/) 9 - u) and A = /)3 _ (3/2)u/)+ w (a 2 = 1 is used in [362] which we follow for convenience but the procedure should work in general with minor modifications - n o t e / ) means/)x and u ---, - u in the development of (1.1)). We have then (cgy - L ) r = 0 with ( / ) t - A ) r = 0 and for the adjoint or dual wave function r one writes in [600] r = -Oyr with r = ~)re* where r - (-/))J (r f ). We modify the formulas used in [599] (and [362]) in taking (cf. [148] - r and r are periodic) r -- e px+Ey+gtt" ( ~ ( U x q-
Vy + Wt, P); q2* =
e-px-Ey-f~t"
~)*(-Ux
-
Vy - Wt, P)
(1.31)
where p = p(P), E = E(P), f~ = f~(P), etc. (cf. (1.3) - (1.4)), which isolate the quantities needed in averaging. The arguments to follow are essentially the same for this choice of notation, or t h a t in [362] or [599]. Now one sees immediately that (r162
= r 1 6 2 + Ox(r162
+/)~(r162
+...
(1.32)
where e.g. L r = ((-1)r/r!)(drL/d(/))r). In particular L 1 = - 2 0 and L 2 = 1 while A 1 = - 3 0 2 + (3/2)u, A 2 = 30, and A 3 = - 1 . We think of a general Riemann surface Eg. Here one picks holomorphic differentials dwk as before and quasi-momenta, quasi-energies, etc. via dp ..~ df~l, dE ..~ d~t2, df~ --~ d ~ 3 , - - , where A ~ k, p = f ~ df~l, etc. Normalize the d~k so t h a t ~ fA d~k -- 0 -- ~ fBj d~tk (cf. remarks after (1.4)); then U, V, W , . . . are real 2g period vectors, and one has BA functions r as in (1.31). As before we look for approximations based on uo(xU + y Y + t W I I ) = uo(Oj,Ik). For averaging, Oj ~ xUj + y ~ + tWj, 1 < j < 2g, with period 2~ in the 0j seems natural (but note 0J, 09+J "~ Uj, etc.). Then again by ergodicity < r > x = limL__,~(1/2L)f~_L Cdx becomes < r > = (1/(27r) 2g f . . . f Cd2gO and one notes that < / ) x r > = 0 automatically for r bounded. In [362] one thinks of r + . . . ) with Cx = ~ Ui(/)r and f . . . f(/)r - O. Now for averaging we think of uo "~ uo( 88
with ( S , I ) .- ( S , I ) ( X , Y, T), /)xS =
5.1.
263
BACKGROUND
U, o ~ S = V, and i)TS = W . We think of expanding about u0 with 0x ~ /)x + e 0 x . This step will cover both x and X dependence for subsequent averaging. Then look at the compatibility condition (8) : OtL - OyA + [L, A] = 0. As before we will want the term of first order in e upon writing e.g. L = L0 + cL1 + . . . and A -- A0 + eA1 + .-- where slow variables appear only in the L0, A0 terms. Details are indicated in [148] and after some calculation (&) becomes OtLo - OyAo "~- [L0, A0] + e{OtL1 - OyA1 -~- OTL -- OyA + [L0,-41] + [L1, A0]} -~- O(e 2) (1.33)
Now the OtLo - OyAo + [L0, A0] term vanishes and further calculation shows that to make the coefficient of e vanish one wants OtL1 - OyA1 + [L0, All + [L1, A0] + F = 0; F = OTL -- OyA - ( L I O x A - A I O x L )
(1.34)
However via ergodicity in x, y, or t flows, averaging of derivatives in x, y, or t gives zero, so from (1.34) and further computation we obtain the Whitham equations in the form < r 1 6 2 > = 0. In order to spell this out in [362] one imagines X, II, T as a parameter and considers L(~), A(~), etc. and r ~ ~(~) (with I ~ (Ik) being any additional factors dependent on the slow times) so that r
= ep(~)x+E(~)y+a(~)t . r
but one keeps ~* = e x p ( - p x p , E , ~ , U, V, W , I fixed in ~*). Riemann surface varies with ~. priate for heuristic purposes at as independent variables. Note
+ Y(~)y + W(~)tlI(~))
(1.35)
gy- ~tt)r V y - W t l I ) fixed in ~ (i.e. assume We recall that one expects Ak = )~k(X, If, T) etc. so the The procedure here is somewhat contrived but seems approleast. Also recall that x, y, t and X, Y, T can be considered also from (1.35) for P fixed (0 ~ x U + y V + y W )
0~r162
= (ibx + / ~ y + ~tt)r162
+ ( U x + Vy + Iidt). r162
(1.36)
+/~. r162
where ] ..~ Of/O~. In [362] one assumes that it is also permitted to vary ~ and hold e.g. the h constant while allowing say the P to vary. Now one computes terms like O~[Ot(r162 averages, and subsequently inserts X, II, T for ~. Taking into account compatibility relations OyU = a x V ; OTU = O x W ; OTV = O y W
(1.37)
after considerable calculation one arrives at the Whitham equations in the form PT = ~ X ; PY = E x ; FIT = ~ y
(1.38)
(cf. Section 3 for general forms). We feel that this derivation from [148], based on [362, 599], is important since it exhibits again the role of square eigenfunctions (now in the form ~*r in dealing with averaging processes (a derivation based on (1.26) also seems important). In view of the geometrical nature of such square eigenfunctions (cf. [~63] for example) one might look for underlying geometrical objects related to the results of averaging. Another (new) direction involves the Cauchy kernels expressed via r and their dispersionless limits (cf. [156, 150]). It is also proved in [148] that dp = < ~ * > d~, d E ,.~ - < ~ * L I ~ > d~, and d~ ~ - < r162 > d~. This shows in particular how the quantity ~r determines the W h i t h a m differentials dp, dE, d~, etc. REMARK 5.1. Since D + D* - 2oo ~ K2, r 1 6 2 is determined by a section of K~ (global point of view) but it is relations based on < r162 >, < r162 >, etc. (based on the Krichever averaging process) which reveal the "guts" of r 1 6 2 needed for averaging and the expression of differentials.
CHAPTER 5. WHITHAM THEORY
264
5.1.4
Dispersionless K P
We give next a brief sketch of some ideas regarding dispersionless KP (dKP) following mainly [158, 160, 168, 555, 902] to which we refer for philosophy (cf. also Section 1.1). We will make various notational adjustments as we go along. One can think of fast and slow variables with ex = X and etn = Tn so that On -+ ~O/oqTn and u(x, tn) --+ (t(X, Tn) to obtain from the KP equation (1/4)uxxx + 3uux + (3/4)0-10~u = 0 the equation Orft = 3(,OXft + ( 3 / 4 ) 0 - 1 ( 0 2 g / 0 T 2) when e --+ 0 (0 -1 --+ (I/e)0-1). In terms of hierarchies the theory can be built around the pair ( L , M ) in the spirit of [160, 163, 902]. Thus writing (tn) for (x, tn)(i.e, x ~ tl here)consider (X)
Le = e 0 +
O(3
(X)
Eun+l(e,T)(eo)-n; Me = EnTnLne -1 + EVn+l(e,T)L-~n-1 1
1
(1.39)
1
Here L is the Lax operator L = 0 + ~ Un+l O-n and M is the Orlov-Schulman operator defined via Cx = M e . Now one assumes Un(e,T) = Un(T)+ O(e), etc. and set (recall L r = Ar
r : [I + O (-~)] exp ( ~
TnAn) = exp (1S(T'A) + O ( I ) )
r=exp(~F(T)+O(1))
(1.40)
We recall that OnL = [Bn, L], Bn = L~_, OnM = [Bn, M], [L,M] = 1, Lr = A~, 0),~ = M e , and r = ~-(T- (1/n:~n))~xp[E7 Tnan]/~'(T). Putting in the e and using On for O/OTn now, with P = Sx, one obtains OO
OO
A = P + E Un+lp-n; P = A - E Pi)k-1; 1 O0
(1.41)
1
O0
.s = E nrnAn-1 + E Vn+l)k-n-1; OnS : Bn(P) ==~OnP = OBn(P) 1
1
where t) ~,, Ox + (OP/i)X)Op and M --+ A/[. Note that one assumes also Vi+l(e,T) = V/+I(T) + O(e); further for Bn = ~ bnm Orn one has Bn -- ~-~ bnm Prn (note also B~ = L n + ~c~ a2L-J). We list a few additional formulas which are easily obtained (cf. [160]); thus, writing {A, B} = OpAOA - OAOpB one has
OnA= {Bn, A}; On.M = {Bn,.M}; {A,.M} = 1
(1.42)
Now we can write S = E ~ Tn An + ~_,~ Sj+IA-J with (~rnSj+l -- 0"?, Vn+l -- --rtSn+l, and O),S = .M (ar~ ---, 5"r~). Further O0
(~gn + l n
]~n "-- /~n _}_ E On~J +l/~-j; 0 S n + l ~'~ - P n 1
(~(~nF n
(1.43)
We sketch next a few formulas from [555]. First it will be important to rescale the Tn variables and write t ' = ntn, Tin = nTn, On = non = n(O/OT~n). Then -~
o'P
=
=
01n)~ : { ~ n )~} ( ~ n " - + apQnOP;
a'Qm
-
a'Q
n
;
= {Qn, Qm}
(1.44)
5.1.
265
BACKGROUND
Think of ( P , X , T ' n ) , n > 2, as basic Hamiltonian variables with P = P ( X , T ' n ) . - Q n (P, X, T~) will serve as a Hamiltonian via p . = dP' ., dX dT,n = (9~n; X n ~-- dT,n = -(gPQn
Then
(1.45)
(recall the classical theory for variables (q,p) involves 0 = O H / O p a n d f9 = -i)H/oqq). The function S(A, X, Tn) plays the role of part of a generating function S for the Hamilton-Jacobi theory with action angle variables (A,-{) where An P d X + QndT" = - ~ d A - K n d T " + dS; Kn = - R n = - - - ;
dA
dTn'
= 2n =
(1.46)
= 0;
(note that J~n = 0 ~ 0~nA = { ~n, ~}). To see how all this fits together we write (gP d X d R = (9~nP -t. . . . dT" O X dT"
This is compatible
OP _. , OQn + ~ - i)Qn + OPOpQn + OPf(~ oA Xn
with (1.45) and Hamiltonians
S~=~;
-Qn-
Furthermore
(1.47)
one wants
(1.48)
Sx = P; 0 " S = Q n - R n
and from (1.46) one has P d X + QndT~ = - ~ d A + RndT~n + S x d X
+ S),dA + O~nSdT~
(1.49)
which checks. We note that (9~nS = Qn = B n / n and S x = P by constructions and definitions. Consider S = S - E ~ AnT~n/n. Then S x = S x = P and S~n = S~n - Rn = Q n - Rn as desired with ~ = S~ = S~ - E ~ Tn~An-l" It follows that ~ ~ Ad - E ~ T-~An-1 = X + E ~ V/+I A-i-1. If W is the gauge operator such that L = W ( g W -1 one sees easily that Me = W
kxkc~ k - 1
w-l~)
=
G +
kxkA k-1
~o
(1.50)
from which follows that G = W x W -1 ---+~. This shows that G is a very fundamental object and this is encountered in various places in the general theory (cf. [160, 163, 990]). R E M A R K 5.2. We refer here also to [158, 168] for a complete characterization of dKP and the solution of the dispersionless Hirota equations (cf. also Section 1.1). and will sketch this here. Thus we follow [158] (cf. also [555, 902]) and begin with two pseudodifferential operators ((9 = i)/(gx), L=O
+
fi
Un+l (9-n ; W =
I +
1
Wn(9 - n
(1.51)
1
called the Lax operator and gauge operator respectively, where the generalized Leibnitz rule with (9- i (9 = (9(9-1 _ 1 applies Oi f = ~
j=o
(~ f)(gi-j
(1.52)
CHAPTER 5. W H I T H A M T H E O R Y
266
for any i E Z, and L - WO W -1. The KP hierarchy then is determined by the Lax equations
(On -- O/Otn), OnL = [Bn, L] -- B n L - LBn
(1.53)
where Bn = L~ is the differential part of L n = L~ + L n_ __ ~ - - ~ ~n0i _~_ ~-~_-1c~ t!n0i" One can also express this via the Sato equation, OnWW
-1 :
- L n_
(1.54)
which is particularly well adapted to the dKP theory. Now define the wave function via oo
r
W e ~ -w(t,A)e~;
(x)
~ - ~-~tnAn; w ( t , A ) = 1 + E W n ( t ) A -n 1
where tl = x. There is also an adjoint wave function r w*(t, A) = 1 + ~ w~(t)A -i, and one has equations Lr = Ar
One = Bnr
(1.55)
1
L*r
= W *-1 exp(-~) - w*(t, A) exp(-~),
Ar
0he*=-Bnr
(1.56)
Note that the KP hierarchy (1.53) is then given by the compatibility conditions among these equations, treating A as a constant. Next one has the fundamental tau function 7(t) and vertex operators X, X* satisfying
r
r
= X(A)T(t) = e{G_(A)T(t) _ e { T ( t - [A-l]). ~(t) ~(t) -~(t) '
~) =
x*(~)~(t) ~(t)
~-~c+(~)~(t) ~(t)
=
=
(1.57)
~ - ~ ( t + [~-~]) ~(t)
where G+(A) = exp(:t:~c(o~,A-l)) with c~ = (01, (1//2)02, (1/3)03, ...) and t + [A - 1 ] A-1, t2 + (1/2)A-2, .. .). One writes also
tnAn
e ~ = exp
E XJ( tl' t 2 , ' " , tj)A j o
(tl +
(1.58)
where the Xj are the elementary Schur polynomials, which arise in many important formulas (cf. below). We mention again the famous bilinear identity which generates the entire KP hierarchy. This has the form
/
r
A)r
A)dA - 0
(1.59)
where ~c~(-)dA is the residue integral about c~, which we also denote Res~[(.)dA]. Using (1.57) this can also be written in terms of tau functions as /c T ( t - [A-1])7(t ' -~-[A-1])e~(t'~)-~(t"~)dA - 0 This leads to the characterization
t-y,
t I --* t + y)
of the tau function in bilinear form expressed
( ~ 0 Xn(-2y)Xn+I (~)e~-~YiOi) T ' 7 ~ - O
(1.60) via (t -~
(1.61)
267
5.1. B A C K G R O U N D
where O~na. b = (om/Osr~)a(tj + s j ) b ( t j - sj)ls=o and c5 = (01, (1/2)02, (1/3)03,---). particular, we have from the coefficients of Yn in (1.61),
In
(1.62)
0 1 0 n T ' T "-- 2 X n + I ( 0 ) T ' T
which are called the Hirota bilinear equations. One has also the Fay identity via (cf. [7, 160, 168, 147] - c.p. means cyclic permutations) E(80
c.p.
S3)T(t d-[S0] nu [ S l ] ) T ( t nu [82] nu [83]) = 0
-- 8 1 ) ( 8 2 --
(1.63)
which can be derived from the bilinear identity (1.60). Differentiating this in so, then setting so = s3 = 0, then dividing by sis2, and finally shifting t ~ t - Is2], leads to the differential Fay identity, T(t)Or(t +
[81]- [82])-
= ( 8 1 1 -- 8 2 1 )
T(t nt- [81]-
[82])OT(t)
[T(t -Jr-[81]- [82])T(t)- T(t nt- [ S l ] ) T ( t -
[S2])]
(1.64)
The Hirota equations (1.62) can be also derived from (1.64) by taking the limit sl ~ s2. The identity (1.64) will play an important role later. Now for the dispersionless theory (dKP) one can think of fast and slow variables, etc., or averaging procedures, but simply one takes tn --* etn = Tn (tl = x ~ ex - X) in the KP equation ut -- (1/4)uzzx + 3uux + (3/4)0 -luyy, (y = t2, t = t3), with On -* eO/OTn and u(tn) --* U(Tn) to obtain OTU = 3UUx + (3/4)O-1Uyy when e --~ 0 (0 = O/OX now). Thus the dispersion term uxxx is removed. In terms of hierarchies we write (2O
Le = eO + ~
Un+l(T/e)(e_O) -n
(1.65)
1
and think of un(T/e) = U n ( T ) + O(e), etc. function with the action S
One takes then a W K B form for the wave
Replacing now On by COn, where On = O/OTn now, we define P = OS = S x . Then ei0iO --~ p i r as e --+ 0 and the equation L r = Ar becomes (2r
,X = P + ~
CX3
Un+lp-n;
P = ,X - ~
1
Pi+I,X -i
(1.67)
1
where the second equation is simply the inversion of the first. We also note from On~/ B n r = E ~ bnm(eO)mr that one obtain~ OnS = Bn(P) = A~ where the subscript (+) refers now to powers of P (note eOnr162-~ OnS). Thus Bn = L~ --* Bn(P) = ~ = E ~ bnm Pm and the KP hierarchy goes to O n P = OBn
(1.68)
which is the d K P hierarchy (note OnS = Bn ::~ OnP = OBn). The action S in (1.66) can be computed from (1.57) in the limit e --, 0 as
s=Er
1
~ OmF ~-rn n-E m 1
(1.69)
268
C H A P T E R 5. W H I T H A M T H E O R Y
where the function F - F ( T ) (free energy) is defined by (1.70)
exp
The formula (1.69) then solves the dKP hierarchy (1.41), i.e. P = B1 = (9S and Sn:OnS~--.~n-
(1.71)
Fnm~_m m
E 1
where
Fnm :
OnOmF which play an important role in the theory of dKP.
Now following [902] one writes the differential Fay identity (1.64) with eOn replacing On. looks at logarithms, and passes E ---, 0 (using (1.70)). Then only the second order derivatives survive, and one gets the dispersionless differential Fay identity E m,n=l
#-m'k-nFmn =l~ mn
(1 -
#
_/~-n p - A
n
/17 /
Although (1.72) only uses a subset of the Pliicker relations defining the KP hierarchy it was shown in [902] that this subset is sufficient to determine KP; hence (1.72) characterizes the function F for dKP. Following [158, 168], we now derive a dispersionless limit of the Hirota bilinear equations (1.62), which we call the dispersionless Hirota equations. We first note from (1.69) and (1.67) that Fin = nPn+l so E/~-nFln 1
n
-- E
Pn+l)~-n = )~ - P()~)
Consequently the right side of (1.72) becomes 1o~,t ~[P(tt)-P(A) ~-,x ] and for # ~ ~ w i t h / 5 _ we have
logP(x)= Z
m,n=l
(1.73)
1
a-m-nY~ _-~ mn
=
n
~Y~n a_j
j mn
O,xP
(1.74)
Then using the elementary Schur polynomial defined in (1.58) and (1.67), we obtain T'(~) = ~ - ~ X j ( Z 2 , . . . , Z j ) ~ - j = 1 + ~-~ F l j ~ - j - 1 ; o 1
a =
~
F~
(z, = 0)
m+n=i m n
(1.75)
Thus we obtain the dispersionless Hirota equations,
Yl~ = Xj+~(Zl = 0, Z2,-.., Zj+I)
(1.76)
These can be also derived directly from (1.62) with (1.70) in the limit e ~ 0 or by expanding (1.74) in powers of A-n as in [158, 168]). The equations (1.76) then characterize dKP. It is also interesting to note that the dispersionless Hirota equations (1.76) can be regarded as algebraic equations for "symbols" Fmn, which are defined via (1.71), i.e.
o~ __F~A_~ 1
(1 .77)
269
5.1. B A C K G R O U N D and in fact
(1.78)
F n m - F m n - Resp[AmdA~]
Thus for A, P given algebraically as in (1.67), with no a priori connection to dKP, and for Bn defined as in (1.77) via a formal collection of symbols with two indices Fmn, it follows that the dispersionless Hirota equations (1.76) are nothing but polynomial identities among Finn. In particular one concludes as in [146] that (1.78) with (1.76) completely characterizes and solves the dKP hierarchy. Now one very natural way of developing dKP begins with (1.67) and (1.41) since eventually the Pj+I can serve as universal coordinates (cf. here [32] for a discussion of this in connection with topological field theory - TFT). This point of view is also natural in terms of developing a Hamilton-Jacobi theory involving ideas from the hodograph - Riemann invariant approach (cf. [160, 153, 382, 555] and in connecting NKdV ideas to T F T , strings, and quantum gravity. It is natural here to work with Qn = (1/n)13n and note that OnS = Bn corresponds to OnP = i)Bn = nOQn. In this connection one often uses different time variables, say Tn~ = nTn, so that O~nP = OQn, and Gmn = Finn~ran is used in place of Fmn. Here however we will retain the Tn notation with OnS = nQn and OnP = nOQn since one will be connecting a n u m b e r of formulas to standard KP notation. Now given (1.67) and (1.68) the equation OnP = nOQn corresponds to Benney's moment equations and is equivalent to a system of Hamiltonian equations defining the dKP hierarchy (cf. [382, 555]); the Hamilton-Jacobi equations are OnS = nQn with Hamiltonians nQn(X, P = OS)). There is now an important formula involving the functions Qn (cf. [158, 156, 555]), namely the generating function of OpQn(A) is given by 1 p ( # ) _ p(A) = ~ OpQn(~)# -n 1
(1.79)
In particular one notes
p(~) _ p ( a ) a . = 0pQ~§
(1.s0)
which gives a key formula in the Hamilton-Jacobi method for the dKP [555]. Also note here that the function P(A) alone provides all the information necessary for the dKP theory. It is proved in [158] that the kernel formula (1.79) is equivalent to the dispersionless differential Fay identity (1.72). The proof uses OpQn
= Xn_l
(Ql,
. . . , Qn_l
(1.81)
)
where x n ( Q 1 , . . ' , Qn) can be expressed as a polynomial in Q1 = P with the coefficients given by polynomials in the Pj+I. Indeed
Xn = det
P P2
-1 P
0 -1
0 0
0 0
".
0 0
P3
P2
P
-1
0
...
0
Pn
Pn-1
"'"
P4
P3
P2
P
-= OPQn+I
(1.82)
and this leads to the observation that the Fmn can be expressed as polynomials in Pj+I Flj/j. Thus the dHirota equations can be solved totally algebraically via the equations Fmn - r P 3 , ' " , Pro+n) where ~mn is a polynomial in the Pj+I so the Fin : nPn+l
270
CHAPTER
5.
WHITHAM
THEORY
are generating elements for the Finn, and serve as universal coordinates. Indeed formulas such as (1.82) and (1.81) indicate that in fact dKP theory can be characterized using only elementary Schur polynomials since these provide all the information necessary for the kernel (1.79) or equivalently for the dispersionless differential Fay identity. This amounts also to observing that in the passage from KP to dKP only certain Schur polynomials survive the limiting process e --~ 0. Such terms involve second derivatives of F and these may be characterized in terms of Young diagrams with only vertical or horizontal boxes. This is also related to the explicit form of the hodograph transformation where one needs only OpQn = X n - l ( Q 1 , . ' . , Q n - 1 ) and the Pj+I in the expansion of P (cf. [158]). Given KP and dKP theory we can now discuss nKdV or dnKdV easily although many special aspects of nKdV for example are not visible in KP. In particular for the Fij one will have Fnj = Fjn = 0 for dnKdV. We note also (cf. [156, 555]) that from (1.81) one has a formula of Kodama (x)
P(#) - P(s
5.2
co
(x)
= ~ O~Q~. -~ = ~ x~(Q). -~ = ~xp( z Q ~ . - ~ ) 1
0
ISOMONODROMY
(1.83)
1
PROBLEMS
Notations such as (3A). (4A), 15A), and (16A) are used. We begin with [906] where the goal is to exhibit the relations between SW theory and Whitham dynamics via isomonodromy deformations. One considers algebraic curves C ---, Co (spectral covering) over the quantum moduli space of z E Co ~ algebraic curve of genus 0 or 1 via (3A) d e t ( w - L ( z ) ) = 0. For the N = 2, S U ( s ) Yang-Mills (YM) theory without matter Co = C P 1 and z = log(h) for h E C p 1 / { 0 , oc} with Lax matrix
L(z)=
bl Cl
1 b2
... "'"
0 0
csh -1 0
0
0
0 h
0 0
...
0
0
... ...
bs-1 cs-1
1 bs
(2.1)
Here bj = glj = d q j / d t with cj = e x p ( q j - q j + l ) and one compares with [499] where qs+l = ql and qj --+ - q j ~ cj ~ c~ 1. The eigenvalue equation (3A) becomes (A 28 = 1-I~cj = e x p ( q l - qs+ 1) h nt- h 2 S h
-1
=
P(w); y2 = p2(w ) _ A2S; h = 2(y + P(w)); P ( w ) = w s + ~
ujw s-j
(2.2)
2
(where A 28 = 1 in [499]). Thus one has hyperelliptic curves as in the case of a finite periodic Toda chain where such curves arise via a commutative set of isospectral flows in Lax representation OnL(z) = [ P n ( z ) , L ( z ) ] ;
[On - Pn, Om -- Pm] = 0
(2.3)
The Pn are suitable matrix valued meromorphic functions on Co. The associated linear problem w e = Lr One = P n r (2.4) then determines a vector (or matrix) Baker-Akhiezer (BA) function. replaces such an isospectral problem by an isomonodromy problem 0r e-~z = Qr
0 h e = Pnff2
Now in [906] one
(2.5)
5.2. I S O M O N O D R O M Y P R O B L E M S
271
The idea is that the t flows leave the monodromy data of the z equation intact if and only if [On -- Pn, e0z - Q] -- o; [On -- Pn, O m - Pro] -- 0 (2.6)
An idea from [339] is to write 9 in a WKB form (e ---, 0)
r =
r +
nCn
Oz . ]
(2.7)
where dS = wdz corresponds to the SW differential. Thus in the leading order term for e0z~ = Q ~ one finds (3B) (0zS)r = Qr and if we identify (3C) w - Sz - dS - wdz then the algebraic formulation gives essentially the same eigenvalue problem as (3A) (where the relation between r and ~ is clarified below). The idea from [339] is that isomonodromic deformations in WKB approximation look like modulation of isospectral deformations. The passage isospectral ~ isomonodromy is achieved via ( 3 D ) w ~ eOz which is a kind of quantization with e ~ h and 9 ~ a quantum mechanical wave function. Now one separates the isomonodromy problem into a combination of fast (isospectral tn) and slow (Whitham -Tn) dynamics via multiscale analysis. The variables are connected via Tn = etn and one assumes all fields ua = u a ( t , T ) . Then writing On ~ O/Otn we have (SE)Onu~(t, Et) = O n u a ( t , T ) + c(Ou~(t,T)/OTn)lT=~t. Further take Pn and Q as functions of (t, T, z) and look for a wave function ~ as in (2.7) with r - r T, z) (say r = r and S = S(T, z). The leading order terms in (2.4) become, for (3F) w = OzS and r = r tn(OS/OTn), w e -- Qr 0 n r Pnr (2.8) which is an isospectral problem with associated curve C defined by det(w - Q(t, T, z)) - 0 which is typically t independent but now depends on the Tn as adabatic parameters. One can then produce a standard BA function r = r tnf~n) (cf. [147, 268] and Section 2) where ~n "- f z d~n. The amplitude r is then composed of theta functions of the form ~(~-] tntTn-[-'..) where (3G) (7n - (fin,l,""" ,an,g)Twith an,j -- (1/27ri)~Bj d~n (we assume a suitable homology basis (Aj, Bj) has been chosen). The theta functions provide the quasiperiodic (fast) dynamics which is eventually averaged out and does not contribute to the Whitham (filow) dynamics. The main contribution to the Whitham dynamics arises by matching r ~-, r and r ~,, r with (3H) OS/OTn = f~n(Z) which in fact provides a definition of a Whitham system via (3I) Odf~n/OTm = Odf~m/OTn representing a dynamical system on the moduli space of spectral curves. Thus one starts with monodromy and the WKB leading order term is made isospectral (via (3F)) which leads to a t independent algebraic curve. Then the BA function corresponding to this curve is subject to averaging and matching to r We ignore here (as in [906]) all complications due to Stokes multipliers etc. The object now is to relate Q and L and this is done with the N - 2, SU(s) YM example. The details are spelled out in [906] (cf. also [760]). We go next to [908] where isospectrality and isomonodromy are considered in the context of Schlesinger equations. Thus in the small e limit solutions of the isomonodromy problem are expected to behave as slowly modulated finite gap solutions of an isospectral problem. The modulation is caused by slow deformation of the spectral curve of the finite gap solution. Now from [908] let gl(r, C ) N ~ @Ngl(r, C) ~ N-tuples ( A 1 , . . . , AN) of r x r matrices. There is a GL(2, C) coadjoint action A~ ~ gAng -1 and the Schlesinger equation (SE) is
Otj
Ai, (1 - 5ij) ti -- tj
5ij
. t~ - tk
(2.9)
CHAPTER 5. WHITHAM THEORY
272
and each coadjoint orbit Oi is invariant under the t flows. Thus (SE) is a collection of non-autonomous dynamical systems on 1-I1NOi. We consider only semisimple (ss) orbits labeled by the eigenvalues 0~a (a = 1 , . . . , r) of A~. Thus these eiqenvalues (and in general the Jordan canonical form) are invariants of (SE). There are also extra invariants in the form of the matrix elements of Ace = - ~1N Ai, invariant via OA~/Oti = 0. Assuming Ace is ss it can be diagonalized in advance by a constant gauge transformation Ai --~ CAiC -1 and then only the eigenvalues 0cr ((~ = 1 , . . . , r) of Ace are nontrivial invariants. One can introduce a Poisson structure on gl(r, C) N via = 6ij (-6zpAi,~r + 6r
(3J) {Ai,af~, Aj,pr
which in each component of the direct sum is the ordinary Kostant-Kirillov bracket. The (SE) can then be written as
OAy ot~
= {&,H~};
Hi =
nr
lT~ M(~) 2 -~
Tr
( AiAj ti - tj )
(2.10)
with {Hi, Hi} = O. Now (SE) gives isomonodromy deformations of (3K) dY/dA = M()~)Y with M(A) = O n e c a n assume for simplicity that (3L) The Ai and A ~ are diagonalizable and Oia - 0 i Z ~ Z if a ~/3. This means that local solutions at the singular points )~ - t l , . . . ,tN, cx~ do not develop logarithmic terms. The isomonodromy deformations are generated by (3M) OY/Oti = - [ A i / ( A - t i ) ] Y and the Frobenius integrability conditions are
EN[A~/(A-t~)].
