Analysis, Manifolds and Physics Part II

ANALYSIS, MANIFOLDS AND PHYSICS Part II This Page Intentionally Left Blank ANALYSIS, MANIFOLDS AND PHYSICS Part II ...

Author: Y. Choquet-Bruhat | C. DeWitt-Morette

174 downloads 1722 Views 18MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

ANALYSIS, MANIFOLDS AND PHYSICS Part II

This Page Intentionally Left Blank

ANALYSIS, MANIFOLDS AND PHYSICS Part II

by YVONNE CHOQUET-B RUHAT Membre de l' Acaddmie des Sciences, Universitd de Paris VI, Ddpartement de Mdcanique Paris, France

CI~.CILE DEWITT-MORETTE University of Texas, Department of Physics and Center for Relativity, Austin, Texas, USA

REVISED

AND

ENLARGED

EDITION

2000 ELSEVIER AMSTERDAM

9L A U S A N N E

9N E W Y O R K

9O X F O R D

9SHANNON

9S I N G A P O R E

9T O K Y O

E L S E V I E R S C I E N C E B.V. Sara Burgerhartstraat 25 EO. Box 211, 1000 AE Amsterdam, The Netherlands 9 2000 Elsevier Science B.V. All rights reserved. This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier Science Global Rights Department, PO Box 800, Oxford OX5 1DX, UK; phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: [email protected]. You may also contact Global Rights directly through Elsevier's home page (http://www.elsevier.nl), by selecting 'Obtaining Permissions'. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (978) 7508400, fax: (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W 1P 0LP, UK; phone: (+44) 171 631 5555; fax: (+44) 171 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier Science is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier Science Global Rights Department, at the mail, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. [First edition 1989] Revised and enlarged edition 2000: ISBN 0-444-50473-7 Set (together with "Analysis, Manifolds and Physics - Part I" (revised edition)): ISBN 0-444-82647-5 Library of Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for.

O T h e paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.

P R E F A C E TO T H E S E C O N D E D I T I O N

Twelve problems have been added to the first edition; four of them are supplements to problems in the first edition. The others deal with issues that have become important, since the first edition of Volume II, in recent developments of various areas of physics. All the problems have their foundations in Volume I of the 2-Volume set Analysis, Manifolds, and Physics. It would have been prohibitively expensive to insert the new problems at their respective places. They are grouped together at the end of this volume, their logical place is indicated by a number in parenthesis following the title. The new problems are: 9 "The isomorphism H (9 H _~ M4(R). A supplement to Problem 1.4 and 1.3 (I. 17)." Its logical place is the seventeenth problem of Chapter I. 9 The problem "Lie derivative of spinor fields (III.15)" belongs to Chapter III. 9 "Poisson-Lie groups, Lie bialgebras, and the generalized classical Yang-Baxter equation (IV. 14)" has been contributed by Carlos Moreno and Luis Valero. It belongs to Chapter IV. Additions to Chapter V on Riemannian and K~ihlerian manifolds include: 9 "Volume of the sphere S n . A supplement to Problem V.4 (V. 15)" 9 "Teichmuller spaces (V. 16)" 9 "Yamabe property on compact manifolds (V. 17)" To Chapter V bis on Connections are added: 9 "The Euler class. A supplement to Problem V bis 6 (V bis 13)" 9 "Formula of Laplacians at a point of the frame bundle (V bis 14)" 9 "The Berry and Aharanov-Anandan phases (V bis 15)" based on notes by Ali Mostafazadeh. To Chapter VI on Distributions: 9 "A density theorem. A supplement to Problem VI.6 on 'spaces Hs.~ (Rm) ' (VI.17)" 9 Tensor distributions on submanifolds, multiple layers, and shocks (VI. 18)" 9 "Discrete Boltzman equation (VI.19)"

PREFACE TO THE SECOND EDITION

A fair number of misprints have been corrected. An updated list of errata for Volume I is included. Naturally more problems are on our drawing boards. We would like to think of them as contributions to a third edition. Most of the new problems were completed during a visit of Y. ChoquetBruhat to the Center for Relativity of the University of Texas, made possible by the Jane and Roland Blumberg Centennial Professorship in Physics held by C. DeWitt-Morette. Help and comments from M. Berg, M. B lau, M. Godina, S. Gutt, M. Smith, R. Stora, X. Wu-Morrow and A. Wurm are gratefully acknowledged.

PREFACE

This book is a companion volume to our first book, Analysis, Manifolds and Physics (Revised Edition 1982). In the context of applications of current interest in physics, we develop concepts and theorems, and present topics closely related to those of the first book. The first book is not necessary to the reader interested in Chapters I-V bis and already familiar with differential geometry nor to the reader interested in Chapter VI and already familiar with distribution theory. The first book emphasizes basics; the second, recent applications. Applications are the lifeblood of concepts and theorems. They answer questions and raise questions. We have used them to provide motivation for concepts and to present new subjects that are still in the developmental stage. We have presented the applications in the forms of the problems with solutions in order to stress the questions we wish to answer and the fundamental ideas underlying applications. The reader may also wish to read only the questions and work out for himself the answers, one of the best ways to learn how to use a new tool. Occasionally we had to give a longer-than-usual introduction before presenting the questions. The organization of questions and answers does not follow a rigid scheme but is adapted to each problem. This book is coordinated with the first one as follows: 1. The chapter headings are the s a m e - but in this book, there is no Chapter VII devoted to infinite dimensional manifolds per se. Instead, the infinite dimensional applications are treated together with the corresponding finite dimensional ones and can be found throughout the book. 2. The subheadings of the first book have not been reproduced in the second one because applications often use properties from several sections of a chapter. They may even, occasionally, use properties from subsequent chapters and have been placed according to their dominant contribution. 3. Page n u m b e r s in parentheses refer to the first book. References to other problems in the present book are indicated [Problem Chapter Number First Word of Title]. The choice of problems was guided by recent applications of differential geometry to fundamental problems of physics, as well as by our personal vii

viii

PREFACE

interests. It is, in part, arbitrary and limited by time, space, and our desire to bring this project to a close. The references are not to be construed as an exhaustive bibliography; they are mainly those that we used while we were preparing a problem or that we came across shortly after its completion. The book has been enriched by contributions of Charles Doering, Harold Grosse, B. Kent Harrison, N.H Ibragimov, and Carlos Moreno, and collaborations with Ioannis Bakas, Steven Carlip, Gary Hamrick, Humberto La Roche and Gary Sammelmann. Discussions with S. B lau, M. DuboisViolette, S.G. Low, L.C. Shepley, R. Stora, A. H. Taub, J. Tits and Jahja Trisnadi are gratefully acknowledged. The manuscript has been prepared by Ms. Serot Almeras, Peggy Caffrey, Jan Duffy and Elizabeth Stepherd. This work has been supprted in part by a grant from the National Science Foundation PHY 8404931 and a grant INT 8513727 of the U.S.-France Cooperative Science Program, jointly supported by the NSF and the Centre National de la Recherche Scientifique.

CONTENTS

Preface to the second edition Preface Contents Conventions

v

vii ix xv

I. REVIEW OF FUNDAMENTAL NOTIONS OF ANALYSIS

.

2. 3. 4. 5. 6. 7. 8. ,

10. 11. 12. 13. 14. 15. 16.

Graded algebras Berezinian Tensor product of algebras Clifford algebras Clifford algebra as a coset of the tensor algebra Fierz identity Pin and Spin groups Weyl spinors, helicity operator; Majorana pinors, charge conjugation Representations of Spin(n, m), n + m odd Dirac adjoint Lie algebra of Pin(n, m) and Spin(n, m) Compact spaces Compactness in weak star topology Homotopy groups, general properties Homotopy of topological groups Spectrum of closed and self-adjoint linear operators II. DIFFERENTIAL CALCULUS ON BANACH SPACES

1. 2.

Supersmooth mappings Berezin integration; Gaussian integrals

1 3 5 6 14 15 17

27 33 36 37 39 40 42 46 47 51

51 57

CONTENTS

3. 4. 5. 6. 7.

1. 2. 3. 4. 5. 6. 7. 8. ,

10. 11. 12. 13. 14.

1. 2. 3. 4. 5. 6.

Noether's theorems I Noether's theorems II Invariance of the equations of motion String action Stress-energy tensor; energy with respect to a timelike vector field

64 71 79 82

III. DIFFERENTIABLE MANIFOLDS

91

Sheaves Differentiable submanifolds Subgroups of Lie groups. When are they Lie subgroups? Cartan-Killing form on the Lie algebra ~ of a Lie group G Direct and semidirect products of Lie groups and their Lie algebra Homomorphisms and antihomomorphisms of a Lie algebra into spaces of vector fields Homogeneous spaces; symmetric spaces Examples of homogeneous spaces, Stiefel and Grassmann manifolds Abelian representations of nonabelian groups Irreducibility and reducibility Characters Solvable Lie groups Lie algebras of linear groups Graded bundles

91 91 92 93

108 110 111 114 114 115 118

IV. INTEGRATION ON MANIFOLDS

127

Cohomology. Definitions and exercises Obstruction to the construction of Spin and Pin bundles; Stiefel-Whitney classes Inequivalent spin structures Cohomology of groups Lifting a group action Short exact sequence; Weyl Heisenberg group

83

95 102 103

127 134 150 158 161 163

CONTENTS

,

8. 9. 10. 11. 12. 13.

o

2. 3. 4. 5. 6. ,

8. 9. 10. 11. 12. 13. 14.

Cohomology of Lie algebras Quasi-linear first-order partial differential equation Exterior differential systems (contributed by B. Kent Harrison) B~icklund transformations for evolution equations (contributed by N.H. Ibragimov) Poisson manifolds I Poisson manifolds II (contributed by C. Moreno) Completely integrable systems (contributed by C. Moreno)

200 219

V. RIEMANNIAN MANIFOLDS. K/~HLERIAN MANIFOLDS

235

Necessary and sufficient conditions for Lorentzian signature First fundamental form (induced metric) Killing vector fields Sphere S n Curvature of Einstein cylinder Conformal transformation of Yang-Mills, Dirac and Higgs operators in d dimensions Conformal system for Einstein equations Conformal transformation of nonlinear wave equations Masses of "homothetic" space-time Invariant geometries on the squashed seven spheres Harmonic maps Composition of maps Kaluza-Klein theories K~ihler manifolds; Calabi-Yau spaces

V BIS. CONNECTIONS ON A PRINCIPAL FIBRE BUNDLE 1. 2. 3. 4.

167 171 173 181 184

235 238 239 240 244 244 249 256 262 263 274 281 286 294

303

An explicit proof of the existence of infinitely many connections on a principal bundle with paracompact base 303 Gauge transformations 305 Hopf fibering S 3 ~ S 2 307 Subbundles and reducible bundles 308

xii

CONTENTS ~

6. 7. 8.

9. 10. 11. 12.

,

3. 4. o

,

7. 8.

9. 10. 11. 12. 13. 14. 15. 16.

Broken symmetry and bundle reduction, Higgs mechanism The Euler-Poincar6 characteristic Equivalent bundles Universal bundles. Bundle classification Generalized B ianchi identity Chern-Simons classes Cocycles on the Lie algebra of a gauge group; Anomalies Virasoro representation of ..~ (Diff S 1). Ghosts. BRST operator

310 321 334 335 340 340 349 363

VI. DISTRIBUTIONS

373

Elementary solution of the wave equation in d-dimensional spacetime 373 Sobolev embedding theorem 377 Multiplication properties of Sobolev spaces 386 The best possible constant for a Sobolev inequality on R n, n > 3 (contributed by H. Grosse) 389 Hardy-Littlewood-Sobolev inequality (contributed by H. Grosse) 391 Spaces Hs,a (I~ n) 393 Spaces Hs(S n) and Hs,a(Rn) 396 Completeness of a ball on Wp in WsP_l 398 Distribution with laplacian in L 2(R n) 399 Nonlinear wave equation in curved spacetime 400 Harmonic coordinates in general relativity 405 Leray theory of hyperbolic systems. Temporal gauge in general relativity 407 Einstein equations with sources as a hyperbolic system 413 Distributions and analyticity: Wightman distributions and Schwinger functions (contributed by C. Doering) 414 Bounds on the number of bound states of the Schr6dinger operator 425 Sobolev spaces on Riemannian manifolds 428 SUPPLEMENTS AND ADDITIONAL PROBLEMS

~

The isomorphism H | H ~_ M4(R). A supplement to Problem 1.4 (I. 17)

433

435

CONTENTS

2. 3.

4. 5. 6. 7. 8. 9. 10. 11. 12.

xiii

Lie derivative of spinor fields (III. 15) 437 Poisson-Lie groups, Lie bialgebras, and the generalized classical Yang-Baxter equation (IV. 14) (contributed by Carlos Moreno and Luis Valero) 443 Volume of the sphere S n. A supplement to Problem V.4 (V. 15) 476 Teichmuller spaces (V.16) 478 Yamabe property on compact manifolds (V. 17) 483 The Euler class. A supplement to Problem Vbis.6 (Vbis.13) 495 Formula for laplacians at a point of the frame bundle (Vbis. 14) 496 The Berry and Aharanov-Anandan phases (Vbis. 15) 500 A density theorem. A supplement to Problem VI.6 "Spaces H s , a ( R n ) '' (VI.17) 512 Tensor distributions on submanifolds, multiple layers, and shocks (VI. 18) 513 Discrete Boltzmann equation (VI. 19) 521

Subject Index Errata to Part I

525 531


CONVENTIONS

(1) {fn}N := { f n : n

e

N}. .f

(2) Commutative diagram x

>y

h~Tg

r

{f'x--~Y, h--go f

g ' Y --+z

Z

(3) Integer part: if d/2 = 3.5, then [d/2] = 3. (4) A \ B and A / B sometimes mean left and right coset, respectively; but usage varies and is determined in each context. (5) Exterior product, exterior derivative, interior product (Or A fl)(l) 1 . . . . .

l)p+q)

--

1 ~-~(signl-l)Fl[ot(Vl

p!q~

X fl(Vp+l,...,

(0/A fl)(l)l, . . . ,

l)p+q)

--

1

(p+q)! X

, ...,

Vp)

H

Vp+q)],

Z(signH)H[ot(Vl .... Vp)~ n

(l)p+ 1 . . . . .

l)p+q)].

When operating on a p-form d - d/(p + 1) and ?v = piv. Note that Kobayashi and Nomizu (Vol. I, p. 35) use what we call A. (6) Riemann tensor, Ricci tensor (Vc~Vr - V~Vc~)v x

m

Ro~xu v/z ,

i.e.

Rur x -- OuFl~x - O~Fu x + F'up

p Fu ~

R~u "-- Ra~au. These conventions agree with Misner, Thorne, and Wheeler and differ from those of our first book Analysis, Manifolds and Physics. XV

xvi

CONVENTIONS

(7) The Dirac representation of the g a m m a matrices Yt, Yv + YvYt, = 2rhzv

~3

0 0 0 -i

0 0 -i 0

0 0 -i 0

0 0 0 i

0 i 0 0 i 0 0 0

Otzv = d i a g ( + , + , + , - )

i 0 0 0

'

0 -i 0 0

9/2-

'

),'4-

0 0 0 1

0 0 -1 0

i 0 0 0

0 i 0 0

0 -1 0 0 0 0 -i 0

1 0 0 0

'

0 0 0 -i

"

Majorana representation of the g a m m a matrices 0u. = d i a g ( + , + , + , - )

Ytz?'v + YuYt, = 2r/uv

, Yl--

, ?'3--

0 0 0 -1

0 0 1 0

0 0

0 0 0 -1

1

0 1 0 0

-1 0 0 0 -1 0 0 0

' 0 -1 0 0

,

, )/2--

1 0 0 0

, )/4--

0 0 0 -1

0 1 0 0

0 0 -1 0 0 0 -1 0

0 0 0 -1 0 1 0 0

1 0 0 0

'

"

Note that in Vol. I, p. 176, we give the Dirac representation of the g a m m a matrices for Ou. = d i a g ( + , , , -).


1. G R A D E D A L G E B R A S

For applications and references see, for instance, Problems II 1, Supersmooth mappings and III 14, Graded bundles. A ~2 graded algebra A is a vector space over the field of real or complex numbers which is the direct sum of two subspaces A+ (called even) and A_ (called odd)

graded algebra

A=A+@A_ endowed with a n associative and distributive operation, called product, such that ArA~-Ar+~(mod.

2),

r,s=0,1,

A 0=A+,

AI=A_.

A Z 2 graded algebra is called graded commutative if any two odd elements anticommute and if even elements commute with all others" ab = (--1)d(a)d(b)ba,

a, b ~ A

where d(a) = r if a E A r is the parity of a. We shall consider in this section only graded commutative algebras, so we shall omit the word "commutative". The algebras we shall use will be endowed with a locally convex Hausdorff topology for which sum and product are continuous operations. For example, t h e exterior (Grassmann) algebra over a finite dimensional vector space X (p. 196) is a graded algebra. A generalization used in physics, which we shall call a (Bryce) DeWitt algebra is the algebra B of formal series with a unit e and an infinite number of generators z ~, I E N, with the usual sum and product laws and the anticommutation property z l z I = -- ZJZ I.

An element a E B is written (notion of convergence is irrelevant) a= ~ pEN

a(p),

graded commutative

a ( p ) = ~.1 a h . . .

lpZ I1 ...zZp .

parity

(Bryce) DeWitt algebra B

I. REVIEW OF FUNDAMENTAL NOTIONS OF ANALYSIS body soul degree

a(0) = aoe is called the body of a, a s = Ep>_1 a ( p ) its soul. The numbers a0, azx...zp are real or complex, azl...zp is totally antisymmetric in 1 1 . . . Ip; the degree of a ( p ) is p. B+ consists of the formal series which contain only terms of even degree, B_ consists of those with only terms of odd degree. B+ is a subalgebra of B, while B_ is not. Show that if ab = 0 f o r all b ~ B+ [resp. b E B_ ] then a = O. Are these properties true in a finitely generated Grassmann algebra? Answer" If ab = 0 for all b belonging to B+, or to the even part of a

finitely generated Grassmann algebra we see that a = 0 by taking b = e. Suppose now ab = 0 for all b E B_. In particular az I = 0 for each z 1, I E N . Suppose a coefficient ai1...i p # 0 . Choose z J j ~ ( z Z l , . . . , zip). We have azl . . . xpz z~ . . . zIpz J # O,

hence

a#0.

If there is a finite number N of generators the hypothesis ab = 0 for all odd b implies only a = cz . . . z N,

c arbitrary numbers.

II

B is endowed with a locally convex, metrizable, Hausdorff topology by the countable family of seminorms (cf. for instance, p. 424) =

The sum of formal series (in particular the decomposition B = B+ 9 B_) and their product have the required continuity. Show that: The partial sums m

am= ~

p---0

a(p)

converge to a, in the B-topology, when m tends to infinity. Answer: If []

is a seminorm on B we have exactly I l a - am[[,,...~p = o

if

m>p.

Let f ( x ) = E~=o cn xn be a numerical series with radius o f convergence p. Show that f ( a ) = ~, c,a ~ is a well-defined f o r m a l series in B, depending continuously on a, if }aol < p.

2. BEREZINIAN

3

Answer: We have a = aoe + as, so n

an

=

~,

P n-PaP.

Cnao

p=0 n-p

Since f(x) is convergent for Ix[ < p, the numerical series E~_>pcnCPao are convergent for 1%1 < P. We denote their sum by ap and we write

~, Cnan = ~, apa~ = ~, b(q). Each term on the right-hand side is well defined: b(q) is obtained by finite sums and products since a term of order q arises from a p only when

p n CC(n, m) _~ M2. (I~) | %0(0, m - n).

(5)

12


These two relations show that all the Clifford algebras are determined through the properly euclidean ones, and their classification depends upon n - m modulo 8. Write the classification table. Answer 4a: Using the result of paragraph 4 with k 2 -- - 1 we obtain:

~ ( 0 , n + 8) _~ ~ ( 0 , 2) | ~ ( n + 6, 0) _~ %0(0, 2) | cC(2, 0) | (C(0, n + 4) ~ - . . %0(0, 2) | %0(2, 0) | 5r 2) | %0(2, 0) | ~ ( 0 , n) ~' M2(R) | I[-]I| M2(R) | ]HI| cC(0, n). The tensor product of vector spaces is commutative, up to an isomorphism. It is easy to see that if Mn (I~) denotes the algebra of n • n real matrices Mn (IR) | mn,(IR) ~-- Mnn'(lt~),

and it is known that (the proof is given in a problem at the end of the book) H | ]HI~_ M4(R); formula (4) follows. Answer 4b: The proof of (5) is analogous"

~ ( n , m) "~" ( |

1))n | ~ ( 0 , m - n) "~ M2, (R) | (if(0, m - n).

The table reads, with d = n + m, abbreviating Mp(R) to 1~(p), and H(p) denoting the algebra of p x p quaternionic matrices" (m - n) rood 8

cff(n,m) (m

n) mod 8

%~

0 I~(2d/z) 4 H(2(d/2)-1 )

1

I~(2(d-l)/Z)~It~(2(d-1)/2) 5

IHI(2(d-l)/2-1)~H(2(d-l)/2-1 )

2 I~(2d/2) 6 IHI(2(d/2)-1 )

3 C(2(d-l)/2) 7 C(2(d- 1)/2)

Note that for d even the Clifford algebra is isomorphic to either the algebra of real matrices or the algebra of quaternionic matrices: for n - - 1, ~ ( 1 , m) admits a real representation for m = 1,2, 3 mod. 8, thus d - 2, 3, 4, 10, 11, 12, etc. Note that ~ ( n , m) and ~'(m, n) are not isomorphic, unless I m - nl = 0 mod 4. The explicit construction of the table is done in [M. Berg et al., see p. 39] where the Clifford algebra is not (1) but YA FB + FB YA -- 2gAB 11. 5. GRADING OF A CLIFFORD ALGEBRA

a) Give a Z2-grading to a Clifford algebra. b) Show that the even parts o f f ( n , m) and %~(m, n) are isomorphic. c) Show that ~ + (p, q) -- (C(p, q -- 1).

4. CLIFFORD ALGEBRAS

13

A n s w e r 5a: A Clifford algebra %0 is a direct sum

%0= %0+ (9 %o_,

(6)

where %o+, called the even part of %0, is generated by the even products of elements of a basis: 11, Y A 1 Y A 2 , ' " ,

YAI""

Ai < A j,

YAp,...

and ~ _ , called the odd part, is generated by the odd products: YA , Y A 1 Y A 2 YA 3 . . . .

A i < A j .

Note Y1 ... Yd e %~+ if d is even; 7'1 ... Y d e %~- if d is odd. We have obviously

~+~+ C~+, c

~-~- C~+, c

(7) (8)

The laws (7), (8) show that the decomposition (6) gives a Z2 grading to ~. Note that this Z2 grading is not Z2 commutative (Problem I 1). A n s w e r 5b: Suppose that ~ ( n , m) is the Clifford algebra generated by and Va, A -- 1 . . . . , n + m, with a product such that YA YB + YB YA - - - - 2 g a B 2 .

Consider a vector space V over the reals generated by 11, elements denoted i y a , A -- 1 , . . . , n + m, and their products defined through the product in the algebra ~ ( n , m) and the products of complex numbers, namely, (i y a ) (i y B ) =~ -- YA YB .

(9)

We endow the vector space V with the structure of an algebra through the product so defined. We deduce from (9) that (iya)(iyB) + (iyB)(iVa) = 2gaB2,

hence V, together with the product we have defined, is the algebra %~(m, n). It results also from (9) that the algebras %~ (m, n) and %~ (n, m) are identical: ~ + (n, m) -- ~ + (m, n).

(10)

A n s w e r 5c: Let 2 and (YA) = (Y0, Ya), a -- 1 ...... p 4- q -- 1 be generators

of ~f(p, q). Then • 2

and the ya'S are, together with ~, generators of ~ ( p , q - 1). On the other

14


hand, ~ + ( p , q) is generated by 11 and the even products ~'OYa, ?'aVb. We deduce the isomorphism

~+(p,q)=~(p,q-

1)

(11)

from the relation (•215215215

= - •215

•

-

•

The equalities (11 ) and (10) imply ~ ( p , q - 1) = ~f(q, p - 1), which can also be read from the table of w

5. CLIFFORD ALGEBRA AS A COSET OF THE TENSOR ALGEBRA

Let V be a real vector space with a symmetric bilinear form V x V --->R given by (v, w) ~-~ b(v, w). Let us denote by ~ - y'~g>o(| k the tensor algebra of V. We say that the subset ~ of the algebra ~ i s generated by the subset {ra, a ~ A family of indices} of ~ if the elements of ~ are the finite sums i = y~ ta |

ra,

t a E ~,

1) Show that the subset ~ of ~ is a left ideal of ~. 2) Let ~ be generated by elements of the form u | u + b(u, u). Show that the coset space ~ \ fs is isomorphic to a) The exterior algebra of V if b = O. b) The Clifford algebra cC(V, g) if b is a nondegenerate pseudo-euclidean scalar product g.

left ideal

Answer 1: It is obvious from the definition that if i ~ ff then t | i ~ ff~ for every t ~ s Thus 3 is a left ideal of s (p. 8). Answer 2: There is a natural map from V into s \ ff~, namely v ~ [v], where [v] is the equivalence class of v ~ V C s in s \ q2. It is easy to see that the equivalence class [v | w] depends only on [v] and [w] and thus defines a product for these elements of ff \ if: [I)][W] de_f [l) @ W].

6. FIERZ IDENTITY

15

To prove the isomorphisms stated in a) and b) we compute

[v][w] + [w][v] = [v | w] + [w | v] = Iv | w + w | v] = [(v + w) | (v + w)] - Iv | v] - [w | w]. On the other hand, by the definition of Iv (~ v] + [b(v, v)] = O;

(1)

thus, since b is bilinear and symmetric, we have [v][w] + [w][v] = - 2 b ( v , w).

(2)

a) If b = 0 relation (2) shows that there is an isomorphism between ~ and the exterior algebra of V. b) If b is an inner product, there is an isomorphism between V, the subspace of ~ defined by {Iv], v E V} and the subspace { 3'o, v E V} of the Clifford algebra ~(V, g).

6. FIERZ IDENTITY

Here we use the generic term spinors for both spinors and pinors. For the definition of spinors and cospinors see pp. 415, 418 and [Problem IV 2, Obstruction]. The following identity is useful in many computations, particularly in supergravity. Let ~p, ~p be arbitrary spinors on a dm = 2p dimensional vector space with pseudo-euclidean product g, and A, fi arbitrary cospinors. P~,ove the equality of the two scalars (juxtaposition denotes the duality product of a spinor and a cospinor) -

1

(a~)(~z~,) = ~

~, (Xr,~)(~zr'~),

(~)

where FI, 1 = 1 , . . . , 2 d denotes a basis 112e, FA, F A B , ' ' ' , El. 9Fa of a representation of the Clifford algebra ~ by 2 P x 2 p complex (or real) matrices and F I = (F1)2F~. Answer" Let S be the (complex) vector space of spinors, S* its dual. We know (Problem 1 4) that the complexified Clifford algebra ~c is isomorphic to the algebra S * Q S of linear mappings S---> S. The left-hand side of (1) defines a quadrilinear map ( S * x S ) x ( S * x S) ~ C by (A, ~_; fi, ~b)~ (A~p)_(fir This is also a quadrilinear map if considered as (A, r 12, ~p)~--~(Ar It can thus be written as

(,~)(~z~) = E ~"(~r,~)(~zr,~), l,J

16

I. R E V I E W O F F U N D A M E N T A L

NOTIONS OF ANALYSIS

where the a is are numbers which we shall determine. The above equality can be written, if a, b denote indices for c o m p o n e n t s in S and S* respectively, as E Ol 1JFla b .l-'jCd = t ~ d ~ b . 1,J

We determine the a H by taking traces of products with various F K, as follows" 2

a

IJl-, b l-,a " K a" l b I"jCd

=

FK

ba

a

c

8dab"

(2)

1,J

We know that tr F K F~ = 0

if K # I

so (2) reduces to

J

a KJ t r ( F K) 2Fj = F K

from which we deduce KJ

re = 0 if K # J , KK r~r,\~ - 1 c~ = / t r t d2 ),.) ,, . Each matrix F K is a product of gamma matrices, and hence has square -+~2p. If

(rx)~= ~x~, then tr(Fx) 2 = 2 % K. The conclusion follows. E x a m p l e d = 4: we take as matrices F~

114, Fa,/"a/3, ~= FoF1GF3, ~Ft~. Then

(114)2= 1],, (F.) 2= -g,,~l] 4, ~2 " - " - ~]4,

(F~a) 2= -g,~,~g~al] 4,

(~2Fa)2 = -gaa

1] 4 .

The formula can then be written"

( X ~ ) ( C o ) = 1 { ( ~ ) ( e ~ ) _ ( X ~ o ) ( e ~ ) _ ( Xro 0 ) ( e r o s ) - (~-~r~ ~ ) ( ~ r ~ ) - (AT~ ~ ) ( ~ r ~ % ) } .

7. PIN AND SPIN GROUPS

17

7. PIN A N D SPIN G R O U P S

To be in agreement with modern physics notation we shall modify somewhat the terminology used (p. 67). We shall give the fundamental properties of Spin(n, m) and Pin(n, m), for arbitrary dimensions. We treat in the first paragraph the case n + m even, which is easier, using a matrix representation. In paragraph two, we treat the general case by using the grading of the Clifford algebra, as done by Atiyah, Bott and Shapiro in the euclidean case. Recall the following definitions. Let GL(V) denote the group of isomorphisms (linear, invertible maps) from a real vector space V onto itself; V has dimension d = n + m and g is a pseudo-euclidean scalar product on V of signature (n, m) (i.e., with n plus signs and m minus signs): Orthogonal group O(n, m) = {L ~ GL(V), g(Lu, Lv) = g(u, v)}. Special orthogonal group SO(n, m ) = {L E O(n, m), det L = 1}. Identity (connected) component of O(n, m): SO0(n, m). In the euclidean case (n or m = 0 ) , SO(n, m ) = SO0(n, m) but in the general case they are not equal. In the Lorentz case, SO(m, 1) and SO(1, m ) = L(m + 1) have two connected components. The Lorentz group O(1, m) has four connected components defined as follows: Let SO0(1, m)=-- Lo(m + 1): Lo(m + 1): 9L ~0 > 0 det L = +1 orthochronous proper Lorentz group 0 < 0 det L = - 1 (t a time reflection) tLo(m + 1): L ~ 0 sLo(m + !)" L0 > 0 det L = - 1 (s a space reflection) 0 stLo(m+l): L 0 < 0 d e t L = + l Lo(m + 1) and stLo(m + 1) are both orientation preserving. 1. CASE d = n + m EVEN. Consider the group of elements A of the Clifford algebra Cr are invertible and satisfy the relation

A%A-~ E ~v,

Vv ~ V

m) which

(1.1)

where 3'0 uA'yA is the element of rr corresponding to v E V and crv the linear subspace of cr isomorphic to V spanned by the %'s. {em; A = 1 , . . . , d} is a basis for V and {YA; A = 1 , . . . , d} a basis for the isomorphic space crv, whose products satisfy =

TA'YB + )'BTA = --2gab 1].

(special) orthogonal group identity component

orthochronous proper Lorentz group


Clifford group

la) S h o w that there exists a non-injective h o m o m o r p h i s m ~ f r o m the group defined by (1.1) into the O(n, m) group o f ( V , g). S h o w that this homomorp h i s m is not surjective if n + m - d is odd. Show it is surjective if d is even, the group is then called the Clifford group F (n, m). A n s w e r l a: Let {CA, A = 1 . . . . , d} be a basis of V; consider the d • d matrix L with elements LB a defined* by

AyA A - I --L~),'B. LB are real numbers, since Aya 1 . . . . . d} is a basis of %%.

A-I

E c~ V

(1.2)

by hypothesis and {yB; B --

The mapping A --+ L is a mapping into O(n, m) because if A satisfies (1.2) for A = 1 . . . . , d, we have also

AyAVC A - l -

(L~yB)(LDyD),

and, using YAYD + YDYA = - - 2 g A O ~ , we find gAC -- L B L cD gBD

i.e.,L60(n,m).

The mapping A --+ L is a homomorphism because A'AyAA

-

-

I A ' - I - - A ' L B y B A ' - I - L tC B LBYC;

thus A ' A w-> L ' L . This homomorphism cannot be injective, since for any k 6 I~, A and kA have the same image in O(n, m). parity operator

It is not surjective if d is odd: the (space time) parity operator P - (pA = - I , P ~ -- 0 if B ~ A) belongs to O(n, m), but there exists no A ~ ~ ( n , m) which satisfies A YAA - I

--

--YA,

A -- 1. . . . . d

since this would imply

A yd + I -Jr-~/d+l A = 0,

Yd + i = V l ... Vd ;

but for d odd we know that Yd+i commutes with A ~ %O(n, m), hence AVd+I = 0 which is impossible since A and Vd+i are invertible. It can be proved the homomorphism is surjective if d is even as follows: let FA be a set of gamma matrices, and let L E O(n, m). Then it is clear that

G -- L rA *Note that (p. 176) we use the convention

LBA YB.

19


is also a set of gamma matrices. Therefore, if n + m is even there exists a matrix M such that"

F~ = M F B M - ' . If d is even, the matrices Fa can be chosen all real or all with quaternion elements. The matrices F~ are of the same nature as the FA, i.e., all real or all with quaternion elements, and so is M, which represents then also an element of the Clifford group. lb) Show that if Yo is invertible, then Yo satisfies (1.1). What is the corresponding O(n, m) transformation? Show that this transformation changes the orientation if d is even.

Answer lb" For all y~ and invertible Yo (i.e., v such that g(v, v ) ~ 0), we have YoYwY o-1

_.

-2g(v

-1

w ) y o - Yw

2g(v, w)

- - -

u

g(v, v) yo

~

yw,

the mapping L" V---~ V corresponding to Yo is given by

w ~ Lw Let for instance v = (1, 0 , . . . ,

-"

2g(v, w)

g(v, v)

V--W.

0), then whatever the signature of g:

( W 1, W 2 , . . . , W d ) ~

( W 1, - - W 2, 9 9 9 , - - w d ) .

Take for instance a Lorentzian signature goo = 1, gmA 3'o = 3'0, then B

Bm~B

L a -- 2goa 8 0 L =

(1 0

=

--1

for A ~-0, and

m

O) 1] = P x t,

t being time reflection.

In all cases, the mapping V---~ g corresponding to y~ is the negative of the symmetry with respect to the hyperplane orthogonal to v:

~ ( y~ ) = Pso ,

P parity operator.

The determinant of the transformation is 1 if d is odd, .and - 1 if d is even. lc) Show that the kernel o f Y(: F(n, m ) - . O ( n , R~, ~ the unit of CO(n, m).

m), n + m = d even, is

20


Answer lc: A E F(n, m) is in the kernel of ~ if ~ ( A ) = 1], the unit of O(n, m) that is A=l,...,d;

AYA = YA A,

hence A commutes with ~(n, m). The result follows [cf. Problem 1 4, Clifford algebras]. We deduce from this result that if two elements A 1, A 2 E F(n, m) have the same image L under ~, then

A 1=cA2,

pinor group

ceil.

ld) If d = n + m is even, ~(n, m) and hence F(n, m) admit a faithful representation by 2 p • 2 p real or complex matrices. The pinor group Pin (n, m) is by definition the subgroup of F(n, m) such that, A being in such a matrix representation, Idet A[ = 1.

Show that Pin(n, m) is a double covering of O(n, m) under the homomorphism ~(" A ~ L. Answer ld: It is obvious that if A (E Pin(n, m), then also - A E Pin(n, m) because det A = d e t ( - A), since A is a 2 p x 2 p matrix. We show that the kernel of the homomorphism O(n, m) is indeed -.+112p, as follows:

~" Pin(n, m)--->

~ ( A ) = 1, implies [cf. lc] A = cl]2P ,

c~R;

and Idet A[ = 1 implies then c

2P

=1,

hencec=-+l.

On the other hand, the mapping ~ : Pin(n, m)--> L is surjective b e cause the mapping ~ : F(n, m)--> L is surjective [cf. la] and because, given M E F(n, m), there exists k E R such that [det(kM)[ = 1,

namely k = - I d e t M 1-1/2P.


21

Therefore ~ : Pin(n, m)---~ O(n, m) is a double covering. The spinor group Spin(n, m) is the subgroup of elements of Pin(n, m) whose image under Y(is an orientation preserving element of O(n, m). It is a double covering of SO(n, m).

spinor group

le) An O(n, m) transformation can be shown to be the product of a certain number of symmetries with respect to nonisotropic hyperplanes [Riesz pp. 74-75, Artin pp. 129-130] L = SolSoz . . . So . S h o w that if A E F(n, m), n + m even, A is a product o f elements To. Denote by F+(n, m) [resp. F ( n , m)] the elements of the Clifford group

which are in the even part ~+ (n, m) of the Clifford algebra [resp. the odd part c~_(n, m)]. S h o w that r ( n , m) = F+ (n, m) U r (n, m). That is, an element o f the Clifford group is either even or odd. Show the same is true for the pinor group and that the spinor group is the even part. A n s w e r le: Let L E O(n, m), n + m even: L = So . . . SoN = ( P S o , . . . PSoN)P N,

since the parity operator P = - 1] = p-1 commutes with each S o. We know that if d is even we have ~(Ya+l) = P,

Ya+x =

YX''"

')td;

hence, due to the results of lb), the element A E F(n, m), given by N

A = %1"'" "YoN'Yd+I' satisfies Y((A) = L. The general solution of the above equation is N

A = c% . . . %Nya+l

=

")Icy I

...

N ')/vN)td+l

9

We see on this formula that an element of F(n, m) is a product of elements in cdv. In particular, we see that A is either even (when N is even) or odd (if N is odd), since ya+l is even. The element L is in SO(n, m) if it is orientation preserving, hence if N is even. If L changes the orientation of V, then N is odd.

(n, m)

22

I. R E V I E W OF FUNDAMENTAL NOTIONS OF ANALYSIS

The pinor group is a subgroup of the Clifford group, hence satisfies also Pin(n, m) = Pin+ (n, m) U Pin_(n, m), where Pin+ (n, m) = Spin(n, m) is constituted of even elements of the Clifford algebra. We have ~(Spin(n, m)) = SO(n, m). Spin §( 1, m)

In the case of Lorentzian signature, one defines Spin+(1, m) as the subgroup of Spin(l, m), whose image by ~ preserves also the time orientation. 2.

CASE

OF m + n = d oF

ARBITRARY PARITY.

Inspired by the results of Section 1, we introduce the automorphism a of C~(n, m) defined by al v = Using the grading of Cr

-

Id.

m) [cf. 14, Clifford algebras], we have

a(A+ ) = A+ a(A_ ) = - A_

ifA+~+(n,m) if A_ ~ rs (n, m).

2a) Show that the elements A E ~(n, m) that are invertible and satisfy the relation a ( A ) % A - 1 =- ad A% E ~v (2.1) Clifford group twisted adjoint representative

f o r m a group called the Clifford group F(n, m). Its representation ad on csv is called its twisted adjoint r e p r e s e n t a t i o n . Show that the Clifford group so defined contains, in the case of d even, the Clifford group defined in 1. They will be shown to be identical in w

Answer 2a: a is an automorphism of Cr

m), namely

,~(A,A~) = a(A1),~(A~), and we have ad(A 1A2) = ad A 1o ad A 2. Therefore F(n, m) defined by (2.1) is a group and ad a representation of this group on csv. We have defined in 1, in the case of d even, F(n, m) as the group of invertible elements of ~(n, m) that satisfy

AvoA-' %,

(2.2)


23

and we have shown that A is then either even or odd. Hence (2.2) implies (2.1). We shall show in w that elements of ~ ( n , m) satisfying (2.1) are also either even or odd, whatever is the parity of n 4- m, hence the two definitions of the Clifford group in the case n + m even are identical. 2b) i) Show that the mapping ~ :

A ~ L defined by

o t ( A ) y A A -1 -- L B y B

(2.3)

is a surjective homomorphism ~:

F(n, m) --+ O(n, m).

ii) Show that the kernel of ~ is R • ~, with R • multiplicative group o f nonzero real numbers. Show that if A ~ F (n, m), it is either even or odd. iii) Show that W ( Y u ) changes the orientation of V whether m + n is even or odd. Answer 2b: i) A proof analogous to the one given in 1 shows that (2.3) imp|ies that

L -- (LAB) ~ O(n, m), and that d@: A w-~ L is a homomorphism. To show that 5@ is surjective, we remark that ot(yl)YA()/1)-I _ {--YA YA

if A - 1, if A # 1.

(2.4)

Thus V1 6 F ( n , m). The transformation of V corresponding to Yl is the reflection in the hyperplane perpendicular to el, and not its negative as in question 1. Applying the same argument to any element of the basis of Z~v we obtain all the orthogonal reflections in hyperplanes of V and thus generate O(n, m). ii) Let A0 6 Ker ~ . Eq. (2.3) implies o~(A0) ~'a = ?'A A0.

(2.5)

We decompose A0 into its even and odd parts A 0 - - A0+ + A0-; then (2.5) reads ( A o + - AO-)VA = YA (Ao+ + A o - ) .

The odd part of this equation yields Ao-VA -- --?'A AO-. However no odd element can anticommute with all the ya'S; therefore A0- -----0.

24

I. REVIEWOF FUNDAMENTALNOTIONSOF ANALYSIS

Therefore A0+ A - A A0+,

for all A ~ %~

m).

We know that the center of ~ ( n , m) is I~31 if n + m is even and I~:l].+ RVd+ 1 if n + m -- d is odd. But if d is odd, V,+l is also odd; hence the only elements in ~ + ( n , m) that commute with ~ ( n , m) are in Ii~31. Those elements which are in F+ (n, m) are of the form c31, c ~ I~ x. We deduce from the above results, by a method analogous to the one used in 1, that every A ~ F' (n, m) is a product of N elements in ~ v N F (n, m) hence is either even or odd. iii) j T ( y , , ) changes the orientation, since it is a symmetry. norm on F

2c) Define a " n o r m " v on F (n, m) by the condition

v ( A ) - - Ig(Vl, lJi)...g(1JU, vU) I

for A -- y,,! "'" YVN

and 1?( C'fl) -- C2. Show that the subset of elements of F (n, m) such that vCA)- I pinor group spinor group

is a subgroup o f F ( n , m). It is called the p i n o r g r o u p Pin(n, m). Show that the homomorphism ~ restricted to Pin(n, m) is a 2-to-I surjective homomorphism onto O(n, m). The s p i n o r g r o u p Spin(n, m) is the subgroup of elements whose image by ~ is in SO(n, m), that is, of elements that are in ~+ (n, m ). Show that if n + m is even the groups just defined are the ones defined in 1. Answer 2c" v is a homomorphism from F (n, m) into I~ • : v(A I A2) -- v(A I )v(A2), hence the subset of elements of F(n, m) such that v(A) -- 1 is a subgroup of F(n, m); we call if Pin(n, m). The restriction of ,~F to P i n ( n , m ) is surjective (cf. w because, given L 9 O(n, m), there exists A 9 F (n, m) such that J ? ( A ) -- L and, given A, there exists k 9 R • such that

v ( k A ) - 1,

namely, k - -+-(v(A)) -I/2"

such k A project also on L by ~ . The previous reasoning indicates that the homomorphism ~

is 2 to 1.


25

To avoid discussions about the uniqueness of v(A), we proceed directly" ~7~(A 1) -- ~ ( A 2 ) We know that the center of ~

if and only if ~ ( A I A ~

l) -- 11.

in F (n, m) is IR2; hence

A1A21 -- c2,

c ~ IR.

v(c2[) - 1 is equivalent to c 2 - 1, i.e., c - 4-1. In the case n + m even, the pinor and spinor groups just defined are the same as those defined in 1 because y 2 _ - - g ( v , v)ll; therefore, if we consider the matrix representation of ~ ( n , m), we have

F 2 -- - g ( v ,

V)J]-2P

I det Fo 12 -- (g(v, v)) 2p. If A -- GI...

GN ,

then Ig(vl, V l ) . . . g ( v N , VN)I 2p-1 9

Id e t A I -

Then the conditions I det A I - 1 and v(A) -- 1 are identical. 3.

EXAMPLES

3a) Construct F ( 0 , 1) and F ( 1 , 0 ) and their Pin and Spin subgroups. Show that Pin(0, 1) ~ Z2 x Z2 and Pin(l, 0) ~ Z4. Show that Spin(0, 1) ~_ Spin(l, 0) ~ Z2. 3b) Construct Pin(0, 2), Pin(2, 0), Pin(l, 1) and their Spin subgroups. Pin(0, n) and Pin(n, 0) are often labelled Pin + (n) and P i n - ( n ) . 3c) Show that Spin(n, m) is isomorphic to Spin(m, n). Pin(n, m) is not necessarily isomorphic to Pin(m, n).

Answer 3a: In Problem I 4, Clifford, we have computed ~ ( 0 , 1) and %~ The generators of Pin+(1) are 11 and Y such t h a t y 2 _ -4-1. Their multiplication tables are y

Y -11 -y

~[ y ly

y ~t

-2 -y

-y -~

It y

__11

y

"1

--y

Pin (0, 1) = Pin + (1)

-y

11 y -It -y y -11 - y 11 -1l - y ~l y -y ~t y -~t Pin(l, O) _= P i n - (1)

Pin +

26


~'2 X 7/2 and 7/4,

which are identical to the multiplication tables of respectively.

0,0 0,1 1,0 1,1

0,0 0,1

1,0 1,1

0 1 2 3

0,0 0,1

1,0 1,1

0

0,1

0,0

1,1

1,0

1

1

1,0 1,1

1,1 0,0 0,1 1,0 0,1 0,0 7/2 x ~'2

2 3

230 1 3 0 1 2

0 1 2 3 2

3

0

7/4

The following table gives ~ ( A ) = L for A E Pin• and makes it possible to select the e l e m e n t s of Pin• which are in Spin• Since +- 73')' - 1 = _+31 w h e t h e r 3, ~E Pin + (1) or 3, ~ P i n - (1), A

T h e elements in Pin• Hence

~(A)

det L

_+4

1

1

+7

-1

-1.

such that ~ ( A ) is orientation preserving are +-1]. Spin -+(1) is isomorphic to ~'2.

A n s w e r 3b" We first show that all invertible elements of a c~, a Clifford algebra CO(n, m) with n + m = 2, that are either even or odd are in its Clifford group. Let )'1 and 3'2 be the generators of CO(n, m) with 2=E1, ~ )'1 '

)'2 2 = ~2 ~ '

~1' ~2 "- --+1

A+ E ~+ ~ A+ = all + b)'l)'2 A _ ~ ~ _ ~ A _ = c)"1 + d ) ' 2. -~ if ce~ + de 2 is not isotropic; i.e., A_ is invertible, with A - ~ = y~l+d,2 c2al + d2~2 # 0. It is easy to check that A+ is invertible if a 2 + b2Ela2 # 0. (Note that both conditions always hold for the algebras c~(0,2) and c~(2, 0)). It is straightforward to check that the same conditions imply the existence of 2 x 2 matrices L-+ such that ~ ( A - + ) = L• while A+)',, t = )'AA+

or

A_)' A = - - ) ' A A _ ,

A = 1,2;

hence that A+ E F+ (n, m), A_ E F_ (n, m). In Problem 14, Clifford we have given basis for ~(0, 2), c~(2, 0), and c~(1, 1) in terms of the Pauli matrices tr I , tr2, tr3. Let )'1, )'2 E q~v(n, m ) , n+m=2. A n e l e m e n t of P i n ( n , m ) , n + m = 2 is of the form either A = all + b)' 1)'2 or A = c% + d)'2, a, b, c, d E R and such that det A = +- 1. 'Yl

~2

')/l'Y2 ~(~/1)= ~a(~/2)

-1

Pin(0, 2) - Pin+(2) Pin(2, 0 ) - Pin-(2) i~1 i~2 i% Pin(l, 1)

0"1 io"2

0"3

0

(o 1) (\ -10 o) 1 1 0 ( o ,)

~(~/2)= ~(~/1)

0

('o -1)0 o)

8. WEYL SPINORS, MAJORANA PINORS

27

For n + m even, both ~7~ and ~ are surjective and they coincide when restricted to Spin(n, m). The elements of Spin(0, 2) and Spin(2, 0) are A -- all + bylY2,

a, b e R,

a 2 q-- b 2 - -

1.

The elements of Spin(l, 1) are A = all + by1Y2,

a, b E R,

a 2 - b 2 = 4-1.

Answer 3c: We have shown that Spin(n, m) is a subgroup of F+ (n, m). The isomorphism Spin(n, m) _~ Spin(m, n) results from ~+ (n, m) = %0+(m, n) [cf. 14, Clifford]. We deduce from the periodicity modulo 8 of the Clifford algebras that

~ ( m , n) --- ~ ( n , m)

if m - n -- 4k,

k e N.

If m - n ~ 0mod4, the group Pin(m, n) and Pin(n, m) may not be isomorphic; we have seen examples in the previous section. In particular, Pin + (n) and P i n - ( n ) are not isomorphic if n r 4k, k e N. References for Problems I 7 to I 11 may be found at the end of Problem I 11. 8. WEYL SPINORS, HELICITY OPERATOR; MAJORANA PINORS, CHARGE CONJUGATION l . WEYL SPINORS. HELICITY OPERATOR

Suppose d = n + m = 2p, Ya ?'B + YB Ya = --2gAB ~. la) Show that the element Y1. . . Y~l o f the Clifford algebra YT(n, m) defines a linear operator ~: S --+ S where S is a space ofspinors; i.e., the space o f a linear representation o f Spin(n, m) by 2 p • 2 p matrices, p = [d/2]. 1b) Show that ~ is an hermitian or antihermitian operator with eigenvalues + 1 or +i. Decompose S as a direct sum o f eigenspaces S=S+OS_. Show that Spin(n, m) preserves S+[resp. S_]. The operator ~ is called the helieity operator, sometimes parity operator.

l c) A Weyl spinor of positive helicity [resp. negative] is defined as an equivalence class (ap, A) with ape S+[resp. S_] and A e Spin [cf. Problem IV 2, Obstruction]. I f ~ is a differentiable field o f Weyl spinors o f some given helicity on an

helicity parity operator Weyl spinor helicity

28


orientable, pseudo-riemannian manifold (M, g), show that

FA VA I[r is a Weyl spinor field of opposite helicity. ld) Give a representation of cC+(n,m) and Spin(n, m) by 2p-1 x 2 p-1 matrices.

Answer 1a: We choose a representation of ~'(n, m) on C 2p by gamma matrices and set =

rl

. . . ru.

is a representative of a spinor-cospinor (p. 418), invariant under a change of spin flames because it anticommutes with the gamma matrices for d even and thus commutes with even products. A - I ~ A -- ~,

A e Spin(n, m).

Answer 1b" We have is transposition and complex conjugation, SO

We know [cf. Problem 1 4, Clifford w

that

~2 -- ( - - l ) P + n l l 2 p ;

thus the image of kse with k 2 - ( - 1 ) n+p by the homomorphism ~ is the parity operator P -- -11; it preserves the orientation since n 4- m is even, hence P e SO(n, m) and st e Spin(n, m). Let 7r e S. Then

implies that ~21/,r _ ( _ l ) P + n l / f _ ) ~

_ )2~;

thus ~.2 = (_l)P+n ~. = + 1

if n + p is even.

~. = +i

ifn + p is odd.

One denotes by S+ [resp. S_] the eigenspace with eigenvalue +1 or + i [resp. - 1 or - i ] . Since ~ is hermitian or antihermitian, S = S+ @ S_.

8. WEYL SPINORS; MAJORANA PINORS

29

Elements F_ E c~_[resp. F+ E c~+] anticommute [resp. commute] with r Thus, if r = A~,, ~:F_ ~ = - F_ ~:~ = - AF_ ~

[resp. ~:F+~ = AF+ ~b].

Hence, c~+S+ = S+,

c~+S_= S_,

c~_S+ = S_,

c~_S_ = S+.

Answer lc: If ~, takes its values in S+, the same is true of VA~,. Therefore, FAvA~, takes its values in S . The massless Dirac equation

I"AVA~ = 0 is meaningful for Weyl spinors of given helicity while the Dirac equation with a mass term

FAVA ~, + m $ = O,

m scalar

is not.

Answer ld (Kirillov p. 278): We choose a basis of S = C 2p adapted to the splitting S = S+ @ S . In such a basis ~: takes the form ~ = k(1]2p-~ 0

0 ) --112p-1 '

k2 = (_l)n+p

and the gamma matrices, anticommuting with ~:, are of the form

"]"A= DA

O '

where C A and D A a r e 2 P - ~ • 2 p-x matrices. Since [cf. Problem 1 4, Clifford w F A = EAFA, EA = _+1, we have

DA = ~A CA" ~+(n, m) is generated by even products of the gamma matrices; hence every element A E ~+(n, m) can be represented by a 2 P • 2 p matrix A with 2 p- 1 • 2 p- 1 blocks A +, A-"

A_(A 0) 0

A ~ A § and A ~ A -

A-"

are two representations of ~'+(n,m), sometimes

called half-spinor representation. The matrices A that are in Spin(n, m) are of the form [cf. Problem 17, Pin and Spin groups]

A = F,,1

.

9 .

F

02q

half-spinor representation

30


with

Fvi ~ uAi l~Ai and satisfy I det A I = 1. Therefore, A = v AI . . . vA2q (CA1 ""0 CA2q

0

)

DAI ...DA2q

and I UAI . . .

vA2q 12t' I det CA1... CA2q II det DA1 ... DA2q I -- 1.

Due to the relation between CA and DA this equation is equivalent to

I1)A! ... l)A2q ]2p-I IdetCA~ ... CA2q ] -- 1; that is, I det A+I = 1

(also I det A-I = 1).

An example (the case of Spin(0, 2)) can be found in [Problem 1 9, Representations]. Odd-dimensional spaces do not admit Weyl spinors; the representation of Spin(n, m) o n C 2t', p -- [(n + m)/2] is irreducible if n + m is odd. 2. CHARGE CONJUGATION. MAJORANA PINORS

Majorana pinors

The group Pin(l, 3), which acts on ordinary space-time pinors, admits, like the Clifford algebra W(1,3), a real representation. If such a representation is chosen, it is meaningful to speak of real pinors: if a representative lp(i) is real, the same is true of another representative A1/z(i) of the same pinor. Real pinors are called M a j o r a n a pinors. It is also meaningful to speak of the complex conjugate 7z* of a pinor ap, represented in each pin frame by the complex conjugate of the representative of 7z. Moreover, each pinor 7z can be written as -- 1//'1 -I-

ilP2

with aPl and aPl Majorana pinors. A Majorana particle has no connection with Majorana pinors. See reference to M. Berg et al. p. 39. 2a) Give a definition, independent of the representation of Pin(l, 3), of the operator C defined in a real representation by C: 7t --+ 7t*. This opera-

8. WEYL SPINORS, MAJORANA PINORS

31

tor is called "charge conjugation". Give its definition w h e n the g a m m a m a t r i c e s are the D i r a c m a t r i c e s (see conventions).