-~j - tj , M ( A ) 0 + A Aj
~//-~ A - t i , i)tj ~ A - t j
=0;
- 0
(2.11)
and these are equivalent to (SE). Next, since A = t l , . . . , tN, cx~ are regular singular points of (3K) there will be local solutions (x)
Y/ -- Y/" ( ~ - ti)oi; Yi - E Y i n ( )~ - ti)n; 0
(2.12)
oo
Yc~=?~.A-~
Ycc=EYo~,nA -n o
where the Yin are r x z matrices, Yio and Yoc,o are invertible, and Oi, O ~ are diagonal matrices of local monodromy exponents. This leads to (3N) Ai Y/0OiY/o I for i = 1 , . . . , N, where Oi = diag(Oil,..., 0~r). The tau function for (SE) can be defined in two equivalent ways ( 3 0 ) dlogT = E N Hidti or O.log~-/Oti = Tr(OiYiolYil and the integrability condition OHi/Otj = OHj/Oti is ensured by (SE) (see [908] for details). The spectral curve is now (3P) det(M()~) - #I) = 0 and this generally varies under isomonodromic deformations (see [9081). =
Consider now Garnier's autonomous analogue of (SE). This is given by 0
Ai
,M(A)
=0;
[0
Ai
0
E+
Aj
-cj
=0
(2.13)
and this is an isospectral problem with det(M()~)- #I) independent of t. An auxiliary linear problem is given by wV=M(A)r
~-
_
Ai r A-ci
(2.14)
5.2. I S O M O N O D R O M Y
PROBLEMS
273
where r is a column vector. This kind of problem can be mapped to linear flows on the Jacobian variety of the spectral curve and the case of r = 2 is particularly interesting since here the Painlev6 VI (and Garnier's multivariable version) emerge. For the geometry one goes now to [5, 73, 440, 441]. Let Co be the spectral curve (3Q) F(A, #) = d e t ( M ( / k ) - # I ) = 0 which can be thought of as a ramified cover of the punctured Riemann sphere 7r : Co ---* C P t / { C l , ' " , C N , CX~} where 7r(A,#) = A. Then generically 7r-1(~) ~ {(~,#~), a = 1 , - - . , r } and the #~ are eigenvalues of M(~). Near )~ = ci one has #a = 0ia/(A - ci)+ nonsingular terms, where the Oia are the eigenvalues of Ai. Similarly near A = oc one has #a = -0o~,aA -1 + O(A -2) where the 0~,a are eigenvalues of Ac~. One can compactify Co by adding points over the punctures. Thus near ci replace tt by /5 = f(A)# where f(A) = I - I N ( A - ci). Then the spectral curve equation becomes (3R) /~(A, fi) = d e t ( f ( A ) M ( A ) fiI) = 0 a n d 7 r - l ( ) ~ ) = {()~,]~a), o~ = 1 , - - . , r } where (3S) fia = f'(ci)Oia + O ( A - ci). Since the Oia, (a = 1 , . . . , r) are pairwise distinct one is adding to Co at ci, r extra points (A, #a) = (ci, f'(ci)Oia) to fill the holes above ci. At c~ one uses t5 = A# with 7r-l(ec) = {(oc,-0c~,a)}. Thus by adding r N + r points to'C0 one gets a compactification C of Co with a unique extension of the covering map ~r to a ramified 7r : C --~ C P t. The genus of C will be g = ( 1 / 2 ) ( r - 1 ) ( r N - r - 2). Taking (3T) M~ = EN[A~ A ~ = diag(Oil,..., Oir) as a reference point on the coadjoint orbit of M(A) one compares the characteristic polynomials via (3U) F ( A , / 5 ) - / ~ ~ = f(A) ~]~=2p~(A)fi r-e where pe(A) = E~0~ hmeA TM with 5e = ( N - 1)t~- N. This leads to
~(~, ~) = ~0(~, ~) + ~ ~ h ~ ~ - ~ ~=2 0
(2.15)
The spectral curve has therefore three sets of parameters (3V) (1) Pole positions c~ (i = 1, . . . , N) (2) Coadjoint orbit invariants 0ia (i = 1,. 9 9 N, c~), and (3) Isospectral invariants hm~ ( t ~ = 2 , . . . , r ; m - 0 , - - - , b e - 1 ) . Now one develops the W h i t h a m ideas as follows. First reformulate the ( S E ) equation via (cf. [9461) 0 0 0 0 Ai Ai
ot~
~ ~ 0-~;
0--~ --* ~b-X;
~ - t~
~ h - T~
(2.16)
Note this corresponds to eti = Ti with Oi = eO/OTi and eA = A with 0h = e0h and A i / ( A - ti) ~ A i / ( A - Ti) so Ai ~ eAi. Then (2.9) becomes
s
0A~
[A~,Aj]
= (1 - 5ij)~ii [ 5 ~
[A~,Aj]
- 5iy -~iZ-~jj
(2.17)
(the notation in [908] is confusing here since the step Aj ~ A j = eAj was not clarified). The auxiliary linear problem is (cf. ( 3 K ) and ( 3 M ) )
OY
OY
e-~-~ = A/I; e OTi -
Ai
A - Ti y
(2.18)
(thus M --* AA and Y --~ Y). Then the e dependent (SE) equation (2.17) can be reproduced from the Frobenius integrability condition
~ - ~-X = 0; [~gT-JJ+ 0 A----2-~'AM(A) ~ [~b-~ 0 + A-T~'~b~j A~ 0 + A-Tj Aj 1 =0
(2.19)
CHAPTER 5. WHITHAM THEORY
274
Now start with (2.17)-(2.19) and introduce first variables written as
ti = e-ZT~ so that (2.17) can be
OAiotj = ( 1 -- 6ij ) Tii[Ai"--AJ]~_ 6ij ~ . Tii[Ai"--Ak]~kk
(2.20)
In the scale of the fast variables the T/can be regarded as approximately constant and hence (2.20) looks approximately like Garnier's autonomous isospectral problem. Of course the Ti vary slowly as do the hmg (but not the Oik). For multiscale analysis following [241] one writes now .A~ - .A~(t, T, e) and eO/OT~ --~ O/Ot~+eO/OT~ in (2.20) (note t~ and Ti are considered to be independent now with t~ - e-lT~ imposed only in the left side of the differential equation (2.20)). Now assume ( 3 W ) A~ .A~ T) + eAl(t, T) +.... The lowest order term in e gives 0 0 0"40 [A~ A~ - 6ij ~ [A~ ~ :,Ak] ~ Otj = (1 -- 6ij ) Ti - Tj k#i For
(2.21)
Ti = ci this is Garnier's autonomous system. The next order equation is 0"41 = (1 - 6ij [A~ A}] + [A~, A~] Otj Ti - Tj -
[A~ k#i
+
(2.22)
+E
Ti - Tk
where E refers to terms in A ~ and their T derivatives. A standard procedure in multiscale analysis is to eliminate the t derivatives of .A~ by averaging over the t space but this involves some technical difficulties. The spectral curve for (SE) is hyperelliptic only if r = 2 and even in that situation there are problems. So another approach is adopted. In addition to the multiscale expression for the Ai one assumes (3X) y = [r176 + r + . . . ] e x p [ e - l S ( T , A ) ] (y and the r are vector valued functions with S a scalar). Then one writes the auxiliary linear problem (2.18) in the multiscale form e~--~ = M(A)Y;
~ i + e~-~/ y = -A_----~/y
(2.23)
The leading order term reproduces the auxiliar linear problem of the isospectral problem
OS o ~,
M0r =
0r ~
0Sr
;-5?T+~-~
where Ad o = z N [ A ~ Ti). Define now (3Y) # so (2.24) becomes (2.14)in the form
, r = M0r
0r = _
Oti
-
A0 A-T~
_~r
OS/OA and r - r176
A~~162
A - T~
(2.24)
ti(OS/OTi)]
(2 25)
Next one compares this r with the ordinary BA function r = Cexp[~ tif~i] where fti fA,u dfti and the vector function r involves 0 functions as before. For matching we want now (3Z) r = r (or more generally r = h(T, A)r and OS/OTi = fti. Thus one proposes (.) It = OS/OA and OS/OTi = f~i (and consequently Of~i/OTj = Of~j/OTi) as the modulation
5.2. I S O M O N O D R O M Y P R O B L E M S
275
equations governing the slow dynamics of the spectral curve ( . - ) det(A/I~ - #I) = O. Details in this direction are given in Section 6 of [908] where it is determined t h a t
Oia
d a i = -------------~dA ( A - Ti) + nonsingular terms
(2.26)
near Pa(A) 9 7r-l(A) (a = 1 , . . . , r). Normalization is achieved via ( - . - )
~Aj df~i = 0 for
j=l,...,g. Next consider the hi for I = 1 , . . . , g as the set of isospectral invariants and let S = S(T, h, A) with dS = #(T, h, A)dA. One defines
d&i = OdS _ O#(T, h , A ) d A = 1 0/5 dA Oh1 c3hi f(A) c3hi
hme (cf. (2.15)) (2.27)
T h e n (&) [OP(A,f~)/Ohmt] = f ( A ) f f ~ r - m and there results
(~) d&mt =-[Atfitr-m/(OF/Ofit)]dA These differentials d~I form a basis of holomorphic differentials on the spectral curve and one shows t h a t
Oaj = --~A OdS = ~A d&)I = A I j OhI j OhI j [b
(2.28)
is an invertible m a t r i x (d&i = ~ A H d w j where the dwj are determined via ~AI d~j = 5Ij). Thus one has an invertible map h --~ a where the aj are the s t a n d a r d action variables of period integrals aj = ~Aj dS. One can also express dS now as dS(T, a) with OdS/OTi = d~i and OdS/Oai = dwi since
OdSoai -- ~.10hJoai OdSohj-- E(AIj)-ld&j
= dwi
(2.29)
One can also introduce a prepotential as in SW theory with
OJc = bi = / B i dS; -~i O~ = Hi Oai
(2.30)
(cf. (2.30) for Hi).
5.2.1
JMMS
equations
We go now to [907], which is partly a rehash of [906] for the Jimbo-Miwa-Mori-Sato (JMMS) equations, giving isomonodromic deformations of the matrix system (r • r) N
d Y = M(A)Y; M(,X) = u + ~ , a - ti ; u d,a 1
1
uaEa
(2.31)
where Ea ~,, 5a/~Sa~ in the (/3,7) position so u = d i a g ( u l , . . . , u r ) where the us ~ time variables of isomonodromic deformations. Evidently ~ = ti are regular singular points and = oc is an irregular singular point of Poincare! rank 1. In order to avoid logarithmic terms one assumes the eigenvalues 0ia of Ai have no integer difference and t h a t the ui are pairwise distinct. T h e n isomonodromic deformations are generated by
OY c3ti
Ai = -~Y;
) k - ti
OY Oua
= (AE~ + B~)Y;
(2.32)
CHAPTER 5. WHITHAM THEORY
276
X-" E s A ~ E ~ + E z A ~ E s Aoo x-'g Bs = - z_., ; = - A.., Ai us -- u~ 1 with Frobenius integrability conditions
[ ff--~i -~ A - ti' 0
M(A)] = 0 ;
(2.33)
- - $Es - Bs,-ff-f - M($)] = 0 This leads to
OAj = [tiEs + Bs, Aj]; Ou~ OAj
=(1-hij)
(2.34)
[Ai, Aj] [ k~ci Ak Ajl ti - tj + hij u + . ti - tk'
There are two sets of invariants (4A) Eigenvalues 0is of Aj 9 (OOjz/Oti) = (OOjZ/Ous) - 0 and (4B) Diagonal elements of A ~ 9 (OAoo,Zz/Oti) = (OAc~,~Z/Ous) - O. The JMMS equation is a nonautonomous dynamical system on the direct product 1-I (.Oi of coadjoint orbits and can be written in the Hamiltonian form
OAj = {Aj, Hi}; OAj = {Aj, Ks}; Hi = Tr Oti Ous
Ks = Tr
( ~ Es
tiAi + y~
AiAj ) uAi + ~ ti - tj ; j#i
EsA~E~A~) u s - u~
(2.35)
The dual isomonodromic problem can be formulated in a general setting where we suppose rank Ai - gi (which is also a coadjoint orbit invariant and constant under the JMMS equation in the generalized sense). Thus write first with r >_ t~i (cf. [440, 441,442] for more details and expanded frameworks)
M()~) = u - G T ( A I - T ) - I F
(2.36)
where F = (Fas) and G = (Gas) are t~ x r matrices, ~ = E t~i, and T is a diagonal matrix of the form (4C) T = ~ g tiDi with Di = Etl+...+t~_l+l +'" "+ Et~+...+t~ (so D1 = E1 + - - - + E~ with D2 -- Et~ + . . . + Et, etc.) In particular Ai can be written as Ai = - G T D i F and Fas, Gas may be understood as canonical coordinates with Poisson bracket
{ Fas, Fbl~} ---- { Gas, Gb~ } -- 0; {Fas, Gb~ } = ~ab~sl~
(2.37)
As an example consider e l = 3 and ~2 = 2 which implies t~ = 5 and take r = 3 so F, G are 5 x 3 matrices with D1 = E1 + ' - " + E5 = / 5 and D2 = E4 + Eh. Then A1 = - G T I h F is 3 x 3 of rank 3 and A2 = - G T D 2 F is 3 x 3 of rank 2. The map (F, G) ~ (A1,"" ,AN) then becomes a Poisson map and the JMMS equation can be derived from a Hamiltonian system in the (F, G) space with the same Hamiltonians Hi and Ks. The dual problem is formulated in terms of the ~:ational ~ x ~ matrix (4D) L(#) = T - F ( # I - u)-lG T where L(#) = T + ~ [ P s / ( # - us)] with Ps = - F E s G T of rank 1 (but this restriction can also be relaxed). The map (F, G) --, (P1, P2,'" ") is Poisson for the Kostant-Kirillov bracket
277
5.2. I S O M O N O D R O M Y P R O B L E M S
(4E) {Pa,ab, Pz,cd} = 5ae(--SbcP5,ad + ~SdaPl3,cb) and one can write the dual isomonodromy problem in the form dZ . d~
.
OZ . . . L(#)Z; Ous
Ps OZ Z; = - ( # D i + Qi)Z I t - us Oti
Qi = - Y ~ . DiPooDj + DjPooD, ; P ~ = - Y ~ _ , P s j~=i ti - tj
(2.38)
Now introduce a small parameter E via (cf. [946]) 0 Oh
0 OA
0 Oti
0 OTi
0 Ous
0 OUs
(2.39)
with Y~. us -+ Y~. UsEa and the JMMS equatins take the form
i
(2.40)
~-5-~ = ( 1 - 5~j ) T~ _ T~ + 5~j U + EaAooEz + E z A o o E s
oAj~ = [TjE~ + B~, & ] B~ = - Z E-5~
U~ - U,
(where ,4i = eAi, Ti = eti, Us = eus, and Bs = eBs). Working as before we assume .4j = A j ( t , u, T, U) and then, using ti = c - 1 T / a n d us = e-lYs, one has e.g.
0Aj
0Aj
0Aj
e OTi --+ Oti + e OTi
(2.41)
as in (2.20)-(2.21), leading to (Aj = A ~ + .4Je + . . . )
0
(2.42)
where B ~ is given by the same formula as Ba but with .4cr replaced by .4 ~ - - ~ N .4~ The slow variables are parameters at the lowest order and to determine the slow dynamics one goes to the next order and one uses here the following approach. The lowest order equations are again an isospectral problem with Lax representation 0-~i +0
A---E-~ ' ' 4 ~ A/t~
[a-~.
]
- 0; A/t~
A0
-- U + ~1 A - Ti;
hE. - Bo , M o = 0
(2.43)
and d e t ( # I - A d ~ = .=.is constant under (t, u) flows..=, depends on (T, U) however and the slow dynamics may be described as slow deformations of the characteristic polynomial of the spectral curve, namely ~ = 0. This spectral curve Co on the (A, #) plane can be compactified to a nonsingular curve C; the projection r(A, #) ~ A : Co ~ C p 1 / { T 1 , . - - , TN, oc} extends to C to give an r fold ramified covering of C over C P 1. One adds points to the holes over
CHAPTER 5. W H I T H A M T H E O R Y
278
A = (Ti, oc) as before. There will be r points (oc, U a ) ( a = 1,..., r) over A = oc. For the ~ write (4F) ~ = f ( A ) # with f(A) = r l N ( A - Ti) so E becomes (4G) F(A, ~) det(fitI- f(A)Ad~ = 0. Add then the r points (A,~) = (Ti, f'(Ti)Oia) (a = 1 , - . . , r) over ~ to obtain a curve of genus g = ( 1 / 2 ) ( r - 1 ) ( r N - 2). To describe the slow dynamics one wants a suitable system of moduli in the space of permissible curves and the slow dynamics involves differential equations for such moduli. Now one can write r
F(A,/2) = F~
5s-1
+ f(A) ~
hrnsAmfit s
~
(2.44)
s=2 m = 0
where (4H) 5s = ( N - 1 ) s - N . F~ 15) is a polynomial whose coefficients are determined by the isomonodromy invariants and (T, U). The coefficients h6~-l,s (2 _< s < r) are also determined by these quantities and the remaining coefficients hm,s for 2 <_ s <_ r, 0 <_ m <_ 8s - 2 give the appropriate moduli of number ~sr=2(Ss - 1) = g. Thus the .A~ are solutions of the isospectral problem with slowly varying spectral invariants hm,s(T, U) and to derive the modulation equations one writes
Y = (r176
+r
x exp[e-ls(r,U,A)]
(2.45)
The lowest order equations are
or
OS o
~,-M
0r ~
OS o
-0-~+~ 0r 0
OS r
o~--2+ ~
which determines r up to a multiplier r into the isospectral linear problem
=
os
r
A-T~
'
(AEa + B~162 ~
___,r
0r _ A~ r #r = M~162Oti--A~-Ti
"= -A; ~ = r
OA~
~ =
(2.46)
U). These equations can be rewritten
0r
= (AE~ + B ~ 1 6 2
(2.47)
(~ t~-~ os + Z ~-~ os)
In particular the characteristic equation d e t ( # I - Ad~ = 0 is satisfied by # - c3AS. One now identifies this r with a corresponding algebro-geometric BA function via (A,#)
(2.48) One must choose here a symplectic homology basis AI, BI so that the W K B solution gives a correct approximation to the isospectral problem and we assume that this has been done (the problem is nontrivial). The meromorphic differentials are characterized now by the following conditions: (4I) dftT~ has poles at the r points in 7r-l(Ti) and is holomorphic outside of 7r-1(~). Its singular behavior at these points is such that dftT~ -- - d # + nonsingular. (4J) dFtv~ has poles at the r points in 7r-1(oc) with singular behavior df~v~ = dA+ nonsingular. (4K) ~AI dftT~ = fiAt dftv~ = 0. Matching the exponential parts in the two forms of r gives then the modulation equations
OS OS OT~ = f~a; oga = ftu~
(2.49)
5.2. I S O M O N O D R O M Y P R O B L E M S
279
These can be rewritten as
OdS I OdS I = df~u, ; dS - #dA OTi A=~ = d~tTi ; - ~ a A=c leading to "classical" Whitham
(2.50)
equations of the form (evaluation at A = c)
Odf~T~ = OT~
Od~T~
(2.51)
.
OTj '
Od~vz Od~v~ Od~v~ Od~Ti OVa = 0----~ ; OTi = OVa A similar multiscale analysis applies to the dual isomonodromic problem with WKB ansatz Z = ( x O ( t , u , T , U , # ) + x l e + ...) • exp[e-IE(T,U,,)] (2.52) This leads to slow dynamics of the dual isospectral problem based on the dual expressions (4L) d e t ( A I - L ~ = 0 of the same spectral curve. The modulation equations are obtained in the dual form (evaluation at # = c)
OdE OdE OTi = d~Ti; OUa = d~v~; dE - - A d p
(2.53)
(cf. [269, 499]). As before one can relate all this to SW ideas as follows: (A) One has a g dimensional period map h = (hms) ~ a = (hi) where ai = fA, dS for I = 1 , . . . , g. One can show that the Jacobian of this map is nonvanishing and one may write h = h(T, U, a) (inverse period map). For a fixed this gives a family of deformations of the spectral curve producing a solution of the modulation equations; varying a one has a general solution. (B) One will have (for suitable holomorphic differentials d~j)
OdS - dwi; /A dwj Oai i
--
(2.54)
5Ij
(C) There is a prepotential 9r(T, U, a) satisfying
O~ = bI = fB dS; ~O/J:= Oh1 i
Hi;
(2.55)
OJz 02~ : ~Ij : fB j dwi ou~ - Ka; cgaiOaj (where Hi and Ka are as before).
5.2.2
Gaudin model and KZB equations
We go to [909] now in order to exhibit connections of isomonodromy problems and the Knizhnik-Zamolodchikov-Bernard (KZB) equations (see e.g. [87, 325, 443, 500, 553, 586. 619, 642, 816, 876, 915] and cf. also [296, 501,779]). Here [909] follows [619] at first and later [501]; given the background in Sections 3 and 4 it is appropriate to follow [909] here instead of the Russian school (to which we go later via [642, 643, 782, 783]). First a generalized Gaudin model is based on a generalization of the XYZ Gaudin model to an SU(n) spin
C H A P T E R 5. W H I T H A M T H E O R Y
280
system following [619]. Let X be a torus with modulus T, i.e. X = C / ( Z + ZT). For integer (a, b) define
O[ab](Z) -- Oa 1 1 b (Z, 7");
n 2~2 n
(2.56)
Okk,(Z, T) = ~ exp[uin(m + k) 2 + 2~i(m + k)(z + k')] mEZ Define n x n matrices (1hA) Jab = g ahb and jab = (1/n)J~b 1 where g = diag(1,~,... , ~ n - 1 ) with w = exp(27ri/n) and h = (hi-l,j); one notes that gh - whg. The J matrices give a basis of su(n) over R and of sl(n, C) over C with Tr(JabJ cd) = 5achbd. As an R m a t r i x define
R()~) =
E Wab(~, ?])Jab @ jab; Web -- O[ab]()~ + r]) (a,b)eZn xZn O[ab](r])
(2.57)
These Boltzman weights are n-periodic so the summation is as indicated. The r-matrix is the leading part in the r] expansion of R at rI = 0, i.e.
r()~) =
E
(a,b)r (0,0)
WabJab @ jab; Wab(Z) = O[ab](/~)OiO0](O) O[ab](0) 0[00] ()~)
(2.58)
where ' ~ d/dz. The sum is now over (Zn x Z n ) / { ( 0 , 0 ) } (alternatively one could take woo(z) = 0). The r matrix satisfies the classical Yang-Baxter (YB) equation
[r(13)(~), r(23)(#)]- _[r(12)()~- #), r(13)(/~)+ r(23) (~)]
(2.59)
where e.g. r(12)(/~) = ~ Wab()~)Jab | jab | I. The generalized Gaudin model is a limit as rl ~ 0 of an inhomogeneous SU(n) spin chain with N lattice sites and the mono d r o m y m a t r i x of the spin chain is (15B) T(,~) = L N ( s t N ) ' ' ' LI(s - tl). The ti are inhomogeneous parameters which eventually correspond to time variables in an isomono d r o m y problem and (15G) Li(A) = ~-~(ab)Wab(/~)Jab ~ fli(Jab) acting on c n | If/ where If/ is the representation space of an irreducible representation (pi, Vi) of su(n). The L operators satisfy ( 1 5 D ) R L L - L L R and the m o n o d r o m y matrix acts nontrivially on C n | V for V = V1 | ... | VN. The leading nontrivial part in the rbexpansion of T(A) at 77 = 0 is ( 1 5 E ) T ( A ) = ~ g ~-~(ab)Wab(/~- ti)Jab | Pi(J ab) and one has the fundamental c o m m u t a t i o n relation ( 1 5 F ) [ T I ( A ) , T2(#)] = - [ r ( A - #), T I ( A ) + T2(#)] where e.g. TI(~) = ~ i ~(ab) Wab()~ -- ti)Jab | I @ pi(J ab) (i.e. the m o n o d r o m y acts nontrivially in the first C n of C n | C n | V). Mutually commuting Hamiltonians Hi (i = O, 1 , . . . , N) of the generalized Gaudin model are defined via
1
N
N
-~TrnxnT(A)2 -- E CiT)()~ - ti) + E Hi~(A - ti) + Ho 1 1
(2.60)
where P ( z ) and ~(z) are Weierstrass functions with modulus r. The Ci are quadratic Casimir elements of the algebra generated by pi(jab), i.e. one has the relations ( 1 5 G ) Ci = (1/2) ~-~(ab)Pi(Jab)Pi(J ab) and ( 1 5 H ) H i = ~ j # i ~ ( a b ) W a b ( t i - tj)Pi(JabPj(J ab) for i > 0 with ~ N Hi = O. One considers now (15I) M(A) = Z1N Z(ab)Wab(A -- ti)JabA ab as a classical analogue of the T operator in ( 1 5 E ) ; the A ab are scalar functions of ti and T (for i = 1 , . . . , N ) and the isomonodromy problem will be formulated as differential equations for these functions. Note the A ab depend on t but the pi(J ab) do not; this is related to the difference between
5.2. ISOMONODROMY PROBLEMS
281
Heisenberg and SchrSdinger pictures (as indicated below). The commutation relations for Pi(jab) can be packed into {M()~) ,o M(p)} = - [ r ( / k - #), M ( ) ~ ) e I + I | M(p)]
(2.61)
Here the left side is an abbreviation for (15J) E{Mab(A),Mcd(p)}Jab | Jcd where M ab are the coefficients in M = ~ MabJab. In terms of the residue matrix (15K) Ai = Res~=t~M(A) = ~(ab) JabAab the Poisson structure is simply the one induced from the Kirillov-Kostant Poisson structure on sl(n, C); symplectic leaves are the direct product O1 • • ON of coadjoint orbits Oi in sl(n, C) on which Ai is living. These leaves become the phase spaces of the nonautonomous Hamilton system described as follows: Using M(A) for T in (2.60) with Ci Casimir elements in the Poisson algebra above, one thinks of Hamiltonians Hi for the classical Gaudin system in the form (15L) Hi = ~-~j#i ~--~-(ab)Wab(ti -- tj)Ai,ab Aab. HO is also a quadratic form in the A ab but is more complicated. The ti here are just parameters of the classical Gaudin system, which is a Hitchin system on the punctured torus X / { t l , . . . , tg}. The nonautonomous Hamiltonian system will be a nonautonomous analogue of this Hitchin system in the form
OA~b = { A~b Hi}; Oti '
=
OT
'-2-~i H o - 771
tiHi
(2.62)
where rh arises via the known transformation law (15M) ~(z + 1) = ~(z) + 7/1. This nonautonomous Hamiltonian system is an isomonodromy problem which can be put in Lax form. The key is a relation
{M()~),-~ITrM(#)2}=
EEWa-b'~"(/-W )(ab) abJ)i"(at-bAabM ' "(l)
(2.63)
(not p r o v e d - one can use function theoretic methods as with the YB equation). Taking residues gives
{M(A), Hi} = -[Ai()~), M(A)]; Ai(s = E Wab(~ - ti)JabA ab
(2.64)
(ab) and after lengthy calculations N
{M(A), H0} = [47riB(A) - r/1 ~
tiAi(A), M(A)];
(2.65)
1
= Z Z Zab( - )JabA?b; t Zab _ 1 (ab)
[ab] (0)
47ri
O[ab](/~)
0' O[ab](O)
The first formula can be written in the form N
{M(~), H 0 - 7/1 ~-~tiHi} = [47riB(~),M(~)]
(2.66)
1
and one obtains the Lax form of the nonautonomous Hamiltonian system as
OM(A) _-[Ai(A) M ( A ) ] - OAi(A) Oti ' 0)~ ;
(2.67)
CHAPTER 5. WHITHAM THEORY
282
OM(A) = [2B(~), M(~)] + 2 ~~(~B'~------a' Or Oh One notes that T derivatives correspond to A derivatives via (15N) (O0[ab]()~)/OT) = (1/47ri)(O20[ab]()~)/O/~2) These equations are integrability conditions for a linear system (~-
-~i +Ai(A)
(2.68)
M(A)) Y(A) = 0;
Y()~) = 0 ;
N
where the last two equations can be interpreted as isomonodromy deformations of the first equation.
(0
Now the elliptic KZ equations of [619] can be written as
)
+ E E Wab(ti -- tj)Pi(Jab)pj(J ab) F(t) = 0; j#i (ab)
(n O
N
(2.69)
)
-~T + E E Zab(ti - tj)fli(Jab)pj(J ab) F(t) = 0 i,j=l (ab)
where ~ = k + n with k the level of a twisted WZW model. These equations characterize N-point conformal blocks with irreducible representations Pi sitting at ti. Following [816] one adds another marked point at A with fundamental representation (C n, id) to obtain
(o
t~
+ E E Wab(/~ -- ti)JabPi(J ab) 1 (ab)
)
G(/~,t) = 0;
(2.70)
)
-~i + E E Wab(ti -- tj)Pi(Jab)pj(J ab) + E Wab(ti -- ~)pi(Jab)J ab j#i (ab) (ab) (
G()~,t) = O;
0 N N -~T + E E Zab(ti - tj)Pi(Jab)PJ(Jab) + E E Zab(ti - A)Pi(Jab)jab+ i,j=l (ab) 1 (ab) + E E Zab()~ -- tj)JabPj(J ab) + E Zab(O)JabJab G(/~, t) = 0 1 (ab) (ab)
Let F = exp(S) be a fundamental solution of (2.69) (cf. [816]) and consider the resulting equations for X = F-1G of the form
) )
+ E E Wab()~ -- ti)Jab Aab 1 (ab)
(0
X()~, t) = O;
- ~ wab(A - ti)JabA ab X(A, t) = 0; (ab)
(2.71)
5.2. I S O M O N O D R O M Y
a-~T + 2 ~
283
PROBLEMS
~
Za6(A - ti)JabA~ b + ~
1 (ab)
Zab(O)Jab Jab
X(A, ~) : 0
(ab)
where ( 1 5 0 ) A a6 - F ( t ) - l p i ( j a 6 ) F ( t ) . This is almost the same as (2.68) except that: (A) The last equation contains an extra term. (B) The A a6 are not scalar functions but operators on V - V1 | | VN. (C) There is an arbitrary parameter k; the previous monodromy (i.e. (2.68)) can be reproduced by setting ~ - - 1 . Now the first descrepancy ( g ) can be removed by gauging it away via X - , [exp(f(T))]X. As for (B) one remarks that the system (2.71) is a quantization of the isomonodromy problem (2.68); this is parallel to the relation between quantized and classical Hitchin systems. The operators A a6 depend on (t, T) but inherit the same commutativity as the J matrices from pi(ja6). The passage from p i ( J a6) to A a6 amounts to a change from the Schr6dinger picture to the Heisenberg picture and classical limit now means replacing these operators by functions on a phase space, thus giving the iso/nonodromy problem. 5.2.3
Isomonodromy
and Hitchin
systems
We will extract here from [782] which gives an excellent survey of many matters (cf. also [500, 642, 643, 644, 783]). However first some remarks about RS and VB are appropriate. REMARK 5.3. In a first version of this survey we made up a number of remarks (seven) on connections, vector bundles (VB), and Hitchin systems following [47, 54, 227, 256, 257, 258, 259, 284, 314, 373, 447, 463, 464, 465, 466, 482, 487, 554, 556, 632, 642, 643, 644, 665, 685, 720, 749, 782, 783, 807, 938]. The result was complete enough but lacked coherence so we will try again using [442] as a springboard. We will first simply indicate, generally without proof, some facts about RS, VB, and sheaves from [467] (for background information cf. [147, 207, 319, 351,429, 487, 732, 807] - we will assume various definitions are known). First a holomorphic section of a line bundle L over E (Tr : L ~ E) is a holomorphic map s : E ~ L such that lr o s = id~ and the space of such sections H ~ L) is finite dimensional. Given a point p 6 E, with U a neighborhood (nbh) of p, V - E / { p } . and z(p) = 0, then, using z as a transition function on U N V, one produces a line bundle Lp which will be useful later. The canonical bundle K is the bundle of holomorphic 1-forms and on E = p1 one defines 59(n) as the line bundle with transition function z n on U N V where e.g. 0 e U and Y = P1/{0}); the dimension of H ~ 1, 59(n)) is n + 1. Given a line bundle L one forms L* - L-1 via transition functions gab (L*) - gab - I ( L ) and there are obvious relations Horn(L, L) _~ L* | L, etc. If 3E is compact then g e n u s ( E ) - g - d i m H ~ K). One defines sheaf cohomology in the standard manner and we recall for a line bundle L, H I ( E , L ) _~ H ~ | L*) (Serre duality). The isomorphism classes of holomorphic line bundles on E are given by elements of H i ( E , 59*) (called the Picard group) and for given degree d, j d ~ line bundles of degree d is a complex torus isomorphic to J a c ( E ) (more on this below). Note also for S = 59(L) = sheaf of holomorphic sections of L, that H P ( E , S ) = 0 f o r p > 1; i f S = C or Z then H P ( E , S ) = 0 f o r p > 2. From the short exact sequence (.) 0 ~ Z ~ 59 ~ 59* ~ 1 with r ~ exp(27rif) one determines a long exact sequence leading to H 1(E, 59) --+ H1 (E, 59* 0 --+ H I ( E ' Z) ) ~ H 2 ( E ' Z) = 0
(2.72)
where H 2 ( E , Z ) _~ Z, H i ( E , 59) ~_ C g, and H I ( E , Z ) _~ Z 2g. The coboundary operator in (2.72) yields (since H i ( Z , 59*) ~_ equivalence classes of line bundles) 6([L]) = deg(L) =
CHAPTER 5. WHITHAM THEORY
284
cl (L) e Z (first Chern class). Thus for L such that 5([L]) = d one obtains Jd(E) _~ Jac(E) One shows next that if a section s E H ~ L) vanishes at Pi E E with multiplicities rni then deg(L) = ~ m~ so if deg(L) < 0 then L has no nontrivial holomorphic sections. For vector bundles V one defines deg(V) = deg(det(V)) = cz(V) where det(V) ~ AmV for rank(V) = m has transition functions det(guw) in an obvious notation. We recall also the Riemann-Roch (RR) theorem which states
dim H ~
V) - dim H i ( E , V) = deg(V) + rank(V)(1 - g)
(2.73)
Many nice results can be obtained from this. Another classical theorem states t h a t any rank rn holomorphic VB V ---, p1 has the form V _~ O ( a l ) O . . . 9 O(am) for ai C Z. One defines direct image sheaves via f,S(U) = S ( f - l ( V ) ) for f : E ~ E and for ,5" = O(L) there results (A) H ~ f,(9(L)) ~_ H~ O(L)) with (B) f,(9(L) = O(E) for E a rank rn holomorphic VB (where m = deg(f) = deg(f*Lp) = # [ f - l ( p ) ] ) and (C) If V is a holomorphic VB on E then f,(9(L @ f ' V ) ~_ (9(E N V) (where E is given in (B)). One obtains then for L and E as indicated deg(E) = deg(L) + (1 - ~) - deg(f)(1 - g) (note here a useful fact in proving theorems is that for V a VB on E H ~ VLp n) = 0 for n sufficiently large or equivalently, on p1, V(n) = VNO(n) will have holomorphic sections for n sufficiently large). Next note t h a t if deg(E) = 0 above then E is trivial if and only if L ( - 1 ) = L | f * ( 9 ( - 1 ) has no nontrivial sections. Concerning Jac(E) one usually defines J g - l ( E ) = Jac(E) and the the image of the holomorphic map (Pl,"" ,pg-1) --~ Lpl ""Lpg_l (tensor product) from E x -.. x E ~ j g - l ( E ) is called the O divisor; it is of codimension 1 in Jac(E). Now going to Lax equations and integrable systems, take a section w E H ~ f*O(n)) where f : E ---, p1 and for U c p1 multiplication by w defines a linear map w : H ~ L) ---, H~ where L(n) = L | f*O(n). One takes here L such t h a t L ( - 1 ) C jg-1/O and f,O(L) = O ( E ) with E trivial (thus deg(E) = 0 and L ( - 1 ) = L | f * ( 9 ( - 1 ) has no nontrivial holomorphic sections). By definition of E this w determines a homomorphism W : H~ E ) ~ H~ E(n)) and thence W:
H~
E) _~ C m ---, H~
E(n)) ~_ C m | H ~
1, O(n))
(2.74)
Thus one has an rn x rn matrix valued holomorphic section of (9(n) so W = A(z) = Ao + Alz + . . . + Anz n. This gives a construction L --~ A(z) and from A(z)si = wsi on sections (via local constructions) one is led to the algebraic curve ( . . ) S : d e t ( w I - A ( z ) ) = w m + al(z)w m-1 + . . . + am(z) = 0 which is an m-fold covering of p l . One can think (rather crudely) of w embedding E as a one dimensional submanifold S of the total space of O(n) (C C 2 locally) and eigenspaces of A(z) correspond to line bundles on S. As for Lax pair equations dA/dt = [A, B] with A(z), B(z) matrix polynomials one sees that the spectrum of A is preserved in such evolutions so the eigenspace bundle moves in a complex t o r t s and if a line bundle follows a linear motion there is a basis for which A(z) evolves as a Lax pair for a specific B depending on A (cf. [467] for details). Finally one comes to completely integrable Hamiltonian systems (CIHS). We recall that symplectic manifolds M 2N = M have a nondegenerate closed 2-form w (and we assume w is holomorphic for complex manifolds M). This provides an isomorphism T M ~ T*M so that for a given function H : M ~ C (Hamiltonian) one has a vector field XH characterized by cz(XH, Y) = Y(H) for vector fields Y. The Poisson bracket is defined via {f, g} = XI(g) = - X g ( I ) and X{Lg } = [XI, Xg]. One speaks of coadjoint orbits of a Lie group G as follows. If ~ E g* then write w~(X, Y) = ~([X, Y]) where X, Y 6 ~ define tangent vectors to the orbit at {. Further for f , g 6 C~(~0 *) one can express df(~) E ~ as a linear form on T ~ * = t)* and write {f, g}(~) = < ~, [df(~), dg(~)] >. One writes also Adg" ~* ~ ~* via < Ad~(~), X > = <
5.2. I S O M O N O D R O M Y P R O B L E M S
285
~ , A d g - l X > (and similarly < ad*x~,Y > = - < ~,[X,Y] >). The m o m e n t u m map is an equivariant map # 9 M ~ ~* and as an example of all this let G = S L ( m , C) and use T r ( A B ) to identify ~ and ~*. Take a product of coadjoint orbits (9 x ... x Ok of elements of "~ ~* with distinct eigenvalues. The moment map is ( R 1 , . . . , Rk) ~ ~ Ri with symplectic quotient M = { ( R i ) / ~ Ri = 0} which has dimension (8) 2N = k ( m 2 - m) - 2(m 2 - 1). Let k p~ k A(z) = p(z) ~ z - a-------~;p(z) = I I ( z - hi); (2.75) 1
1
det(w - A(z)) = w m + a2(z)w m-2 + . . . + am(z) = p(z, w) The eigenvalues )~k of A(ak) = Rk are fixed in advance and p(aj, w) - det(w - Rj) - 0 for w - Aj so the polynomial coefficients hi(Z) are not arbitrary. It is shown in [467] how Lax pair equations arise in this example upon fixing the ai subject to constraints (cf. also [?]). In any event one defines a CIHS to be a symplectic manifold M 2N with N Hamiltonian functions Hi such that {Hi, Hi} - 0 and AgdHi ~ O. The bigger picture developed in particular in [256, 258, 259, 463, 464, 465, 466, 467, 642, 643, 644, 685, 782, 783, 874] can be illustrated now by considering R(z) = ~ [ R j d z / ( z - aj)] with Rj the residues of a matrix valued differential on p1. The fact ~ Ri = 0 corresponds to the vanishing of the moment map for symplectic reduction. Now replace R(z) by a 1-form on an arbitrary compact RS with values in E n d ( V ) for an arbitrary holomorphic VB V. The numerical miracle now involved consists in obtaining in suitable circumstances precisely the right number N of Hamiltonian functions. Thus e.g. in the example above one requires the number of independent coefficients ai in (2.75) to be N (cf. (&)). In order to obtain a suitable structure in general one considers the space T~ of equivalence classes of stable holomorphic rank m vector bundles V over Eg. Stable has various definitions (which we omit) and the facts used here are only that the only global endomorphisms of V should be scalars and that 7~ is a complex manifold with dim(Tr = 1 - m 2 ( 1 - g). Then T*7~ is a symplectic manifold and since TT~ at IV] is H I ( E , E n d ( V ) ) Serre duality gives T*T~ ~ H~ E n d ( V ) | K). Thus locally a point in T*Tr is (up to equivalence) a VB V and a holomorphic section A of E n d ( V ) Q K. Therefore A is an m • m marix with values in K and its characteristic polynomial is p(z, w) = d e t ( w - A ( z ) ) = w m + a l W m - l + . . "+am with holomorphic coefficients ai E H ~ Ki). One can show that d i m H ~ = g and d i m H l ( E , K ) = 1 so by RR deg(K) = 2 g - 2 (g > 1) and by Serre duality again H I ( E , Ki) * ~ H ~ K l - i ) . But K 1-i has negative degree and no sections for i > 1 so RR for K i implies dim H ~ K i) = ( 2 i - 1 ) ( g - 1). The number of degrees of freedom in choosing the characteristic polynomial p(z, w) is then g + E ~ n ( 2 i - 1 ) ( g - 1) = 1 - m 2 ( 1 - g) and this is exactly the dimension of T~ or (1/2) • dim(T*7~). This will then give rise to a CIHS based on T*7~ and p(z, w) - 0 determines a spectral curve S in the total space of K over E; A is determined by a line bundle on S and the Hamiltonian flows are linear (see below for more on this). We go now to [782] and let Eg be a RS of genus g and F B ( E , G) be the space of fiat vector bundles (VB), V = Va (G = G L ( N , C)), with smooth connections ,4. Flatness means (16A) ~A = d A + (1/2)[A,A] - O. Fixing a complex structure on Eg one has A ~ (A, ft.) with a system of matrix equations (0 + A ) r = 0; (c~ + .2.)r = 0
(2.76)
(linearization of (16A)). One modifies this via a parameter ~ E R (the level) and uses n0 instead of 0 in the first equation of (2.76). Now let # be a Beltrami differential # C
C H A P T E R 5. W H I T H A M T H E O R Y
286
~ ( - 1 , 1 ) ( ~ g ) (cf. [487]). Then in local coordinates (16B) # = #(z, 2)Oz | d2 and one can deform the complex structure on Eg to produce new coordinates
& w=z-e(z,
2.); ~ = 2 ;
#=
1-0e
(2.77)
(note this should be called - # to agree with Ow/Ow but we retain the notation of [782]). Then 0~ = 8 + p0 annihilates dw while 8 annihilates dz (note ((9 + p O ) ( z - e) = -Oe + p #0e = 0). In the new coordinates (16A) has the form (16C) $-.4 = ((5 + Op)A - ~Ofii + [fi~, A] = 0 and one arrives at the system (n0 + A)r = 0; (0 + #0 + A)~b = 0
(2.78)
Write now # ~ea= 1 t a p 0 w h e r e p0 .. , p0 is a basis in the tangent space to the moduli space Adg of complex structures on Eg (here g = 3 g - 3 for g > 1). Note that pc9 measures the deviation of 0~ from 0. Fix now a fundamental solution of (2.78) via ~b(z0, 50) = I. Let 7 be a homotopically nontrivial cycle in Ng such that (zo,5o) E 7 and set (16D) Y(7) r163 = P e x p f ~ A (monodromy - P ~ path ordered product as in [54] for example where many basic ideas about connections etc. are spelled out). The set of matrices Y(7) generates a representation of HI (Ng, z0) in GL(N, C) and independence of the monodromy Y to deformation of complex structure means
CgaY = O
a = l, . . . , g; cga =
It follows that (2.79) is consistent with (2.78) if and only if
OaA = 0; 0ft. = l A p ~ (a = 1 , . . . , g ) ts
(2.80)
Further this system (2.80) is Hamiltonian where one endows F B ( E , G) with a symplectic form ( 1 6 E ) w ~ = frog < 5A, hfi. > where < , > ~ Trace; Hamiltonians are defined via (16F) Ha = (1/2)frog < A , A > p0 (a = 1 , . . . , g ) . Consider now the bundle P over A/fg with fiber F B ( E , G); the triple (A, A, t) can be used as a local coordinate and one thinks of P as an extended phase space with a closed 2form (16G) w = co~ (1/~) ~ a 5HaSt. Although w is degenerate on P it produces equations of motion (2.80) since w~ is nondegenerate along the fibers. Gauge transformations in the deformed complex structure have the form
A --, f - l n c g f + f - l A f ;
7t ~ f-l(oq + pO)f + f - l ~ f
(2.81)
The form w is invariant under such gauge transformations (the set of which we call ~) but not w~ or Ha independently. If one writes now (16H) f t . ' = f i , - (1/~)pA and uses (A, A') as a connection then (16I) co = fr.g < hA, hA' >. A gauge fixing via (2.81) plus the flatness condition (16C) is in fact a symplectic reduction from the space of smooth connections to the moduli space of flat connections F B ( E , G) = S M ( E , G)//G where / / m e a n s (16C) plus a gauge fixing are in force. The flatness condition is called the moment constraint equation. Now fix the gauge so that the ft. component of,4 becomes antiholomorphic, i.e. (16J) 0L = 0 v i a / , = f - l ( O + p O ) f + f - l f f t f . This can be achieved since (16J) amounts to the classical equations of motion for the W Z W functional S w z w ( f , fit) with gauge field f in the external field A (cf. [87, 619]). Let L be
5.2. I S O M O N O D R O M Y PROBLEMS
287
the gauge transformed A, i.e. L = f - l a O f + f - l A f , and then (16C) takes the form (16K) (/)+ #O)L + [L, L] = 0. Thus the moduli space of fiat connections FB(E, G) is characterized by the set of solutions of (16K) along with (16J) and this space has dimension (16L) dim(FB) = 2(N 2 - 1 ) ( g - 1) (for g > 1). After gauge fixing the bundle 7) over Aag becomes 75 with F'-'B as fibers and the equations (2.78)-(2.79) become (n0 + L ) r = 0; (c~ + #0 + L)r = 0; (t~Oa-t- Ma)~) - 0
(2.82)
where we have replaced r by f - 1 r and Ma = aOaf f-1. Note that a is not involved in the W Z W result and thus a seems to be arbitrary in (2.82). The gauge transformations do not spoil the consistency of this system and from (2.82) one arrives at the Lax form of the isomonodromy deformation equations
OaL - aOMa + [Ma, L] = 0;
(2.83)
aOaL - #~ = (0 + #O)Ma - [Me, L] These equations play the role of (2.80) and the last equation in (2.83) allows one to find Ma in terms of the dynamical variables (L, L). The symplectic form w on 75 is 1
w=
< 5L, 5L > - - ~ g
I';
SHaSta; Ha = -~ a
< 5L, 5L > #o
(2.84)
g
and we introduce local coordinates (v,u) in F B via ( 1 6 M ) ( L , L ) = (L,L)(v, u,t) with v = (Vl,..., VM) and u = ( u l , " . , UM) for M - ( g 2 - 1 ) ( g - 1). Assume for simplicity now that this leads to the canonical form on F B
~o = f_ < 5L(v, u, t), 5L(v, u, t) > = (hv, 5u)
(2.s5)
J2_5 g
where ( , ) is induced by the trace. On the extended phase space one has then
w = (hv, 5u) - 1 E 5Ka(v, u, t)hta
(2.86)
a
and variations in the Ka take the form
(2.87) J2.~ g
Now because of (16K) the Hamiltonians depend explicitly on times and one considers the Poincar@-Cartan integral invariant (cf. [42])
0 = 5-1w = (v, h u ) _ _1 ~ Ka(v, u,t)hta t~
(2.88)
a
Then there exist 3 g - 3 = dim(My) vector fields (16N))2a = ~Oa + {Ha," } (a = 1,..., ~) that annihilate O and one can check that (Ka ~ Ha here and below)
~OsHr - ~OrHs + {Hs, Hr}wo - 0
(2.89)
Note from [642] that ~ ~ E(AaO~ + B~Ou~) + Oa and Pa E ker(O) means (Oa ~ i)/Ota)
OHa
Aa + ~ = 0 ;
OHa
,B a+~-0;
(2.90)
C H A P T E R 5. W H I T H A M T H E O R Y
288
OHb _ B ~ OHb _ ObHa + Oa Hb = 0
so that ( 1 6 N N ) ~2a = -(OHa/OUi)Ovi + (OHa/OVi)Oui + Oa Thereby they define the flat connection in P and these conditions are sometimes referred to as the Whitham hierarchy. Thus for E E 3dg the Whitham equations determine a flat connection in F B ( E , G). This gives an entirely new perspective for the idea of Whitham equations; it is based on geometry and deformation theoretic ideas (no averaging). For a given f(v, u, t) on P the corresponding equations take the form
df (v, u, t) Of (v, u, t) =n +{Hs, f} dts Ots
(2.91)
called the hierarchy of isomonodromic deformations (HID). Both hierarchies can be derived Fu,t~ s s from variations of a prepotential F on 75 where ( 1 6 0 ) F(u, t) = F(u0, to) + E Juo,to with s u,t) = (v, O s u ) - Ks(v, u,t) the Lagrangian (Osu = 5 K s / S V - note F ~ S is better notation). F then satisfies the Hamilton-Jacobi (HJ) equation
~ Os f + H s \ -g-~u u , t
)
= 0
(2.92)
and F ~ log(T) where 7- corresponds to a tau function of HID (the notation is clumsy however). Note that equations such as (2.92) arise in [658] with period integrals ai as moduli (cf. also Section 3 - recall that the Toda type Seiberg-Witten curves for massless S U ( N ) have N - 1 = g moduli with a 1 - 1 map to the ai). Singular curves are important for low genera situations but we omit this here (cf. [782]). Consider next the moduli space 7~ = ~g,N of stable holomorphic GL(N, C) vector bundles V over E = E 9 (cf. [466, 554, 807] for stable and semistable VB and note that in [642, 644, 782] one develops the theory for E ~ Eg,~ with n marked points but this will be omitted here. ~ is a smooth variety of dimension 9 = N 2 ( g - 1) + 1. Let T*T~ be the cotangent bundle to 7~ with the standard symplectic form. Then Hitchin defined a completely integrable system on T*7~ (cf. [463] - strictly speaking the Hitchin system lives on the moduli space of semi stable Higgs pairs). The space T*7~ can be obtained by a symplectic reduction from T*T~Sg,N = {(r where fi~ is a smooth connection of the stable bundle corresponding to 0 + A and r is a Higgs field r E f~~ End(V) | K) with K the canonical bundle of E (K ~ holomorphic cotangent bundle - cf. [147]). One has a symplectic form (16P) w ~ = fr~9 < 5r 6!i > which is invariant under the gauge group 6 = C ~ ( E , GL(N, C)) where ( 1 6 Q ) r ~ f - l C f and .2, ~ f - l O f + f - l A f with 7~g,N -- ~Sg,N/G. Let now PskOkz-~ | d2 be ( - k + 1, 1) differentials (Psk E H ~ ( E , F k-~ | K) where F k-1 ~ k - 1 times differentiable sections, and s enumerates the basis in H i ( E , F k-1 | K) so p~,2 ~ ps). By Riemann-Roch dim(HI(E, F k-1 | K)) = ( 2 k - 1 ) ( g - 1) and one can define gauge invariant Hamiltonians
Hs,k = -~ 1 fr~ < Ck > Ps,k (k = 1 , - . . , N ; s = 1 , . . - , ( 2 k - 1 ) ( g - 1)) where the Hamiltonian equations are
(
Oa(9-- 0 0 a
0
= ~ a ; a = (s, k)
) ; OaA = c k - l fls,k
(2.93)
(2.94)
The gauge action produces a moment map #" T * ~ s ~ g-l*(N, C) and from ( 1 6 P ) - ( 1 6 Q ) one has (16R) # = / ) r [.4, r The reduced phase space is the cotangent bundle T*~g,N
5.2. I S O M O N O D R O M Y P R O B L E M S
289
T*T~S//G = #-1(0)/{~ (gauge fixing and flatness) and finally the Hitchin hierarchy (HH) is the set of Hamiltonians (2.93) on T*T~g,N. Note also that the number of Hs,k is Y~lN(2k1 ) ( g - 1) = N 2 ( g - 1) + 1 = ~ = dim(T*7~g,g). Since they are independent and Poisson commuting (HH) is a set of completely integrable Hamiltonian systems on T*T~g,N. To obtain equations of motion for (HH) fix the gauge of ft. via fi~ - f ~ f - 1 + f L f - 1 so that L = f - l C f is a solution of the moment constraint equation (16S) 0 n + [L, L] = 0 (flatness condition). The space of solutions of (16S) is isomorphic to H~ End(V)| ~ cotangent space to the moduli space 7~g,y. The gauge term f defines the element Me - Oaf f - 1 E gl(N, C) while the equations ()~ + L ) Y = 0; (Oh + Ma)Y = 0;
(~ +
Z
s,k
(2.95)
+ L)V - 0
are consistent and give the equations of motion for (HH). To prove this one checks that consistency follows from (16S) and in terms of L the equations (2.83) take the form
OaL + [Me, L] - 0 (Lax); O a L - OMa + [Ma, L] - Lk-lps,k
(2.96)
where a = (s, k). The Lax equation provides consistency of the first two equations in (2.95) while the second equation in (2.96) plays the same role for the second and third equations in (2.95); this equation also allows one to determine Ma from L and L. Due to the Liouville theorem the phase flows of (HH) are restricted to the Abelian varieties corresponding to a level set of the Hamiltonians Hs,k = Cs,k. This becomes simple in terms of action-angle coordinates defined so that the angle type coordinates are angular coordinates on the Abelian variety and the Hamiltonians depend only on the actions. To describe this consider
P()~, z) = det(A - L) = )~N _~_bl)~N-1 + . . . + Dj)~N-j _~_. . . ~_ bN
(2.97)
where bj = E M i n j where M i n i ~ principal minors of order j, and bl -- Tr(L) with bN = det(L). The spectral curve C c T*E is defined via (16T) C = {P(A, z) = 0} and this is well defined since the bj are gauge invariant. Since L E H~ End(V)| the coefficients bj E H~ KJ) and one has a map (16U) p" T*T~g,N --+ B = |176 KJ). The space B can be considered as the moduli space of the family of spectral curves parametrized by the Hamiltonians Hs,k; the fibers of p are Lagrangian subvarieties of T*T~g,N and the spectral curve C is the N-fold covering of the base curve E" ~ 9 C ~ E. One can say that the genus of C is the dimension .~ of ~ g , N (recall ~ = N 2 ( g - 1) + 1). There is a line b u n d l e / : with an eigenspace of L(z) corresponding to the eigenvalue )~ as a fiber over a generic point (A, z); thus (16X) s C ker()~ + L) C zr*(Y). It defines a point of the Jacobian Jac(C), the Liouville variety of dimension [I = g(C). Conversely if z E Eg is not a branch point one can reconstruct V for a given line bundle on C as (16Y) Vz - |163 Let now a2j where j = 1 , . . . , ~, be the canonical holomorphic differentials on C such that for the cycles o~1,... , O~.~,/~1,''', ~ with a i . a j =/~i 93j = 0 one has ~a~ wj - 6ij. Then the symplectic form ( 1 6 P ) can be written in the form N
w~
< 5L, 5L > = y ~ 5)~jb~j 1
(2.98)
Here ~j are diagonal elements of S L S where S L S -1 = diag()~l,...,)~y). There results (16Z) w~ = fcb)~5~. Since A is a holomorphic 1-form on C it can be decomposed as
CHAPTER 5. WHITHAM THEORY
290
)~ = E011ajwj and consequently (I) w~ = E~ 5aj f wjh{. The action variables can be identified with (II) aj = ~c~ )~" To define the angle variables one puts locally ~c = Olog(~); then if (Pm) is a divisor of r fc~jh~
=
~m f Po~
wjlog(r
The Cj are linear coordinates on Jac(C) and (III) w~ = ~
5r
(2.99)
5ajhOj.
R E M A R K 5.4. For completeness let us give here some comments on the above following [256, 258, 259]. As before take 7~ for the moduli space of stable holomorphic GL(N, C) vector bundles Y over E = Eg with dim(n) = ~ = N 2 ( g - 1 ) + 1 (note rank(V) = N and write d = deg(Y). The cotangent space to 7~ at a point Y is T~T~ - g ~ 1 7 4 K) (note H ~ 1 7 4 K) ,.., H~ E n d ( V ) N g ) ) and Hitchin's theory says T*7~ is an algebraically completely integrable system (ACIS). This means there is a map h 9 T'74 ~ B, for a .0 dimensional vector space B, that is Lagrangian with respect to the natural symplectic structure on T*7~ (i.e. the tangent space to a general fiber h - l ( a ) for a E B is a maximal isotropic subspace relative to the symplectic form). Then by contraction with the symplectic form one obtains a trivialization of the tangent bundle Th-l(a ) "~ Oh-l(a ) @ T a B . This gives a family of Hamiltonian vector fields on h - l ( a ) , parametrized by T ' B , and the flows generated by these fields on h - l ( a ) all commute. Algebraic complete integrability means in addition that the fibers h-l(a) are Zariski open subsets of Abelian varieties on which the flows are linear (i.e. the vector fields are constant). In a slightly more general framework let K be the canonical bundle as before with total space K = T * E g and think of a K valued Higgs pair (V, r V ~ V | K) where V is a VB on Eg and r is a K valued endomorphism. Imposing a stability condition this leads to moduli spaces 7~K (resp. 74~) parametrizing equivalence classes of semistable bundles (resp. isomorphism classes of stable bundles). Let B = BK be the vector space parametrizing polynomial maps Pa " K ---, K N where Pa = x N - ~ - a l x N - 1 - t - ' " + aN w i t h ai E H ~ | (i.e. B = B K = |176174 The assignment (V, r --. d e t ( x I - r gives a morphism hK" ~ K ~ BK (to the coefficients of det(xI-r Then the map h is the restriction of hK to T ' T / w h i c h is an open subset of T/~ and dim(B) = ~0 (miraculously- see Remark 5.3). The spectral curve E = Ea defined by a E BK is the inverse image in K = T*E of the zero section of K | under Pa " K ~ K N. It is finite over E of degree N and for Ea nonsingular the general fiber of hK is the Abelian variety Jac(E). R E M A R K 5.5. We omit discussion from [463, 464, 465, 466] since the notation becomes complicated and refer also to [47, 256, 272, 258, 259, 412, 482, 554, 685, 807, 874, 938] for more on all of this ([720] is an excellent reference for symplectic matters). REMARK 5.6. There are two other versions of the Hitchin approach related to physics, namely those of [256, 257] and [314, 373]. Both serve as a short cut to some kind of understanding but both are incomplete as to details (cf. [259, 685] for more detail but with a lack of concern regarding connection ideas). Let us look at [314, 373] as the most revealing, especially [3731. Thus let E be a compact RS of genus g and let G be a complex Lie group which can be assumed simple, connected, and simply connected. Let A be the space of ~ valued (0, 1) gauge fields ft. = A2d2 on E and take for T*A the symplectic manifold of pairs (.2., r where r Ozdz = Cdz is a 0 valued (1, 0) Higgs field. The holomorphic symplectic form on T*A is ( X X I V ) fz Tr 5r 5A where Tr stands for the Killing form on suitably normalized. The local gauge transformations h E ~ - Map(E, G) act on T*A via
fI --~ ~h = h~h-1 + hSh-1; r --~ ch = hCh-1
(2.100)
291
5.2. I S O M O N O D R O M Y P R O B L E M S
and preserve the symplectic form. The corresponding moment map # 9T*.A ---, ~* ~ A2(E)| takes the form ( X X V ) #(ft., r = c5r + .2.r + r (note here that Ad2 A r +r A fiid2 [.~, r A dz so one has adjoint action). The symplectic reduction gives the reduced phase space ( X X V I ) P = # - I ( { 0 } ) / G with the symplectic structure induced from that of T*A. As before T' can be identified with the complex cotangent bundle T*Af to the orbit space Af = ~4/G where Af is the moduli space of holomorphic G bundles on E (of course the identification here really should be restricted to gauge fields .2. leading to stable or perhaps semi-stable G bundles but [373] is rather cavalier about such matters and we are delighted to follow suit). Now the Hitchin system will have 7) as its phase space and the Hamiltonians are obtained via ( X X V I I ) hp(fit, r - p(r = p(Oz)(dz) dp where p is a homogeneous Ad invariant polynomial on ~ of degree dp. Since hp is constant on the orbits of {~ it descends to the reduced phase space ( X X V I I I ) hp" P --~ H~ Again K is the canonical bundle of covectors proportional to dz and H ~ dp ) is the finite dimensional vector space of holomorphic dp differentials on E. The components of hp Poisson-commute (they Poisson-commute already as functions on T*A since they depend only on the "momenta" r The point of Hitchin's construction is that by taking a complete system of polynomials p one obtains on 7) a complete system of Hamiltonians in involution. For the matrix groups the values of hp at a point of 7) can be encoded in the spectral curve (: defined by ( X X I X ) d e t ( r ~) - 0 where E K. This spectral curve of eigenvalues ~ is a ramified cover of E; the corresponding eigenspaces of r form a holomorphic line bundle over C belonging to a subspace of Jac(g) on which the hp induce linear flows. For example for the quadratic polynomial P2 = ( 1 / 2 ) T r the map hp~ takes values in the space of holomorphic quadratic differentials H~ This is the space cotangent to the moduli space A/t of complex curves E. Variations of the complex structure of E are described by Beltrami differentials 5# - 5pZOzd2 such that z t - z + 5z with O~bz = 5# z gives new complex coordinates (see below for connections to the notation in (2.77)). The Beltrami differentials 5# may be paired with holomorphic quadratic differentials ~ ~ d z 2 via ( X X X ) (~, 5#) = fE ~ 5#. The differentials 5# = v5(5~) for 5~ a vector field on E describe variations of the complex structure due to diffeomorphisms of E and they pair to zero with/3. The quotient space H I ( K -1) of differentials 5p modulo cw is the tangent space to the moduli space ~4 and H ~ 2) is its dual. The pairing ( X X X ) defines then for each [5#] E H I ( K -1) a gamiltonian ( X X X I ) h ~ - hp25 # and these commute for different 5#. R E M A R K 5.7. In (2.77) one has # -- #(z, 2)Oz | d2 with w - z - e(z, 5), ~ - 5, and # -- 0 e / ( 1 - 0 e ) (which should be # = - 0 e / ( 1 - 0 e ) = Ow/Ow) while from [373] the notation is z' - z + 5z and 5# - 5#ZOzd2 with (~5(5z) - 5# z. Thus 5z ~ - e and one surely must think of 5# z ~ #(z, 2) in which case -che ~ # (adequate for small/)e). Thus for small 0~ the notations of (2.77) ~ [782] and [373] can be compared at least. To compare ideas with the Kodaira-Spencer theory of deformations (cf. [556, 749]) one can extract from [487] (cf. also [280,447, 466]). Thus the space of infinitesimal deformations of E is determined by H i ( E , (9) where (9 is the sheaf of germs of holomorphic vector fields on E. This in turn can be identified with the tangent space T0(T(E)) of the Teichmiiller space T(E) (or Tg) at the base point (E, id.). We further recall that the moduli space A/~g corresponds to T ( E ) / M o d ( E ) where M o d ( E ) is the set of homotopy classes of orientation preserving diffeomorphisms E ~ E (modular group). Recall ~4g is the set of biholomorphic equivalence classes of compact RS of genus g with dimension 3 g - 3. Now, more precisely, going to [487] for notation etc., for a given RS E consider pairs (R, f) with orientation preserving diffeomorphism f " E --~ R. Set (R, f ) - (S,g) if g o f - 1 . R ~ S is homotopic to a biholomorphic map h" R ---, S. Then write [(R, f)] for the equivalence class and the set of such [(R, f)] is T(E). For _
_
_
C H A P T E R 5.
292
WHITHAM THEORY
[(R, f)] E T ( E ) work locally: (U, z) ---. (V, w) with f ( U ) C V, and write F = w o f o z -1 with #f = F2/Fz - OF/OF. This is the Beltrami coefficient and one can write # = c3f/Of in a standard notation. There are transition functions for coordinate changes leading to an expression # / = # ( d 2 / d z ) w h e r e # / is a ( - 1 , 1) form. In terms of the space M ( E ) o f Riemannian metrics on E one has T ( E ) ~ M ( E ) / D i f f o ( E ) and .My ~ M ( E ) / D i f f + ( E ) where D i l l + ,,~ orientation preserving diffeomorphisms of E and D i f f o ,'~ elements in D i f f + homotopic to the identity. Next one writes A2(E) for the space of holomorphic quadratic differentials r = r 2 and r E .A2(E)I if [[(~[[1 -- 2 fr~ [r < 1 (cf. below for the factor of 2). Then T(E) is homeomorphic to .A2(E) (and hence to R6n-6). One notes again the natural pairing ( X X X ) of .A2 with Beltrami forms via (r # f ) = fr~ # r formally (recall for z = x + iy and 2 = x - iy one has dx A dy = (i/2)dz A d5 so strictly 2 f g dxdy - i f g dz A d2). Now it is easy to show that there is an isomorphism 5*"
-H~163176 OH~176176
--~ H I ( E , O )
(2.101)
and O = O ( K -1) with A2(E) = H~ ol'~ (recall K ~ holomorphic cotangent space of E). We note that H ~ 1 7 6 ~ Beltrami differentials { # j ( d 2 j / d z j } while v E H~176176 corresponds to a C c~ vector field v ~ {vj(O/Ozj} so cgv = { # j ( d 2 j / d z j } where pj -- Ovj/O2j. Consequently there is a canonical isomorphism ( X X X I I ) h o (5*) -1 : H i ( E , O) ---. A2(E) where A[#](r = f z # r for # e H ~ 1 6 3 1 7 6 and r e A2(E). This gives the isomorphism H i ( E , O) _~ T0(T(E)) mentioned above. These facts will help clarify some remarks already made above (note H I ( K -1) ~ H i ( E , O) and its dual H ~ 2)
A2).
We return now to [782] again and consider HID in the (scaling) limit ~ ~ 0 (critical value). One can prove that on the critical level HID coincides with the part of HH relating to the quadratic Hamiltonians in (2.93). Note first that in this limit the A connection is transformed into the Higgs field (A ~ r as s ~ 0 - cf. (2.81) and recall L = f - i C f ) and therefore F B ( E , G) --. T*TP~g,g (perhaps modulo questions of stability). But the form w on the extended phase space 7~ appears to be singular (cf. (16G) and (2.84)) and to get around this one rescales the times (IV) t = T + t~tH where t H are the fast (Hitchin) times and T the slow times. Assume that only the fast times are dynamical, which means (V) 5#(t) = s ~ s # ~ H where one writes #s~ = Ons. After this rescaling the forms ( 1 6 G ) and (2.84) become regular. The rescaling procedure means that we blow up a vicinity of the fixed point #s~ in fl/lg,n and the whole dynamics is developed in this vicinity. The fixed point is defined by the complex coordinates (VI) w0 = z - ~ s Tses(Z, 2) with @0 = 2. Now compare the BA function r of HID in (2.82) with the BA function Y of HH in (2.95). Using a WKB approximation we assume (VII) r = g2exp[(S~ S 1] where 9 is a group valued function and S ~ S 1 are diagonal matrices. Put this in the linear system (2.82) and if ( V I I I ) c3S~ = 0 = OS~ H then there are no terms of order ~-1. It follows from the definition of the fixed point in the moduli of complex structures (VI) that a slow time dependent So emerges in the form
S~176
(2.102)
In the quasiclassical limit put (IX) OS ~ = A. so in the zero order approximation we arrive at the linear system of HH (namely (2.82)) defined by the Hamiltonians Hs,k (k = 1,2) and the BA function Y takes the form (X) Y = O e x p [ ~ s t H ( o s ~ The goal now
5.3.
293
WHITHAM AND SEIBERG-WITTEN
is the inverse problem, i.e. to construct the dependence on the slow times T starting from solutions of HH. Since T is a vector in the tangent space to the moduli space of curves A4g it defines a deformation of the spectral curve in the space B (cf. ( 1 6 U ) ) . Solutions Y of the linear system (2.95) take the form Y = Oexp[E~tHat,] where the ~t~ are diagonal matrices. Their entries are primitive functions of meromorphic differentials with singularities matching the corresponding poles of L. Then in accord with (X) we can assume that (XI) cgdS/cgTs = d~ts so the Ts correspond to W h i t h a m times (along with the ts from (2.89), (2.91), and (2.92) - note ts = Ts + ~t H so ts and Ts are comparable in this spirit). These equations define the approximation to the phase of r in the linear problems (2.82) of HID along with (XII) OdS/Oaj = dwj. The differential d S plays the role of the SW differential and an important point here is that only a portion of the spectral moduli, namely those connected with Hs,k for k = 1, 2, are deformed. As a result there is no matching between the action parameters of the spectral curve aj (j = 1 , . . - , t)) in (II) and deformed Hamiltonians (cf. [908] and Section 3). Next the KZB equations (Knizhnik, Zamolodchikov, and Bernard) are the system of differential equations having the form of non-stationary SchrSdinger equations w i t h the times coming from A4g,n (cf. [297, 328, 500]). They arise in the geometric quantization of the moduli of flat bundles F B ( E , G). Thus let V = V1 | | Vn be associated with the marked points and the Hilbert space of the quantum system is a space of sections of the bundle Sv,~quant(Eg,n) with fibers F B ( E , G) depending on a number t~quant; it is the space of conformal blocks of the W Z W theory on Eg,n. The Hitchin systems are the classical limit of the KZB equations on the critical level where classical limit means that one replaces operators by their symbols and generators of finite dimensional representations in the vertex operators acting in the spaces Vj by the corresponding elements of coadjoint orbits. To pass to the classical limit in the KZB equations ( X I I I ) (tcquant~ s -+- H s ) F - 0 (note these are kind of flat connection equations) one replaces the conformal block by its quasi-classical expression ( X I V ) F = e x p ( j r / h ) where h = (tcquant) -1 (where ~; = ~quant/h as in [642]) and consider the classical limit lr quant ---* 0 w h i c h l e a d s t o HID as in (2.92) so ( X I I I ) is a quantum counterpart of the Whitham equations). One assumes that the limiting values involving Casimirs C~ (i - 1 , . . . , r a n k ( G ) ) and a = 1 , . . . , n), corresponding to the irreducible representations defining the vertex operators, remain finite and this allows one to fix the coadjoint orbits at the marked points. In this classical limit ( X I I I ) then is transformed into the HJ equation for the action jr = log(r) of HID, namely (2.92) (note t~quantcgsF ---* t~Osjr etc.). The integral representations of conformal blocks are known for W Z W theories over rational and elliptic curves (cf. [314, 326, 497]) so ( X I V ) then determines the prepotential j r of HID. The KZB operators ( X I I I ) play the role of flat connections in the bundle 79quant over A4g,n with fibers Sv,~q~ant(Eg,n) (cf. [325, 464]) with [l~quan t Os -~- H s , tcquan t Or -~-/-?/r] -" 0
(2.103)
In fact these equations are the quantum counterpart of the W h i t h a m hierarchy (2.89).
5.3
WHITHAM
AND SEIBERG-WITTEN
We mention first that a general abstract theory of Whitham equations has been developed in [269, 472, 473, 474, 600, 601, 602] and this was summarized in part and reviewed in [148, 150, 152]. One deals with algebraic curves Eg,N having punctures at points P~ (~ I,...,N) and a collection of Whitham times TA and corresponding differentials d~A is constructed. The Whitham equations (XXXVIII) OAd~B -- ~Bd~A persist as in (1.38).
294
CHAPTER
5.
WHITHAM
THEORY
Some further remarks on this appear in Remark 5.15 but we do not pursue this point of view. Rather we will follow the approach of [282, 283, 287, 393, 394, 499, 760, 903] (summarized as in [148, 150, 151, 152]). The most revealing and accurate development follows [394, 903] (as recorded in [150, 151]) and we will extract here from [150].
5.3.1
Basic variables and equations
We take a SW situation following [115, 148, 150, 151,256, 257, 287, 393, 394, 472, 473, 499, 602, 698, 760, 855] and recall the SW curves X]g (g - N - 1) for a pure S U ( N ) susy YM theory d e t N x N [ L ( w ) - A] -- O; P ( A ) : A N ( w + 1 ) ; N
N
ukA N - k = I--I(A - Aj); Uk = (--1) k
P ( A ) = AN - ~
(3.1)
2
1
~
Ail""Aik
il<...
Here the Uk are Schur polynomials of hk = (l/k) ~1N Ak via the formula (A) log(A - N P ( A ) ) - Y ' ~ k ( h k / A k ) ; there are g = N - 1 moduli Uk and we refer to [499] for the Lax operator L. Thus u0 = 1, Ul = 0, u2 = h2, u3 = h3, u4 = h4 - (1/2)h 2, etc. (hi = 0 for S U ( N ) ) . One also has the representation y 2 = p 2 ( A ) - 4A2N; y :
(3.2)
A N (w-1)
giving a two fold covering of the punctured Riemann sphere with parameter A. Such Toda chain curves are characterized by a function (B) 2 A g w = ( P + y). Note also from (3.1) (3.2) one obtains 5w ~ P + P ' h A = N P ~ l o g ( A ) + y--~-; ~ P = - ~
AN-k~uk;
p~
:
OP OA
(3.3)
On a given curve (fixed Uk and A) dw
w
P~dA = ~ ;
y
dw AdP dSsw = A-- =
w
OdSsw I _ A y - k dw OUk w=c P~ w
where the dv k are g - N -
y
A g - k d A - dvk; k = 2, 9 9 9, N
(3.4) (3.5)
y
1 holomorphic one forms with ai = i
dSsw;
6r i k - ~
dvk=
OUk
i
and dwi = ( a i k ) - l d v k are the canonical holomorphic differentials with (C) ~Ada~j = Sij; J~B~dwj = B i j . Note in (3.5) it is necessary to assume w is constant when the moduli uk are varied in order to have O d S s w / O u k = dv k holomorphic. Now the periods ai - ~A~ d S s w define the ai as functions of Uk (i.e. hk) and A, or inversely, the uk as functions of ai and A. One proves for example (see (3.39) below) Ouk Olog(A)
Ouk = kuk - h i - Oai
(3.7)
and generally A and T1 can be identified after suitable scaling (cf. [89, 115, 150, 287, 394]). This is in keeping with the idea in [287] that Whitham times are used to restore the homogeneity of the prepotential when it is disturbed by renormalization (cf. also [150]).