2b) Is it p o s s i b l e to e x t e n d the definition o f charge c o n j u g a t i o n to an arbitrary 0 ( 1 , d - 1) group ? A n s w e r 2a: The operator defined in a real representation of %~

3) by 7z --+ 7r* in each pin frame is an antilinear (i.e., ()~7r + #~o)* - )~*~p* + /z*~p*) map S --+ S, which is also involutive (i.e., its square is the identity). If we look in an arbitrary representation of %~ 3) for an antilinear operator C 4 --+ C 4 by 7z w-~ 7z c = CTr such that 7r c is a representative of a pinor when ~p is, we must have (1)

CA~--A~C--AC~;

that is C must commute with pin transformations. An antilinear operator on 7t is a linear operator D on ~ * CTt - D ~ * ,

D 6 ~ ( S , S).

(2)

A--0,...,3.

(3)

Equation (1) is equivalent to DF,~--FAD,

Since the F~ are a set of gamma matrices of O(1; 3) if the FA are, we know there exists such an operator D, determined up to product by a complex number #. If we want C to be involutive, we must have ( gr c ) c =_ D ( gr c ) * -- D D * llr -- llt ,

i.e.,

D D * -- 11.

(4)

Once D is known for a choice (FA) of gamma matrices it exists and is determined for another choice/~A" if we have /~A -- MFA M -1

hence

/~,~ -- M* F2 M *-1

the operator /9 -- M - 1 D M * ^

satisfies (3), for the matrices FA, and (4). In the Majorana representation of the gamma matrices we have D -- 1. In the Dirac representation D--

0 0 0 1

0 0 -1 0

0 -1 0 0

1 0 0 0

"

charge conjugation

32

I. R E V I E W O F F U N D A M E N T A L

NOTIONS OF ANALYSIS

The operator of charge conjugation C: qJ ~ DqJ* has, in any representation, eigenvalues +_1. The eigenspace with eigenvalue 1 is the space of Majorana pinors. The space S is the direct sum of the two eigenspaces S = Sl(~S

2.

In the Dirac representation, qJ is a Majorana pinor if I~1 = 1~r -~ (1]/4) * .

Remark 1: In our O(1, 3) case the only pinor that is both Weyl and Majorana is the zero pinor: a real pinor cannot be in an eigenspace of the helicity operator, ~:, since for O(1, 3) the eigenvalues of ~: are --+i. Answer 2b: We can take as the definition of a charge conjugation operator C, in the general O(n, m) case, n + m even, the conditions (1) (equivalently (3)), (2), and (4). If qJ is a pinor, qjc= CqJ will then also be a pinor and charge conjugation will be an involutive operator. For arbitrary n and m, it is not always possible to find such a D. We know that D is defined up to the product by a complex number ~, so if D D * > 0 we can rescale D to obtain D D * = 1]; but if D D * < 0 we can only obtain DD* = - ~, since/z/z* > 0. If the F A are real, it follows from (3) that a possible choice of D is D = 1]. There exists a charge conjugation operator with representative C: qJ~ q,* in this real representation. The eigenvalues of C are - 1 , as in the O(1, 3) case. The eigenspace with eigenvalue 1 is the space of Majorana pinors. The space S is the direct sum of the two eigenspaces S--

Sl(~S

2.

With the given definitions, Pin(n, m) and Pin(m, n) do not always simultaneously have charge conjugation operators. We shall now restrict the charge conjugation to spinors. We can then treat also the case n + m odd, as well as enlarge the group, which admits a charge conjugation operator, namely the group Spin(m, n), if Spin(n, m) admits one. The operator C: q / ~ Cq/= q/c must satisfy (1) for A E Spin(m, n); we still set Cq/= Dq/*,

D a linear operator.

(1) is now satisfied, since A is even, if we have

ora r , = rA r, o; that is

Dr] with~=+lorE=-l.

=

WAD

(5)

9. REPRESENTATIONSOF SPIN(n, m), n + m ODD

33

The involutivity of C still gives D D * = ~..

(6)

The existence of D determined up to product by # ~ C satisfying (5) is again a consequence of the fact that the ~ F~'s are a set of gamma matrices if the FA'S are; but, again, it is not always possible to satisfy condition (6). However, if D exists for Spin(n, m), it also exists for Spin(m, n) - as expected, since these groups are isomorphic- because if FA are gamma matrices of O(n, m), FA = iFA are gamma matrices of O(m, n) and if the FA satisfy (5) the flA satisfy ^

D/~/~ -- -~/~A D. Examples: The Clifford algebra %0(3, 1) admits a representation by pure imaginary matrices, generated by f'a -- iFa, FA the gamma matrices of the Majorana representation of ~ ( 1 , 3). The corresponding operator D is X. A pinor is Majorana if it is real. We give a table, taken from Coquereaux, listing the representation spaces for Dirac (i.e., general), Weyl or Majorana spinors in hyperbolic signature and dimension d = 4 to 11. Coquereaux' table includes representation spaces for Majorana spinors in dimensions d -- 8 and d -- 9 because he combines the results obtained for ~ ( n , m) and 5r n). d Dirac Weyl Majorana Weyl-Majorana

4 5 C4 C4 C2 / 1~4 / / /

6 C8 C4 / /

7 C8 / / /

8 C16 C8 / /

9 C16 / / /

10 C32 C 16 N32 I~ 16

11 C32 / R32 /

References for Problems I 7 to I I 1 may be found at the end of Problem I 11. 9. REPRESENTATIONS OF SPIN(n, m), n + m ODD 1. Show that the Clifford algebra %~ m) can be identified with the subalgebra o f ~ ( n , m + 1) defined by the relation ( A + nt- A - ) Y n + m + l

A=A++A_6~(n,m+I)

=

gn+m+l (A+ - A_),

= ~+(n,m+l)@~_(n,m+l).

(1)

Answer 1: Let 11, ga, A = 1 . . . . . n + m, and ~/n+m+l be generators of the Clifford algebra ~ ( n , m + 1). Then ]t and VA are generators of %~(n, m). Relation (1) expresses that A 6 W(n, m + 1) is in the subalgebra generated by 11 and YA since

34

I. R E V I E W

OF FUNDAMENTAL

NOTIONS

OF ANALYSIS

n§

A = c ~ + ~, (

fAil

9 . . Aiq

")lA iq) "Jl-d

")lA i l , , ,

q= 1

Aiq

" 9 " Aiqo/A

t

1

, . . ~/ A i q ")tn + m + l

satisfies (1) if and only if d

All

9

Aiq

=0.

2. Define Pin(n, m ) , n + m odd, as t h e s u b g r o u p o f Pin(n, m + 1), whose image by the twisted h o m o m o r p h i s m Y( is the subgroup o f O(n, m + 1), which leaves invariant the axis en+m+ 1. Give a representation o f Pin(n, m), n + m = 2p - 1, by 2 p x 2 p matrices. A n s w e r 2: An element A E Pin(n, m + 1), whose image by Y( leaves invariant the axis en+m+X, satisfies -

a(A)~/AA 1 a(A)y,,+,,,+l A-a

B = LA~/B ,

A , B = n, . . . , m + n,

= 'Yn+m+l"

(2) (3)

(3) says that A ~ Cr m), as identified in (1), while (2) expresses that A E Pin(n, m). From a representation of Pin(n, m + 1), n + m + 1 = 2p, by 2 P • 2 p matrices we deduce a representation of Pin(n, m) by 2 p x 2 p matrices. 3. Deduce f r o m a representation o f Spin(n, m + 1), n + m + 1 = 2p, by 2 p-1 x 2 p-1 matrices a representation o f Spin(n, m) by such matrices. A n s w e r 3" In a basis adapted to the splitting of the representation space C 2p= S+ O S_, we have seen [Problem I 8 , Weyl spinors] that the representative of an element A ~ Spin(n, m + 1) C cr (n, m + 1) reads

A=(A + 0) 0 Awhere A § and A - a r e 2 p-1 x 2 p- ~ matrices, while % +m+ ~ = YZp is represented by

0) An element of Spin(n, m) is represented by a 2 P - 1 • 2 p-1 matrix, for instance A § such that, together with its canonically associated A- [cf. Problem 1 8, Clifford], it satisfies A+Czp=C2pA - ,

A-C2p = C2pA +.

In the case of euclidean signature, we have A = A [cf. Problem 14, Clifford]. The second relation is the hermitian transpose of the first.

35

9. REPRESENTATIONS OF SPIN(n, m), n + m ODD

Example: Consider the representation of ~ ( 0 , 2) with generators F 2 - cr2. Then -- El F2 -- -io'3

--

-i

(1 o) 0

- 1

E l - - o'1,

"

is written in a basis adapted to the splitting of helicities. Spin(0, 2) is represented by 2 • 2 matrices A -- all + b~ such that [ d e t A I - 1; i.e., a 2 + b 2 -- 1. The representatives A+ [resp. A_] are the complex numbers a - ib [resp. a + ib], with a 2 + b 2 - - 1. It was shown (p. 68) that the element of SO(2) corresponding to a -- cos qg, b - sin ~o is the rotation of angle 2~p, in accordance with the fact that Spin(0, 2) is a double covering of SO(2). Spin(0, 1) is represented by elements such that the corresponding SO(2) transformation leaves e2 invariant, hence is the identity (it is also easy to check that A F2 -- F2 A implies in this case b - 0). Spin(0, 1) is isomorphic to Z 2 .

4. Show that when n + m is odd the group O(n, m) is the direct product O(n, m) = SO(n, m)

x Z2.

Define P (n, m) as a double covering of O(n, m). Answer 4: When n + m is odd, we can use the parity operator P = -11 of O(n, m) to write L 6 0 ( n , m) as a pair L = (L+,~) with L+ 6 SO(n, m), e - +1 if L -- L+, and ~ - - 1 if L r SO(n, m). If L1 -- (LI+, el) and L2 -- (L2+, ~2) we have

L1L2 -- (LI+L2+, El~2) because P commutes with SO(n, m). The same argument cannot be used for n + m even because P is then orientation preserving; P cannot be replaced by a reflection because these transformations do not commute with SO(n, m). When n + m is even, O(n, m) is a semidirect product SO(n, m) ~< Z 2 (see reference M. Berg et al., p. 39). Since O(n, m) - SO(n, m) • Z2, n + m odd, we define a double covering by setting Pin(n, m) = Spin(n, m) • Z2; the corresponding homomorphism ~:

Pin(n, m) --+ O(n, m)

is given by (s+, e) w-~ (L+, e),

~ = 4-1, s+ 6 Spin(n, m).

I. R E V I E W O F F U N D A M E N T A L N O T I O N S O F A N A L Y S I S

36

10. D I R A C A D J O I N T

We have seen that a choice among the isomorphic spaces of pinors attached to a pseudo-euclidean vector space V is defined by a pair (p0, e), where P0 is an orthonormal frame in V and e is the unit of the group Pin(n, m), homomorphic to the group O(n, m) which leaves invariant the scalar product on V. We consider the hyperbolic case n = 1, m = d - 1 and label the gamma matrices Fo. . . . , Fd- 1" 1) We say that qJ is a spinor if it is an equivalence class of triples (~(i),.p(i)~dA(i)) with ~(A(i))=+L(i ) when p(i)=L(i)P(o) and A(i)~_ Spin*(1, 1), that is, L(i ) (E L (1, d - 1), so p(~) has the same orientation and time orientation as P(o).

,Show that = r

Dirac adjoint

s c

. = [d/2].

where ~ denotes transposition and complex conjugation, are the components of a covariant spinor (a cospinor), called the Dirac adjoint of tp and denoted tp. m

2) Extend the definition of the Dirac adjoint when ~ is a pinor. 3) Define the Dirac adjoint of a pinor field on a manifold, when possible.

Answer 1" Using the definition and the hermiticity properties of the gamma matrices (see Problem 14, Clifford w d iii) we find"

FoFAFol=-FA,

m=0,...

,d-1.

We deduce from the definition of an element A of Pin(l, d - 1) that

Thus

j- rora A

little

algebraic

= La ror ro

manipulation

shows

that

/'A

A-1Fol ~I -1Fo, so A-1Fol-~-aFo = aA ~2P,

a A E C.

Taking the determinant of this equation, we find that (aA) 2p-" 1.

commutes

with

37

l 1. LIE ALEBRA OF PIN(n, m) AND SPIN(n, m)

Since a A takes discrete values, it is constant for A in a connected component of the pinor group, and it is equal to 1 in the connected component of unity. For such A's

AFo-- FoA-~. This relation shows that ~, with representatives (P~i~Fo, is a covariant spinor if 7t, with representatives 7t~i~, is a spinor.

Answer 2: In order to obtain the relation ~(i) --

@(.j)A-1

when

l~r(i ) - -

A~(.j)

we set, for any A 6 Pin(l, d - 1), -

ro.

Note that a A 1 for A 6 Pin+(1, d - 1) (the set of elements projecting on orthochronous Lorentz transformations) because a A 1 for A -- e and A -- F0, which projects on the space reversal transformation. aA - --1 otherwise. a A is a representation of Pin(l, d - 1) in Z 2 - { 1 , - 1 }. - -

- -

Answer 3: It will be possible to define the Dirac adjoint of a pinor field on a hyperbolic manifold when it admits a time orientation. A pin frame (Pb IV w is future [resp. past] oriented if it projects onto a future [resp. past] oriented Lorentz frame; to each pin frame z(i) is associated the number az(i) 1 if it is future oriented, az(i) otherwise. The Dirac adjoint of the pinor field ~ (with representative O(i) in the pin frame z(i~) is then the copinor field - -

- -

- - 1

fit(i ) -- az(i) l~(i) Fo.

References for Problems 1 7-11 can be found at the end of Problem I 11.

11. LIE ALEBRA OF PIN(n, m) AND SPIN(n, m)

We perform in arbitrary dimensions the exercise done (p. 176) in four dimensions, since it is not obvious that one obtains the same numerical factors in all dimensions. P i n ( n , m ) and Spin(n,m) are d ( d - 1)/2 dimensional Lie groups d = n + m , 2-fold coverings of O(n, m) and SO(n, m) by a homomol~hism Jd ~. They have the same Lie algebras, homomorphic to the Lie algebra


38

tY(n, m) and in fact isomorphic to it, since the derivated mapping ~ is an isomorphism between the tangent spaces at the unit elements of the groups. We have [Problem 1 7, Pin] Jr-c~ Spin(n, m) ~ SO(n, m)

by A w-~ L

with

ot(A)FA A-1 = AFAA -1 -- L A B F B ,

~ %~+,

A

Id e t A I -

1.

Thus ~ ( 1 1 ) is determined by the mapping between tangent spaces: ~r

(~)" rRPin(n,m) ---+ rRo(n.m )

by A ~-+I

with IAB - - _ _ I B A

),.FA -- FA),. -- IAB I-'B,

tr)~ = 0.

(1) (2)

With FAFB + FBFA -- - - 2 g A B $ the general solution I of (1) is )~-- l l A B F A F B

+ IZ ,

where/~ is a solution of the homogeneous equation UFA -

rau

= O.

Thus/z is such that # =all,

a ~C.

Since (2) implies tr/z = O, we have a = O, and/z = O. The isomorphism between the Lie algebras Ypin(n, m) and tY(n, m) is therefore )~ = l lAB FA FB

where l

=

(l AB) E

tY(n, m) and ,k 6 J p i n ( n , m).

REFERENCESFOR PROBLEMSI 4--1 1 1 E. Artin, Geometric Algebra (lnterscience, New York, 1957). M.F. Atiyah, R. Bott and A. Shapiro, "Clifford Modules", "Topology", Vol. 3, Sup. 1, (1964) pp. 3-38. 1Note that the sign of the solution depends on the choice of sign for the Clifford algebra, and the choice between L a B/-'B and L B A FB.

12. COMPACT SPACES

39

M. Berg, C. DeWitt-Morette, S. Gwo and E. Kramer, "The Pin groups in Physics: C, P and T" (to be published). M. Cahen, S. Gutt, L. Lemaire and P. Spindel, "Killing spinors". Bulletin de la Soci6t6 Math6matique de Belgique, XXXVIII (1986) 75-102. C. Chevalley, The algebraic theory ofspinors (Columbia Univ. Press, New York, 1954). Y. Choquet-Bruhat, "Einstein Cartan theory with spin 1/2 sources", in Geometry and Relativity Volume in honor of I. Robinson, ed. W. Rindler and A. Trautman (Bibliopolis, 1987). R. Coquereaux, "Spinors, reflections and Clifford algebras", in Spinors in Physics and Geometry, eds. G. Furlan and A. Trautman (World Scientific, Singapore, 1988). A. Kirillov, Elements of the theory of group representations (Springer, Berlin, 1976). A. Lichnerowicz, "Champs spinoriels et propagateurs en relativit6 g6n6rale", Bull. Soc. Math., France, 92 (1964) 11-100. A. Lichnerowicz, "Champ de Dirac, champ du neutrino et transformations ~ , ~ , f f sur un espace temps courbe", Ann. I. H. E 1 (1964) 233-290. A. Lichnerowicz, "Killing Spinors", in Spinors in Physics and Geometry (loc. cit.). T. Regge, "The group manifold approach to unified gravity", in Relativitd, Groupes et Topologie H, eds. B.S. DeWitt and R. Stora (North Holland, Amsterdam, 1984). M. Riesz, "Clilford numbers and spinors", Lecture Series #38, University of Maryland (1958). J. Tits, private communication. H. Weyl, Classical groups (Princeton University Press, Princeton, 1939).

12. COMPACT SPACES The proof z of the Tychonoff theorem (p. 21) is based on the following criteria which use closed subsets to establish fundamental properties of compact spaces often stated in terms of open subsets. Let fY be an open cover of X and let ~ be the family of complementary closed sets of X. Then

u~-x~n@-O. 1) Show that a topological space X is compact if and only if every family of closed subsets of X with empty intersection has a finite subfamily ~n with empty intersection.

2) Show that X is compact if and only if every family ~ of closed subsets of X has a nonempty intersection provided every finite subfamily ~n C of closed subsets of X has a nonempty intersection. Answer 1" By definition, X is compact (p. 15) if and only if for every family fY of open sets which covers X there is a finite subfamily fYn which covers X. Let ~ be a family of closed sets with empty intersection" the 1 See, for instance, Casper Goffman and George Pedrick, Fimt coume in functional analysis (PrenticeHall, Englewood Cliffs, N.J., 1965).

40


family ~3 of complementary open sets covers X, and thus admits a finite subfamily ~dn which covers X. The family ~n C ~ of complementary closed sets then has empty intersection. Suppose X is compact. Then

n

=O

=>

u

3n, u

=x

.=x

no%=O. (1)

Similar arguments show the converse implication.

Answer 2: a) The implication X compact

=>

{ O,~,, = O, Vn

::~

n , ~ = O}

follows from the first criterion. Indeed, if there is a family ~ of closed subsets of X with empty intersection such that every finite subfamily ~ has nonempty intersection the space X cannot be compact by (1). b) Conversely, assume X noncompact; then there exists ,~ such that n , ~ = 0, and all ~:~ are such that O ~ # 0 , which is contrary to the hypothesis. For an example of compactness see the next problem 1 13.

13. COMPACTNESS IN W E A K STAR T O P O L O G Y

Let X be a Banach space, and Y = X* its dual. A basis of the weak star topology on Y is given by the finite intersections of open sets of the type

Ux,' = {y E Y; (x, y ) E I}, where x is an arbitrary element of X, ( , ) is the duality between X and Y and I is an open set in R. Denote by Z = II(Rx; x E X) the topological product (cf. p. 20) of copies R x of R indexed by x ~ X. 1) Show that Y can be identified with a subset of Z and that its weak star topology is the one induced by the topology o f Z. 2) Show that the subspace K of Z defined by K = {z; z x is compact.

Izxl Ilxll)

3) Show that the ball of Y, strongly closed and bounded in the norm topology

[lyll----c is compact in the weak star topology.

13. COMPACTNESS IN WEAK STAR TOPOLOGY

41

A n s w e r 1: An element y of Y is a continuous linear form X---> R given by y" x---> y x = ( y , x ) E ff~ and can be identified with the element o

z = II(yx; x E X ) ~ Z. The weak star topology of Y is the trace of the topology of Z, by the definition of their respective bases.

A n s w e r 2" Each of the subsets IZxJ 0 and pair of elements of X an element y E Y such m

[(Y, Xl)

Zxl[ X by x ~ (fl + f2)(x) (fl + f2)(X) = fl( 2xl, X 2 " ' ' ,

xk),

0 - < X 1 -
-

llxll =,

x

E D(A).

We shall see applications in [Problem VI 15, Bounds]. REFERENCES

H. Br6zis, Analyse fonctionneUe, th~orie et applications (Masson, 1983). M. Schechter, Spectra of Partial Differential Operators (North-Holland, Amsterdam, 1971). M. Reed and B. Simon, Methods of Modern Mathematical Physics, I Functional Analysis (Academic Press, New York, 1972).

II. DIFFERENTIAL CALCULUS ON BANACH SPACES

1. S U P E R S M O O T H M A P P I N G S

We have given (p. 71) the definition of the (Fr6chet) derivability of a mapping f between locally convex vector spaces, and (p. 79) the definition of C k mappings between Banach spaces. If X and Y are Banach spaces the space ~ ( X , Y) of continuous linear maps X--~ Y is a Banach space, but if X or Y is only a locally convex topological vector space the space ~ ( X , Y) does not have a good topology: it is known for instance that the dual of a Fr6chet space (p. 26) is never a Fr6chet space. It is therefore more convenient to use a weaker notion of differentiability, the Gateaux differential, and to give new definitions for C ~ and C k mappings (cf. for instance, Milnor, p. 1024). Definition" Let X and Y be locally convex spaces. Let f" U---~ Y be a continuous map from an open set U E X into Y. Let x ~ U and v ~ X. The (Gateaux) derivative of f in the direction v is the vector

f ' ( x ; v) =- f ' ( v ) = lim t=o

f(x + to)- f(x) t

if the limit exists. The mapping f is C 1 in U if this limit exists for each x ~ U, each v E X and is continuous as a function: U x X---~ Y. It can be proved that if f is C a in this sense, f'~" v ~ f'x(V) is a linear m a p and the derivative of a composition of C 1 maps obeys the usual chain rule. Moreover f~ = 0, x E U, U connected, implies f is constant in U. On the other hand the inverse function t h e o r e m (p. 90) and the existence and uniqueness theorem for ordinary differential equations (p. 95) are in general n o t true if X and Y are not Banach spaces. Definition" mapping

A map f" U ~ U--~ Y

Y is C 2 if it is C 1 and if for each v 1 ~ X the by

x~--~f'(x,

u1)

is C 1,

that is, if this mapping admits a (Gateaux) derivative in each direction v 2 E X: 51

Gateaux derivative

52


lim t=0 second derivatives

f ' ( x nt- tv2, Vl)

--

f ' ( x , Vl)

t

where the limit, denoted fx'(Vl, v2) and called the second derivative in the directions (Vl, v2), depends continuously on x 6 U, l) 1 E X , v 2 E X. Higher derivatives are defined similarly. Fundamental properties of derivatives are: If f is of class C r in U, then for each x 6 U the mapping XxXx...xX-+X

Taylor formula

by

(vl

9. . ~

Vr) t+ f(r)(v I ,i x

~ . . . ~

Vr)

is a symmetric r-linear map. Any composition of C r maps is a C r map. If f is C n+l in U, and if the straightline segment joining x and x + h lies entirely in U then the Taylor f o r m u l a holds:

1 f(n)

f ( x + h) - f ( x ) + f/r(h) + I f/~'(h, h) + . . . + -~. Jx

(h . . . . . h) + Rn

with remainder Rn given by 1

R n - - ~1

f (1 - t ) n j xf(n+l) + t h (h . . . . .

h)dt.

0

We now consider the special case of Z2 graded commutative algebras, [Problem I 1, Graded algebras], with locally convex Hausdorff topologies for which sum and product are continuous operations.

1. HYPERDIFFERENTIABLE MAPPINGSf: A ~ A left right hyperdifferentiable

Let A = A+ 4- A_ be a Z2 graded algebra. A mapping f : A ~ A is called left [resp. right] hyperdifferentiable at a ~ A if it is differentiable (in the sense just defined) and if there exists an element u ~ A such that the linear mapping .f~: A ~ A, by h ~ fat 9h is the left [resp. right] product by u: .f" . h = uh

[resp. hu].

Show that the C 2 and left [resp. right] hyperdifferentiable maps in an open set @" C A are locally affine mappings in ~ if A satisfies the following hypothesis (i) A admits a unit e (ii) cb = 0 f o r all c ~ A _ implies b = O. A n s w e r 1" Suppose that for a 6 ~ ' f ' . h - u(a)h.

I. SUPERSMOOTH MAPPINGS

53

If f is C 2, the second derivative f ~ is a symmetric quadratic from

(k, h) v-+ f~'(k, h) - (u'(a) . k)h. By the hypothesis it is A-linear in h, so, by symmetry, also A-linear in k. Therefore

f 'a( k

~

h) -- v(a)kh

Since A is a graded algebra we can have v(a) symmetric in h and k only if v(a)hk - 0 for h, k 6 A_. It follows from the hypothesis that v(a)h -- 0 a n d v ( a ) - - 0 , i.e f ' - - 0 a6~" "~

a

~

"

2. SUPERDIFFERENTIABLE MAPPINGS A mapping f ' A ~ A is called superdifferentiable [resp. G l ] at a 6 A if it is differentiable [resp. C 1] and there exist elements u, v ~ A such that

Vh-h++h_

f~(h) - h+u + h_v,

6A+|

(1)

a) Show that u is uniquely defined if A admits a unit e, and that v is uniquely defined if h _ b -- 0 for all h_ ~ A _ implies b = O. b) Show that when B is a DeWitt algebra, superdifferentiability is equivalent to the requirement that f~, is B+ linear. !

Answer 2a: Take h+ -- e, the hypothesis on A shows that fa" (e + h_) -u + h_ v define u and v uniquely. Answer 2b: If f~ is given by (1), it is obviously B+ linear. Conversely, if fa, is B+ linear, then f ~ ( h + ) - - . f ~ ( e ) h + . We show now that f~ (h_) = h _ v ( a ) . For x, y, z 6 B_, we have f~ (xyz) = - f ~ ( y x z ) since yx = - x y . But, since xy and yz ~ B+, we have also f ~ ( x y z ) - - f ~ ( x ) y z . Thus f a ( x ) y z = - f ~ ( y ) x z . This being true for all z 6 B_, we have for all x, y 6 B_ f~ (x) y = - f~i (y)x.

(2)

In particular -

f~ (x)x

- - O,

x ~ B_.

Thus taking x = z I, one of the generators, we find that there exists an element q(/) E B such that

f~(z I) - q(t)z I

(no summation).

superdifferentiable

II. D I F F E R E N T I A L C A L C U L U S ON B A N A C H SPACES

54

The element q(z) is defined modulo the ideal generated by z z. We show that we can construct a q such that f'~(z') = qz',

Vl.

Let q~ be the solution of f ' ( z 1)

= ql z l

which does not contain Z 1. We have by (2) ( f ' ( z 2) - qlZ2)Z i = 0,

i=1,2.

Thus there exists v 1 ~ B containing neither z 1 n o r z 2 such that f ' ( z 2) - ql z2 = vlzlz 2. We choose 1

q2 = ql + VlZ 9

We can construct v n and qn by induction on n" ( f ' ( z n + l ) - qnZ"+l)Z i = 0 ,

i= 1,2,...,

n + 1

implies the existence of on such that f ' ( z n + l ) -- q n z n+l = VnZ 1 . . . Z n+l

and we set qn+ l = qn + Un Z1

9

.

zn

.

.

We have f'~(zi) = qn + 1zi

9

i= 1

if this same equation held for i = 1 , . . . , have

9

9

9

"

9

n + 1

9

n with qn+l replaced by qn. We

qn+l = ql + Vl z l + U2 Z2 "+" "'" "+" Un Z1

zn

The limit q, when n tends to infinity, exists in the topology of formal series [cf. Problem I 1 ( G r a d e d algebras)] since qn and qm, m-> n, differ only by terms of order ->n. 3. G P-MAPPING A

C p mapping is called G p if f ( a P ) ( h l , . . . ,

hi+,...

, hp+, hi_,...

, hp_

hp)

with coefficients in A.

is a polynomial in

1. SUPERSMOOTH MAPPINGS

55

a) Show that a C p and G' mapping is also G p if A admits a unit and f(P) (a) vanishes when restricted to ( A _ ) p. Show that if A is a DeWitt algebra B, then a C p and G l mapping is always G p. A G p mapping, with p some convenient number is called supersmooth. A mapping which is G p for any p is called G ~176 b) Prove that a polynomial on A #'l

P" a -+ ~

Cpa p,

Cp ~ A

p=0

is a G ~176 mapping.

c) Let B be a DeWitt algebra. Prove that a mapping f" U --+ B defined by a formal series f (a)-

E

with U - - {a 6 B; la0l < K }

cnan

is G ~176 on U if K is such that the numerical series ~ IcnlK n converges.

d) For U open in B, a mapping f" U ~ B given by the formal series f (a) - E

cnan '

a ~ B,

Cn ~ C. (or R)

can obviously be written

thus when f is restricted to B_ it is an affine map f (a_) -- co -4- cla_,

(3)

co, Cl E C or R.

Prove that conversely if a C 2 mapping from an open set U C B into B is G 1 then the restriction of f to B_ is of the form (3), but with arbitrary co, cl E B. Does this last result hold for an arbitrary graded algebra? Answer 3a: Analogous to proofs of 1). For details see references. Answer 3b" an

- - a +n +

na~r-la_

n--I (a n)'

(h) -- ~

a n- 1-q ha q _

h +na

n - 1 -Jr

h _ n a +n - - I

.

q=0

Answer 3c" The mapping f is well defined (cf. above). For a, a -4- h 6 U, a straightforward computation using the fact that a _ h _ + h _ a _ - - 0 and f (a) -- ~ Cn (a~_ -4- na~_-la-) shows that

supersmooth

56


lim f (a + th) - f (a) 1 = 7 t=O t

+th)"

= h+ Z c n n a n - l

-

an

+ h_ Z c n n a ~ _ - I

= h+(O+f)(a) + h _ ( O _ f ) ( a ) . O+ ~)_

The mapping f is thus G l with the formal series O+ f fined in U.

(O+f)(a) = ~

Cn#'la n - l

(O_.f) - ~

,

and O_f well de-

cnna~_-1 .

We deduce from these formulas that fa' vanishes on (A_) 2, that is

(O_O_f)(a) - 0 ; thus f is G 2. An inductive proof shows that f is G c~, with partial derivatives (oP+f ) ( a ) - Z Cnn(n - 1 ) . . . (n - p + l)an-p,

(op-lo_f)(a) - ZCnn(n-

l)...(n-p+

l)a+ p

All partial derivatives which involve two or more a_ are zero.

Answer 3d" If a, b ~ A and f is G 2 in A we have, by the Taylor formula I

,

f(a)=f(b)+fh(a-b)+~

,f

)2 h,+t(a-h) ,, (a_b)Zdt

(l-t

0

where the action of f " on (a - b) 2 is multiplication in the algebra. Taking b - a+ gives (a - b) 2 -- (a_) 2 = 0 and f ( a ) = f (a + ) + ./~i+ a _. If A is a DeWitt algebra every G l and C 2 mapping is also G 2. 4. SUPERDIFFERENTIABLEMAPPING

.f :

A n --+ A p

Extend naturally the definition of superd~fferentiability to mappings A n --+ A p, or more generally A n'+• P xA_." Answer: A C l mapping f " ~2 --~ A p • A q_, with Y2 an open set of A~ • An_, is called G l if ./~ 9h is obtained from h by the action of an (m + n) x (p + q) matrix with elements in A" when h ~ A~ x An_, 117

II

i=l

.j=t

57

2. BEREZIN INTEGRATION; GAUSSIAN INTEGRALS The elements ( a + i f ) ( a ) and ( a _ j f ) ( a )

of A are called partial derivatives

off.

partial derivatives

5. S h o w that a G "+1 m a p p i n g f : A m + X A"_ ---> A is o f the f o r m m

f ( a ) = fo(a+ ) + ~

where f0, f , . , . . . , f l . . . , Answer:

fiia _a i _j + . . .

fi(a+ )a i_ -t- Z

i=a

+ f l . . . ,, (a + ) a x. . . .

a_"

i#j

are G "+1 functions on A+.

Use the Taylor formula.

We thank R. Schmidt for calling our attention to Gateaux derivatives. REFERENCES

B.S. DeWitt, Supermanifolds (Cambridge University Press, 1984). A. Rogers, J. Math. Phys. 21 (6) (1980) 1352-1365. A. Jadczyk and K. Pilch, Comm. Math. Phys. 78 (1981) 373-390. Y. Choquet-Bruhat, "Supermanifolds and Supergravities", in Geometrodynamics proceedings, A. Prastaro ed. (World Scientific, 1985). J. Milnor, "Infinite Dimensional Lie Groups", in RelativitY, Groupes et Topologie H, B.S. DeWitt and R. Stora ed. (North-Holland, Amsterdam, 1984).

2. B E R E Z I N I N T E G R A T I O N ; G A U S S I A N I N T E G R A L S

I.

DEFINITIONS

a) Let B be a DeWitt algebra B = B+ @ B_. Let y ~ f ( y ) be a supersmooth mapping from B_ into B. T h e n (cf. P r o b l e m II 1, Supersmooth mappings) f(Y) = f0 + faY

with

f0 and fl in B.

By definition the Berezin integral of f on B

f

is

f ( y ) d y = cfl,

B_

where c is a constant independent of f, chosen equal to 1 by Berezin, to (27ri) a/2 by DeWitt, and (2~r) ~/2 in section ii of this problem. Now let f be a supersmooth mapping B q- ---->B by y ~ - - > f ( y ) , y = (yl,... , y q ) . Then f ( Y ) = fo + f a Y ~ + " "

+ fx . . . q y l . . . yq,

fo, L , ....

Berezin integral on B.

58

II. D I F F E R E N T I A L CALCULUS ON BANACH SPACES

and the Berezin integral of f on B q_ is by definition

f f ( y ) dy

= cqfl..,

q.

Bq

Let Y be an invertible q x q matrix with elements in B§ It determines an isomorphism B q ~ B q by ~ ~ y = Y~. Show the change of variables formula f f ( y ) d r = ~ f ( Y ~ ) ( d e t y)-X d,f. Bq

Bq

More generally it can be proved for instance by induction on q (cf. Vladimirov and Volovich, p. 756) that if )3 ~ y = F(.9) is a supersmooth mapping B q ~ B q such that F ' ( 0 ) is invertible (since F is supersmooth F ' is a q • q matrix with elements in B, in fact B+ since F' is a linear map from B q into Bq_), then

If(y) dy = f (fo F)()~)(det Oq_ integrable integral On B+

F'(33)) -a d)3.

Bq_

b) Let f be a mapping from B§ into B. We call it integrable if its restriction to R9 the body of B, is a B-valued integrable function for the Lebesgue measure of ~. Its integral is then the element of B"

f f(u) du = f f(ua) duB, B+

uB ~ R.

R

Analogously the integral of a mapping f : B P+~ B is, if the right-hand side is defined:

f

B P+

f(u) du

= f f(uB) dUB,

u=(u 1 9

~

~

~

9

u p)

9

du B = d u n

~

~

~

du~

~

RP

Give a formula for the change of variables in B P+, u = q~(fi), with ~p a supersmooth bijective mapping B p ~ B p such that u B = q~(fiB). c) Let f be a supersmooth mapping B P+• B q ~ B. Define its integrability and its integral. Give a formula for the change of variables

u = ~(~),

y = a(a) + Y(a)~,

u ~ B p+, y ~ B _ q

where ~ is a supersmooth mapping BP+~ B p such that ~(fiB) = u B and preserves the orientation, a is a supersmooth mapping B p ~ B q and Y(fi) is a q • q matrix with elements in B+ , each a supersmooth function on B P.

2. BEREZIN INTEGRATION; GAUSSIAN INTEGRALS

59

d) Let f be a supersmooth mapping from U C B+ into B and y a C 1 path in U, mapping t ~ y (t) from an interval [or, fl] of R into U, and such that y(ot) = y8 (or) = o~, y(fl) = YB(fl) = ft. 8§ The integral I• ( f ) of f over y is naturally defined by l• r(t) _ BB_ ~ a

B

- f f(y(t))g'(t)dt

with y'(t) an element of B+ depending continuously on t.

Give sufficient conditions on y and U to have the equality: I•

- I• ( f ) ,

where Yo is the soulless path [tx, fl] ~ R by the identity map, i.e. I•

(f) - ] f (r) dr. O/

Answer 1a: We deduce from the expression of f , using the antisymmetry y~y~ - -y[~y~ that the term of order q in f(Y~) is f (YY)l...q -- fl...q(detY). Thus, using the definition of the Berezin integral, we find

f

f f(Y~)(detY) -1 dy--cqfl...qf(y)dy. Bq_ 8 ~_ Note that if (yl . . . . . yq) is just a permutation Jr of @ l . . .

~q)

then the

previous formula may also be written

f f (y) dy -- (-1)sign~r f g(~) d~, B~

g--fort.

B~

Answer lb" By definition

f f(u)du=f f(us)dus. Bp

IRP

If uB -- ~o(fin) is a diffeomorphism of R p the classical formula gives

f ( f ogo)(fiB)l(Jgo)(uB),d~tB = NP

f NP

f(uB)duB.

integral over y


60

If ~0 is orientation preserving, i.e. the restriction of J~0 to R p is positive, we can suppress the absolute value, and since the body of B is a subalgebra of B we have

~_ (fo q:,)(~)(Jq~)(~)d~ B'+

-

fB'_,.

f(u)du.

Note that this formula can be generalized to mappings f whose restriction to R p has support in an open set U of 0~p and ~0 is an orientation preserving diffeomorphism U ~ V.

Answer lc: If (u, y) ~ f(u, y)

f is a supersmooth mapping B~ x then

Bq_--->B

by x =

f(u, y) = fo(U) + f~ (u)y" +... + fl... q(u)Y 1... Yq. If the usual Lebesgue integral of the right-hand side exists we can define

f

fdudy=cqff~...q(us)duB.

BP+•

aP

Under a change of variables of the indicated type we have, denoting x = (u, y ) = X(~), ,~ = (fi, ,f) ( f o X)(fi, )3) = (f0 o ~o)(fi) + - . . +

(fa...qO

q~)(fi) det Y ( f i ) ~ . . . ~9q,

thus, using previous results,

_f

f

(foX) BerX(fi,~)dfidy=

BP+• q

f(u, y) dudy

BP+xB q

where (cf. Problem 12, Berezinian) Ber X = (Jtp)(det y ) - l .

Remark:

A more general theory of integration on "foliated submanifolds" of B P+ x B q- can be developed, using the idea of question 3, i.e., defining integrals over immersed submanifolds of B P+, considered as mappings G ~ BP+ where G is an open subset of R P; we considered the particular case G = R p and the identity mapping. More general formulas for the change of variables can then be obtained [DeWitt, p. 41]. See the references.

Answer

l d: We have v(t) =

,/s(t).

61

2. B E R E Z I N I N T E G R A T I O N ; G A U S S I A N I N T E G R A L S

We suppose that the real function t~-+r = yB(t) is a diffeomorphism [a, fl]--->[a, /3]. By making the change of variables in the classical integral we find

I t ( f ) = J f(r + ys(r))(1 + Ys(r)) dr. of

We expand the integrand in a formal series of functions of r, using the Taylor formula, valid if r + Ays(r)E U for ,t ~ [0, 1] (hence y can be continuously deformed to 3'0 in U) " ) ) (1 + ys(r)) f(r + ys(r))(1 + y s ( r ) ) = ( ~ _o ~1 f(")(r)ys(r and we find, since Ys E B+"

f(r + ys(r))(1 + ys(r)) = f(r) +

:

(n + 1

!

f(n)('r)Ts+l('r)

;

from which the result follows since Ys(a) = Ys(/3) = 0. Note that the result obtained here is analogous to a result in path integration in the complex plane. For more general formulations using superdifferential forms and Stokes formula, see Vladimirov and Volovich, Rogers, or Bruzzo. II.

COMPUTATION OF GAUSSIAN INTEGRALS

a) A quadratic form Q on B P+ x B q

Q(x) = xiMijx j is defined by a matrix with elements in B, Mij, i, j = 1 , . . . , matrix M is supposed to be even, i.e., of the type

p + q. The

M = C(Da , B 1 elements of A and D [resp. B and C] even [resp. odd]. The action of M on B~ x B q- then preserves the parities.

c \

! I

An even matrix of order (p, q) preserves the parity of the components of x ~ B ','+q, while an odd matrix of order ( p , q) inverses their parity. The shaded areas cover even elements. T h e matrix on the left is even, the one on the right is odd.

62


An even graded matrix M of order (p, q) is said to be supersymmetric if X i Mijx

i -- x j Mijx i

for every

x

i

,

x .i,

no summation.

(1)

Show that A M-(C

B D)

is supersymmetric if and only if A - A,

i.e., Aij - A.ii; D - - -s

C- -B.

(2)

Show that if M is supersymmetric and invertible, then q is even. Answer 2a: (2) follows from (1) by using the fact that the elements of A and D are even, the elements of B and C are odd, (x i , . . . , x P) are even, (x p + l , . . . , x p+q) are odd and the fact that odd elements anticommute. M is invertible if A and D are invertible; the antisymmetry of D implies then that q is even. b) Let M be an even invertible supersymmetric matrix of order (p, q).

Show, by appropriate change of variables, that I -

f

exp(- 89

--(2rc)(P+q)/2(BerM)-l/2.

itJ

B ~ x B q_

(d p'qx stands for the du dy of the previous section, with the notation u (x ! . . . . . xP) ~ B p and y - (xP+! . . . . . x P+q) ~ Bq__.

Answer 2b: The linear change of variables x - X~ by X--

(

llp D _ l/~

O) Ilq

thati s

u-t~ y - - D -1/~fi+.~

block diagonalizes M. Indeed if x - (u, y) 6 B p • B q we have

Q(x) =_ Q(u, y ) -

Tu(Au + By) + Ty(--Bu + Dy).

Thus, under the indicated change, using the symmetries of A and D (Q o X)(.~) = (Q o x)(t~, ~) - Tt~(A + BD -1B)Ct + TyDy with

2~,l_ ( fi, 0

O)

D

wi th fi~ _ Z + B D - l [~

The Berezinian of the change of variables is unity, therefore

--

.~il~i.j.~.J

2. BEREZININTEGRATION;GAUSSIANINTEGRALS

/

e x p ( - 1 X i Mi.jx "j) dP'qx

-

f

-

63

e x p ( - 1 fci mi.jx .j) dP'q;.

8~• We shall further reduce ,~, a symmetric matrix, and D, an antisymmetric one, to canonical forms. It is known that there exist orthogonal transformations which reduce real matrices of such types respectively to diagonal or block diagonal ones. On the other hand we can always reduce symmetric or antisymmetric invertible matrices to their bodies because we have the identity, for any such matrix N

N =-- (I[ + NB1Ns)I/2n'I/2A'I/2( "'B

+ NB 1NS) 1/2

We perform all appropriate transformations by considering the linear mapping B p x B q --+ B~_ x B q given by s L~ with

o

L - ( O1 (~-p q- AB10As)l/2

)

02(tlq -+-D-~ 1DS)I/2 ' where O1 and 02 are such that det O 1 -- 1 and det 02 -- 1, and O 1A B (~ 1 -d i a g ( a l . . . a p ) , ai E R

02DBO2--diag((

0 -dl

0). .(0

dl

0))

We have then

P

(Q o X o L)-I(Ft, y) -- Z a i ( u i ) 2 -at- 2 d l y l y 2 q - . . . -I- 2dq/2yq-ly q i--1 while BerL -1

det(ll p + f~-~lAs)-l/2det(11 q q- D-B1Ds) 1/2.

Our integral therefore is

':

f

e x p ( - - I ~ ai(Fti)2-dl~l~2 . . . . .

dq/2yq-ly q)

x Ber L-1 dz/dy. We expand the exponential to find the term in y l . . . Yq, then we integrate over I~P, after replacing t~i by a real variable: the integral exists in the usual sense if all ai > 0, and (taking c -- ~ in the Berezin integral) it is

I -- (-1)q/2(2rc)(P+q)/2(Ber L - 1 ) ( a l . . . a p ) - l / 2 d l ...dq/2.

64


We note that al ...ap

--detAs,

dl . . . d q / 2 -

(detDB)l/2;

thus, using the expression of Ber L -1 , and the evenness of A and D" I -- (2zr)(P+q)/Z(detA)-l/Z(detD)l/2,

i.e., I -- (2rc)(P+q)/Z(Ber

M) -I/2.

REFERENCES F.A. Berezin, The method o/'second quantization (Academic Press, New York, 1966). B.S. DeWitt, Supermani?blds (Cambridge University Press, 1984) and Appendix of "The spacetime approach to quantum field theory", in: Relativitg, Groupes et Topologie II, eds. B.S. DeWitt and R. Stora (North-Holland, Amsterdam, 1984). L. Fadeev and R. Slavnov, Introduction to gauge theories (Benjamin, 1980). V.S. Vladimirov and I.V. Volovich, "Superanalysis" I, II, Theor. Math. Phys. 59 (1984) 3; 60 (1985) 743. A. Rogers, J. Math. Phys. 27 (1985) 710 and references therein. U. Bruzzo, in General Relativity and Gravitational Physics, eds. U. Bruzzo, Cianci and Massa (World Scientific, 1987).

3. NOETHER'S THEOREMS I

In this problem we consider systems with a finite number of degrees of freedom (questions 1-5) and systems with an infinite number of degrees of freedom (question 6). In the first case a dynamical variable is a path q: [ a , b ] ~ M, in the second case a mapping u: V--+ M; V and M are finite dimensional smooth manifolds. We prove the two fundamental Noether theorems. In questions 3 and 4 the lagrangian is taken to be invariant under a Lie group G of diffeomorphisms of M; this implies the existence of conserved quantities (identified with elements of the dual of the Lie algebra of G). In question 5 the action admits an infinite dimensional invariance group; this implies the existence of identities. Let L: T M • [a, b] --+ R be the lagrangian (p. 169) o f a dynamical system with configuration space the smooth manifold M. The action S is the mapping C 1([a, b], M ) --+ R by q w-> S(q) with b

S(q) --

f a

L ( q ( t ) , Cl(t), t) dt.

65


1) Show that a diffeomorphism ~" M---~ M induces a diffeomorphism T~" TM ~ TM. 2) Show that if qo" t ~ qo(t) is a critical point of S and ~ a diffeomorphism -1 of M then r o = ~( qo) is a critical point of S o~ . 3) Recall that a necessary condition for a smooth (C 2) mapping q to be a critical point of S is that it satisfy the Euler equations

Euler equations

d - L o ( q ( t ) , ~t(t), t) + Lq( q ( t ) , q ( t ) , t ) = O, $L -= - -dt P

l

!

!

where L o denotes the partial derivative along the fibre TqM and t q the partial derivative on M. One calls a solution of the Euler equations on a connected interval of R a trajectory of the dynamical system. Suppose that L is invariant under the 1-parameter group T~s of diffeomorphisms of TM induced by the 1-parameter group of diffeomorphisms q~s of M, that is

L(~p~o q, q~'. Vq, .) = L ( q , Oq, .),

Yq ~-. M, Vq

trajectory

tangent to M at q. (1)

Let the duality between TM and T * M be denoted by a dot. Show that the numerical function defined by ,

0

t~--~ Lo( q(t ), q(t), t) . -~s q~s(q(t)) is constant along each trajectory of the dynamical system. Show that the conclusion is the same if (1) holds but the trajectories are those of the dynamical system with lagrangian L1= L +

dr(q) d---7-'

where f is an arbitrary smooth function on M. 4) Suppose that the lagrangian L is invariant under a Lie group G of diffeomorphisms of M. Define the momentum mapping as a map from the space of trajectories of the dynamical system into the dual ~* of the Lie algebra of G. 5) Let q - I T M be the bundle with base [a, b] and fibre Tq(t)Mat t ~ [a, b]. We recall that the derivative of the mapping q ~ S( q) acting on a smooth section h of this bundle with compact support in (a, b) is b

S'( q). h = J ~L( q)" h dt. a

momentum mapping


66

Suppose that S'( q). h =-0 for all h of the form h=Du with D some linear differential operator mapping smooth sections u of a vector bundle E over (a, b) into smooth sections of q-ITM. Find an identity satisfied by ~ ( q). 6) Extend the previous results to lagrangians defined for smooth mappings

u: V---~ M, with V and M finite dimensional manifolds. Answer 1" If ~ is a C x map, its derivative ~'(q) at q is a linear map TqM'* Tq,(q)M. If q~ is a diffeomorphism (of class CP), ~0'(q) is an isomorphism and T~o'(q, Oq)~--->(~(q), ~o'(q). Oq) is a (fibered) diffeomorphism of TM of class C p-1. Answer 2: The mappings S o - 1 is defined by b

( S o t p-1) ( r ) = f (Lo Tip - 1)(r, i;, t) dt. a

We have (letting primes denote derivatives of maps) (So ~-x)'(r) = S'(~o-X(r)) 9(~o-')'(r); t h u s ( S o ~ --1 ) ' ( r o ) = 0 i f S t( qo)=0.

Answer 3" (a/as)~os(q(t)) is a tangent vector to M at ~,(q(t)). We set ,

d

0

A - Lq(~O,(q(t)), -dt q~s(q(t)), t).--~s ~(q(/)); we have dAd(,

d

dt = ~

L q(~Os(q(t)), -dt ~os(q(t)), t) ,

d

)

0

9~ ~os(q(t))

d

0

+ Lq(~(q(t)), -~ ~(q(t)), t). -dt -~s tp~(q(t)). On the other hand 0

d

,

a~

d~

O---sL(~s( q(t)), -dt ~'( q(t)), t) = tq(~Os( q(t)), --~ (q(t)), t) " ~ ,

d

(q(t)) ~ dq~

+ L q(~,(q(t)), ~ ~(q(t)), t ) ' a s

dt

67

3. NOETHER'S THEOREMS I Finally, since d / d t and O/Os commute dA = g'L

dt

0

0

d

" --~s q~( q(t)) + -~s L(q~,( q(t)), -dr q~s(q(t)), t).

Therefore d A / d t = 0 if L is invariant under q~,, that is if

L ( q~sOq, Tq~, o dl, .) = L ( q , cl , .), and if ~gL = 0; thus A is constant on a trajectory. If L 1 = L + ( d / d t ) f ( q ) and L is invariant, the conclusion still holds for the trajectories of L1, since L 1 and L have the same Euler equations.

A n s w e r 4: We have just seen that to the generator ~: = (O/Os)q~s(q(t))l,=o of a 1-parameter group of diffeomorphisms of M which leaves L invariant, and to each trajectory q of the dynamical system, there corresponds a number P

L q(q(t)' dl(t), t ) ' ~ which does not depend on t. If G is a Lie group of diffeomorphisms which acts effectively on M, the generators ~: of the 1-parameter subgroups are the elements of a vector space isomorphic to the Lie algebra ~d of G. Given a trajectory q, the linear mapping !

~ L q(q(t), q(t), t). defines an element a of ~d*. The mapping q ~ a is called the m o m e n t u m mapping.

A n s w e r 5" If S ' ( q ) . h = f b ~ t ( q ) . D u d t = O for all u" t ~ u ( t ) E E with compact support in (a, b) we have

u,

then

for

b t"

S ' ( q ) . h = 1 D* g'L (q)" U dt = 0 t/

with D* the adjoint operator of D, mapping sections of q - ~ T * M sections of E*. As a consequence

D*TgL(q) = 0

into

Vq.

A n s w e r 6" Let u" V---~ M be a smooth mapping between smooth finite dimensional manifolds V and M. The differential D u , or u', is a section of the vector bundle T*V | u - 1TM (cf. Problem V 11, Harmonic maps), with base V and typical fibre at x ~ V the vector space T*xV @ Tu(x)M. A lagrangian function of derivatives up to first order is obtained by composi-

momentum mapping

68


tion of a smooth map L" M x (T*V (~ T M ) x V---~R with (u, Du), and the action S(u) is obtained by integrating the lagrangian over V, assumed to be oriented, and 1 relatively compact with boundary OV"

S(u) = f L(u(x), (Du)(x), x)r(x) v

where r is a volume form on V. The derivative at u ~ C2(V, M) of the mapping S" CI(v, M)---~R given by u ~ S(u) is the linear map on sections h of the vector bundle u-ITM with base V and fibre Tu(x)M at x given by (we omit x in the right-hand side for brevity)

S'(u). h = f ~L(U)" hz + f to, v

(2)

av

where ~L(U), the Euler operator on u, is given by

~L(U) = L'(u, Du, .) - div(L~u(u, Du, .)). For each given u, ~L (U) is a section x ~ (~L (U))(X) of the vector bundle u-aT*M, the dual of u-ITM; the dot on the right-hand side of eq. (2) denotes the duality between these finite dimensional vector bundles. The divergence is relative to the volume form ~-; the explicit expression in local coordinates x i of V and y~ of M where u = (u~(x')), L has expression l_,(u~, t~iua, xi), and T = P(xi) d x l . . , dx d, is

o Ox

o O(Ou,/Ox,)

9

The d - 1 form to has for its differential dto = d i v ( L 'Du 9h)z with L/~u a section of TV @ u-~T*M--~ V. This can be written

to = ixz,

X = h. L~u,

a vector field on V,

and in coordinates (cf. p. 207) 1 to---

(p-

OL

a dxJ2 1)! a(a,u ~ h %2... Jd

dxJd

Remark" The addition to Lz of an exact differential does not modify the Euler operators: L and L 1 = L + div f(u) 1If V t.J OV is c o m p a c t - i.e., if V is relatively c o m p a c t - the integral exists without hypotheses other than Smoothness on L and u; in other cases further hypotheses have to hold for the integral to exist.

69


have the same ~L(U). The derivatives S ' ( u ) and S~(u) are equal on h's with compact support in V (i.e., vanishing on a V ) . Let % be a 1-parameter group of diffeomorphisms of M which leaves L invariant, that is, L ( ~ o u, ~ "

v u, .) = L ( u , v u, .)

Vu E M, v u ~ TuM.

We replace the scalar A of answer 3 by the vector field on V aq~ J~(x) = L ~ u ( ~ ( u ( x ) ) , D ~ ( u ( x ) ) , x). ~ (x); in coordinates the components of J~ are d L i) (~)[3(U~/(xk)) aj~#(U~/(xk)) X i) a~s(Xk) J~(xh) -- ~(aua/ax s ' s ' aS

"

We have div Js = div(L~u). --~s + LDu. D a----s A computation analogous to that in the one-dimensional case leads to && aL(~o~o u, D ( ~ o u), .) = 0. div ,/~ = ~ " ds + as

Thus div J~ = 0 if ~L = 0 and L is invariant under ~s. O n e writes J = J,]~=0, and calls J a conserved current associated to the 1-parameter group ~,. More generally if G is a Lie group of effective transformations of M and ~: is an infinitesimal generator of G, i.e., an element of ~d, the linear mapping ~:~ L~-

conserved current

~:

defines a mapping between the space of solutions of Euler equations and ~3" valued conserved currents on M. Such a mapping is called a m o m e n tum map.

The property of current conservation still holds for lagrangians L 1 whose sum with a divergence L = L 1 + div f ( u ) is invariant, since L and L1 have the same Euler equations. Of course J must be constructed with L. R e m a r k : A n o t h e r proof in the present context of Noether's theorem on current conservation is to consider the derivative S ' ( u ) . h. If L is invariant under the 1-parameter group ~s of diffeomorphisms of M the same is true of S ( u ) for any open submanifold U C V, and the corresponding derivative in the direction h = a~s/asls= o is zero. Thus

momentum map

70

II. D I F F E R E N T I A L C A L C U L U S O N B A N A C H S P A C E S

3%

f $L(U)" h'r + f to = O, U

if u satisfies

h = - ~ -s

s=0

aV

g~L(U)= 0 we therefore have

f

for all boundaries

to=O

OV.

av

Thus d00 -- 0~

that is, 0R

,

div X = O,

where X =

h. LDu, and h = -~s ~=o'

i.e.

X=J.

A section h such that

S'(q).h=O infinitesimal invariance

is called an infinitesimal invarianee. Infinitesimal invariances of the type Do, with v an arbitrary section of a bundle over V, come in general from infinite dimensional invariance groups of the lagrangian L, or of L plus a divergence in V relative to its volume element (the two corresponding actions have the same derivative when restricted to h's with compact support in V). When M admits an infinitesimal invariance h = Dv the proof given in the case of dimension V = 1 of the identities D* SL(V ) = 0 ,

Vv

applies without change. D* and D are adjoint for the L 2 scalar product on V with volume element r.

Example: v = A is a Yang-Mills potential (p. 403), assumed to be globally defined on V. The Yang-Mills action is S(A) = ~ F~' . F~,~,"r,

Fa~, = O~A~, - a~,A,~ + [A,~, A ~,].

v

If h is a ~d-valued 1-form with compact support in V,

S'(A). h = f ~VM . h'r= - f (~F v

v

~aF ~" = 7~F "" + [ A~, F "" ].

h~,)'r,

71

4. N O E T H E R ' S T H E O R E M S II

S(A) is invariant under a change of gauge: A---', U-1AU +

U -1

dU,

U: V--> G,

so

S'(A).h=O

if

h=Vu-du+[A,u],

u'V--.q3

f V,FX '.V ,ur = -f v

v

and we find the identities These can also be derived as a consequence of the Bianchi identities. References can be found at the end of Problem II 5, Invariance.

4. N O E T H E R ' S T H E O R E M S II

We consider as in the previous problem (question 6) lagrangians for smooth mappings u" V---~M between finite dimensional manifolds. These lagrangians are defined by mappings

L" M x ( T*V (~ TM) • V---~R and a volume form r on V. The corresponding action, for a C 1 mapping u" V-~ M, with differential Du is the integral of the d-form Lr"

S = f L(u(x), (Du)(x), x)z(x). v

We shall consider in this problem invariance under diffeomorphisms of V (questions 1, 2) or V x M (question 3).

1) Let f" x ~ y = f(x) be a diffeomorphism of V. The lagrangian Lr is said to be invariant under f if we have the equality of forms L(u, f ' Du,

f-1)f-l*'r

--

L(U, Du, .)r.

a) Justify this definition by constructing the image of the function x ~ L(u(x)(Du)(x), x) by the diffeomorphism f, extended to a diffeomorphism

M x (T*V (~ M) x V--->R.

invariant

72


b) Suppose that the lagrangian Lz is invariant under a 1-parameter group of diffeomorphisms f~ of V. Obtain the corresponding infinitesimal invariance law. c) Show that this infinitesimal invariance implies the conservation law J = -L~uX(Du ) + LX,

div J = 0,

where X is the generator o f the 1-parameter group f~ and u is a C 2 solution of the Euler equations of the lagrangian Lr. Give an example. d) Let V be a product S x R and let (x, t) E V. Let fs be the 1-parameter group of translations of V:

fs" x~--~ ~ = x,

energy

t---~ t = t + s.

Give the necessary and sufficient condition for a lagrangian Lr to be invariant under fs. Give the corresponding conserved current J. The energy o f the field u is the quantity (when the integral exists) Eu(t ) =

J~ Sl

S t = S x/ {t}, "'

where trt is the ( d - 1)-form induced on S t by r and jo the component of J in the "time" direction ff~. Show that when S is compact (without boundary) the energy Eu(t ) is independent of t, when u satisfies the Euler equations. Discuss the case of a non-compact S. e) Consider the lagrangian on scalar functions on a manifold V= S x R with pseudo-riemannian metric g of hyperbolic signature, S x (t} being space-like and {x} x R time-like, with associated contravariant tensor g : L~" = ( g ~(Du, Du) + m2u)T, stationary

~- volume element of g.

Suppose g is invariant under the 1-parameter group of diffeomorphisms o f V defined by (x, t)~-+(x, t + s); such a metric is called stationary, Construct the corresponding conserved current. Give another example of a lagrangian L z on scalar functions on a manifold V with a metric g, where g and Lz are invariant by a 1-parameter group of diffeomorphisms of V. Construct the corresponding current. Answer la: The lagrangian Lz is a d-form on V, L is a function on V obtained by the composition of maps" V---~ M x ( T*V (~ TM) x V---~ R x ~ (u(x), (Du)(x), x) ~ L(u(x), (Du)(x), x).

4. N O E T H E R ' S

THEOREMS

73

II

If f" W---~ V is a diffeomorphism, its derivative defines a diffeomorphism f* of T*V on T* W. We consider then the following sequence of maps:

W---~ V---, M • ( T * V @ TM) • V---~ M • ( T * W @ TM) x W---> R by (recall that W = V)

y ~ x = f ( y ) ~ (u(x), Du(x), x) ~ (u(x), f*(x)Du(x), f - l ( x ) ) L(u(x), f* (x)Du(x), f -

X(x))

which justifies the definition. A computation in coordinates also gives the result: let

,p(x j) = s

4u~(xJ), x j)

with s the representation of L in coordinates x i in V and y~ of M, and let x i= fg(.;cj) represent locally the diffeomorphism of V. We then set _

02 t

(o(2 j) = L(u'~( fi(2])), ~t(u'~( fJ(2J))) ax---i, fJ(2k)); we have

~(~) = r

if

x =fl~).

We denote by v~(2 j) the function u~(fi(2J)) and we have

~(~) = s ,

~,u~(~j) a~' 0x---7, f ( ~ ) ) .

The action S(u) on a domain of a chart of V with image a subset U of R a was

S(u) -- _] (~D(xJ)fl(xi) u

dxX . . .

dx a E R;

by the theorem on the change of variables the value of this integral is equal to

S(u) = ~(v) =

f

~(~J)~

~

i~")

f-l(u)

The lagrangian L(v, D r , .)§ under the integral sign is the transform of the original one. They are the same lagrangian if and only if

L(v, Dv, .)§ = L(v, Dv, .)r which gives the indicated invariance law.


74

Example" Consider the action on a ball B of R 2" S(u) .

m 2.u 2 d--

-+-

.

3X 1

.

-~- h x I O~u t~X 1

OX 2

+ /~X2 0~~ u

O~X2

dx I dx 2

B

and a rotation B --* B; x = f~ (y)" 1 1 y2 X = y COSS+ sins, 2 _yl y2 x = sin s + cos s.

The lagrangian is invariant by the rotation if and only if A =/z. A n s w e r l b: If L is invariant under a 1-parameter group of diffeomorphisms fs of V with generator X = afs/Osls= o, then L(u, f* Du, f~l)(f-~l*r)

(1)

is i n d e p e n d e n t of s. The corresponding infinitesimal invariance law is obtained by taking the derivative of (1) with respect to s and putting s = 0. Since f,]s--0 = Identity, fs - Identity, then 0

a s L I,--0 = x ,

and

~ss f~*l,-0 = D X .

The result is, up to multiplication by z X,(L)-

OL O(Du) D u . D X -

X . O L - L div X = 0,

where a L is the partial derivative of L with respect to its explicit dependence on x and - d i v X~" is the Lie derivative of z with respect to X; that is in local coordinates div X =- Vi X i -

1

0

p ax ~ (PX~)'

0L aiXJaju ~ - xiai L - L V~X i = 0 X , ( L ) =- O(o~iua) A n s w e r lc: The components of the vector J, in a natural flame on V are ji = _

OL

O~( Ifti u a ) x Jtftj U a -I- t X '.

We have, therefore

%Ji= -x%u~

u

-a(a,u

+ L V ~ X ~ + X ~D i L ,

(aW ju + x'a, aju ) (2)

75

4. NOETHER'S THEOREMS II

where D iL is the derivative" 0L

OL

D iL = OiL + O(u~) OiUj q- ~OU~ Oiu '

Uj = OIU .

(2) can therefore be written

Vi Ji ~ - X r ( Z

) -- XJOjU~

(U),

~L,a

(U)

~

( ~r (U))a

which implies Vi Ji = 0 when ~L(U) = 0 and X , ( L ) = O.

A n s w e r l d" If the diffeomorphism fs" S x R---> S x R is :~ = x the

mapping fs* is the

and

identity

i= t+ s

mapping.

Therefore,

a lagrangian

L(u, Du, .)r is invariant by f~ if and only if in coordinates x " = (x ~, x ~ adapted

to

the

product

structure

of

S • R,

the

9 9 L- ( u ,a D ~ u , x i, x 0 ) p ( x ,i x 0 ) does not d e p e n d exphcltly on

X 0.

lagrangian The corre-

sponding conserved current is jO~ _.... __

0L

0 ( 0 , , . u a) O~

a

~

+'5~ L,

a =O, 1 . . .

d-1.

The conservation law is 1

D " J~ - -p a~ (J~p) = 0 which implies for any d o m a i n / 2 of V with b o u n d a r y 0/2"

o

(3) g2

where cr is the ( d -

Oil

1)-form A

cr = p dx ~ ^ . . . ^ dx'* ^ " " ^ dx d-l, where dx ~ is suppressed. If we t a k e / 2 = S • [t 0, tl] and S is c o m p a c t we have 0 0 = Sto t_J St1 and (3) reads

f

St 1

f jop dx 1 . . . dx d - 1 = I jop dx I . . . dx d- 1. St0 /

If S is not compact, but J~' has c o m p a c t support we have the same formula. If J~ is a limit of functions with compact support we m a y have the same formula. Otherwise s u p p l e m e n t a r y terms can appear.

76


Answer le" In local coordinates, x ~ E R, x i, with i = 1, 2 , . . . ,

d-

1,

coordinates on S,

L = g ~ O~u O~u + m2u 2. L does not depend explicitly on x ~ (since O0g ~ = current is jo = L - 2g ~~ O~u OoU = -g~176

0), and the conserved

2 -I- gi] 0i u t~jU Jr m2u 2,

j i = _2g~i O~u OoU. The energy density jo on S t is positive. Note" C o m p a r e the current o b t a i n e d here with the relations on pp. 513-514.

Example: The lagrangian for scalar functions Lz -- ( al3 t~aU OoU Jr m2u)z,

z = dx ~ . . . dx d-1

with the Minkowski metric ~ / ~ ' = d i a g ( - 1 , + . . . + 1) is invariant under Lorentz transformations, and so is L. The conserved current corresponding, for instance, to the infinitesimal boost

X = x~

- x~ao

is

j1 = 2rlal aau X u + x~ j0 = 2r/a0 aau X u - xlL, ji = 2rlai aau Xu, i = 0, 1. 2a) Consider the action defined by the lagrangian Lz"

S(u) = f L(u(x), (Du)(x), x)<x). v

Suppose that L admits the infinitesimal invariance X , ( L ) = 0 (cf. l b ) , with X a tangent vector field to V, tangent.also to OV. Show that S admits the infinitesimal invariance h = X ( u ) = - X j aju. 2b) Give an identity satisfied by the Euler operators when S is invariant by

all diffeomorphisms of V, hence, when S'(u). h = 0 for all h of the form

X(u).

Answer 2a" We have

S'(u). h = f (L;h + V

Oh)

.

4. NOETHER'S THEOREMS II In local coordinates, and for h =

S'(u).

h =

77

X(u)"

~

O(O,u ~)

ai(X j aju '~) r

v

which can be written, using the expression of computation

X,(L)

and an easy

S'(u). h = f (X,(L ) + D,(LXi))~ ". v

We have

f

Di(LXi)'r

= f LXinitr

v

since X is tangent to

OV.

= 0

av

Therefore,

S'(u) . h = ~ X, (L )r. v

Note that h =

X(u)

is the Lie derivative in the direction X of the mapping

U.

Answer

2b: We restrict X to be of compact support in V, h is then also with compact support, and we have

- vf

- vf

If S'(u). h = 0 for all X we have therefore ~gL,~(U)ajU~ = 0. The Euler operators satisfy the identity (vanishing of a section of T'V) ~'L,~ (U) Du~ ----0.

Example" Consider on a 2-dimensional manifold V the lagrangian L~with two unknown functions u I and u 2 (i.e., M = R 2) L - f f(u~, u2) dul ^ du2= f f(u1, u2)(OluXt92u2- t92UlOlU2)dxl V

dx 2.

V

A straightforward computation checks that

~L,lt~l ul + ~L,Et~l U2 5 0 '

Remark:

~L,1 t~2ul + ~L,20~2u2-~-0.

When the unknown is not a mapping V---~ M but a section of some more general fibre bundle over V the invariance of the lagrangian by all diffeomorphisms of V leads to differential identities on ~L(U) (cf.

78


an example in Problem II 7, Stress energy tensor), if the Lie derivative of u with respect to X is a differential operator on X. 3) Let fs" V x M ~ V • M be a 1-parameter group of diffeomorphisms of

V x M with generator Y =dL/dsl,=o. Denote by X ( ~ ~ the direct sum decomposition of Y corresponding to the product structure of_the image; in local coordinates (x i) on V, (y~) on M, the representative f~ of fs is f~ " (xi, Y~ ) ~ ( L-j (x i,y a ), L71~(xi, Y~ )), -

the vector Y has components X i, (;~ with

x'

dL

aff

= -d-7 , - o '

"ds ,-o"

The lagrangian for mappings u: V--) M, L = i L(u(x), (Du)(x), x)r v

admits the infinitesimal invariance Y if Y,(L) =- X , ( L ) + r

= O,

where X , ( L ) is given in l b and (cf. Problem H 3, Noether I) ~(L) = L'u" ~ + L;,~ .D~.

Give the corresponding conservation law for a solution of Euler's equations. Answer 3" A straightforward computation analogous to previous ones shows that div J - ( ~ -

X(u)). ~L(U) + Y~(L)

with J = L ~ , , . ( ~ - X(u)) + L X , i.e., ~L

ji __ ( ~a _ gjt~juOt ) t~(t~iu~ ) "[- z x i " Thus div J = 0

if

~'L (U) = 0.

Remark: We also have div J = 0 when *L(U)= 0 if we suppose only that Y, (L) = 0 when ~'L (U) = 0. References can be found at the end of Problem II 5, Invariance.

5. INVARIANCE OF THE EQUATIONS OF MOTION

79

5. INVARIANCE OF T H E E Q U A T I O N S OF M O T I O N

1) Show that an arbitrary infinitesimal invariance of the action is also an infinitesimal invariance of the equations of motion.

Answer 1" Let S(u)= fv L(u(x), Du(x), x)~-be the action defined by a lagrangian L [Problem II 4, Noether II]. The equations of motion are by definition the Euler equations ~L (U(X))= 0. The Euler operator ~L (U(X)) is such that the derivative of the action with respect to u at an arbitrary u is, for all h with compact support on V,

S'(u) . h =- f ~L (U(X)) " h(x)r.

(1)

v

S admits the infinitesimal invariance k if

S'(u). k = 0 for all u.

(2)

We deduce from (1) that the second derivative

S"( q). (h, k) -- f (~L(q(x))" k(x)))" h(x)r.

(3)

V

where ~L(U(X)) is the Jacobi operator (denoted Jq, p. 87). But by (2) s"(u) .

(h,

k ) = S"(u) . ( k , h ) = O.

Thus

f ( ~ 'i.(q(x)). k(x)) . h(x)r = 0

V h with compact support.

v

Therefore

~L(q(x))" k(x) = 0.

(4)

Remark: An infinitesimal invariance- for instance, a Killing vector field of a Lie group action on Y - does not in general have compact support. Thus, in general

S'( q)" k = f (~L (U(X))" k(x) + div K(x))r

(5)

v

with

K= k. OL/a(Dq).

(6)

2) Conversely, let k be an infinitesimal invariance of the Euler equations.

80

on,

off shell


a) Show that if k has a compact support, or the boundary term in (5) vanishes, the action S admits this infinitesimal invariance, when v is a solution of S'(v) = O. Such an invariance valid only when v is a solution of S'(o) = 0 is said to be on sheU 1. b) Example" Let L be the lagrangian for a free Dirac particle" m

L =

f( 89

T

a~ dx k

a~ k ax k

9

qJ is the Dirac ad]oint of q~ [Problem 1 10, Dirac]. Let G be the group of transformations f,; qJ ~ q6 defined by ~ = ~ + sq~

with ~ a given spinor such that ~ L ( ~ ) = 0.

(8)

Show that ~L(6) is invariant under G. Show that the action S(~, (O)= L dx is not invariant under G; show that, on shell, S is invariant under G. Compute the generator Xy of the group of transformations defined by (8) and the corresponding conservation law. Answer 2a" The derivative of S'(u). k is S"(u).(k, h), which can be written as (3) for all h with compact support. Thus, if (4) holds and k has compact support or is such that boundary terms vanish S"(u) . (k, k) = 0

Vu.

The property S ( f , v ) = S(v) is then a consequence of the Taylor formula (expansion in s) and the property S ' ( v ) = 0, satisfied when v is a critical point of S.

Answer 2b" The Euler-Lagrange equations for a Dirac particle are O= ~gL(~)= T k a~/~ + m~, ax k

(9a)

O=~L(~)=

(9b)

and its conjugate 34' k m~. ~x k "Y -

One checks readily that if ~ and ~3 solve (9a) and (9b), respectively, ~ L ( ~ , ) - ~gL(~b)

and

~L(~b,) = ~L(~')"

1The expression on (the mass) shell [resp. off (the mass) shell] was used first for properties of real [resp. virtual] particles which hold only when (mass)2= (energy)z- (momentum)2 [resp. even when the mass-energy-momentum relation is not satisfied].

81

5. INVARIANCE OF THE EQUATION OF MOTION

Alternatively, one can check that the following condition (derived in paragraph 1) is satisfied:

L(~I'(X))" k(x) = O, where g L(~,(X)) is the Jacobi operator along a trajectory of L. The action is invariant only on shell; indeed,

s(~,,, r

s(~,, ~,) + ~

~ ~

ox--7

ox----~~ -

~ ,.

The generator of the group of transformations defined by (8) is 0

= ~~

0

~ - ~ + ~ ( x ) e$o

and the quantity,

OL

i

_

= ~($~)~-

OL

~r162

D

is conserved for g, and q, solutions of the Euler equation (9):

2 0 i

a X""~ g ~p --"

(0____~_~i) r x i ")l

= (O~l

i

- i) 0__~_~ 0~

~[~ -~- ( ~ll')l

aXi

-

0x----7~/ - m ~ ) q~- ~

(,y

a X i ( ,)/ i~ ) _ ~

( i 01//) ")/ a X i

0111

i--+m~)t~xi

since ~0, q5 are solutions of the Dirac equation. Hence div J~0 = 0 when g' and ~ are solutions of the Dirac equation.

Note: In this question the manifold V is R 4, the fields and their derivatives are square integrable on R 4, the boundary terms vanish for this reason. Many other developments related to Noether theorems are of interest in physics: lagrangians depending on higher order derivatives, equivalence between invariance and conservation laws for non-degenerate lagrangians, etc. They can be found in the references or their bibliography. We thank N. Ibragimov for useful comments. REFERENCES J. Goldberg, "Conservation laws", in General relativity and gravitation, A. Held, ed. (Plenum Press, 1980).

82

II. D I F F E R E N T I A L CALCULUS ON B A N A C H SPACES

N.H. Ibragimov, Transformation groups applied to mathematical physics (Reidel, Dordrecht, 1985). Y. Kosmann Schwartzbach, "The Noether theorem", in Geometrie et Physique, Y. ChoquetBruhat, B. Coil, R. Kerner, A. Lichnerowicz eds. (Hermann, 1986).

6. S T R I N G A C T I O N

string

A string is a 1-dimensional object which sweeps out an oriented 2dimensional submanifold W ("world sheet") of the spacetime in which it moves. Let (M, g) be this spacetime, a d-dimensional C ~ manifold with riemannian or pseudo-riemannian metric g. Let h be a metric on W, d ~ ( h ) the volume element defined by h, f an embedding W---~ M, and let

S(h, f, g) = f (h (~ g) 9(Vf Q Vf) d/x(h) w

I

h abg~,~a~f~'ab f~ d/x(h)

w

be the energy o f this mapping. 1) Show that S is invariant under conformal rescaling of h. 2) Give a necessary condition for h to be a critical point of S, when f and g are given. Show that under this condition the action is equal to S = 2 (dp,(h). w

Answer 1" k = Ah implies k ab = A-lh "b d/.~(k) = A d/x(h)

~

Answer 2" For h to be a critical point of S it is necessary that S~ = T = 0, with

Tab

=

g~,~O~ftzO b f" - 89 hab (hCdoc f u O d f~g~,~).

The condition T = 0 is satisfied by h = Af ' g ,

A arbitrary function.

If W is endowed with the metric f * g we have

h abg,~a af"a b f u =2

7. S T R E S S - E N E R G Y T E N S O R

83

and

s = 2 f d~(h). w

7. S T R E S S - E N E R G Y TENSOR, E N E R G Y WITH RESPECT TO A

TIMELIKE V E C T O R FIELD

I. LAGRANGIANS DEPENDING ON A METRIC

Let S:g ~ S(g) = fv L(fl(g), x) d/x(g) be a C 1 mapping from the space ~ / o f C ~ riemannian metrics on a given compact smooth manifold V into R; we denote by jk(g) the k-jet of g, i.e., the set of partial derivatives of g of order - k . The derivative S'(g) of this mapping at a point g E ~ / i s a linear map:

jk( g) k-jet

Tg J/l ---->~ .

The tangent space Tg~ is the space of C k symmetric 2-tensor fields over V, since the space ~ of C k metrics over V is an open set of their vector (Banach) space. SuppOse that S ' ( g ) . h can be written

S'(g)'h = f ~L(g)" h(x)dtz(g),

(1)

v

where the dot on the right-hand side denotes duality between ((~TxV) 2 and (QTx*V) 2. The quantity ~L(g) (Euler operator for g, cf. Problem II 3, Noether) will now be denoted by T and called the stress-energy tensor. We can also write OS/Og = T (2) with the convention that OS/Og is a linear mapping on h given by the scalar product with volume element d/z(g)"

L 2

S'( g). h = (T, h) = f T(x). h(x) d/x(g(x)). v

1) Show that if L is invariant under isometries of (V, g), that is, diffeomorphisms f: V--->V and replacements of g by f ' g , then T satisfies the identities (where V is the riemannian covariant derivative) V" T - O ,

i.e., V~T ~ - = 0 .

stress-energy tensor

84

II. D I F F E R E N T I A L C A L C U L U S O N B A N A C H SPACES

2) Show that if L is invariant under conformal rescaling of g, i.e., under g ~ Ag, then trgT-g

T~=0.

3) Comment on the extensions of these properties to noncompact V and pseudo-riemannian metrics.

Answer 1" If S(g) is invariant under isometries of g then it admits the infinitesimal invariance (where ~ denotes the Lie derivative and X is an arbitrary vector field)

h=.~xg. Thus f

S'( g) . exg - f T(x)..~xg d / z ( g ) - J T~t~(V~X ~ + V~X ") d/z(g) = 0 v

v

which implies, since V is compact and T symmetric,

f x~ Vo T~a d/z(g) = 0

VX~

v

and therefore V~T ~ =0.

Answer 2" The infinitesimal invariance corresponding to conformal rescaling is

h = pg,

p an arbitrary function

Thus

f

r(x).

p(x)g(x) d

(x) = 0

Vp,

v

and therefore T- g = trg T = T,,a g ~ = 0.

Answer 3: If V is noncompact, some hypothesis must be made to insure the convergence of S(g). However, the derivative S'(g) is still given by the L 2 scalar product (1) on V if we demand that h have compact support in V. We then have the same definition and properties for T. The signature of the metric is irrelevant.

7. S T R E S S - E N E R G Y

TENSOR

85

II. LAGRANGIAN DEPENDING ON THE METRIC AND OTHER FIELDS

1) Suppose now that the lagrangian L depends not only on the metric g on V, but also on some other field u, a section of a vector bundle E over V. Suppose that the action does not depend explicitly on x ~ V S( g, u) - f L(Jkg, Jtu) d/z(g), v

and that the derivative of S at the point ( g, u) can be written S'(g, u ) . ( h , v ) = f ( ~ t ( g ) " h + ~gt(U)" V) d ~ ( g ) , v

where ~gL(U) is a section of the dual bundle to E over V and ~ t ( g), denoted by T as in I, is a contravariant 2-tensor field on V. Suppose that S is invariant under diffeomorphisms of V, that is

S(g, u) = s(f*g, f* u), where f * u denotes the field on V induced from u by the diffeomorphism

f. Show that if u satisfies $L(U)= O, the stress-energy tensor T satisfies the equation V. T = 0

i.e., V~T ~t3 = 0 .

2) Examples" compute T and V. T in the following cases: a) the nonlinear scalar field, with lagrangian Z = - g at3o uo u + u p + l , b) the Yang-Mills field, with lagrangian

L = F~O.F~r where the dot denotes the scalar product in the Lie algebra (p. 403) in which the field F = D A takes its values, and F = D A means F~o = V~A~ -V~A~ + [A s, A~].

Answer 1: Since we have assumed that diffeomorphisms induce transformations of the field u, we have a definition of the Lie derivative of u with respect to a vector field X on V. The invariance of S under diffeomorphisms implies the infinitesimal invariance under generators of 1-parameter groups of diffeomorphism, that is S'(g, u). (h, v ) = 0

if

h = ~ x g , v = ~ x u.

(3)


86

Eq. (3) reads

S' (g, u) - f (T ...~xg 4- ~L (u) . oL~'XU)dtt (g) -- 0. t,I

V

It reduces to eq. (1) when OPL(u) -- 0, hence V.T--0

if

OPL(U)--0.

More precisely we can compute V 9T in terms of ~L (u) as follows. We already know that (~xg)~[~ = V~X~ 4- V/~X~. We write .LFxu -- DuX. where Du is a linear partial differential operator on X, depending on u; since ..~.xu, like u, is a section of E, the operator Du maps vector fields, i.e., sections of T V, into sections of E. We denote by D u, the L2-adjoint operator, mapping sections of E into sections of T V. Eq. (3) gives

f

(-zv

9T + DuEL(U)). X dtt(g) - 0

'v'X.

v

We therefore have the identity

- V . T + 89Du*gL (u) -- 0.

(4)

If Ca(u) - - 0 , i.e., d~ ( u ) - 0 is the zero section of E, then D*oPa(u) - 0 , and V . T = 0 2) Examples" a) For a scalar field u we have f * u = u o f - l and .Ld,xu _ X . O u -- X~3~u. Thus Du" X ~-+ X ~ 3~u contains no differentiation of X and D*" v t-+ Du*v is the product by 3u, since it is defined by

f

DuXvdtt

-f

X~Oo~uvd#

-f

(Duv)~ X ~ d # ,

v a scalar field.

An easy computation gives (recall that d # ( g ) also contains g and that 8gOt/3 =-g~

Tal~ -- (~L (g))al~ -- 3auO13u - I g~(OXuO;~u

_

l,l P q - 1 )

~ ( u ) --= 2V ~ Ootu + (p + 1)u p. Formula (4), or a direct calculation, gives

VotT ~ - 3l~u(V~

4- 89(p 4- l)u p) --0;

thus V~ T ~/~ -- 0 if u satisfies the equation of motion of the scalar field, 2V ~ Ootu + (p + 1)u p -- O.

7. S T R E S S - E N E R G Y T E N S O R

87

b) Computation gives (cf. Yang-Mills field, p. 403) T,~t3 = i g,~t3(F*'* 9F,,., ) - 2F,~, 9Ft3*. D" F - D , , F '~t3-=V,~F "t3 + [A ~, F '~ ]

gL(A) = - 2 D . F, and

V ~ T ~ = D~F*~ . F~ - 2D~F~* . F~* - F~, . D~F~ ~. By changing names of indices and using the antisymmetry of F we obtain: V~T ~t3 = ( D ~ F ~" + D~'F t3A + D~Fm') 9F~, - 2 D ~ F ~A. Ft3~. Thus V~ T ~t3= 0 modulo the Bianchi identity D F = 0 (which reduces to the closure of F in the abelian, in particular Maxwell, case) and the Yang-Mills equation D . F = 0. III. HYPERBOLIC METRIC G ON A MANIFOLD V O F SIGNATURE ( + , - - , . . . , - - ) .

We suppose that V is a product V = M x ~, with the submanifolds M, = M x {t} spacelike. The e n e r g y - m o m e n t u m vector associated to a stress-energy tensor T and a timelike or null vector field X is defined to be

p~ = X~ T '~t3. 1) The energy o n M t with respect to the vector field X is defined by the integral (when it exists)

E(t) = f P~n~

dl.tt,

Mt

where n is the unit normal t o M t and d p , t is the volume element of the metric induced on M, by g.

Suppose that T satisfies the equation V~ T ~ = O. Study the dependence of E(t) on t. 2) Determine E(t) for the scalar field of H-a and for the Yang-Mills field

of ll-b. Answer 1" If V~ T ~ = 0, for a symmetric tensor T "~, we have V~P~ -

T ~ (V~X~ + V~X~). 89

(5)

Thus for any compact domain K C V with smooth boundary a K, and smooth T and X, we find


88

f

f P~'n,, do-= 89J T~t~(V,,Xt3 + Vt3X~)d/z,

aK

(6)

K

where d/z is the volume element of g, do- is the volume element of the induced metric on OK, and n is the unit normal of OK. We consider a domain D = M x [to, tl] of V, and we suppose that T ~t~ is smooth and that its support is contained in a fixed compact set Z for each t E [ t o , tl]. We apply formula (6) and obtain, denoting by d/z, the volume element induced on M, by g, l1

fP n d t,-fP n dt to= 89 Mq

Mto

Xt3 + Vt3X~) d/z.

tO M t

The right-hand side is zero if X is a Killing vector field of the metric g. In that case E(t 1) = E(t 0). The above formula is valid if T is smooth and has support in a set Z x [to, tl] , ~ compact in M. It will remain valid for any tensor T which can be approximated by a sequence of tensors 7", of the previous type in such a way that the integrals written for the 7','s converge to the integrals written for T. 2) Examples" a) If u is the non-linear-scalar field of II-a, we have

P~n~ = -~X] ~n~(a~uO~u) + X~Oaun~O~u + 1San~u p+I. We define a quadratic form y by

Y

=

X n~ + (X~n ~" + x~'n )

and we see that

P n~ = l y A .OAuO~.u + ~ X a n~u p+I. The quadratic form Y is positive definite" this can most easily be seen by choosing coordinates such that X ~ 1, X ~ = 0 , n i = 0 (since M, is a submanifold x ~ t, constant). Then n o = (g~176 n i = giO(gOO)-l/2, and

y

oo

=g

oo n o,

y

io

=0,

q _ giJn Y = o.

P~n~ is therefore positive for all u -> 0, and positive for all u if p + 1 is an even positive integer; the energy density P~n~ of the scalar field u is then positive. The energy E(t) is defined if P~n~ is integrable o n Mt; this will

7. STRESS-ENERGY TENSOR

89

be the case if the restriction of Ou to M t is square integrable in the appropriate norm and volume and the restriction of u p+I t o M t is integrable. b) The energy density of the Yang-Mills field is

P n,~ = 8 9

n,~-2X

n F,~x.F~ x.

To simplify the writing we take time lines orthogonal to the space-like sections, so g0~ = n ~= O, and Pn~,'~ reads

P'~n,~ = ( 89F~J" F,j - Fo~ " F ~ = ( E 2 + HZ)n0,

H2

=

89

E2

_.

- g ij.~O0~ ~ "~o~. Foj.

E 2 and H 2 are positive if the scalar product in ~ corresponds to a positive quadratic form on ~.


III. D I F F E R E N T I A B L E M A N I F O L D S

1. S H E A V E S

Let X be a topological space and F a surjective mapping from the family of open sets in X onto a family ~- of sets. Suppose that for each pair (U, V) of open sets in X with U C V there is a mapping, called the restriction mapping, r w" F(V)---~ F ( U ) , where F(V) and F(U) are sets corresponding to U and V. The family ~ together with the mappings r~ is called a sheaf over X if: 1) r ~ o r w = r w for open subsets U C V C W; 2) If {Ui} is a collection of open sets in X with U = U~Ui then a) if f, g E F ( U ) and r uu, f = r vi v g, V1. then f = g, 9U i " UI b) if fiEF(U~) and Vi, j, then there exists f E F ( U ) such that r ~ f = f/, Vi. If the elements of ff are groups, modules, graded algebras, etc., and the V r U are homomorphisms of these structures, the sheaf ~ over X is called a sheaf of groups, modules, graded algebras, etc.

sheaf

ruinu}fi=ruinujfj,

Show that the family of smooth functions on the open sets of a manifold X is a sheaf of rings over X.

sheaf of groups modules graded algebras

Answer" The set F(U) associated to the open set U C X is the set of smooth functions U--> [~. Let V and U be two open sets in X such that U C V. The mapping r~/associates to f: V--> ~ in F(V) its restriction to U. It is straightforward to check that it satisfies the listed properties.

2. D I F F E R E N T I A B L E S U B M A N I F O L D S

Definition" N " C M m is said to be a smooth submanifold of M m if for every x E N" there exists an open neighborhood U of x in M m and a diffeomorphism f: U ~ V open in R" 91

smooth submanifold

92


such that for R" defined as the set R n x {0} m-n

f-l(v

inherited structure

n a')

= u n N', (xl,...,

x n, o , . . . , o) ~ ~n.

Show that if { (Ui, q~i)} is an atlas of M m, and fi satisfies the above property on each Ui, then the set {(U i N N n, f/)} is an atlas for N". This differentiable structure is said to be inherited from the differentiable structure of M m. Answer: One checks that { U i t3 N n} covers Nn; the differentiability of

n uj n N")-->

n

vj n N")

is a consequence of the differentiability of the diffeomorphisms f~.

Show that the submanifold N C ~2 defined by x 2= IX1[ cannot inherit the differentiable structure of R 2 Answer: If N inherits a differentiable structure from ~2 there exists a diffeomorphism f from a neighborhood U of zero in R 2 onto V in R 2 such that f - 1. V---~ U

by

(yl, y2) ~ (x 1, x 2) is a diffeomorphism

and U M N is the curve of R 2 (x ~, x 2) = f - l ( y ~ , 0),

y~ E V N R.

This curve is differentiable at each of its points, while the submanifold [XI[ = X2

is a curve which is not differentiable at the origin.

3. S U B G R O U P S OF LIE G R O U P S . W H E N A R E T H E Y LIE S U B G R O U P S ?

We have defined (p. 242) a Lie subgroup H of a Lie group G as a subgroup o f G which is also a submanifold of G. We have defined a submanifold as a regular embedding (p. 239). Show that a Lie subgroup H o f a Lie group G is closed in G. Give an example o f a subgroup of a Lie group which is not a Lie subgroup. It can be proved that, conversely, any closed subgroup H of a Lie group G is a Lie subgroup.

4. CARTAN-KILLING FORM ON THE LIE ALGEBRA ~3

93

Let ~d and ~ be the Lie algebras of G and H. We shall show first that there exists an open n e i g h b o r h o o d V o of O in ~ such that the exponential mapping (p. 160) is a diffeomorphism from V o onto a neighborhood V e of e in G, and

Answer:

exp(V o n

x ) = v.

n

H.

Select a neighborhood N O of O in ~ such that exp is a diffeomorphism from N O onto N e, a neighborhood of e in H. Since H is a submanifold of G there exists a neighborhood U e of e in G such that N e = U e n H . We restrict U e t o an eventually smaller neighborhood V e such that the exponential mapping is a diffeomorphism from V o o n t o V e. If X E V o n c N O we have exp X ~ exp V o and exp X E H. This gives the required property for V o . Let ( h n ) C H be a sequence which converges in G to an element g E G. Then h , g -1 converges to e" for every neighborhood We of e there exists an integer N such that for n-> N, h, g-1 C We. We choose W e small enough to have ( x , y ) E W e X W e => x y - 1 E v e .

Then for n > N,

hnhN 1 ~. V e n H,

so

h,h~v 1 = exp X ~ ,

X~ ~ V o n ~.

We may always choose V o bounded. Then there is a subsequence of the sequence X n which converges to an element X E ~ ; h , h / v 1 converges to exp X E H, and h~ converges to h N exp X ~ H. Thus H is closed in G. Example of a subgroup of a Lie group which is not a Lie subgroup. Let G be the 2-torus T 2 G '- S 1 x S 1-,- U(1) x U(1). Let H~ = {(exp(2i~rt), e x p ( 2 i T r A t ) ) , t E R . Then H is a subset and a subgroup of G. It is not a submanifold if A is irrational; the closure of H in G is in this case the whole of G.

4. C A R T A N - K I L L I N G F O R M ON T H E L I E A L G E B R A ~d O F A L I E G R O U P G

The mapping ~ d : ~d---~~(~d, ~d) (p. 167) is defined through the derivative at e ~ G of the mapping Ad: G ~ .T(~, ~d). Thus, if X E ~3, ~ d X is a

94

III. D I F F E R E N T I A B L E

MANIFOLDS

linear mapping q 3 ~ ~. We denote by tr the trace of such an endomorphism and define the Cartan-Killing symmetric bilinear form on ~ by

B(X, Y) = tr(Md X Md Y),

X, Y E ~d.

An automorphism or of ~d is a bijective linear map or ~ G L ( ~ ) which preserves the Lie bracket (i.e., or is an isomorphism ~d---~~d) ~[X, Y] = [ orX, or r].

1) Show that the Cartan-Killing form B is invariant under automorphisms of c~, that is B(X, Y ) = B(o'X, crY). Show that B is invariant under the adjoint maps A d ( g ) , g E G (p. 167). 2) Prove the formulae

B(X, [Y, Z]) = B(Y, [Z, X]) = B(Z, [X, Y]). 3) Give the expression of the Cartan-Killing form in the basis e i of ~d.

Answer 1" If o- E G L ( ~ ) we have Md(o'X) = or o Md X o or -1 . Indeed, (cf. p. 167) the action of Md X on ~3 is

Md X: ~---~ ~

by

Y~-~IX, Y], by hypothesis,

so Md(crX): ~--~ ~ by Y---~[crX, Y] = or[X, cr-~Y], since cr preserves the bracket. We therefore have tr ~dd(o'X)Md(cr Y) = tr(cr Md X Md Y or

-1

) = tr Md X Md Y.

II

The invariance of B under the adjoint maps is a consequence of the fact that LgR~I" h ~ ghg-~ is an isomorphism of G and the following lemma: Lemma " Let f: G--, G be an isomorphism of Lie groups, that is, a diffeomorphism such that"

f(xy) = f ( x ) f ( y)

V x, y E G.

(1)

Then f determines an isomorphism ~3~ q3 of their Lie algebras.

Proof: A left invariant vector field X on G has an image under f which is a left invariant vector field, since relation (1) implies that f commutes

5. DIRECT AND SEMIDIRECT PRODUCTS OF LIE GROUPS

95

with the left translations: fo L x = Li~x) Of

and, by derivation f ' o Lx = Li(x ) of', LxX = X

implies

P 2= I'x= f'O LxX= L'I(x) o f ' X =

L 'I(x)X.-

The proof is completed by using f ' [ X , Y] = [ f ' X , f ' Y ] (p. 135). Answer 2" The mapping ~dd" ~d--->GL(~3) is a homomorphism of Lie algebras with the usual bracket in the Lie algebra of matrices, that is ~r

Y] = [ ~ d X , ~ d Y ] - sgd X ~Id Y -

~ d Y ~ d X.

Thus, using the fact that tr A B = tr B A and tr(A + B) = tr A + tr B tr(~Cd[X, Y] ~ d Z) = tr(~Cd X ,~d Y ~ d Z - ~ d Y ~ d X ~ d Z) - t r ( ~ d Z ~ d X ~ d Y - ~ d X ~ d Z ~ d Y) = tr(~Cd[Z, X] ,rid Y). Answer 3: We have (p. 168)

( ~ d e,) /

l --'Ci

k~

thus B(e i ej) = c i ' k Cj k l" ~ cj~t = 0 when i # j, then B is negative. If Cgk~"--"--Cikt a n d Cik It can be proved that the Cartan-Killing form is nondegenerate if and only if G is semi-simple, i.e., admits no invariant abelian subgroup. The Cartan-Killing form is negative definite if G is semi-simple and compact.

5. D I R E C T A N D S E M I D I R E C T P R O D U C T S OF LIE G R O U P S A N D T H E I R LIE A L G E B R A

1) Direct products: Let G be a Lie group which is the direct product o f two Lie groups A and B. Show that the Lie algebra ~ is the direct sum - ~ (~ ~ with [~, ~ ] = O. Give an ~gd G invariant scalar product on

Answer" G = A x B means that G is the direct product of the manifolds A and B, with the product law

semi-simple

96

III. DIFFERENTIABLE MANIFOLDS (al, b2)(a2, bz) = (ala2, bib2), aa, a 2 E A;

b 1, b 2 ~ B.

A left invariant vector field X on G is a pair of tangent vector fields to A and B, X = (X~, X~) such that

L g X = X; but

L gX = ( L ~Xa , L bX~ ) . Thus ~J = ,.d x g3 ~ , . d 0 g~

by identifying the subspace of ~d of pairs (X~, 0 8 ) [resp.(O~, X~)] with ~r [resp. ~ ]. The Lie bracket in ~3 is [X, Y] = ([X~,, Y~], [X~, Y~]), as is true in general for the bracket of vector fields on a product manifold; this can be checked using the definition on p. 134. The scalar product of a direct sum def_

(Xa + X ~ ) ( Y a + Y e ~ ) = X a Y a + Xe~Y ~ is Ad G invariant if the scalar products in A and B are respectively Ad A and Ad B invariant. II 2) Semidirect products: Let A and B be groups, and t: (B • A ) - * A be a group of automorphisms of A indexed by B:

tb: A ~ semidirect product

A

by

a ~ tb(a ) = t(b, a).

The semidirect product G = A | B is the space of ordered pairs (a, b) with the product law (a~, bl)(a2, b2)= (altbl(a2), bib2).

(1)

If A is a Lie group and t a Lie group of diffeomorphisms of A defined by B, then G - A @ B is a Lie group. Show that the Poincard group is the semi-direct product of R 4 with the Lorentz group. Show that the Euclidean group [E,- R ' | SO(n). 3) Write the Galileo group as a semi-direct product. Obtain it by a limiting procedure from the Poincar~ group.


97

4) Let A ' = A x { ~ ) and B ' = { ~A} X B and let a ' = (a, ~B) and b ' = ( ~A, b). Show that with the semidirect product law (1)

b'a'b '-1 - (tb(a))'.

(2)

Similarly for the Lie algebras, let M'=M@O~

and

~'=O~@~;

Xf~= (Xa, O~)

and

X~ = (O~, X~).

Let %" M ~ M be such that t b(exp X a ) = exp(% (Xa)),

Show that (% (Xa)) ' = Ad(b')(X~). 5) Write down a group multiplication of G = ~ | where R+ is the abelian multiplicative group of positive numbers. Compute the Lie bracket of its Lie algebra. The calculation given in the answer is used in the construction of the quantum observables of a system whose configuration space is R+ (see Isham, and Klauder). 6) Show that the Lie algebra ~3 of G - A | B is ( ~ - M | ~ , where the semi-direct product Lie algebra is defined by [(Xl, Y1), (X2, Y2)] = ([Xl, X21 + Md(Y1)X2 - Md(rE)Xl, [ r l , rE]),

XEM,

YE~.

(3)

where M d ( Y ) X = [Y, X].

Answer 2: A Poincar6 transformation (a, A) of the four-dimensional Minkowski space X is defined by (a, A)" X---~ X

by

y~ = A~'~x ~ + a ~',

Poincar6 transformation

A E SO(3, 1).

Performing another transformation gives the product law (a2, A2)(al, A1)= (a 2 + A2a 1, A2A1). Therefore the Poincar6 group is the semidirect product of the four dimensional translation group and the homogeneous Lorentz group SO(3, 1). A euclidean transformation of R ~ is defined similarly, leading to the same product law for A ~ SO(3).

Answer 3" A Galileo transformation g = (b, a, v, R) of R 3 x R is defined

euclidean transformation

98


by g: R 3 x R ~ [~3 • R with (x, t ) ~ (y, s), where y i = Rijx j - vet + a i

s=t-b. It follows that the product gxg2 = g is g = (b 1 + bE, a 1 + b21o1 -t- R l a 2 , 0 1 -~- Rll)2, R I R 2 ) . This is the group law of (R 1 x R 3) | n:3 with the group of automorphisms of R 1 • R 2 indexed by n:3 defined by

EE(b, a) = (b, bu + Ra),

E = (u, R) E [E3, (b, a) ~_ ~ • [~3.

Note that the inverse element of g is g

-1

=(-b,-R

-1

(a-bv);-R

-

lv, R ~). -

Set x ~ ct in the Poincar6 transformation and let c tend to infinity. Let A 0 tend to zero in such a way that Aoc tends to v ~, and a ~ tend to infinity in such a way that a~ tends to b. The corresponding limit of the Poincar6 transformation is a Galileo transformation.

Answer 4: The inverse of a pair (a, b) is the pair (a, b) -1 such that (a, b)(a, b) -~ =(1] A, 1In). Hence it is given by (a, b) -1 = (/b-l(a-X), b - l ) . Let a' = ( a , 1In) and b ' = (1] A, b). Then

b'a'b '-1 = ( t b ( a ) , b ) ( t b l ( 1 ] ~ ) , b - a ) = ( t b ( a ) l ] a ,

1]B)=(tb(a))'.

(4)

Let exp(rbX ~) = t b(exp X~ ). It follows from (4) that

( e x p ( % X ~ ) ) ' = b'(exp X ~ ) ' b ' - k Hence

(Zb(X~))' = A d ( b ' ) X ~ . Answer 5" Note that the multiplicative group R+ is isomorphic to the additive group R, with the isomorphism given by exp: ~--* R+. A group law of R | R+ is defined by an automorphism of R indexed by R+. Let g = (v, A) ~ R | R +, consider the following left actions of [~ | ~ +: Lg(x, y) = (x + log )t, y - A ego) defined on R x R,


99

and L g ( X , y ) = (ax, ya - 1 - v) defined on ~+ x R.

These two actions define the same group law

(5)

(u2, /~2)(u1, /~.1)~ (o2 + u1'~.2-1 , a2al).

Let b E ~(ff~), the Lie algebra of ff~, and r E ~ ( E + ) , the Lie algebra of R+. It is easier to compute [(b 2, r2), (bl, rl) ] by decomposing (b, r ) = (b, 0) + (0, r) and computing separately the four Lie brackets [(b2,0),(bl,0)] , [(b2, 0), (0, rl)], [(0, r2) , (b l, 0)] and [(0, r2), (0, rl) ]. We shall do the calculation by two different methods: i) Algebraic method using the Campbell Hausdorff formula (see appendix) together with the group law (5): ii) Computing the brackets of the left invariant vector fields on G = R | [~ + (representation theory). i) We spell out the computation of only two of the four Lie brackets which make up [(b2, re) , (bl, rl)]: exp(tb 2, 0) exp(sb I , 0) e x p ( - tb 2, 0) e x p ( - s b I , 0) = exp( 89 ts[(b 2, 0), (bl, 0)1).

(6)

By virtue of the group law (5) the left-hand side is equal to (0, 1)= exp(0, 0). Identifying the left- and right-hand side of (6) yields [(b 2, 0), (b I , 0)] = (0, 0). Similarly, exp(tb 2, 0) exp(0, s r l ) e x p ( - tb2, 0) exp(0, - s r l ) = (tb 2 - e-Srltb2, 1) = exp(tb2(1 - e -srl ), 0) = e x p ( s t b 2 r I + higher order terms, 0), which yields [(b2, 0), (0, r 1)1 - (b2rl, 0). A similar calculation gives [(0, r2), (bl, 0)1 = ( - r 2 b l , 0) and [(0, r2), (0, r,)] = (0, 0). Finally [(b 2, r2), (b~,

rl)l =

( b 2 r , - r 2 b , , 0).

ii) The set of left invariant vector fields L x on G can be considered either

100


as the Lie algebra ~?(G) of G or as a representation of the Lie algebra obtained in the previous paragraph from the exponential map from ~ ( G ) into G. The left invariant vector field L x on G is the generator of the one parameter group of transformations {Rg(t); dg(t)/dtl/= 0 = X}" d d Lx(h ) = ~ (Rg(oh)[t-o = ~ (Lhg(t))lt-o = LhX. We shall compute hg(t) with h = (v, A) for g(t) in the two following cases

g(t) = exp tX = exp(tb, 0) = (tb, 1)

(7)

g(t) = exp tY = exp(0, tr) = (0, e").

(8)

and

In the first case, using the group multiplication (5)

hg(t) = (v, Z)(tb, 1) = (v + h-~tb, A), d L(b.o)(V, A)= ~-~ (hg(t))[t= o = (A-lb, 0 ) = A-lb ~ v + 0 a a--A'3

Lx(v, h)= k - ' b a/dv.

(9)

In the second case

hg(t) = (v, A)(0, e t') = (v, A err), L(o,r)(V ,

A)= ~d (hg(t))[,. o = ( 0 ,

A r ) = 0 -~v a + hr--~, (9

i,e,

Lv(v, A) = Ar a/dA.

(10)

We can consider L x = h - l b a/Or as the representative of X = (b, 0) and Ly = hr O/aA as the representative of Y = (0, r). The brackets [Lx, Lx] , [Lx, Lv], [Ly, Lx] and [Ly, Lr] are the representatives of [X, X], [X, Y], [Y, X] and [Y, Y] respectively. It follows from (9) and (10) that

[L x, [L x, [Lv, [L v,

Lx] Ly] Lx] Lv]

=0 = b2rlh-la/ao = - r2blh-lo/av =0

hence hence hence hence

[X, [X, [Y, [Y,

X] = (0, 0) Y] = (bErl, O) X] = ( - r 2 b l , O) Y ] - (0, 0)

Finally, [(b E, rE), (b 1, rl) ] = (bEr I - rEbl, 0).


101

Answer 6: As in answer 5, we decompose (X, Y ) = (0, Y ) + (X, 0). We shall show that

[(Xl, o), (x~_, o)] = ([x,, x:], o) [(o, Y1), (o, Y:)] = (o, [Y,, Y:]), [(Xl, 0), (0, Y2)]-- (-"~d(Y2)Xl, O) [(0, Y1), (Y2, 0)] = ( ~ d ( Y , ) X 2 , 0), which combine to give (3). Let ai(t ) = exp(tXi),

bi(t ) = exp(tYi),

i = 1, 2.

Then

[(x,, 0), (x~, 0)] = ~r

0))(x~, 0)

d = - -dt( A d ( a l ( t ) ) ( X

2 , 0)]1t =0

(p. 167)

d d (al(t)1] )(a2(s ), 1] )(al 1 =dtds , B ~ (/), 1 ~)1,=,=0 d

d (al(t)a2(s)a_l

= dt ds

= (~d(X,)X~,

1 (t), 1]B)lt=~=0

o) =

([x~, X~l, 0).

A similar calculation gives the other brackets. For instance the calculation of [(X 1, 0), (0, Y2)] requires the computation of d

d

dt ds ((al(t)' 1]B)(l]a' b2(s))(al

_~

(t), 1]B))lt=s= 0

d d 1 dt ds ax(t)tb2(s)(al (t), b2(s))lt=s= o "- ( - 6 ~ d ( Y 2 ) x 1 , 0).

REFERENCES See also references at the end of [Problem III 7, Homogeneous]. C.J. Isham, "Topological and Global Aspects of Quantum Theory", in Relativity, Groups and Topology H, ed. B.S. DeWitt and R. Stora (North-Holland, Amsterdam, 1984) pp. 1141, 1163, 1171. J.R. Klauder and E.W. Aslaksen, "Elementary models for quantum gravity", Phys. Rev. D2 (1970) 272-276.


102

6. H O M O M O R P H I S M S AND A N T I H O M O M O R P H I S M S OF A LIE A L G E B R A INTO SPACES OF V E C T O R FIELDS

Given a faithful left action o f a Lie group G on a manifold X by trg" X---> X , let K ~ be the Killing vector field ( p. 165) on X generated by d KVx = ~ (trgr

o with g ( t ) = e x p ty.

(1)

Show that the mapping F: y---, K v is an anti-isomorphism, i.e., [ K v~, K ~] = K[~2,~,l.