5.3. W H I T H A M AND SEIBERG-WITTEN
295
For the prepotential one goes to [499, 760] for example and defines differentials (D) ds _[_ O(1))d~ for n _> 1 w i t h ~Ai df~n = 0 (pick one puncture momentarily). This leads to the generating function of differentials (~-n-1
oo
W(~, ~) = ~ n4n-ld~dfln(~) = O~Or
~)
(3.8)
-ld~ + O(1)
(3.9)
1
where E is the prime form (cf. [319]) and one has
d~d~
~
W(~, ~) ~ (~ _ ~)-----7+ O(1) = ~
d~
n~-f~
n
(cf. here [150, 156] for comparison to the kernel K(#,A) = 1 / ( P ( p ) - P(A)) of Section 2.4). One can also impose a condition of the form (E) Od~/O (moduli) - holomorphic on differentials d~n and use a generating functional (note we distinguish dS and dSsw) oo
g
oo
dS =- E Tnd~n --- E aidwi + E Tnd~n 1
1
(3.10)
0
(we have added a Tod~o term here even though To = 0 in the pure SU(N) t h e o r y - cf. [287, 773]). Note that there is a possible confusion in notation with d~n since we will choose the d~n below to have poles at oc+ and the poles must balance in (3.10). The matter is clarified by noting that (D) determines singularities at ~c - 0 and for our curve = 0 ~ c~+ via ~ = w ml/N which in turn corresponds to P(A) 1/N for A = 1 (see (3.19), (3.28), and remarks before (3.16) - cf. also (3.30) for ds Thus in a certain sense d ~ here must correspond to ds + + d~t~ in a hyperelliptic parametrization (cf. (3.30)) and Tn ~ Tn+ = T~ (after adjustment for the singular coefficient at ~ + ) . The periods ai fA~ dS can be considered as coordinates on the moduli space (note these are not the c~i of [148, 760]). They are not just the same as the ai but are defined as functions of hk and Tn (or alternatively hk can be defined as functions of ai and Tn so that derivatives Ohk/OTn for example are nontrivial. One will consider the variables ai and Tn - -(1/n)Res~=o~ndS(~) as independent so that
OdS OdS = d~i; = d~n (3.11) (Oo~i OTn Next one can introduce the prepotential F(a~, Tn) via an analogue of a D - O.T/Oa~, namely OF Oai
~B i
OF dS; OTn
---
1 Reso~_nd S 2~in
(3 12)
Then one notes the formulas
02F 1 i)Tmi)Tn -- 27ri----~Res~
OdS
1 =. 27ri-----nRes~
1 - 27ci----~Res~
(3.13)
(the choice of ~ is restricted to w • in the situation of (3.2)) and factors like n -1 arise since ~-n-ld~ - -d(~-n/n). Below we use also a slightly different normalization df~n +w• -- (N/n)dw +n/N near oc+ so that residues in (3.12) and (3.13) will be multiplied by N / n instead of 1/n. Note also that in accord with the remarks above Res~=o will correspond to the sum of residues at oc+ involving ~ - w :]:l/N which in turn corresponds to P(A) 1/N. By definition F is a homogeneous function of ai and Tn of degree two, so that
OF OF 02F 02F 02F 2F = ai-~a i + Tn-~-~n - aiaj OaiOaj + 2aiTnoaiOTn + TnTmOTnOTm
(3.14)
296
CHAPTER
5.
WHITHAM
THEORY
Note however that F is not just a quadratic function of ai and Tn; a nontrivial dependence on these variables arises through the dependence of dwi and d~n on the moduli (such as uk or hk) which in turn depend on c~i and In. The dependence is described by a version of Whitham equations, obtained for example by substituting (3.10) into (3.11). Thus OdS = d ~ n + Tm - -
OTn
Oue OTn
(since ~A, df~n = 0). Now (cf.
= d~n :=~
mOTh
= -
~ Oue
d~n
(3.15)
[148, 499]) to achieve (E) one can specify Od~m/Ou~ -
~)-]~/~dwj so ~&(Od~m/OU~) = ~Ai ~ t 3 ~ d w j = /3~m and since d~n - d~tn + ~ c~da;j of necessity there results (F) ~ ( O u e / O T n ) ~Tm~'~m = ~ ( O u e / O T n ) a a = - c n which furnishes the W h i t h a m dynamics for Up in the form Oup/OTn = - ~ cna ~p (cf. [148, 499]- the formulas in [394] and copied in an earlier version of [150] were too rushed).
Now the SW spectral curves (3.2) are related to Toda hierarchies with two punctures. We recall (3.1) - (3.2) and note therefrom that (G) w +1 = ( 1 / 2 A N ) ( P -4-y) ~ ( 1 / A N ) p ( ) ~ ) ( 1 + O ( ) ~ - 2 N ) ) n e a r A = c~+ since from y2 = p2 _ 4A2N we have ( y / P ) = [1 - (4A 2N/P2)]1/2 = (1 + O ( / ~ - 2 N ) ) s o w • • ( P / 2 A N ) [ 1 =k. (y/P)] = ( P / 2 A N ) [ 2 + O(A-2N)]. One writes w(A = oo+) -- ce and w(A = oo_) = 0 with ~c ~ w~=l/N (i.e. ~c = w - U N ,.,., A-1 at oc+ and = w 1/y ~ A-1 at c~_). Near A = =kc~ by ( G ) one can write then w +I/N ~ P(A) 1/N (for A = 1) in calculations involving W :kn/g with n < 2N. The w parametrization is of course not hyperelliptic and we note t h a t (D) applies for ~ = w + l / y with all d ~ n and later only for ~ -- A-1 in certain differentials d~n. It would now be possible to envision differentials d ~ and d ~ but d~n = d ~ + + d ~ n is then clearly the only admissible object (i.e. d ~ n must have poles at both punctures); this is suggested by the form d w / w in (3.4) and the coefficients of Wn/g at co+ and of W - n / g at c~_ must be equal (see also remarks above about d ~ n in (3.10)). This corresponds to the Toda chain situation with the same dependence on plus and minus times. Moreover one takes differentials d~n for (3.2) (A - 1 here for awhile to simplify formulas) d~n = Rn(A) d__w_w_ p ~ / N (A) -dw w
(3.16)
w
These differentials satisfy (E) provided the moduli derivatives are taken at constant w (not A) and we can use the formalism developed above for ~ = w T1/N. Note the poles of d~n balance those of d ~ n = d~n - (Y]~gl(~A3 d ~ n ) d w j and we have
dS = ~ Tnd~n = ~ Tndftn + ~
1
where we keep n < 2 N for technical reasons (of. simply d S s w = d ~ l , i.e. dSIrr~=Sn,1 = d S s w ;
Tn
[9O3]). The
J
OF
- 27tin
:
N
27tin
(3.18)
OliIz~--5~,~ = ai; c~D ITn=Sn,1 = aD
~ p ( A ) U N , with d ~ n as
(Resoo+wn/NdS -t" Res(x) w - n / N d s ) -
-
(3.17)
SW differential d S s w is then
W i t h this preparation we can now write for n < 2N, where w • indicated in (3.16), OTn
df~n rico3
(3 19)
:
"
w --
5.3.
297
WHITHAM AND SEIBERG-WITTEN
N2 - iTrn 2 E TmResc~ (P+/N()i)dpn/N()t)) = m
N2 iTrn2 E TmResc~ ( p n / N ( / k ) d p + / N ( ~ ) ) m
One can introduce Hamiltonians here of great importance in the general theory (cf. [150, 274, 394, 499, 658]) but we only indicate a few relations since our present concerns lie elsewhere. Then evaluating at Tn = 5n,1 (3.19) becomes OF N2 N OTn = -iTrn 2 Resc~pn/N ()~)d)~ = i~n ~]'~n+l
(3.20)
where Tln+l = - - - R e s c c P n/ydA = ~ (--1)k-1 k! n k>l hn+l
k-1
E hi1 "'" hik = il+...+ik=n+l
n ~ h i h j + O ( h 3) 2N i+j=n+l
(3.21)
This can be rephrased as OF _-- fl E mTmT-(m+l,n+l : ~ T l ~ ' ( n + l OTn 2win m 2~in
where
N
~-~m+l,n+l ~- - ~ R e s ~ mn
(p n / N d p y / N ) :
~'~n+l ~ ~'~n+l,2 -'- - N R e s ~ p n / N d A n
+ O(T2, T3, "" ")
--7-~n+ 1 re+l;
(3.22)
(3.23)
= hn+l + O(h 2)
Now for the mixed derivatives one writes 02F fB 1 O(~iOTn = d~n = ~ R e s o ~ - n d w i 2~in
=
(3.24)
=
g (Resc~+wn/Ndwi + R e s ~ w-n/Ndwi) -- . N R e s c ~ p n / N d w i 2~in ~Trn Next set (H) p n / g = ~ _ ~ PnkA N k SO that (I) Resc~P n/g dwi = ~-~n_c c PnkRes~,kkdw~. N Then e.g. )~N - k d~ = y(~) = ~jk p(~) (1 + O =
=
: --0"~ 1 0 log P()~) OUk d)~
(1 + O(/~ -2N) )
(3.25)
From (A) and a ~ 1 = OUk/Oaj one obthins then Ohn d,~ Ohn+ l ds dwj(A) (1 + O()~-2N)) = E 0"~10U k An -- E Oai /~n+l n>2 n>l
(3.26)
so for k < 2N, (J) Res~)~kdwi = Ohk+l/Oai. Further analysis yields (K) R e s ~ w n / N d ~ i = ResccPn/Ydwi = 07-ln+l/Oai leading to 02F _ N_N_Res~P()~)n/gdwi = N 07-~n+ 1 Oo~iOTn irn i~n Oai
(3.27)
CHAPTER 5. W H I T H A M T H E O R Y
298
For the second T derivatives one uses the general formula (3.13) written as
02F 1 N \(Res~+ w n/N df~m + Resr = --Reso~-ndf~m = 27tin OTnOTm 27tin
W-n/N d~m)
(3.28)
while for the second a derivatives one has evidently O2F
f
OaiOaj
= ~b dwj = Bij JS
(3.29)
Note that one can also use differentials d~ defined by (D) with ~ = / ~ - 1 (not ~ recall oc+ ~ (4-, A --, oc) in the hyperelliptic parametrization. This leads to
w
-_ N dpn/N + . . . n
-
wT1/N);
(3.30)
n
N n n kPnNAk-idA +
--
N n
. . . . .n E
1
kPnNkd~
1
Putting (H) and (3.30)into (3.28) gives then
02F cOrmOTn =
N2 m iTrmn ~-~PNt Resc~wn/Nd~e 1
(3.31)
where d~t = d~{ + d~-[. Further analysis in [394] involves theta functions and the Szeg6 kernel (cf. [319, 394]). Thus let E be the even theta characteristic associated with the distinguished separation of ramification points into two equal sets P(A)4- 2A g - 1-IIN(A- r~). This allows one to write the square of the corresponding Szeg6 kernel as 9 ~(A, #) = P(A)P(#) - 4A 2N + y(A)y(#) dad# 2V(X)y(,) ( ~ _ ~)2
(3.32)
We can write (cf. [394])
nAn-ld# n>l
#n+l
(1 nt- o ( r - l ( . ) )
~ ( ~ ) = P 4- ydA = ~ (1 + O(A-2N))dA 2y [ 0()~ -2g dA
;
(3.33)
near oc+ near oc T
and utilize the formula
02
~E(~, ~)~-E(~, ~) = W(~, ~) + dwi(~)dwj(~) OziOzj logOE(O[B)
(3.34)
(cf. [319, 394]). Here one uses (3.26) and (3.8) to get (1 _ n < 2N, ~ ~ 1/p)
dwj(#) = ~
n>l
nd# ( l Ohn+i ) d ~ ( A ) An-i~+(A) Pindwi(A); #n+l n Oaj ; = -
d~n(A) = An-i(~2+(A)+ ~ 2 - ( A ) ) - 2pindwi(A)
(3.35)
5.3. W H I T H A M AND SEIBERG-WITTEN
299
where (L)P/n = (1/n)(Ohn+l/Oaj)O~4l~ calculation 02.~
OTmOT n Next one notes
k
9 From this one can deduce with some
2N 07"tm+i 07-ln+i ) N 7-Lm+l,n+l-k mn Oai 7tin Oaj 021~ OE(O B)
(3.36)
Ouk Olog(A)
(3.37)
) fA OdSsW ai--c
i
OUk
/A OdSsw = 0 ~-
i 01 og (h)
Then there results
Ouk Oai _ _ ~ OdSsw fA P' P dw w _- _ N fA i Pd)~ y -_ O lo9(A) Ouk -- JA O log(A) = - N i k
(8.88)
= - - N /A P + Y d ) ~ = - 2 N A N fA wd)~ ---- - 2 N A N fA~ wdvN Y
~ Y
Here we take dSsw = Adw/w and using (3.3) in the form (hP = 5w - O) 5dSsw/5 log(A) 5)~(dw/w)/5 log)A) = (YP/P')(dw/w). Then (3.4) gives gPd)~/y and the next step involves ~A~ dA = 0. Next ( S ) is used along with (3.5). Note also from (M))~dP - )~[N)~g - 1 ~ ( N - k)uk)~g-k-i]d)~ -- NPd)~ + ~ kuk)~Y-kd)~ one obtains via (3.4), (3.5), and (3.38) (middle term)
Ouk -~Olog(A)
=
Oai f Pd)~ Ouk = g f A ~ Y =
(3.39)
Oai - ai - E k U kOuk --
Z
which evidently implies (3.7). Now from (3.38) there results
Ouk Oai = 2NA N fn wd)~ = N fn P + Y d;~0 log(A) Ouk i Y i Y
Oh:O log eE(61B)
- 2Nfa ~ from which
(3.40)
Ouk
0log(A)
= _2NOUk Ou2 02log OE(6IB) Oai ~aj
(3.41)
Here one can replace Uk by any function of Uk alone such as hk or ~ n + i (note u2 = h2). Note also (cf. [150, 287, 394, 473, 698]) that identifying A and Ti (after appropriate rescaling hk ~ Tikhk and 7~k ---+TlkT~k) one has (/3 = 2N)
OFsw = /3 (T2h2) O log(n) 27ri
(3.42)
(this equation for OF~O log(A) also follows directly from (3.20) - (3.22) when Tn - 0 for n > 2 since 7-/2 = h2). Finally consider (3.3) in the form (N) P'6)~-~k/~N-k6uk for 6hi = 0 (cf. (3.38))
5ai = fA 5)~-dw i
W
=
NP5 log(A). There results
dw + N5 log(A) fA p, P dw ; ~ 5Uk /A "kN-k p--T--k
i
W
i
W
(3.43)
300
C H A P T E R 5. dv k k
Ouk 0 log(A) a=a
~
= -N
p, ~
w
WHITHAM
THEORY
A PdA
:-N
On the other hand for ai = Tlai + O(T2, T 3 , . . . ) f dw 5ai = ai5 log(T1) + T1 ~b 5A - - + o ( r 2 , ra, 9..) JA i w
(3.44)
so for constant A with Tn = 0 for n > 2 (while a~ and Tn are independent) 5c~i - 0 implies k
~
Olog(T~)~=~
=
TI--
~ ~
(3.45)
Since A d P = N P d A + ~ k kuk AN-kdA it follows that (cf. (3.7))
I
OUk = Ouk I -- kuk -- --hi Ouk 0 log(T1) ~=c 0 log(A) a=~ Oai
(3.46)
(cf. (3.7) - note the evaluation points are different and ai = Tlai + O(T2, T 3 , . . . ) ) . This relation is true for any homogeneous algebraic combination of the Uk (e.g. for hk and 7-/k). We will return to the log(A) derivatives later. For further relations involving W h i t h a m theory and A derivatives see [89, 115, 150, 151,287, 394, 472,473, 499] and references there. We summarize in S U M M A R Y 5.8. Given the RS (3.1)- (3.2) one can determine OF/OTn from (3.20), 02F/OaiOTn from (3.27), 02F/OTnOTm from (3.28) or (3.36), and 02F/OaiOc~j from (3.29) (also for Tn = 6n,1, ai = ai as in (3.18) with A = 1). Finally the equation (3.42) for OF/O log(A) corresponds to an identification T1 ~ A and follows from (3.20)- (3.22) when Tn = 0 for n >_ 2. Therefore, since F ~ F s w for T1 - 1 we see that all derivatives of F s w are determined by the RS alone so up to a normalization the prepotential is completely determined by the RS. We emphasize that F s w involves basically Tn - 5n,1 only, with no higher Tn, and ai = hi, whereas F involving ai and Tn is defined for all Tn (cf. here [151, 394]). In fact it is really essential to distinguish between F s w = F s w and general F -- F W = F w = FWhit and this distinction is developed further in [151]. The identification of A and T1 is rather cavalier in [394] and it would be better not to set A = 1 in various calculations (the rescaling idea is deceptive although correct). Thus since F sW arises from F W by setting Tn = 51,n it is impossible to compare AOAF sW with TIO1F W directly. Indeed a statement like equation (3.42) is impossible as such since T1 doesn't appear in F sW. In [151] we took a simple example (elliptic curve) and computed everything explicitly; the correct statement was then shown to be TIO1F w = AOAF W. In addition it turns out in this simple example that the Whitham dynamics lead directly to homogeneity equations for various moduli and the prepotential.
5.3.2
Other points of view
The formulation in Section 5.3.1, based on [394], differs from [287, 499, 760] in certain respects and we want to clarify the connections here (for [472, 473, 600, 601,602] we refer to the original papers and to [150, 152]). Thus first we sketch very briefly some of the development in [760] (cf. also [87, 148]). Toda wave functions with a discrete parameter n lead via Tk = etk, Tk -" etk, To = - e n , and aj -- ieOj (= J~Aj dS) to a quasiclassical (or averaged) situation where (note 2P does not mean complex conjugate) g d S - ~-~ aidwi-t- E Tndf~n § E Tnd~n (3.47) 1 n>0 n>l _
m
5.3.
301
WHITHAM AND SEIBERG-WITTEN
y=~
~J~-j~j+ E
n>0
(3.4s)
T~~n + E ~n~ n>l
where df~n ~ df~+n, dfin "~ d~2n, Tn ~ T - n , and near P+
I
df~+ =
OO
--nzmn-1 -- Z
qmnZm-1
dz (n > 1);
(3.49)
1 6nOZ-1 -- ~
df~ n =
rmnZ m-1
dz (n > O)
1
while near P_ df~+ =
-~n~
Z rmnzm-1 1
df~ n =
_ n z -n-1 _ ~
dz (n > 0);
(3.50)
OO
qmn zm-1 d z ( n > _ 1)
1
Here dfl0 has simple poles at P+ with residues =t=l and is holomorphic elswhere; further df~ + = df~ o = df~0 is stipulated. In addition the Abelian differentials dgt~ for n > 0 are normalized to have zero Aj periods and for the holomorphic differentials d~vj we write at P+ respectively d&j = - Z grJmzm-ldz; dcdj = - ~ ~ j m z m - l d z (3.51) m>l
m>l
where z is a local coordinate at P+. Further for the SW situation where (g - N dS =
AdP Y
AP'dA
= ~ ; Y
1)
N-2
y 2 _ p 2 _ A2N; P ( A ) = A N + Z UN-k/~k 0
(3.52)
(cf. (3.1) where the notation is slightly different) one can write near P+ respectively dS-
(E -
nTnz-n-1 + T~
Z
_
OFzn-1) az;
(3.53)
-~n
n>l
-- _ - ~ n z
n> l
dz
leading to
F=~
~-~i j d S - Z
(3.54)
TnRes+z-ndS-
n>l
\ - y~ Tnnes_z-ndSn>_l
To[nes+log(z)dS- nes_log(z)dS]]
/
(the 2ri is awkward but let's keep i t - note one defines a D - ~B~ dS). In the notation of [760] one can write now (.) h = y + P, h = - y + P, and hh = A 2N with h -1 ~ z g at P+ and z g ~ [z-1 at P_ (evidently h ~ w of Section 2). Note also (**) h + (A 2N/h) - 2 P and 2y - h - (A 2N/h) yielding y2 _ p2 _ A2N and calculations in [760] give (o 9 9 d S =
302
C H A P T E R 5. W H I T H A M T H E O R Y
)~dP/y = )~dy/P = Adh/h (so h ~ w in Section 7.1). Further the holomorphic dw~ can be written as linear combinations of holomorphic differentials (g - N - 1) dvk=
)~k_ l d)~
(k=l,...,g);
y2=p2_l=
y
2g+2 1-I(A-Aa) 1
(3.55)
(cf. (3.5) where the notation differs slightly). Note also from (3.55) that 2ydy = E~ N 1--Iar Aa)dA so dA = 0 corresponds to y = 0. From the theory of [760] (cf. also [148]) one has then W h i t h a m equations Odwj Odwi. cOdwi cOd~'lA cOd~B i:C)d~'lA cOai = Oa--T ' OTA = Oai ; OTA = OTB
(3.56)
along with structural equations OdS = d~i;
Oa~
OdS
OTn
= d~+;
OdS
OTn
= d~ n
OdS
; OTo
= d~o
(3.57)
Finally we note that in [760] one presents a case (cf. also [150]) for identifying N - 2 susy Yang-Mills (SYM) with a coupled system of two topological string models based on the A N - 1 string. This seems to be related to the idea of t t fusion (cf. [275]).
5.4
SOFT SUSY BREAKING
AND WHITHAM
The idea here is to describe briefly some work of Edelstein, Gdmez-Reino, Marifio, and Mas about the promotion of Whitham times to spurion superfields and subsequent soft susy breaking Af = 2 --~ Af = 0. 5.4.1
Remarks
on susy
We begin by sketching some ideas from [55, 56] where a nice discussion of susy and gauge field theory can be found (cf. also [474]). In particular these books are a good source where all of the relevant notation is exhibited in a coherent manner. One says that a quantum field theory (QFT) is renormalizable if it is rendered finite by the renormalization of only the parameters and fields appearing in the bare Lagrangian (for renormalization we refer also e.g. to [55, 56] and the bibliography of [150]). We denote by A the renormalization scale parameter. One notes that renormalization of bare parameters occurs as a quantum effect of interaction and the shifts thus generated are infinite, which means that the bare parameters were also infinite, in order to produce a finite measured value. Further in order to implement gauge invariance in weak interactions for example one must find a method of generating gauge vector boson masses without destroying renormalizability. Any such mass term breaks the gauge symmetry and the only known way of doing this in a renormalizable manner is called spontaneous symmetry breaking. This arises when e.g. when there are nonzero ground states or vacua which are not invariant under the same symmetries as the Lagrangian or Hamiltonian. Once such a vacuum is chosen (perhaps spontaneously by the system "settling down") the symmetry is broken. One can then define new fields centered around the vacuum which have zero vev (vacuum expectation value) and the Lagrangian expressed in the new fields will no longer have the same symmetry as before. Such new fields (with nonzero vev) have to be scalar (not vector or spinor) and are called Higgs fields; this kind of spontaneous symmetry breaking is nonperturbative (the vevs are zero in all orders of perturbation theory). In the case of continuous global symmetry in the Lagrangian there
5.4. SOFT S U S Y B R E A K I N G AND W H I T H A M
303
can be a subspace of degenerate ground states and massless modes called Goldstone bosons arise. In any event the idea now is to break a local gauge invariance spontaneously in the hope that the break will induce gauge boson masses while the (now hidden) symmetry will protect renormalizability. This is referred to as the Higgs mechanism and what happens is e.g. that the Goldstone bosons are "eaten" by the gauge transformed massive boson field and a scalar Higgs field remains (we recall that massive terms are quadratic in the Lagrangian). In the case of nonabelian gauge theories one includes Yukawa couplings of fermions to the scalar fields in order to have fermion masses emerge under spontaneous symmetry breaking. Further magnetic monopoles may arise in spontaneously broken nonabelian gauge theories and instantons arise in general (which are classical gauge configurations not necessarily related to spontaneously broken symmetry); we omit any discussion of these here. Now, turning to susy (following [56]) one introduces a spinor geometry to supplement the bosonic generators of the Poincar(~ group. This leads to a natural description of fermions and in a susy theory the vanishing of the vacuum energy is a necessary and sufficient condition for the existence of a unique vacuum. Further, every representation has an equal number Of equal mass bosonic and fermionic states. Generally the nonzero masses of observed particles are generated by susy breaking effects so one looks first at representations (and their T C P conjugate representations) of the N = 1 susy algebra that can be realized by massless "one" particle states. This leads to supermultiplets for N = 1 involving (~ ~,, helicity) (A) chiral: quarks, leptons, Higgsinos ()~ = 1/2) with squarks, sleptons, Higgs particles for )~ = 0 (scalar particles) (B) vector: gauge bosons ()~ = 1) with gauginos (~ = 1/2) (C) gravity: graviton (~ = 2) with gravitino ()~ = 3/2) along with TCP conjugate representations (A)' ~ - - 1 / 2 a n d ) ~ = 0 (B') ~ = - 1 andS=-1/2 (C') ~ = - 2 and ~ = - 3 / 2 . For N = 2 s u s y o n e has supermultiplets involving 4-D real representations of U(2) (in the absence of central charges) (D) vector: )~ = 1, double ~ = 1/2, and ~ = 0 (E) hypermultiplet" )~ - 1/2, double )~ = 0, and )~ = - 1 / 2 (TCP self-conjugate) (F) gravity: ~ = 2, double )~ = 3/2, and = 1 along with T C P conjugations for (D) and (F). One goes then to superfields S(x, 8, ~) with Grassman variables 8, ~ (for which a nice discussion is given in [56]). There will be expansions i
-
i ~#Oa~O~
1
_
10~0~r
for chiral superfields where ~b ,,~ left handed Weyl spinor, r F ,,, complex scalar fields, and a ~ = /2 with a i -,- Pauli matrices (i = 1, 2, 3). It is useful to note also that 5F is a total divergence under susy transformations. Similarly vector superfields can be written _
_
- = C(x) + i8x(x) - iO2(x) + yi88 [ M ( x ) + i N ( x ) ] - --~ i80 [M(x) - iN(x)] + (4.2) Y(x, 8, 8) i 1 [D +e(.~'~V,(x)+iee~[~(x)+ ~i ~o~.x(x)]-i#Oo[~(x)+ ~,o,~(x)] +~0000 ~ - ~0,1o,c] and ~D will be a total divergence (along with ~(O~O'C)). Here X, A are Weyl spinor fields, V~ is a real vector field, and C, M, N, D are real scalar fields. Generally one refers to the coefficient of 88 in product expansions (I)i(I)j o r (~i(~j(~ k for example as an F term and the coefficient of 8888 as a D term. Then susy Lagrangians::involving chiral superfields will have
CHAPTER 5. W H I T H A M THEORY
304 the form
~:
E[e~e~i]D'~-([W(e~)]F-'~ - H C ) ~
f d40~-~.(~(~i"~- (f
d20W(g2)-k - H C )
(4.3)
(HC ~ Hermitian conjugate) where W is called a superpotential and involves powers of ~i only up to order three for renormalizability. The latter equations arises since the superspace integration projects out D and F terms (recall f dO0 = 1, f dO = O, f d2000 = 1, (d/dO)f(O) = f dOf(O), etc.). Note that, apart from a possible tadpole term linear in the (I)i, one will have W(O) 1 -~mijd2id2j q- -~1 )~ijk d2idPjd2k (4.4) =
There is then a theorem which states that the superpotential for N - 1 susy is not tenormalizable, except by finite amounts, in any order of perturbation theory, other than by wave function renormalization. Regarding susy breaking one must evidently have this since we do not see scalar particles accompanied by their susy associated fermions. To recognize when susy is spontaneously broken we need a vacuum 10 > which is not invariant under susy (or alternatively 10 > should not be annihilated by all the susy generators). As a consequence whenever a susy vacuum exists as a local minimum of the effective potential it is also a global minimum. For the global minimum of the effective potential (physical vacuum) to be non susy it is therefore necessary for the effective potential to possess no susy minimum. In theories of chiral superfields one needs < OIFilO > # 0 for spontaneous susy breaking where the tree level effective potential is V = F~Fi = IFil2 and F/t = - 0 W ( r 1 6 2 (here Fi is an F term arising in (2.3)). Once spontaneous susy breaking occurs a massless Goldstone fermion appears which for Fi will be the spinor r in the supermultiplet to which Fi belongs. When global susy becomes local susy in supergravity (sugra) theories the Goldstone fermion is eaten by the gravitino to give the gravitino a mass. For theories involving vector superfields there is also another possibility when there is a D term with < 01D(x)]0 > = ~b =/=0. Finally one notes that the renormalized coupling constants necessarily depend on the mass scale A but the physics described by the bare Lagrangian is independent of A so the coupling constants must "run" with A. The RG equations specify how these coupling constants vary. Now to couple with sugra one recalls first the Noether procedure for deriving an action with a local symmetry from an action with a global symmetry. For example given So : i f d4x(~'yuOur invariant under the global symmetry ~ ~ exp(-ic)~2 one lets ~ : ~(x) and considers ( & ) r ---, exp(-ie(x))r Then 6S0 : f d4x~'yur : f d4xjt`Ot`~ where Ju = ~'Yt`r is the Noether current. To restore invariance a gauge field At` is introduced transforming under (&) as (tb) A t, ~ A u + Ot`c and a coupling term is added to So to obtain S = So - f d4xjUAu = f d4xi~'Tu(Ou + iAu)~b. This S is invariant under (&) and (tb). One can use this technique to construct a locally susy action from the global susy action for the sugra multiplet. Next one extends the pure sugra Lagrangian of the graviton and gravitino to include couplings with matter fields. Recall that whereas in global susy theories susy breaking manifests itself in the appearance of a massless Goldstone fermion, in locally susy theories the corresponding effect is the appearance of a mass for the gravitino which is the gauge particle of local susy. The most general global susy Lagrangian for chiral superfields is
f-,Glob : ] d4OK(O t, O) + f d20(W(O) + HC)
(4.5)
where K is a general function (since nonrenormalizable kinetic terms cannot be excluded in the presence of gravity). Similarly the superpotential may contain arbitrary powers of the Oi. The sugra Lagrangian turns out to depend only on a single function of r and r
5.4. SOFT SUSY BREAKING AND WHITHAM
305
namely (-) G(r r = J(r r + log ]W[2; J = -31og(-K/3), where G (or J) is called the K~hler potential. Note G is invariant under ( . . ) J ~ J + h(r + h*(r W ---, exp(-h)W. The sugra Lagrangian /: may be written as s = EB + s + /:F where EB contains only bosonic fields, ~FK contains fermionic fields and covariant derivatives (supplying the fermionic kinetic energy terms), and /:g has fermionic fields but no covariant derivatives (see [56] for details). Now regarding spontaneous susy breaking, for theories with local susy breaking the vacuum energy is no longer positive semidefinite. There are in particular the following possibilities. First at least one of the fields in the theory must have a vev not invariant under susy. Under certain assumptions one has then an F term generalization involving OW/Or + r # 0 or a D term generalization involving G i ( T a ) i j C j 7s 0 where G i - (r -[- (1/W)OW/Or (here ( T a ) i j ~ generators of the gauge group in the appropriate representation). There are also other possibilities (cf. [56]). For string theory both IIA and IIB are unsuitable to describe the real world but the heterotic string is perhaps tenable, with the extra 16 left mover dimensions providing the gauge group for the resulting 10-D theory (upon compactifying on a 16-D torus or variations of this). Further compactification of 6 dimensions is then still necessary leading at first to an N = 4 susy theory if toroidal compactification is used (which is unsuitable since N >_ 2 susy models are nonchiral). However orbifold compactifications will yield a 4-D theory having N = 1 susy and then some symmetry breaking must be induced. One can also compactify on Calabi-Yau (CY) manifolds and modular invariance can be achieved. We leave [55, 56] in what follows in order to concentrate on material more directly related to the projects at hand. In [667] one takes now an N = 1 Yang-Mills (YM) action
l ~ [Tf d4xf d2OTrW~W~l_ 327r OyM2 f d4xTrF m n F m n 7 t-
8---~
(4.6)
+ ~ f d4xTr [-4F,~nF'~n-i)~a'~V,~X + 2D2 ] where (Jbtb) T =
(OyM/27r)+ (47ri/g2). Here the
Wa = -iAa(y) +
i
OaD(y) - Z-~(am~no)a(OmVn-- OnVm)(y) + (O0)am~OmA~(y)
W ~ = -iAa(y) + where
Wa are chiral spinor superfields of the form (4.7)
O~D(y)+ Of~a~naFmn(y)- (O0)~mZ~VmA~(y)
~mn = (1/2)cmnpqFpq (dual
field strength) and
Finn = OmVn - OnVm + i[vm, Vn]; Vm ~ - Om~ -~- i[vm, ~ ]
(4.8)
We omit the background considerations involving the Wess-Zumino (WZ) gauge etc. (cf. also [56]). For N = 2 susy one thinks of 0 and 0 with Da --~ D a , / ) a and f d20 ~ f d20d20, etc. Then an N = 2 chiral superfield is an N = 2 scalar superfield which is a singlet under global SU(2) and satisfies -
D ~ ( x , O, O, O,
-
=0; D~=0
(4.9)
where e.g.