Let v ~L and v R be respectively the vector fields on G generated by the left action Lg(t) and the right action Rs(,), g(t) = exp ty. Show that the mapping F: y~--~v~L is an anti-isomorphism and F: y ~ v vR is an isomorphism.

i

Answer: Let Cjk be the structure constants of G, i.e., [,/j, ,/,,1 = Cjk%. ,

It has been proved (p. 164) that [

] = _

In other words [F( yj), F( yk)] = -- C'jkF( Yi). The mapping F is linear, since K~+~ = d dt (crso)(x))]t--~ d = dt (cry( g(t)))lt= o = ~

+

with g ( t ) = exp t(% + Y2) with Crx(g(t)) - crso)(x ) - or(g(t), x)

Yz)= K~' + K~2.

F is a linear mapping between finite dimensional spaces and satisfies

[F(yj), F(Yk) ] = F ( - C~kY i i ) = F ( - [ y j , Yk])" Hence F is an anti-isomorphism. It has also been proved (p. 164) that [V~j,L V~k]L =--CjkV~L

(right invariant vector fields)

i R [RTj, irk] --Cjk~),yi

(left invariant vector fields).

and

103

7. HOMOGENEOUS SPACES; SYMMETRIC SPACES

It follows by a similar argument that

F" y--+ vyL

is anti-isomorphism

F" y--+ v vR

is an isomorphism.

Various conventions are used to ease the fact that F: y --+ K • is an antiisomorphism. For instance, Isham defines g(t) -- e x p ( - t y ) . Note that he also denotes without comma the Lie bracket [yjy~] of o~(G) and with comma the commutator [KYJ, K • ] of vector fields. With his conventions [ K yl ,

K Y2]

-

-

K[Y1Y2].

Note also that labels L and R are sometimes used to denote left invariant and right invariant vector fields rather than generators of left and right translations. 7. H O M O G E N E O U S SPACES; SYMMETRIC SPACES Let G be a connected Lie group and H a closed subgroup of G. The quotient space G / H of left cosets g H is called a homogeneous space. It can be proved (Chevalley I p. 114) that G / H can be given the structure of a differentiable manifold.

homogeneous space

l) Show that G / H is not necessarily a Lie group. Define an action of G on G/H. 2) Let ~ and ~ be the Lie algebras of G and H. Show that ~ subalgebra of ~. 3) Show that if G is compact there exists a subspace ~ = ~

@ ,/~'.,

~

C'l ~

= 0,

is a Lie

C ~ such that

A d ( H ) J ~ / C -/N'~.

(1)

A Lie algebra (Y and a Lie subalgebra ~ satisfying these properties is called a reduetive pair. The corresponding homogeneous space is said to be reductive. For examples see Problem V 9, Invariant geometries.

reductive pair

4) Assume that H is a compact, invariant, Lie subgroup of the compact Lie group G, i.e., ghg -1 ~ H for every h ~ H and g E G. Show that there exists a Lie subalgebra 1C of fg such that ~, __ ~

@ jc/,

j(r n ~

= 0,

Ad(H)~,U C/C,

[ ~ , Jc/] - 0. (2)

5) A s y m m e t r i c space is an homogeneous space G / H with an involutive a u t o m o r p h i s m a on G (i.e., a map a" G --+ G satisfying a ( g k ) --

symmetric space involutive automorphism

104


0-(g) 0-(k) and 0-2 = Id), such that Or

or

G OC H C G ,

(3)

where G Or is the closed subgroup of G consisting of all the elements left fixed by 0- and G O is the connected component of G Orwhich contains the identity. Show that SO(n + 1 ) / S O ( n ) is a symmetric space. 6) Let t be a transitive left action of a connected Lie group G on a differentiable manifold M, t: G x M ~ M

by

t(g, x) - ts(x ) - gx,

tglg2 = t~tg 2

(4)

and such that

t(e, x) = x. G-space homogeneous

The manifold M is called a transitive G-space or a homogeneous space. Show that M can be identified with a homogeneous space in the sense defined above. 7) Show that the cotangent bundle T * S 1 is a ~.2-space, where ~-2-RE Q s o ( 2 ) is the euclidean group whose product is defined by (m 2, n 2, ~bE)(m1, n 1, ~bl) = (m, n, (~b2 + ~bl) mod Err), with (m)=(m2) ( c~ n n 2 + sin~b 2

symmetric

riemannian space

-sin~22)(ml ) cos n~ 9

A riemannian manifold M is said to be symmetric if: there is an isometry group G which acts transitively on M, there is a subgroup Hp C G which leaves p ~ M invariant, there is an isotrometry Sp E Hp of M onto itself which reverses the geodesics through p. Show that such riemannian symmetric space is a particular case o f a symmetric space in the sense defined above in question 5. 8) a) b) c)

Answer 1" The product in G does not induce a product in G / H unless H is an invariant (normal) subgroup, i.e., gHg -1 = H for all g E G. Indeed if gl, g2 E G and hi, h E E H we have glhl g2h2 J~- gl g2H unless h lg2 -" g2h~ for some h~ E H. If H is a normal subgroup, the space of left cosets G / H , equal to the space of right cosets G \ H , is a Lie group. One can define a left action L k of k • G on G / H by

L k g H = (kg)H.


105

One cannot define a right action unless H is a normal subgroup. A n s w e r 2: Y( is a linear subspace of q3 and is given the structure of a Lie

algebra by [X, Y ] x - [X, Y]v E ~

for X, Y ~

since H is a subgroup of G. A n s w e r 3" Since G is compact, it admits at least one bi-invariant (left

invariant and right invariant) positive definite metric. Hence ~3 admits at least one scalar product invariant under the transformations {Ad(g); g E G}, i.e., (X[ Y) = (Ad( g)XlAd (g) Y). Let ~ be the orthogonal complement of Yt' in this scalar product. Then

~3=~|

~n~=0.

On the other hand A d ( h ) g = Y( since H is a Lie group and Ad(h)A/is orthogonal to A d ( h ) W because the scalar product in ~ is Ad(g) invariant. Hence A d ( h ) J / / C ~ for every h ~ H. A n s w e r 4: Conditions (2) are a special case of conditions (1). Since H is

invariant in G, [X, Y] E ~ for every X E Y( and Y E cg,

i.e., [ ~ , ~] C W.

Let Y = Y~ + Y2,

Y~ E 2g,

Y2 E g'g.

[X, Y~] E ~, and [X, Y2] E Y(; in fact, [X, Y2] = 0, since Y2 E Y{ implies Ad(h) Y2 ~ YL Hence [~, Y(] = 0. It follows from the Jacobi identity and the previous property that

[~, ~] c ~, i.e., :7~ is a Lie subalgebra of q3. It is isomorphic to the Lie algebra of the group G / H . A n s w e r 5" Let --1 g ~ sgs " where

G = S O ( n + 1) and H = S O ( n ) .

Let

or" G---~G

by


106

S--

-1

0 )

0

qqn

lln the unit element of SO(n)

'

(5)

It is easy to show that the map o- is an involutive automorphism of G" ~r(gk) - cr(g)cr(k) and cr (or (g)) - g. The subgroup G cr C G consisting of all the elements left fixed by a is the set of matrices of the form 1 0

0) h '

h 6SO(n)

(6)

The group SO(n) is connected, hence G ~ is connected, and G ITo m G ~ 9 we can identify G ~ with H -- SO(n). The homogeneous space SO(n + l ) / S O ( n ) is symmetric. See also [Problem III 8, Examples]. isotropy, stability, stationary, little group

A n s w e r 6: Let Hx C G be the isotropy group of x (aliases: stability subgroup, stationary subgroup, little group), i.e., the subgroup of G which leaves x invariant. If the action of G on M is transitive, the isotropy group of different points are isomorphic. Indeed, let y = gx. The isotropy group Hy at y is then conjugate to the isotropy group Hx, at x: Hy -- g H x g -1,

y - - gx.

The isotropy group is a closed subgroup of the connected Lie group G, so G / H x is a homogeneous space. There is a natural identification of M with G / H x . Let y = gx be the point in M obtained (4) by the action tg of g 6 G on x. Then g determines y, but y does not determine g since y = (gh)x

for any

h 6 Hx.

There is a diffeomorphism between M and G~ Hx, and one writes M = G/H,

where H is isomorphic to Hx.

A n s w e r 7" Let (l, 7r) be the coordinates of x ~ T * S l - - R x S 1. Let (m, n, qS) be the coordinates of g 6 E2, and set tg(1, ~ ) "-- (l + m sin(4~ + 7r) - n cos(q5 + 7r), (4~ + 7r) mod 2rr).

One checks that tg is a left action of E2 which acts transitively on T*S 1. The isotropy group Hx C E2 which leaves (l, 7r) 6 T*S l invariant is such that

th(l, ~r)--(l, ~t)

for

h 6 Hx,

hence it consists of the elements of E2 such that 4) -- 0, n / m -- tan and is isomorphic to R.

107


Hence T* S 1 = N:2/R.

A n s w e r 8: Let M be a symmetric riemannian manifold. It follows from answer 6 that M is a homogeneous space. Moreover, let cr( g) = sp o g o sp ,

g ~_ G

we shall check that cr is an involutive automorphism of G such that or

GoCHpCG

or

,

where g E G ~ C G implies or(g) = g and Go is the identity component of Gor.

First, the relations cr(gk) = cr(g)cr(k) and or(or(g)) = g follow from the fact that Sp o Sp = 1. Next we show that if h E Hp then h E Gor; that is, we show that the isometries or(h) and h of M into M are identical, or(h) = h. Indeed or(h) and h coincide at p cr(h)(p) = p

and

h ( p ) = p,

i.e., (cr(h)oh-1)(p) = 1

and their differentials coincide on T p M (cr(h))'(p) = - 1 o h ' ( p ) o ( - 1 ) = h ' ( p ) ;

i.e., (or(h))' o h'-llr~M - 1.

It follows that tr(h)o h - l = I in a normal neighborhood of p (neighborhood admitting a system of normal coordinates at p). Since M is connected, it can be covered by overlapping neighborhoods and o'(h)~ h -1 = 1 . Hence if h E Hp, then h E G", i.e., Hp C G '~. Finally, G o and Hp have the same Lie algebra. Since, by definition, G o is connected, it is included in Hp, and O"

or

or

GoCHpCG

or

.

Remark: In cosmology one sometimes says that a space is homogeneous if it admits isometries which are only locally transitive, that is, transitive between points in the same neighborhood. REFERENCES Elie Cartan, Lecons sur la geometrie des spaces de Riemann (Gauthier-Villars, Paris, 1946). English translation: Geometry of Riemannian Spaces (Math. Sci. Press, Brooklyn, 1983). C. Chevalley, Theory of Lie Groups (Princeton University Press). S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces (Academic Press, New York, 1978). S. Helgason, "Lie Groups and Symmetric Spaces", Battelle Rencontres 1967, eds. C6cile M. DeWitt and John A. Wheeler (Benjamin, New York, 1968) pp. 1-71.

normal neighborhood

108


s. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. 2 (Interscience, New York, 1969). A. Lichnerowicz, "Commutativit6 de l'alg6bre des op~rateurs diff6rentiels invariants sur un espace sym6trique", in Battelle Rencontres 1967 (loc. cit.) pp. 73-83. S.G. Low, "Path Integration on Spacetimes with Symmetry" Ph.D. dissertation, University of Texas at Austin (1985). A.O. Barut and R. Raczka, Theory of Group Representations and Applications (PWN Polish Scientific Pub., Warsaw, 1980); 1986 edition (World Scientific, Singapore). A. Lichnerowicz, Geometrie des Groupes de Transformations (Dunod, Paris, 1958). Steven Weinberg, Gravitation and Cosmology (Wiley, New York, 1972). M. Reed and B. Simon, Functional Analysis (Academic Press, New York, 1972).

8. E X A M P L E S OF H O M O G E N E O U S SPACES, STIEFEL AND GRASSMANN MANIFOLDS

1. Show that O(n + 1 ) / O ( n ) can be identified with the sphere S ~. Answer 1: O(n + 1) acts transitively on the unit sphere S". The isotropy group of some given point x0 is isomorphic to O(n), so S" can be identified with O(n + 1 ) / O ( n ) . (See Problems III 7, Homogeneous spaces answer 6, and V 10, Invariant geometries.)

2. Stiefel manifolds, a) A k-frame in the vector space R" is a set of k linearly independent vectors {Vk}. Show that the set of k-frames in R" can be identified with the Stiefel manifold V'n,k V'~,k = GL(n, R) / GL(n, k, R), where GL(n, k, ~) is the subgroup of GL(n, R) leaving fixed k vectors. b) Show that the set of k-frames orthonormal in a given euclidean scalar product on R" can be identified with the Stiefel manifold V,,k:

V,, k = O(n) /O(n - k). c) Show that V,, k can be identified with the set of orthonormal k - 1 frames tangent to S ~-1. d) Show that V,, k is a fibre bundle with base V,,k_ 1, typical fibre S "-~ and structure group O(n + 1 - k).

Answer 2: a) GL(n, R) acts transitively on the set of k-frames, and the subgroup (isotropy group) leaving fixed each vector of a given k-frame is

8. EXAMPLES OF HOMOGENEOUS SPACES

109

isomorphic to GL (n, k, R). Thus

V~, k -- GL(n, R ) / GL(n, k, IR). b) The same type of reasoning applies to orthonormal k-frames, but the subgroup of O(n) which leaves fixed k orthonormal vectors is identical with the group of rotations of the space spanned by the (n - k) orthogonal complement, thus isomorphic to O(n - k). c) Translate each k-frame along its first vector to the end point of this vector on S n-l" this gives a k - 1 frame of vectors tangent to S n-1. This is a reversible process. d) To define the bundle structure we consider a natural inclusion i: H ~--+ H, with H ' = O(n - k), H ---- O(n + 1 - k). We define the projection p: Vn,k --+ Vn,k-1 by the mapping of left cosets

p(gH') = gi(H'),

g ~ O(n).

The fibre at a point g H e Vn,k-1 is the set in Vn,k with projection g H , that is the cosets of the form g~H ~ with gZ -- gH. It is therefore isomorphic to H / H ' ~ S n-k. One defines a right action of O(n + 1 - k) on Vn,k, which preserves fibres, as the natural action of H on the cosets which constitute the fibre of Vn,k. 3) Grassmann manifolds. Let Grass(n, k) denote the set of k-dimensional linear subspaces (k-planes through the origin) of a space R n. a) Show that one can identify Grass(n, k) with the homogeneous space Grass(n, k) - O ( n ) / ( O ( k ) • O(n - k)).

The projective n-space is Grass(n + 1, 1). b) Define the manifold Grass + (n, k) of oriented k-planes. c) Show that Vn,k is a bundle over Grass(n, k) with group and fibre O(k).

Answer 2: a) We endow ~n with a euclidean scalar product so that we have a group O(n) operating on R n. This group operates transitively on k-planes. Let Pk be a given k-plane, and Pk-n its orthogonal complement. The subgroup of O(n) leaving Pk fixed is the direct product of the subgroup of O(n), isomorphic to O(n - k), leaving each of its elements fixed, and another subgroup, isomorphic to O(k), leaving fixed the elements of Pn-k. Grass(n + 1, 1)is the space of lines through the origin in ] ~ n + l , o r equivalently the space of pairs of antipodal points on S n .

110

III. DIFFERENTIABLE

MANIFOLDS

b) The group SO(n) acts transitively on oriented k-planes. Clearly Grass+(n, k) = S O ( n ) / ( S O ( k ) x SO(n - k)). c) Identify V,,,k with O ( n ) / O ( n - k).

9. ABELIAN REPRESENTATIONS OF NONABELIAN GROUPS

commutator subgroup commutant

Let G be a nonabelian group and [G, G] its commutator subgroup, [eommutant] i.e., the group generated by elements of the form (g~gE)(gEg~) -~, gx, g2 ~ G. 1. Show that G/[G, G] is an abelian group. 2. Examples: Let G = GL(n, 1~). Compute [G, G] and G/[G, G]. Do the same for G = GL(n, C). 3. Construct the one-dimensional unitary representation of GL(n, ~) and GL(n, C). Answer 1: a) H----[G, G] is a subgroup of G by definition. It is appropriately called the commutator subgroup of G because a generating element reduces to unity if g~ and g2 commute. Its Lie algebra ~ ( H ) is spanned by the elements [X i, Xj] where Xi, Xj E ~ ( G ) . b) Next we show that H is a normal subgroup of G. Indeed set h(gl, g2) = glg2(g2gl) -~ then V g E G .

.

.

.

1

-

1

gh(g~, gE)g ~ = g g l g lggEg lggl g Igg~ g = (ggag-~)(gg2g-~)(gglg-~)-~(gg2g-~) -1 E h. Since H is a normal subgroup of G, the quotient space G / H is a group. c) Finally we show that the group G / H is abelian. An element [k] E G / H is the set kH. The group multiplication in G / H is defined by

( k l H ) ( k 2 H ) = (k~k2)H. The two sets klk2H and k2k~H are identical since for every h 1 ~ H there exists h 2 ~ H such that

k2klh 2 = klk2hl, namely

h 2 = (k2k~)-~(k~k2)h~.

10. IRREDUCIBILITY AND REDUCIBILITY

111

Answer 2: Examples: For G = GL(n, ~), [G, G] = SL(n, ~), G/[G, G] = R • • {1, - 1 } where ~• is the space of strictly positive numbers treated as a multiplicative group, called ~+ in [Problem III 6, Direct]. For G = GL(n, C), [G, G] = SL(n, C), G/[G, G ] = C x • S 1 where C x is the space of complex numbers without the origin, treated as a multiplicative group.

Answer 3: A one-dimensional unitary representation (i.e., a character) of G is a homomorphism X: G--*CX with characters are { U,}, t ~ ~, where U,: ~• --~ C •

by

Ix(g)l--1.

characters

For G = R x the

r~-~ U t ( r ) = r it.

The characters of Z 2 = {1, - 1} are { Un} where

Un: Z2--~C •

by

Un(z )= z n,

n E Z.

The characters of S 1 represented by 0 ~ R mod 27r are { U,} where U n" $1--~ C •

by

Un(exp(i0)) = exp(in0),

n~ Z

Characters of GL(n, ff~) and GL(n, C) are {U(t,n)} where

U(t,n )" g~--~ [det glit( [det d e t gg[ )~ '

t E [~, n E

77.

It can be shown that { U(,,n)} is the complete set of characters of GL(n, R) and GL(n, C).

Remark" A group which is equal to its commutator subgroup is called perfect. The abelianization of a group G is G/[G, G].

perfect abelianization

10. I R R E D U C I B I L I T Y A N D R E D U C I B I L I T Y

Definition" A representation T" x--~ T x of a topological group G in a Hilbert space H is said to be irreducible if it has no proper invariant closed subspace [i.e., there is no closed (linear) subspace H a C H, H~ ~ {0} and H~ ~ H, such that T~u ~ H~ for every x E G if u ~ H~]. 1) Show that the restriction of T to an invariant subspace H 1 is a representation of G. It is called the subrepresentation of G on H 1 and is denoted H1T.

irreducible

subrepresentation

112


2) Show that if T is a unitary representation the orthogonal complement H 2 o f a subspace H 1 is invariant if and only if H I is invariant. 3) Denote by P1 the projection operator H---~ H 1. Show that when T is unitary, n I is invariant if and only if P1Tx = TxP1 f o r every x ~ G. Give a counterexample with G = ff~ and H = R 2 with the euclidean scalar product to show that if T is not unitary the orthogonal complement o f an invariant subspace is not in general invariant.

indecomposable

4) A representation T of G in H is said to be the direct sum T = E i ( ~ Zi, of representations T i of G in H i if H i are invariant subspaces of H such that H = E i @ Hi (topological sum) and if each T i is a subrepresentation of T. One says that T is fully reducible if it is the direct sum of irreducible representations. A reducible but not fully reducible finite dimensional representation is called an indeeomposable representation. Show that every finite dimensional unitary representation is fully reducible. 5) Let T be a finite dimensional irreducible representation of G. Show that the only operators which commute with all T x are scalar multiples o f the identity. A n s w e r 1: Vx E G, T x restricts to a continuous linear operator on H1, HiT x ~ ~q~ nx) , which satisfies the properties required of a representation.

unitary

Answer 2: T x unitary is equivalent to (Tx)* = ( L ) - ' ; i.e., since T is a representation of G (Tx)* = L - , ;

therefore denoting ( , ) the scalar product in H

( L v , u) = (v, ( L ) * u) = (v, L - , u ) ,

Vu, v E H ,

x E G.

If H a is invariant, u E H 1 implies Tx-lU E H I. Thus if v E H 2 we have (o, Tx-lU ) = 0, so (Txo, u) = 0 Vu E HI; thus Txv E H 2. The converse statement follows immediately since H 1 is the orthogonal complement of

A n s w e r 3: Let H 1 be invariant. Then TxPI u E H a for every x ~ G, u E H, that is p*

* .._

1ZxP1

Txel,

Vx E G.

By taking the adjoint we obtain P I T x 9P1 = PT T*" x,

10. I R R ED U C I B I LI T Y A N D R E D U C I B I L I T Y

thus, since

P1 "-P~' and

113

Tx* = Tx-,

P1 r x - l P1

V x - 1 ~ G.

= P1Tx-~,

Comparing (1) and (2) gives the result. Conversely if rxP 1 --P~ T x V x E G then, since u = P~u for u E H~, we have T x u = P1 Tx u E H 1. C o u n t e r e x a m p l e " Represent R by 2 • 2 nonunitary matrices acting on R 2 T" x~--~ T x =

The subspace (0'), ua E R, of ment (~ u 2 E R, is not.

(1 x) 0

~2 is

1 '

xER.

invariant, but its orthogonal comple-

A n s w e r 4: If H a is a proper invariant subspace of H then the same is true

of the orthogonal complement H 2, while H = H a + H 2. If H a [resp./-/2] contains a proper invariant subspace, then we decompose it again into two invariant subspaces. We obtain a decomposition of H in irreducible invariant subspaces by a finite n u m b e r of steps if H is finite d i m e n s i o n a l otherwise we have problems of convergence. 5: D e n o t e by K the kernel of the mapping S: H---~ H. If S commutes with T x, we have

Answer

S T x K = T x S K = O.

Thus T x K C K is an invariant subspace of H. Since T is irreducible, this implies that K = 0 or K = H, that is, S is an isomorphism of H or vanishes identically. If S commutes with all T x, the same is true of S - hl, where Z is an eigenvalue of S. But S - h i is not an isomorphism of H, and thus S-hI=O. The result just proved extends in part to the infinite dimensional case in a form known as Schur's lemma" Sehur's lemma: Let T and T' be two irreducible representations of G in H and H ' respectively, and S; H----~ H ' a bounded linear map such that S T x = TxS,

V x E G.

Then either S is an isomorphism of H onto H ' , or S = 0. The proof (cf. for instance, R e e d and Simon) makes use of the spectral decomposition of the hermitian operator S ' S , after observing that, since T is unitary, TxS* = T x S * ,

References: cf. Problem III 7.

V x E G.

Schur's lemma

114

III. DIFFERENTIABLE MANIFOLDS 11. C H A R A C T E R S

1. Show that every irreducible unitary representation of an abelian group G on a Hilbert space H is one dimensional. character

2. A one dimensional unitary representation 2 of G is called a character of the group. Show that the set of characters of G forms a group under the usual multiplicative law of function =

Vx

c.

Answer 1" Given a fixed y E G, Ty commutes with every Tx, x ~ G, since TxTy = Txy = Ty x = TyT~,

Vx, y E G.

Thus (cf. Problem III 10, Irreducibility) Ty = A(y)I, every one dimensional subspace H 1 of H is invariant. But since T is irreducible, H 1 coincides with H.

Answer 2: If x ~ 2~ (x) and x--> 22(x ) are one dimensional representations of G, i.e., X~Xa(X) [resp. 22(x)] is a continuous mapping G ~ .~([~, R ) = ~ which satisfies 2~ (xy) = 2 a(x)2a (y), then

.~a.l~2" X~--~f(,I(X)X2(X )

Vx, y E G

satisfies the same relation.

References: cf. Problem III 7.

12. S O L V A B L E LIE G R O U P S

commutator commutant

1) The element q = xyx-ay -~ is called the commutator of the elements x, y of the group G. The set Q of elements of G which are finite products of commutators is called the commutant of G. Show that Q is an invariant subgroup of G. See [Problem III 9, Abelian] where Q = H = [G, G]. 2) Consider the chain of commutants

G = Qo D Q1 D . . . D O~ D Qi+l ~ ' " , solvable

where each Q~ is the commutant of Qi-1. The group G is said to be solvable if the chain terminates, i.e., if for some n we have that Q, is abelian, and hence Q , + ~ = 1].

115

13. L I E A L G E B R A S O F L I N E A R G R O U P S

Show that the Poincard group in one space dimension is solvable.

It can be shown that every solvable connected Lie group can be represented as a product of one dimensional subgroups G=Tlx...•

where each G k = Tk+ 1 X ' ' "

is an invariant subgroup of G.

X r n

Answer 1" If q E Q, i.e., q = q l . . . VgEG g - 1 qg = g - 1 X l Y l X ? l

Y-( 1 . .. x , y , x , - 1 y ,- 1 g = q l, . . . r

w i t h q i'

q, with qi = x~yix~ ly7,1

-1

t

we have

q,t

-ly,

the commutator of x i = g xig and yi = g ~g. Note that Q is not closed in general. See [Problem III 3, Subgroups].

Answer 2" The Poincar6 group in one space dimension is the space of

pairs (a, L) with a E R 2, L E O(1, 1) with the multiplication (cf. Problem III 5, Direct) (a, L ) ( a ' , L ' ) = (a + La', L L ' ) .

The inverse of (a, L) is ( - L - ~ a , L - l ) . The Lorentz group in one space dimension is commutative. The commutator of x = (a, L) and y = (a', L ' ) is computed to be q=(a-L'a+

La'-a',I)

Thus 01 is the commutative group R 2. REFERENCE cf. Problem III 7.

13. LIE A L G E B R A S OF L I N E A R G R O U P S

1) Show that the space ~W(E, E ) o f linear endomorphisms o f a finite dimensional vector space E together with the mapping ~W(E, E) x ~ ( E , E)---> ~ ( E , E)

by

(A, B),---->[A,B]= A B - B A

is a Lie algebra. A n s w e r 1" The space ~ ( E , E) is a vector space. The operation [, ] is

obviously anticommutative. It is easy to check that it satisfies the Jacobi identity.

116


2) Let G be the group GL(E) of linear automorphisms of E. Show that its

Lie algebra is isomorphic to the space ~ ( E , E), with the Lie bracket defined above. Answer 2: The left action L x of X E GL(E) on GL(E) is given by the product of automorphisms, i.e.,

Lx: Y~-->L x Y = XY. The space GL(E) is an open set of the vector space ~ ( E , E). The tangent space at Y ~ GL(E) is therefore isomorphic to ~ ( E , E). A vector field V on GL(E) is left invariant (p. 155) if

V(X) = LxV(e ),

e the unit of GL(E),

that is,

V(X) = XV(e). Consider two such vector fields Va and VB

Vm( X ) = X A ,

VB(X ) = XB,

where A and B are given elements of ~ ( E , E). Let f: GL(E)---~ R, by X----~f(X) be a C | function on R. Choose a basis (E~) in ~ ( E , E). Then ,, ~ Of

VAf= X ~

OX,ot

and a straightforward computation gives V

(VAVn - VoVa)f = X~, (A~,B ~'

~

It

of

B~A~' ) OX,/

Therefore

IVA, VB](X ) = X ( A B -

B A ) = X[A, B],

where [A, B] denotes the bracket in the Lie algebra ~ ( E , E). Weyl basis

3) a) Take as a basis, called Weyi basis, in ~e(n, R), the Lie algebra of GL(n, R ) - GL(R ~) the n x n matrices E ij whose elements are k

= 8 i 8' k

i"

Compute the structure constants of c~ta(n, R). b) Determine the Lie algebra of SO(n). Prove that the structure constants are totally antisymmetric.

13. L I E A L G E B R A S

OF LINEAR GROUPS

117

Answer 3) a" By the previous result, ~d((n, R) is the space of n x n matrices, and the Lie bracket of elements of the Weyl basis are i Elk]=cijhmlk [Ej,

E hm

with Cij

hm

h ~ m ~j I ~ h ~ m ~k i k -j 9

lk=~i

Answer 3) b: The group SO(n) is the subgroup of GL(n, R) = G L ( ~ n) of n x n invertible real matrices X such that TXX

--

I,

where I is the n • n unit matrix and TX the transpose of X. The tangent space TzSO(n) is a subspace of LP(R~, ~n), the space of n x n matrices, spanned by tangents at t = 0 to curves t ~ X(t), X(O)= I, in SO(n), that is such that

TX(t)X(t) = I. Thus, if we set A = dX(t)/dt[,=0 TA + A = O . The Lie algebra 6eO(n) is therefore the vector space of antisymmetric n x n real matrices, a Lie subalgebra of ~t'(n, R). A basis for 5eO(n) is constituted by the antisymmetric matrices ~

i

j

E~j= E j - E/. By straightforward calculation using the results of a), [ E i j , E l k ] = Cij~ hmlkEhm , ~ hm

where C~j ~k is the sum of eight terms

Ci j~ hmlk ---- 1 (6hakm~jl- ((i, j)+->(h, m ) ) - ((i, j)+->(l, k)) + (l, k ) ~ (h, m)} - 89 {m h). Therefore the structure constants C~v of 9~ ric 3-tensor (where a Greek index, an is substituted by a pair of latin indices).

are a totallyantisymmetordinary vector index,

4) It can be proved that every compact Lie group G admits at least one faithful linear representation either by orthogonal matrices or by unitary matrices (Chevalley, Theory of Lie groups, pp. 176, 211).

Show that the structure constants of a compact Lie group have a vanishing trace: C,f ~ = O.

118

III. D I F F E R E N T I A B L E MANIFOLDS

A n s w e r 4: If G is a subgroup of an orthogonal group O(n) its Lie algebra ~d is a Lie subalgebra of the Lie algebra ~(n). Therefore the structure constants are totally antisymmetric, and in particular have a vanishing trace. If G is a subgroup of U(n), its Lie algebra is represented by antihermitian matrices. A complex n x n matrix can be written as a 2n x 2n real one. Indeed, U = A + iB acting on C " = [~" + iR" is written as

a//=(A

B

-B

A)"

If U is antihermitian then A = --TA, B = TB, and 0-//is thus antisymmetric. The same conclusion follows on the structure constants. 14. G R A D E D B U N D L E S

In this problem we study vector bundles over an ordinary C ~, ddimensional manifold M where the typical fibre is a graded (commutative) algebra A, as defined in Problem I 1. In supersymmetric theories, before quantization, the fermionic fields are odd sections and the bosonic fields even sections of such bundles. In superstring theories, the fibre is not a vector space. 1. KOSTANT GRADED BUNDLES

The first requirement is the definition of graded C = functions on M as an extension of the algebra of ordinary C = functions. The extension can be defined by considering C = mappings from M into A. But it is possible to give a more general definition. Kostant has used sheaf theory (Problem I 1) to construct graded C = functions on open sets of M, whose local algebra is an extension of the local algebra of ordinary C = functions on these open sets. It is also possible to define a local algebra of graded functions as local sections of a vector bundle over M with typical fibre A if the product can be defined independently of the choice of trivializations. We give the following definition" Kostant graded bundle

A Kostant graded bundle K over M is a topological space with 1) A continuous projection p: K--> M 2) A covering of M by open sets U i together with homeomorphisms ~i" P - I ( u ~ ) --~ U~ • A ,

A a graded algebra with locally convex topology

14. G R A D E D

BUNDLES

119

which commute with projection, that is, q3i(Z ) -- (X, (.~i,x(Z)),

(~i,x(Z) E A ,

x = p(z)

and which have the properties that if x E U i D Uj then the mapping q~i,xO ~Oj,x A---~ A

(i) is a continuous linear mapping (ii) commutes with the product in A, that is, l

-I

,

(~Oi,x(Z)('~i,x(Z ) --" ((43i,x o (~ j,x)(~,x(Z)~gj,x( Z ))"

The bundle is of class C k if the mappings -1 are of class C k. U~ D Uj ~ L ( A , A ) by x ~ (P~,x~ r j,x

a) S h o w that the C p sections ( p < - k ) o f a C k K o s t a n t b u n d l e f o r m an algebra.

b) Give a possible f o r m o f transition m a p p i n g s B ~ B in the case that B is a G r a s s m a n n or D e W i t t algebra. A n s w e r l a: i) A Kostant bundle is a vector bundle" the fibre K x = p - l ( x )

is endowed with a vector space structure by the homeomorphisms q~. Indeed, we can define a y + / 3 z E K~ by qgi ,x ( Oty + ~ Z ) -- a q~~,x ( y ) + fl q~i ,x ( z ) ,

a , / 3 E C (or ~).

y,z~Kx,

-1

9

This definition does not depend on the trivialization" since q~j,x~ q~i,x Is a linear mapping we have -1

(%, o q~,,x )(aq~,,x(Y) + J~(~~

"-- ~176

-1

(~i,x)r

+ ~(r

-1

~ qgi,x )r

= aq~j,x(y) + fl%,x(Z).

The space of sections over an open set of M is therefore also a vector space. ii) The product of two sections f and h is the section x ~-->( f h ) ( x )

defined for x E U i by (fh)(x)

-1

= q~i,x ((~i,x ( f ( X ) ) (~i,x ( h ( x ) ) ) .

The definition is independent of the index i due to hypothesis (ii). A section f is called even [resp. odd] if ~ o i , x ( f ( x ) ) ~ A + [resp. A _] for all x E M. The space of even and odd sections form a graded algebra, but addition must be restricted to sections of the same type.

120


l b" We suppose that B is a Grassmann or DeWitt algebra and look for linear, invertible maps f: B--+ B which commute with the product, i.e., maps such that

Answer

(i)

(ii)

f(aa

Va, b E B,

+ ~b) = af(a) + ~f(b), f(ab) = f(a)f(b)

a , / 3 E C (or JR)

V a , b ~ B.

We first take a = e and write (ii) f(b)(e

- f(e)) = 0

Y f ( b ) E B.

Therefore f ( e ) = e and f ( a e ) = a e , that is, f necessarily preserves the body of B. If (i) and (ii) are satisfied we must then have, for all a E B,

(aoe+ a,,z'..,

z")= a0e +

p-~l

Therefore f is known on B if it is known on the generators. In particular f ( z t z J) = - f ( z ' z 1) = f ( z t ) f ( z

~) = - f ( z J ) f ( z l ) .

Hence f must map generators into odd terms in B. We also see that the restriction to B(1) (terms of degree 1) of the image of f(B(1)) must be surjective for f to be invertible, since by (ii) and the bodylessness of f(B(1)), the image under f of a term in B ( p ) , p > 1, is never in B(1). These remarks show that f must be of the form /

f(z 1)= Llz

J

+ Z

I

f l, . . . lpZ

Jl

. . . z Jp,

(1)

p>l

where the mapping B(1)--+B(1) by ( z t ) ~ + ( L jz 1 ~ ) is continuous and invertible. A possible choice of f satisfying the required conditions is IZJ

f ( z 1) = Lj

,

a formal series in the generators

if the corresponding mapping f, determined on B by conditions (i) and (ii) is well-defined, continuous and invertible. Let a ~ B be the formal series a = aoe + _

-~. at~ . . . ~pz t~ . . . zIp.

We must have 1

i~

f ( a ) = aoe + ~ , -~. a,1. .. l p L j

p>_l

~

. . . L.pzJ1. . . z

je

.

Therefore f ( a ) is defined as a formal series, and also continuous in a, if

14. GRADED BUNDLES

121

and only if each ordinary series aI~ . . . IpLI'~ . . . L ~;

is convergent for any numbers a t t . In particular, for each J the t 1"-p . series, a z L j must be convergent for any choice of numbers a t, this will be I true if and only if, for each J9 L j ~ 0 for only a finite number of indices I. The corresponding transition function U i N Uj---~ L(B9 B ) , X ~ f x is determined by

L ( z I) = L I (x) ZJ. In the case of a finite number of generators the Kostant bundles thus obtained are isomorphic to ordinary exterior bundles. M. Batchelor's theorem, proved in the context of Kostant's original definition, says that the category of Kostant bundles (when locally trivial, which is included in our definition) is isomorphic to the category of exterior bundles if A is a finitely generated Grassmann algebra. This theorem can perhaps be proved in our context by using the fact that if transition functions U i A Uj ~ L ( B , B ) by x ~ f~ are given by the general formula (1)9 the obstruction to their extension to M lies entirely in the first term. c) A graded manifold is a pair (M9 K), with M a C~ ordinary manifold, and K a Kostant bundle over M9 also called the fundamental bundle. A graded chart is a triple (Ui9 ~bi9 ~oi) where (U i, cki) is a chart of M and ~i a local trivialization of K over U i. A graded function f of M is a section of the fundamental bundle. Give the definition o f a C p graded function f. Give the necessary and sufficient conditions f o r f to be C p when A is a Grassmann or D e W i t t algebra. A n s w e r lc: The representative of f in a graded chart is the mapping from an open set of R d into A fi ~ (49i~f ~ ~ ~t 1 9

fi" r

'''~ A "

A graded function is C p (with p--< k if K is C k) if each of the f~ is C". If A is a Grassmann or DeWitt algebra B9 a representative f~ f is -

1

f = foe + ~' -~. fll''

"It, z,'

" " " zip

where each of the fo, fll lp are ordinary numerical functions on an open set O of Rd; the function f: J2---~B is C p if and only if each of the functions f o , . . . , fl~... Ip is a C p function.

graded manifold fundamental bundle graded chart graded function

122

2.

III. D I F F E R E N T I A B L E M A N I F O L D S GRADED VECTOR OR AFFINE BUNDLES

Let P---~ M be an ordinary, C k, principal fibre bundle over M with ,f,,, projection ~r, Lie group G and transition functions ~ij " ::~

u, f) uj--, c (recall that oSj(x) E G defines an automorphism of G by its left action on

G). The vector bundles associated to P can be extended as follows to graded bundles. a) Let r be a linear representation o f G on C" or R". Define an action o f r( g) on A", when A is an arbitrary locally convex vector space over C (or

R).

b) Let E - - . M be a vector bundle with typical fibre C" (or R") associated to P through the representation r. Construct a graded bundle ~g~ M with typical fibre A", associated to P through the same representation.

c) Let K---, M be a Kostant bundle with typical fibre a Grassmann or DeWitt algebra B and transition functions which preserve the unit of B. th Define K"---~ M as the vector bundle with base M and fibre at x the n direct product o f fibres, K"~, and construct a K-graded extension IgK of the vector bundle E ~ M. Define the K-graded tangent bundle to M.

A-valued sections

d) When K is the trivial bundle K = M x A , the sections of the corresponding graded extension of a vector bundle E will be called A-valued sections o f fgr. Show that A-valued tangent vectors define a linear endomorphism o f the algebra o f C =functions. Study its properties. Consider the case where B is a DeWitt algebra. e) Define A-valued covariant vectors, metrics, spinors, connections. Answer 2a" r ( g ) E G L ( n , C) is an n x n matrix with complex elements rj.i It acts o n a = ( a i ) , i = l , . . . , n E A " b y t h e o r d i n a r y l a w 9

.

r( g)a = (r'ja'). Answer 2b: Since G acts on the left on A" through the representation r, it is possible to follow the general procedure given on p. 383 to define the graded bundle ~' ~ M as the quotient of P x A" under the action zg of G" rg" P x A" ~ P x A"

by

( p , a) ~ (Rgp, r-l(g)(a)).

Answer 2c: Since K x is a vector space over C (or R), the product of

14. GRADED BUNDLES

123

y E K x by a E C (or R) is well defined, and the action of r on y E Kx~ also" if y = ( y l , . . . , y " ) C K ~ r ( g ) y = (r j i iy)~-gx.

n

We give below the construction of the graded extension g'~, following essentially the same ideas as on p. 383, but with the fibre F replaced by K x, which itself depends on x. The K-graded extension g'K of E is ~K = U ~ M ~'x with ~x the set of equivalence classes { ( p, r) } , p E G , r E K x (p,r)=(p',~")

if

p'=gp,

r ' = r ( g ) r for some g ~ G.

~ r projects onto M by ,IT" ~x-----~ X. Each ~gx is a vector space. To endow ~K with a topology and a vector bundle structure, we choose a local section of P with representative Pi over each open set U i domain of a chart of a graded atlas of M over which P is trivializable. The following mapping ~i is bijective: (~i

9

"IT -

1 (Ui)

Ui x A n

..... ~

defined for y E ~x by y ~--~(x, ti),

where ( Pi ,

((~ i,x - 1 )nti) is the representative with first e l e m e n t Pi of the equivalence class defining y. (Note that ( - 1i,x ) , acts on each element in A by ~oi,-1x when t i E A " . ) The topology of g'r is defined to be such that each of these maps is a homeomorphism: the consistency of the definition rests on the fact that U~ D Uj is open in U i. The vector bundle structure of ~K is defined by the family of local trivializations ~i, with the resulting transition functions 9

--1

n

Si j (q~ j o ~i) r(orfi). n

(Note that since r is a linear map, and (~Pi ~ - q ~ i I, I the unit matrix, r commutes with r(o']i). ) A change of choice of the local sections Pi of P gives an equivalent vector bundle structure to g'K. It is possible to define even-valued or odd-valued extensions by replacing the vector space A " by A~ or A n . If the bundles K and E are C k the same is true of the graded extension g'K and we can speak of its C p sections ( p - < k).

Remark"

K-graded extension


124

graded tangent cotangent

In agreement with the preceding definitions we have" A graded tangent [respectively cotangent] vector at x ~ M to the graded manifold (M, K) is an equivalence class (p, V) where t9 is a linear frame in the ordinary tangent space at x and V E Kx~ and the equivalence relation is [respectively

even odd A-valued tangent vectors components

V'=g-IV V' = gV

if if

p'=pg, p' = gp,

gEGL(d,R) g E GL(d, R)]

The vector is called even if V E (K~+ )n, odd if V ~ (K x )n.

Answer 2d: W h e n K is the trivial bundle M x A, the A-valued tangent vectors have a representative V E A d in each linear frame on M. The o s, a = 1 , . . . , d, (u s) = V E A d are called the components of o in the frame 19. A C ~ A-valued tangent vector o to M (i.e., a C ~ section of the graded extension with K = M x A of the vector bundle TM, assumed to be C ~ defines an endomorphism of the algebra C~(M, A) of A-valued C ~ functions on M: in a chart (U, 4)) of M,

v ( f ) = v"e9s ( f o 6),

#s = #/#x~;

the result is chart independent due to the definitions. The endomorphisms #s are a basis of the A-valued tangent space TxM. This endomorphism is additive and obeys the graded Leibniz rule, when v and f have a parity (i.e., are either odd or even)

v ( f h ) = v ( f ) h + (-1)d(I)a(~ and is zero on constant functions"

v(a) = 0

B-supertangent vector

if a is the constant map x ~ a ~ A.

Conversely, if A is a DeWitt algebra B, every endomorphism of C"(M, B) which enjoys these properties is a B-valued C ~ vector field on M (see for instance the proof in [3]). A B-supertangent vector is an endomorphism of C~(M, B) which satisfies the above properties except o(a)= 0; this is replaced by v(ke) = 0

for every constant map x ~ ke, k E R.

The endomorphisms a/az I defined by 0

1

Oz J f = iz" f = ~ ( p - 1)! f J ' ~ are B-supertangent vectors.

tp_~

dztl

..-

dz/p_l

14. GRADED BUNDLES

Answer 2e: The differential, also called gradient, of an A-valued

125 C 1

function f on M is a covariant A-valued vector, with components in a chart a s (fo ~b-1). When A is a DeWitt algebra B it is easy to see that a B-valued covariant vector is an exact differential if and only if each of its projections on a subspace of B generated by a given subset of its generators is an exact differential. Analogous definitions apply to tensor field and scalar densities of various weights. In particular, an A-valued metric on M is a covariant symmetric A-valued 2-tensor field g which is nondegenerate, that is, such that at each x ~ M, the mapping TxM---~ T*xM by o ~ u = gx(O, .) is an endomorphism. This will be the case if and only if in a local chart at x (and hence in all charts) the A-valued components g ~ determine an isomorphism of vector spaces by the mapping

(v~)~(u~ = g~v~),

Ad---~A d.

The inverse linear map is a matrix with A-valued elements g ~ , which is a representative in the chart of a contravariant tensor g~. A B-valued metric g, with B a DeWitt algebra, is nondegenerate if and only if its body is nondegenerate. Indeed, the mapping is degenerate if the body of g is degenerate, since the body of g maps the body of v into the body of u" =

The body g(0) = go e, with go a numerical metric, is nondegenerate if go is nondegenerate, and the same is then true of g" the inverse if# of a representative ~ is given by (g~t3) = ~ # = ~ o l ( e + Z ( - - a ) n ( g s g o 1 ) n ) 9

Each g ~ has a meaning as a formal series in the generators z I.

Remark: In a DeWitt algebra a matrix X is simultaneously left and right invertible and the inverses are equal" XX

-1 "- X - 1 X -

le.

Graded spinor fields on an ordinary pseudo-riemannian manifold admitting a spin structure are sections of the graded extension of the usual spin bundle. Graded connections can be defined as graded Lie algebra valued 1-forms

Graded spinor fields Graded connections

126


on the principal bundle P with the appropriate equivariant property, or as a collection of local sections of a graded affine bundle. In particular, an A-valued connection on M associated with the principal bundle P ~ M with group G is a family of 1-forms toi taking their values in ~ @ A, with ~ the Lie algebra of G; s i is defined on U i, and satisfies in

u/nuj toj(x) = md(tr~l(x))toi(x) + (trTiOMc)(x)e, where Ad(trj~l(x)) is the linear map ~d @ A ~ ~d Q A canonically de* 0MC is the pullback on M of the duced from the linear map on qd, and crji M a u r e r - C a r t a n 1-form on G; the unit e of A is assumed to exist.

Remark: The affine bundle of connections on U C M is not associated in the usual way to P; Or j*i involves the derivative of the mapping crji" Ui N Uj---~ G. Such an association can be described in terms of the first jet of P.

REFERENCES [1] B.S. DeWitt, Supermanifolds (Cambridge University Press, 1984). [2] B. Kostant, "Graded Manifolds", in Differential Geometrical Methods in Mathematical Physics, ed. K. Bleuler and A. Reetz, Lecture Notes in Mathematics 570 (Springer, 1977) pp. 177-306. [3] Y. Choquet-Bruhat, "Mathematics for Classical Supergravities", in Differential Geometric Methods in Mathematical Physics, Ed. Garcia and Perez Randon, Lecture Notes in Mathematics (Springer, 1987). [4] M. Batchelor, "The structure of supermanifolds", Trans. Amer. Math. Soc. 253 (1979) 329-338; "Two approaches to supermanifolds", Trans. Amer. Math. Soc. 258 (1980) 257-270.


1. C O H O M O L O G Y . D E F I N I T I O N S A N D E X E R C I S E S

For applications and generalizations, see for instance (IV 2, Obstruction; I V 4 , Cohomology of groups, IV6, Short exact sequences, V bis 6, Euler-Poincar6 characteristic). The building blocks of p-chains can be either simplices or rectangles in R p. In the book Analysis, Manifolds and Physics we used rectangles (p. 217); in this problem we use simplices and we recall first the basic definitions. See for instance [Patterson], [Singer and Thorpe] or [Nash and Sen] for introductions to simplicial complices. 1. HOMOLOGY

A geometric open m-simplex tr m = ( P 0 , . . . ,

Pm) ( m E t e ) is a set of

points x in some space R n, n--> m, defined in terms of m + 1 linearly independent points P o , . . . , Pm by tn

geometric simplex or" - (P0 . . . . . P,.)

m

x = 2 o tp~,

x, =

i.e.

a =0

2 o it p ~ ,

i=l,...,n,

t~ = 0

where t~ E R ,

~t~=l

and

0SO(n, m)

by

x ~ Lis(x )

with p(i)(X) = Lis(x)P(s)(x )

in

U i ('] Uj .

Let us denote by z(i ) = (x, A(i)(x)) the image of a point z E p - l ( x ) C 6P under the trivialization of 6e over U~. The transition function Yis is "Yi]" X~'> Yij(x) --

-1

A(i)(x)A(s)(x)"

2. CONSTRUCTION OF SPIN AND PIN BUNDLES

137

The hypothesis (2) implies that (1) is satisfied, and conversely. The projection H of z ~ p - ~ ( x ) C ~ onto p E SO is defined in local trivializations by

(II(z))(i) = ~(A(~)(x));

II

it is independent of the trivialization. Examples" See for instance [Geroch, Isham, DeWitt].

If (M, g) is a non-orientable riemannian manifold, it does not admit an SO(n, m) bundle linked to g. Indeed (p. 386), the existence of a metric g implies that the principal bundle of linear frames on M is reducible to an O(n, m) bundle linked to g. If M is not orientable, the O(n, m) bundle linked to g is not reducible to an SO(n, m) bundle. In the following, we use the short exact sequence 0---~ Z 2 ~ Pin(n, m)---> O(n, m)---> 1] ; hence, we use the surjective homomorphism from P i n ( n , m ) onto O(n, m), labelled ~ in [Problem 1 7, Clifford], now labelled Y ( - reserving ~ for the bundle homomorphism. We shall now consider an O(n, m) bundle sc of orthonormal frames over (M, g) and investigate under which conditions it admits an Ygextension sc. The definitions of pin frame, pin bundle, pin structure are similar to the definitions of spin frame, spin bundle, spin structure, O(n, m) replacing SO(n, m). Pinors, like spinors, are defined via a linear representation of the group, now the Pin group. Another notation (p. 416) is Spin[resp. Spin0] for what is called here Pin[resp. Spin]. Whereas Spin(n, m) and Spin(m, n) are isomorphic, we have established in [Problem 1 7, Clifford] that Pin(n, m) is not always isomorphic to Pin(m, n). We set Pin+(n) = Pin(0, n)

and

Pin-(n) = Pin(n, 0).

We use the convention TATB + TBTA = --2gAB'l] 9 If a property or a discussion is valid for both Pin+(n) and Pin-(n), we write Pin(n). Let sc be an O(n) bundle over (M, g); let {gi/} be the set of its transition functions relative to a covering ~ = { Ui} of M. The set { Yi/} of Pin(n) valued functions on U i f] U~ is a set of transition functions of a Pin(n)-bundle over M if, for any triple (i, j, k) such that Uij k := U i f3 ujn #O, ij(x)'rjk (x) =

(x) ,

x

uij

.

pin frame pin bundle pin structure pinors


138

The concepts and theorems of cohomology developed in the previous problems can be adapted to the problem of determining whether or not one can construct a set of transition functions defined on a cover ~ of a manifold M. 2. DEFINITIONS abstract simplicial complex K abstract p-simplex /th face

An abstract simplicial complex K is a collection of finite subsets of an ordered, countable set # subject to the following condition" if cr E K, then every subset of or is also in K. An element orp - { i 0 , . . . , ip}, i 0 , . . . , ip E ,9, of K is called an (abstract) p-simplex. A simplex is oriented, or ordered, by the ordering of its defining set. The j t h face of or p is given by Ojo'=-- { io, . . . , ~,, . . . , i t , ) .

vertex

A zero-simplex is called a vertex. Exercise:

boundary

Show

t h a t i f j Pin(n) such that ~ o A~= A~, and

f~/"U~/~ Pin(n)

by

f~/(x)= 3'~(x)[~.i(x)yi/(x)~./(x)-] 2

1 -1

.

Then ~(fij(x)) = e ,

the unit element of O(n).

Hence for the reasons given in paragraph 3a), f/j(x) defines a Z2-2-cochain f on N. The coboundary df of f satisfies

dr(i, j, k)= f(j, k)[f(i, k)l-~f(i, j) . The desired equality, namely p l ( p 2 ) - I = d f follows from

p~j,(x)[p~k(x)]-' = fjk(X)[f~k(X)]-lf,/(X) which itself follows from the definitions and the fact that ~(f~j(x)) and ff((Pijk(X)) a r e in the center of O(n). I d) Show that an O(n) bundle over M admits a Pin-bundle extension if and only if [p] is trivial.

Answer 3d" Assume that the Pin-bundle extension exists. Then there are liftings y~/of gi/such that p(i, j, k) = 1, the unit element in 7/2; hence, the cohomology class of p is trivial. Conversely if [p] is trivial, then either

p(i, j, k) = 1 for all 2-simplices, or there is a 1-cochain f with values in 7/2 such that p(i, j, k)(df(i, j, k)) -1 -- 1 for all simplices, i.e., f(j, k)[f(i, k)l-lf(i, j) = p(i, j, k)

for all simplices.


141

Define hi.j" Uij --+ Pin(n)

by

hi.j(x) -- yi.j(x) f -1(i, j ) .

It is easy to check that h i k ( X ) -- h i . j ( x ) h . j k ( X )

and also that ~)~ (h i.j ) - Jd~ (-l-Yij ) - gij ; that is, {hi.j } is a set of transition functions of a Pin-bundle extension of the O (n) bundle. II R e m a r k : One can generalize (cf. Greub and Petry) the results derived in paragraphs a) through d) when O(n) is replaced by a topological group G and Pin(n) by a topological group F' such that there is a continuous homomorphism Jd~: F' --+ G with discrete kernel K contained in the center of F'. A continuous homomorphism satisfying these properties is said to be central.

central homomorphism

Remark: We shall assume in the following that M admits a simple cover, and we write H2(M, Z2) instead of H2(jV ", Z2). 4.

STIEFEL-WHITNEY CLASSES

We shall indicate why [p] = to2 for Pin + (n) bundle and [p] = w 2 q- Wl L.) wl for Pin-(n) bundle where wl and w2 are the first and second StiefelWhitney classes of the O(n) bundle. The Stiefel-Whitney classes of an O(n) bundle [Hirzebruch] are, by definition, the Stiefel-Whitney classes of the associated vector bundle under the natural action of O(n) on I~n . The Stiefel-Whitney classes of such vector bundles can be defined by the following set of axioms [Milnor and Stasheff, p. 37]. A x i o m 1: To each vector bundle ~ with base B (~) there corresponds a sequence of cohomology classes

Wi(~) E H i (B(~); Z2),

i -- 0, 1, 2 . . . . ,

called the Stiefel-Whitney classes of ~. The class w0(~) is the cohomology class of the constant cocycle 1 (which is also the identity element for cup products - See IV 1, p. 133)

1 6 H~

~]2),

and wi (~) equals zero for i greater than n if ~ is an n-plane bundle. Here Z2 is a ring, hence Z2 = {0, 1 }.

Stiefel-

Whitney

IV. I N T E G R A T I O N ON M A N I F O L D S

142

A x i o m 2: Naturality. If f: B( sc)----~B(rt) is covered by a bundle map from sc to r/, then wi( ~) = f*wi(rl) . A x i o m 3" The Whitney product theorem. If sc and r/ are vector bundles over the same base space, then k

|

Whitney sum

=

u

i=0

where the cup product has been defined in [Problem IV 1, Cohomology] and where the Whitney sum of two bundles ~: and r/over the same base B space is defined as follows. Let d" B---~ B x B denote the diagonal embedding. Then sr ~ ) r / = d*(~: x 77). Note that each fibre of ~ O r / is canonically isomorphic to the direct sum of the corresponding fibres of sc and r/. The Whitney product theorem says: |

=

+ w,(,7),

w2(~: O7/) = w2( sr + w,(~:) U Wl(r/) + w2(r/), etc. A x i o m 4. For the twisted line bundle over the circle, the first StiefelWhitney class is nontrivial.

a)

First Stiefel-Whitney class of O(n) bundles. i) Show that the determinants of the transition functions gu(x) E O(n), x ~ U u C M o f an O(n) bundle over M define a cocycle g on the nerve of M with values in 7/2. ii) Show that the cohomology class [g] = wl(O(n) bundle). Answer 4a-i: Since Uu is connected, and we can define

det

gij(X) does not depend on x ~ Uu

g" (i, j ) ~ det gu(x) E Z 2 ,

x E Uu .

It is easy to check that dg(i, j, k ) = g(j, k)(g(i, k))-lg(i, j ) = 1 ; that is, g is a cocycle with values in 7/2. Answer 4a-ii: We shall check that [g] satisfies the four axioms of a Stiefel-Whitney class. A x i o m 1" The cohomology class [g], represented by the cocycle g, is independent of the choice of transition functions of the O(n) bundle.


143

Indeed, let {g~j} and {g2.} be two sets of transition functions of an O(n) bundle; it is straightforward to check that there is a zero-cochain f such that g1(g2)-1 = d r .

Hence, [g] ~

HI(M,

7/2).

Axiom

2: Naturality of [g]. Let )7 be a bundle morphism from an O(n) bundle ~:a, over Ma with transition functions gl: U~---~O(n) to an O(n) bundle ~:2 over M 2 with transition functions g2: U 2_.~ O(n). Denote by f the induced mapping M 1--~ M E. We shall show that f , [ g 2 ] = [gl]. The bundle map f: ~:1__~sr which covers f: M 1--~ M E is fibre preserving: n

of = f o

.

It is easy to prove [see, e hg., Steenrod p. 12] that given a system of local cross sections 0-/2: U 2 ~ sr there exists one and only one system of local cross sections o-~" U~1 ~ sc 1 such that ~ focr

1

or2 o f ,

--

i.e., (r~(x)

=

fi-,( O r 2, r ( i ) ( f ( x ) )

(3)

)

where r(i) is defined as follows: 2

f(U~ ) C Uj

for some j which is set equal to r ( i ) .

Systems of local cross sections define trivializations by A

r

= e,

the identity in the structure group G,

(4)

which, in turn, define transition functions A A_ 1 -- ~i,x o ~i,x ,

gii(X)

We

shall

X E

Ui. i .

now establish the relationship It follows from (4) that

2 g,(i)~(j)(f(x)).

A1

1

(x)) =

A2

g~j(x)

between

and

2

It follows from (3) that ~,x(

f~ - 1 (~ 2

A2 2 ~o~(i),f(x)(O',(i)(f(x))),

i.e.

(~i,x A1

o

J~- 1

A2 "-- ~)'r(i),f(x)

and

~ fo A1 (~i,x)

--1

A2

--- ((~'r(i),f(x))

- 1

;

hence, (5)

144


and f,[gZ] = [g~], where f*: H2(d~f2,772)--+ H 2 ( N 1 , 7 / 2 ) f: Mx----~ M 2.

is

the

mapping

induced

by

A x i o m 3" The Whitney product theorem. Let

~:1 be a principal

O(m) bundle with transition functions

g~j

~2

~:

O(n) bundle O(n) (~ O(m) bundle

g2 ij gij

~:a

O(n + m) bundle

gij

(6)

A

over the same base, with 0

g

and g ,A= ho&j. The identity inclusion h A: G - O(n) (~ O(m)----~ G A-= O(n + m) maps a block diagonal matrix consisting of an n x n matrix and an m x m matrix into the same matrix now thought of as an (n + m ) x (n + m) matrix. It is straightforward to check that [ga]=[gl]U[g2]

.

A x i o m 4: [g] (M6bius b a n d ) # 0 . Hence, [g] = w 1. This construction of wl shows that w~ is the obstruction class for orientability, w 1 = 0 r M is orientable; w 1 = 1 r M is not orientable.

b) In section 3, we established that [p] E H2(M, Z2). We shall prove that [p] satisfies the second, third and fourth Stiefel-Whitney axioms. This is not, however, sufficient to prove that [p] = w 2 because [p] fails to vanish on all line bundles whereas, according to the first axiom, w 2 = 0 on line bundles. i) Naturality o f [p]. With the notation of section 3 and paragraph 4a-ii, let Yij2 be liftings of the g2ij and set 1

2

2. C O N S T R U C T I O N

145

O F SPIN A N D P I N B U N D L E S

1

It is easy to check that the yij are liftings of the g~j. Indeed ~(')t]j(X))

2

= ~(]/r(i),r(j)( f ( x ) ) )

2

-- gr(i)r(j)( f ( x ) )

1

= gi.i(X) .

Thus the obstruction cocycle pl is given by 1

1

pl(i, j, k)= y jk(X)('r, ik(X))

-lyl

1

ij(X) for

2 2 = "yr(y)r(k)(f(x))('y~.(i)~.(k)(f(x)))

=p2(7"(i), 7"(j), r(k))

for

1

1

x ~ Ui A Uj N Uk -1

2 "Y~.(i),(i)(f(x))

x ~ f ( U ~ ) A f ( U j ) A1 f ( U k ) . 1

Hence [pl I = f , [ p 2 ] , where f* is the cohomology mapping induced by f: M 1

M E.

ii) The Whitney sum product. With ~:L, ~ and scx defined by (6) and ~1 a principal

Pin(n) bundle over scl with transition functions

Yij1

Pin(m) bundle

Yij2

over

~2

Pin(n) @ Pin(m) bundle over s

Yij

Pin(n + m) bundle over ~

~/ijA

where all y's are lifts of the corresponding g's: ~ ( y ) =

0)

Y= )t

~ '~

g;

'

~

A, in contrast with A, is not ncc~,ssarily the identity inclusion, not even necessarily an homomorphism, but A" F = Pin(n) @ Pin(m)---,/'~ = Pin(n + m) is such that the following diagram is commutative"

F

G

2 ~Fa

l

(7a)

a_.~G~

and where

A(AkA ) = A(Ak)A(A ) whenever A k is in the kernel of F.

(7b)

146

IV. I N T E G R A T I O N ON MANIFOLDS

One can use the graded tensor product defined in [Problem 1 3, Tensor] to construct a mapping A, satisfying (7), as follows: Let V1 and V2 be two real vector spaces, Q1 and Q2 two quadratic forms on V1 and V2, respectively; let iQ be the canonical map from V into the Clifford algebra C(V, Q). The mapping

f: VI @ V2~ C( V1, Q1)(~ C ( V2 , Q 2) onto the graded tensor product of the Clifford algebras C(V~, Q~) by

f(x,, x2) = io,(Xl) (~ 1] + 1](~)i02(x2) induces an isomorphism of the 7/2-graded algrebras

C(V1 Q V2, 01 G Q 2 ) ~ C(V1, Q1) (~ C(V2, Q2). deg A

Let deg A = 0, 1 according to the parity of A [Problem 1 3, Tensor]; and let the dot in deg A x .deg A 2 denote numerical multiplication. The mapping

A" (A x, A2)~'>(--1)degA~'detA2AXQ A 2

(8)

satisfies conditions (7). 4b ii a) Let A - (A1, A 2) E Pin(n) @ Pin(m). Show that

A(AA') = (--1)d*gA"A'2A(A)A(A'),

A - (A 1, A2), A' - ( A '~, A'2).

(9) Show in general that if A satisfies (7), it is a homomorphism from the kernel of F into the kernel of F ~. Answer 4b ii a" )~((A x, AE)(A , ~, A,2)) = )~(A~A' ~, AEA'2) = (--1)~d~gA'+d'gA"~'~d~gA2§

~ A 2 A '2

(8)

whereas

~t(A', A2)A(A '', A '2) = (--1)d~gA"d~sA2+a~gA'"d~gA'~(A1 (~ A2)(A '1 (~) A '2) and by the definition of the graded tensor product of graded algebras (A' (~) A2)(A '' (~) A '2)

=

(--1)degA2degA'lA1A'l ~ A2A '2 .

Equation (9) follows readily from the last three equations.

II

Let A k = (k 1, k 2) be in the kernel K of Pin(n)Q Pin(m) with k I = +_.e1 in the kernel of Pin(n) and k 2= +__ez in the kernel of Pin(m). It follows from (7a) that

147

2. CONSTRUCTION OF SPIN AND PIN BUNDLES (fit'* o A)(k 1, k 2) = (Ao ff()(k 1, k 2) = X(ele 2) for e 1, e 2 the unit elements of O(n), O(m), respectively = e the unit element of O(n + m ) .

Hence, A(k 1, k 2) = me the unit element of Pin(n + m) and A" kernel of Pin(n)0~)Pin(m)---~ kernel of Pin(n + m ) . It follows from (7b) that A is a homomorphism from the kernel of F onto the kernel of F*. I 4b ii fl ) Let the non-homomorphism of A be diagnosed by the properties

of o,j (x) __ 2( ,)/ij(X))2(,)/jk(X))2((,)/ij(X),)/jk(X))

1).

(10)

Show that {Ook(X)}, (i, j, k) in the nerve W of the cover ~ of M, defines a 2-cocycle 0 on W with values in Z 2. Show that if A is given by (8) then

O(i, j, k) =

(-1)~~

~x)d~ ,y*~x) .

(11)

Let p = (p', p2) and p , be the obstruction cocycles of ~ and ~, constructed in section 3; show that p ~k(X) = Oijk(X) A( pok (X)) .

(12)

Answer 4b ii /3" Following the argument developed in paragraphs 3a), 3b), and 4a-i), we check that "-- e E G ~ Oijk(X) = +_e ~ F ~.

ff(A(Oijk(X))

Hence, Oijk(X) defines a 2-cochain on W with values in 7/2,

O(i, j, k)ede=~Oijk(X) . The 2-cochain 0 is a 2-cocycle since (dO)(/, j, k, l) = O(j, k, l)(O(i, k, l))-lo(i, j, l)(O(i, j, k)) -1 by the definition of d = 1 by the definition of 0. Inserting (8) into (10) yields (11).

148

IV. I N T E G R A T I O N

ON MANIFOLDS

Applying A to -1

Yik = P qk Yii5%

and using (7b) gives * "Y ik

(13)

= A(p,k)-~A(%j%k).

On the other hand (14) ~/ik = P qk Equation (12) follows from (13), (14) and the definition (10) of Oijk. It follows from (12) p*(i, j, k ) = O(i, j, k)A(p(i, j, k)) ,

where we use the symbol A also for maps of cocycles defined by elements Pqk(X) E F. I Inserting 0 given by (11) and using the definition of paragraph 4b), we get p*(i, j, k) = (--1)d~g vb(~)'~8~k(~)i~(p~(i, j, k)p2(i, j, k)) .

Now, because the restriction of A to the kernel ( e , - e } homomorphism

(15)

of F is a

A-(pI(i, j, k)p2(i, j, k ) ) = p~(i, j, k)p2(i, j, k) .

(16)

Inserting (16) into (15) and applying the isomorphism between the multiplicative group S o and the additive group Z 2

r (1, -1}--, (0, 1}, we get ~b(p*(i, j, k)) 2 = deg y q1( x ) d e g yjk(X) + ~b(p~(i, j, k)) + r

j, k)) (17)

= r

g~j(x))r

gEjk(X)) + ~b(pl(i, j, k)) + ~b(pZ(i, j, k)).

Hence,

[p,] = [ga] u [g2] + [p,] + [p2]. and [p] satisfies the Whitney product theorem.

(18) II

iii) The nontriviality o f [p] for the twisted line bundle over the circle can be checked by explicit construction. The fact that [p] ~ H2(M, Z2) satisfies the naturality condition [pl] = f,[p2] for f: M 1--->M 2, the Whitney product theorem, and the non-triviality condition imply that

2. CONSTRUCTION OF SPIN AND PIN BUNDLE [p] = aw 2 + bw I U w I

where a, b

149

~ 27 2 ,

and depend only on the group. We shall compute a and b for the Pin groups [See P r o b l e m I 7 , Clifford], Pin+(n) and P i n - ( n ) . It will be sufficient to c o m p u t e [p](Pin-+(1)) and [p](Pin-+(2)). Indeed, if n >-2, there are natural maps" (Pin-+ (n) --> Pin-~(n + 1) which cover the natural inclusion O(n)---> O ( n + 1); the obstruction classes of the (Pin -+(n + 1)) bundles are the same as the obstruction classes of the (Pin+-(n)) bundles. Now, according to A x i o m 1, w E - 0 for a line bundle, hence [p](Pin-+(l)) = bw I t..Jw 1 and we determine b for [ p ] ( P i n + ( 1 ) ) and for [ p ] ( P i n - ( 1 ) ) as follows" Pin+(1)

= 7/2 x Z 2

and P i n - (1) = 2z4 9 The short exact sequence 0--> :~2 --> Pin+-(1)--> 0(1) --> 1 induces a long exact sequence 0--> HX(M, 7/2)--> H i ( M , Pin-+(1))---> HI(M, ~'2)---> HE(M, ~'2)--~ 9 9 9 The long exact sequence splits for P i n + ( 1 ) = 772 X 7~2, namely

0--> Hi(M, Z2)----> Hi(M, 7/2 x Z2)---> Hi(M, 7/2)--*0----> H2(M, 7/2)--->.... Thus, [p](Pin § (1)) = 0. The long exact sequence does not split for P i n - ( l ) and the Bockstein map B" Hi(M, 7/2)---> H2(M, Z2) maps W1~

W1 U w I .

Thus [ p ] ( P i n - ( 1 ) ) - w~ U w 1. It can be proved that one can construct* a P i n - ( 2 ) bundle but not a Pin§ bundle over the real projective plane R P ( 2 ) for which w I U w~ = 1 and w 2 = 1 [see, for instance, Milnor and Stasheff]; thus it follows that [p](Pin § [p](Pin

[p](Pin §

= w2 ,

( 2 ) ) - [ p ] ( P i n - ( n ) ) = w 2 + w I U w 1.

We refer the r e a d e r to [Karoubi] for the construction of the obstruction class of Pin(n, m) w h e n both n and m are different from zero. For a 9Use the well-known triangulation of RP(2)- see for instance [Patterson p. 100]-and construct [ p] explicitly.

150


physical consequence of the fact that the obstruction classes of the Pin+(n) bundles and the Pin-(n) bundles are different, we refer the reader to [Carlip, DeWitt-Morette]. Acknowledgements and References may be found at the end of Problem IV 3. Sections 3 and 4 use results and arguments from [Greub and Petry]. However, Greub and Petry assume Pin(l) = 0(1) x 0(1); hence, they investigate only the obstruction for Pin+(n). We are indebted to S. Carlip for this remark. 3. I N E Q U I V A L E N T SPIN S T R U C T U R E S

INTRODUCTION

We have given in Problem IV 2 a necessary and sufficient condition- the vanishing of the second Stiefel-Whitney c l a s s - for the existence of a Spin(n) bundle ~:, extension of a given SO(n) bundle ~: over a manifold M. This e x t e n s i o n - not necessarily u n i q u e - has been obtained by lifting the transition functions g~j: U~ tq Uj ~ SO(n) of g to transition functions Yij: Ui f3 Uj ~ Spin(n). The elements t7 of ~ are then the equivalence classes = [i, x, A~],

x ~ U~,

A, E Spin(n)

with the equivalence relation (i, x, A~) - (j, x, Aj)

if x E U~ N Uj,

Aj = yj,(x)A~,

if x ~ U~ N Uj,

L i ~_ SO(n) Lj = gji(x)Li.

while the elements of ~: were p = [i, x, L,], (i, x, L,) - (j, x, Lj)

There is a mapping ~ : st--->~: defined by Sf(/~) = p ,

L, = ~(A,) ;

the definition does not depend on the chosen representative of $7 if the ),~j are lifts of the g~i, i.e., such that ~(Tii(x))= g~](x), x ~ U~ N Uj. The mapping ~ is a bundle morphism, i.e., commutes with the right actions of the relevant groups" ~(/~A) = ~"(/~) ~ ( A ) .

3. INEQUIVALENT SPIN STRUCTURES

151

M o r e g e n e r a l l y a pair (~:, Y() with ff a Spin(n) bundle over a manifold M and ~ a bundle morphism from ~: onto an SO(n) bundle ~: over M is called a spin structure over ~:. See for instance [Milnor] for alternative definitions. Two spin structures (~:~,, N~,) and (~:~, No) on ~: are said to be equivalent if there exists an isomorphism f" ~:~ ~ ~a such that N~, of = N0"

spin structure equivalent spin structures

1) A n example o f inequivalent spin structures on a trivial SO(n) bundle. a) S h o w that any Spin(n) bundle on a solid torus T diffeomorphic to R N x S 1 is trivial f o r n > 2. A n s w e r la: to showing if it admits construct a

To show that a Spin(n) bundle over T is trivial is equivalent that it is trivial over S 1. A principal bundle is trivial if and only a global (continuous) cross section (p. 133). It is possible to global cross section on a Spin(n > 2) bundle over S 1 because

{

S~ Spin(n > 2)

is a 1-complex and is a path connected group.

Indeed, the 1-complex S 1 consists of a 0-skeleton (the 0-simplices) and a 1-skeleton (the 1-simplices). The spin-bundle restricted to the 0-skeleton admits cross sections. Let us choose one of them. It is possible to extend this cross section to a cross section over the 1-skeleton if and only if there is a path between any 2 points of Spin(n). II More generally one can show [Problem V bis 8, Universal Bundle, paragraph 3] that a G-bundle over X is trivial if X is an n-complex and G is (n - 1) connected, that is if I l k ( G ) is trivial for k -< n - 1. Remark: A trivialization of a Spin(n) bundle ~: determines a trivialization of the SO(n) bundle sr Indeed

s~ "~ ( ~: • SO(n)) /(equivalence defined by ( u A , a) --- (u, g g ( A ) a ) . (1) u E Spin(n) bundle, A E Spin(n), a E SO(n), Y(" Spin(n)--+ SO(n). Since s~ = S 1 • Spin(n), the isomorphism (1) provides a trivialization of sc. b) S h o w that it is possible to construct two inequivalent Spin(n), n > 2, structures on a solid torus T diffeomorphic to R ~v x S 1. A n s w e r lb" We give here a rough, heuristic construction; the correct one requires an atlas over S 1 and is done in paragraph c).

n-connected group

152


Let s = [0, 1], F~" S~ • Spin(n)---~ S 1 • SO(n) by (exp 2 ~ris, A) ~ (exp

2 zris, Y((h~ (A)A))

N

and a similar definition of F~, with h~" [0, 1]---~ Spin(n) h a" [0, 1]---~ Spin(n)

such that such that

h~(0) = e, h~(1) = e h~(0) = e, h~(1) = - e .

We shall show that there is n o i s o m o r p h i s m f such that/~t3 o f = / ~ . the preimages by F~ and F a of the set S i x (e} C S I • respectively

#21(S Ix

{e}) = (S 1 • {e}) U ($1 x { - e } ) C

Indeed are,

S 1 x Spin(n)

and

F;'(S' x {e})= {exp(21ris), (ha(s)) -1} U {exp(21ris), -(ha(s))-'),

s E [ 0 , 1].

The first preimage is disconnected whereas the second one is connected, hence there is no isomorphism f such that F a of = F~. A correct construction using an atlas to cover S' is not expected to change this result and we expect to be able to construct at least two inequivalent spin structures on a solid torus. Spin (n)

Spin (n)

-e

~-'(S'

-e

x {e})

~ g - ' ( S ' x (e~)

Two different spin structures correspond to two different prescriptions for patching the pieces of the double covering.

Remark: We establish for use later on that F(gA) = F(/7/~(A),

F = F~ or F o .

Indeed let t7 = (exp 27ris, A)I), then

F(,ff ) ~(A) = (exp 2aris, ~'(h(s)A~ )) ~(A) -

F(#4).

(2)

153


c) Construct inequivalent spin structures ( ~, ~ )

on a solid torus T.

A n s w e r lc: Let 0-//= {U1, U2 ' U=} be a simple cover of S 1. Let (U i, q~i) [resp. ( U i, ~bi)] define local trivializations of a Spin(n) bundle sr[resp.

SO(n) bundle] over S 1 such that A

A

A

~i ~ ~ ; l l x --" ")tij(X) ,

fi((Vq(X))

=

A

~i ~ ~ ; l l / --- gij(X)

(3)

gq(x).

(4)

Denote by h a family {hi) such that h i" Ui---~ Spin(n), h j ( x ) A j = +--yji(x)hi(x)Ai,

x E Uij =- U i fq Uj .

(5)

Define A,

,~,

by ( ~-

- - 1 )(X) A i) = d?i- ' ( x , ~ ( h ~ ( x ) A , ) ) o (~i

x E Ui,

,

(6)

where {h~'} satisfies (5) with definite signs. Eq. (6) makes sense on the bundle ~: whose elements are the equivalence classes ff = [i, x, A~], x ~ U~, because ~

~

( ~ ( o ~i

-1

)(X, Ai) - (~(~a o ~1

-1

)(x, Aj) ,

Aj -= ~ji(x)Ai .

Indeed -I -1 ~p ]-l(x, Eg(h~.(x)Aj))= #p, o ~pi o ~ j (X, f f ( ( •

by (5) by (3) and (4). II

--~i-l(x) E((+h"(x)A~))~

SO( n)

Spin ( n h(s)

----4---

...

uh(s)

- -

A!

~ s

Rh(s) I I \

--~-ZT.

-1P" " "

pa(s)l -.~.-~

', Ra~s)

I

i

|

s)

(-

s)

a(s)


154 The pair (f, ~ )

is a spin structure.

2) Show that the number of inequivalent spin structures on an SO(n) bundle ~ is equal to the number of elements in the first Z2-cohomology group HI(~, 3~'2)"

Answer 2: Let 6 0 = yu(x)hj(x)Aj(h~ (x)Ai) i(~/ij(x)h ~j (x)Aj(h~(x)Ai) Ot~

Ot

Ot

--

1

)- 1

=- 6~a(i, j ) e .

(7)

The 1-cochain 6 "~ on the nerve N of S 1 defined by q/with values in 7/2 = { 1, - 1} is a cocycle. Indeed

d6"'(i, j, k ) = 6~a(j, k)(6"a(i, k))-16a~(i, j) by the definition of d = 1 by (7) and property of {Yu}" The 1-cocycle 6 ~ defines a cohomology class 6(a,/3) E HI(N, 7/2). We shall show that, given a spin structure (~, ~'~) and w E H~(N, 7/2), there is one and only one spin structure (~:, ~'a) such that 6(a,/3) = w. To prove uniqueness we shall prove in paragraph i) 6(a, V) = 6(a,/3) r ~ = ~v or equivalently

a(v,

0

~'~ = ~'~

=>a(y,/3)=0.

- -

(8) (9)

To prove existence we shall prove in paragraph ii) that given ~a and w ~ Hi(N, Z=), there exists ~a such that 6(a,/3) = w. i) Uniqueness. To prove (8) we note that if 6(a, 13)= 0, then either 8"a(i, j ) - - 1 ,

for all pairs (i, j);

(10)

or there is a 0-cochain A on N with values in 7/2 such that

6~~

j)(dA(i, j))-~ = 1

6~'(i, j) = A(j)(A(i))-'

for all pairs (i, j), i.e., for all 1-simplices (i, j).

(11)

We note that (10) is a particular case of (11) when A(i)= 1 for all i. We shall show that if 8 ~ ( i , j ) = A(j)(A(i)) -~ for all 1-simplices then ~e = ~ , . Indeed, inserting (11) into (7) we get

"),q(x)A( j)h ~.(x)Aj( A(i)h~.(x)Ai) -1 = Zj(x)h~(x)Ai(h~i(x)A~) -1

(12)

155

3. INEQUIVALENT SPIN STRUCTURES Hence for all/

(13)

' ~((A(i)hT(x)Ai) )

(14)

i - -l(x, (og~'oA)(h~(x)Ai)) -~b

(15)

-(y(oAo~b~ )(x,A~)

(16)

h~i (x) = A(i)hT(x)

and, by (6) ~ ~ ~)"-1 (fill3 i ) (X ,

A i)

= ~) i- l ( x

~

~

--1

with obvious definitions for Y(o A and Y(~ o A. II

Thus G = X~. ~

~

,~

Now assume ~ = ~ o A, reverse the sequence of eqs. (13)-(16); inserting (13) into (7) gives 6~t3(i, j ) = A(j)(A(i)) -1. ii) Existence. Let a be a 1-cocycle in N with values in ~2 such that y~i(x)h~(x)A/(h~(x)Ai) - 1 = a(i, j)y~j(x)h~. (x)A/(hT(x)A~) -~ .

It is straightforward to check that {h~(x)} defines a spin structure (st, ~0), namely one checks that Nr satisfies (6) with 4) and ~b satisfying (3) and (4). ~,

~

~

N

3) S h o w that there are only two inequivalent spin structures on an SO(n) bundle over a solid torus T. A n s w e r 3: We shall count the elements in H I ( s I , ~ 2 ) . Consider the simple cover of S 1 consisting of U~, U 2, U 3 such that the only nonempty intersections are U~2 ~ 12, U23 - 2 3 , U3] - 3 1 . The representatives of the elements of H I ( s 1, Z2) are the maps

{1 {1 {1 i12 {1 {12{_1 {12 i1 {12{1

f~" 23--~ 31

f4: 23--, 31

1, 1

f2" 23--* 31

1,

L" 2 3 ~

-1

fT: 23---* - 1 , 31 1

31

fs" 23----~ 31

1, 1

-1, 1

f3" 2 3 - - - , - 1 , 31 1

f6" 2 3 ~ 31

1, 1

1 1.

The maps f~, fs, f6, f7 belong to the same cohomology class, namely 0 and the maps f2, f3, f4 and f8 belong to the same cohomology class, namely 1. Let us check for instance that [fl] "-[fT] but [fl] ~ [f8].

IV. I N T E G R A T I O N O N M A N I F O L D S

156

fi { 1

i 12

1 1 then fT(dh) -1" 23 -'--> 1 . -1 21 1 On the other hand there is no A such that 12 { - 1 dh" 23 --+ 1. 31 1 Set A"

II

Remark" One can check that if f(i, j ) = A(i)(A(j)) -1, then [f] =0.

Since there are 2 elements in H~(S 1, 7/2) there are 2 inequivalent spin structures on an SO(n) bundle over T. 4) Construct inequivalent spin structures on nontrivial bundles. Answer 4" Having rephrased in paragraph 2a) iii) the construction of

inequivalent spin structures given in paragraph 2a) ii), we can generalize readily the construction of inequivalent spin structures when the Spin(n) bundle is trivial to the case when it is not. 5) Construct inequivalent connections on the Spin(n) bundle ~ over T corresponding to a given connection to on ~. Answer 5" Different spin structures determine different connections on

the Spin bundle (p. 419). Indeed let (~, ~ ) be a spin structure over ~: and let to" T~ ~ .~(SO(n)) be a connection 1-form on ~:. The corresponding connection on ~ is (p. 419) o',~ = ~ ' - ~ ( e )

o ~* co,

where ~*" T*sr

T*r

~ to is the 1-form on ~: with values in ~(SO(n)) pull back of to. Under the map ~"-~(e) it becomes a 1-form % on g with values in ~(Spin(n)). For the computation of the pull back t~ of % by local sections f/on Ug canonically defined by the local trivialization 4)i we refer the reader to p. 419. ACKNOWLEDGEMENTS FOR PROBLEMS IV 1, 2, 3

Remarks of Steven Carlip and critical reading by Steven Blau of the first drafts of the three problems in this series are gratefully acknowledged. Much is due to Gary Hamrick, whose comments and examples


157

h e l p e d grasp la substantifique moeUe of s e v e r a l theor~mes fins a n d m a k e it u n d e r s t a n d a b l e to n o n - s p e c i a l i s t s . REFERENCES FOR PROBLEMS IV 1, 2, 3

Atiyah, M.F., R. Bott and A. Shapiro, "Clifford Modules", Topology 3, Suppl. 1, (1964) 3-38. Avis, S.J. and C.J. Isham, "Vacuum solutions for a twisted scalar field", Proc. R. Soc. Lond., A363, 581-596 (1978). Avis, S.J. and C.J Isham, "Quantum field theory and fibre bundles in a general spacetime", in Cargese 1978 Lecture Notes Recent developments in gravitation, eds. M. Levy and S. Deser (Plenum Press, New York). Bichteler, K. "Global Existence of Spin Structures for Gravitational Fields" (J. Math. Phys. 9, 813-815, 1968). Carlip, S. and DeWitt-Morette, C. "Where the sign of the metric makes a difference". Dabrowski, L. and Percacci, R. "Diffeomorphisms, Orientations, and Pin Structures in 2-dimensions", ISAS (Trieste) Preprint 103/86/EP. DeWitt, B.S., C.F. Hart and C.J. Isham, "Topology and Quantum Field Theory", Physica 96A (1979) 197-211. Geroch, R. Spinor Structure of Space-Times in General Relativity, I, II, J. Math. Phys. 9 (1968) 1739-1744; 11 (1970)343-348. Greenberg, M.J. and J.R. Harper, Algebraic topology, a first course (Benjamin/Cummings, Reading, MA, 1981). Greub, W., S. Halperin and R. Vanstone, Connections, Curvature and Cohomology (3 volumes) (Academic Press, New York 1972, 1973, 1976). Greub, W. and H.R. Petry, "On the lifting of structure groups", in Differential Geometrical Methods in Mathematical Physics H, eds. K. Bleuber, H.R. Petry and A. Reetz, Lecture Notes in Mathematics, No. 676 (Springer-Verlag, Berlin, 1978) pp. 217-246. Hirzebruch, F. Topological Methods in Algebraic Geometry (Springer-Verlag, New York, 1966). Isham. C.J. "Twisted quantum fields in a curved spacetime", Proc. R. Soc. Lond. A362 (1978) 383-404. Isham, C.J. "Quantum Field Theory in Curved Spacetimes; A General Mathematical Framework", in Mathematical Physics II, eds. K. Bleuler, H.R. Petry and A. Roetz, Lecture Notes in Mathematics, No. 676 (Springer-Verlag, Berlin, 1978). Isham, C.J. "Spinor fields in four dimensional space time", Proc. R. Soc. Lond. A364 (1978) 591-599. Isham, C.J. and S.J. Avis, "Lorentz gauge invariant vacuum functionals for quantized spinor fields in non-simply connected spacetimes", Nucl. Phys. B156 (1979) 441. Isham, C.J. and S.J. Avis, "Generalized spin structures on four dimensional space-times", Comm. Math. Phys. 72 (1980) 103. Isham, C.J., "Topological and global aspects of quantum theory", in Relativity, groups and topology H, eds. B.S. DeWitt and R. Stora, Les Houches Session XL, 1983 (NorthHolland, Amsterdam, 1984). Isham, C.J, and C.N. Pope, "Nowhere vanishing spinors", Class. Quantum Grav. 5 (1988) 257-274. Jiinich, K. Topology (Springer-Verlag, New York, 1984). Karoubi, M. "Algebres de Clifford et K theorie", Ann. Scient. Ec. Norm. Sup. 4 serie 1 (1968) 161-270. KiriUov, A.A. Elements of the Theory of Representations (Springer-Verlag, Berlin, 1976).


158

Lichnerowicz, A., Bull. Soc. Math. 92 (1964) 11-100; see also Ann. Inst. Henri Poincar6 13 (1964) 233-290. Maclane, S., Homology (Springer-Verlag, Berlin, 1963). Milnor, J., "Spin Structures on Manifolds", L'enseignement Math6matique 9 (1963) 198-203. Milnor, J.W. and J.D. Stasheff, Characteristic Classes (Princeton University Press, 1974). Patterson, E.M. Topology (Oliver and Boyd, Edinburgh, Second Edition 1959). Petry, H.-R., "On the non uniqueness of spin structure in superconductivity", in Differential Geometrical Methods in Mathematical Physics II, loc. cit. under Greub) pp. 247-254. Petry, H.R. "Exotic spinors in superconductivity", J. Math. Phys. 20 (1979) 231-240. Seiberg, N. and E. Witten, "Spin Structures in String Theory" (Preprint, Inst. for Adv. Study, Princeton 1986). Spanier, E.H., Algebraic Topology, (McGraw-Hill, New York, 1966).

C.W. complexes

Often used in these references, C.W. complexes have not been used here. C.W. complexes are built in stages, each stage being obtained from the preceding one by adjoining cells of a given dimension. Considered as a C.W. complex a polyhedron frequently requires fewer cells than a simplicial triangulation [e.g., Spanier, p. 400].

4. C O H O M O L O G Y OF G R O U P S 1. DEFINITIONS AND EXERCISES

cochain

Suppose that a group G acts on a group M by a left action tr: G x M--* M denoted by tr( g, m) - trg (m) =- gm. An n-dimensional coehain is a mapping c: G x G x . . . x G(n + 1)factors-~ M such that c(ggo, g g l , . . . , g g n ) = g c ( g o ,

for allg, g i E G .

gl,...,g,)

(1)

a) Show that the space C(G, M ) of cochains c: C"+1---~ M forms a group if trg commutes with the product in M. Show in particular that trg is not a free action. Answer la" Let Cl, c 2 ~ C(G, M ) and let ClC2 be the product defined pointwise. The product ClC2 is also in C(G, M ) if it satisfies (1). We have Cl( ggO,

" " " , ggn)C2(

ggO,

"'"

, ggn)

~" O ' g ( c ~ (

go'

" " " ' g,,))

x %(c~(go,..., g,)). Thus if o-g(ma)O.g(mz) = o'g(mxmz) ,

(2)

4. COHOMOLOGY OF GROUPS

159

then ( C l C 2 ) ( g g 0 , . . . , gg,) = trg(clc2( go, . . . , gn)). It follows from (2) that for all g E G trg(m)trg(1) = trg(m) , i.e., ag(1)=l

for a l l g E G .

O-g leaves the unit element of M invariant; it is not a free action.

b) The coboundary operator d

coboundary operator

d: C"(G, M)----> C n+ 1(6, M ) is defined by dc(go,...,

g~+l) = c ( g l , . . . , g~+l) x (C(go, g 2 , . . . , gn+l)) -1

~

~

~

• (c(g0,..., g~,..., g,§

• (c(g0,...,

g~)) (-~r+x

(3)

Check that d d c - 1. A cochain c is called a coboundary of the cochain b if c = db. A cochain c is called a cocycle if dc = 1.

coboundary cocycle

Show that if M is an abelian group the n-dimensional cocycles f o r m a group Z " ( G , M ) ; show that the coboundaries o f ( n - 1 ) - d i m e n s i o n a l cochains f o r m a normal subgroup Bn(G, M ) o f Z ~ ( G , M ) . A n s w e r lb: Let c I and c 2 be two cocycles; then d(ClCE)(go," " , gn+l) = c l ( g l , ' ' ' , g n + l ) c E ( g l , • (cl(g0, g 2 , . . . ,

gn+l)CE(go, g 2 , . . . ,

.. . , gn+l)

g~+l)) -1 . . . .

The given properties follow from the fact that if M is abelian this equality can be written d(ClC2) = (dCl)(dc2) .

II

The n-eohomology group of G with values in the abelian group M is the factor group H"(G, M) = Z"(G, M)/Sn(G, M) .

cohomology


160

c) Let c be an n-cochain, and define ~: G • t ~ ( h l , . . . , h,,) = c(e, hi, h l h 2 , . . . ,

• G(n times)---~ M by h l h 2 . . , hn).

(4)

Conversely show that given ~ relation (4) and the cochain condition (1) determine a cochain c. Show that when M is abelian a 2-cochain ~ is a cocycle if and only if ~ satisfies t~(hlh2) = ~(hl)(hl~.(h2)).

(5)

Answer lc: From (4) and (1) we deduce C(go,.. . , g , ) = go~.(go~ gl, g l l g 2 , . . . ,

g-~llg,).

(6)

Let c be a 1-cochain. Then the cocycle condition dc = 1, written in terms of ~" gives 1 = dc(e, h I , hlh2) = c(h 1, hlh2)(c(e, hlh2))-lc(e, h 1) = hl~.(h2)(~.(hlh2))-l~'(hl)= dc'"(hl, hE).

homogeneous

The last equality defines dc and the cocycle condition d c - 1 gives equation (5) when M is abelian. II The cochains c are homogeneous and the t~, also called cochains, are not. We give in the next section an example of cochains easier to state in terms of ~" than in terms of c. 2) Cocycle condition for mappings f: G x X---~ K where the group G acts on a space X by left actions and f is a mapping on G x X with values in an abelian group K. For an example see the following problem (IV 5 Lifting). a) Use the definition o f cochain (2) and coboundary operators (5) to define a cocycle condition for mappings f : G x X---~ K. Answer 2a: Let M be the space of mappings from X to K. The mappings f: G • X----~ K can be considered as mappings F: G---~ M, by setting (F( g))(x) =- F(g)(x) = f ( g, x) E K . The group operation in K induces a group structure on M by (ulu2)(x) = Ua(X)U2(X) ,

Ua, u 2 E M, x ~ X

(7)

5. LIFTING A GROUP ACTION

161

The action of G on X, denoted x ~ gx, defines an action lg of G on M by 9=

u( gx) ,

u

M; u(x), u( gx)

r .

(8)

By analogy with the construction in paragraph lc), the mapping F: G---~ M together with the action lg will be called a cochain and it will be called a cocycle, by analogy with (5), if F(glg2) = F( gl)(lg F( g2)) , i.e., (9)

f ( gig2, x) = f ( gl, x ) f ( g2, g l x ) 9

cocycle condition

A mapping f satisfying eq. (9) is said to satisfy the eoeyele condition. References for Problems IV 4 and 5 are listed at the end of Problem IV 5.

5. LIFTING A G R O U P A C T I O N Given a K-principal bundle P with base space X and an action of G on X, one may wish to "lift" the G action on the base to a fibre preserving G action on the bundle. The questions arise as to the existence and uniqueness of such lifts. For example, let the base X of P be a Lorentz bundle over a manifold M. Let the typical fibre K of P be Z 2. We shall set up the problem so that the existence of a spin bundle over M is stated_ in terms of the existence of appropriate lifts R k of the right action Rg of the Lorentz group L ( n ) on the Lorentz bundle X. A right action R k . P-.--~P ,

i.e. such that R k l ~ R k 2 = Rk2k ~,

(1)

is said to be a lift of the right action Rg: X---~ X

if ~

N

"n"o R k = Rg o "a" .

(2)

Equivalently R k is a lift of R g if the following diagram is commutative:

162


ON MANIFOLDS

P

r-p.

Rg

X----* X . In the case where X is itself a principal fibre bundle with base manifold M, and Rg is a fibre preserving right action on X, for instance the right action of its structure group G, we say that the lifting to P is appropriate if the projection on X of R k(p) and p lie in the same fibre of X.

Consider a local trivialization of P over a fibered open set of X with k = ~'2 = {1, - 1 } : 9 " U ~ V x Z2,

V = "n'~ (v) C X,

v C M.

Show that the most general appropriate lift R k of Rg is such that A

9 (Rk(P)) = (Rs(x), f(g, x ) ~ ( p ) ) ,

p ~ P, 7 r ( p ) = x,

(3)

where f: G x X---~ Z 2

satisfies the cocycle condition (Problem IV 4, Cohomology eq. (9) for the cocyle condition of a left action) f ( g l g2, x) = 3"( g2, glx)f(gl, x ) .

(4)

Answer" In a local trivialization of P we have 9 (Rk(p)) = (y, q),

y = ~r(gk(p) E X,

qEZ

2

and moreover if the lifting is appropriate

1rM(y) = 7rM(x),

with x = 7r(p).

Hence there exists g • G such that y = Rg(x). We define f(x, g) ~ Z 2 by A

~

f(x, g ) = q ( ~ ( p ) ) - x = ~ ( R k ( p ) ) ~ - l ( p ) , and obtain the formula (3). We shall now look at the conditions to impose to f in order to satisfy (1). Since k E 7/2, it is not necessary to distinguish right and left action. We shall nevertheless write the factors in the "right" order. L e t Rg2o Rg 1 = Rg3, i.e., g3 = glg2"

6. SHORT EXACT SEQUENCE; WEYL HEISENBERG GROUP

Rk, P

Rk 2

p

--

Rq~

X

~g

P

2

X A

163

~

X A

A

~

f(g3, x) = f ( g l g 2 , x) = ~(RkE(P))CIf l ( R k l ( p ) ) ~ ( R k l ( p ) ) ~ = f(g2, ggx)f(g,,

A

-

1(

p)

x).

This is the cocycle condition for the right action R k to be a lift of a right action Rg. Note that in [Problem IV 4, C o h o m o l o g y , eq. (9)] we established the cocycle condition for the lift of a left action. It is then straightforward to check~ that the lift R k of Rg on X can be identified with the right action R A of a spin bundle over the manifold M, base of the bundle X. REFERENCES FOR PROBLEMS IV 4 A N D IV 5

Kirillov, A.A. Elements of the Theory of Representations (Springer-Verlag, Berlin, 1976). Maclane, S. Homology (Springer-Verlag, Berlin, 1963). Isham, C.J. "Topological and global aspects of quantum theory" in Relativit6, Groupes et Topologie II, eds. DeWitt and Stora (North-Holland, Amsterdam, 1984), pp. 1208, 1229, 1233, 1239. Faddeev, L.D. "Operator anomaly for Gauss law", Physics Letters 145B (1984) 81-84. Stora. R, "Algebraic structure of chiral anomalies", in New Perspectives in Quantum Field Theories. XVI GIFT International Seminar, eds. J. Abad, M. Asorey, A. Cruz (World Scientific, Singapore, 1986). Libermann, P. et C.M. Marie, Geom6trie symplectique, Bases th6oriques de la m~canique Tome III, (Publications Math6matiques de l'Universit6 Paris VII, p. 170. Diffusion UER Math6matiques Tour 45-55, 5 6tage. 2 Place Jussieu F-75251 Paris Cedex 05 43362525 poste 3761).

6. S H O R T E X A C T S E Q U E N C E ; W E Y L H E I S E N B E R G

GROUP*

1) A sequence A ~ B ~ C of vector spaces A, B, C with linear mappings f, g, is said to be exact if the kernel of g is equal to the image of f. S h o w that the short sequence *Written in collaboration with Humberto J. LaRoche.

exact sequence

164


O.__>A r_~B g-.~C--->O

(1)

is exact only if f is injective and g is surjective. Answer 1" Follows from the definitions.

2) If A , B, C are groups, the element.O in the definition o f kernels will be replaced by the neutral group element 1 and linear maps by homomorphisms. Let H be a normal subgroup o f G. Show that the following sequence o f groups is exact: i

qr

1 --> H---> G - - * G / H - - > 1 ,

where i is the inclusion mapping and zr the natural projection. Answer 2: Straightforward.

extension

3) Let 1---->G 0---->G---> G 1--->1 be an exact sequence of groups. The group G is often called an extension of G1 by G o. In general the group G cannot be reconstructed merely from G O and Ga. Additional information is needed. It can also happen that an extension of G~ by G Ois not possible. In the following problem we shall construct an extension G of the group of translations z " o n ~2 by R known as the (Weyl) Heisenberg group. We choose a notation appropriate to the role of the Heisenberg group in quantum mechanics; we denote points in R 2 by (q, p ) - s E R

2.

The group z 2 of translations of R 2 acts as follows, l(a.b)( q, p) = ( q + a, p - b) ,

(a, b) =- A E R 2 .

a) Let yA be the generator o f the translations by (at, bt). Let or = dq A dp be the symplectic f o r m on R 2. Show that ira or is an exact differential; i.e., show that ~,a is a Hamiltonian vector field (p. 269) and [Problem IV 11, Poisson]. Answer 3a" The Lie derivative of or with respect to yA vanishes, since or is invariant by translation"

d(q + at) ^ d ( p - b t ) = dq ^ dp . Hence (p. 269 and Problem IV 11, Poisson) the vector field yA is locally Hamiltonian.

6. SHORT EXACT SEQUENCE; WEYL HEISENBERG GROUP

165

A locally Hamiltonian vector field o n ]R2 is globally Hamiltonian; hence

izAa -- dPa

(2)

for some function PA o n ]t~2 determined by ya up to a constant function. b) Show that the set ~ of possible functions PA together with the Poisson bracket { PA, PB } defines a Lie algebra which is not isomorphic to the Lie algebra oSx'(r2) of the translation group r 2. See also [Problem IV 11, Poisson].

Answer 3b" In natural coordinates 0

g (a'h) - - a

Oq

0 b~ Op

(3)

and

dPa

--

iyAa

--

(4)

a dp + b dq.

Hence the function PA is determined up to an additive constant k. Set

PA,k -- ap + bq + k,

k ~ N,

A E ]~2

(5)

and, setting PA~,k, -- P1, {P1, P2} ~

OP10P2

OP] OP2

Oq Op

Op Oq

(6)

= bla2 - alb2.

{P1, P2} is ofthe form (5) with a - 0 , b - 0 , k - bla2 - alb2. Because Poisson brackets satisfy the Jacobi identity, the set ~@ together with the Poisson bracket {, } is indeed a Lie algebra. As a vector space ~@ is ]~2 • ]~. It cannot be isomorphic to oL~(r2), which has ]R2 as underlying vector space. Moreover, the subspace ]t~2 • {0} o f ~ (i.e., k - 0) is not a Lie subalgebra of ~ ; it cannot be isomorphic to o~(r2). Note that S ( r 2) is the trivial Lie algebra:

[yA1, yA2]

_ 0,

VA 1 A 2 E ]R2

o

c) Show that the Lie algebra S ( G ) of the Heisenberg group G defined by the group law (al, bl, kl)(a2, b2, k2) -- (al + a2, bl + b2, kl + k2 + 1 (bla2 - alb2))

(7) is isomorphic to the Lie algebra ~,~ equipped with Poisson brackets. Answer 3c" The Lie algebra S ( G )

of the Heisenberg group G

on

I[~2 • ]R

Heisenberg group

166


can be obtained from the group law. Let (or, fl, K) e ~-CF(G) and exp(e(ot, fl, to)) = (a, b, k). Using exp(e~) exp(e() e x p ( - e ~ ) e x p ( - e ( ) -- exp(e2[~, (] + higher order in e) we obtain [(Oil, ill, KI), (Ct2,152, K2)] -- (0, 0, fllOt2 --Otlfl2).

central

(8)

The map: Pa,b,k -~ (or, fl, ~c) is a Lie algebra isomorphism. d) An extension G of a group G l = G / H by a group H is said to be central if f embeds H into the center of G in the exact sequence

1--+ H f

G --% G / H --+ 1.

The sequence 1 --+ R --+ G --~ r2 --+ 1 where i(k) --+ (0, 0, k) and

yr(a, b, k) = (a, b) is exact; r 2 = G / R but G is not the direct product r 2 x R, since the Lie algebra S ( G ) , given by (8) is not trivial. Show that the Heisenberg group G defined in c) is a central R-extension of z"2.

Answer 3d: The Heisenberg group is a central extension of the translation group since (0, 0, k) 9 I~ commutes with all its elements. II In summary the extension of the group of translations provides an isomorphism between .2V(G) and ~ { PA,k, A 9 R 2, k e I~}. If the basic classical observables of a system are the functions of q and p in the set ~ , then a unitary irreducible representation of G on square integrable functions yields natural quantum observables. If all the unitary irreducible representations of G are unitarily equivalent then the quantum observables are uniquely defined by the classical observables. =

e) Construct a unitary representation of the Heisenberg group and show its relationship with the quantum mechanical representation of the operator and ~ with [1 - multiplication by q, ~ - - i h d/dq. Answer 3e" Quantization means finding irreducible weakly continuous unitary representations ~ ' of a group G, on the space L2(I~) if the configuration space of the system is I~.

7. COHOMOLOGY OF LIE ALGEBRAS

167

Set ~ ' (a, 0, 0) = U (a), ~ ' (0, b, 0) = V (b) and ~ ' (0, 0, k) -- exp(i/zk). Then, it follows from the group law that U(al)U(a2) = U(al + a2) V(bl)V(b2)

=

V(bl +

b2)

U(a) V(b) = V(b)U(a) exp(-itxab).

This representation is called the Weyl representation of the canonical commutation relations. If we set U(a) - exp(+ia/3), V(b) -- e x p ( - i b 0 ) , then [c~, 0] -- 0,

[/3,/3] -- 0,

[0,/31 -- i#.

Weyl representation

II

4) The standard procedure for extending a group G1 by R is to obtain a new Lie algebra S ( G ) defined on the set .L~(G1) @ R by the Lie bracket [(A1, kl), (A2, k2)] = ([A1, A2], z(A1, A2)),

A1, A2 ~ S ( G 1 ) , k 6 N,

(9) where z(A 1, A2) is an antisymmetric bilinear map and [, ] the Lie bracket in S ( G 1 ) . Show that (9) defines indeed a Lie algebra structure on .2~(G1) @ R if and only if z(A, [B, C]) + z(B, [C, A]) + z(C, [A, B]) -- 0.

(lo)

Answer 4: Straightforward, (10) expresses that (9) satisfies the Jacobi identity.

We shall show in the following problem (IV 7, Cohomology, paragraph 4) that z is a cocycle on the Lie algebra with values in R. References for Problems IV 6 and IV 7 can be found at the end of Problem IV 7. 7. COHOMOLOGY OF LIE ALGEBRAS* See [Problems V bis 11, Cocycles, V bis 12, Virasoro] for applications. 1) Let fY be a Lie algebra over a field ~ . A f~-module is a vector space M over ~ together with a homomorphism fY --+ L (M, M)

(linear transformations of M);

*We acknowledge fruitful discussions with M. Dubois-Violette and I. Bakas.

f~-module

168


we shall denote the linear transformation corresponding to 3' E ~ by m ~--~'y . m , cochains on q3

m E M .

The n-dimensional cochains on ~ with values in M are the n-linear alternating functions o n (~n with values in M" f: ~d x . . . • ~ (n-factors)---~ M linearly in all factors

(1)

and f ( y a , . . . , 3',) totally antisymmetric in y a , . . . , y,. S h o w that the set Cn( ~, M ) o f n-cochains makes up a vector space over ~ . A n s w e r 1" Straightforward from the definitions. 0-cochains

2) The space of 0-eoehains C~ M) is identified with M. Let f be an n-dimensional cochain on q3 with values in M and 3' ~ c~; define 3" f as the n-linear function on q3 with values in M: ( y . f ) ( ~ r ~ , . . . , a',,)= y - ( f ( c r ~ , . . . , o-,,)).

(2)

a) S h o w that y . f is an n-cochain. b) Let f be an n + 1 cochain, and let f~ be the n-cochain defined by L(~,,

. . . , ~,)=

f(~,

~,,

. . . , ~,)

.

S h o w that y'f~=(y.f)~ A n s w e r 2:

a) y. f is a multilinear alternating function. b) Straightforward. coboundary operator

3) The coboundary operator d" C"(~, M) ---> C" + ~(~d, M) is defined by n

(df)(~,o,..., ,,,,1 = ~ (-a)',/~" f(~,o,..., 5,~,..., ~',) i=0

+ E (-1)P+qf([Yp,

p

% ) = ~ (-1)i[yi, f ( ) ' o , . . . , 4/i,..-, )'n)] i=O + ~ (--1)P+qf([)'p,)'q]') ' 0 , ' ' ' , ~ / p , ' ' ' , ~ / q ' ' ' ' , )'n)" p T M (denoted vdf on p. 269) inverse of the isomorphism T M-+ T * M defined by Y~.

Poisson bracket

186


Let ( O ~ ) be the matrix of components of O in local coordinates and let O ~ be the elements of its inverse. Then

(i o12),~ = v'~O~ ,

1"1 = 89

dx '~ ^ dx ~ .

Thus if 0 is a 1-form

(,,o)

=

and an easy calculation gives

{f, g} = O'~O,~fO~g. By the Darboux theorem (p. 282) there always exists a local coordinate system (x ~, y j) in which the symplectic form O reads

= ~ d y~ A dx i" i=l

the corresponding Poisson bracket is then {f, g } = ~ ( O f

Og

ox' y,

hamiltonian vectO~ fields

Of Og )

ylox'"

(2.2)

It is obvious that this Poisson bracket is bilinear and antisymmetric. It is clear, for instance, in expression (2.2), that it satisfies a), b) and c). If ( M , / 2 ) is a symplectic manifold the hamiltonian vector fields are defined as vector fields v such that (p. 269) ioD is an exact differential the hamiltonian vector field v g associated to the hamiltonian function H is such that

ioHO = - d H ,

(2.3)

Vn = - ~ d H .

(2.4)

also written

Then, for all f E C=(M, R) we have

OHf = {H, f } as can easily be checked, for instance, in local coordinates since

VHf=-12~'~O~HO,~f = {H, f } . ~'(M, O)

Casimir functions

Thus the two definitions of the space ~ ' ( M , / } ) of hamiltonian vector fields coincide. The kernel of the homomorphism of Lie algebras C| R)---->~ ( M , ~ ) , called a set of Casimir functions, is in this case the set of functions with a vanishing gradient, i.e., the constant functions if M is connected: we can

187

11. POISSON MANIFOLDS I express this fact by saying that O--->a --> C~176 a ) ---~ ~ ( M, 0 ) ---~0

is a short exact sequence (cf. Problem IV 6, Short). 3. DUAL OF A LIE ALGEBRA Let ~ be the Lie algebra of a Lie group with Lie bracket [, ]. Let ~* be the dual o f f~. Let f and g belong to Coo(f~*, R). a) Show that one can define a Poisson structure { , } on c~,, by denoting ( , ) the duality between (#* and c~ and setting

{f, g}(x) = (x, [dr(x), dg(x)]) ,

xE~*.

(3.1)

Give the expression o f the Poisson bracket in coordinates. Show that there exists an homomorphism from the Lie algebra c~ into the Lie algebra o f hamiltonian vector fields. b) A bilinear antisymmetric mapping ~9" q3 • ~---~ R is called a cocycle [cf. Problem IV 7, Cohomology eq. (6)] if

~9(X, [X, Z]) + O(Y, [Z, X]) + ~9(Z, [X, Y]) = O,

VX, Y , Z ~

(g.