D~ = O~ + 2iam~O~Om; D~ = -0~
(4.10)
CHAPTER 5. WHITHAM THEORY
306
Set also (&&&) ~m = x TM + iOo.m~ + iOo.m~. Then expanding an N = 2 chiral superfield in powers of 0 the components are N = 1 chiral superfields. Thus
= e(~. 0) + ~ v ~ " w . ( ~ . 0) + ~ga(~. 0)
(4.11)
(Wa is an N = 1 chiral spinor superfield). For N = 2 YM, if one forgets about renormalizability, there arises ~ : -1 ~
~ [] d4x f d:Od2~Tr.T(9)]
(4. 12)
with .T (1/2)T~ 2 and constraints on 9 of the form ( ~ ~ ) (DaC~Db)q2 = (DaDbC~)~t where a,b are SU(2) indices. Writing .Ta(0) = O.T/OOa and .Tab = c92.T/OOaOCDb the Lagrangian (4.12) can be written in terms of N = 1 superfields as =
12- - 1- ~1 2
/ d20.Tab(~) waaw. b + /d40(d2te2V)a.T a (d2)]
(4.13)
where V will be clarified below. 5.4.2
Soft s u s y b r e a k i n g a n d s p u r i o n fields
We go now to [282, 283, 284, 684] and one works upon the foundation of [394] sketched in Section 5.3.1. The idea of soft susy breaking goes back to [388] for example and was developed in a form relevant here in [23, 24, 25, 683]. From [282, 283, 684] one extracts the following philosophical comments: Softly broken susy models offer the best phenomenological candidates to solve the hierarchy problem in grand unified theories. The spurion formalism of [388] provides a tool to generate soft susy breaking in a neat and controlled manner (i.e. no uncontrolled divergences arise). To illustrate the method, start from a susy Lagrangian L((I)0, (I)l,...) with some set of chiral superfields, and single out a particular one, say (I)0. If you let this superfield acquire a constant vev along a given direction in superspace, such as e.g. < (I)0 > = co + 02F0, it will induce soft susy breaking terms and a vacuum energy of order IF012. Turning the argument around, you could promote any parameter in your Lagrangian to a chiral superfield, and then freeze it along a susy breaking direction in superspace giving a vev to its highest component (the Fn terms below). In the embedding of the SW solution within the Toda-Whitham framework we have obtained an analytic dependence of the prepotential on some new parameters Tn (the Whitham or slow times). Then these slow times can be interpreted as parameters of a non-supersymmetric family of theories by promoting them to be spurion superfields. In [23, 24, 25, 683] this program was initiated with the scale parameter A and the masses of additional hypermultiplets mi as the the only sources for spurions. To deal with the times Tn now one writes
Tn - TnTln; uk = Tkl uk; oLi(uk,Tn) = Tlai(uk,A = 1 ) + O(Tn>l);
(4.14)
a~ = ~ ( ~ k , T 1 , 2 n > l = 0) = Tla~(~k, A = 1) = a~(ek, A = T1)
Then the Whitham times Tn (or Tn) are promoted to spurion superfields via
sl = -ilog(A); Sn = -iTn; S1 = Sl + 02F1; V1 =
(4.15)
1 2-2 D10202; Sn = 8n + 02Fn; Yn = -~DnO
h = ~xp(i~l); ~1 = y~7 ;
~ = ~0 + 74~i ;
A2N ~ exp(27riv)
(4.16)
5.4. S O F T S U S Y B R E A K I N G A N D W H I T H A M
307
(note the 0 in T has a different meaning from the Grassman 0 in the superfields). One also
writes ~-~m+l,n+l = Tm 1 +n az t,.m+l,n+l and in the manifold Tn>l = 0 the Tn are dual to the 72/n+1 via _
03 c
0log(a)
-
= 7tin
N ~2;
(these are special cases of more general formulas below and in Section 3.1 - cf. (3.20)). Further the ~ n + l are homogeneous combinations of the Casimir operators of the group; this means that one can parametrize soft susy breaking terms induced by all the Casimirs of the group and not just the quadratic one (associated to A). In this way one extends to Af = 0 the family of Af = 1 susy breaking terms first considered in [41]. Note the uk in the SW curve (3.1) can be written as uk =< Ok > for Ok -- ( 1 / k ) T r Ck+ lower order terms (basic observables for a complex scalar field r as in [684] for example- see Section 5.6); the 7-lm,n are certain homogeneous polynomials in the Uk and the (Casimir) moduli hk of (3.1) refer basically to a background Toda dynamics (cf. [499]). Thus there are at least two points of view regarding the role of the Tn" (1) The philosophy of (4.15) implies that Tn or Tn correspond to coupling constants while (2) The philosophy of looking at moduli dynamics of Uk or hn depending on Tn puts them in the role of deformation parameters. We should probably always treat Tn and aj in parallel, either as coupling constants or deformation parameters (this does not preclude treating aj = aj(Tn) however). We indicate now some formulas arising from [282, 394] which are discussed further in [903] in connection with [658, 682, 743]. Thus, in the notation of [282, 283], one defines Hamiltonians 7:lm+l,n+l = T~n+nT-lm+l,n+l with 7-/m+l = 7-/m+1,9 (homogeneous polynomials in the s (or hk) via
7-lm+l,n+l---Resc~ mn
(Pm/YdP+/Y ) =7-tn+l,m+l;
7-ln+l----Resc~ n
(Pm/gd)t )
(4.17)
and setting s D = OF/Osn write
02F
02F
02F
; j 7-?--O0~i08n; T ran= 08mOSn Tij = ~OOliOL
(4.18)
There results
~ I~2q-i E m S m ~ ' ~ m + l - - E mSmSn~-~mWl,n+l] ; m>2 m,n>_2 fl ~-~n+l "4- i E SnD__ -- 27rn
m>2
T/n -
m Sm ~m +l 'n+l ; T 1 = 2 - ~
fl 0~n+X T11 27rn Oa----~; =
Tln:--2T:T;OT, jlogOE(OIT); T m n :
(4.19)
-~a i q- i Z Sn Oa i n>_2
_2~.l~.lG,jlogOE(O{E); fl ~'~m+l,n+l- 2T?T?OT, jlOgOE(OIT)
27ri
Here O E designates
OE(OIT ) -- O[(:~, fi](tVlT ) ----E exp[iTrTijninj + itVini - i~ ~ n i] where Vi - O u 2 / O a i and ~ = ( 0 , . . . , 0) with f i - ( 1 / 2 , . . . ,
1/2).
(4.20)
CHAPTER 5. WHITHAM THEORY
308
ATTENTION. Henceforth we assume all ~-~p, ak, ui, etc. have hats but we remove them for notational convenience. Now the spurion superfield S1 appears in the classical prepotential as (D) ~ = (N/Tr)S17-[2 and one obtains the microscopic Lagrangian by turning on the scalar and auxiliary components of $1. Next the remaining Sn are included and one expands the prepotential around s2 . . . . . SN-1 = 0; the Dn and Fn will be the soft susy breaking parameters (more on this below). The microscopic Lagrangian is then determined by
= _N 7r
(with
~1 1
_1Sn~~n+l-at. n
N
E SnSmT-~n+l,m+l m,n>2
~
(4.21)
sn - 0) and one is primarily interested in 7 e d = N ~ 1 !$n7_l.+ 1 n
1
(4.22)
n
which in fact is the relevant prepotential for Donaldson-Witten (DW) theory. Note 0 2 7 ed OSmOSn
_-82--...---8N_1"- 0
2_N2 mmn
10TijlogOE(OiT ) cOaj iTr
07"~m+l O~'~n+l
Oai
(4.23)
are essentially the contact terms of [658] (cf. Section 5.5.2). Expanding (4.22) in superspace one has a microscopic Lagrangian and this gives an exact effective potential at leading order for the Af = 0 theory, allowing one to determine the vacuum structure. Detailed calculations for SU(3) theory are given in [282, 283]. So are we dealing with coupling constants Tn or deformation parameters? One answer is "both" and we refer to Section 5.5 for more on this. The spurion variables parametrize deformations of the SW differential and the Tmn and T/n of (4.19) have nice transformation properties under S p ( 2 ( N - 1),Z); the Sn behave like the aj in many ways. Promotion of Tn ---* Sn for susy breaking should however correspond to a coupling constant role; the Tn are parameters of a non-susy family of theories; thus role of Sn is parallel to c~j. The prepotential determines the Lagrangian via S'A, ~AB, 2FABC, the Dn and Fn, A, r gluinos, r = scalar component of Af = 2 superfield, etc. Thus Z: = Lkin -t- Lint with Lkin
1
= ~--~-~ [(Vttr
a+
i(V#~))ta~tt~)bJc~ -
(4.24)
- i T a b ~ " ( V . ~ ) b - ~1a b , -(F . ~a- -Fbt~v + iF~,[~bt,v)l;
1 [ ((r Lint = --~r.~ ~FABFA(F*) B -- -~abC 1 + _1"," 2 J - A B JnAr-,B J 1.1 + ig (r
.J
+ (AaAb)F C + iv/-2(~2aAb)DC) + V/2 {'~'*)0 [W )a ~ao/,b bit " --
~) ] (~/)~)a.~'aj/
Here A and r are the gluinos and r is the scalar component of the Af = 2 vector superfield. The f~c are structure constants of the Lie algebra. The indices a, b, c , . . . belong to the adjoint representation of SU(N) and are raised and lowered with the invariant metric. Indices A , B , . . . run over both indices in the adjoint and over the slow times (m, n , . . - ) . Since all spurions corresponding to higher Casimirs are purely auxiliary superfields the
5.5.
309
RENORMALIZATION
Lagrangian can be simplified as follows. Set D a = -(bclas - 1 s ) a c ( ( b d a s s )mDm + ~(gCbf~ca.F 9 b a )) with F a : -(bdass) -1 ac ( bdass )cm Fm where the classical matrix of couplings VABhClassis defined via bclass -- (1/47r)T dass where
Tclass b --
mn T~ab; Tclas s :
0;
," class ~m ~7" )a ~-
,~,~dass N ~,, ~m+l ?rim 0r a
N
~ T r
?rim
(r
-~-""
(4.25)
where the dots denote the derivative with respect to r a of lower order Casimir operators. This leads to 1
= f-.Ar=2- Bclmanss FmF* + ~ D m D n O(Tdass)r~
+ ~'~
5.5
)
e
+ f~c(b
class m
)a (b
class -
-
)aelDmCbr
(4.26)
[ ( ~ ) a C b ) F m nt- (/~a)~b)F m -~- i v / 2 ( ) ~ a ~ ) b ) D m ]
0r
RENORMALIZATION
We extract here from [150] where a number of additional topics concerning renormalization also appear. In particular we omit here the work in [472, 473] (sketched in [150] and partially subsumed in the formulation of Section 3.1) and other work of various authors on the Zamolodchikov C theorem. The formulas in Section 5.3.1 involving derivatives of the prepotential all have some connection to renormalization of course and to indicate this briefly we refer to the original elliptic curve situation (cf. [151,287, 393, 499, 698, 760, 855] for example). Thus consider (CC) y2 = ( ~ _ A2)(/~ + A2)()~_ u) for example with a = (v/2/r) fhA~[()~- u)/(/k 2 -- A4)]l/2d~. One will have then for F ~ F sW 2iu 2 F = aFa - ~ ; 7~
AFA =
2iu
= -8~ibl u
(5.1)
where bl - 1/4~ 2 is the coefficient of the l-loop beta function. This is the only renormalization term here and (in the more enlightened notation of Section 5.3.1) A F ~ - TIF1W shows that an important role of the Whitham times is to restore the homogeneity of F W which can be disturbed by renormalization (cf. [150, 151, 287] for more on this - and see below for more details about renormalization). We remark also that beta functions for this situation are often defined via
~(~) = hOA~l~:~; za(~)= hOA~la=~
(5.2)
where T is the curve modulus (e.g. T = F/aW - cf. [89, 115, 151]). 5.5.1
Heuristic
coupling space geometry
In any event renormalization is a venerable subject and we make no attempt to survey it here (for renormalization in susy gauge theories see e.g. [89, 115,243, 244, 245,472,473, 635,698, 823, 856, 872]. In particular there are various geometrical ideas which can be introduced in the space of theories - the space of coupling constants (cf. here [214, 244, 245, 246, 247, 248, 634, 775, 885, 890]). We extract here now mainly from [243] where it is argued that RG (= renormalization group) flow can be interpreted as a Hamiltonian vector flow on a phase space which consists of the couplings of the theory and their conjugate "momenta", which are the vacuum expectation values of the corresponding composite operators. For theories with massive couplings the identity operator plays a central role and its associated coupling
CHAPTER 5. W H I T H A M T H E O R Y
310
gives rise to a potential in the flow equations. The evolution of any quantity under RG flow can be obtained from its Poisson bracket with the Hamiltonian. Ward identities can be represented as constants of motion which act as symmetry generators on the phase space via the Poisson bracket structure. For moduli g regarded as coupling constants one could obtain beta functions via an(Og/Oan) = Ong for Tn = log(an) and cO/OTn = an(C~/Oan). W h i t h a m dynamics on moduli spaces corresponds then to RG flows and one obtains e.g. Ohk/OTn or Ouk/OTn which could represent beta functions/3nk. Thus following [150, 243] (revised for [155]) we may consider the moduli space of uk or hn as a coupling constant space AJ of elements g a (space of theories) and interpret RG flows as a Hamilonian vector flow on a phase space T*Ad (in this context cU should also be regarded as a deformation parameter but we will ignore it here). Take T = log(a) and set ~a(g ) = aO,~ga (note, corresponding to A = exp(isl) in (4.15) with is1 = log(A) one would obtain ~a AOAga and recall that = AOAT is a standard beta function). We have now a tangent bundle TAd ~ (ga,/3a) with T*Ad ~ (ga, Ca) where Ca Ow(g,t)/Og a for some free energy W = -log(Z) with W ~ f wdDx. Here Z could correspond to f Z)r162 and 1 = f I ) r 1 6 2 W] implies d W = < dS > = f Z)r162 + W] (and S ~ f s for some Lagrangian so an underlying D-dimensional space is envisioned). The approach here of [243] is field theoretic and modifications are perhaps indicated for the Af = 2 susy YM theory; thus the constructions here are only intended to be heuristic. Now one can display a Hamiltonian ( H H ) H(g, r }-~/3a(g)r +/3r(g, F ) r which governs the RG evolution of (ga dpa) via ~,~
-
-
dg a OH] dec dT = OCa g ; dT =
OH] Oga r
(5.3)
Here F (cosmological constant) is a coupling associated with the identity I whose conjugate m o m e n t u m is the expectation value of the i d e n t i t y - in fact one can take heuristically 9 r = F; /3 r(9, F) = dT dr = -Dr
+ Ur (g); Cr =
aD
(5.4)
where T = log(a) could refer to any Tn (~ an). One has then a symplectic structure and a Hamilton-Jacobi (H J) equation
O---~+H g,
=0=
~-~+~--]~
(g)r
(55)
For/3 a = tiC(g, T) one could take T as an additional coupling and work on A~4 = (ga w i t h / 3 T = 1 and CT = OTW = - H ( g , r (T ~ log(n) implies aO~T = / 3 T = 1). Now i d e n t i f y w ~ F + F a
F, T)
D so CT = WT = F T + F D a D so that (5.5) says e.g. (with
ga ~ ha) OF
cOhk OF +
OT
uF nO = 0
(5.6)
(note r ~ ( - D F + u r ) a D ~ - D F a D + Ura D and the DFa D term cancels). We know by W h i t h a m dynamics that hk = hk(c~j, Tn) satisfies some homogeneity equations (A) ~ aj(Ohk/Oaj)+ F, TnOnhk = 0 so a typical equation like (EE) 2 F = ~ cu(OF/OcU) + TnOnF (cf. (3.14)) plus (5.6) for T ,,~ Tn implies (we think now of Wn = F + F a D for fixed F while U r is also held fixed for different Tn)
OF Ohk)
OF
5.5. R E N O R M A L I Z A T I O N
311
= y ~ TnOnF - E Tn ( - O n F - U r ~ D) = 2 E TnOnF + ( Z Tn~D) Ur
(5.7)
This seems to say (for ~ M Tnexp(DTn) = 7")
F - ~
(5.8)
TnOnF = 1 T U r
where F = F(Tn, aj) and hk = hk(Tn, aj) with Tn and aj independent allow us to set aj = aj(hk) for Tn fixed. In particular from (3.14) and (5.6) with say Tn = ~n,1 and 2'1 ~ A (there is an implicit switch here to the notation of (4.14)) we see that (3.14) says 2F = AOAF + ~ aj(OF/Oaj) (and F = F sW now) while (5.6) says (for T ~ log(A) as in (4.15)) and no other Tn (C) AOAF + ~ A(Ohk/OA)(OF/Ohk) + UrA D = 0. This means (cf. (5.1) with more variables inserted)
OF aJ-~a j = A O A F =
2F- ~ 9
-A ~
2iu2
= -UrA D-
(5.9)
OF OF Ohk _ _ U FAD _ E ~ Ohk Ohk OA -
Note u2 ,,~ h2 ~ 7-/2 (cf. below for more such notation) is a Hamiltonian and we see that it has the form ( H H ) (after shifting W ~ F + FAD). In addition U r is determined as
A D u r = --u2 2i OF - ~/3 k 7r Ohk
(5.10)
One notes that HJ equations reminiscent of (5.5) appear in [658] in the form O~"
0t-
-H
a,
-b--~a '
t
(5.11)
in connection with Lagrangian submanifolds (with H somewhat unclear) and this is surely connected to formulas such as (2.92) referring to W h i t h a m flows. Thus our heuristic picture of renormalization based on [150, 243] seems suggestive at least, modulo various concerns over terms like F, W = f wdDx with w ~ F + Fn D for t ~ log(n), etc. We remark that in [243] one has a dictionary of correspondence between Q F T or statistical mechanics with RG and classical mechanics. This has the form QFT ~ Statistical
Mechanics
Couplings ~ ga(t) Beta f u n c t i o n s ~ ~a(t) vevt s ~ Ca(t) Bare couplings ~ (g3, r Generating functional ~ w(g(t), go, t) H(g, r t) - l~a(g, t)r [~a
OH ~a "-~ Odpa ; -V(g, t)= 3 A + de _ - d U dt -
OH Oga
DA
-
, -6y , t - 0 N o explicit ~ dependence in .'~ Anomalous d z m e n s w n s RG invariant {0, H } = 0
Classical Mechanics
Coordinates ~ qa(t) Velocities ~ (~a(t) M omenta ~ Pa (t) Initial values ~ (q~, pO) Action ~ S(q(t), qo, t) H = ~-~ 1 gab (q)PaPb+ U(q, t) (ta
OH -- -~a ; iga =
s, +.
U(q,t) = -dU
OH Oqa
, 3-4, t ) - 0 Conservative s y s t e m P s e u d o - forces (Coriolis) Constant o f motion {0, H} = 0
(5.12)
CHAPTER 5. WHITHAM THEORY
312
In [798] one also finds a dictionary of comparisons between QFT and magnetic systems based on the effective action and we expand on this idea of effective action as follows. R E M A R K 5.9. In fact the effective action is isolated by A. Morozov in [746] as the source of integrability (cf. also [393, 394, 399, 420, 499, 747]). Indeed effective action of the form exp[S~f:(tlr = Z ( t l r /DCexp[S(tlr (5.13) in matrix models for example will correspond to a tau function of integrable systems such as KP and the "time" variables t can be thought of as coupling constants. In [380] one looks at diffeomorphism action and shift groups in the moduli space of coupling constants as providing indications of hidden group (integrability) structure in QFT. Further investigation of related generalized Kontsevich models (GKM) leads to effective action involving both KP times tn along with "Whitham" times Tk which have a rather different origin. Thus for D = n 2, modulo a few details (C is defined below)
ZCKM(LIVp+I ) = C j dDX exp[Tr(-Vp+l(X) + XL)]
(5.14)
where L is an n • n Hermitian matrix, Vp+l is a polynomial of degree p + 1, and Wp = V~+1 is a polynomial of degree p. Further C in (5.14) is a prefactor used to cancel the quasiclassical contribution to the integral around the saddle point X = A, namely, modulo inessential details, C = exp[Tr(Vp+l (A)) - T r (AVe+ 1(A))]detl/2[O2Vp+l (A)] (5.15) The time variables are introduced in order to parametrize the L dependence and the shape of Vp+l via
Tk - kTr(A-k); Tk = kTr(A-k);
(5.16)
P
L = Wp(A)=/~P; tk = k ( p - k)Res'Wl-(k/p)(P)d# Then one can write ZGKM
--
exp[--JCp(Tkltn)]Tp(Tk + tk)
(5.17)
where Tp is a p-reduced KP tau function and 1
3Cp = -~ ~ Aij(t)(Ti + ti)(Tj + tj); Aij = ResccWi/p(,k)dWJ+/P(A)
(5.18)
is a quasiclassical (Whitham) tau function (or the logarithm thereof). In addition such a GKM prepotential is a natural genus zero component in the SW prepotential (cf. [394] and compare (5.18) with (3.23) for example). R E M A R K 5.10. It is possible now to apprehend how the Whitham times can play two apparently different roles, namely (1) coupling constants, and (2) deformation parameters. We follow the Russian school of Gorsky, Marshakov, Mironov, and Morozov ([393, 394, 399, 400, 401, 603, 692, 693, 734, 746, 747]) and especially the latter as in [746, 747] (cf. also [420, 499, 658]). The comments to follow are mainly physical and heuristic and the matter seems to be ~ummarized in a statement from [746], namely: The time variables (for SW theory), associated with the low energy correlators (i.e. the renormalized coupling constants) are Whitham times (the deformations of symplectic structure). Thus both roles (1) and (2) appear but further clarification seems required and is possible. One relevant theme here is described in [499, 746] roughly as follows. Given a classical dynamical
5.5. R E N O R M A L I Z A T I O N
313
system one can think of two ways to proceed after exact action-angle variables are somehow found. One can quantize the system or alternatively one can average over fast fluctuations of angle variables and get some effective slow dynamics on the space of integrals of motion (Whitham dynamics). Although seemingly different, these are exactly the same problems, at least in the first approximation (nonlinear WKB). Basically the reason is that quantum wave functions appear from averaging along the classical trajectories - very much in the spirit of ergodicity theorems. In string theory, the classical system in question arises after some first quantized problem is exactly solved with its effective action (generating function of all the correlators in the given background field) being a tau function of some underlying loop-group symmetry for example. The two above mentioned problems concern deformation of classical into quantum symmetry and renormalization group flow to the low energy (topological) field theory. The effective action arising after averaging over fast fluctuations (at the end point of RG flow) is somewhat different from the original one (which is a generating functional of all the matrix elements of some group); the "quasiclassical" tau function at the present moment does not have any nice group theoretical interpretation. The general principle is in any case that the W h i t h a m method is essentially the same as quantization, but with a considerable change in the nature of the variables; the quantized model lives on the moduli space (the one of zero-modes or collective coordinates), and not on the original configuration space. Note that in higher dimensional field theories the functional integrals depend on the normalization point # (IR cutoff) and effective actions describe the effective dynamics of excitations with wavelengths exceeding #-1. The low energy effective action arises when # ~ 0 and only a finite number of excitations (zero modes of massless fields) remain relevant. Such low energy effective actions are pertinent for universality classes and one could say that Af = 2 susy YM models belong to the universality class of which the simplest examples are (0 + 1)-dimensional integrable systems. We could supplement the above comments with further remarks based on the fundamental paper [393] (some such extractions appear in [150]). Let us rather try to summarize matters in the following manner, based on [393, 746]. First think of a field theory with partition function Z(tlr ) = f4)o DCexp[S(tlr as in (5.13). The dynamics in space time is replaced by the effective dynamics in the space of coupling constants g-2, 0, and ti via Ward identities which generate Virasoro conditons and imply that Z is the tau function of some KP-Toda type theory. The parameter space is therefore a spectral curve for such a theory and the family of vacua (~ r is associated with the family of spectral curves (i.e. with the moduli space). Now the averaging process or passage to W h i t h a m level corresponds to (A) Quantization of the effective dynamics corresponding to some sort of renormalization process creating renormalized coupling constants Ti for example, and (B) Creating RG type slow dynamics on the Casimirs hk of the KP-Toda theory; then via a 1 - 1 map (hk) --~ (urn) to moduli one deals with moduli as coupling constants while the Tn are deformation parameters. In this spirit it seems that in [282, 283, 284] both features are used. Promoting Tn to spurion superfields corresponds to a coupling constant role for the Tn (as does treating the prepotential as a generating function of correlators) while dealing with the derivative of the prepotential with respect to the T~ in the spirit of renormalization involves a deformation parameter aspect. REMARK 5.11. We remark that there is also a brane picture involving RG flows and W h i t h a m times (cf. [396, 397, 398, 400, 427, 401, 603] for example) in which W h i t h a m dynamics arises from the motion of certain D - 4 branes. This dynamics generates conditions for the approximate invariance of the spectral curve under RG flow and provides the validity of the classical equations of motion in the unperturbed theory if suitable first order
314
C H A P T E R 5. W H I T H A M T H E O R Y
perturbation is allowed. It is also conjectured that both Hitchin spin chains and Whitham theories can be regarded as RG equations (Hitchin times are to be identified with the space RG scale- t ~ log(r) - and the fast Toda-Calogero system- motion of certain D - 0 branes - corresponds to RG flows on a hidden Higgs branch, providing susy invariant renormalization for the nonperturbative effects, while the Whitham system is operating on the Coulomb branch). In this spirit the spectral curve is the RG invariant and the very meaning of integrability is to provide the regularization of nonperturbative effects consistent with the RG flows. R E M A R K 5.12. Another arena (no renormalization here) where Whitham times play an important role involves the structure of contact terms in the topological twisted Af = 2 susy gauge theory on a 4-manifold X (cf. [282, 283, 286, 658, 682, 684, 743, 903]). We will not try to cover the background here. An earlier version of this section (written in 1998) made some attempt at this but it would have to be considerably enlarged and revised to be instructive now. Since enlarged versions already exist in the references above there seems to be no point in an inadequate sketch. Thus we follow mainly [682, 684, 903] here for the S U ( N ) theory (as in Sections 3 and 4) and go directly to the u-plane integral (for 52+ = 1)
Zu=/~
[dadS]A(uk)XB(uk)aexp(~-~.pkuk + S2~-~fkfmTk,m)
(5.19)
Coulomb
Here S E H2(X, Z) and the Tk, m a r e so called contact terms. We omit discussion of the other terms. The contact terms were originally derived via blowup procedures for certain observables (cf. [658]) and this introduces conjecturally the tau function of a periodic Toda lattice. One expects then (notation as in Sections 7 and 8)
Tk+l'm+l
"-
4N 2
OTkOTm
(5.20) Tn>2=0
(cf. (4.23)) where ~red is defined in (4.22). The key idea is that the slow times 5bk are dual to the ~/k+l in the subspace Tn>2 = 0 (cf. ( D U A L ) after (4.16)); note also ~k+l = uk+l + O(uk). Further the fk in (5.19) are proportional to the fast Toda times tk. The development in [682, 684, 903] shows how this is all a very natural development.
5.6
WHITHAM, WDVV, AND PICARD-FUCHS
We add a few comments now relating Whitham times to the fiat times of Frobenius manifold theory and the WDVV equations. This involves connections to TFT, Landau-Ginzburg (LG) models and Hurwitz spaces, and Picard-Fuchs (PF) equations. 5.6.1
ADE
and LG approach
Connections of TFT, ADE, and LG models abound (cf. [32, 160, 225, 269, 601, 602, 902, 990]) and for N = 2 susy YM we go to [497] (el. also [89, 288, 776]). First we extract from [497] as in [150]. Thus one evaluates integrals ai = ~A~ Asw and a D = fB~ ASW using PicardFuchs (PF) equations. One considers PR(u, xi) = d e t ( x - OR) where R ~ an irreducible representation of G and OR is a representation matrix. Let ui (1 _< i _< r) be Casimirs built from OR of degree ei + 1 where ei is the i th exponent of G (see below). In particular Ul quadratic Casimir and u~ ~ top Casimir of degree h where h is the dual Coxeter number of
5.6. WHITHAM, WDVV, AND PICARD-FUCHS
315
G (h = r + 1 for Ar). The quantum SW curve is then
PR(X,Z, Ui)--PR
x, ui+hi,r
z+
--0
(6.1)
where #2 = A2h with A ~ the dynamical scale and the ui are considered as gauge invariant moduli parameters in the Coulomb branch. This curve is viewed as a multisheeted foliation x(z) over C P 1 and the SW differential is Asw = x(dz/z). The physics of N = 2 YM is described generally by a complex rank(G) dimensional subvariety of the Jacobian which is a special Prym variety (cf. [257, 693]). Now one writes (6.1)in the form #2 Z -~- - - -~- Ur -- ~/R(x, U l , ' ' ' , Ur-1) Z
(6.2)
For the fundamental representation of Ar for example one has (we have organized the indexing to conform more closely to [256, 394, 693, 760]) w r +A l~
__ x r + l
-- Ul xr-
1
--
....
(6.3)
Ur-lX;
and setting (SP) WR(x, ul " " 9~ U r ) - I~R(x ~ Ul ~ " " 9~ U r - 1 ) - Ur it follows that W mr+l is the r fundamental LG superpotentials for Ar type topological minimal models (cf. also [225, 269, 288]). The ui can be thought of as coordinates on the space of T F T . We will concentrate on Ar but Dr and other groups are discussed in [497]. For comparison to [394] we recall (cf. [150]) that for a pure S U ( N ) susy YM theory
dety•
-- A] -- O; P ( A ) = A y ( w + 1 ) ;
N
=
(6.4)
N
- Z
= H(
2
-
(-1)
1
il~"'
Here the Vk are Schur polynomials of hk = ( l / k ) ~ N Ak via the formula (A) log(A -NP(;~)) -~k(hk/)~k); there are g - N - 1 moduli uk and we refer to [499] for the Lax operator L. T h u s v 0 - 1, V l - 0 , v 2 - h 2 , v 3 = h 3 , v 4 = h 4 - ( 1 / 2 ) h 2,etc. ( h l = 0 f o r SU(N)). One also has the representation y2=p2() 0.4A2N;
y: AN (w-1)
(6.5)
giving a two fold covering of the punctured Riemann sphere with parameter A. Thus r + 1 N~h, g=N-1 A~x, anduk~Vk+l withur~vywhileP~W r + l - W AgN - 1 where ' Ar AN-1 "~ S U ( N ) or S L ( N ) . To clarify the curve correspondence write = z + #2z-1; @ = P(x); y = z - #2z-1; y2 = p2 _ 4p2
(6.6)
with #2 = A2N leading to the equivalence of curves y2 = p 2 _ 4A2N. Recall also from Section 7 (3.4), etc., that dSsw = )~(dw/w) = )~(dP/y) = )~(dy/P). For the moment we ignore relations between w and @ and refer to [256, 693] for discussion of the algebraic geometry picture for the @ parametrization. Now in 2-D T F T of LG type Ar with superpotential ( S P ) the flat time coordinates for the moduli space are given via
Ti
ci ~ dx W~(x, u) e'/h (i = 1,..., r) J
(6.7)
316
C H A P T E R 5. W H I T H A M T H E O R Y
(the ei will be discussed later and recall h = r + 1). Note that no Riemann surface is involved here since the integral corresponds to a residue calculation for x ~ p at oc (cf. Remark 5.14). Thus the T F T here depends only on WR(x, u) with x a formal parameter. These are residue calculations for times Ti (with no reference to Whitham theory) which will be polynomials in the uj (the normalization constants ci are specified below. One defines primary fields = cOWR(x'u) (i = 1 ... r) 0T/ ' '
r
(6.8)
where CrR = 1 is the identity ~ puncture operator. The one point functions of the gravitational descendents an(C/R) (cf. [32, 225, 269]) are evaluated via
< O'n(r R) >= bn,i ~ ?~ij / W g ( x , ?.t)(ej/h)+n+l (It -- O, 1,...) 1
(6.9)
for certain constants bn,i (cf. [497] for details). The topological metric ~ij is given by
o2/
~ij = < r
> = bo,r cOTicOTj
WaR(x' u)l+(1/h)
(6.10)
a n d l]ij -- 5ei+ej,h can be obtained by adjustment of ci and bn,i. The primary fields generate the closed operator algebra
(6.11)
1 where
o wg(x) OTiOTj
= cOxQR(x)
(6.12)
(for details on (6.11)- (6.12) we refer to [32, 225, 226, 289, 290, 439, 978]). Note again x ~ p is a formal parameter and X ~ Tr would be a natural identification. Now the structure constants ckij are independent of R since cij k -- Cij?]gk is given v i a Cijk(T ) --,( ~)RcRd/)R i j k > which are topologically invariant physical observables. In 2-D T F T one then has a free energy F such that Cijk = c33F/cOTicOTjOTk (cf. [32, 167, 225, 269]). Now for the Picard-Fuchs (PF) equations, the SW differential Asw ~ A = (xdz/z) can be written as (el. [150, 394]) )~sw = [xOxW/v/W 2 - 4#2]dx (for W ~ W~) and one has then, for # fixed
cO)~sw
1
cOW
v/W2 - 4p 2 -0-~d x
0Ti
+d
x cOW) v / W 2 - 4# 2 cOTi
(6.13)
(total derivative terms will then be suppressed). Suppose W is quasihomogeneous leading to
qi T iO- -W = h W (i.e. W(tx, tqiTi) = thW(x, Ti))
zOxW + 1
(6 14)
O~
(qi = ei + 1 is the degree of Ti). Then ,~sw-
1
qi~
O,~sw 0Ti
=
hWdx v/W 2 - 4# 2
(6 15)
5.6. WHITHAM, WDVV, AND PICARD-FUCHS
317
and applying the Euler derivative E qjTj(O/OTj) to both sides yields (after some calculation)
qiTi-~i - 1
)~sw - 4# 2h 2 02"~sW ~ =0
(6.16)
The calculation goes as follows; first
y~ qjTjOjA - (~--~qjTjOj) ~ qiTiOi)~ = hdx ~ qjTjOj v/W2W_ 4# 2 =
(6.17)
hWdx =_~ q,qjT~Tj~j + E V(V - 1)Tj~j - Z qjrj~j + ~ - ,/W~ - 4.~ = =-hdx
~
qjTjWj
v / W 2 _ 4#2
+ W2 E qj Tj Wj 1 = ( W 2 - 4p2)a/2
W
- ~ q~qj T~T~~ j + ~ qj (V - 2)Tj ~j + ~ = h~x Next note that
hd
(
v/W2 _ 4# 2
v/W 2 - 4#2
+
4 # 2 ( h W - xWx) (W 2 - 4#2)3/3
v/W2 _ 4# 2 - (W 2 _ 4#2)3/2
so the right side of (6.17)can be written as 4Wu~h~dx/(W2-4U~)3/~+hd
I
.
(xW/v/w~ - 4,~).