Let 19 be a cocycle on ~. Show that one can define a Poisson structure { , } o on ~* by setting

{f, g}o(x) = ( x , [df(x), dg(x)]) - ~9(df(x), dg(x)).

(3.2)

Answer 3a: If f ~ Coo(~*, R), then dr(x) is a 1-form on T x ~*, identified with ~* since ~* is a vector space. Thus dr(x) can be identified with an element of ~3, as well as dg(x) and therefore also [df(x), dg(x)]. Formula (3.1) is meaningful. The mapping so defined is obviously bilinear and antisymmetric, and obeys the Leibniz rule. It can be checked to satisfy the Jacobi identity by i using this identity for the Lie bracket as follows" let c jk be the structure constants of ~3, and x i be the components of x in a basis of ~*:

i Of Og [df(x), dg(x)] = c jk OXj OXk ' thus {f' g}-

of

Og

xicijk OXj OXk "

Let X be an element of q3. We associate to X the C = linear function on ~* defined by H(x)=(x,X)

thus

dH=X,

cocycle

188


that is, in dual basis of = ( A d g X , x > ,

x E ~*.

We have seen, on the other hand, that the symplectic leaves are trajectories of hamiltonian vector fields. In the case that we consider these vector fields are Vi.l.xh = (x, [dH(x), dh(x)]> ,

h ~ C~

R)

and can be proved to be invariant under the coadjoint action of G on ~d* (for details see [6 or 7]). A n s w e r 9b" The tangent space TeG w

to G w at the unit

1

is

0

spanned by the matrices dg(t) [,=o , where g', g2, g3 are arbitrary functions of t, i.e., by the 3 • 3 matrices of the form: 0 X 1 X 3) 0 0 X2 9 S 0 0 0 The Lie algebra qd can be represented by these matrices; a basis of q3 is . _ _

(01!)

E,=P= 0

0

0

0

,

(00i)

E2=Q=

0

0

0

0

,

E3=R=

(00i) 0

0

0

0

.

The Lie brackets for this basis are [P, Q] = P Q - Q P = R ,

[P, R] = [Q, R] = O .

(0 0i)

The dual ~d* of ~d can be represented as the space of transposed matrices" X--

X1

0

x2

x2

with the duality between q3 and ~* given by

<X, X> "- Six1%- g2x2 + X3x3 . *There is also another definition which can be found in [Problem V bis 12, Virasoro].


196

(11 ,()(0

The adjoint action of the linear group G w on ~ is Adg X =

gXg-

o x O 0

1

o 0

o 0

1/ 0

0/\0

1

2X1 7223 -4- g l X 2 ) . X1 - g 0 0 0 The coadjoint action on qd~, is such that =

i

(x, Adg X ) = ( A d ; x, X ) , therefore

( 0

0 i) .

Ad~ x = xl - g2x3 0 x3 x2 + glx 3 The Poisson structure on ~d~, is

{f, h } ( x ) = (x, [df(x), dh(x)]) , dr(x) = (af/axi) dx i and dh(x) = (• dx i are identified with elements X and Y of qd, and [X, y]k = C~kj X iyj with Cjik the structure constants of ~w; i.e.,

[X, Y ] ' = [X, y ] 2 = O,

[X, y ] 3 = X , y 2 _ X 2 y 1 .

thus,

Of ah

{ f , h } ( x ) = x 3 0x 1 0x 2

Of Oh )

ax2 axl

.

The Poisson structure on q3~ has rank 2, except on the plane x 3 = 0. Each other plane x 3 =constant is a symplectic leaf, with symplectic form x 3 d x l ^ dx 2. These leaves are the Kirillov orbits under the coadjoint action of G on ~*, as can be seen on the expression of Adg. 10. LIE ALGEBRA OF OBSERVABLES

Let ( M , / 2 ) be a symplectic manifold, and G be a connected Lie group which acts on M by symplectomorphisms, i.e., diffeomorphisrns leaving invariant the symplectic form 12. a) Show that the Killing vector fields of this group action are locally hamiltonian vector fields (i.e., such that i o / 2 - 0, cf. p. 269).

197

11. POISSON MANIFOLDS I

b) Suppose that the group G of symplectomorphisms acts effectively on M. Show that there is an isomorphism between the Lie algebra of Killing vector fields and the Lie algebra fg of G. c) Suppose that the Killing vector fields of the action of G are (globally) hamiltonian" it is always the case if all closed 1-forms on M are exact, i.e., the cohomology vector space (Problem IV 1, cohomology) Ha(M, R) = O. To A fi ~ associate the hamiltonian vector field VA, image of A by the isomorphism q3---~(Killing vector fields). Since v m is (globally) hamiltonian there exists a function H A E C~176 R) such that ioaO = - d H A . H A

is the hamiltonian of DA = VHA and we have [VA, VB] = V[A,B ]

by the isomorphisms qd--> Y~"

and, by previous results OH[A,B] -- [VA , VB] = V {HA, HB }

.

But these equalities imply only {HA, HB } = H[A,B] + z ( A , B)

with z ( A , B) some constant. See, for instance, [Problem IV 6, Short w Show that z" ~ x ~---->R defined by (A, B)----> z ( A , B) is a 2-cocycle on (g, that is an antisymmetric bilinear map such that z ( A , [B, C]) + z(B, [C,

AI) +

z(C, [A, B]) = O.

Show that if H A are other hamiltonian functions corresponding to the Killing vectors v a and z ' ( A , B) = {HA, H'B} -

i

H[A,B ]

then z and z' differ by a 2-coboundary, i.e., a mapping c~ x ~--~ R of the form z ( A , B ) - z ' ( A , B ) = ( d, [A, B]) with d E ~* and ( , ) the duality. Show that if the second cohomology class defined by z is z e r o - in particular if H2(C~, R ) - - - 0 - t h e n there exists an isomorphism P: A---> PA from c~ onto the Lie algebra C~176 R) with the Poisson bracket ( , ) . Answer 10a" A locally hamiltonian vector field v (called hamiltonian on p. 269) is such that diog2

-

O,

198


equivalent to, since O is closed 3?o0 = 0 . Therefore the flow of a locally hamiltonian vector field is a 1-parameter group of symplectomorphisms of (M, O). Conversely a Killing vector field v of a group G of symplectomorphism of M is such that ~o O = 0, thus dioO = 0, and v is locally hamiltonian. A n s w e r 10b: This result is true in all cases of the Killing vector fields of an action of a Lie group G on a manifold M (p. 163). A n s w e r 10c: z ( A , B ) is linear in A and B, since A--~ H A is a linear map. It is antisymmetric and satisfies the cocycle identity because of the Jacobi identity satisfied by [A, B] and {H A, H B}. If H A and H~t are hamiltonian functions for v a then d(H A -

HA) = O,

thus HA-HA=C

A e R.

The application ~3--->R by A---~ C A is linear. Therefore there exists C E ~* such that CA= PA = H~4 is an isomorphism of Lie algebras. This mapping is sometimes called a momen-

11. POISSON MANIFOLDS I

199

turn map. The functions PA are taken as a set of observables in the quantization on (M, O) (see [Isham]). They are determined up to addition of a linear form (d, A) with (d, [A, B]) = 0 VA, B ~ ~d. These elements are 1-coboundaries, the arbitrariness of the observables PA is therefore classified by the elements of H I(~, R).

Remark: If G is a semi-simple Lie group then both H2(~J,R) and H~(~, R) vanish (cf. Jacobson, Lie algebras, Interscience, pp. 93-96). REFERENCES [1] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sterneiheimer, "Deformation theory and quantization", Ann. Physics 111 (1978) I pp. 61-111, II pp. 111-152. [2] A. Weinstein, "The local structure of Poisson manifolds", J. Diff. Geom 18 (1983) pp. 523-557. [3] B. Kostant, Lecture Notes in Maths. 170 (1970) 87. [4] C.J. Isham, in RelativitY, Groupes et Topologie II, B. DeWitt and R. Stora, eds. (North-Holland, Amsterdam, 1984). [5] A. Lichnerowicz, "Les vari6t6s de Jacobi et leurs alg6bres de Lie associ6es", J. Maths Pures et appl. 57, (1978) pp. 453-488. (with F. Gu6rida) "G6om6trie des alg/~bres de Lie de Kirillov", J. Maths Pures et appl. 63, (1984) pp. 407-484. [6] P. Libermann and C.M. Marie, "G6om6trie symplectique et bases th6oriques de la m6canique", Pub. de l'Universit6 Paris VII [English translation" Reidel, Dordrecht]. [7] R. Abraham and J.E. Marsden, "Foundations of mechanics" (Benjamin, New York (2nd edition), 1981). [8] J.M. Souriau, "Structure des Syst/~mes Dynamiques" (Dunod, Pads, 1970). [91 A.A. Kirillov, Russian Math. Surveys 31: A (1976) 55. [101 A.A. KiriUov, "Elements of the theory of representations" (Springer, Berlin, 1976) Further reading on physical applications and the infinite dimensional case: V. Guillemin and S. Sternberg, "Symplectic techniques in physics", (Cambridge University Press, 1984). C.S. Gardner, Jour. Math. Phys. 12 (1971) 1548-1551. J.E. Marsden and Ph.J. Morrison, "Non-canonical hamiltonian field theory and reduced MHD", Contemp. Math. 28 (1984) pp. 133-150. D.D. Holm, J. Marsden, T. Ratiu and A. Weinstein, "Nonlinear stability of fluid and plasma equilibria", Phys. Rep. 123 (1985) 1. Ph. J. Morrison, Phys. Lett. 80A (1980) 383. J.E. Marsden and A. Weinstein, Physica 4D (1982) 394. G. Segal, "Unitary representations of some Infinite Dimensional Groups", Comm. Math. Phys. 80 (1981) 301-342. D.D. Holm, "Hamiltonian systems", in "Relativistic fluids", C.I.M.E. (1987) M. Anile et Y. Choquet-Bruhat, eds.


200

12. POISSON M A N I F O L D S II* INTRODUCTION

This Problem gives further properties of Poisson manifolds. A first set of questions concerns the local structure of a Poisson manifold, the transverse Poisson structure to its symplectic leaves and the characterization of the latter in the linear case, ~d*. The second set of questions involves the notions of Realization of a Poission structure, Function groups on a symplectic manifold and Dual Pairs. The first two notions are reciprocal in some sense, and the third is a very important by-product of them. Hereafter only finite dimensional Poisson manifolds will be considered, although similar notions can be defined in an infinite dimensional context. See papers [1, 2], which bring out the importance of this notions in applications. This problem closely follows Weinstein [3]. Theorems related to those involved in this problem have been stated by Lichnerowicz in the general framework of Jacobi manifolds [4, 5]. 1. POISSON MAPS AND POISSON SUBMANIFOLDS

Definition. Let M 1, M 2 be two Poisson manifolds, and let U be an open set in M 1. A differentiable map ~o" U--* M 2 satisfying {Fo ~; Go ~}M,(m) = {F; G}M2(~(m)), local Poisson map

VF, G U C~(M2), Vm E U, is called a local Poissan map. 1) Let H be a C=(M1) function, let VH be the corresponding hamiltonian field, and let ~t be the local 1-parameter group of local diffeomorphisms generated by v H. Prove by straightforward calculations that ~t is a local Poisson map (local Poisson diffeomorphism).

Answer: The following equality holds locally

(VH{F; G})(~t(m))= {VHF; G}(~Pt(m))+ { F ; VHG)(~t(m)). Using d

(VHF)(~Pt(m)) = -~S F(~s(~t(m)))ls---~ we easily obtain d d-~ ({F; G } ( r *Contributed by C. Moreno.

{Fo(~t; Gor

,

Vt.

II

12. POISSON MANIFOLDS II

201

2) I f J: M 1--->M 2 is a surjective Poisson map, and if H E C~176 prove that the trajectories of the vector field Vlr are the images by J o f the trajectories o f the vector field VHoJ.

Answer: Letting C: I--~ M 1 be an integral curve of VHoj, we consider the map j o C: I--~ M 2. If F ~ C~176 we have d (F (J dt

- -

0

0

C))(t)

---

d -~t(F J)(C(t)) 0

~

--

{F J ; H O J } M I ( C ( t ) ) 0

= - {F; H } M2((J~ C ) ( t ) ) . I---~ M 2 is therefore an integral curve for VH and any integral curve of v H is of this form, since J is surjective. II

j o C:

Definition. Let (Mi; A;), i = 1,2 be Poisson manifolds, and suppose Ma is a submanifold of M 2. We will say that (Mi; Am) is a Poisson submanifold of (M2; A2) if the inclusion i" M~ ~ M 2 is a Poisson map. 3) Let (ml; Am) be a Poisson sub manifold of (M2; A2). Prove that A~ is unique and rgAa(m) = rgAzi(m), V m E M~.

Answer" Vf, g E C~

we have

A2i(m)((df)i(m) ; (dg)i(m))= A,(m)(d(f~ i)m ; d( go i)m) " T * M 1 is generated by d ( f o i),,, f ~ C~(M2). A 1 is therefore unique. Let A2~(~~" Ti(m)M2 ---~ Ti(m)M2, Alm" T*M~---~ TraM ~ be defined as in Problem IV 11, question 4, and let T,,.i" TraM ~--. T,m>M 2 be the tangent map to i at m. We thereby obtain ~'

~

o

Im A2i(m ) -- Tmi ~ Im A 1( m ) . As Tmi is injective, rgAlm--rgA2i(m ) .

II

4) Let M 1 be a submanifold o f the Poisson manifold (M2; A2). Prove that Poisson structures A 1 such that (M1; A1) is not a Poisson submanifold o f (M2; A2) can exist on M 1.

Answer: For example, let (M2, 0)2) be a symplectic manifold, and let M a be a submanifold of M2, i: M 1 ~ M 2, such that i , w 2 - o)1 is not degenerate. Thus, (M1; Wl) is a symplectic manifold. It is a Poisson submanifold of (m2; w2) if and only if dim M 1 = dim M2, but this is not always the case for such a M 1. i

Poisson submanifold

202


ON MANIFOLDS

5) Prove that (M1; A1) is a Poisson submanifold Im A2i(,,) C Tmi. TmMa; i" M1--9 M 2 inclusion.

o f ( M 2 ;

N

A2) if and only if

Answer: Let ( x ~ , . . . , x"' ; y~,

. . . , y n2) be local coordinates in M 2 such that the local equation of M 1 is y ' = 0; i = 1 , . . . , n 2. (X 1, . . . , x n~) are therefore local coordinates in M~. Let F, G be functions of the local coordinates (x k) and let F, G be functions of the local coordinates (x k" y J) such that P = F o i" G = G o i If (M~; A1) is a Poisson submanifold of (ME; A2), we get ~

9

{F; G}M ~ = i , { F ; G}M 2 . The left-hand side depends only on x k. Therefore, the right-hand side does not contain terms of the form OF

.

OG

O-~yk)(x;o)'

(x,0)

Thus {F; G}m2(x, 0 ) = ~'~ A2(~;o) (vr)'x;~ = ~'~ az(x;~

~

(OF)(x;O)(

aG

(x;O)

(x;0)

v r is therefore tangent to M 1. Conversely, if every hamiltonian vector field on M 2 is tangent to M 1, we have (x; 0) '

and then

{ xk" Y J} M2 = { Y', Y J} M2 = 0

,

Only the components A ab(X" < a, b < n 1 are therefore nonzero, and 2 ~0) ~ 1 ~ thus A2(x; O)= Alx. A look at the local expression of the Schouten-Nijenjhuis bracket (Problem IV 11, question 4b) will enable us to write [A2, A2](x. y)= 0=> [Az; A2](x;0) = 0r

A,] x = 0 .

(M1; A1) is therefore a Poisson submanifold of (ME, A2). 2.

INDUCING

A POISSON

STRUCTURE

ON

A SUBMANIFOLD

II

M 1 OF A POISSON

M A N I F O L D (M2; A2)

If M 1 and (M2; A2) satisfy certain relative conditions, it is possible to define a Poisson structure induced by (M2; A2) on M 1. With this induced


203

Poisson structure G, if dim M 1 ~ dim M2, Poisson manifold (M1; G) will not be a Poisson submanifold of (ME, A2). 1) Let ( TxM 1)0 C T* M 2 be the orthogonal space o f T~M 1 in T * M 2. Prove that if (TxM1) ~ n ker Ax = {0}.

(1)

then ~

TxM 1 + I m A x = TxM 2 . ~

Consequently, seeing that Im A x = TxS (Problem IV 11, question 7) if S is a symplectic leaf o f M 2, we have TxM 1 + TxS = TxM 2 . M 1 and S are therefore transverse submanifolds at every point o f the set M 1 n S, which is then a submanifold o f M 2. Answer: Obviously, (ker Ax) ~ Ax . The hypothesis is then equivalent to ( T x M 1 ) ~ (Ira Ax) ~ {0}. From linear algebra, we have (TxM~) ~ n (Ira Ax) ~ = ( T x M ~ + I m Ax) ~ ,

and ({0}) ~

TxM 2. We thereby obtain (1).

I

Remark" What has been proved is actually the following equivalence (TxM~) ~ n ker Ax = {0} r

TxM 1 + Im Ax = T, M 2 .

2) Space TxM 1 n A x T * M 2 is then the tangent space at x to the submanifold M 1 n S. Let 0 be the 2-form induced on by the symplectic f o r m o f M 1 n S. Prove the following equality

ker O x = A~((TxM1) ~ O TxM 1 .

(2)

Answer: Let X~ be a vector in Ax((TxM1) ~ N TxM 1. Then X x E T x M 1, and there is some tOg E ( T x M 1 ) ~ T*xM 2 such that Xx = A~tOx. We thus

have ~

~

a x ( g x ; TxM 1 n AxT*xM2) = tOx[Ax(T*xM2) n L M 1 ] = 0.

The~ inclusion _~ is therefore proved. Conversely, if X x ~ TxM 1 n A x T * M 2 and X~ E ker ~ , the latter equality is satisfied by some ~o~ E [Ax(T~M2) n T~M~]~ Therefore X~ = A~o~ E .,i~(LM~) ~ Consequently X x E TxM 1 n Ax(TM1) ~ The inclusion C is thereby also proved. I ~

transverse submanifolds

204


3) Let us n o w suppose that equality (1) is satisfied, and that in equality (2), ker 12x = {0}, i.e., (TxM1) ~ N ker Ax = {0}

(1') VX (~ M 1 .

Ax(TxM1) ~ N TxM 1 = {0)

(2')

Prove that, in this case, Ax(TxM~) ~ ~ T x M 1 = TxM 2 .

(3)

Answer: Let w~, w x be two vectors in (TxM1) ~ such that Ax ~~ = Axwx 9As a consequence of (1'), ~o~ = w'. Therefore dim A~(T~MI) ~ dim(TxM1) ~ giving us

dim A~ (T~M a)~ + dim T x M ~ = dim T~M z . This equality and (2) imply (3).

II

We will suppose equalities (1'), (2') (and therefore equality (3)). Let T M M 2 be the restricted bundle to M 1 of the vector bundle T M 2. Equation (3) yields the splitting of vector bundles TM1M 2 = ~ ( T M 1)o ~[~ TMI ,

and the surjective morphism of vector bundles 7r: TMIM 2 ~ T M 1 .

Let ~r*" T * M 1 sequence

T ~ I M 2 be the dual morphism of zr, and consider the

r* 9 T M1M2 , ~.~ TM1M 2_~ T M 1 T* M 1--->

Let us define a bundle morphism, G" T*M~--o TM~ as G =_~roAoTr*. ' ~ T*M~, we have (o~'M~; GO.}M~) = ( ~r* wM~ With WM~, ~ M~ ' ; A ~ * WM1) = A ( T r * WM~, " 7r* a~M~ ' ) . An antisymmetric contravariant 2-tensor G is therefore defined by the relation G(WMa; w M1) ' = A(zr * WM,," 7r * ~OM, ' ) Poisson induced structure

(4)

Obviously [G; G] = 0. G is therefore a Poisson structure on M 1, called the induced structure by A, when M 1 and A satisfy relations (1') and (2'). The symplectic leaves of (M1; G) are the symplectic manifolds M 1 A S x, x E M1; where S x is the symplectic leaf of M 2 through x; M 1 = t_J geM1 (M~ A Sx), and question 3 merely put these symplectic structures together in a contravariant way.

205

12. POISSON M A N I F O L D S II

3.

LOCAL STRUCTURE OF A POISSON MANIFOLD

Definition. Let (Mi; Ai) , i = 1,2 be two Poisson manifolds. On the manifold M 1 x M2, the 2-tensor A = A 1 9 A 2 defines a Poisson structure. The Poisson manifold (M; A) will be called the product Poisson manifold of (Mi, Ai) , i = 1, 2. 1) Prove that the foregoing Poisson structure A is characterized by the following two properties a) Projections "tri: M 1 • M 2 ~ M i are Poisson maps. b) q'gi*C~176 i= 1, 2 are commuting Lie subalgebras o f (C=(M) ",A). Answer: Let (xi; ya) be local coordinates on M adapted to the product M 1 x M 2. We thus have o

{ Xi" X]} M = A liix 9

{ ya, y b } M = A2by "

{X i. ya} M = 0

The result follows.

I!

2) The aim of this question is to prove the following theorem about the local structure of a Poisson manifold. Let x o be an arbitrary point o f a Poisson manifold (M; A). There will be

an open neighborhood U o f point Xo, a product Poisson manifold S x N, (where S is a symplectic manifold with the same rank as Ago, and N is a Poisson manifold), and a diffeomorphism 4) = dPs x dpN: U ~ S x N such that the rank of N at CN(X0) is zero. Moreover S and N are unique up to local isomorphisms. Answer" Let us suppose ran Axo#O. There is a P ~ C = ( U ) such that (re)go ~ O. By "straightening out" re, we can find a neighborhood of x 0 and a function Q E C=(U) such that v v ( Q ) = ( P ; Q} = 1. Therefore [Vp; re] = O. According to Froebenius's theorem, ve and vo generate an integrable sub-bundle of TM. This means, in particular, that if we write p = P(x) and q = Q ( x ) , there are local functions ( y 2 , . . . , ym), m - d i m M, such that ( p , q, Y2,. 9 9, Ym) is a set of local coordinates x 0. In these coordinates, r e = O/Oq; v o = - O / O p , and the matrix of A is

(0 1

)

0

G ( p ; q; y) where G

ab(p; q; y) = { yb., y, }

product Poisson manifold


206 Note that

OGab(P; q; Y) = {p; Gab(p; q; y)} = {p; {yb. ya}} = 0 aq

,

~

(Jacobi)

.

Gab(y) is thus independent of both p and q; and obviously [G; G] - 0. G therefore defines a Poisson structure of rank r g A - 2 on a manifold of dimension m - 2. We see that, by iterating this procedure, we can define an open neighborhood U of x 0, a symplectic manifold S through x 0 which is isomorphic to a symplectic open of R 2", rgAx0 = 2n, a Poisson submanifold N through x 0, rgNxo = 0, which is diffeomorphic to an open set in R m-En, and a Poisson diffeomorphism t h - ~bs x ~bN satisfying the conclusions in the question. This proves existence. Unicity is proved when we observe that if S 1 x N 1 = S 2 x N2, and (Si; Ni), i = 1, 2 has the properties desired in the question, then S 1 and S 2 are locally isomorphic because they are of the same dimension. Manifolds N~ and S~ intersect transversely and only at a point ~b~(x0); a lemma from Weinstein [3], p. 531 proves that there is an automorphism of (M; A) such that the image of any neighborhood of ~bl(x0) in N1 by this automorphism is another (Poisson isomorphic) neighborhood of t~E(X0) in N 2. II A natural representative for S is the symplectic leaf through x 0, but there is no natural representative for N. Weinstein's lemma nevertheless_ applies to any submanifold N which is transverse to S and such that N N S is a single point. The transverse Poisson structure to S is therefore properly defined as the equivalent class of the Poisson manifold (N; G). 3) Show that the Poisson structure of the submanifold N of M is the

Poisson structure induced by the method developed in question 3 of the previous section. Answer: We need to prove that equalities (1'), (2') also hold in this case. (TxN) ~ N ker A~ = {0}( r

TxN + Im A~ = T~M),

A.(T.N) ~ (q T.N = {0} ;

Vx ~ N .

Using the notations in question 1. we have

T . N = lin{a/ay ~} ;

(T.N) ~ = lin{dq'; dp,}

and

A-.(T.N) ~ lin{a/aq'; a/ap~} = T.S ;

x E U.

These relations show equalities (1') and (2'). The Poisson structure induced on N is thus defined by

(1') (2')

12. P O I S S O N M A N I F O L D S II

T*N--~ T~vM

TNM

207

TN,

dy a __~dy a __, ~ Gab ab b, (y)O/t~y b---) ~ G(y)a/tgy

(see question 3 of previous section). We can see that the Poisson structures on N coincides with the one obtained in question 1). i Consequently, every symplectic leaf of M in the neighborhood of a given symplectic leaf S is locally the Poisson product of S and a symplectic leaf of the Poisson manifold N. 4.

TRANSVERSE POISSON STRUCTURE TO THE COADJOINT ORBITS IN c~,

1) Let G' and G be connected Lie groups, each with Lie algebra ~. Let us suppose G' is simply connected. Show that the coadjoint orbits of G and G' coincide. Answer: G is a quotient of G' by a discrete subgroup in the center of G'. The coadjoint representation on this subgroup is trivial. Thus, the result holds, m

2) Let X be a vector in ~; the corresponding generator in the adjoint (coad]oint) representation written X ~ ( X ~ . ) , is defined by d X ~ ( Y ) = ~ Ad(exp tx). rlt=o ;

d X~.(/x) = ~-] A d * ( e x p - tX)./xlt= o ;

Y ~ ~, /x E ~*

Show that: X ~ ( Y ) = ad X . Y,

X~,(/z) = - a d * X . / x . Answer: Straightforward.

i

3) In Problem IV 11, question 9, we proved that the coadjoint orbits of G on ~d* are the symplectic leaves of the canonical Poisson structure of fg*. The purpose of the question is to identify the transverse Poisson structures to the orbits in ~*. Suppose/z E ~3", and let G , C G be the closed (Lie) subgroup defined by A d * G ~ . / z =/x. Let ~d~ C ~d be the Lie algebra of G~, and ~ , the dual space of ~d,. 3i) As in Problem IV 11, question 3), we let T~ ~d* and ~* be identical

208


ON MANIFOLDS

with T:~3 * and ~3. Consequently, the Poisson tensor A on ~3" defines a map A+," ~3--->~3"

X--> A~,X by

(A~,X; Y) = ( / z ; [ X ; Y]) = - ( a d * X . / z ; Y ) .

(1)

ker A~ = { X E ~1 ad* X - / z = 0}.

(2)

Therefore

From (1) and (2), we obtain ker A~, = ~3+, .

(3)

3ii) The tangent space at ~ to the orbit Ad* G . / z can be identified as T~,(Ad* G ' / z ) = {ad* X ' / z l X E q3} ; clearly o T~,(Ad* G . / z ) C q3~,,

where ~3~ is the subspace in q3* which is orthogonal to q3~,(in the duality). But dim q3~, = dim T~,(Ad* G . / z ) and therefore o

T~,(Ad* G . / z ) = ~ . . o

3iii) Let V be a subspace in ~* supplementary to ~3+,, i.e., 0 ~3" = ~3~,~V.

(4)

The affine submanifold /z + V therefore intersects the orbit A d * G . t z transversely, only at the point/z. As shown in section 3, question 2 the submanifold /z + V carries the transverse Poisson structure to the orbit. We now want to identify the linear approximation to this structure. We must now return to section 2, question 2. In a neighborhood of/z we found local coordinates on M, (p;, q;, ya). In the case under consideration, manifold N is an open neighborhood of/z in/z + V and S is an open neighborhood o f / z in Ad* G . / z . If

i: N C Iz + V---> ~*=- M is the inclusion, functions Ya o i are local coordinates on N. The sequence (1) in section 2, question 3 will be written with the following notations

T*(la, + V)

T~,+v

+v

--,

+ V).


d( y,, o i)---~ dy, --->~ Gab(y) -~yb -..->s G ab( y)

209

aYb

Here (dYa)~,+a , tz + A (E N annuls every tangent vector to the manifold ya = ya(A + /s a = 1 , . . . , dim V at the point (A + / z ) . If h = 0, the submanifold ya = ya(/x); a = 1 , . . . , dim V is on the orbit Ad* G . / ~ , (dy~)~, 0 annuls every vector in T~, ( A d * G 9 = ~d~,. But if A ~ 0 the tangent space to the submanifold y ~ = ym(A + t z ) ; a = 1 , . . , dim V at (a + / z ) is not in 0 general equal to ~d~,. Therefore (dYa) x +~, E ~ = ~ , (~ V ~ and (dYa)a+~, = x0 + Xl ;

X0 (~ ('~/z ; Xl ~ V0"

The transverse Poisson structure at tx is defined by (see section 2, question 3, (4))" G , + a ( d ( y a o i); d(ybo i)) = A~+x((dYa)~,+, ; (dyb)~,+,) = (ad*(dya)~,+a(~ + A); (dYb)~,+,) . From the above sequence, we get ad*(dy,)~,+a(/x + A)E T~,+,(/., + V)-= V . If we now write (dyb)~,+, = Yo + Yl; Yo E ~d~,; Yl E V ~ we obtain

G~+a(d(ya~176

= (Iz + A;[x o + Xl; Yo])

= (~; Ix0; yo]) + (~z; [xa;yo]) + (x; [Xo; yo]) + (a; [x~; yo])

0//

%0

= ( a; [Xo; Yo]) + (a; [x~, Yol)We have ~d*= ~ ,o G V. Therefore , V ~ cg,/q3o , and the following map ;

A E cg,/ca0

is a bijective map and an isomorphism, when the affine s p a c e / x + V ~ /x + ~*/q3 ~ carries the canonical vector structure with point /x as zero vector. We then obtain /x + V ~ ~ ,* ~ V consequently, the term (A; [Xo; Yo]) in the expression G~,+x is precisely the canonical Poisson bracket on ~d~, because A E V ~ ~d~,. If, following Weinstein [3] we now consider the linear approximation to

210


Poisson structure at a point where this structure is of rank zero, we see that (A; [x0; Y0]) is precisely the linear approximation to G,+~ at the point/x. We thereby obtain the Theorem" "The linear approximation to the transverse Poisson structure at/z at the orbit Ad* G . / z is isomorphic to the canonical Poisson structure of ~ * " [3 9] The point/z ~ ~* is regular if rank A, is maximal. In the neighborhood of/x, the rank of A is also maximal. The symplectic leaves of ~* in this neighborhood have maximal dimensions, and the rank of the transverse Poisson structure at all points of this neighborhood is therefore zero. Thus G~, +a is zero on N; which implies that the linear approximation at/z is also zero, i.e., ( A; [x0; Y0]) = 0; A E N; x0, Y0 E ~ . It is therefore zero for every A E ~ (N is open) and ~d~ must be abelian; (regular/z ~ ~, abelian). If point/x is such that [~,; V ~ C V ~ we have ( Al[Xl, Y0]) = 0; and at these points the transverse Poisson structure to the orbit Ad* G . / x is therefore itself isomorphic to that of ~d~,. 9 However this cannot be the general case, because, if it were true that, for every /x, the transverse Poisson structure at/~ was isomorphic to ~ , ,9 all points/x, at which ~, is abelian, would be regular. But we know that, if ~3 is an arbitrary Lie algebra, abelian ~, does not imply regular/x. However, abelian ~, does imply regular/x if ~ is semisimple ([10], section 1.11). /~,

,

9

5. REALIZATIONS OF POISSON MANIFOLDS AND FUNCTION GROUPS

(See the Introduction) (full) realization of a Poisson manifold

Definition. A realization of a Poisson manifold (M; A) is a pair (S;

J)

where S is a symplectic manifold and J" S--> M is a Poisson map. The realization is full if J is a submersion. I If a realization is full, mapping J is locally surjective, and every hamiltonian flow on M has a convenient form in the coordinates of S (Clebsch variables). See [1, 2]. Realizations require consideration not only for their theoretical interest, but for their many useful applications as well.

1) Let J: S--->M be a realization of (M, A). Show that J* C=(M) is a Lie subalgebra of C~ I Definition. Let U C ~2n be a symplectic open subset of II~2n. Let ~i, i = 1,...,r be independent elements of C~ i.e., d~b1 ^ . . - ^ d~br(m ) # 0. Consider the map ~b" ~b1 x . . . x ~br- U--~ R r, and let V C R r be the image of U by th. ( V - I m ~b.) Let us define the set ~ = {Fo 4~ ~ C~176

~

C~176

9

211


If ff is a Lie subalgebra of Coo(U), we will call ~ a local function group generated by (q~l,. 9 9, thr). Function groups are useful on account of the following property, and particularly its converse, stated below.

local function group

2) Show that a local function group ~ canonically defines a Poisson structure on V.

Answer: Functions ~bi are in o%, because ~b~ = 1ri o ~b where ,r i E C~176 is the ith projection. Thus, { ~bi, ~bj}v ~ "~. Hence, there are Wij ~ C~176 such that

(~i') ~j}U ~--- W/j ~ (~1 =

W/j(r

X-''

X ~r)"

From Jacobi identity on U, we obtain the property that the antisymmetric contravariant 2-tensor with coordinate Wij in the basis (O/Odpi)~ of ToV satisfies [W; W]v = O. I Conversely, every Poisson manifold has a full realization locally and is therefore defined by a local function group. See [3]. 3) Definition. A global function group on a symplectic manifold S is a foliation q~ of S such that the set of leaves S / ~ is a manifold, and if It" S---~ S / ~ is the canonical projection, the space ~r*Coo(S/~) is a Lie subalgebra of C~176 3i) The following bracket on Coo(S/~) obviously defines a Poisson structure on S/q0" {F; G}s/~ o ~r = { F o r r ; Gorr}s ;

F, G~_ C = ( S / ~ )

(cf. question 2)), and ~'" S---~ S / ~ is a Poisson map. 3ii) Let J" S ~ M be a realization of M, and let us suppose that the fibres J - l ( m ) , m E J(s) define a foliation q0 of S in such a way that S / ~ is a manifold. The space ,r* C~176 of functions on S which are constant on the fibres of J, coincides with the space J* C~176 Since the latter space is a Lie subalgebra, so is 7r* Coo(S/@). therefore (S; q0) is a global function group. 3iii) We will see in this question that each global function group on S,

written ~ = 7r*Coo(S/clg), is in canonical association with another global function group (called its polar) end that this correspondence is involutive. Let Tq~ be the subbundle of the tangent bundle TS whose fibre at point x E S is the subspace Txq~ C TxS, of vectors tangent at x to the leaf through x, q~x. Let T x ~ • be the subspace of TxS, whose vectors are

global function group

212


orthogonal symplectic vectors to T x q~, i.e.,

X x ~ Tx ~ • r

Tx~ ) =0.

Finally, let T ~ ~ be the sub-fibre bundle of TS with fibre T x x~S.

• at point

Prove that the Hamiltonian vector fields on S with Hamiltonian in ~ are sections of T ~ ~, and that T x 9 • is generated by the values of these sections of x. Answer: If F E ~ , and w E F T ~ is a section of Tq~, we have tO(OF; W) = w(F). But the trajectories of w are tangent to the leaves of q~, and F is constant on these leaves; therefore o F ~ F T ~ ~. Furthermore, if dim S = 2n, and if r is the dimension of the leaves of 9 , the dimension of T x ~ • is 2 n - r. But q~ is locally defined by 2 n - r functions F 1 , . . . , F2,_ r in ~ such that d F 1 ^ - - . ^ dF2,_r(x ) 4= 0. Hence hamiltonian vector fields Oei are linearly independent local sections of Tqb • and T x ~ • is generated by ((OF)x;i= 1 , . . . , 2 n - r). II Show that ~;~ is a Lie subalgebra o f C~(S) if and only if T ~ • is an integrable sub-bundle of TS. Answer: If F, G, {F; G}s E o%~, we have [or; vc] = - v{r;c} , where v F, v a, V{F;c} ~ F T ~ • T ~ ~ is thus an involutive sub-bundle, and therefore integrable. Let q~ ~ be the corresponding foliation, and let us suppose that the space of leaves defined by q~ • is a manifold S/c19• If ,~, is not a Lie subalgebra of C| Tq~ ~ cannot be an integrable subbundle because, if v{p;C}s~Fq ~ where F, G E o ~ we must have

{F;

polar

Consequently, since Tq~ is integrable, , ~ , l is a Lie subalgebra of C~(S), and ~r: S---~ S / ~ ; I r * C ~ ( S / ~ L) = ~ : ~ , is a function group on S. II The correspondence 9 ~ 9 • is obviously involutive. Functions groups on S exist by pairs, ~ and ~ ~, and we will say that each function group in a pair is the polar of the other. Show that ~ is the space of functions on S commuting with the functions in ~ . Therefore, ~ A o%~ is the (commun) center of the Lie subalgebras ~ and ~ , and hence the set of Casimir functions of the Poisson manifolds S / ~ and S/q~ •


213

6. DUAL PAIRS Let SJ---~ Mi, i = 1, 2 be two realizations of the symplectic manifold S. Let us suppose that define global function groups on S (cf. 3ii) question 5). Let ~*i = J*Coo(Mg) be the associated Lie subalgebras of C~176 (The function groups.) J2

J1

Definition. We will say that the diagram M 2 ~--S ~ M 1 is a dual pair if the function groups o~,, are polar to each other. If J~ is a surjective submersion, we will call the dual pair full. J2

J1

6i) To say that M 2 C| is a homomorphism of Lie algebras. Answer" Straightforward calculations.

II

2) In this question, let us suppose the action of G on S is free, i.e., if g . x = x for some x ~ S and some g E G, then g = e, e being the identity of G. We also suppose that the action of G has an equivariant momentum mapping J and defines a foliation q0 (the common dimension of the leaves is dim G, because the action of G is free) in such a way that the set of leaves is a manifold and H: S ~ S / G submersion.

Show that H*C| is a Lie subalgebra of C| and hence that S/G has a canonical Poisson structure such that 1I" S ~ S~ G is a Poisson map. Answer: If F, G ~ H*C| there are unique F, 0 ~ C~176 such that F = F - H ; G = G - H . For ~ ~ ~d let ~s = Xj(~) be the hamiltonian vector field with hamiltonian J(~:). We then have Xj(~){Po_ ~r; 0 o ~r} = 0; since the trajectories of Xj(e) are on leaves of q0 and _F~ t3_o / / a r e constant on each leaf. G is connected and acts freely {Fo H; G o II}s is therefore constant on the leaves of q0, consequently there is a unique function {F; G}s/o ~ C| such that

{po n ; Oo rt}s = (P; O}s, O rl. {F; G)s/G is obviously a Poisson bracket on S/G.

I

215


3) We have now defined a function group II*C~176 on the symplectic manifold S; the purpose of this question is to find the corresponding polar

function group. Let T~ q0 _L be the subspace of T~S which is o r t h o g o n a l - - s y m p l e c t i c to the subspace T~(G. s), itself tangent at s to the leaf through s of the function group II*COO(S/G). The subbundle Tq0 • is integrable because H*COO(S/G) is a Lie subalgebra of C~176 (see 5, question 33) ). Let q0 • be the corresponding foliation. 3i) If v s E T s 9 1, show that TsJ.v~ = 0 and conversely.

A ns wer: V~:~ ~ ;

~Os(( ~:,)~; v~) = d J( ~:)~" v~.

But dJ(~:)," v~ = (T~J" v,)" ~ . Hence

TsJ. vs = 0r

I

v~ E T~O • .

3ii) Show that the momentum mapping J takes the same value on the points of a connected leaf of 9 •

Answer: Let q0s• be a leaf through S, and let C" I--* q0 s• be an arc on that leaf. Then tE I ;

C'(t) = TtC. ( d-dt t

Therefore (3i)

t ~

rc(t) ~ ~S "

(d)

0 = Tc(t)J" C ' ( t ) = Tt(J~ C) ~

t"

(3ii) Show that the mapping J" S----> ~* is a Poisson submersion.

I

I

From this result, we see that j-1(/s ~ (~ J(S) is a submanifold of S and a union of connected leaves of q0 1 (see 3ii). We thus obtain the dual pair

*s The equivariance of J implies J - ~ ( A d * G 9,u,) = G 9J-~(,u,).

216


Hence H J - ' ( A d * G - / z ) - - H G . j-l(lj,)

,

And H J - ~(IX) is, by (6iii) a union of connected symplectic leaves of S / G. 3iv) I f S, S' ~ J-~(l~) and S' = gS for some g ~ G, show that, g E G,, where G,, C G is the subgroup defined by Ad* G , , . / z =/z, and conversely, G , acts o n 1-1(]/,). Answer: Straightforward

II

From this result, we see that H j - l ( / z ) ~ J - l ( t z ) / G t z . And therefore, J - l ( / z ) / G , carries the symplectic structure induced by this diffeomorphism. If F, H E C~176 we have As/c(dF; d H ) = tOs/6(VF; VH) , where tOs/G is the symplectic form on the leaves of S / G . Therefore As/a(dF; a l l ) = As(d(Fo H ) ; d(Ho H)) = w(vp. n ; VHoU) , where to is the symplectic form on S. Hence

v.)=

v..,,)

vr, n, VH,u are tangent to J-~(/x) and VF, Vg are their projected vector fields on J - ~(/z) / G~,. We thus see that the symplectic manifold J - 1(/x) / G~, coincides with the reduced phase space of the Marsden-Weinstein theorem. A particular case of the preceding one provides another example of a dual pair. It has been considered in [2] in an infinite dimensional context as well. Let G be a connected Lie Group and T * G its cotangent fibre bundle endowed with canonical symplectic structure. G acts freely and symplectically on the left and on the right on T* G. More precisely L" G x T'G----> T * G ( g; oth)---->(TghLg-~)* a h and R" T * G x G---~ T * G ( ah ; g) ~ ( ThgR g- , ) * an. Here

Lg and

Rg are respectively left and right translations on G.


217

Each of the foregoing actions has a momentum mapping. More precisely JL : T* G---~ ~* , a~"-->(TeRg)* % ,

and JR: T* G---~ qd*, ag-"> ( TeL g ) * Otg , i.e., a right translation to the fibre at e E G , T * G =- ~* for the left action e

and a left translation to the fibre for the right action. 4) Show the following equalities by straightforward calculations J L ( L ( g ; ah) ) = Ad* g-a" JL(ah) , JR(R(ah; g ) ) = Ad* g - l " Jr(ah) , i.e., m o m e n t u m mappings JL and JR are equivariant. Answer: Straightforward.

5) Let ~+* denote the canonical Poisson structure J, on ~* and ~*z ' its opposite. Show that the m o m e n t u m mappings T* G ~ ~g* and T* G--~ ~ * are surjective Poisson submersions. I 6) Show the following equivalences by straightforward calculations JL(ah) = JL(ah,)r

ah, = R(ah; h - l h ' ) ,

JR(ah) = JR(ah,)CZ~ah, = L ( h ' h - 1 ; ah).

I

Therefore, two points of T * G have the same image under JL(JR), if one of them is obtained by right translation (left translation) by an element of G. Sets R \ T * G and T * G / L are Poisson manifolds (see 5, question 2). Hence the diagram nR T* nL R \ T * G *-G--~ T* G / L , is a full dual pair. Likewise, the diagram

JL JRc~ , c~+ * ~-T*G-'-> is also a full dual pair. We therefore have the following commutative

diagram

218


nR T , nr R\T* G *--- G---->T* G / L

,.X J,,.,.\ /,.

where Poisson isomorphisms JL and JR are the quotient mappings of JL and JR respectively. The foregoing general facts about dual pairs can be stated for this example as follows: If 6~+ is an orbit in ~3+, * n r J Z ' (~7,,) + is a symplectie leaf in T * G / L which is isomorphic to a reduced phase space JZ~(it)/G,,. Also, n,

+

) =

) ,

where ~7~, is the orbit of ~ in ~d*. Likewise, --

+

is the reduced phase space G ~ \ J ~ ( / ~ ) which is isomorphic to a symplectic leaf of R \ T * G .

8. FOR FURTHER STUDY

Much attention is currently being paid to both the theory and the applications of the mathematics of Poisson manifolds. On the theoretical side Lichnerowicz has introduced and enlarged upon the notion of Jacobi manifolds, a generalization of both Poisson and Pfaffian manifolds. Specifically a Jacobi manifold is stratified (has generalized foliation). Each leaf of odd dimension is a pfaffian manifold, while leaves of even dimension are conform globally or conform locally symplectic manifolds. A theorem of local structure of Jacobi manifolds has been proved [4]. Their algebra of infinitesimal automorphisms and Chevalley cohomology have been determined in [4, 5]. The Poisson structure on ca* induces a Jacobi structure on a sphere of qd*. The corresponding leaves are the orbits of G under the quotient coadjoint representation [7]. A theorem by Lichnerowicz [7] states that every proper contact homogeneous space is a covering of an orbit of odd dimension, and every locally conform hamiltonian symplectic homogeneous space is a coveting of an orbit of even dimension. These theorems must be considered in the same light as the KirillovKostant-Souriau theorem for homogeneous symplectic spaces and coadjoint orbits.

13. INTEGRABLE SYSTEMS

219

REFERENCES [1] J. Marsden and A. Weinstein, "Proc. of the Conference on Order in Chaos". Los Alamos (1982). [2] J. Marsden, T. Ratiu and A. Weinstein, Trans. Amer. Math. Soc. 281 (1984) 147. [3] A. Weinstein, J. Diff. Geom. 18 (1983) 523. [4] A. Lichnerowicz, J. Math. Pur. et Appl. 63 (1984) 407. [5] A. Lichnerowicz, J. Math. Pur. et Appl. 57 (1978) 453. [6] P. Olver, "Applications of the Lie Groups to Differential Equations", Graduated Texts in Mathematics (Springer, Berlin, 1986). [7] A. Lichnerowicz, J. Math. Pur. et Appl. 65 (1986) 193. [8] P. Libermann and Ch. Marie, "Geometrie Symplectique. Bases Theoriques de la Mechanique", Pub. d l'Universit6 Paris VII. (English Translation, Reidel, Dordrecht) to be published. [9] A. Weinstein, J. Diff. Geom. 22 (1985) 255. [10] J. Dixmier, "Alg6bres Enveloppantes", (Gauthier-Villars, Paris 1974).

13. COMPLETELY I N T E G R A B L E SYSTEMS*

We will now proceed to define classical completely integrable hamiltonian systems. The reason for the choice of this name will become apparent from the underlying symmetries group and the classical method for integrating such systems. We will prove Arnold's theorem on the topology of the submanifolds generated by the trajectories of the system for the simplest case; then some recently stated theorems on functions in involution on coadjoint and adjoint orbits of connected reductive Lie groups, and the important Adler-Kostant-Symes theorem which generalizes the geometrical framework of the non-periodic Toda system. Finally we will indicate some recent developments in research on this subject. 1. DEFINITIONS AND ARNOLD'S THEOREM

We will speak of the flow of a vector field X on a manifold M when referring to 1-parameter local pseudogroup M ~ M generated by X. Prove that, if X1, X 2 are two vector fields on M such that [XI; X2] = 0, their flows commute, i.e., q~l ~ = q~2~q~l for each q~l •flow(X1); q~2~ flow(X2). The image of a trajectory of X 2 (resp. Xa) by an element of the flow of X 1 (resp. X2) will thus also be a trajectory of X 2 (resp. X1). *Contributed by C. Moreno.

220


Anwer: If [Sl; S2] = 0, then X 1 is invariant under the flow of X 2 (p. 150); therefore, the flow of X 1 commutes with the flow of X 2 (p. 147). II Let (M; A) be a Poisson manifold. Let F, H E C=(M) be the corresponding hamiltonian vector fields (p. 269 and Problem IV 11). We say 1-parameter that Xr generates a 1-parameter group of symmetries of the hamiltonian group of system defined by H (the differential system defined by XH) if symmetries [XF; S h r] = 0. (See [1] for more general definitions of symmetries of differential systems). first F is called a first integral of Xn if {E; H} = 0, i.e. (d/dt)F(C(t))= O, integral for each trajectory t---~ C(t) of Xn. We know that {F; H} = 0 implies [Xr; Xn] = 0 (Problem IV 11).

1) Show that, conversely, if X r generates a 1-parameter group of symmetries of the differential syste_m defined by X H, there is a function F E generalized C~(M;R) such that (d/dt)F(C(t))= 0. (We will call P a generalized first first integral integral of the system Xn. ) Answer: The function F(x; t ) = F ( x ) - {H; F}(x)t satisfies the stipulated conditions. II Remark. If we define, for each t E R, the following hamiltonian vector field on M (Problem IV 11): Xp(t) = a ( d F ( . ; t)),

we get Xp(t)= X e ,

whereby this field is easily seen to be independent of t. 2) Let F be a first integral of X n. Show that there exists a hamiltonian system (called a reduced system) on a Poisson manifold of dim M - 2 such that every trajectory of X n can be obtained from one of its trajectories by quadrature.

Answer: Choose local coordinates on M as in Problem IV 12, 3, i.e., p = F(x); q = Q(x); y = ( y l , . . . , y,,,-2)., m = d i m M. In these coordinates, the components of the tensor field A are A(p; y)=

0 1 0

-1 0 - a T ( p ; y)

0 ) a(p; y) a o ( p ; y)

since X r = a/Oq. For each p, A0(p; y) defines a Poisson structure on a manifold of dimension m - 2, for which ( y X , . . . , ym-2) are local coordinates. Hamilton's equations are then

13. INTEGRABLE SYSTEMS dp(t) =0; dt

221

dq(t) dt OH

m-2 OH c~p ( p' y) + j~= l a j( p; y) ~~yJ ( p"' y)( =-- A ( p '" y)) '

dyi(t) = m~- 2 71'~(p; y) ~O H (p" y)" dt j=l Oy j ' '

i=1,...

,m-2.

The latter, a hamiltonian system of dimension m - 2, is a reduced system. If y i _ yi(t ) is any solution to this system, the trajectories of the original system defined by X a are then p(t) = Po ;

q(t) = j A(p0; y; t) d / ;

yi__ yi(t ) ;

which are clearly obtained by a quadrature from the trajectory y i = yi(t). (See [1-3] for further information.) I Let (M; A) be a symplectic manifold, and G a connected Lie group defining a symplectic action on M with equivariant momentum mapping J. If H E C~ is G-invariant, the vector field X n is projected on the hamiltonian vector field Xa; H = H o//; H: M---> M / G , of the Poisson manifold M / G (if this exists). The vector field X a is tangent to the symplectic leaves of M / G (just as in Problem IV 12, 7) ~t* M / G can be shown to be a dual pair, so that each connected symplectic leaf of M / G is isomorphic to some connected component of the reduced phase space J - I ( t z ) / G ~ . Consequently X n induces a hamiltonian system on this space. We know from Lie that, if the isotropic group G , is solvable, the trajectories of X n can be obtained by using quadratures from the trajectories of the induced system X~ on J - X ( i z ) / G ~. See [1]. w

3) Let (M; A) be a symplectic manifold o f dimension 2m. Let F i E C~176 = 1 , . . . , n be functions in involution, i.e., {Fi; F j } = 0 and independent, i.e., d F 1 A ' ' ' A d F , ( m ) # 0. We suppose the flow q~i o f XFi to be complete. For each ( t l , . . . , t k) ~ R k, let us define the following map M---->M ,

( t l , . . . , tk)(m)-'-'> qh(tl)" " " ~Pk(tk)(m ) .

(1)

Show that these maps define a symplectic action on M o f the abelian group R k with the equivariant m o m e n t u m mapping M J__~Rk ( F l ( m ) ,

.

.

.

,

.

222


Also prove that J-l(tz)/G~, ; tx ~ J(M) are the reduced phase spaces of this action. Answer: Map (1) defines an R k action on M, since flows ~; i = 1 , . . , n commute (See question 1). The Lie algebra of ~k is identical with R k, and the coadjoint action of the group ~k is trivial, since this group is abelian. F i is a first integral for XFj, consequently J is an equivariant momentum mapping. Since dFx A ' ' "^ dFk(m ) ~ 0 every point/~ ~ J(M) is regular (i.e., TmJ is surjective, J(m) = I~). R k acts on j - l ( / ~ ) , so that J-~(l~)/R k is a reduced symplectic phase space for each/~ E J(M). I1 In view of the above, the trajectories of XF, are obtained by quadratures from trajectories of the induced field on J - l ( / ~ ) / R k. If k = n, we know that dim J-X(/~)/Rn = 0 , so that the system XF~0, i 0 E [ 1 , . . . , k] is completely integrable by quadratures

completely integrable by quadratures. 4) Let (M; to) be a symplectic manifold, dim M = 2n; let Fi E C~(M); i = 1 , . . . , n be in involution in such a way that dF 1 ^ . . . A dFn(m ) ~ 0 V m ~ U C M; and let (qi. Pi) be canonical coordinates on M, i.e to = E dp~ ^ dq ~. We define 0 = E p~ dq ~ and N c = {m E U IFi(m ) = Cz} "~

a) Prove that ( q~; Fi) are local coordinates on M, that the induced l-form i*O,c i~" N~--+ M is closed, and that consequently i*~o~= O. b) Let S( qi; Fi) be some local function-such that Pi = aS( qi; Fi )/aqi. We also define Q i = oS(qi; Fi)/~F i. Prove that (Qi; Fi ) are canonical local coordinates on M. c) Obtain the trajectories of Xr~o; i o E [ 1 , . . . , n]

Answer a: According to the stated hypothesis hamiltonians Fi, i= 1 , . . . , n are independent. The transformation F ~= F~(qk', Pk), q~= qi is thus a change of local coordinates on M. We now write pi = 0~( k., Fk ) and i*O = Oi(q k" Fk)dq i. We then have {Pi" Pj} = {Oi(qt~" Fk)" g ( q k Fk)} = 0 which is equivalent to ,90,( qk; Fk ) Oq j

OOj(qk; Fk ) OF~ '

i.e. i ' 0 is closed. ~

r

Answer b: Since i*c0 is closed, there is a locally defined function S( qk; Fk)


223

such that Pi = OS(q ~'', Fk)/Oq ~. A straight-forward calculation gives us

dqi ^ dpi = ~, d Q i ^ dF i . Answer c: Consider the hamiltonian system defined by Fioio E [ 1 , . . . , n]. In coordinates (Qk; Fk ) Hamilton's equations are dQ i dt

8Fio( Qk; Fk ) OFi '

dp i dt

8Fio ( Qk; Fk ) OQ ~

Obviously we obtain,

dQi(t) dt

and consequently

Qi(t)

=

=

~ioi ;

~ioi "JI-Qi(to) ;

dei(t) dt

0

Pi(t) = Pi(to).

The system has thus been "integrated by quadratures".

I

On a symplectic manifold (M; to), dim M = 2n there is generally no set of n independent functions /71,... , F~ ~ C~(U) such that {F i ; F j } = O. Arnold's theorem states some of the conditions required for the existence of such a set. 5) We start from the same hypothesis as in question 3), where K = n. a) Let I~, be a connected component o f j-a(i/,)" Prove that the R"-action on M defined in question 3) is a transitive action on I~. I~ ~ ff~n/H will thus be a homogeneous manifold where H is some discrete subgroup o f R n. We accept the following result, proved in [4] p. 274.: Every discrete subgroup H of R n determines a unique positive integer h and (n - h) linearly independent vectors, ( a h + l , . . . , an), of R" such that

i=h+l

b) Prove t h a t J - l ( / z ) ~ h • T n-h, where T n - h - " ~ n - h / 7 ~ n - h torus. If I~ is compact it will therefore be a torus.

is

the (n - h)

Answer a: From the definitions in question 3, it is clear that R n acts on I~. The R" orbit of a point m E I , is then closed in I , . The map

(tl,...,

tn)---~ ~1(t2).-. r

is a local diffeomorphism Vm E I , , since its tangent mapping at every

224


point ( 4 , . . . , t , ) E R n is surjective by reason of the independence of the vector fields XFi;i= 1 , . . . , n. Therefore the image of R n is open (and closed) in I,,. The R n action is then transitive. There is thus a closed sub-group H C [~", such that I~, is diffeomorphic to Rn/H. H is necessarily discrete because n = dim R" = dim I~,.

Answer b: This point can easily be proved using the above-mentioned result on H. I See Arnold [4] for further information on his theorem. 2.

CONSTRUCTION OF FUNCTIONS IN INVOLUTION AND THE LAX EQUATION

In the preceding section, we saw that, as a general rule a hamiltonian system on a symplectic manifold is not completely integrable. Whittaker [15] analyzes most of the classical completely integrable systems. The aim of this section is to show how to obtain systems of functions in involution on orbits of the coadjoint representation of a Lie group. These methods have recently been described in connection with the classical and quantum integrability of the systems considered by Calogero, Sutherland, Moser, Toda, etc. [11, 12]. They have been tested in the development of the scattering inverse method, or Lax method, which proves the classical and quantum integrability of infinite-dimensional hamiltonian systems, (Korteweg-de Vries (KdV), Kadomtsev-Petviashvili, Zakharov-Shabat, Dubrovin, etc., equations). In particular, we will see that the equations of hamiltonian dynamical systems on adjoint orbits of a reductive Lie group have Lax form.

1) Let G be a connected Lie group acting symplectically on the symplectic manifold (M; ~o) with equivariant momentum mapping J. We assume that we can define the dual pair (cf. IV 12, 6) ~d* J~_M H..~M / G and the reduced phase spaces j-a(lz)/G,, , tx ~ J(M).

Let f, g E C| be G-invariant functions, and f~,, g~,, the functions induced by f and g, respectively, on J - ~ ( ~ ) / G . . Prove that, if f and g are in involution on M, f . and g. are in involution on J - l ( l ~ ) / G . . Answer: Let

be functions such that f = f o 7r. We have

{f; g}M = ( f ;

o


225

If So is a symplectic leaf of M / G and if i: S o ~ M / G is the inclusion, we have _

m

{foi; ~Oi}s0 = {f; ~}M/o o i . Since rr is surjective, these two equalities prove that {f; g } M = O ~ { f ~

g~

= O.

We now observe that S O is isomorphic to a connected component of J - ~ ( t z ) / G , , , and f o i , ~oi correspond respectively to f~, and g~, in this isomorphism. II The following result was obtained and used by Mischchenko and Fomenko [5] in their study of integrable hamiltonian systems on Lie groups, and was generalized by Ratiu [6], to prove the involution of first integrals of the motion of the N-dimensional Lagrange top. 2) Let f, h be two Ad* G-invariant functions on ~*, i.e., f(Ad* g . / x ) = f(/a,); tz ~ ~*, g E G. Let s, r be two real numbers and let A be fixed in ~*. Let us define the following two functions on c~,

fs(/Z) " - f ( ~ + SA);

hr(la, )

=

h(~

+

rh).

Prove that f~ and h r are in involution on the Poisson manifold c~,, and consequently on any symplectic leaf, i.e., on the orbits o f Ad* G. Answer: We have f(Ad* t X . Ix) = f(Iz); VX E ~, t E R. If we now derive at t = 0 , we get (df)~,-ad* X/z = 0 . ad*(df)~, 9 = 0 ((df)~, ~ q3**--- q3). If r ~ s, we can write

/x =

r r--s

(/z + s h ) +

s s--r

Therefore

(/x + r a ) ,

and calculate

(fs; hr}.d.(/z)= (/z; [(dfs)tz ; (dhr)g]) ((/z + SA); [(df)lz+sa; (dhr)~])

r-s

S

+

s--r r

r-s

((/z + rh); [(df,)~,; (dh)~,+ar]) (ad*(df)~+~x-(/z + sA); ( d h r ) . ) + . . . .

If r = s, the result holds by continuity.

O. II

226


3) In the following question we set out the main idea involved in the Lax method. We will see below that the hamiltonian systems on the adjoint orbits of reductive Lie groups provide one part of its geometrical foundations. This method has been very useful for obtaining first integrals in involution for hamiltonian systems of arbitrary (and even infinite) dimension. Let (t; X)---~ B(t; X ) be a continuous map from R x Mk(C ) to the space Mk(C ) of complex matrices of order k, such that for every (t;X), B(t;X)=-B*(t;X), where * signifies "adjoint for matrices". Let t---~ L(t) ~ Mk(C ) be some solution of the non-linear equation

dL(t) dt = [B(t; L(t)); L(t)],

(1)

where [ ; ] stands for "anticommutator for matrices" (cf. Problem III 14 ). Let t---~ U(t) ~ Mk(C ) be the solution to the Cauchy problem

dV(t) = B(t; L(t)). V(t); dt Prove that U(t)*. U(t)= I and

V(0) = I .

L(t) = U(t)L(O)U*(t).

Lax pair Lax equation

(2)

Quantities Tr L"(t) will then be time-independent. (L; B) is called a Lax

pair, and (1), a Lax equation.

Conversely, let t - . U ( t ) E Mk(C ) be a given map such that U(t)*. U(t) = I. Let us define L(t) by relation (2), where L(O) is arbitrary. Prove that L(t) satisfies (1) when B ( t ) = (dU(t)/dt). U*(t). Answer: The proof is straightforwardly obtained deriving (2). (See [7] for the original application of the Lax method to the KdV equation.) II

As an example, consider the system [8] (p. 444) da(t) = -g(t; a(/); b(t)), dt db(t) dt = g(t; a(t); b(t)) . A Lax pair for this system is L(t)=

b(t)

a(t)

a(t) )

-b(t)

;

o

B(t) = ( _ 89 g(. . . )

Quantity Va2(t)+ b2(t) is time-independent whatever the continuous

227


function g may be. If g = 1, the system is that of the 1-dimensional harmonic oscillator. In the following tWO questions, we will see that hamiltonian equations on adjoint orbits of a reductive group are of the Lax type. 4) Let G be a connected Lie group and let ~ be the corresponding Lie algebra. ~* will be the dual space of ~3, t?a = Ad* G . fi and Ix E ~ . a) Prove that the tangent space to ~ at the point tz is the subspace of ~* given by T~,Ga={ad*s b) Prove that the following Ad* G-invariant differential 2-form on tog(Ad* g-1. fi,)(ad* ~:-Adg_ 1 /2,; ad* r/ 9Adg_l/2 ) = -Ad* n], defines a symplectic structure on ~;,. c) Let us suppose (~ to be reduetive, i.e., that there is a bilinear, nondegenerate form K on f~ such that K(Adg-~;Adg.~l)=K(~;~l);

~,Tq~;

gEG

(invariant by Ad G). Let q~" ~---~ (g* be the isomorphism defined by ~b(sc).r/= K(sC; r/). Prove that (~b*~) = d~; ~ = th-l(/2) is a symplectic 2-form on the adjoint orbit Ad G. ~, invariant by Ad G, and is given by &~(Ad g. sC)(ad r/. Ad g~:; ad ~'. Ad g~:) = - K ( A O g. so; [r/; ~']), where

is the tangent space to the orbit (72 at ~ ~ ~ . Answer (Indications) a) The arc Ad* exp tsr IX; t E R; ~: ~ ~3 defines ad* ~:.ix as a tangent vector to the orbit G~z at Ix. We can also see that ad* sr Ix = 0r ~: E ~ , where ~3~ is the Lie algebra of the Lie subgroup G~ defined by Ad* G~,. Ix = Ix. b) This point can be proved by a straightforward calculation. c) The invariance of K implies (by derivation): K(ad ~'- ~:; r/) + K( ~; ad ~'- r/) = 0.

reductive

228


And ~b(Ad g. sc) = Ad* g-X-

1~(~).

Since ~b is linear, the latter relation implies &(ad 77. ~:) = - a d * 77. t~(~:). The expression for d~ is obtained by straightforward calculations from these equalities and from its own definition, which is d ~ - (T~4~)*t%(~). II We now introduce some of the notations from [9]. It is very convenient to keep in mind the isomorphism e" ~---~ ~3"* defined by e(~:)/3 = / 3 ( ~ ) , /3 E q3*. If H • C00(~3"); ( d H ) , ~ ~3"*; /z E ~3", then e-'(dH)~, E cb. If ~" ~3--~ ~3" is the isomorphism defined by K (see the preceding question), we have E = H o 4~ E C~176 (d&)r : ' = 6( ~:');

Vsr ~:'E

and

(dH),(~) = e(~b-'(dE)r gradient of E

We will speak of the gradient of E when referring to the vector field on q3 defined as follows (grad E)~ = (h-' (dE)~ = e-'(dH)6(~). 5) Let H be a Coo function on f~; let I71 be the restriction of H to any orbit ~_;, in ~3"; and let E be the restriction of E = H . r to any orbit ~ ; sr = r in c~. Obtain the expressions of the hamiltonian vector fields X~I and Xg. Conclude that the flow of Xg satisfies a Lax-type equation.

Answer a" X~I is defined by (i(X/~)to)~, = (dH)~, ;/z E (Ta. For any rt ~ we must then have m

~o~((X~)~ ;ad* 77./z) = (dH)~. ad* r/./z = ( d H ) . . ad* ,1./z = - / z [ e - ~ ( d H ) . ; ~]. We thus get (X~q)~, = ad* e-~(dH),, ./z = ad*(grad E)v-,(~,) 9 .

Answer b: X~ is defined by (i(X$)&)e = (d/~)e; ~: ~ ~$; sr = r any ,7 in ~ we then have d~(X~; ad ~7" ~:) = (d/~)e. ad 77" ~: = ( d E ) e . ad 7/. ~: = (dH)~(e) 9r r/. ~:) = k(~:; [e-l(dH)~(~); ,7]).

For

229

13. INTEGRABLE SYSTEMS We thus obtain (Xg)e = - a d ( g r a d E ) e . ~.

Answer c" If q~(t; ~0) ~ ~ is an integral curve of X~, it satisfies d dt ~p(t; s~o)= - a d ( g r a d g)q~( t. sr q~(t; ~:o) = - [ ( g r a d E)~(t. r r sr If we transform (grad E),p( t. ~0) and q~(t; ~:0) into antisymmetric matrices (into anti-self-adjoint operators) by means of a finite (infinite) dimensional unitary representation of G, the latter equation will be of Lax type exactly as described in question 3). II The flow of the non-periodic Toda system satisfies the foregoing equation on some adjoint orbit of the group of lower-triangular matrices with positive diagonal entries. By analyzing the geometrical framework of this system, Adler-Kostant-Symes obtained, in particular, the theorem explained below. Our exposition closely follows papers [9; 10]. Let W be a Lie subalgebra of ~, and let ~ be a complementary subspace of W in c~. We write ix 9/ ~ q3 and ix" ~{'~ q3 for the inclusions of W and Y{, respectively, in ~. H x :i q3---~W and H x" ~---~ y~cwill be the projections defined by the splitting ~ = Y{ ~2r We thus have @*= Y{*GW*. If /x ~ ~*; /z =/Zlx + / x l x , where (I) represents restriction, then ~[x ~ Y~'* and txlx E W*. p E Y{*(v E W*) defines an element of q3*, also written p(v), if we impose p(W) = 0; v(Y() = 0 and linearity. Let ix." Y{*---~ ~*, ix." N*--. ~* be the inclusions of Y(* and W*, respectively, in q3* and let Hx." ~3"--* X*; I-Ix." ~3"--. W* be the corresponding projections defined by Hx.tz = / x l x and Hx. ~ = tzlx. The dual mappings i~c: ~3"---~W*; i~: q3*---~3'{* then satisfy i~ = Hx, and i~c = Hx,. If ad and ad signifies the adjoint representations of W and q3 respectively, we obviously have ad x sc = a d sc. i x and adX*sc = Hx, ad* sc. Consequently, if 9 G is a connected Lie group with Lie algebra c~ and N is the connected Lie subgroup of G with Lie algebra W, we have AdUg = Ad g. ix ;

AdN*g = I I x , . Ad* g ;

g ~ N.

Let ex" Y(---~ Y(**; ex" W---~W** be the canonical isomorphisms. If f ~ C~176*) then (df)~ ~ c~**; ~: ~ q3. We also have c~** = y(** ~ W** y~c~ ~c, for the isomorphism ex ~ ex. The partial differentials of f are then (dX*f)e = (df)elx, ~ Y(** ;

(dX'f)r = (df)elx. ~ X * * .

230


If v ~ X* we have ( E2F~.~

~ E - l ( d l ) / j ; 1/) --- ( 1];//2~ .~ E - I ( d I ) ~ )

= ( ly; s162

) --" (df)~ 9ly.

We thus get (dX'f)e = ek~

~e-'(df)e

and similarly (dX'f)e = e x ~

~e-'(df)e .

6) Let ~7~ be the Ad N" N-orbit of a point ~ ~ Ac*. Let f, g be functions in C=(~ *) and fo ix.; h o ix., their restrictions to ~ . Obtain the expressions for the hamiltonian vector fields X ~ ; Xh~,~. and for the Poisson bracket (foix.;hoix.)

x" v

9

Answer a: From question 5) and the above notations, we obtain (Xt.-U:x.),, = ad x" e~. x d ( f o i x.),,, v

= adX'e~cl((df)ix.vo(dix.)~)v = adX'enX(dX'f)~ 9v = adX'Hxe-X(df)v "v = / / x . ad*

rGe-

(dl).

9v .

Answer b" From question 4, b) we get { f o ix,; h o i x , I f = -v[nxo

*

,he,

= r ~ (X/,-~x.; Xh,-:~.)

e-~(df)~; Hxo e-'(dh)~].

II

7) (Adler-Kostant-Symes theorem). Let ~ = Y( ~ N be as above; let Y~ and ,/f be two Lie subalgebras of ~. We now consider the set A = {f•

C|

* G - / z ) = f ( / z ) ; / z E q3*},

of Ad* G-invariant functions, on ~*. a) Prove that for each ~ E N* the set ~ a = { f o i x . E C|

") l f E A } ,

is in involution on ~?f'. Consequently the set B=(foix.~C| is in involution on the Poisson manifold N*. b) Prove that the vector field N on ~ , with hamiltonian fo ix. is (X~)~

=-ad*(lI~re-l(df)~).v


231

A n s w e r b" We saw that f ~ AC:>ad* e - l ( d f ) ~ 9v = 0 ; v ~ ~d, thus ad* U ~ c e - ' ( d f ) ~ . v + ad* H x e - l ( d f )

. v = O.

Therefore the required expression for (XT77:~.)~ may be obtained from question 6. A n s w e r a: We have {fo ix.; ho i x . } x'~ = - ( a d * / / w E - l ( d f ) l , ) " v " [ I x e - l ( d h ) ~ =-ad*

Hxe-l(dh)

. v 9H x e - l ( d f ) ~

= a d * I l x e - a ( d h ) ~ 9v . H x e - l ( d f ) v = -v[llxe

-1

(df)~; H x e

-1

(dh)~] = 0.

The latter equality holds because v ~ W * C ~* and the bracket is in yd. Therefore { fo ix *; h o is*}x* = 0 holds on the Poisson manifold W*. II Let us now suppose that we can define a bilinear form on K as in question 4c, and let 4," ~d---> ~d* be the isomorphism defined by K. Obviously ~ b - l ~ * = W 1 and ~b-lN * = ~ 1 , where W 1 and Y~cI are the K-orthogonal subspaces to Yg" and X, respectively. We have ~ = Yg'" @ W 1. It is easy to see that th - l o H x. = H x i o ~b-1 and that the A d N~ action of N on W* translated to Yf 1, by means of ~b is A ( n ) = th - l ~ AdN*n o th = / / k i o A d n-1 ," A(T1) = - I I k ~ ad r/;

n ~ N

77 ~ At .

By means of these relations and th, we can read question 7 on Y{• The A d l e r - K o s t a n t - S y m e s theorem often appears in the following form in specific problems of mathematical physics. 8) Hypothesis and notations are as in question 7). L e t us consider the set C = {F E C~(c~)IF(Ad g-7/) = F(rt); , / E ~} o f Ad G-invariant functions on ~g. Prove that f o r each ~ E Y~ ~ the set o f functions C$ = {Fo i x . ~ C = ( A ( N ) ~ ) ] F E C}

232


is in involution on A ( N ) ~ . Consequently the set

D = {Fo i x . ~ C~(Y{ ~)[F ~ C} is in involution on the Poisson manifold ~r• Prove that the vector field on A ( N ) ~ with hamiltonian Fo ix~ is

(X~)~

= [/-/k(grad E)~; so];

sr E ~ •

I

3. FOR FURTHER STUDY a) The observation that the systems Calogero, Sutherland, Toda, etc. are hamiltonian systems defined on symplectic manifolds which are reduced spaces of some other hamiltonian systems having obvious first integrals in involution and such that they induce a complete and independent set of first integrals in involution on the reduced system, led to a fuller understanding of the proof of their complete integrability and of the manner of effecting the integrations themselves. This insight is systematically developed in the review [12] (where the intimate connection between integrability and the structure of simple Lie algebras [11, 13] is also proved) and can be verified in all the involutions theorems given in [9]. Another approach developed by Adler and van Moerbeck involves the systematic use of Kac-Moody algebras. This method has proved to be more suitable for dealing with the integrability of a richer class of systems [9, 10, 13]. This is currently an actively developing field of research. b) Many other integrable systems have recently been obtained, in particular by the method of "dressing transforms" [14] in connection with the Kadomtsev-Petviashvili equation. Understanding these systems requires some change in the geometrical framework. Their symmetries groups have a Poisson structure (closely related to the classical YangBaxter equation). The geometrical framework within which the above systems can be understood invariably includes a description of the Poisson structure of their symmetries groups. This implies, in particular, that the dual structure of the Poisson structure of the system must be taken as fundamental. Many other new aspects of these systems are revealed when they are quantized. This is also a fast-growing field of research. REFERENCES [1] P. Olver, "Applications of Lie Groups of Differential Equations" Graduated Texts in Mathematics (Springer, Berlin, 1986). [2] R. Abraham and J. Marsden, "Foundations of Mechanics" (Benjamin, New York, 1981).


233

[3] P. Libermann and Ch. Marie, "Geometrie Symplectique. Bases theoriques de la Mecanique", Pub. de l'Universit6 Paris VII (English translation, Reidel, Dordrecht) to be published. [41 V. Arnold, "Methodes Math6matiques de la M6canique Classique" (Editions Mir, Moscou). [5] A. Mischenko and A. Fomenko, Math. USSR. Izvestija 12, (1978) 371. [6] T. Ratiu, Ind. Un. Jour. Math. 29 (1980) 609. [7] P. Lax, Comm. Pure Appl. Math. 21 (1968) 467. [8] Lectures Notes in Physics, No. 38 (1975). [91 T. Ratiu, Lectures Notes in Mathematics, No. 775 (1980). [~0] W. Symes, Physica D1 (1980) 4. [11] D. Kazhdan, B. Kostant and S. Sternberg, Comm. Pure Appl. Math. 31 (1978) 481. [12] M. Olshanetsky and A. Perelomov, Phys. Rep. 71 (1981) 313. [13] M. Adler and P. van Moerbeke, Adv. in Math. 38 (1980) 267-317, 318-379. [14] M. Semenov-Tyan-Shansky, Publications RIMS Kyoto Univ. 21. No. 6 (1986). [15] E. Whittaker, "A treatise on the analytical dynamics of particles and rigid bodies" (4th ed. Cambridge Univ. Press, Cambridge, 1944).


V. RIEMANNIAN MANIFOLDS. KAHLERIAN MANIFOLDS

1. N E C E S S A R Y A N D SUFFICIENT CONDITIONS F O R LORENTZIAN SIGNATURE 1) Show that the following statements are equivalent" a) g is hyperbolic, i.e., of signature (1, d - 1 ) = ( + , - , . . . , - )

and

a/ax ~ is time-like. b) goo is positive and g ijv~vj is negative definite. Answer: a) ~ b); indeed goo = g( a/Ox~ O/Ox~ > 0

since O/ax o is timelike.

Since goo > O, we can write

(

ds2 = g ~ dx~ dx~ = V'~0o dx ~ + ~

goi

)2 dx ~ + 3'o dxi dxj

(1)

with ~/0-=-go~goj/goo + go. We note that (3'o) is the inverse matrix of (g/Y): we have gij')/jk-'t~k.i Indeed g ~t3gt3~ = 8 ~t3 implies

l

gOg, k + gi~

e g gjo + gi~

~ - O,

i.e., g

~o

-

1 goo g ~176

Inserting this value of g~0 in the previous equation gives

i

8 k = gij

( gjk

1

goo gj~ gko

) = gij'Yjk

9

(2)

Next we note that the quadratic form defined by Y0 is negative definite. Indeed, we deduce from the identity (1) that -3'o dx' dx j admits a decomposition as a sum of squares of d - 1 linearly independent one forms: d-1

"YOdxi d x J = - ~ (aij dxJ) 2i=1

Therefore 3'ij is negative definite, namely 3,ouiuJ< 0 for all u ~~ 0 . The 235

236

V. RIEMANNIAN MANIFOLDS. K,s

MANIFOLDS

same is true of its inverse g ijv~vj since they coincide in an orthonormal basis. Answer: b):ffa); indeed implies a / a x ~ timelike.

0 < goo = g( O/ax~ a/ax~

Since g00 > 0, the quadratic form ds 2 can be decomposed as in eq. (1). If g ijv~vj is a negative definite quadratic form, so is the inverse quadratic form Yo dx' d x ( It follows from (1) that g is hyperbolic of signature (1, d - 1). 2) Show that the following statements are equivalent: a) g is hyperbolic o f signature (1, d - 1) and {O/Ox i} is spacelike. b) gOO is positive and gijviv j is negative definite. Answer: We note that g00= g(n, n) where n is the gradient of the "time function", x ~ Indeed for such an n ~= 0 ,

n i = ax~

for all i ,

and n o = 1, so g(n, n) = g~tJn~nt3 = gOO.

(3)

To prove a)::> b), we prove first that g hyperbolic together with a/ax i spacelike imply g00> 0. Let the quadratic form g ~ u ~ u ~ be decomposed as follows g,~tju'~u t3 =-- U(u~ 2 - ~] (aih uh + hiU~ 2 ,

(4)

i

with E a i h a i k -" --ghk ,

(5)

~ a i h A i = --ghO ,

(6)

U = g0o + ~] (h,) 2.

(7)

i

i

i

Since all the a/ax ~ are spacelike, the quadratic form defined by --ghk is positive definite. Equation (5) is the decomposition of a positive definite symmetric matrix G = A A . For such a decomposition, A is real, i.e., the aih are real. Since ghk is definite, det ghk r 0 and det a~h r 0; the matrix (aih) is real and invertible and h~ is real by (6).

1. LORENTZIAN SIGNATURE

237

Inserting in (7) the value of )t; given by (6) yields

U = goo + ~

h,i,k

(a-1)hi(a-a)kighOgkO

= ( gOO)-l[ gOOgoo-- ~ ( ghk)-lghOgkOgO0 ] h,k

= (g~176

(8)

- '~ gho(g h~ +(ghk)-lgkogO0)] h,k

= (gO0)-I . We have U > 0 since g is of signature (1, d - 1), hence gOO> 0, i.e., n timelike. b ) ~ a). Indeed gqviv j negative definite implies that the hyperplane generated by the O/Ox' is spacelike. Since gOO> 0 the quadratic form ds 2 can be decomposed as in eq. (9), and g is hyperbolic of signature (1, d - 1). 3) Show that the following statements are equivalent. a) g is hyperbolic of signature (1, d - 1), goo and gOO are positive. b) g is hyperbolic of signature (1, d - 1), 8 /Ox ~ is timelike, O/Ox ~ spacelike. c) g is hyperbolic of Signature (1 , d - 1), gqviv j and g qviv j are negative definite.

Answer: a)c:>b). Indeed goo = g( O/Ox~ O/Ox~ >Or ~ timelike. gOO= g(n, n) where n is normal to the hyperplane generated by the vectors a/Ox ~ (see answer 2). g oo > 0 r n timelike r hyperplane generated by O/Oxi spacelike. b)~c). goo >0=> g qViV j gOO> O~ gijviv j

negative definite (see answer 1) negative definite (see answer 2).

c) ~ b) by decomposing

g ~ v ~ v ~ and g~v~v~ as in (1) and (2). Remark" a/3x ~ timelike and 8/3x ~ spacelike, i.e., goo > 0 and g, < 0 do not imply that the metric is hyperbolic. For example

238

V. RIEMANNIAN MANIFOLDS. KA,HLERIAN MANIFOLDS ds 2 = (dx~ 2 - (dx')2 _ (dx2)2 _ (dx3)2 _ 4dx 2 dx 3 = (dxO)2 _ (dx 1)2 + 3(dx2)2 _ (dx 3 + 2 dx2) 2

has signature (2, 2); the metric is not hyperbolic.

Remark" goo > 0 and giJvivj negative definite imply that the metric is hyperbolic but do not imply that O/Ox~is spacelike. For example ds 2 = (dx~ 2 + 4 dx ~ dx I + 3(dx x)2 = (dx ~ + 2 dx ~)2 _ ( d x ' ) = .

H e r e g0o > 0 , g,,~ = '

g ViVj

=

1

(1 2) 2

3

'

-1

< 0 but g ~ = 3 > 0 and O/&xi is not spacelike.

2. F I R S T F U N D A M E N T A L

timelike null, spacelike

2

FORM (INDUCED

METRIC)

Let f : Xd--'> yd+l be an embedding. Let n be a normal vector to the hypersurface f ( X ) C Y. Let g be a Lorentzian metric on Y. The hypersurface f(X) is said to be timelike [resp. null, spacelike] if its normal is a spacelike vector [resp. null, timelike], i.e., g(g*~nv, gaSn~)< 0 [resp. =

0,>0].

Show that a) if fiX) is timelike, then the first fundamental form f*g is hyperbolic, b) if f (X) is spacelike, then f*g is negative definite, c) if f (X) is null, then f*g is degenerate. A ns wer: Xd

Let {e i}, i = 1, 2 , . . . , f(X) at y = f(x)

f

~

yd*l

d be a basis for TxX. Let n be the normal to

(n, f'v} =0

for all v~E TxX.

(1)

3. KILLING VECTOR FIELD

239

Let e 0 E Ty Y be the contravariant vector corresponding to n"

e o = g at3nt3 a

Assume first that e o JE' Ty f(X). Then {e0, f'e~) is a basis for Ty Y, and the components of g in this basis are got ~

g(eo, eo) ( f* g)(e i, ej)

since g(e o, f'ei) = 0 by (1). First of all if eoJ~Tyf(X ) then f ( X ) cannot be null, i.e., g(eo, eo) cannot vanish. Indeed g(eo, e 0 ) = 0 together with g(eo, f ' e i ) = 0 implies g(e o, v)= O, Vv E TyY, which contradicts the fact that a metric is not degenerate. a) If f(X) is timelike, g(eo, e 0 ) < 0. Since g has signature (1, d), the first fundamental form f*g must have signature (1, d - 1); it is hyperbolic. b) If f(X) is spacelike, g(eo, e0) > 0 and f*g must have signature (0, d); it is negative definite. c) Consider now the case where e o E Tyf(X) ~nd let w ~ TxX be the vector such that f'w =

r

f(x) .

Then f(X) is a null hypersurface since by (1) g(e o, f'v) = O, Vv ~ TxX. In particular 0 = g( f'w, f ' w ) = g(eo, eo) . On the other hand, since

0 = g(f'w, f ' v ) = (f*g)(w, v), there is a nonzero vector w such that f*g(w, v) = 0 for all v E T~X, and the first fundamental form is degenerate.

3. KILLING V E C T O R FIELDS

Let g be a riemannian metric of arbitrary signature, and ~ be a Killing vector field for this metric. Prove the following formula, useful in particular in the study of Killing vector fields with given asymptotic properties: V~V~ ~ = R a ~ " ~ , .

(1)

240


Answer: A Killing vector field $ for the isometry group of g satisfies

(p. 152) Va$~ + V/~$~ --O.

(2)

On the other hand, by the definition of the Riemann tensor VoeV/~se), - V/~V~se)~- Roq~)~uSu.

(3)

We deduce from (2) that V~V/~$;~ + V/~V~$;,- Vc, V/~$x - V/~Vx$c,

(4)

Thus, using (3) and (4) V~V~$x + V~V~$x

-- V ~ V ~ $ x - R/~)~u$ u - V x V ~ $ ~

= V ~ V ~ $ x - R/~)~lxSt, + R)~oq~uSu + V ~ V x $ ~ =

+

(5)

Adding (3) and (5) and using the algebraic identity for the Riemann tensor (p. 309) gives (l).

4.

SPHERE

Sn

A supplement to this problem entitled "Volume of the sphere S n'' can be found near the end of the book. The sphere S n is the submanifold of R n+l given by the equation E 7 2 1 1 ( x i ) 2 - - 1.

l ) Polar coordinates: Show that the mapping given by x n + l __ COS O n

x " = sin O n COS 0 n - 1

x n- 1 = sin On sin On- I cos 0 '7-2

(1) x 2 - sin 0 n ... sin 0 2 COS 0 ] x I _ s i n 0 'z sin O n - I sin 0 n - 2 ...sin01 is an analytic diffeomorphism f r o m the open set S-2 Q R n"

aQ"

0 ~,"+1 which is an analytic diffeomorphism onto its image V: .~,0 = T = tan -~ u + tan -a v ,

with - I r / 2 < tan -~ u < ~r/2.

(1)

m

ki=

A xi r

with A = r - ~ { l ( 1 + uZ)(1 + 12)[ a'2

- ( 1 + uv)) , (2)

where u = t + r,

v = t-

r,

r = (,~(x')2) 1/2

2) Denote by ~ the metric on q~(R"+x), which is the pullback by dp-1 of the

Minkowski metric ~l on R "+a,

i= Show that ~ is conformal to the canonical metric g on .,~,,+1 defined in Problem V 5

g = 02 . 3) Since dp is an analytic diffeomorphism []~n+l_._> V C ,a~n+ 1 it defines an analytic map T*V---~ T*N "+ 1. Determine this map and show that it extends to an analytic map T* ~, "+ 1--. R" +1.

8. C O N F O R M A L T R A N S F O R M A T I O N

O F WAVE E Q U A T I O N S

257

4) Denote by ff an R iv valued function on V, i.e., an N-multiplet of scalar functions on V, and, by u its pullback by 4~ on R "§ 1. Denote by

the d'alembertian of u, and by r-left the d' alembertian of ff in the metric g. Show that

(nu)(x) = (Dia)(p),

p = 4,(x).

What is the analogous equality .for the operator Vlu - f(u, Ou, 02u), where Ou is the gradient of u, 02u the set of its second derivatives and f a mapping

NN(n+2)(n+3)/2....~NN?

5) Use the preceding results to transform the problem of the global solution on R "+1 of the equation

[']U- f(u, c]U, t ~ 2 U ) " - - 0 into a local problem on R n+l involving the operator O g. 6) Find conditions on f and n under which the preceding local problem extends to ~ n+1. Answer 1 ~ .~ n+ 1 __ {R X P} is represented in the coordinates x,-i 2 0 by the whole of ~n X ~: the formulae (1), (2) define a mapping ~b" R "+ 1_.., ~ , + 1. It is obvious, except perhaps at r = 0, that this mapping is analytic. The mapping is analytic even at r = 0 because A

2

r

l+t

2

when r = 0. The image tk(R "+1) is the open set of .~"+ 1 ~ ( R n + 1) =

{2 tan -1 ~P - 7r< T < 7 r - 2 t a n

i=1

(2')

1

.

(3)

~b is bijective onto its image: its inverse is the analytic mapping t=~

tan

2

+tan

2

'

r=~

tan

2

2

'

(4)


258

with

(5)

o r - - 2 t a n -1 fi 2

(Note that (2) is equivalent to f(i _ fix i / r , ~ -- A ( t , r), while (5) gives fi 2r u - v tan c~ -= = , ot -- t a n - 1 u - t a n - 1 v). 1 -/52/4 1 + uv 1 + uv A n s w e r 2: Let 17 = - E i =n I (dxi) 2 + dr2 -- - d r 2 - r2 dw2 + dt2 with dw 2

the metric of the sphere S n - 1 in polar coordinates. A straightforward computation using the results of 1) gives ~) _ r

1,17 _ (COS T + c o s or) - 2 ( - d o t 2 - sin 2 ot d w 2 + dT2).

That is, if g is the canonical metric of Z n+l _ ~-2g, if2 -- cos T + cos c~. Note that ~2 > 0 on V - r (I~n+l) C , ~ n + l In the coordinates (T, X t) s -- cos T + fi,

and ~ - 0 on 0 V 1 --/32/4

fi --

1 +/32/4

.

(2.1)

If we make analogous computations in Z n+l - {I~ x N}, in the coordinates ( T , X i), with N the north pole of S n we find analogous results with 1 -p2/4 ~2 - cos T - a, fi -(2.2) 1 + p2/4" The functions defined by (2.1) and (2.2) coincide in Z n+l - I K x {P tO N} (recall (p/:3) 2 -- 4), they define an analytic function on Z n+l . Note that we can also write 17 -- r -- (S'2 o r )-2r where I2 o r

is the expression of s

in coordinates (t, x i) (valid only

o n I~ n+l )"

(S-2 o r - 1)2 =

4 (1 + u 2) (1 + v2) '

u-t+r,

v-t-r.

A n s w e r 3" The map r associates to each covector u at p 6 V a covector at x -- r ( p ) (5 ]~n+l, by a linear map E ( p ) =_ r This mapping E ( p ) is represented in the natural frame of the coordinates )~-U __ (T, ~ i ) o f z~ n + l and x a of I~n+l by the matrix with elements

Ox, ,

-r

8. CONFORMAL TRANSFORMATION OF WAVE EQUATIONS

259

Using the expression of th given above, we find:

E,~ = a I ~ + K~'A~ with i 0"- COS T ,

il'._. (1

i 0-- i 0 -- 0 ,

/~0= sin T ,

/~i= _~.i,

~ 0 = sin T ,

.4~ = - 89 (1 + d ) X ~ .

+ p2/4)8i, -

l gigj

,

In the frames of the coordinate X of 2 , + 1 we find analogous formulae. In the c o m m o n domain of the two charts, E~-

OX ~" _

OF(~, E ~" .

We deduce from these formulae that ~b* extends to an analytic mapping E: T*Z"+I---> R ~+~ (since T*R ~+~ is a trivial bundle we can identify all its fibres): if t~(p) are the c o m p o n e n t s in the natural frame of a covector u ( p ) at p ~ Z n+ 1, then ~ = E ( u ( p ) ) is the element of R ~+1 with components

~,, = E ~ ( p ) f f ~ , ( p ) . Note that the E~,, for fixed a, define a tangent vector to ~"+a', when p = ~b(x) this vector is the image ~'O/Ox ~ of the frame vector O/Ox ~ of ~n+l Note also that we have g(E0,

E0)

= - g2 2

E o, = -

X~ sin

T

,

Answer 4" D e n o t e by ff an R"-valued function on V, and by u its pullback on R n+ 1 u = ~b*u = tTo ~b,

i.e., u ( x ) = i f ( p ) ,

The differentials of u and t7 at x E R ~§

p = ~(x).

and at p = ~b(x) are linked by

(du)(x) = E ( p ) V f f ( p ) , where V is the covariant derivative relative to ~. The second covariant derivatives with respect to the metrics r/ and g = ~b- 1,7/ are linked by (a 2u)(x) = ( E ( p ) |

E(p))(f72a(p)).

That is, in coordinates X ~ on V m

m

~

N

260

V. R I E M A N N I A N

MANIFOLDS.

KtkHLERIAN

MANIFOLDS

which implies the equality of the scalars

(Vlu)(x) = (Vlgff)(p). Thus (Vlu)(x)= f(u(x), Ou(x), OZu(x)) is equivalent to (I--igtTl(p) =/~(p, tT(p), (VtT)(p), (V2tTl(Pl) with the definition /~(p, if(p), ~rtT(p),

V2tT(p))=f(ff(p),

E(p)V2ff(p)).

E(p)V~(p), E(p)|

Answer 5" Let t7 be an R N valued function on V and set ff

=

~"~(n-1)12V.

Since g = O2g we have (cf. p. 352, with change of sign convention for R) n-1

l--lgv

(

4n Rv = g2 -(3+")/2 Vl~t7

n - 1 /~t7) 4n "

But

( g flat) (scalar curvature of .~"+a).

R=0

R= n ( n - 1)

Thus the function u: x ~ u(x) with U(X) ~--" ( a ( n - 1 ) 1 2 v ) ( p )

,

p = ~b(x)

will satisfy the original equation on R "+1 if and only if v satisfies on V [-']gO --

(n - 1)2 o =

V(p,

o, Vo, V o).

We have ~

-1

o

Thus V/~/~ -- ~-~(n- 1)/2

V

+

2

g2 V. Ov

,

+ O_~[-n + 1 (V~,aV.v + V . a V , , v ) - g~.g~ { 2

n - 1 -2[ n + 1 2 0 2 V~,g2V~g2 - g~,~gP*V, g2Vp 0

]

8. CONFORMAL TRANSFORMATION OF WAVE EQUATIONS

261

and with the notations of section 4:

F(p, v(p), Vv(p), VZv(p))= O(p)-(3+")/zf(O("-a)/Zv(p), ffl(n-1)/ZE(p) ( Vv + n -2a ,I"~-Iv,~"~V) $2(n_l,/2E( p)| E(p) ,

x

(V v +.--))

where VZv + . . .

is the expression in parentheses above.

Answer

6" The operator [-Ig-((n- 1)/4) 2 is defined on ,v,+l. The equation obtained will extend to ~"+a if the same is true of the right-hand side. g2 is an analytic function on Y,"+x, but not necessarily positive. The expression above can be smooth only if O appears with a non-negative integer power. We first show that E ( p ) ~ - l v ~ and E(p)@E(p)~-lv2~ extend to analytic functions on ~,+1. a straightforward computation gives, in the coordinates X ~ [resp. X ~']

e

o.a=ar

with Y~ the set of scalars

v

Y0 = a sin T Yi' = - 89cos T(1 +

E EoV V.a= a(an

a)X i

- V Vo)

[ r e s p . - d sin T] [ r e s p . - 89cos T(1 + [resp. -

d)f( i] dg2rl~t3- 0 V,~Vt~]

with V~ the set of scalars [resp. = - ( 1 + d cos T)] [resp. 89sin T(1 + d)f(i].

V0 = - ( 1 - a cos T) V1 = 1 sin T(a + a)X i

We deduce from this result that F can be written

F(p, o, Vv, V'v)- (gl)-(3+")/2f(O("-l)/20, 0("-1)/20', 0("-1)/2v"), where o' [resp. v"] is a linear sum in o, Vo, [resp. o, Vo, VEo] with coefficients which extend analytically to ~,+1. We suppose that f is a smooth function of its arguments vanishing together with its derivative at

(0,0,0)

f(0, 0, 0 ) = 0

and

dr(0, 0, 0 ) = 0.

Then there exists a smooth function h such that

f(f~(n-1)/ 2u, g2(n-1)/2O,

g2(" -1)/2 w) = g2"-lh(g2 ( n - l ) /2 U, V, W)

(1)

262


and F extends to a smooth function on ~ "+1 if n - 1 - (3 + n)/2 = ( n - 5)/2 is a non-negative integer, i.e., if n is odd and n-> 5. Under the above conditions the global existence of a solution of the Cauchy problem on M,+a for the equation

Du = f(u, au, a Zu) results then from the local existence up to the time T = ~r of the equation o n "~n+l

4

v = F ( p , v, Vv, V2v).

This existence is insured for small enough data in appropriate Sobolev spaces: Vlr-0 • n , ( s " ) , 00Vlr_.0 E n , _ l ( S " ), s > n/2 + 2. This appartenance implies fall-off properties of the Cauchy data at space infinity. The corresponding solution of M,+ 1 has decay properties at time and null infinity (for details see the reference). REFERENCE

D. Christodoulou, Comm. in Pure and Applied Maths 39 (1986) 267-282.

9. MASSES OF " H O M O T H E T I C " SPACE-TIME 1) Show that if the metric g satisfies R ( g ) = 0 and is asymptotically euclidean in the chart (x ~) the same is true o f the metric ~ = A 2q~,g where is a diffeomorphism given in the chart (x i) by s Ax i, and A is a constant. mass of a

metric

2) T h e " m a s s " of a metric g on a 3-dimensional asymptotically euclidean

manifold is given by (cf. for instance the review article "Positive energy theorems" in "RelativitY, Groupes et Topologie II" B. DeWitt and R. Stora eds, North-Holland, 1983)

,fs

m = ~

(Oigu - O/gii)Ejkt dxk ^ d x l ,

where S is the 2-sphere at infinity of R 3. Show that the mass th o f ~ is rh = Am.

Answer 1"

~ij(~h)._ i~2 t~.~ axl O~X a2]m gtm(xh) = gq(Xh) = gi/( A-1xh) ,

10. INVARIANT GEOMETRIES ON THE SQUASHED SEVEN SPHERES 263

thus

gij(xh) = ~ij + Oij(r-I)

gij(~'h) ~-- ~ij + ~

implies

9

On the other hand

~h~ij(~h) ._ ,~-lr

F ilj(x, h )

'

~ij(~h) ._ gij(,~-l~.h) - 'r

=

ilj( ,~ - ljl~h )

,

,

R(g) = A-ER(g). Answer 2"

rh -- 89f ,~- 1(r gij - Ojgii)(s ) ejktd.~h A ds k , s Example

(,

set ~= Ar, t = At

s2:(1

(,

r

r

2m)-1 r

)d,2 (1)

r h = Am.

d r 2 _ r2( d 0 2 + sin 2 0 d ~ 2) ,

-1 r

d r 2 - r 2 ( d 0 2 + sin 2 0 dq~2) .

One concludes from these results that if there exists a space-time with m < 0, there exist space-times with arbitrarily large negative mass.

10. INVARIANT GEOMETRIES ON THE SQUASHED SEVEN SPHERES* The (round) seven sphere is the submanifold of the 8-dimensional euclidean space 0:8 (i.e., R 8 together with the euclidean metric) defined by 7

(x ~)2__ 1,

x E0:8 .

/z=0

1) Show that the ( r o u n d ) seven sphere is the 7-dimensional reductive homogeneous space SO(8)/SO(7) (see Problem III 7, Homogeneous spaces). Answer 1" Let 57 be the (round) seven sphere, thus labelled in anticipation of the "squashed" seven spheres, which will be labelled ~17, $~ and 537 (see paragraph 6). *Based on the unpublished work of G.S. Sammelmann. Discussions with L.C. Shepley, P. Spindel and S.G. Low are acknowledged.

264


The action of the connected group SO(8) on E 8 induces an action on ST. Indeed, if x 6 S 7 C E 8, g 6 SO(8), and y - g x by matrix multiplication, then y 6 S 7. A straightforward calculation shows that this action is transitive: any point on the sphere can be taken to any other point by a rotation. The subgroup of SO(8) which leaves a point of ~7 fixed is isomorphic to SO(7). We shall use the same notation for the group SO(7) and for a subgroup of SO(8) isomorphic to SO(7) whenever the context makes the meaning clear. For instance, let N 6 ~7 C E 8 be the point of coordinate x ~ -- 1, x i - O. We shall use the same notation h for the matrix h 6 SO(7) and for the matrix (01 0) in the subgroup of SO(8)isomorphic to SO(7) which leaves N invariant. Let N 6 S 7 and g 6 SO(8); a point x - g N ~ $7o is identified with the left coset gSO(7), where SO(7) C SO(8) leaves N fixed. Hence ~7 _ SO(8)/SO(7). The identification of S 7 with SO(8)/SO(7) depends on the choice of N. The homogeneous space SO(8)/SO(7) is reductive, since SO(8) is compact. 2) S h o w t h a t $70 is p a r a l l e l i z a b l e , a n d c o n s t r u c t e x p l i c i t l y left a n d r i g h t s i e b e n b e i n s on ~7. parallelizable

quaternions octonions

2: An n-dimensional manifold is said to be parallelizable if it admits n linearly independent smooth vector fields. We shall construct left and right siebenbeins by using the following theorem. For the proof see, e.g., [Brickell et al.] If there exists an l~-bilinear mapping v" I~k+l • ~n+l ~ i ~ n + l such that i) v ( v , z) = 0 implies that v = 0 or z = 0; ii) there exists e 6 I~~+l such that v ( e , z) = z for all z 6 I~n+l then ~n admits k independent vector fields. One can check that such mappings v exist for k = n = 1,3, 7, namely

Answer

n = 1,

v determined by the product on C

n -- 3,

v determined by the product on Q

(quaternions)

n = 7,

v determined by the product on 9

(oetonions, Cayley

numbers). The space O is a vector space of 8 dimensions over the real numbers. Let {e0, ei}i=l...7 be a basis for this real vector space; the left product function v on O is denoted by v ( v , z) - v . z . It is bilinear and such that

10. I N V A R I A N T

GEOMETRIES

ON

e o 9e o = e o ,

THE

eo

SQUASHED

e i 9e

9e i =

o

=

SEVEN

SPHERES

265

ei ,

e i 9 ej = - t~ije 0 + a i j k e k ,

(1)

k

where aijk = aij are real numbers totally antisymmetric in ijk and equal to 1 for ijk = 123, 516, 624, 435, 471, 673, 572

and zero for triples which are not permutations of the above ones. It may be c h e c k e d that u ( v , z ) = v . z = 0 implies v = 0 or z = 0 and thatv(e 0, z) = e 0 9(Z~ + z i e i ) = Z for all z. O n e defines: The complex conjugate of z" z * = - Z ~ z i e i . The { e i } are called the imaginary elements.

The Cayley inner product: (x l y ) -

89

y + y*. x).

imaginary

elements

The Cayley norm:

C a y l e y inner

Iz[ 2 --= (z [z) = ~

product

(z ~')2e 0

Cayley norm

~=0

It may be verified by direct calculation that for z - x . y

Izl-Ixl l yl. For z = x . y

it follows from bilinearity that with (/3p)N~ = (ep- e~ ]e~,)

z ~' = B ~'~ ( x ) y ~ = ( B p ) ~ ' , , x P y ",

One can check that the matrices Bp satisfy BpBp=

l]

BpB,~ + B,~Bp = O ,

(2)

p ~ tr

=

where B d e n o t e s the transpose of B. Explicitly, it follows from the definition (1) that x

x X

(Bp)"~x p = B ". ( x ) :

complex conjugate

x X

0

1 2 3 4 5

X 6 X 7 X

--x

x X --x X --X X ~X

1

0 3 2 7 6 5 4

--x

-x X x X X --X --X

2

3 0 1 6 7 4 5

--x

x --X x --X X X --X

3

2 1 0 5 4 7 6

--x

-x --X x X --X X X

4

7 6 5 0 3 2 1

--x

x --X --x X X --X X

5

6 7 4 3 0 1 2

--x

-x X --x --X X X X

6

~x 7

5

X4

4

5 X X6

7 2

--X 1

1

mX2

0 3

--X X

3 0

(3)

266

V. RIEMANNIAN MANIFOLDS. KA,HLERIAN MANIFOLDS

The first column of B(x) consists of the components of x. If x ~-0 the remaining columns consist of 7 smooth orthogonal vector fields li(x ) on

E

(0). x = v(eo, x) = e o ' X = x~

+ xiei

and l i ( x ) =- v(ei, x ) = (left) siebenbein

e i .x-

+ x~

+

aijkxJek .

(4)

If Ixl = 1, the set {li} is a particular (left) siebenbein on $70 . Alternatively we could have used the right product function btr(V , Z)

right siebenbein

-xieo

"-- Z " V

7

and constructed a particular right siebenbein (r~} on H=8, and thus on S 0" X = Vr(e0, X)

and

r i ( x ) = t'r(ei, X) "- x ' e i = - x i e o

+ x~

aijkXJek 9

(5)

We identify e o ~ O with the point N ~ ~0 left fixed by SO(7). Remark: Formulae (4) and (5) are particular cases with a = e 0 of la(X) = (e i 9a). (a*. x) and rT(x) = (x. a*). (a. x) which depend on a, an arbitrary octonion of unit norm, because the algebra of octonions is not associative. Remark: Let J ~ be the 28 linearly independent vector fields defined by J o(x) = x e~ - xtJe~ , ot

a, ~ = 0 , . . . , 7 .

a

ot

The left and right siebenbein can be expressed in terms of the J ~ as follows: I i = J~ i + l a q k J ] k , ri=

J~ i -

89

k.

3a) Construct the Lie algebra ~ ( S O ( 8 ) ) in terms o f the left and right siebenbein vector fields on ~o. 3b) Show that . ~ ( S O ( 8 ) ) = ~ ( S O ( 7 ) ) ~ d ~ for tilt isomorphic to the tangent space Tlv~7o, where N E ~7o is left invariant under the action o f S 0 ( 7 ) on $7. A n s w e r 3a: We shall identify an element y E ~ ( S O ( 8 ) ) with the Killing vector field vv on [E8 generated by y. Recall (p. 164, and Problem III 6, Homomorphisms of a Lie algebra) that if

10. I N V A R I A N T G E O M E T R I E S

[%, y~ ] = c L , y~

ON THE SQUASHED

SEVEN SPHERES

for ( y~ } a basis of ~ ( S 0 ( 8 ) )

267

(6a)

then v , ] = -cL

v, .

(6b)

The vector fields {li} do not satisfy eq. (6b), but we note that [[li, lj], lk]

=

48 km q lm ,

[[1 i, lj], [l k, 11]]

=

6 ki. 6 m j __ 6 j k6 i m

6 qk m =

48qkm[l m , lt]--481m[lm q , It].

(7)

Thus the set { 89 ~[li, lj]} satisfy eq. (6b) and can be identified with 28 Killing vector fields v~, a = 1 , . . . , 28. Each vector field v,, can be identified explicitly with an element y~ ~ ~ ( S O ( 8 ) ) as follows. The action of SO(8) on E8 is an isometry, i.e., the Lie derivative of the euclidean metric g with respect to 'the Killing vector field v~ corresponding to a generator of SO(8) vanishes o~/3 a t~

O= (~.,..g)~...-

t~ /3 a v

Ox ~ + , g x . .

The vector field v~ defines an antisymmetric 8 x 8 matrix y~ _- a v 2 ( x )

0x v

.

(8)

/ 3 1 , / 3 2 , . . . , /37 be equal to /1, 1 2 , ' " 9 , /7, respectively; then 3'1, Y2,. 9 9 3'7 are equal to B1, B E , . . . , B7, respectively, where the matrices {B0, B1, .... , B7) can be read off from (3). We now Check that if {/3~} satisfy (6b), then the {y~ } defined by (8) satisfy (6a). Indeed, /3~(ct- 8 , . . . , 28) are identified with [li, lj] and Let

0[l,(x), lj(x)]~/ax "= - [ B , , Bj]'~, .

(9)

A n s w e r 3b: Since SO(8)/SO(7) is a reductive homogeneous space (see answer 1)) ~(SO(8)) = ~(SO(7)) ~ ~ / ,

(10)

where ~(SO(7)) N ~t = {0)

and

~ac(~(SO(7)))J/t C ,/R.

First we check t h a t {l[li, ly]} is a basis for ~(SO(7)). Indeed ~(SO(7)) = ~(Spin(7)). Now, the 8 matrices Bp defined by (Bp)~'vx p= B~'v(x), where B(x) is given by (3) generate a representation of the Clifford algebra cr 0) (p. 65 and [Problem I 4, Clifford algebra]). The equation B 0 = 1] and (2) together with B i - - B i yield

268

V. RIEMANNIAN MANIFOLDS. K,~HLERIAN MANIFOLDS

B, Bj + BiB i = - 2 6 0 ,

i, j = 1 , . . . , 7.

(11)

Given an explicit representation of r162 0), the construction of ~(Spin (7)) is straightforward: Let ( A , ( t ) } , be a one-parameter subgroup of Spin (7), i.e.,

Aa(t)BiA-al(t) = a/(t)Bj ,

(12a)

det Aa(t ) = 1.

(12b)

The derivative of (12a) with respect to t at t - 0 is f

AaB i - BiA a = or/By,

{

Aa

dAa(t) /dtlt-o ,

a/=da/(t)/dtl,=o

.

The solution of this equation is easily shown (p. 177 and [Problem I 11, Lie algebra]) by repeated applications of (11) and use of (12b) to be equal to

Aa = _ 1 ot qBiBj ,

ot ij

__ Ol/ .

The transformation B~--~ A B A -1 is an isometry; hence the matrix a is orthogonal and a is antisymmetric. To form a basis for Ze(Spin (7)) we need 21 such linearly independent h a. An obvious choice is the set { - I [ B i, Bj]}, which by (9) can be identified with the Killing vectors

(111,i, 1,]}

i, j = 1 , . . . , 7 .

We now exhibit the isomorphism between ~ and TN5 7. A basis for ~ is the set {3%, a = 1 , . . . , 7} defined by (8). Note that A d ( S O ( 7 ) ) ~ C ~ as a consequence of eq. (7). We have already shown the correspondence between {%,, a = 1 , . . . , 7} and { 89 But t h e / i ' s are seven everywhere independent vector fields, so { 89 is a basis for TN5 ~. This result can be understood in a more general context as follows. Let 7r denote the projection map zr" SO(8)---~ S O ( 8 ) / S O ( 7 ) - ~0

by g~--~ g N .