We note also that by degree counting (see below for more on this)
0 Oui 0 _ OTr = E OTr Oui -
0 OW OUr =~ ~ = 1
(6.19)
since OW/Our = - 1 (note here OUk/OTr = --hkr =~ (6.19)) and thus from (6.13) (Wr = 1, Wxr = 0 )
0)~ OT~
= -
dx v / W 2 - 4~2
+ d
(
x v / W 9. - 4~2
)
;
(6.20)
02A Wdx ( x ) Wdx OT2 = (W 2 _ 4 p 2 ) 3 P . + d Or V/W2 _ 4p 2 ~ (W 2 _ 4#2)3/9 Consequently the right side of (6.17) is equivalent to 4#2h202)~/OT2 and (6.17) becomes _4#2h 2 02A
02A
+ E qiqjTiTj OTiOTj + E qJ(qJ - 2)rj ~O)~ + )~ = 0
(6.21)
which is equivalent to (6.16). We note that the second term in (6.16) represents the scaling violation due to p2 = A2h since (6.16) reduces to the scaling relation for Asw in the classical limit #2 ~ 0. Note that )~sw(T~, #) is of degree one (equal to the mass dimension) which implies ( ~ i b ~ ) ( ~ qiTi(O/OTi)+ h#(O/O#)- 1 ) ~ s w = 0 (see R e m a r k 5.14 below) from which (6.16) can also be obtained. In this respect we note that
4x # Wxdx
4h p2 x Wxdx
0u)~ = (W 2 _ 4#2)3/2 ::* (~--~ qiTiOi- 1),~ = -h#O,)~ = - (W ~ _ 4p2)3/2 Then, using (6.18) / r---~
\
\
/
{,~_~ qirioi - 1) A = -hpOuA = -
-
4hp2xWzdx = 4#2)3/2
(W2
(6.22)
318
CHAPTER 5. W H I T H A M THEORY = hd
xW v/W2 - 4 # 2 ) - ( v / w W h d x 4 # 2 )
(6.23)
and this implies, via (6.20) - -
REMARK
_ ~
-h#d
4h2# 2Wdx #2 02 ~ (W 2 _ 4#2)3/2 ~ 4h 2 OTr2
0 u v/W2 _ 4P 2
(6.24)
5.13. In connection with scaing we note from
W(tx, tn+lun) - tNx N -- ult2(xt) N-2 . . . . . that (corresponding to (6.14))
xOxW + E ( n + l)un
tNuN-1 = tNW(x, Un)
(ow) ~Un
- NW
(6.25)
(6.26)
Alternatively we can write W(x, un) = x N - u2 x N - 2 . . . . . UN with W(tx, tnun) tNW(x, Un) leading to [xOx + ~nun(O/OUn)]W - N W . Similarly we see that Wx = Nx N-l( g - 2)u1 x / - 3 . . . . . UN-2 implies W x ( t x , tn+lun) = t N - 1 W x ( x , Un) and c o n s e q u e n t l y for )~ = x W x d x / v / W 2 - 4# 2 one gets
txt N-1Wxtdx )~(tx, t n+lun, tN#) = (t2gw2 _ 4t2Np2)l/2 = t)~(x, Un, p) ==~ =~ (XOx + Y~(n + l)Un~un + N#O,) ~ = s
(6.27)
Next from (6.7) we have (x ~ t~) T i ( t n + l u n ) = ci / d x W ( t n + l u n ,
x) ei/N =
(6.28)
OTi = qiTi = cit ei+l f d~W(un,~) = tqiTi(un) =~ E ( n + 1)Unhurt Now to confirm (6.14) write W(tx, tqiTi#) ~ W(tx, tn+lun, #) = t g w ( x , Ti, #) (via (6.25) and (6.28)). Further note (via (6.27)
A(tx, tq'Ti, tNp) ~ A(tx, tn+lun, tNp) = t)~(x, Ti, p) =~ (xOz + ~ qiTiOi + NpOu) )~ = )~ Let us try now to derive the relation ( a a a ) , n~mely, (E (aaa) - (6.22) ~nd thence (6.23) which means
(~-~qiTiOi-1) ) ~ -
(6.29)
q~T~O~+ N~O, - 1)~ = 0. Th.s,
Whdx v/W 2 _ 4#2
(6.30)
and this corresponds to (6.15) which we know to be true. This shows only however that an integrated form of (tblbtb) is valid (i.e. (lbtblb) is valid modulo h d ( x W / v / W 2 -4p2). Another set of differential equations for )~sw is obtained using (6.11), to wit
02 02 OTiOTj )~sw = ~ C~(T) orkOrr Asw k
(6.31)
5.6.
WHITHAM,
WDVV, AND PICARD-FUCHS
319
To see how this arises one writes from (6.11), (6.12), and (6.13) (using (6.8)) 02 )~ ~OTiOTj
Wij dx r Cj W dx nt_ v/W 2 - 4# 2 (W 2 - 4#2) 3/2
dQij
(~
c~jCk + Q~jw~)Wdz k
v / W 9 _ 4p2
= E
(6.32)
( W 2 - 4p2)a/2
Cikj Wk W dx ( w 2 _ 4~2)3/2 - d ( v/W2 _ 4p9
)
Then observe that from (6.13) WWkdx 02 )~ ,-,-, (W 2 - 4#2)3/2 OTkOTr
(6.33)
resulting in (6.31). Then the PF equations (based on (6.24) and (6.31)) for the SW period integrals II = ~ )~sw are nothing but the Gauss-Manin differential equations for period integrals expressed in the flat coordinates of topological LG models. These can be converted into uk parameters (where Ouk / OTr = --hkr ) as
f~oYI--
s
(~
qiui~
- 1
) 2
II -
42-202YI p h ~ - 0;
(6.34)
02II ~L~ 02II ~L~ OH = OuiOuj FAijk(U) OUkOUr + Bijk(u)-~u k -- 0 1
where Aijk(U) =
~-~OTmOTnOuk 1
1
Oui Ouj OTe Cemn(U); Bijk(u) = -
~ 02Tn Ouk 1
OuiOuj OTn
(6.35)
which are all polynomials in ui. One can emphasize that the PF equations in 4-D N - 2 YM are then essentially governed by the data in 2-D topological LG models.
5.6.2
Frobenius algebras and manifolds
We sketch here very briefly and incompletely some basic material on Frobenius algebras (FA) and Frobenius manifolds (FM) following [269, 271] (cf. also [598, 677, 678]) with special emphasis on LG models and TFT. This is not meant to be complete in any sense but will lead to a better understanding of Section 5.6.1 and serve simultaneously as a prelude to WDVV. Generally one is looking for a function F ( t l , . . . , tn) such that cc~z.~ = 03F/Otc~OtZOt'Y satisfying (C) r/c~,~= Cla3(t) is a constant nondegenerate matrix with 7/a3 - - ( ? ] c ~ 3 ) - 1 ; (D) %3 ~ -rfl%mz(t) determines a structure of associative algebra At " e a . e z - c ~aze~ where el, 99 9 en is a basis o f R n with el ~ unity via ~ a = 5~, and (E) F ( c d l t l , . . . ,cd"t n) -- cdFF(tl, . . . ,t n) which corresponds to s = E a O a F = dF" F for E = EaOa with E a = dat a here (we use t k ~ tk and later, for a certain LG model as in Remark 10.2, t k ~ Tk - cf. (6.41)). In [269] one looks at dl = 1 and physics notation involves da = 1 - qa, dE = 3 - d, qn = d and qa + qn-a+l = d. The associativity condition in (D) reads as (WDVV equations) 03 F 03 F 03 F 03 F Otc~Ot~ Ot ;~r l ~ Ot'~Ot ~0~ = Ot'YOt~ Ot ~ r]~ Otc~Ot~Ot u
(6 36)
C H A P T E R 5. W H I T H A M T H E O R Y
320
Next one defines a (commutative) FA with identity e via a multiplication (F) (a, b) ---,< a,b > with < ab, c > = < a, bc >. Here i f w E A* is defined b y w ( a ) = < e,a > then we have < a, b > = w(ab). The algebra A is called semisimple (ss) if it contains no nilpotent a (a m = 0). Given a family At of FA one often identifies At with the tangent space T t M at t to a manifold M (t E M). M is a Frobenius manifold (FM) if there is a FA structure on Tt M such t h a t (G) < , > determines a flat metric on M, (H) e is covariantly constant for the Levi-Civita (LC) connection V based on < , >, i.e. Ve = 0, (I) If c(u, v, w) =< u .v, w > then (Vzc)(u, v, w) should be symmetric in the vector fields (u, v, w,z), and ( J ) There is an Euler vector field E such that V ( V E ) = 0 and the corresponding one parameter group of diffeomorphisms acts by conformal transformation on < , > and by rescaling on TtM. One shows t h a t solutions of W D V V with dl # 0 are characterized by the FM structure (0~ ~.~ O/Ot ~, f~E as in (E)) (6.37) and s dE,O,
= dFF + Aaztat ~ + Bat s + c (where the extra terms in s d E - d a ~ O, and d E - d a - d 5 ~ O ) .
can be killed when
R E M A R K 5.14. The case of interest here is based on M = {W(p) = pn+l + anpn-1 + 9"" + al; ai E C} where T w M ~ all polynomials of degree less than n and A w on T w M is A w = C / W ' ( p ) where ' ~ d/dp and < f , g > w = Rescc[f(p)g(p)/W'(p)]. T h e n e ~ O/Oal and E = ( 1 / ( n + 1)) E ( n - i + 1)ai(O/Oai). This should be compared to Section 11.1 where W = x r + l - u l x r-1 . . . . . Ur so ak ~ --Ur-k+l and e ~ - 0 / 0 u ~ w i t h n = r. Note the indexing leads to
W(tp, tn-k+2ck) = tn+lw(p, ak) =a pop + y ~ ( n - - k + 2)ak-~a k
W - (n + 1)W
(6.38)
and ( n - k + 2 ) a k ~ --(n--k+2)Un-k+l ~ - - ( m + 1)urn (cf. (6.26)). One checks the vanishing of the curvature for the metric < f, g > w = Rescc[fg/W'] as follows. Consider p = p ( W ) inverse to W = W(p) obtained via Puiseaux series
p=p(k)=k+
1
(~r__~
n+l
+
t n-1 ~
+.--+
tl) ~
+O
(
1 )
(6.39)
where k n+l - W; this determines the coefficients ti(ak) where
P(k) n+l + anp(k) n-1 + " " + al = k n+l = W
(6.40)
will determine the expansion (6.39). There is then a triangular change of coordinates (L) ai = - t i + fi(ti+l, ... , t n) for i = 1 , - . . , n. Evidently
ta = _
n+ 1 n-a+l
Reset W
n--a-}-I awl
(p)dp
(6.41)
(cf. (6.7)) which suggests that ei = n - i + l and ci = - ( n + l ) / ( n - i + l ) in Section 10.1 and we identify t k and Tk). One can verify (6.41) by looking at dp = d k - (1/(n + 1))[(tn/k 2) + " " + (ntl/kn+l)]dk + O ( d k / k n+2) with W = k n+l. To prove that the t a are flat coordinates one uses the thermodynamic identity
Oa (Wdp) lp=c = -Oa (pdW) l w : c
(6.42)
5.6.
WHITHAM, WDVV, AND PICARD-FUCHS
321
which follows from W ( p ( W , t ) , t ) = W via OaWlp=c + (OpW)Oaplw=c - 0 which says Oa(Wdp)]p=c + Oa(pdW)lw=c = 0 since OpW ~ dW/dp. Next we have for 1 < a < n
Oa(Wdp)lp=c = - [ k a - l d k ]
(6.43)
+
where [fdk]+ = [f(dk/dp]+dp. To see this note k = W 1/(n+l) = p + O(1/p) via (6.40) and from (6.40) and (6.42) we have
-Oa(Wdp)lp=c = Oa(pdW)lw= c = (cOap)dkn+l = [
(6.44)
1 1 (k__~)]dkn+l=ka_ldk+o(dkk) n + 1 k u-a+l + 0
The left side is polynomial in p and [O(1/k)dk]+ = 0 since dk = dp + O(1/p2)dp while ( l / k ) = O(1/p) so (6.43) is proved. Now use (6.44) in a standard formula (cf. [269])
< Oh, O~ > = Resoo Oa(W(p)dp)O~(W(p)dp)" dW(p) '
(6.45)
(Wdp)Oz(Wdp) (Wdp)
c a ~ - Resc~
dpdW(p)
which yields < 0a, c9~ > = Resp=c~
ka-lkZ-ldk 1 dkn+ 1 - n + 15a+~,n+l
(6.46)
so the t a are flat coordinates since rlaZ is constant. The corresponding F arises via 1
n-t-~+2
OaF = (a + 1)(n + a + 2) Resp=ooW n.l dp
(6.47)
(based on [225, 269]). Thus one will have some W D V V equations based on the LG model corresponding to those specified by W ~ in Section 10.1. Strictly speaking there is no W h i t h a m theory here - only a LG model based perhaps on a dispersionless K d V hierarchy (see Remark 10.3). 5.6.3
Witten-Dijkgraaf-Verlinde-Verlinde
(WDVV)
equations
We go first to [497] and continue the context of Section 5.6.1. The main point is to show that the W D V V equations for SW theory (involving variables hi) arise from those of the corresponding LG model (this is quite cute). Thus one takes z + (#2/z) - WG(x, T 1 , . . . , Tr) as in (6.2)- (6.3) but WG is now expressed in terms of flat coordinates as in (6.7) (again only An is considered). We tentatively identify Ta and t a in (6.7) and (6.41), where n - r. One writes r ~ cR as in (6.8) with Cr = 1 and flatness of rlij in (6.10) (where P ~ Cr - 1) implies (6.12), namely, OxQij(x) = OiOjW(x) with ~ij - < r >-- ~ei+ej,h. Associativity of the chiral ring (cf. (6.25)) r 1 6 2 (r162 implies that C ~ j C ~ - cjkc' e m or (M) cj] 0
=
From F,j,, = Cfj,7,,,
obtains th n W D V V in
Firl-lFj = Fj~-IFi; (Fi)jk = Fijk
form
(6.48)
This is all based on T F T for the LG model with W ~ P in (6.2) - (6.4) or in R e m a r k 6.2. Now look at the SW theory based on W with (N) #2 _ A2N//4 where N ~ h ~ r + 1. We used #2 _ A2N in Section 6.1 (as in [497] (9712018)) but switch now to ( N ) plus
CHAPTER
322
5.
WHITHAM
THEORY
)~ = /~sw = (1/27ri)(xdz/z) (instead of A = x d z / z in Section 6.1) in order to conform to the notation of [497] (9803126). The P F equations (6.34) - (6.35) can be written in fiat coordinates as (cf. [497] or simply integrate ~ ,~sw in (6.24) and (6.31)) qiTiOi - 1
l:0H =
c~jn = o ~ o j n -
II - 4#2h 202H = O;
(6.49)
0
1
c~o~o~II = o (o~ ~ -5~)
Now one makes a change of variables T/ --* ai = ~A~ )~ (philosophy below) so t h a t (Oi =
O/OT~)
(i)iaiOjaj_~-:cki)kaiOrad)
02yI + 101 Oai OH = 0 OalOaj
(6.50)
1
where P / = OiOjai - ~-~flC~OkOraI. Since ai satisfies ~-,ijai - 0 one knows t h a t II - ai satisfies (6.50) and Pi~ = 0. Next take II = a D = &T'/OaI to get the third order equation
for ~: ~ Y ~
(P~5 = O)
fzijk = ~
1
CeijfZerk; Jzijk = OiaIOjajOkaK.T'IJK; ~ I J K =
039r(a)
Oaii)ajOaK
(6.5~)
Defining a metric by ~ij - fi'/jr one has ( 0 ) 9 ~ / = Ci~ for 9~/ = (~)jk - .Yijk and from c o m m u t a t i v i t y of the Ci there results
ff:Yi~-- l ffvj = ffYj~-- l ff2i
(6.52)
Hence the G- 1~'i commute and consequently the matrices (P) fi'~-1 ~ = (~-l.)~'k)-i ~ - 1.~-'i also Commute for fixed k. Therefore we obtain (Q) 9~ifi'~-ig~j = 9~jfi'k-l~i and removing the Jacobians Oai/OTi from (Q) implies the general W D V V equations
(6.53) as in [89, 115, 688, 733, 734, 744] where an association ai "~ dwi ~ holomorphic differential is used in the constructions and proof (cf. also [150, 167, 269, 389, 496, 600, 601,602, 677, 678] for W D V V ) . Here 9~ ~ F sW and the corresponding W D V V equations are a direct consequence of the associativity of the chiral ring in the An LG model (hence of W D V V equations of the form (6.36)). Note there is no flat metric present so connections to Frobenius manifolds are lacking here. Regarding philosophy note we have assumed no a priori connection between F and 9~. F comes from the T F T for the LG model while 9~ is defined via a D = O,Y/Oai = ~B~ )~sw and (general) W D V V for ~ follows from the P F equations. In particular F is not related to F W ~ W h i t h a m prepotential for the SW curve. There is also a W D V V theory for a W h i t h a m hierarchy on a RS involving the LG type Tk for 1 <_ k _< n plus other variables including the ai (see R e m a r k 5.17 below) and we note that the metric is quite different from insofar as the ai are concerned. REMARK 5.15. In this direction one can make the following comments. In (6.7) or (6.41) we have a formula (AS) Tm -- - [ ( n + 1)/(n + 1 - m ) ] R e s Wl-[m/(n+l)](p)dp for LG times based on a superpotential (AT) W = p n + l _ qlpn-1 . . . . . qn (qk ~ uk and
5.6.
WHITHAM, WDVV, AND PICARD-FUCHS
323
ak = --qn-k+l). This expresses the Tm as functions of the qk and one requires m - 1 , . . . , n so there are n primary times and n coefficients qk. Such objects W as above often arise from the dispersionless form of n-KdV situations for example and one can refer to t h e m as dispersionless times for a T F T of LG type. Note that diagonal coordinates (Riemann invariants) are obtained via ui = W(pi) where W'(pi) = 0 (i = 1 , . . . , n). Further in [269] for example one shows how this extends, in the context of Hurwitz spaces, to a new class of W h i t h a m times as follows (note this differs from Section 3 and see also [600, 601, 602]). Consider g-gap solutions of KdV problems based on L - 0 n+l - qlO n-1 . . . . . qn and let My,n+1 be the space of such g-gap solutions. This moduli space AJ = My,n+1 is the moduli space of algebraic curves Eg (with fixed homology) whose ramification is determined by the meromorphic function ( A T ) . We recall by Riemann-Roch that [r 2 g - 2 and for p ~ 1/z a pole pndp corresponds to - z - n - 2 d z so this has order n + 2 (not n). Hence d W will have N = 2g - 2 + (n + 2) = 2g + n zeros Pi where dW(p~) = 0 and one defines u~ = W(p~) (i = 1 , . . . , N ) as local coordinates at the branch points p~. The averaged K d V hierarchy (or W h i t h a m hierarchy) then has the form ( A U ) Omdf~l - Oxdf~m (or more generally Omdf~s = Osd~tm) where the d~m corresond to d W m/(n+l) + regular t e r m s as p ~ c~ with ~As d~tm = 0 (see b e l o w - note for z ~ 1/p near cx~ we want to deal with dftm ~ - m z - m - l d z + regular terms. Further one identifies p here with f~l, i.e. dp - dftl, and defines a flat metric via (AV) gii = Resp~(dp2/dW) as ds 2 = ~ N gii(u)du~ and the corresponding flat times ti (1 < i < N - 2g + n) for ds 2 have the form ti--(n
+ 1)Res~
tn+a ~- ~ i
W(p)l-~/(~+~) n+l-i
pdW; tg+n+a-
dp ( i -
1 , . . . , n);
dp ( a -
(6.54)
1 , . . . , g)
(as indicated earlier t i ~ ti ~ Ti for 1 < i < n). Then (6.46) holds, < On+a, Og+n+Z > = ha,Z, and otherwise < , > is zero. Thus the set of T F T times ( A S ) is enlarged to contain 2g additional primary times and this is a basic sort of W h i t h a m theory (cf. also [150, 269, 600, 601,602] for further enlargments). The associated W h i t h a m - L G theory for An now involves a coupling space A / / = Mg,n+l with flat coordinates T 1 , ' " , TN as above and primary fields r ~ -dp, r (~N of the form ( A W ) r - - ( n + 1)dfti (i = 1 . . . , n) with Cn+a = dwa and Cg+n+a = daa ((~ = 1 , . . . , g ) where the dwa are holomorphic differentials with ~A~ dwk = 27ribjk and there are multivalued holomorphic differentials daa - dcr~ satisfying
AB~daa = - d W
and fAk daa = 0 (where /kBadaa = d a a ( P + Ba) - d a a ( P ) ) . Further the
LG potential is W = W ( p ) where p ~ f~o dp ~ f ~ d ~ l and
O(Wdp) I
Ota
p=c
= - C a - - O(pdW)
Ota
with correlation functions (note the minus sign in r < r
>-naZ
= ~
(6.55)
W=c
above)
Resdw=o r dW '
< r162162 >= ~ Resdw=or162162 dWdp = The chiral algebra cTaz(t) has the form ( A X ) CaCz - cazr
(6.56)
caz~(t) modulo d W divisible dif-
ferentials and one will have an expansion (AY) p d W - (n + 1)dftn+2 + ~]N taCa for the
C H A P T E R 5. W H I T H A M T H E O R Y
324
multivalued differential pdW. Writing k n+l = W one has also
p d W = k d W - ~ tak ~-1 + O(k -1)
(6.57)
1
along with (s = 1 , . . . ,g)
JA
PdW = 2~ritn+s; AA 8(pdW) = 0; AB8 (pdW) = tn+g+sdW
(6.58)
8
In keeping with d~tm ~ d W m/(n+l) as indicated and ( A W ) one has Cm = ( -kin-1 -~O(k-2))dk with ~A., Cm = 0. One can also determine a partition function or prepotential F such that cazv = ,30 F/OtaOzOv. This development is an obvious ancestor to the SW theories of more recent vintage. The analogous SW theory is best developed in a two puncture Toda framework with two points c~+ corresponding to p --~ c~ (cf. [150, 394, 472, 473, 600, 601, 602, 760]); the analogous W term in not polynomial but logarithmic which severly limits the number of primary Ti times. We will not develop this further here. R E M A R K 5.16. The Whitham formulation of Krichever, developed in great detail with Phong (cf. [600, 601, 602]), involves a RS Eg with M punctures P~. One picks in an ad hoc manner two Abelian differentials dE and dQ having certain properties and sets dS = QdE as a SW type differential. Moduli space parameters are constructed and suitable submanifolds of a symplectic nature are parametrized by Whitham times TA with corresponding differentials d~A. With suitable choices of dE and dQ the formulation is adequate for SW type situations and T F T with WDVV equations will arise as well.
Chapter 6
GEOMETRY AND DEFORMATION QUANTIZATION We feel compelled to say something about an emerging class of papers on non-commutative geometry in connection with deformation quantization. The papers [194, 195, 328, 597] related to path integrals and renormalization were mentioned before but we will not attempt to summarize that here since the exposition would be too complicated. There are a number of other, sometimes related, papers involving deformation quantization in connection with branes, gravity, gauge theories, susy, and geometry about which (in some cases) something can be said (in Section 6.2) in a briefer format and we mention in this connection e.g. [45, 81, 91, 92, 201,176, 177, 178, 199, 202, 216, 220, 261, 262, 267, 334, 360, 368, 370, 371, 372, 387, 493, 508, 516, 523, 533, 595, 596, 628, 656, 629, 657, 668, 731,780, 847, 923, 983] with apologies for omissions. Another interesting theme involving stability of algebraic structures is developed in [948]. Some examples of quantum spaces and small quantum algebras are given in Section 6.1.4 and 6.2.4.
6.1
NONCOMMUTATIVE
GEOMETRY
Notations such as (TA), (UA), and (B1) will be used. At a number of places in Chapters 3 and 4 for example we have encountered noncommutative geometry and we give here a brief and incomplete sketch of some of the ideas. As one sees from Connes' book [193] there is an enormous amount of mathematics involved and there exist some other nice introductions (cf. [627, 669, 942] for example). Thus we certainly do not attempt to cover everything but try only to give some of the features which ae useful in establishing connections with Moyal quantization. Some useful papers in this direction are [81, 180, 184, 185, 196, 197, 198, 220, 261,334, 360, 372, 490, 628, 668, 820, 848, 857, 941,943,983]; we will not dwell on q-calculi and q-deformations as such (see e.g. Chapters 3 and 4) but mention also [180, 229, 230, 231, 232, 233, 234, 970]. We will select material here from [627, 669, 942], often condensing and occasionally expanding the exposition. Thus first from [627] one gets a global view as follows. The commutative Gelfand-Naimark theorem states that there is a complete equivalence between the category of (locally) compact Hausdorff spaces with (proper and) continuous maps and the category of commutative (not necessarily unital) C* algebras with 9 homomorphisms. Any commutative C* algebra can be realized as the C* algebra of continuous functions on some (locally) compact Hausdorff space. In this spirit 325
CHAPTER 6. GEOMETRY AND DEFORMATION Q UANTIZATION
326
a noncommutative C* algebra is to be thought of as the algebra of continuous functions on some "virtual noncommutative space". The emphasis is shifted from spaces which may be a mess or not exist concretely to algebras of functions. There is a new calculus based on the idea of a real spectral triple (A, l-f, D, J) where j[ is a noncommutative 9 algebra (not necessarily a C* algebra), 7-/is a Hilbert space on which j[ is realized as an algebra of bounded operators, and D is a suitable operator on 7-/which contains (almost all) the geometric information. The antilinear isometry J on 7-/will provide a real structure for the (,4, 7-/, D). For example for any closed n-dimensional Riemannian spin manifold M there is associated a canonical spectral triple with .A = C~176 7f = L2(M, S) (the Hilbert space of square integrable sections of the irreducible spinor bundle), and D is the Dirac operator associated with the Levi-Civita connection. This triple gives back the usual differential calculus on M and J is the composition of the charge conjugation operator with complex conjugation. Yang-Mills (YM) and gravity theories arise from the notion of connection (gauge or linear) on vector bundles (VB). The extension of these notions to the realm of noncommutative geometry relies on the Serre-Swan theorem which states that there is a complete equivalence between the category of (smooth) VB over a (smooth) compact space with bundle maps and the category of projective modules of finite type over commutative algebras with module homomorphisms. The space F(E) of (smooth) sections of a VB E over a compact space M is a projective module of finite type over C(M) and any finite projective C(M) module can be realized as the module of sections of some bundle over M. With a noncommutative algebra ,4 as the starting point the (analogue of) vector bundles will be projective modules of finite type over ,4 (not entirely straightforward) and one can develop a full theory of connections culminating in the definition of a YM action (Connes-Lott theory for example - cf. [197]). More recently in [198] Connes proposed a pure geometrical action which for a suitable noncommutative algebra A (noncommutative geometry of the standard model) yields the standard model Lagrangian coupled with Einstein gravity (cf. also [198]). Further, algebraic K-theory of an algebra .4 as the study of equivalence classes of projective modules of finite type over .4 provides analogues of topological invariants of the corresponding virtual spaces and cyclic cohomology provides analogues of differential geometric invariants. The connection of K-theory and cohomology via the Chern character has applications in physics
(cf. [77]).
6.1.1
Background
We follow here [942] and [595] is also excellent as an introduction. There are two contravariant functors from the category of locally compact Hausdorff spaces to the category of C* algebras (classically one thinks of commutative C* algebras). First C takes a compact space X to the C* algebra C ( X ) of continuous complex valued functions on X and takes a continuous map X ---, Y to its transpose C f : h ~ h o f : C(Y) ----, C(X). If X is only locally compact the associated C* algebra is Co(X) (continuous functions vanishing at oc) and the continuous maps f must be proper (the preimage of a compact set is compact) in order that h ---, h o f take Co(Y) ~ Co(X). The other functor goes the other way taking a C* algebra A onto its space of characters (i.e. nonzero homomorphisms # : A ---, C. If A is unital M ( A ) is closed in the weak 9 topology of the unit ball of the dual space A* and hence is compact. If r : A ---, B is a unital 9 homomorphism the functor M takes r to its transpose M e : # ---, # o r M ( B ) ---, M(A). One writes X + = X t2 {oc} (whether or not X is compact) and A + = C x A for the algebra A with an identity adjoined via the rule (A, a)(#, b) = (A#, Ab + #a + ab) (whether or not A is unital). Then C ( X +) ,-~ Co(X) + as
6.1. N O N C O M M U T A T I V E
GEOMETRY
327
unital C* algebras. This is all very classical and functorial; we will assume it is generally known. One notes that any ideal of Co(X) is of the form Co(U) where U C X is an open subset (the closed set X/U being the zero set of this ideal). Further for ,4 - CC~(M) a character on ,4 is a positive distribution (i.e. #(a'a) -- I#(a)l 2 >__ 0) and such a measure extends to C(M) so ,4 determines the point space M ~ M(A) (A ~ Co(M)). Another classical object involved in the study of a manifold M involves the category of (complex) VB E -~ M with bundle maps ~- 9 E -, E ~ satisfying 7r~ o 7 = v defining fibrewise maps Tx " Ex -~ E~x (linear). One writes F(E) - COO(M, E) for the space of smooth sections and if T" E -~ E ~ is a bundle map then FT" s ~ T o s" F(E) -~ F(E t) satisfies ( T A ) [~T(sa)(x) : wx(s(x)a(x)) -- Tx(S(X))a(x) -- ( F v ( s ) a ) ( x ) so FT(sa) - F T ( s ) a where a E .A. Thus FT 9 F(E) --, F(E') is a morphism of right .4 modules. Now one notes t h a t if E = M x C r then F ( E ) = `4r is a free `4 module. Since M is compact one constructs a partition of unity via r Ca with ~ r = 1 such t h a t E is trivial o v e r the set Uj where Cj > 0. Let fij be transition functions Ui M Uj --, GL(r, C) with f i k f k j = fij. T h e n a section s E F ( E ) given locally via smooth functions si = f i j s j can be regarded as a column vector s = ( r CqSq) T E C ~ ( M ) qr satisfying ps = s where p E M q r ( A ) is defined via Pij = qAfijCj and obeys the rule p2 = p with ~ P i k P k j = Ply. Hence one can identify F ( E ) and p A at. The Serre-Swan theorem says t h a t any (right) A module of the form p A m for an i d e m p o t e n t p E M m ( A ) is of the form F(E) = C ~ ( M ( A ) , E) where the fiber at # E M ( A ) is the vector space p A m | ( A / k e r # ) . Such modules p A m are called finite projective modules (i.e. finitely generated projective modules) and one can now think of n o n c o m m u t a t i v e vector bundles as finite projective modules over a not necessarily c o m m u t a t i v e algebra A which is a dense subalgebra of a C* algebra A (where A ~ C ( M ) here in which A is dense).
Any complex VB can be endowed in various ways with a H e r m i t i a n metric. Traditionally one defines a positive definite sesquilinear form ( ] )x on each fiber Ex which must vary s m o o t h l y with x. The n o n c o m m u t a t i v e point of view is to eliminate x leaving a pairing g' x g" ---, A on a finite right ,4 module t h a t is .4 linear in the second variable, conjugate symmetric, and positive definite. Thus
(rls + t) = (rls) + (r]t); (rlsa) = (rls)a; (rls) = (sir)*; (sis) > 0 (s :/: 0)
(1.1)
Note t h a t (rbis) - b*(rls ) for b E `4. W i t h this structure $ is called a pre-C*-module or prehilbert module. One can complete it in the norm []]sll] = v/ll(sls)l] where ]], IJ is the n o r m in A. T h e resulting Banach space is a C* module (not generally a Hilbert space). If is any right .4 module the conjugate space $ is a left `4 module and by writing ~ = { ~, s E $ } one defines aS = (sa*). For $ = p`4m one writes ~ = fi~p with row vector entries for A. Now consider the ket-bra operators on $ of the form ]r > < s[" t --, r(sit ) 9 $ --, $. Since r(sita) = r(slt)a for a E `4 these operators act on the left on $. Composition ( T B ) ]r > < s I 9Ir > < u I = Ir(slt) > < u I = Ir > < u(tls)l implies finite sums of ket-bras form an algebra B - E n d A ( E ) . W h e n E = p Jr m one has 13 = p M m ( A ) p so ~" becomes a left/3 module and $ is called a B - .A bimodule. One can also regard E n d A ( E ) as $ | ~ via ]r > < s ,--, r | ~. F u r t h e r one can form $ | $ with is isomorphic to A as an A bimodule via ~ | s ~ (ris) and this is an example of Morita equivalence (M equivalence). Generally two algebras A and B are M equivalent if there is a B - ~4 bimodule $ and a . 4 - B bimodule $ such t h a t (TC) $ | 9v ~ 13 and ~v | $ "" A. This is actually a very i m p o r t a n t idea. If there is a M equivalence of two algebras .4 and B as indicated then the f u n c t o r s / - / ---, ~r | 7-I and /-/~ ---* 9v | implement opposing correspondences between representation spaces of .4 and B. T h u s if one studies an algebra A via its representations then simultaneously the M equivalent algebras must be studied. In particular the c o m m u t a t i v e algebra C ~ ( M ) a n d
328
C H A P T E R 6. G E O M E T R Y A N D D E F O R M A T I O N Q U A N T I Z A T I O N
the noncommutative algebra M n ( C ~ ( M ) are packaged together for the purpose of doing geometry. We describe now some structure based on more or less classical ideas and some references are given; however much of this should be familiar to graduate students today. First recall t h a t the Clifford algebra CLn associated to the vector space R n is the complex algebra generated by selfadjoint objects 7 a satisfying ( T D ) Va-)'z + VZ7a = 2g az. The 7 a can be represented by N • N Dirac matrices with N = 2 [n/2] where [n/2] is the largest integer in n/2 here. If n is even Cln ~ MN and otherwise Cln ~ two copies of MN. The spinor representation of SO(n) is defined by the map A ~ S(A) where S(A) is the N • N matrix determined by ( T E ) A ~ - y z = S - I ( A ) v Z S ( A ) with det S(A) - 1. This is two valued since • are solutions. For A near the identity in SO(n) (i.e. A~ ~ 5~ + )~) there is a unique solution to ( T E ) given by ( T F ) S(A) ~ 1 + (1/4)AaZVa7 z. The set of all S(A) is the compact Lie group Spin(n) and S(A) --~ A defines an epimorphism spin(n) --~ SO(n) (double covering). For M an n-dimensional Riemannian manifold with a metric g on T M one builds a Clifford algebra bundle Cl(M) --~ M whose fibers are full matrix algebras over C as follows. If n - 2m one has Clx(M) - Cl(TxM, gx) | C ~ M2m(C). If n = 2m + 1 then one takes only the even part of the Clifford algebra, namely Clx(M) Cleven(Tz M, gx) | C "~ M2m(C) (thus losing the Z2 grading of C1). However in all cases the bundle C l ( M ) is a locally finite field of finite dimensional C* algebras. Such a field is classified up to equivalence by a (~ech cohomology class 5 ( e l ( M ) ) E H3(M, Z) (DixmierD o u a d y class). Locally one finds trivial bundles with fiber Sz such that CIx(M) ~ End(Sx) and 5(Cl(M)) is the obstruction in the C* category to patching t h e m together. For a R i e m a n n i a n manifold with A = Co(M) and B = Co(M, Cl(M)) one can say t h a t T M admits a spin c structure if and only if T M is orientable and 5(Cl(M)) - O. In that case a spin e structure on T M is a pair (e,S) where e is an orientation on T M and S is a B - A equivalence bimodule (see [636] for spin e and spin structures). Such a pair (e, S) is called a K orientation on M. The equivalence bimodule $ must be of the form F(S) for some complex VB S ~ M that also carries an irreducible left action of C l ( M ) and S is called the spinor bundle displaying the spin e structure. Here one calls S = F(S) = C~ S) the spinor module; it is an irreducible Clifford module of rank 2 m over C ~ (M) for n - 2m or n=2m+l. Once we have a spinor module the Dirac operator can be introduced as a self adjoint differential operator ~ defined on the space 7-l = L2(M, S) of square integrable spinors, whose domain includes the smooth spinors S = Cc~(S). If M is even dimensional there is a Z2 grading of F(Cl(M)) which in turn induces a grading of the Hilbert space 7-I - 7-I+ | 7-/-. T h e grading operator will be denoted by X so X2 - 1 and 7-/• are its +l-eigenspaces. Thus the Riemannian metric g = (9ij) defines isomorphisms T x M ~ T ~ M and induces a metric g-1 = (gij) on the cotangent bundle T*M. With this isomorphism we can redefine the Clifford algebra as the bundle with fibers Clx(M) - CI(T~M, gz-1 ) | C (replacing C1 by Cl even when d i m ( M ) is odd). Let A I ( M ) = 7 ( T ' M ) be the A module of 1-forms on M. T h e spinor module S is then a B - .A bimodule on which the algebra g - F ( C l ( M ) ) acts irreducibly and obeys the anticommutation rule ( T G ) {V(a), ~(/3)} = -2g-1(c~,/3= 2giJaigj for a, g E .AI(M). The action 7 of F(CI(M)) on the Hilbert space completion ?-/ of $ is called the spin representation. Now the metric g-1 on T * M gives rise to a canonical Levi-Civita connection Vg 9A~(M) ~ A ~ ( M ) | A~(M) which obeys the Leibnitz rule ( T H ) vg(wa) = Vg(w)a + w | da, preserves the metric, and is torsion free. The spin
6.1.
NONCOMMUTATIVE
GEOMETRY
329
connection is then a linear operator V S 9r ( s ) ---, r ( s ) | vS(Oa)
=
vs(~b)a + ~b | da; vs(v(w)~b)
A I ( M ) satisfying v(Vgw)~b + V(w)vszb
=
(1.2)
for a E .4, w E .41(M), and ~b E S. Given the spin connection one defines the Dirac operator as the composition V o V s or more precisely (TI) ~ = "~(dxJ )Vo/ox s j . This is independent of coordinates and defines ~ on the domain S c 7-/. This operator is symmetric and extends to a self adjoint operator in 7-/ which is Fredholm if M is compact. Since ker:p is finite dimensional one may define the compact operator ~ - 1 on its orthogonal complement. The Dirac operator may be characterized more simply by its Leibnitz rule. Indeed since the algebra .4 is represented on the spinor space 7"/by multiplication operators one may form ~(a~p) for a E .A and ~p E 7-/leading to (WJ) ~ ( a r = 7(da)~b + a ~ p . Equivalently this may be written as ( T K ) ~ , a] = v(da). In particular since a is smooth and M is compact the operator II[P, a]ll is bounded and its norm is simply the sup norm Ilda]lcc which equals the Lipschitz norm of a defined via
IlallL~p =
8ltpp~q
(1.3)
la(p) - a(q)]
d(p,q)
where d(p,q) is the geodesic distance. This might appear to be unwelcome because of introducing points but in fact one can simply stand this formula on its head by writing
d(p,q) = sup{la(p ) - a ( q ) [ ; a e A;
IlallL~p <_ 1}
-
(1.4)
= s u p { [ ( ~ - ~ ) [ ; a e .4; [l~,a][[ < 1} Thus the metric on the space of characters M(`4) is entirely determined by the Dirac operator. With this the ingredients for a reformulation of commutative geometry in algebraic terms are almost in place. They are briefly: an algebra .4, a representation space 7-t for .4. a selfadjoint operator ~ in 7~, a conjugation operator J (still to be discussed), and in even dimensional situations a Z2 grading operator X on 7-/. This package or 4 or 5 terms is called a real spectral triple or a real K-cycle or more simply a geometry. 6.1.2
Spectral triples
We continue to follow [942]. The basic data for a K-cycle consists of (.4, 7-/,~), sometimes accompanied by other data F, J, X with compatibility conditions. If F is present the Kcycle is even and otherwise it is odd. If J is present the K-cycle is real and otherwise it is called complex. The real case will be emphasized. D E F I N I T I O N 6.1. An even, real, spectral triple or K-cycle consists of a set of 5 objects (A, ~ , D, J, F) of the following types: 9 A is a pre-C* algebra 9 7-/is a Hilbert space carrying a faithful representation 7r of .4 by bounded operators 9 D is a selfadjoint operator on 7-/with compact resolvant 9 F is a selfadjoint unitary operator on 7-/with F 2 -- 1 9 J is an antilinear isometry of 7-/onto itself
330
C H A P T E R 6. G E O M E T R Y A N D D E F O R M A T I O N Q U A N T I Z A T I O N
In the present context a pre-C* algebra ,4 is defined to be a dense involutive subalgebra of a C* algebra A that is stable under the holomorphic functional calculus. This means t h a t for any a E ,4 and any function f holomorphic in a neighborhood of the spectrum sp(a) the element f(a) belongs to A and can be determined via
f(a) = ~
l j; f(~)(~ -
a)-ld~
(1.5)
where -), is any circuit winding once around sp(a). This is standard functional analysis and discussions can be found in many places (see e.g. [143, 144, 145, 146, 147, 171]). One supposes here that ,4 has a unit - otherwise `4+ and A + can be used. This situation is realized whenever `4 = ClOAk where the Ak for a decreasingly nested sequence of Banach algebras with continuous inclusions Ak ---* A and in the present circumstances where A = CCr with M a spin manifold one can take Ak = Dom(~2k). A major consequence of stability under the holomorphic functional calculus is that the K-theories of `4 and A are the same, i.e. the inclusion j : .4 ~ A induces isomorphisms j , : Ko(.4) ---+ Ko(A) and j , : KI(.A) ~ KI(A). Thus one may use the technology of C* algebraic K-theory with the dense subalgebra (cf. [193, 965]). One also recalls that compact resolvant for D means t h a t ( D - A)-I is compact whenever A r R. Equivalently D has a finite dimensional kernel and the inverse D -1 (defined on the orthogonal complement of the kernel) is compact. In particular D has a discrete spectrum of eigenvalues of finite multiplicity. On the basis of the distance formula (1.4) one interprets the inverse of D as a length element and since D may not be positive IDI = (02) 1/2 may be preferred. In any event one writes ds = ID1-1 The grading operator F available for even K-cycles splits the Hilbert space as 7-/= 7/+ | 7 / - . In this case one can suppose t h a t the representation of .4 on 7-/is even and t h a t the operator D is odd (i.e. aF = Fa for a E .4 and D F = - F D ) . This can be expressed as 7r(a)=
(a0) 0
a
; D-
(0
D+
0
(1.6)
The real structure J must satisfy j2 = -4-1 and commutation relations J D = + D J with J F = + F J . Its adjoint is j t = j - 1 = + j where (TL)7r~ = JTr(b*)J t defines an antirepresentation of .4. It is convenient to think of 7r~ as a true representation of the opposite algebra .40 consisting of elements {a ~ : a E .4} with product a~ ~ = (ba) ~ in which case ( T L ) becomes b~ = Jb*J t. It is important to require ( Y M ) [a, b~ = a, Jb* jr] = 0 for a, b E .4. One now can enunciate the axioms needed for these data to underlie a geometry. Thus 9 A x i o m 1. D i m e n s i o n There is an integer n, the dimension of the K-cycle, such t h a t the length element ds = ]D] -1 is an infinitesimal of order 1/n. Infinitesimal order one means simply a compact operator T on 7/. Thus T must be an operator with discrete spectrum with any nonzero )~ E sp(T) having finite multiplicity. The singular values of T are the eigenvalues of the positive compact operator IT I = ( T ' T ) ~/2 and they will be arranged in decreasing order p0 > pl > "" ". One says that T is an infinitesimal of order a if ( T N ) pk(T) = O(k -a) as k ~ cr In this respect the infinitesimals of first order have singular values forming a log divergent series
#k(T) = 0
(1) ~
=V aN(T) = ~
#k(T) -- O(log(N))
(1.7)
k
The dimension axiom can then be reformulated as: There is an integer n for which the singular values of D -n form a log divergent series. The coefficient of log divergence will be denoted by :f--]DI -n where ~ denotes the noncommutative integral (see below).
6.1. N O N C O M M U T A T I V E G E O M E T R Y 9
331
A x i o m 2. O r d e r o n e . For all a, b E .A the commutation rule ( T O ) [[D, a], Jb* jt] = 0 holds. This could be rewritten as [[D, z], b~ -- 0 or more precisely [[D, ~(a)], ~~ = 0. Using ( T M ) and the Jacobi identity one sees that this condition is symmetric in the representations 7r and 7r~ since ( T P ) [ a , [D, b~ = [[a,D],b ~ + [D, [a, b~ = - l i D , hi, b~ - 0. In the commutative case the condition ( T O ) simply expresses the fact that ~ is a first order differential operators via ( T Q ) [~, a], Jb*J t] = [~, a], b] = [~/(da), b] = 0. Note here that a second order operator like the Laplacian satisfies [[A,a], b] = - 2 g - l ( d a , db) :/= 0 in general. One can interpret ( T O ) as saying that the operators 7r~ commute with the subalgebra of operators on 7-/ generated by operators ~(a) and [D, ~(a)] and this gives rise to a linear representation of the tensor product
7rD(a @ al @ . " @ an) = 7r(a)[D, 7r(al)][D, 7r(a2)]... [D, 7r(an)]
(1.8)
,.., aiD, al] [D, a2]'-" [D, an] In view of the order one condition we could replace the first entry a by a | b~ E A | A and write
rcD((a @ b~ @ al |
| an) = rc(a)Tr~
7r(al)]... [D, 7r(an)]
(1.9)
Moreover C n ( A , A N A ~ = (A N A ~ @ A | is a bimodule over A and this recipe represents it by operators on 7-/. The elements are called Hochschild n-chains with coefficients in the A bimodule ,4 @ .4 ~ 9 A x i o m 3. R e g u l a r i t y . For any a E .4, [D, a] is a bounded operator on 7-/and both a and [D, a] belong to the domain of smoothness N ~ D o m ( 5 k) of the derivation 5 on s given by 5(T) = [[D[, T]. In the commutative case this corresponds to saying that a has derivatives of all orders, i.e. A C C~176 (proof using pseudodifferential operators). 9 A x i o m 4. O r i e n t a b i l i t y . There is a Hochschild cycle c E Zn(A, .4 @ A ~ whose representative on 7-/is ~(c) = F (n even) and ~(c) = 1 (n odd). Here c is a Hochschild n-chain with ~(c) given by (1.9). One says that c is a cycle if its b o u n d a r y is 0 where the b o u n d a r y operator is
b(mo@al @ . . ' @ a n ) = m o a l | 1 7 4 1 7 4 1 7 4 +(-1)n-lm0
+'"+
(1.10)
| an-lan + (-1)nanmo | al | "" | an-1
| al |
Here m0 E A | A ~ This satisfies b2 = 0 and makes C.(A, A | A ~ a chain complex whose homology is the Hochschild homology H.(.A,.A | A~ One notes that if x = (a | bO) @ al | | an then by telescoping .
7rD(bx) = (-1)n-l(ab~
al]...[D, an_l]an-anab~
al]...[D, an_l])
(1.11)
The cycle c above is the algebraic equivalent of a volume form on a noncommutative manifold (see [942] for details). 9 A x i o m 5. F i n i t e n e s s The space of smooth vectors ~oo = N ~ D o m ( D k) is a finite projective left A-module with a Hermitian structure given by (TS) -f--(~Jr/)ds n = < ~lrl >. Note (~Jarl) = a(~[rl) so ( I T ) f-- a(~Jrl)ds n = < ~lar] >. To see how ( I T ) defines a Hermitian structure implicitly note that whenever a E .A then ads n = alDJ -n is an
CHAPTER 6. GEOMETRY AND DEFORMATION Q UANTIZATION
332
infinitesimal of first order so t h a t the left side of ( T T ) is defined provided (~lr]) E .4. As a finite projective left .4-module 7-/oo ~_ .Amp with p = p2 = p, in some Mm(`4) so ~ E 7-/oo can be written as a row vector (~1,"" ,~m) satisfying ~ j p j k = ~k with ( T U ) (~lrl) = ~ n ~ j ~ E .4. One should note also that
a(~l~)ds ~ - < ~la~ >=< a*~lTI >= ~ (a*~l~7)dsn - ~-(~lT1)ads n
(1.12)
and the axiom implies that ~ (.)ID[ -n defines a finite trace on .d; this implies t h a t the von N e u m a n n algebra .4 ~ also has a finite normal trace so it cannot have components of types I~, I I ~ , or IIicr (cf. [193] and see below for ~ ) . 9 A x i o m 6. P o i n c a r 6 d u a l i t y . The Fredholm index of the operator D yields a nondegenerate intersection form on the K-theory ring of the algebra .4 | .40. Recall here t h a t Poincar6 duality is based on formulas like ( T V ) (a, r/) --~ fM a A 77 with the right side depending only on cohomology classes and vanishing if a or r/is exact. Here each .4k (M) carries a scalar product induced by the metric and orientation given by ( T W ) a A ,/~ = ek(al~)~ where ek = • or + i and ~ is the volume form on M. In view of the existence of isomorphisms between K" (M) | Q and H" (M, Q) given by the Chern character one could hope to reformulate this as a canonical pairing on the K-theory ring. This can be done if M is a spin c manifold and we refer to [193, 942] for more details on K-theory and intersection forms. 9 A x i o m 7. R e a l i t y . There is an antilinear isometry J : 7-/ --, 7-/ such t h a t the representation ~r~ = Jr(b*)J t commutes with 7r(a) and j2 = =[=1 with JD = i D J and J F = • where the signs are expressed in terms of n mod8 (see [942] for a table). In s u m m a r y one says that a noncommutative geometry is a real spectral triple (.4, ~ , D, J, F) or (,4, 7-/, D, J), according as the dimension is even or odd, satisfying the 7 axioms above. R i e m a n n i a n manifolds provide the commutative examples and various n o n c o m m u t a t i v e examples are given in [193, 627, 669, 942]. We have omitted many things here and refer to [595] for a revealing discussion of the axioms; our interest involves connections to Moyal quantization, developed below following [942]. 6.1.3
The
noncommutative
integral
Early a t t e m p t s at noncommutative integration used the ordinary trace of operators on a Hilbert space as an ersatz integral. For example in Moyal quantization one computes expectation values via Tr(AB) = f WAWBd# where WA, WB are Wigner functions of operators A, B and # is the normalized Liouville measure on phase space. However on the representation space of a noncommutative geometry one needs an integral t h a t suppresses infinitesimals of order higher than 1 so the trace will not do; in fact it diverges for positive first order infinitesimals since oo
TrlTI = ~ #k(T) = n---~OO lim an(T) = r o
(an(T) -- O(log(n)))
(1.13)
Now the algebra/(: of compact operators on a separable infinite dimensional Hilbert space contains the ideal 121 of trace class operators on which IITII1 = TrlT I is a norm (recall the operator norm is denoted by IITII = #0(T)). Each partial sum of singular values O"n is also
6.1. NONCOMMUTATIVEGEOMETRY
333
a norm on K: via ( T X ) a n ( T ) = sup{[[TPn[[1; Pn is a projector of rank n}. an(T) <_n#o(T) = n[[T[[. In fact one can show that (eft [942] for proof)
an(T) = inf {[[R[[1 + n[[SI[; R, S E/C; R + S - T}
Note t h a t (1.14)
One can think of (Tn(T) as the trace of IT[ with a cutoff at scale n and in fact for any A one can define a~(T) = inf {I[R[I1 + A[]S[[; R , S E 1C; R + S = T} (1.15) It is easily checked t h a t for A = n + t with 0 _ t < 1 one has ( T Y ) a~ (T) = (1 - t)an (T) + tan+l (T) so A ~ a~(T) is a piecewise linear, increasing, concave function on (0, oc). Further there is a reverse triangle inequality ( T Z ) a~(A) + a,(B) < a~+,(A + B) for A, B > 0 and hence a sandwich of norms (A, B >_ 0)
ax(A + B) <_a~(A) + a~)B) <_a2~(A + B)
(1.16)
DEFINITION 6.2. T h e D i x m i e r ideal. The first order infinitesimals can now be defined as the following normed ideal of compact operators 121+ = ~ T E K:; [ITIII+ = sup a,x(T) < oc~ ~>E log(A) ( J
(1.17)
Evidently/21+ D/21. If T E/21+ the function A ~ a~(T)/log(A) is continuous and bounded on [e, oc), i.e. it lies in the C* algebra Cb[e, oc) and one can form the Ceshro mean
1
~oo ha(T)du
= Zog( ) .o
log(
(1.18)
)
T h e n A ~ T),(T) E Cb[e, oo) as well with upper bound [[T[[I+. Consequently ~-~ will be asymptotically additive on positive elements of/21+ (via (1.16) - see [942] for details). From this one gets an additive functional as follows. Let ~(A) be the class of A ~ %(A) in the quotient C* algebra B - Cb[e, c~)/C0[e, oo). Then "/- is an additive positive homogeneous map from the positive cone of/21+ into B and -i-(UAU-1) - -/-(A) for unitary U. Hence the map extends to a linear map ~" /21+ ~ B such that ~(ST) = ~-(TS) for T E /21+ and any S. Finally one follows "/- with any state co (i.e. any normalized positive linear form co" B ~ C). The composition is the Dixmier trace ( U A ) Try(T) - co(~(T)). In order to produce now a noncommutative integral one notes that a function f E Cb[e, oc) has a limit lim~__.oef(A) = c if and only if co(f) = c does not depend on co. One says then t h a t an operator T E/21+ is measurable if A ~ %(T) converges as A ---, oc, in which case its limit is denoted by Trw(T). T h e n one denotes by ~ T the common value of the Dixmier traces, i.e. ( U B ) -f--T = lim~__.oe%(T) if the limit exists. One notes that if T E /C and an(T)/log(n) converges as n ~ oc then T E/21+ and is measurable. E X A M P L E 6.3. There are many examples in [942] for example which we have skipped but it would be nice to see an example here. Thus let A = - ~ 02 in the (commutative) torus T n = Rn/(27rZ) n. The eigenfunctions are C k ( x ) = exp(ik, x) for k E Z n. Discard the zero mode and look at A as an invertible operator on the orthogonal complement of the constants in L2(Tn). The multiplicity m~ of the eigenvalue A = IkI 2 is the number of lattice points in Z n of length Ikl and A -s is compact for any s > 0. Let Nr be the total number of lattice points with Ikl < r so Nr+dr - Nr ~ ftnrn-ldr and Nr ~ n-lf~nr n where (VC) f ~ = ( 2 ~ / 2 / r ( n / 2 ) ) = vol(S~-~). For N = NR one can estimate
aN(A -s) =
~ l_<[k[_
[k1-2s ~
/1
r-2S(Nr+dr - Nr) ~ an
]1
rn-2s-ldr
(1.19)
334
C H A P T E R 6. G E O M E T R Y AND D E F O R M A T I O N Q U A N T I Z A T I O N
Since log(Nn) ~ nlog(R) we see that a~(A-2S)/log(A) diverges if s < n/2 and converges to 0 if s > n/2. For the borderline case s = n/2 one gets a~(A-n/2/log(A) ~ f n / n so TA(A -n/2) "~ f n / n as well and hence ( U D ) :f-- A -n/2 = (fin~n) = 7rn/2/F[(n/2) + 1]. Since = 7(dxJ)Oj on I n satisfies ~p2 = A one has :f--ds n = ~- ~Pl-n = f n / n . We recall now that a pseudodifferential operator (PSDO) A of order d on a manifold M is an operator between two Hilbert spaces of sections of Hermitian VB over M that can be written in local coordinates as
Au(x) = (27r) -n / / ei(X-Y)'~a(x,~)u(y)dnydn~
(1.20)
where the symbol a(x, ~) is a matrix of smooth functions whose derivatives satisfy the growth conditions IO~O~aij(x,~)l <_ Ca#(1 + I~1)d-'#l. We simplify a little here be assuming that a is scalar valued and the bundles are line bundles. Then A is called a classical PSDO. written A 6 ~ d ( M ) , if its symbol has an asymptotic expansion of the form ( U E ) a(x, ~) ~-]~ ad-j(x,~) where each ar(x,~) is r-homogeneous in ~ (i.e. ar(x,t~) - tra~(x,~)). The principal symbol ad(x,~) is globally defined as a function on T * M except possibly on the zero section and the operator A is called elliptic if ad(x, ~) is invertible for ~ =/= 0. We refer to [145, 171, 480, 636] for PSDO. Generally the product A B = C of P S D O has symbol 9 ( U F ) c(x,~) ~ ~a~Nn[(--i)lal/av)OaaO~b and for A e q2d(M) with B E ~ r ( M ) one has
ABe
~d+r (M) and for P = [A, B] e ~ - ~ 1 7 6
a-------{ff-. (O~aO~b - O~bO~a)
~ E
(1.21)
I~l>0
For M an n-dimensional manifold the term of order - n of the expansion ( U E ) has a special significance. It is coordinate dependent so fix a domain U c M over which T * M is trivial and consider a-n(X,~) as a smooth function on T*U/U (i.e. one omits the zero section). T h e n ( U F ) a = a-n (x, ~)d~l A . . . A d~n A dx I A . . . A dx n is invariant under dilations ~ t~. Thus if R = E ~j(O/O~j) is the radial vector field generating these dilations one has diRa = s = 0 so i r a is a closed ( 2 n - 1) form (here s = iRd + diR is the Lie derivative). Writing dnx = dx I A . . . dx n one has n
i r a = a_n(X,~)a~ A dnx; a~ = E(-1)J-l~d~Cl3 A . . - A d~j A .. . A d~n
(1.22)
1
Of course a~ restricts to the volume'form on the unit sphere [~[ = 1 in each T~M. Upon integration of i r a over these spheres one gets
/]~,=1CZ-n(Y' rl)a~ = [detr
j],=l a_n(X, ~)a~
(1.23)
under coordinate changes x ~ y = r and ~ ~ r / = r with a-n(X,~) ~ 5-n(y,~). As a consequence of (1.23) one obtains a 1-density on M denoted by wres(A) whose local expression is
wresx(A) =
1=1 a
,x
,1 4,
This is the Wodzicki residue density and integrating over M one obtains the Wodzicki residue
Wres(A) = /M wres(A)
=
IS * M i R a = I s
*M
a-n(X'~)~r~dnx
(1.25)
6.1. N O N C O M M U T A T I V E
GEOMETRY
335
where S * M = {(x,~) E T ' M : I~l = 1} is the cosphere bundle over M. The integral may diverge so the domain will be specified below. It turns out t h a t Wres is a trace on the algebra 7) of clasical PSDO, i.e. W r e s ( [ A , B]) = 0, provided d i m ( M ) > 1 and the crucial property of Wres is that, up to scalar multiples, it is the unique trace on P. A little calculation yields
Wres(A) = f
JS *M
Tra_n(X,~)cr~dx 1 A ... A dx n
(1.26)
as a unique (up to multiples) trace on the algebra of classical P S D O whose coefficients are endomorphisms of a given VB over M. In fact one can show t h a t (cf. [942])
§
1
A = n(27r)--------~Wres(A)
(1.27)
Now for M an n-dimensional spin manifold with Dirac operator ~ the operator ~Pl -n is'a first order infinitesimal and is also a P S D O of order - n . Indeed ~ acts on spinors with the symbol i~/(~) where ~/denotes the spin representation so ~p2 has symbol g - l ( ~ , ~ ) l g where g denotes the Riemannian metric. This is a scalar matrix whose size N is the rank of the spinor bundle (recall N - 2 m for n = l m or n = 2m + 1). The symbol of ~1 -n = (~2)-~/2 is then given locally by v/det g ( x ) l ~ l - n l y . More generally when a e Gee(M) is represented as a multiplication operator on the spinor space 7-/the operator a~Pl -n is also a P S D P O of order - n with symbol a-n(X, ~) = a(x)v/det g ( x ) l ~ l - n l g 9 Since the Riemannian volume form is ~t = v/det g(x)dx 1 A . . . A dx n one uses (1.26) - (1.27) to obtain
r _
1 n(2r) n
1
a~[ - n = n ( 2 ~ W r e s ( a ~ l - n =
(1.28)
fs*M Tra_n(X,~)a~dn x - 2[n/2] n/Ma(x)f~ n(2~) n
Thus the commutative integral on functions determined by the orientation [~[ is
/M
{ af~ =
m'(27r)m ~-a~-2m
d i m ( M ) = 2m d i m ( M ) = 2m + 1
(2m + 1)!!Trm+l ~-a~1-2m-1
(1.29)
One should point out also that if a positive compact operator A has eigenvalues ~k(A) then the zeta function of A and Wres are connected via
~A(S) = ~
1
Ak(A)S; ReSs=I~A(S) =
n
1 (27r)nWres(A)
(1.30)
In view of (1.27) it should be possible to prove directly t h a t this zeta residue actually coincides with ~ A and in fact this can be achieved via known Tauberian theorems (cf. [942] for some discussion). 6.1.4
Quantization
and the tangent
groupoid
One wants to explore now the extent to which noncommutative methods permit contact with the quantum world. As one knows from the Connes-Lott theory a perfectly noncommutative geometry may be employed to produce Lagrangians for physical models at the classical level (so noncommutative geometry ~ quantum theory). Nevertheless the foundations of QM
C H A P T E R 6. G E O M E T R Y A N D D E F O R M A T I O N Q U A N T I Z A T I O N
336
do throw up noncommutative geometries (cf. Chapters 3 and 4). In conventional QM the simplest bridge is Wigner-Weyl-Moyal quantization which deforms an algebra of functions on phase space to an algebra of operator kernels and this leads to the tangent groupoid. First let X be a smooth symplectic manifold, p an appropriate multiple of the associated Liouville measure, and 7-/a Hilbert space. A Moyal quantizer for the triple (X, p, 7-/) is a map A of the phase space X into the space of selfadjoint operators on 7-/satisfying ( U J ) Tr A(u) = 1 and Tr[A(u)A(v)] = 5(u - v) for u, v E X such that {A(u); u E X} spans a weakly dense subspace of s Here 5 ( u - v ) means the distributional reproducing kernel for the measure #. Given a Moyal quantizer all quantization (and dequantization) problems are solved in principle via a
---, Q(a) = I x a(u)A(u)dp(u); A ---, WA - Tr[AA(u)]
(1.31)
where A E s Here automatically W1 = I and from (U J) one has WQ(a) = a with Q(WA) = A by irreducibility (in particular Q(I) = I or f x A(u)dp(u) = I). Thus ( U J ) can be reformulated as
Tr Q(a) = I x a(u)d#(u); Tr[Q(a)Q(b)] - / x a(u)b(u)dp(u)
(1.32)
for real functions a, b E L2(X, d#). Moyal quantizers are essentially unique and difficult to obtain (cf. [408]). The standard example is given by T * ( R n) with canonical symplectic structure. One has coordinates u = (q,~C) and d#(q,~) = (27rh)-ndnqdn~. For 7 - / - L2(R n) the Moyal quantizer on T * ( R n) is given by a family of symmetries Ah(q,~) which in the Schrhdinger representation have the form ( U K ) [Ah(q,~)f](x) = 2 n e x p [ 2 i ~ ( x - q ) / h ] f ( 2 q x). The twisted product a • n b of two elements a, b of ST* (R n) is defined by the requirement that Q(a • b) = Q(a)Q(b) and one has evidently
(a • Lh=
b)(u) : / / Lh(u, v, w)a(v)b(w)dp(v)dp(w);
(1.33)
T~[~h(~)~h(~)Ah(~)]= 22n~xp { ---~2i[~(~,v)+~(v,~) + ~(~,~)]}
where s(u, u') = q~' - q'~ is the linear symplectic form on T * R n. Note that f a • h b - f ab. Then by duality the quantization rule can be extended to large spaces of functions and distributions (cf. [407, 822]). The reader is invited here to compare this development of Moyal theory with other discussions in Chapters 1, 3, and 5. First one remarks that K-theory is fairly rigid under deformations of algebras. For C* algebras a 9 homomorphism r A ~ B gives rise to a group homomorphism r 9Ko(A) Ko(B) and in fact it is not necessary to ask for 9 homomorphisms in order to have Ktheory maps since asymptotic morphisms of the C* algebras will suffice. Thus for h0 > 0 an asymptotic morphism A --, B is a family of maps T - {Th; 0 < h _< h0} such that h ~ Tn(a) is norm continuous on (0, h0] for each a E A and satisfies the following norm limits (we use h now for h again)
limh$o[Th(a) + ATh(b) - Th(a + Ab)] = 0; limh$o[Th(a)* -- Th(a*)] = 0; limh~o[Th(ab) -- Th(a)Th(b)] = 0
(1.34)
Two asymptotic morphisms T, S are equivalent if limh$o(Th(a)- Sh(a)) -- 0 for all a E A. In practice one usually has only preasymptotic morphisms Th from a pre-C*-algebra .4 to a C* algebra B satisfying the above conditions and these will give rise to suitable asymptotic
6.1. NONCOMMUTATIVE G E O M E T R Y
337
morphisms (see [942] for details). Now one determines an important asymptotic morphism from Co(T*R n) ----,K:(L2(R n) via the Moyal deformation
[Th(a)f](x) = (27rh) -n /R n a ( x +2 y ' ~) ei((z_y)/h f(y)dyd~
(1.35)
(substituting ( U K ) in (1.31)). Here a E C ~ ( T * R n) or in the Schwartz space S(T*Rn). In t h a t case the operators Th(a) are in fact trace class and uniformly b o u n d e d in h. It also follows from the theory of Fresnel integrals for example that (UL) limhloe-nexp[ixy/e] = (27r)nb(x)5(y) so limhtoLh(u,v,w) = 5 ( u - v ) 5 ( u - w) and the twisted product reduces to the ordinary product in the limit h ~ 0. One can extend these mappings to obtain real asymptotic morphisms Co(T*R n) ~/C(L2(Rn)) (but one does not necessarily obtain compact or even bounded operators). We go now to groupoids and refer to Section 3.9.1 for complementary information; there will be some duplication. By definition a groupoid G --~ U is a small category in which every morphism has an inverse. Its set of objects is U (often written G (~ or Go) and its set of morphisms is G. In practice this means t h a t we have a set G, a set U of "units" with an inclusion U ~-~ G, two maps r, s : G --. U, and a composition law G (2) --. G with domain ( U Z ) G (2) = {(g,h) : s(g) = r(h)} C G x G subject to the following rules (cf. Section 3.9.1): 9 (i) f(gh) = f(g) and s(gh) = s(h) if (g, h) E G (2) 9 (ii) If u E U then r(u) = s(u) = u 9 (iii) f(g)g = g = gs(g) 9 (iv) (gh)k = g(hk) if (g, h) E G (2) and (gh, k) E G (2) 9 (v) each g E G has an "inverse" g -1 satisfying gg-1 = r(g) and g-lg = s(g) Thus any group G is a groupoid with U = {e} and any set X is a groupoid with G = U = X and the trivial composition law x. x = x. Less trivial examples include graphs of equivalence relations, group actions, and fiber bundles with fibrewise addition. A basic and i m p o r t a n t example is the double groupoid of a set X where G = X x X and U = X, included in X x X as the diagonal subset A x = {(x,x) : x E X}. Define r(x,y) = x, s(x,y) = y and then (x, y ) - i = (y, x) with composition law (VA) (x, y ) . (y, z) = (x, z). On the strength of this example one calls U the diagonal of G in general. A further embellishment now is to define a smooth groupoid as a groupoid G -~ U where G, U, and G (2) are manifolds (possibly with boundaries) such t h a t the inclusion i : U ~ G and the composition and inversion operations are smooth maps with r, s : G ~ U submersions. Thus the tangent maps Tgr and Tgs are surjective at each g E G so rank(r) = rank(s) = dim(U). One can add further structure as needed. Now functions on groupoids can be convolved as follows. Suppose t h a t on each r-fiber
G u = {g E G; r(g) = u} there is given a measure Au so t h a t At(g) = gM(g) for all g E G. Such a family of measures will be called a Haar system for G. The inversion map g ~ g-1 carries each Au to a measure ~u on the s-fiber Gu = {g E G; s(g) = u} and the convolution of two functions a, b on G is defined by
(a 9b)(g) = fh
k=g
a(h)b(k) = /C ~(g) a(h)b(h-l g)d)ff(g)(h)
(1.36)
338
CHAPTER 6. GEOMETRY AND DEFORMATION QUANTIZATION
This leads one to define the (reduced) C* algebra of the smooth groupoid G --~ with a given Haar system as the algebra Cr(G ) obtained by completing C~(G) in the norm I l a l l SUpuevllTru(a)[I where Zru is the representation of C~(G) on the Hilbert space L2(Gu, At) determined via
[Tru(a)~](g)= fh k--g a(h)~(k) = fcr(g ) a(h)~(h-lg)dAr(g)(h)
(1.37)
The more canonical method of half-densities is not needed for the examples considered here (cf. Section 3.9.1). When G = M x M with M an oriented Riemannian manifold one obtains just the convolution of kernels (VB) (a 9 b)(x, z) = fM a(x, y)b(y, z)du(y) where du(y) = v/detg(y)dny. Here Cc(M x M) is the usual algebra of kernels of smoothing operators on L2(M) and the C* algebra Cr(M x M) is the completion of C ~ ( M x M) acting as integral kernels on L2(M) so Cr(M x M) ~ 1C. When G = T M is the tangent bundle with the operation of addition of tangent vectors and U - M is included in T M as the zero section with r and s being the fibering ~- 9 T M --~ M then C~(TM) is the completion of the convolution algebra
(a 9b)(q,v) = f a(q,u)b(q,v - u)v/detg(q)dnu Jrq M
(1.38)
where one may take a(q, .) and b(q, .) in C~(TqM). The Fourier transform $- replaces convolution by the ordinary product on the total space T*M giving an isomorphism from C* (TM) to Co(T'M) (also called $'). Thus $'a(q, ~ ) = / T q M e-i~Va(q'v)Idetg(q)dnv;
(1.39)
~-lb(q, v) = (27r) -n f ei~vb(q,~)det-1/2g(q)dn~ JT; M One writes here d#q(V) = detU2 g(q)dv and d#q(~) = (27r)-ndet -U2 g(q)d~. Now the asymptotic morphism involved in Moyal quantization can be given a concrete geometrical realization and a far reaching generalization by the concept of a tangent groupoid. Thus given a manifold M one first forms the disjoint union G ~ = M • M • (0, 1] of copies of the double groupoid of M parametrized by 0 < h <__ 1. Its diagonal is U ~ = M • (0, 1] and one writes G" - T M with diagonal U" = M. The tangent groupoid of M is the disjoint union GM = G ~ G " with UM = U ~ U " as diagonal with composition law ( V C ) (x,y,h).(y,z,h) = (x,z,h) for h > Oandx, y,z E M w i t h (q, vq).(q, wq) - (q, vq+Wq) for q E U and Vq, Wq C TzM. Also (x, y, h) -1 = (y,x, h) and (q, Vq)-1 - (q,-Vq). The tangent groupoid can be given a structure of smooth groupoid GM with b o u n d a r y so t h a t G ~ contains the interior of the manifold and G" is the nontrivial boundary. To see this recall t h a t the normal bundle over a submanifold R of a manifold M can be defined as follows. If j 9 R --. M is the inclusion the tangent bundle of M restricts to R as the pullback j*(TM) = TM[R. This is a VB over R with T R as a subbundle and the normal bundle is the quotient NJ = j * ( T M ) / T R . When M has a Riemannian metric one can identify the fibre N j with the orthogonal complement of TaR in TqM and thus regard NJ as a subbundle of TMIR. At each q E R the map r is 1-1 from a small enough ball in Nq3 into M. If the submanifold R is compact then for some e > 0 the map (q, Vq) --. expq(Vq) with [Vq[ < e is a diffeomorphism from a neighborhood of the zero section in NJ to a neighborhood of R in M (tubular neighborhood theorem). Now consider the normal bundle N/x
6.1. N O N C O M M U T A T I V E G E O M E T R Y
339
associated to the diagonal embedding A 9 M ~ M • M. One can identify A * T ( M • M) with T M 9 T M and thereby ( V D ) N A = {(A(q), (1/2)Vq,-(1/2)Vq); (q, Vq) e T M } giving an isomorphism between T M and N A. As in the tubular neighborhood theorem one can find (if M is compact for example) a diffeomorphism r 9 V1 ~ V2 between an open neighborhood V1 of M in N A (considering M C N A as the zero section) and an open neighborhood V2 of A ( M ) in M • M. Explicitly one can find r0 > 0 such that (VE) r Vq,-Vq) - (expq(Vq/2),expq(-Vq/2)) is a diffeomorphism provided Vq e Nq~ and IVql < r0; then take V1 to be the union of these open balls of radius r0. Now to define the manifold structure of GM give the set G / the usual product manifold structure with outer b o u n d a r y M • M • (1}. In order to attach G" to this as an inner b o u n d a r y consider ( V F ) U1 = {(q, vq, h); (q, hvq) 9 V1} which is an open subset o f T M • [0, 1]; indeed it is the union of T M • {0} and the tube of radius ro/h around A ( M ) • {h} for each h 9 (0, 1]. Therefore the map ~ " U1 --* GM given by ( V G ) r Vq, h) = (expq(hvq/2), expq(-hvq/2), h) for h > 0 and r Vq, O) = (q, Vq) is 1-1 and maps the boundary of U1 onto G". The restriction of 9 to U~ - {(q, vq, h) 9 U1; 0 < h < 1} C T M • (0,1] is a local diffeomorphism between U~ and its image U~ C M • M • (0, 1]. One checks that changes of charts are smooth and thus even if M is not compact one can construct maps ( V G ) locally and patch them together to transport the smooth structur from sets like U1 to the inner b o u n d a r y of the groupoid GM. Now one brings the Gelfand-Naimark maps into play. A function on G M is first of all a pair of functions on G ~ and G" respectively. The first one is essentially a kernel and the second is the (inverse) Fourier transform of a function on T* M. The condition that both match to give a continuous function on GM can be seen as the quantization rule. For clarity one considers first the case M = R n and for a(x, ~) a function on T * R n its inverse Fourier transform gives a function on T R n namely ( V H ) 9r-la(q, v) = [1/(27r) n] fR n exp(i~v)a(q,~)d~ where o n a n the exponentials are given by x = expq(hv/2) = q + (1/2)hv and y = e x p q ( - h v / 2 ) q - (1/2)hv. Thus one can solve for q = ( 1 / 2 ) ( x + y ) and v - ( 1 / h ) ( x - y ) and to the function a one associates the family of kernels
1 n /'R ~ a ( x +2y ' ~)ei(x_y)~/hd~ ka(X, y, h) -- h-njc-la(q, v) = (27rh)
(1.40)
which is precisely the Moyal quantization formula (1.35). The factor h -n is the Jacobian of the transformation after ( V H ) . The dequantization rule is then given by Fourier inversion
a(q,~) =/l~n ka(q + (1/2)hv, q - (1/2)hv)eiV~dv
(1.41)
If M is now a Riemannian manifold one can now quantize any function a on T * M such that ~ ' - l a is smooth with compact support say Ka. For h0 small enough the map r of ( V G ) is defined on Ka • [0, h0) and for h < h0 the formulas ( V H ) and (1.40) must be generalized to a transformation between T M and M • M • {h} whose Jacobian must be determined. One follows here [631] and takes ~/q,tv(8) -- ~/q,v(ts) for the geodesic on M starting at q with velocity v with an affine parameter s. Locally (VI) x - 7q,v(S) and y = 7q,v(-S) with Jacobian [O(x, y)/O(q, v)](s). Set /
J(q, v,
=
g(~qv(-S)
V
detg(q)
a(x,y)
O(q, v) (s)
(1.42)
Thus one has a change of variables formula
M
•
F ( x , y)d~(x)dv(y) =
(1.43)
340
CHAPTER
6. G E O M E T R Y
AND DEFORMATION
Q UANTIZATION
= / M / T q M F(Vq,v(1/2), "yq,,(-1/2))J(q, v, 1/2)dpq(v)dv(q)
The quantization/dequantization recipes are now ka(x, y, h) = h - n j - U 2 ( q , v, h / 2 ) ] : - l a ( q , v);
(1.44)
a(q, ~) = ~[gl/2(q, v, h/2)ka(x, y, h)] where (x, y) and (q, v) are related by (VI) with s = hi2. One can check that g(q, v, hi2) 1 + O ( h 2) and that this defines a real preasymptotic morphism from C o ( T ' M ) to ~ ( L 2 ( M ) ) . Further the tracial property (VK)Tr[Th(a)Th(b)] = fT. M a(U)b(u)dph (U) is satisfied and the corresponding map in K-theory T. 9 K ~ --. Z is an analytical index map which in fact is the index map of Atiyah-Singer theory. Note we are assuming here that the basic features of Atiyah-Singer theory are known (cf. e.g. [46, 171, 277, 724]). From the point of view of quantization theory this is not the whole story of course. For example for a given value of h0 not every reasonable function on T * M can be successfully quantized. In particular for compact symplectic manifolds there are cohomological obstructions (see [320, 321]). A modern point of view (cf. [277]) on this is to consider that quantization is embodied in the index theorem and further one might replace the idea of polarization by the use of Dirac type operators (cf. [121]). Thus e.g. take a smooth manifold M with spin e structure, construct prequantum line bundles L according to some prescription as in [824], and let DL be a twisted Dirac operator for L. Then a quantization of M is the (virtual) Hilbert space TID,L -- ker D + - ker D L whose dimension is given by the index theorem. Under suitable circumstances one can also do this in the equivariant category. Moyal quantization on the other hand is a tool of choice for the proof of the index theorem (cf. [294, 320, 321] and remarks in Section 3.8). Thus Moyal quantization seems absolutely fundamental and we have tried to indicate many of its features and some connections to index theorems in this book (cf. Chapters 1, 2, 3, 4, and 6). We have given a few examples in Chapter 3 (Example 3.6 and 3.7- cf. also Example 6.5 in Section 6.2.4) and will indicate here a few more examples of quantum spaces and algebras in a simplified form. Thus for symbols x, y, z (cf. [825]) EXAMPLE
6.4. Typical small situations are"
9 The quantum q-plane (q ~ e x p ( a h ) ) arises from the associative algebra generated by x and y with xy = qyx. 9 the q-Heisenberg algebra Hq is generated by x, y, z satisfying (B83) x z = qzx, zy = qyz, and x y - qyx = z. 9 The first quantum Weyl algebra is obtained from Hq by adding the relation (84) x y q-lyx)z = 1 = z(xy - q-lyx). 9 The q-enveloping algebra Uq(sl(2)) of s/(2) is defined via (B85) x z = qzx, zy - qyz, and x y - y x = (z - z - 1 ) / ( q - q-l). Let us expand a little now on Mq(2) from a different point of view following [680] (note for any full understanding of all this one should consult [187, 672] in addition to Chapter 3)'Thus(usingq-linplace~176176176
( ac db)
(a, b, c, d are symbols- see below) defined via (B87) Mq(2) - C < a, b, c, d > / I where
6.2. GAUGE T H E O R I E S
341
I ,.~ (ab - q-lba, ac = q-lca, cd = q-ldc, bd = q-ldb, bc = cb, a d - da = (q-~ - q ) b c ) and one can prove that monomials aabZc~d ~ form a basis of Mq(2). The quantum plane is (defined by) the ring (B86) Aq21~ = C < x, y > / ( x y - q-lyx) where C < x, y > is the associative algebra freely generated by x, y. This is a deformation of the x, y plane where q = 1. For A a ring an A point of Mq(2) is a quadruple (a, b, c, d) E A n satisfying (B87). One can show that if (a, b, c, d) and (a', b', c', d') are A points of Mq(2) then
c d
c' d'
E Mq(2); D e t q [ M M ' ] - DetqMDetqM'
(1.45)
and Dq = DetqM = a d - q - l b c = da-qcb commutes with a, b, c, d. Further if Dq is invertible then
M - 1 = Dql I - q -dl c
-qb a )
(1.46)
is an A point of Mq-l(2). Formally to define now noncommutative group spaces one writes e.g. (B88) GLq(2) ~ Mq(2)[t]/([t,a], It, b], It, c], It, d], tDetq - 1) and SLq(2) Mq(2)/(Detq - 1) . One can consider e.g. A~ '~ as the noncommutative space upon which GLq(2) acts as an analogue of the of the fundamental two dimensional representation. These matters are all essentially covered in Chapter 3 in a more sophisticated manner and we refer to [187, 672, 680] for a complete discussion.
6.2
GAUGE
THEORIES
Notations ( W A ) , ( X A ) , ( Y A ) , and ( Z A ) will be used. An interesting and important connection of Fedosov type deformations and noncommutative geometry arises in the study of gauge theory for YM fields. In particular D-brane worldvolume theory in a constant Bfield background is described by a so-called noncommutative YM theory whose multiplicative product is the Moyal-Weyl product (cf. [29, 45, 64, 174, 199, 201, 202, 216, 261, 262, 368, 371, 445, 478, 493, 502, 508, 510, 674, 846, 847, 857, 914, 923]). We will sketch below the second paper of [45]; this gives many explicit calculations and provides in addition details about Fedosov deformations omitted in Section 3.8 (see Section 6.2.3 for a perhaps more intuitive point of view). First however we want to sketch some more conceptual background material and here [202, 216, 508, 510, 595, 668, 780, 923, 857] are especially useful. 6.2.1
Background
on noncommutative
gauge theory
The obvious starting point is [857] where noncommutative gauge theory is developed from coordinates in R n with (YA) [x k, xJ] = iOaj for real 0. Then one deforms the function algebra on R n to a noncommutative associative algebra ,4 such that f 9g = fg + (i/2)okJokfO)jg + 0(02) with coefficients being local bilinear differential expressions in f, g. The essentially unique solution (cf. Section 3.8) is given by the Moyal product
(f 9g)(x) = e(i/2)Ok~(O/O~k)(O/O~J)f ( x + ~)g(x + ~) ~=~=0
(2.1)
and one can assume N x N matrix valued f, g, with 9 corresponding to the tensor product of matrices plus the 9 product of functions as in (2.1). For suitable f, g as in (2.1) vanishing at oc one can show that (YB) f T r ( f 9g) = f T r ( g . f). For ordinary YM theory now one writes the gauge transformations and field strength equations as
5;~Aj = Oj)~ + i[A, Aj]; 5~Fkj = i[)~, Fkj]; Fkj = OkAj - (9jAk - i[Ak, Aj]
(2.2)
342
C H A P T E R 6. G E O M E T R Y A N D D E F O R M A T I O N Q U A N T I Z A T I O N
The Wilson line is W(a, b) = Pexp[i fb A] where in the path ordering A(b) is to the right. Under the gauge transformation (2.2) one has (YC) 8W(a, b) = i,~(a)W(a, b ) - i W ( a , b))~(b). For noncommutative gauge theory one uses the same formulas for the gauge transformations and field strength with matrix multiplication replaced by t h e , product. Thus the gauge parameter ~ takes values in A tensored with N x N Hermitian matrices for some N and the same is true for the components .~j of the gauge field A. Thus
Fkj -- OkAj - Ojfik -- iAk * Aj + iAj 9 Ak
(2.3)
The theory obtained in this manner reduces to conventional U(N) YM for 0 ~ 0 but because of the way that the theory is constructed from associative algebras there seems to be no convenient way to get other gauge groups. The commutator of two infinitesimal gauge transformations with generators ~1 and ~2 is a gauge transformation generated by i(~1 * ~2 - ~2 * ~:). Such commutators are nontrivial even for the rank one case, i.e. N - 1, although for 0 = 0 the rank one case is the Abelian U(1) gauge theory. For rank one, to first order in 0 (2.3) reduces to
~,4j = ojS, - OkeOkS,0eAj +
0(O2); ~P~j
= -ok%5,oeP~ +
-f'ij = c?iAj - OjAi + OkeOkAiOeAj + 0(02)
0(o2); (2.4)
Note that the opposite of a noncommutative YM field is called an ordinary YM field rather that a commutative one since for nonabelian gauge groups to speak of ordinary YM fields as commutative would be confusing (cf. also [844]). The discussion in [857] indicates that ordinary and noncommutative YM fields arise from the same two dimensional field theory regularized in different ways. Consequently there must be a transformation from ordinary to noncommutative YM fields that maps the standard YM gauge invariance to the gauge invariance of noncommutative YM theory. This should also be local in the sense that to any finite order in perturbation theory in 0 the fields and parameters are given by local differential expressions in the ordinary fields and parameters. At first sight a local field redefinition ,7t = fi(A, OA, 0 2 A , . . . ; O) seems indicated but this must be relaxed. If there were such a map intertwining the gauge invariances it would follow that the ordinary YM gauge group would be isomorphic to the gauge group of noncommutative YM theory. This cannot be the case since for rank one the gauge group acting by 8Aj = OjA is Abelian while the noncommutative gauge invariance acting by ~Aj - Oj A + iA * Aj - iAj * A is nonabelian. These cannot be isomorphic so no redefinition of the gauge parameter can map the ordinary gauge parameter to the noncommutative one while intertwining with the gauge symmetries. What one needs for the physics is less than an identification between the two gauge groups; rather one only needs to know when two gauge fields A and A' should be considered gauge equivalent. Thus it turns out that one can map A to ~i in a way that preserves the gauge equivalence relation even though the two gauge groups are different. In practice this means that one can find a mapping from ordinary gauge fields A to noncommutative gauge fields ft. which is local to any finite order in 0 and has the following further property. Suppose that two ordinary gauge fields A and A' are equivalent by an ordinary gauge transformation U = exp(i)~). Then the corresponding noncommutative gauge fields .4 and .4' will also be gauge equivalent by a noncommutative gauge transformation gr = exp(is However s will depend on both ~ and A. If ~ depended only on ~ the ordinary and noncommutative gauge groups would be the
6.2. GAUGE T H E O R I E S
343
same but for ~ = ~()~, A) there is no well defined mapping between the gauge groups while there is an identification of the gauge equivalence relations. Thus there will be a well defined equivalence relation but the equivalence classes are not the orbits of any useful group or are such orbits only on shell. Certain limitations of this point of view are pointed out in terms of nonperturbative theory. In the first paper in [45] certain apparent mathematical ambiguities are displayed in this point of view but one can dismiss them if a physical input is required such that a noncommutative gauge field would be defined only up to field redefinitions; in that case the physics does not generally depend on paths in 0 space.
6.2.2
The Weyl bundle
In the second paper of [45] one gives a construction of some noncommutative YM theories with a nonconstant B-field background from the point of view of purely world volume theory. The idea is as fo~llows. The Moyal-Weyl product appears originally in the deformation quantization of a 2n and this scheme is generalized to the quantization of any symplectic or Poisson manifold via a star product. If one regards this not as a quantized space, but as noncommutative geometry, a field theory with such a product is the generalization of the noncommutative YM theory. The deformation quantization of a Poisson manifold (M, {, }) was first defined and investigated in [624] (cf. Section 3.8). Thus let Z = C~(M)[[h]] be a linear space of formal power series of the deformation parameter h (again we use h in place of h); a typical element is then (WA) f - ~ hkfk and deformation quantization involves an associative algebra structure on Z with ( W B ) f 9 g = ~ hkMk(f, g) where the Mk are bidifferential operators such that Mo(f,g) = f g and M l ( f , g ) - Ml(g, f ) = - i { f , g } (cf. Sections 3.8 and 3.9 - there will be some duplication). Two star products "1 and *2 are equivalent if there is an isomorphism of algebras T : (Z, *1) ~ (Z, *2) given by a formal power series of differential operators T = To + hT1 + . . . . The problem of existence and classification is discussed in Section 3.8 (cf. [68, 88, 219, 179, 624]) and here one considers the symplectic case only, following [320, 321], since the geometrical constructions based on the Weyl algebra bundle are relatively simple. The idea is that the tangent space of a symplectic manifold is a symplectic vector space and can be quantized by the usual Weyl-Moyal product. This gives a bundle of algebras - sort of a quantum tangent bundle - on which a flat connection is constructed involving some quantum corrections to the usual aKine connection. The flat sections of this connection can be identified with Z = C~(M)[[h]] so the product on fibers induces a 9 product on Z. One then shows that automorphisms of the Weyl algebra bundle can be regarded as (infinite dimensional) gauge transformations relating equivalent star products. The subalgebra of automorphisms which preserve a star product corresponds to so-called noncommutative gauge transformations and the constructions in [45] are explicit. Thus, with some repetition from Sections 3.8 and 3.9, we consider a symplectic manifold (M, ~t0) of dimension 2n as the base space for gauge theories. The formal Weyl algebra bundle with twisted coefficients involves specifying a symplectic VB (L,~) over M of dimension 2n which is isomorphic to T M with a fixed symplectic connection VL satisfying ( W C ) 0 : T M ~ L; and 5 : L* ~ T*M. A local symplectic frame ( e l , . . . , e2n) Of L and a dual frame (el, ... , e 2n) of L* correspond to local vector fields X j and one forms 0i giving bases of T M and T * M respectively, via
(2.5) < ~.~j >L=< e~.e(x~) >L=< ~(~). xj >~.=< r
>VM= ~
C H A P T E R 6. G E O M E T R Y A N D D E F O R M A T I O N Q U A N T I Z A T I O N
344
The symplectic form w on L is m a p p e d to T M to give a nondegenerate 2-form on M via ( W D ) ~t0 = - ( 1 / 2 ) w i j 0 ~ A oJ and one assumes dFt0 = 0. Further introduce a complex VB g" with a connection Vc and a coefficient bundle H o m ( $ , $ ) = .,4 which is treated as a U(N) gauge bundle. One quantizes the fiber Lx by the standard Moyal-Weyl product so Wx(L,.A) is defined as a noncommutative algebra with unit with elements ( W E ) a(y, h) = ~-~2a+p>_O,a>_o(ha/p!)aa,i~,...,ipyi~... yip where y = ( y l , . . . , ygn) ~ coordinates on the fiber Lx and ak,il,...,ip E .,A.x. The product is then oc 1
aob= ~
0
~.
o2ili1., . . . o.)in ,in., 0 Oyil
-
__0
__0
Oyin a OyJl
-(Oyjn -0t ,
(2.6)
One assigns degree 2 to h and 1 to yi so each term in ( W E ) has degree 2k + p _> 0 (note also t h a t o preserves degree). Let now W ( L , A ) = UxeMWx(L,A) be the formal Weyl algebra bundle (W-bundle) with sections a(x,y,h)
-
hk E -~.ak,il,...,ip(X)y il ...y~P 2k+p>O,k>O
(0,.7)
where ak,il,...,i p is a section of ,4 so here it is an N • N matrix valued symmetric covariant tensor field. The space of sections C ~ ( M , W ( L , A ) also forms an associative algebra but with fiberwise o product and this will also be denoted by W(L,.A). Note t h a t there is a natural filtration W(L,.A D WI(L,.A) D W2(L,.A) D . . . with respect to the degree 2k + p assigned above. The center Z of W(L, .,4) consists of sections which do not depend on yi and have values in multiples of the identity in .A (i.e. they are diagonal U(1) valued) and one identifies Z with CCr A differential form on M with values in W(L, .,4) is a section of the bundle W(L, .,4) | A (where A ~ AT*M) expressed locally via
a(x,y, h) =
21t
2k+p>O,k>>O
Z
l
9
9
A... A
(2.8)
q=O
Here ak,il,...,ip,jl,...,jq is a covariant symmetric tensor in the ils and an antisymmetric tensor in the jls. The sections of W(L, A) | A form an algebra with multiplication defined by the wedge product of the oJ, the o product of polynomials in yi, and the matrix product of coefficients. Let deg(a) be the rank of the differential form a. Then W(L, .4) | A is formally a Z • Z graded algebra with respect to o and the graded c o m m u t a t o r is ( W F ) [a, b] = a o b - (--1)deg(a)deg(b)b o a. The central p-form in W(L,.4) | A is ( W G ) Z | A -- {c C W ( L , A ) | AP; [c,a] = 0} for all a E W ( L , A ) | A (thus it has no yi dependence and a diagonal U(1) valued p-form); the filtration is preserved. Now the connections ~7L and ~'E generate a basic connection V = V L | V$ on W(L,.A) and its induced covariant derivative V : W ( L , A) @/k q --+ W(L,.A)A q+l can be expressed via
Va = da + -~ -- d a -
E
2k+p> 1
Fijy~y j, a + [Fc, a] =
(2.9)
" "Y~POJl " hk Ec~ ( p _ 11)!q! F -m ~P A ak'm'il'""ip-ljl'""JiyZl'" A . . . A 0 jq + [FE, a] 0
where d = dx"O, is an exterior derivative, Fij = wikF k are local connection 1-forms for VL, and FE is a local connection 1-form for VE. In particular ( W H ) V ( a o b) = Va o b +
6.2.
GAUGE THEORIES
345
(--1)deg(a)a o V b and V L ( a o b) = V L a o b + (--1)deg(a)a o VLb (cf. [45] for further details on the calculations). There are two other canonical operators on W ( L , A ) expressed locally via
cO " W p ( L , A ) | A q ~ W p _ I ( L , A ) | Aq+l; 5 = 0 i A ~yi
5-1=
1
i
V4-~Y A0 I ( X i )
(2 10)
pp ++ qq = > O 0 " W p ( L , A | A q ---+ W p + I ( L , A ) | A q-1
where I(X~) is the interior product, p is the number of y's, and q = deg(a). Further ( W I ) 52 = 0, (5-1) 2 = 0 < and 5(hob) = 5aob+(-1)deg(a)aohb with a = 5 5 - 1 a + 5 - 1 5 a + a o o where a00 is the component in Z | A~ The last relation is similar to Hodge but note that 5 is purely algebraic containing no derivative in x so ( W J ) 5a - -(i/h)[a~ijyiO j, a]. Now the main idea of Fedosov quantization is to construct an "Abelian" connection on the Weyl bundle for which flat sections are identified with the q u a n t u m algebra C ~176 (M)[[h]]| ,4. In this direction, given a linear connection V as in (2.9) on W ( L , A ) | A one considers more general n o n l i n e a r c o n n e c t i o n s D on W ( L , ,4) | A of the form ( W K ) D a = V a + (i/h)['y,a] where the 1-form "), is a global section of W ( L , A ) | A 1. D is clearly a graded derivation with respect to the o product (i.e. D ( a o b) = D a o b + (--1)deg(a) o Db) and one notes that 7 is determined up to central 1-forms because it appears in the c o m m u t a t o r (in [320, 321] this ambiguity is fixed by the normalization condition to be 0 but here it is not fixed but denoted by C~). Simple calculation gives ( W L ) D ~ a = (i/h)[f~,a] for a E W ( L , ,4) | A where fl is the curvature of D given by (2.11)
with Rij a symplectic curvature of V L and RE a field strength of Ve (note a symplectic connection VL - i.e. VLa~iy = 0 - satisfies Rij = Rji). Now D is called an Abelian connection i f D 2 a = 0 for a l l a C W ( L , A ) N A , i.e. f~is a c e n t r a l 2 - f o r m f~ E Z | 2. In this case the Bianchi identity implies t h a t Df~ = df~ = 0 or f~ is closed. The condition f~ E Z | restricts ~/since Fedosov proved that for a given V there exist Abelian connections of the form i i yiO j -F- r, a]; D a = V a - 5a + -~[r, z] = V a + -~[Cdij (2 . 12) :
~ 0 nt - ~ l ;
1
~ 0 -- ---~CdijOi AO j
where f~0 is a symplectic form on M and f~l is a closed central 2-form which contains at least one power of h. More precisely for any choice of f~ = f~0 + O(h) C Z | and # E W ( L , ,4) with deg(#) >_ 3 the conditions in (2.12) that r give an Abelian connection are ( W M ) 5r = V(wijyiO j) + R - ~1 + V r + ( i / h ) r o r with 5 - 1 r = #. Using H o d g e - d e R h a m ( W I ) this is equivalent to ( W N ) r = 5# + 5-1(V(aJijyiO j) + R - f~l) + 5 - 1 ( V r + ( i / h ) r o r). Since V preserves the filtration and 5 -1 raises it by 1, this equation can be solved uniquely by iteration. Therefore Abelian connections are the family parametrized by the d a t a V, fi, and # (i.e. by wij, 0 i,Fj,i F g , ~~1, and #). One regards this process as n o n c o m m u t a t i v e deformation and simply refers to background fields. If r is further decomposed to r = rs + ra (where s ~ symmetric) we have ( W O ) rs = ~2k+e>_2(hk/g!)rk,(il...i~,j)Y[ "" "YieoJ
CHAPTER 6. G E O M E T R Y AND DEFORMATION Q UANTIZATION
346
whee (i1," "', i~, j) means the symmetrization of indices. ( W M ) and ( W N ) can accordingly be rewritten in terms of rs and ra as i
6-1rs = #; 6% = 0; Vsa = Va + -s 6ra = V(coijyiO j) + R -
i ~1 + Vrs -~- -s
a]; 6-ira = 0;
i o rs + Vsra + -s
(2.13)
o re
This implies rs = ~2k+e>2(hk/g!)#k,il,...,i~,jy i1"'" yieOJ and
re : 6 -1
(V(coiyy~O3) + R -
,
f~l + Vrs + -s
o rs
+
Vsra + -s
o ra
)
(2.14)
Thus roughly speaking p completely determines rs corresponding to nonlinear (quantum) corrections to V L (cf. [295]). On the other hand ra corresponds to a nonlinear correction to Fc and 6; note also that f~l appears only in the combination ihRE + f~l so t h a t it can be regarded as the correction to the U(1) part of RE. For an A b e l i a n c o n n e c t i o n D one now defines the space A WD of all flat sections in W(L, .4) | A by ( W P ) A WD = {a E W(L, ,4) | A; Da = 0} = ker(D) A W(L, .4) | A. Since D is graded, ADw is a subalgebra of W(L,.4NA, namely hob E WD. One also defines APWD to be the space of flat p-forms and for p = 0 this is written WD = A~ . Fedosov proved t h a t WD can be identified with C~(M)[[hl] | which is the q u a n t u m algebra of observables. In the present case C~[[h]] | A is regarded as a space of fields (as a C~(M)[[h]] bimodule). In fact let a be the projection ( W Q ) a: W ~ C~174 defined by a ~ a(a) = aly=0. T h e n for any ao C CCC(M)[[h]] | ,4 there is a unique section a E WD such t h a t a(a) = a0. This is seen by rewriting da = 0 as ( W R ) a = a0 + 6 - 1 ( D + 6)a which can be uniquely solved by iteration. Therefore a produces an isomorphism between WD and C ~ (M)[[h]] | as vector spaces. Moreover let Q be the inverse of a so ( W S ) Q : C~(M)[[h]] | .4 ~ WD with ao ~ Q(ao) = a; then this isomorphism induces a noncommutative associative algebra structure on C ~(M)[[h]] N `4 which is a 9 product. This is defined through the o product in WD as ( W T ) a0 * b0 = a(Q(ao) o Q(bo)) (examples are given in [45]). Now consider two connections D and D ~ and the corresponding isomorphism between
(WD, o) and (WD,, o) which also induces an isomorphism between (CCC(M)[[h]] | `4, ,) and (C~(M)[[h]] | .4, ,'). One begins with an automorphism of W(L, .4) | A via an extension W + of W(L,.4) with elements cc
hk
U : ~ ~ -~Uk,il,...,ipY i l . . . y*P; Uk, i E C ~ ( M ) @ .A i=0 2k+p=g "
(2.15)
(note there may be negative powers of h). Further introduce a group of invertible elements of W + with leading term 1 having the form
U = exp
( i ) s -~H3 =
0
~.
-~H3
k
; H3 E W~(L,A)
(2.16)
where W~ (C W3) consists of elements whose diagonal U(1) part is in W3 and whose SU(N) part has at least one power of h in W3 (this restriction is needed to give an a u t o m o r p h i s m of W(L, A) | A - but not W + N A). By the BCH formula such elements form a group. Now consider the map ( W U ) a ~ U -1 o a o U = ~ ( 1 / k ! ) ( - i / h ) k [ H 3 , [ H a , . . - , [/-/3, a],.--]] as a fiberwise automorphism of W ( L , A ) | A (note this map preserves the filtration but not
6.2.
GAUGE THEORIES
347
the degree - i.e. a -+ a + O(h, yi)). One could also consider more general automorphisms of W ( L , Jr) | A which move the supports of sections. Thus for a diffeomorphism f 9M --~ M its symplectic lifting to L and its lifting to C, one defines an automorphism ( W V ) A 9a f , ( U -~ o a o U) for a E W ( L , A) | A where the liftings, pullback f*, and pushforward f , are defined via h i ( x ) 9 Lz ~ L/(x); (hi(x) )kwk t ( f ( x ) ) ( a / ( x ) ) j t = ~ij(x) (2.17)
v(x) " Sx --+ E/(=); f~
y, h) = a ( f ( x ) , a / ( x ) y , h);
f* a(x, y, h) = v ( x ) - l a ( f (x), a / ( x ) y , h)v(x) = v(x) -lf0*a(x, y, h)v(x) f , a ( x , y, h) = ( f * ) - l a = v ( f - l ( x ) ) a ( f - l ( x ) , = f~
y~, h ) v ( x ) - l ) ; f~
(a/(f-l(x)))-ly,
h ) v ( f - l ( x ) ) -1 =
y, h) = a ( f - l ( x ) , ( a f ( f - l ( x ) ) ) - l y ,
h)
If f is an identity then v(x) is a usual U ( N ) gauge transformation and (Tf(x) is a local Sp(n) transformation which is an analogue of the local Lorentz transformation in the gravity theory (i.e. in Riemannian geometry). A fibrewise automorphism A is a quantum correction of both transformations. In general v(x) acts only on the fiber while a / ( x ) is a necessary part of the pullback f0,. Therefore one treats them differently as follows (see below for more development). Since v(x) acts on W ( L , A ) | A in the same way as U we may include v(x)'s in the space of U so that the automorphism is reexpressed as
d " a ~ f~
o a o U); U = e x p ( h )H2
= f i l ~. ( h )H2 k
(2.18)
0
for a E W(L, j r ) | A and/-/2 e W~(L, ,4) where W~(L, A) consists of the sum of W~(L,M) and hT-/with 7-I being a section of A. In fact W~(L, .4) forms an (infinite dimensional) Lie algebra as a linear space which includes su(N). If f is an identity and its symplectic lifting is trivial (i.e. a/(x)} = 5~) then the corresponding automorphism A will be called a g a u g e t r a n s f o r m a t i o n on W ( L , .4) ("gauge transformation" in [45]). For an A b e l i a n c o n n e c t i o n D an automorphism (2.18) defines a new connection called an image of D via ( W W ) D'a = A D ( A - l a ) which is also an Abelian connection since (D')2a = A D 2 ( A - l a ) = 0. Restricting the domain of A from W ( L , A ) | A to AWD, any automorphism A defines an isomorphism A : A WD ~ A WD,. In fact if Da = 0 then D'a' = D'(Aa) = A(Da) = 0. Moreover this isomorphism immediately induces an equivalence of two 9 products via the map T : C~(M)[[h]] | A A ~ C~(M)[[h]] | defined as ( W X ) T : ao ---* Q-1A-1Q'(ao) (note a = Q-1 on WD (cf. ( W S ) ) . One obtains then (cf. (WW)) the relation ( W V ) a0 *' b0 = T - l ( T a o 9 Tbo) showing that T determines an equivalence of 9 products. Note here
A - l a ( x , y, h) = U o fO, a(x, y, h) o U -1",
(2.19)
A - l ( a o b) = ( A - l a ) o (A-lb); T -1 = ( Q ) ' -1AQ In more detail one can rewrite an Abelian connection D in a somewhat different form as i
Da = VLa + ~[~/T, hi; ~ T -- -ihFE + wijy~O3 + r;
1
-- -~Rij
yiyj
i
"~- V L ~ / T + - ~ / T o "YT; a E W ( L , A )
| A
(2.20)
348
C H A P T E R 6. G E O M E T R Y A N D D E F O R M A T I O N Q U A N T I Z A T I O N
Thus FE and 3' are treated together. One sees that "TT is nothing but a g a u g e field with respect to the g a u g e t r a n s f o r m a t i o n U (and 3' is the gauge field of U in (2.16) while FE is that of v(x)). In fact ( W W ) is expressed by these variables as v'L =
=
o
o u -
o vLu
(2.21)
+ hc.,)
where the first line is adopted canonically. C.~ E Z | A 1 is an ambiguity of ~/T coming from its center which is harmless in the graded commutator (note that from the construction of ~'T (or r) one has deg(TT) _> 2 so its central part has at least one power of h). In the case of a g a u g e t r a n s f o r m a t i o n (i.e. fo = id case of A) Vr is mapped as required for a gauge field. Further it can be read off following relations from the second term in (2.21), namely i
D U = U o ~(f~
- 7 T -- hC~); fo, f~,_ f ~ _ hdC.y - 0; f
0,
f~0 = f~0
(2.22)
where the first line is given by rewriting (2.21), the second by operating with D on the t i j first equation, using RijY y = ft,0 (Rijyiyj), and the third is the leading term of the second equation in h. These equations designate the conditions for existence of an automorphism of the form (2.18) when two arbitrary Abelian connections D, D ~ are given. The last equation in (2.22) means that the map f should be a symplectomorphism on M with respect to f~0. The second equation in (2.22) states that f~ and f0,f~ should be in the same second cohomology class. The first equation in (2.22) is equivalent to ( W Z ) U = a(U) + 6-1((O + 6)U - (i/h)U o (f~ r - ")IT - - hC.~)) which determines U uniquely by iteration for given o(V), fo, "YT, V~. In fact it is proved in [321] that there exists a fiberwise automorphism of the form (2.18) if the curvatures f~, fff belong to the same cohomology class and their leading terms in h coincide. This shows that any two equivalent 9 products or algebras (C~(M)[[h]] | .4, ,) are related by a combination of symplectomorphisms and gauge transformations. Although one has seen that any automorphism A of W ( L , . A ) | induces an isomorphism A : A WD ----* A WD, we are interested in the particular case that is also an automorphism of AWD, namely (XA) Da = A D A - l a . In this case a 9produce is clearly invariant, i.e. ( W X ) - ( W Y ) become (XB) T = Q - 1 A -