Recalling that ~ ( S O ( 8 ) ) is naturally isomorphic to TeSO(8 ), the derivative of this equation gives a map 9r~" ~ ( S 0 ( 8 ) ) - - > TN~7o . This is a linear map between vector spaces, so ~ ( S O ( 8 ) ) -- Ker rr' @ Im zr'.

(13)

We will show that Ker 7r' = SO(7) and that Im 7r' is spanned by {li(N)}. First consider Ker 7r'. Write v E TeSO(8 ) as v = ( d / d t ) g ( t ) l t = o for a

10. INVARIANT G E O M E T R I E S ON THE SQUASHED SEVEN SPHERES 269

one-parameter subgroup g(t). Then d

77"0 = -dt ( g ( t ) N ) l t = ~ = 0

if and only if g ( t ) N = N near t = 0.

In that case, g ( t ) is in the isotropy group SO(7) of N, so v E ~(SO(7)). Now consider Im 7r'. Recall that N is identified with the octonion e 0, and that for an infinitesimal rotation, Be, = too~,~e, with w,~ = - % ~ , . So ( d / d t ) ( g ( t ) N ) ] , = o = tooie i. But l i ( N ) = e i . e o = e i , s o { l i } spans Im 7r'. The decomposition (13) then corresponds to (10). R e m a r k : The isomorphism of the Lie algebra ~ ( S O ( 7 ) ) and ~(Spin(7))

does not imply the isomorphism of the groups. In particular, here, the group action of S O ( 7 ) C SO(8) on So7 has a fixed point" and the group action of Spin(7)C SO(8) on S 7 is transitive. The homogeneous space SO(8)/Spin (7) is isomorphic to the round seven sphere with antipodal points identified. 7

4) Construct SO(8) invariant metrics on ~o. A n s w e r 4: Given an SO(7) invariant tensor on TN$7O one can construct an

SO(8) invariant tensor field on T $ 7 by the left action Lg: N - .

g ~ SO(8)

x = gN ,

!

L g ( N ) " TN~7O---~ Tx~g 7

with V N ~ V x

and similarly for higher-order tensors. For instance, given an SO(7) invariant scalar product g on TN~ 7, the metric on ~0 defined by 9gx(Vx, Wx) = gN(VN, WN)

where x = g N

and

Vx = L ' g ( N ) V N

(14) is SO(8) invariant by construction. Conversely, on any homogeneous space G / H , uniquely determined by its (H-invariant) value at H. We shall now show that modulo a scale factor, product o n TN "~7o invariant under the full group 7-dimensional vector space, namely

g(VN ,

=

8qVN W NJ ,

a G-invariant tensor is the point identified with there is only one scalar of rotation SO(7) on a

i= 1,... ,7.

(15)

More generally, we show that up to a multiplicative constant, there is only one SO(n) invariant second rank tensor on R n, namely the symmetric tensor ~ . Indeed, let

270


T = T~,~(x) dx ~ dx ~ and let v(x) be the Killing vector field on R ~ corresponding to y E SO(n). The tensor T is SO(n) invariant if its Lie derivative with respect to v vanishes

(~o T) ~,,, = T~,,,,pvP + L ,,v p,~ + T~,pU p, z, "--0. The Killing vector field v vanishes at the center of rotation, 0. Hence

(,~,,T),,,,(O) = Tp,,(O)v p,~, + T,,p(O)v p,~ = 0 . Let T and V be the matrices of components Tp~ and v ~./.L respectively. Then (16) can be written -VT

+ TV=O

(16) = -

v ~'.p~'

at O.

Schur's lemma and the fact that the matrix T is symmetric imply that T~,~(0) = c6~,~. It then follows from (15) and (14) that modulo a scale factor, there is only one SO(8) invariant metric on 5 7. Let { A~} be the dual siebenbein defined by ,

tj(x))

=

a'j .

(17)

Eq. (17) can be solved in terms of the Cayley inner product

Ai(x) = (li(x)[ ) = (e i 9x[ ).

(18)

The invariant metric on 507 is

g(x) = auA'(x)A J(x). invariant connection

(19)

5) A connection on a manifold M is said to be G-left invariant if it defines a covariant differentiation which commutes with a left action of G on M. In general, a connection which is compatible with a metric g is uniquely determined by g together with a once contravariant, twice antisymmetric covariant tensor called the torsion tensor (p. 308" There T is assumed to be z e r o - the theorem remains true for an arbitrary T). If the metric and the torsion tensor are left invariant, so is the corresponding connection.

Show that left invariant torsion tensors on a symmetric space are identically zero; show that SO(8)-invariant connections on the round seven sphere 570 have no torsion. Answer 5" A symmetric space [see Problem III 7, Homogeneous spaces]

10. INVARIANT GEOMETRIES ON THE SQUASHED SEVEN SPHERES 271 is a homogeneous space G / H with an involutive automorphism. The round seven sphere with the metric (19) is a riemannian symmetric space [Problem III 6, Homogeneous spaces question 8]. Hence there is an isometry s N E H N, where H N is the isotropy subgroup which leaves N E 57 fixed, such that

y~ ~o

s N(exp Vu) = e x p ( - VN), and such that the induced mappings are

~;,(x): Tx~ - - , r ~ ,-l(x): SN

Tx, ~ o __..~

~o

by Vx-->-V~No0

9 7 T,N(~)So

by w,, --~ -

O.)$N(X ) ,

If we use the symbol s N to designate the action on an arbitrary tensor induced by the isometry s N (rather than explicitly writing the appropriate combination of Sly(X) and s}-a(x)), then at the point N

SNT(N) = ( - 1 ) P T ( N )

if T is of degree p .

A tensor is invariant under the isometry s N if

s~ T(x) = r(,~(x)). Hence there are no invariant tensors of odd degree on $7. The same argument applies to arbitrary symmetric spaces. 6) Let G 2 be the group of automorphisms of the algebra of octonions which leave e 0 invariant. The two following sequences of inclusions

~(SO(8)) = ~W(Spin(8)) D ~(Spin(7)) D ~(Spin(6)) D ~(Spin(5))

(20)

and ~(SO(7)) ~ ~(Spin(7)) D ~(G2) D ~ ( S U ( 3 ) ) D ~ ( S U ( 2 ) )

(21)

have been used to identify the following reductive homogeneous G / H , where G and H are compact and simply connected, with manifolds $7, diffeomorphic to the round sphere 57, with increasingly smaller isometry groups Spin(7) G2 57 = Spin(6) = SU(4)

su(3) Spin(5) SU(2)

su(3) '

272

V. R I E M A N N I A N

MANIFOLDS.

KA, H L E R I A N

MANIFOLDS

For a study of ~71 see [Englert, Roman, Spindel]; for a study of ~ see [Awada, Duff and Pope; Duff, Nilson and Pope]; for N~ and a detailed study of all the squashed seven spheres N7i above see [Sammelmann] " a) Give a basis for the Lie algebras in (20) in terms o f the left siebenbein { ~ei}. b) Choose a basis for ~ ( S O ( 7 ) ) from which one can read off the inclusion

e(so(7))

c) Show that the homogeneous spaces ~7 are 7-dimensional. Answer 6a: Using the basis for ~ ( S O ( 8 ) ) computed in 3a we have:

Basis for ~(SO(8)) ~(Spin(7)) ~(Spin(6)) ~(Spin(5))

dimension { 89 ~[1~, lj]}, i, j = 1 , . . . , 7

( -~[1i,/j]}, i, j = 1 , . . . , 7 { 88 i,/j]}, i, j = 1 , . . . , 6 ( 88 [l~,/j]}, i, j = 1 , . . . , 5

28 21 15 10

Answer 6b: We note that the difference of a left siebenbein and the corresponding right siebenbein has a zero e0-component: !2(li - ri) -- aij k x ' e k

9

The vector fields 89 i -- ri) correspond under the identification defined by (8) to seven (7 • 7) antisymmetric matrices 1 a (lki _ rki) = (ai)jk = (a,)jk E ~ ( S O ( 7 ) ) i, j, k = 1 7 2 ax j ' ""' " The remaining fourteen (7 x 7) antisymmetric matrices necessary to make a basis for ~ ( S O ( 7 ) ) can be chosen to be the generators of the group of automorphisms G 2 of the space O of octonions which leave e 0 invariant. Under the identification (8) they are the matrices corresponding to G,j= - 88

kl

[lk, ll] ,

where kl_zc. 1 m 1 e l i kl = 1 8 i ] a i] akin 9

The 21 matrices corresponding to G O satisfy the 7 equations i]

ak Gii = 0 ;

14 of them are linearly independent and form a basis for ~W(G2).

10.

INVARIANT

GEOMETRIES

ON

THE

SQUASHED

SEVEN

SPHERES

273

Remark: G~j =

PijklJkl

,

where

Jk,(x) = X%, -- X'ek . Remark: See [Gfinaydin and Gursey] for further study of ~ ( G 2 ) and the sequence of inclusions: Answer 6c: We have the following table: ~(G2) is of dimension 14 ~(SU(3)) is of dimension 8 ~(SU(2)) is of dimension 3 which together with the table of answer 6a gives d i m ( ~ ) -

7.

d) Compute the number n of families of G-invariant metrics on $7, using the following information: with ~ defined by ~ ( G ) = ~ ( H ) O ~ , representations of H on All are not, in general irreducible (representations of H on ~ ( H ) are of course irreducible since H is simple in all our examples); the dimensions of the irreducible subspaces of ~ are as follows. ~ dimensions of irreducible subspaces of ~

Spin(7)/G 2 Spin(6)/SU(3) 7

6, 1

Spin(5)/SU(2) 4, 1, 1, 1

Answer 6d: We have shown in Answer 5) that there is an isomorphism between the H-invariant scalar products o n TN ~7 and the G-invariant metrics on 57 In Answer 3b), we have constructed an isomorphism between TN57 and ~ . In the case ~ = Spin(7)/G 2, the representation of G 2 on u/// is irreducible, and the argument developed in Answer 5) shows that there is only one Spin (7)-invariant metric on ~71, modulo a scale factor, i.e., there is a 1-parameter family of Spin (7)-invariant metrics on 5~. In the case ~ = Spin(6)/SU(3), we can scale independently the SU(3) invariant scalar products defined, respectively, on the 6-dimensional irreducible subspace and on the 1-dimensional irreducible subspace of ~ . 7 There is a 2-parameter family of Spin (6)-invariant metrics on $2. In the c a s e 5 72 - - - Spin(5)/SU(2) , let {v~} ' a = 1 ' 7 be a basis for ' ' Tu5 ~ and consider an action of SU(2) on TN~ ~ which leaves invariant the three 1-dimensional subspaces generated, respectively, by v5, v 6, v 7. The 6 scalar products (v~lvj), i, j = 5, 6, 7, define 6 components of a Spin (5) 9 9 lnvanant metric9 on ~a7 which can be scaled independently of each other "

"

274


and independently of the unique (modulo scale) Spin (5)-invariant metric obtained from the scalar product 8uV i V,j i, j = 1 , . . . , 4. Hence there is a 7-parameter family of Spin (5)-invariant metrics on 5~. REFERENCES Sammelmann, G.S., Ph.D. dissertation. Awada, M.W., M.J. Duff and C.N. Pope, "N = 8 Supergravity Breaks Down to N = 1", Phys. Rev. Lett. 50 (1983) 294-297. Brickell, F. and R.S. Clark, Differentiable Manifolds, an Introduction (Van Nostrand Reinhold, London, 1970). Cahn, R.N., Semi-Simple Lie Algebras and Their Representations (Benjamin/Cummings, Menlo Park, CA, 1984). DeWitt, B.S. and P. Van Nieuwenhuizen, "Explicit construction of the exceptional superalgebras F(4) and G(3)", J. Math. Phys. 23 (1982) 1953-1963. Duff, M.J., B.E.W. Nilsson and C.N. Pope, "Superunification from Eleven Dimensions", Physica A124 (1984) 219-248. Englert, F., M. Rooman and P. Spindel, "Symmetries in eleven-dimensional supergravity compactified on a parallelized seven sphere", Phys. Lett. B130 (1983) 50-54. Giinaydin, M. and F. G/irsey, "Quark structure and octonions", J. Math. Phys. 14 (1973) 1651-1667. Hurwitz, A., Math. Werke II 41 565-571. Jensen, G.R., "Einstein Metrics on Principal Fiber Bundles", J. Diff. Geom. 8 (1973) 599-614. Lain, T.Y., The algebraic theory of quadratic forms (Benjamin, New York, 1973). Lambert, D. and M. Kibler, "An algebraic and geometric approach to nonbijective quadratic transformations", Preprint LYCEN/8642, Physique Math6matique, 2 Chemin du Cyclotran, 1348 Louvain la Nueve, Belgium (1986). Sagle, A.A., "Some Homogeneous Einstein Manifolds" Nagoya Math. J. 39 (1970) 81-106. Van Nieuwenhuizen, P., "An introduction to simple supergravity and the Kaluza-Klein program", in RelativitY, Groupes et Topologie II, eds. Bryce S. DeWitt and Raymond Stora (North-Holland, Amsterdam, 1984). Weyl, H., The Theory of Groups and Quantum Mechanics (Dover, New York, 1950). Wolf, J., "The Geometry and Structure of Isotropy Irreducible Homogeneous Manifolds", Acta Mathematical (1968) 59-148.

11. H A R M O N I C MAPS

Harmonic maps between riemannian or pseudo-riemannian manifolds are the direct generalization of the usual Laplace or wave operators to nonlinear fields, meaning here fields which then take values not in a vector space but in some manifold N. Physical examples are the so-called or-models, for instance the one which considers four mesons fields fa constrained by the relation


275

4 2 ( f a ) X __ constant; a=l

they constitute a field taking its values in the three sphere. Harmonic maps appear in General Relativity, in the definition of generalized harmonic gauges, or in the construction of classes of solutions with given symmetries. Other classical examples of harmonic maps of importance in geometry and physics are given by the extremal (with respect to volume) submanifolds of a pseudo-riemannian manifold: geodesics, minimal (or maximal) surfaces; in this case the metric of the source is not a priori given, but is the pull back of the metric of the range manifold. Let us remark, finally, that the dynamics of a string is governed by an harmonic map from a 2-dimensional hyperbolic manifold into space-time. There is a wealth of r e s u l t s - a n d also of open p r o b l e m s - a b o u t the existence of an harmonic map f: (M, g)--->(N, h) in a given homotopy class of maps M---> N, in the case where (M, g) and (N, h) are properly riemannian manifolds (cf. [1, 2]). In many examples, however, (or-models, harmonic gauges) the metric g of the source manifold is of hyperbolic signature. The natural problem for hyperbolic harmonic maps is the Cauchy problem, determination of the map from its values and the values of its first derivatives on space-like submanifolds S C M of the source (cf. [3, 4]). 1. DEFINITIONS Let (M, g) and (N, h) be two smooth riemannian manifolds of arbitrary signature. Let

f: M--->N be a smooth map. The differential of f at x E M is a linear map (p. 121)

f'(x)" TxM'---~ Tf(x)N, it is therefore an element of T*xM | Tf(,,)N. The differential itself, f', is a mapping x ,--->f'(x), that is a section of the vector bundle with base M and fibre at x the vector space T*x M | Tr(x)N. This vector bundle - the bundle of one forms on M with values at x in Tr(x)N- is denoted T*M | f-~TN. If (x i) and (y~) are respectively local coordinates in M and N, and f is represented in these coordinates by

y'* = f"(x i) the derivative f ' is represented by

differential

276

V. RIEMANNIAN MANIFOLDS. K,~,HLERIAN MANIFOLDS

Ox j The metrics g on M and k on N endow the fibre at x of the vector bundle E = T* M | f - 1TN with a scalar product G(x) - g ~(x) | h(f(x)), where g~ is the contravariant tensor canonically associated with g. In coordinates, if u and v are two sections of E,

G(x)(u, v)= giJ(xk)h~( fa(xk))U~(xk)v~(xk) . The scalar function on M

e(f)energy density

G(f', f')

is called the energy density of the mapping f. In local coordinates

e ( f ) = giJ(xk)h~(fa(xk)) ~Of~ (X k) Of OX-'I~ 7--" (xk) " This can also be written

e ( f ) = tr s f * h . The vector bundle E =- T*M | is endowed with a linear connection V, mapping sections of E into sections of T*M | E, by the usual rules (cf. p. 303)" if t is a section of T*M and s of f - I T N we have Vo(t| s ) = sVot| s + t| f* hVos with sV the riemannian covariant derivative in the metric g and f* hv o defined by pull back of the riemannian connection of h. In local coordinates if (xi)~--~ (uT(xi)) is a section of E, we have

V, uj (x

=

o, uj(x j) + - Fitj(xk)u?(xk),

(1)

where Fa~a and F~tj denote respectively the riemannian connections of g and h. The derivation law for tensor products sections of (| TM) p | q is deduced as usual. Check that VG = O.

Answer 1: V o(g~ | h) = gVog ~ | h + g ~ | f* hVoh -~ O. harmonic

2) The mapping f is called harmonic if trg Vf' = 0

also written as

a) Write eq. (2) in local coordinates.

8 df = 0

or

V. df = 0.

(2)


b)

277

Show that f satisfies (2) if it is a critical point of the "energy functional" E( f ) = f e(f) d/x(g). M

d/x(g) is the volume element of M which is supposed for instance compact so that the integral makes sense for all smooth maps.

Answer 2a: Applying formula (2) we obtain gq ~i

69j fa =_ gij(a2.fa _ l~ik.t~k f a jr_ t~i f'~Oj f~I'aa~z) .

(3)

Answer 2b: The energy functional is f~--~E ( f ) = f G(f', f')d/J,(g)= f gq(xk)h,,,~(f'~(xk)) M

M

X t~i fat~j f ~ d / z ( g ) .

The space of smooth maps f: M--~ N is not a vector (or affine) space if N is not a vector space, but it is an infinite dimensional manifold ~ . The energy is a mapping ~t--~ R and its derivative at f E ~ a linear map E'(f)" Tf~t--~ R. The tangent space to ~ at f is the space of smooth sections of f-ITN, i.e., mappings x~--~u(x)E Tf(x)N. Then

E'( y) " u ~

giJ(xk)

[ aaY h ~~ ( f~(xk))u ~(Xk)a, f~aj f~

M

+ 2h~(fX(xk))~) i u~j f~] d/x(g) which gives, after integration by parts and some computation

E'(f) u = f u~gqVi aj f~ d / x ( g ) ,

us =

h~u ~ ;

M

thus if f is a critical point of E, that is if

E ' ( f ) . u = 0 for all u, we have

. .

trg V f ' ~

g"ViOj f~ = O .

The section trg Vf' of f-~TN is called the tension field of f; a map is harmonic if its tension field is zero. 3) Show that if f is a C 2 harmonic mapping (M, g)-->(N, h) then it satisfies the equation trg vEf ' - Ricc(g). f ' + trg(f* Riem(h) 9f ' ) = 0

tension field harmonic

278


that is, in local coordinates giYViVjO k f ~ -

R{ Oj f'~ + Rx~al30k f;~gii3i f~'t9i f ~ = O.

Answer 3" Straightforward computation by derivation of eq (3) and use of the Ricci identity. 4) Consider now the case where the metric g on M is not a priori given but is the pull back of the metric h of N,

g =f*h. a) Show that the numerical value of E(h), for any given f, is the total

volume of M in the metric g (up to a constant factor). b) Show that the mapping f is a critical point of the mapping f ~ E ( f ) ,

with E ( f ) depending on f now also through g, if and only if it is a harmonic map from (M, g) into (N, h). Answer 4a" e ( f ) = (g* | h)(f', f ' ) = trg f*h = trg g = d or in coordinates,

e ( f ) = g i/h.~O, f"Oj f~ = g ugu = d ,

dimension of M

SO

E ( f ) -- d vol M .

Answer 4b" The derivative of the mapping f ~ E ( f ) , with g depending now on f, is the linear mapping u~-->(E'/+ E'sog'/).u with E'f(f) the linear mapping denoted E ' ( f ) in the previous question. Denoting by 6g = gfI 9u we have, since gu = h~a a i f ~ a j f a ,

8 gii = ha~ ( ai u~ai fa + O, f~ai u and a straightforward computation gives

(E'sog'/).u =

fM 8g ,j(haoOi fOj

= Mf Sg~ g'J -

- 89

d g~j) d ( g )

=

fO t

)d/z(g)

f d - 2 g~ 2

dtx( g)

M


279

which gives, using a previous formula

(E'gog}).u=

d-2 2

E'~.u.

Thus finally d E'( f ). u = -~ E'r . u . Thus f is a critical point of E( f, g ( f ) ) if and only if E~ = 0, i.e., if f is a harmonic map. 5) a) Suppose M is 1-dimensional, show by a direct computation that its harmonic imbedding is a geodesic of N. b) Let f: M---~ N be an imbedding and g = f * h . Show that V dr, 2covariant tensor on M with values in TN, can be identified with the second fundamental form k(p. 312) of M as submanifold of (N, h). c) Suppose N = S x R, M is diffeomorphic to S and the embedding M ~ N is given by a mapping M~SxR

by

y~(x,t),

where the mapping M ~ S by y ~ x is a diffeomorphism. Write the partial differential equation expressing that the imbedded submanifold M is extremal for the volume element induced by the metric of N. Answer 5a" If M is 1-dimensional the volume element induced by the metric of N on the image of M is the length element. Thus if the embedding f is harmonic, the image of M is a geodesic (p. 321). We give now a direct computational proof: if M is one dimensional with local coordinate t, the embedding in N with local coordinates x '~ is given by X

-~"

"

the induced metric on the image of M is df '~ df t3 dt 2 = g dt 2 ds 2= h~t3 dt " dt The equations for the tension field read as the geodesic equations _

g

dt 2

2 dt

dt

a

dt

dt

~___

g

because d

d---s = g

-112 d

dt"

~

ds 2

q-

~a

ds

ds

=0

280

V. R I E M A N N I A N M A N I F O L D S . K A H L E R I A N M A N I F O L D S

Answer 5b" If M is an imbedded submanifold of (N, h), the second fundamental form at x E M is the covariant symmetric 2-tensor k x on M, with values in Trig)N, the subspace of TI(x)N orthogonal to f ' ( T x M ) in the metric h, given by (p. 313) kx(u

, Vx) = (Vi,

' fV)(x

,

Ux, Vx

L,

where V is the covariant derivative on (N, h) and _t_ the projection onto _1_ TI(x)N. We have, by a straightforward generalization of pp. 312-313

Vi,~f'v=f'~v+(Vf,uf'V)

~,

u

where V is the covariant derivative on (M, g). We compute in local coordinates x ~ on M and x a on N:

k ~ = k(a~, a~). If u = 0~ and v = 0~ then f ' u = a,, faO~, f ' v = a~ fbo3b thus

V,o = F~h~Oh ,

f'~v

= F~h~Oh fba b

while

Vi,uf'v

= aa fa(,Jaal3 f b + Fabcal3fC)o~t, c

fo

and therefore as announced k~a = (V~ 0~ f f ) O b . In particular we remark that Vf' is orthogonal to Tr(,,)f(M ), so

h ~ V~ a~ fbO~ f f = O ,

Va, [3, y ;

this formula can also be proved directly through integration by parts and permutation of indices. The tension field of (M, f ' h ) imbedded in (N, h) is the trace of the second fundamental form of this submanifold; if the imbedded manifold is extremal the mapping is harmonic, that is the trace of the second fundamental form vanishes.

Answer 5c" Suppose N = S • R with S diffeomorphic to M and, in adapted coordinates x ~ R and coordinates (x i) in M identified with coordinates in S by the diffeomorphism, f is represented by

fO

=

fi

= xi ;

then, the vanishing of the tension field reduces to the equatio~

12. COMPOSITION OF MAPS (trg Vf') ~

281

g ij ( v i 6 ~ + l-'o~ Oi tp6 q9 + 1-TiOo6 tp

+ r?oe,~ +

~;)= o

the Christoffel symbols being those of h. Remark" if ~(x i) = 0 (i.e., M is the submanifold x ~ well-known formula: ,

ij

0) then we find the

1

trg(Vf )o = g /-~oij= E no t r g K . Cf. references Problem V 12.

12. COMPOSITION OF MAPS

This problem uses definitions of the preceding problem (V 11 Harmonic). The composition of maps is particularly useful to transform problems about mappings between manifolds into problems for mappings which take their values in the vector space ff~q: it is a fundamental tool in the treatment of harmonic maps between properly riemannian manifolds by EeUs and Sampson [1]. We indicate here an application to the case where the source manifold is of hyperbolic signature; for further study of this case see [3, 4]. 1) Show that if f: M---~ N and F" N---~ Q are arbitrary smooth maps between pseudo-riemannian manifolds (M, g), (N, h) and ( Q, q) one has the identity (a dot is a scalar product in h)" V d(Fo f ) -= d F . V df + V dF- (df | d r ) ,

(1)

that is, if (x~), (x a) and (x A) are respectively local coordinates on M, N and O

g~l~(Fo f )A = OaF AV~ Or3fa + V~ebF AO~ faoa fb . Answer 1" By definition we have

V,, O~(Fof) a = a ~: ( F o f ) A _ F~t~Ba(Fof) A + ['sAc

• oo(Fof)"o~(Vof)~. By the law of derivation of a composition map we find ea (Fo f )a = O,,FAOt3i f , fa + OaeAO~ 2 fa , 2 ( F o f ) A =Oq2 abF A.,

a~fbe~

282

V. RIEMANNIAN MANIFOLDS. K A H L E R I A N MANIFOLDS

which shows that ratiO( F o f ) A ~tJaF A ( a ,2, ~ f " - F,,*~a, f" + Fb"~a,,fba~ff) 2 A c + (OabF - F a bOc FA + FBAcaaFBabFC)o,~fb~13fa;

the two terms containing Fb~c add to zero and we thus have the given formula. 2) A mapping f: (M, g)--+(N, h) is called totally geodesic if V d r = 0 .

Show that the composition of two totally geodesic maps is totally geodesic; show that if f is harmonic and F totally geodesic, then F of is harmonic. Answer 2: The first statement follows readily from formula (1). We deduce from this formula the tension field of For r(Fo f ) = dF . r( f ) + V dF . g(df , df)

(2)

and see that r ( F o f ) = 0 (i.e., For is harmonic) if r ( f ) = 0 and V d F = O. It is not sufficient for r ( F o f ) = 0 that r ( f ) = 0 and r ( F ) - O. 3) Suppose that F: (N, h)--+(Q, q) is a pseudo-riemannian immersion,

i.e. h=F*q.

(3)

a) Give the relation between the densities of energy e(Fof) and e(f). b) Give the relation between r ( f ) and r(Fof). c) Show that the map f: M--+ N is harmonic if and only if r(Fof) is

perpendicular to F(N) at each point, and F(N) is nonisotropic (which it will never be if h is properly riemannian). Answer 3a: e(F of ) = qaag ~aa,,(Fof)Aaa(Fof) n --qAB,gaF abF g ,,13a,~faa~fb = hob g~aa~ f~ fb = e ( f ) . _

A

a

Answer 3b: Using (2) we obtain r ( F o f ) A = aaFAz( f ) a + g"lJa,, faalj fbVa abFA .

Each tangent vector Xta ) to Q, with components aaFA is tangent to F(N),

tot get


283

and so is aaFAZ(f) a, while Y~ab)= (VaabFA) is orthonormal to F(N) if F is a pseudo-riemannian immersion (cf. Problem V 11, Harmonic, answer 4b). Thus r ( f ) is the projection of r(Fof) on the tangent plane to F(N) (subspace of the tangent plane to Q).

Answer 3c" Follows from 3b). 4) An arbitrary smooth manifold N can always be embedded in a manifold R Q for Q large enough. Let F be this embedding. The euclidean metric of R Q induces on N the metric 3' given in local coordinates by (x ~ coordinates on N, x A on R Q) Tab : aaFAcgbFBSAB

9

Set taA=

TababF A ,

then

acFBt aA ~AB

= T abt~b F A a c F B S . 4 B : ~ ac 9

a) Show that the quadratic form q A B = 8 A B -- Tab taAtbB q- h a b t

.aA.bB t

(4)

defines a metric q in a neighborhood U of F(N) which induces on N the metric h. b) Show that the geodesics orthogonal to F(N) in the metric q coincide

with the geodesics orthogonal to F(N) in the flat metric of R Q. Answer 4a: By the previous relations the metric F*q induced on N by q is hab =

qABOaFAObFB

.

Answer 4b" Let n denote the dimension of N. Consider the field over F(N) of hyperplanes T y F ( N ) o f dimension Q - n orthogonal to F(N), i.e., to TyF(N) at y. Consider also a smooth field of orthogonal (in the euclidean sense) Q-bein with its first n vectors at y in TyF(N) and the remaining Q - n vectors e a, A = n + 1 , . . . , q in TyF(N). If U is an appropriately chosen neighborhood of F(N) in R ~ each point X in U has one and only one euclidean projection y = H(X) on F(N), that is it belongs to one hyperplane T y~F(N) We define local coordinates yA for X~Uby

284

V. RIEMANNIAN MANIFOLDS. KAHLERIAN MANIFOLDS yA=yA

1,

A=a=

~

9

~

n

~

where yA, A - 1 , . . . , n are local coordinates of F - l ( y ) , while the coordinates y a , A = n + 1 , . . . , q are given by the decomposition (note that X - y is orthogonal to F ( N ) ) e ~

X-y=

yAe A .

A=n+l

In these coordinates the mapping F" N---~ R Q is given by FA(x1,.

. . ,xn)=x

F A (X 1,

9

9

~

~

A

A=l,...,n

Xn) = 0

A=n+I,...,Q.

Thus aa FA = 6 Aa

a,A=l

~F A =0,

A=n+I,...,Q

....

,n

~ b = ~ab

qAB = h m B ,

A,B=a,b=l,.

.. , n

q~A = O,

a=l,...,n,

qAB = 8 A n '

A,B=n+I,...,Q

A=n+I,...,Q

that is, in these coordinates q

q = ha b dx a dx b+

~

(dx A)2

Afn+l

recall that h ab depends only on x 1,. .. , x". The geodesics orthogonal to F ( N ) in the euclidean metric at y are the straight lines x A =tC A , a

x =y

a

,

C A some constants

A=n+l,...,q,

a=l,...,n,

y

A

constants,

they are orthogonal to F ( N ) in the metric q since qab = in the metric q because FacB = 0 if C = n + 1 , . . . , Q.

O,

.

and geodesics

5) In this question and the next the definitions are the same as in question 4, but we identify N with its image F ( N ) in U C R e. Show

that

a

mapping

W: M---~ U C [Re,

is

a

harmonic

map


285

(M, g)---~ (N, h) if and only if it is a harmonic map (M, g)---~ (U, q) and it takes its values in N. Answer: A harmonic map W: (M, g)---~ (U, q) satisfies the equations VaVaW

A ~__ gat3(o~2 W A -

A

o.w

+ r

aco

w"o

w

) ,

(5.1)

where the FBAc are the Christoffel symbols of the metric q. In the adapted coordinates of section 4 w e have

FBA = 0

if

C

FBmc=ffbac

A=n+l,

A=a,

if

9

9

~

B=b9

9

Q

C=c,

a,b,c=l,...,n9

where Fb~c denote the Christoffel symbols of h. W takes its values in N if and only if, in these coordinates

W m = O,

A = n + 1,...,

q.

(5.2)

Eqs. (5.1) and (5.2) imply that W is a harmonic map (M, g)---~(N, h) because they imply g,~:~o,~ .,2 W"

- r , ~oo~ W" + r~ "ca~W ba~w c) = o.

(5.3)

Conversely (5.3) and (5.2) imply (5.1). 6) Show that if the metric g is hyperbolic on M = S x R, with S space-like and if W[s takes its values in N and aoWls takes its values in W l s 1TN then W, solution of (5.1) takes its values in N.

Answer: To show that W takes its values in N is to show that in adapted coordinates on U C R q, W A = 0, A = n + 1 , . . . , q. Since W satisfies (5.1) and since for A = n + 1 , . . . , q we have FBAc = 0 we have that W satisfies ga~Z~ 2 W A A A) = o, A=n+I,...,Q. By the uniqueness theorem for the Cauchy problem for hyperbolic equations (p. 521) we have W A = 0 if WAIs = 0 and aoWAis = 0 ; the conclusion follows.

Remark: The adapted coordinates x A, A = 1 , . . . , q are not globally defined on U; they identify U with a manifold N x I q-n, with I some interval of R, and N has in general no global coordinate system. These coordinates are an intermediate step in the proof that the problem of

286

V. RIEMANNIAN MANIFOLDS. KJkHLERIAN MANIFOLDS

constructing a harmonic map (M, g)---> (N, h) can be reduced to a system of ordinary P.D.E.: this is done by taking global coordinates in U, for instance, the ones induced by the cartesian coordinates of R q. Another procedure for operating this reduction to a usual system is by considering N as a submanifold of (R q, e), with the induced metric, and using Lagrange multipliers (cf. [3, 4]). REFERENCES [1] J. Eells and H. Sampson, Amer. J. Maths. 86 (1964) 109-160; J. Eells and L. Lemaire, Bull. London Math. Soc. 10 (1978) 1-68. [2] A. Lichnerowicz, Symposia Matematica, vol. III (1970) 341-402. [3] J. Ginibre et G. Velo, Ann. Phys. 142 (1982) 393-415. [4] Y. Choquet-Bruhat, Ann. Inst. Poincar6, 46 (1987) 97-111. [5] Gu Chao Hao, Comm. on pure and app. Maths XXXIII (1980) 707-737. [6] Y. Nutku, Ann. Inst. H. Poincar6, A21 (1974) 175-183.

13. K A L U Z A - K L E I N THEORIES

INTRODUCTION

The rigorous nonlinear equations of a Kaluza-Klein model on a manifold Vd are equivalent to the usual Einstein-Yang-Mills equations coupled with a scalar multiplet; a cosmological constant appears in Einstein's equations in the case of a nonabelian gauge group. To enlarge the group without enlarging the dimension of the manifold V d the theory can be extended to the case where the fibre is no longer a Lie group, but rather a homogeneous space G / H (cf. Coquereaux and Jadczyk); the class of exact solutions satisfying the original Kaluza-Klein ansatz for the group G - that is, of metrics on Vd invariant under G - is then greatly restricted. This difficulty can be avoided for physical applications in two ways: either by considering that the full Kaluza-Klein ansatz governs only the ground state (cf, for instance, Duff, Nilsson and Pope (2)), or by relaxing this ansatz in the full nonlinear model by requiring only that the metric on V d projects on I/4 and defining a G-connection on V4. In this latter approach, the equations are no longer taken to be the vanishing of the Einstein tensor of the metric ~ on V d, but are deduced from a "reduced" lagrangian on V4, obtained by integration over the fibres of V d of the

13. KALUZA-KLEIN THEORIES

287

scalar curvature of ~ (cf, for instance, Percacci and Randjbar-Daemi or Duff, Nilsson and Pope). In this problem we establish the equations of the original Kaluza-Klein theory. We take V d to be a fibre bundle over V4, with fibres the orbits of G, and with a metric ~ on V d invariant under the right action of G. It is also possible to start from a manifold V d with a metric ~ with a group G of isometries, but in the study of such a metric one is led to make hypothesis on the orbits; the simplest is the one we use here. 1. Let V d be a d-dimensional differentiable principal fibre bundle whose base is a 4-dimensional differentiable m a n i f o l d V 4 and whose structure group is a Lie group G.

a) Suppose that V d can be e n d o w e d with a pseudo-riemannian metric ~, which is invariant under the right action o f G on V d (p. 129): S h o w that ~, determines i) a G-invariant metric ~ on each fibre (as long as the orbits o f the action o f G are not null f o r ~); ii) a connection on Vd; and iii) a pseudo-riemannian metric g on V 4. b) S h o w the converse. c) Write the metric ~, in local coordinates adapted to the fibred structure and the isometries o f V d. A n s w e r lai): A G-invariant metric ~ on the fibre Gx at x E V4. Suppose that V d admits a metric ~ which is invariant under the fight action of G. Let 2 be a point in V d lying over x in the base space V4. Since the right

action R e (y ~ G) is transitive on the fibre at 2, the fibre may be viewed as the orbit G x of G through 2. Such an orbit is a submanifold of V d diffeomorphic to G. If G x is non-null with respect to ~, then ~ induces (p. 290) a metric ~: = i*~ on G x. It is right-invariant, since the inclusion_ mapping i" G x ~ V d commutes with R e and ~ is invariant under R e. ii) A connection on Vd, defined by horizontal vector spaces H~, x E V d. Recall that (p. 359) a connection on a principal bundle may be defined as a field of horizontal vector spaces invariant under Ry. Since the orbits G x are not null, they admit at each point 2 an orthogonal hypersurface H~ of dimension 4. We shall use the invariance of the orbits and of the metric under Ry to show that these hyperplanes are invariant. The tangent space Vi to G~ at 2 is isomorphic to q3, the Lie algebra of G, and is spanned by the Killing vector fields of the action Ry (p. 360). The space H~ is the orthogonal complement of Vi, i.e., the space of tangent vectors u to V d at 2 such that

288

V. R I E M A N N I A N MANIFOLDS. K A H L E R I A N MANIFOLDS

$~(u, v) = 0

Vv e y~.

Now, f

Ry V i = VRy~

and since ~ is invariant,

that is, ! f L ( u , v) = ~ayf,(Ryu, RyV) .

These inequalities and the definition of H i imply that R y H i = Hhy~ .

Further, the projection zr: Vd---~ V 4 given by ~ x = 7r(~) defines an isomorphism H~ ~ T x V 4, since the tangent space to V e at ~ is spanned by Vi and H i, and 7r'Vi = 0. Thus the field H i of vector spaces meets the requirements (p. 359) to define a connection on Vd. iii) The pseudo-riemannian metric g on V4 is given by

g~(u, o)= ~(;,, ~) ,

u, v e L V 4 ,

where ~ and 6 are the horizontal vectors which project onto u and v, respectively. Due to the invariance of ~ and H~, this definition is independent of the choice of $ E ~r-~(x). The metric g determines the "distance" between orbits of G. The field of 4-hyperplanes H~ is not in general completely integrable: (V4, g) cannot be identified with a submanifold of ( V d, ~).

Remark:

A n s w e r lb" Conversely, an invariant metric on each fibre, a field H~ of

horizontal subspaces of a G-connection on Vd and a metric on V4 determine an invariant metric on V d. To define this metric, decompose each tangent vector o to V d into its horizontal and vertical components o H and v v, and set

~(~v, av)= ~(Ov, av), ~(t~v, a.) = 0, g(aH, 6H)= g(~r't~H, 7r't3H) 9 lc: To write ~ in adapted coordinates we consider a local trivialization r of V d

Answer

4)'7r

-1

(U)--> U x G ,

U the domain of a chart in V4

13. KALUZA-KLEIN THEORIES

289

and we choose as local coordinates, in ~r- I(U), XM -- (local coordinates x ~, a = 0, 1, 2, 3 in U, and local coordinates ym, m = 1 , . . . , d - 4 in G}. We take a moving frame ( ~ , ~m) such that e m a r e the vertical vector fields z~ ~.m=r m=l,...,d-4, where e m denotes a basis of right-invariant vector fields on G, and such that ~ are the horizontal vector fields (i.e., fields orthogonal to ~m) which project onto a basis in V4" I^ r 7re~=Tr~a~Or

=e ~

with 7rcan" U • G ~ U by (x, y ) ~ x. We can take for instance the natural basis e~ = a / d x ~, or an orthonormal basis e~. In any case we have r

"~

m

e a = e~ - A ~ e m .

Since r e~ has a zero canonical projection on U and is invariant under the right action of G, the A m depend only on the coordinates in U. A m em gives the connection 1-form (cf. p. 361) on U determined by ~. The identification of A m em with the connection 1-form differs from the usual one, which is done with left-invariant vector fields. R e m a r k : The vector fields em are invariant under the right action of G" Ryem(fC ) = ~.m(Ry.~) .

The Killing vectors X of the right action of G (i.e., the vector fields generating the transformations 2~-->Ry,~, (cf. p. 154)) are such that, if t~--~ y ( t ) is a one parameter subgroup of G, d(Ry(t)02) dt = X(/?Y(0"f) " The images of ~,, and X in a trivialization are the right-invariant and the left-invariant vector fields e m and X on G. We have [X, era] "-0, SO [fit', em] = q~'-l[ X, em] = O. The coframe dual to ( ~ , em) (P" 136) is found to be ~a

= q~.,(. O a = r

r

a

,

tb m = r

(

0 m + A ~mt o

a)

,

where (to e) is the dual of (e~), and (0m) is a basis of right-invariant 1-forms on G dual to (era). In this coframe the metric ~ reads ~S 2 - "

g ~ o % , fl +

~mn(O m -Jr- A m t o a ) ( o m + A ~nt o l 3 ) ,

290

V. R I E M A N N I A N MANIFOLDS. K.~,HLERIAN M A N I F O L D S ~1r

scalar multiplet

ot

ot

where we have identified "/;'can(.O and t o . A necessary and sufficient condition for this metric to be invariant under the right action of G is that g,,/3, ~:m, and A m depend only on the coordinates (x/3) of V4, since -~xO = 0 if X is a Killing field of the right action of G. In the trivialization and the local coordinates discussed above, the space-time metric is g ~ w aO)/3, the G-connection is represented by the ~3-valued 1-form A = Am~o"em, and the induced metric on the fibre x ~ = const is ~m,0"~0". In a given trivialization, the ~:,,, are scalar fields on U, i.e., they behave like a set of scalar functions under a coordinate change in U. For this reason, ~: = (~:,,,) is sometimes called a scalar

multiplet.

Let us now consider the effect of a change of local trivialization of V d over U] N U. Such a change is described by a transition mapping (p. 126) U] n U - - . G, x ~ h ( x ) E G; h ( x ) acts by left translation. Under such a change, the metric g is invariant, while A changes to A d ( h - ] ) A + h*OMc = h - l A h + h -~ d h , (see pp. 365-366 for notation and proof). The scalar multiplet ~: changes to Ad(h)| as we now show. For simplicity, take G to be the linear group GL(n, R). The index m is then a pair (i, j) (i, j = 1 , . . . , n). The right-invariant 1-form 0ji is determined from its value at the unit element e of G" 0~j(e) =dy~j by right translation" oi /( y ) = g y* l Oij(e) = d y ' , y -ll j .

Under a change of trivialization determined by left multiplication by h ( x ) ~ G,

Y't

= hi j ( y ) y '/! 9

In matrix notation (h ~ = h y~-,= y

etc.), we then have

0 = d y y - ] = d ( h y ' ) ( h y ' ) -1 = d h h -] + h d y ' y ' - ~ h -1 or

O= hO'h -1 + d h h -] .

The transformation law for ~ then follows.

291

13. KALUZA-KLEIN THEORIES 2) Take the metric ~, in local coordinates to be o f the f o r m n

g, = g~t3 dx~ dxt~ + ~mn(Om + Am dx'~)( 0" + At3 dxt3), where the 0 m are a basis o f right-invariant 1-forms on G and g~t~, ~ m n ' Am d e p e n d only on the f o u r coordinates x ~. C o m p u t e the Ricci tensor ~MN o f ~,. Let SM1v = [~mu -- 89 g,mN ~. Compare the equations R~m = 0 and S~t~ = O, respectively, with the Yang-Mills and the Einstein equations on V4. A n s w e r 2: It is straightforward to compute the riemannian connection of relative to the frame tb ~ = dx ~, tb m= 0 m + A m dx ~. The structure coefficients (~meo are defined by d~M

_ 89

=

&P

pQ

Q

^ ~

.

We have d& '~= 0, so ^1,1~

C

MN=O~

m

If C pq are the structure constants of G, then the right-invariant 1-forms 0 m satisfy dO m= I C m pqO p ^ Oq

~

Thus dt~m= 1C m pq(t~)p - APe dxa)(t~ q - Aql3 dx ~) + a~A m dx ~ ^ dx ~ . Reading off the C's, we find d: m Am

Pq

= -- C m

C ,~q " - C

m

pq

pqAa

and C" m ao = O , ~ A

m ~1 _ O t ~ A , ~ m _

C m

p

pqAap A q~= - F

m, ~ ,

where F is the curvature of the connection defined by A. The coefficients of the riemannian connection of ~ are given by the formula (p. 308) ,,M

to eo = i gMN(@gma + OOgNe_ O~Vgl,O) -- l(~.mOe + gMUgzpCLoN + gmUgLQCLeu ) .

For M, P, Q = 0 , 1, 2, 3, they are identical to the coefficients of the connection of the space-time metric g~" (j~ tlt

0v

.-.

to 0v"

For M, P, Q all corresponding to coordinates of G,

292

v. RIEMANNIAN MANIFOLDS. KJkHLERIAN MANIFOLDS ~t) m

Pq

-- r m

Pq '

m

where (to pq) is the riemannian connection of the metric induced by ~ on the orbit G x. The remaining coefficients are

~o~Em = ~ a mE = l F m aE ~om ~ = - 1 F m ~ , m

s

'

E n = 1 ~ m P D /3 ~np + C

(~om nE = 1 ~ m P D E

m

r rn A E '

~np ,

= - 89

.

Here D E is the covariant derivative with respect to the metric A"

DE~mn

~lmClpnaPP3 --

= r

~ln C

l

p

pmAE 9

The components of the curvature tensor are given by the general formula N ^N ^S ^N ^S -- C R ^N N (~)NpM M QP "- to PRr QM -- to QRtO PM pQ(.O RM "[" t~p(~) Q M - - JQ "

We are concerned with the Ricci tensor with/~MN = /~MeNP whose components are found to be, using the notation for symmetrization f Tx, such that j2 = _ 1. 1) Show that the linear operator J: T x ---> T x has eigenvalues +-i, o f multiplicity n. We shall omit the index x when the context makes the meaning clear. 2) Let T*x | C be the complexified cotangent space to V2,, i.e., the set o f complex valued R-linear forms on T x.

K,~HLER MANIFOLDS - CALABI-YAU SPACES

295

An element co ~ Tx* | C is said to be of type (1, O) [resp. of type (0, 1)] if Vv ~ Tx (co, J r ) -

[resp. (co, J r ) - - i ( c o ,

i(co, v)

type (1, O) type (0, 1)

v)].

Let 0 a, a = 1 . . . . . 2n be a basis of T*. Show that the elements ~.a of T* | C defined by (xa, v ) - (0 a, (J + i)v) are of type (1,0). Define by a similar procedure elements )~a of type (0, 1). Show that exactly n of the )a [resp. )a] are linearly independent on C. Show that if ;k~ ( k - 1 . . . . , n) are such a linearly independent set, then ()~, 2 ~) (k - 1 . . . . . n) are a basis of T* | C. Suppose that (V2n, J) defines a complex manifold, i.e., the almost complex structure J of g2n is derived from a complex structure. Give a canonical basis of T* | C associated to a natural basis of T*. Answer 1" j 2 _ --2, SO J v - - Xv

implies -v--kJv--k2v.

Thus ~2 _

-1,

)~ -- +i.

Further, if J v - - i v , then J(Jv)----i(Jv), so the eigenvalues i and - i occur an equal number of times. Note that this proves again that an almost complex manifold must be even dimensional, since complex eigenvalues of a real operator (J) appear in conjugate pairs, each of multiplicity n. Answer 2" We have, by the definition of )a, (X a, J r ) -- (0 a, (J nt- i ) J v ) -

(0 a, - v -4- i J r )

= i(Oa, (J + i ) v ) - i()~a, v).

Thus )a is of type (1, 0). Let * denote complex conjugation. Then ()a, v ) * - ( 0 a, ( J -

i)v).

Hence if we define ~a by (~,a, l)) - - ( ~ a , 1))*

complex manifold

296

V. R I E M A N N I A N MANIFOLDS. K A H L E R I A N MANIFOLDS

it follows that (~a, Jv)

.i(O a, . (J .

.

i ).v )

i( A,v)-a

and A a is of type (0, 1). Since - i is an eigenvalue of J of multiplicity n, the vector space (J + i)v, v E T x, has complex dimension n, and so has the space of 1-forms Aa. The same is true of the space of 1-forms A~. The space A 1,0 of 1-forms of type (1, 0) and the space A0,1 of 1-forms of type (0, 1) are both vector spaces over C, they are subspaces of dimension n and intersection 0. Thus T* | the vector space over C of dimension 2n, is the direct sum T* |

= A1, o (~ Ao, 1 .

If V2, admits a complex structure (p. 331) it admits local coordinates (x i, y~) such that J represented by J=

(0

1],

0

"

An element of T~* | is represented in these coordinates by to = a~ dx ~ + bj dy j , where a,, bj are complex numbers, and an element v E T x by v = v ~ ( a / a x ~) + tOJ(a/OxJ). We have ( tO, J o ) = - a i vi + b j w j .

Thus to is of type (1, 0) if - a i Oi + b ] w j = i ( a i w ~ + bjvJ) ,

i.e., b i = ia i .

Hence tO = a i d z i ,

where d z i = dx ~ + i dy ~ .

Analogously we find that tO is of type (0, 1) if tO = a i d z *i ,

dz*i = dx i _ i dy i .

A basis of 1-forms on T~* | consists of the forms dz ~, i = 1 , . . . , n of type (1, 0) and the forms dz *~ of type (0, 1). We can link this result to the general construction: the 1-forms h i corresponding to dx ~ are such that , ax j

( Ai . a / /= . d x / , ( J +. i ) '

OyJ

ax j a.

, ayJ =

dx/,

o

ax j

,

o/

ax j F i - -

Oy j

9

14. KAHLER MANIFOLDS-CALABI-YAU SPACES

297

Thus /~i _._ i(dx i + i dyi).

Similarly the 1-form A'i corresponding to dy* is found to be A' ~= dx ~ + i dy ~ . n of these forms, for instance the A' i = dz ~, are C-linearly independent. The /~,i a r e dZ i. It follows from (/~,i, V ) : ( ,~,i U > * that A" = dz'* Hence dE ~= dz i*. 3) a) Let f be a complex-valued differentiable function on V2,. Its differential d f at x is an element of Tx* | C. Let Ak be a basis of A 1,0 and let ~k be the corresponding basis of A0,1, i.e., the basis such that ( ,~k, V } = ( Ak, V ) *. We have d r = akX k +/3~,~ k . Dolbeault operators

One defines the Dolbeault operators 0 and 0 by a f = a~;t k ,

af= 13~ ~

and one sets

ak f = ak ,

ak f =13~ .

m

f is said to be holomorphic if af = O.

Show that the definition coincides with the usual one if J is derived from a complex structure on V2,. b) A complex-valued exterior form to on V2, of degree m, is a section of the vector bundle (A T* | m, with base V2, and fibre at x the space (A Tx* | m. The form to is said to be of type ( p , q) if m = p + q and 1

to = - ~ .

where

( ' O i l . . . ip "[p+ l . . . ip + q

: (.Oil . .. tptp+l . .. tp+q

~ A ' ' ' A X~ a X i ~ . . . A X.i~+~

are complex-valued functions of x E V2,.

Show that the type of to does not depend on the choice of the basis (A i, ~i). c) The exterior differential of to is a section of (AT* ( ~ C ) m+l d e f i n e d the usual formulas

dto= ~.) 1 { dtoil.. + 2

" i p ' i p + l ' ' " ip+q A A i 1 " ~ 1 7 6A A--"tp+q

('Oil... ip...

d" ( - - 1 ) m - 1 / ~ i l

ip+q

A "'"

(dh il

A /~i2''"

A d~ip+q}.

by

298

V. RIEMANNIAN MANIFOLDS. KA,HLERIAN MANIFOLDS

S h o w that if (V2,, J) is derived f r o m a c o m p l e x structure, then do) is the s u m o f a f o r m o f type ( p + l , q ) denoted 0o2 and a f o r m o f type ( p , q + 1), denoted Oo)"

dw = &o + ,~o. Dolbeault differential

is called the D o l b e a u l t differential o f to. Define a Dolbeault cohomology and a Dolbeault complex analogous to the De Rham cohomology and complex defined for real valued forms (pp. 223,297). A n s w e r 3a: If Ak = d z k = d x k + i d y k and 0f = 0, then df = a k(dx k + i dyk) ,

ak E C

which is the definition of derivability of f with respect to the complex variables z k. A n s w e r 3b" The type does not depend on the choice of Ai and A~ because

Tx* | = A~,0~ A0,1, the Ai span Ax, 0 and the )~i span choice/z ~, fi' respects the splitting.

A0,1. Any

other

A n s w e r 3c: Let us take coordinates (x ~, yi) on V2, adapte d to the complex structure, and choose as a basis in T* | C the 1-forms dz', d z i. A form of

type ( p , q) is o) =

(p

1

+

q)!

~

o)i, . . . ,p,p+, . . . Z,+q

d z q ^ . . . ^ d z i p . . . ^ d~ip+q

where the ~%... ~prp+~...Zp+q are functions of x i y~. Using the previous definitions of 0k and ak for functions, we have the decomposition of d~o into a (p + I, q) and a (p, q + I) form do)

"-

1 ( p + q) !

( ~kO)il

. . . ip . . . ip+q

"~" Ok ( ' O i l . . . ip . . . ip+q

dzil...

dz k A

dz i 1 ' ' '

^ di, k . . .

^

^ d z ip+q

dzip+q}

since dz i = d x i + i d y i and d~?~= dx ~ - i dy ~ have a zero differential. The definition of do) as well as its type, is coordinate independent: the result must therefore hold for any coffame /~i, ~. We check this easily by verifying it for a one form; a form of type (1, 0) reads necessarily Ai

=

aii d z i .

Thus

dh~ =

3kdi

j

dzk

^ dz

j + a- k a ij

dz k Adz

j

is the sum of a form of type (2, 0) and a form of type (1, 1).

14. KJkHLER M A N I F O L D S - C A L A B I - Y A U SPACES

299

Remark: If (V2n, J) is only almost complex, it is possible that dA i contains a term of the form 89 cjik d)~j ^ d,~. Thus forms of type ( p 1, q + 2) and (p + 2, q), could also appear in do. If (V2n, J) is a complex structure it is clear that a 2= 0 and ~2= 0. One can define a Dolbeault cohomology analogously to the De Rham cohomology (cf. p. 223) and a Dolbeault complex (p. 297). If, moreover, (V2,, J) is endowed with a hermitian metric (p. 334 and see below) it is possible to define a metric adjoint operator to d, a laplacian, an elliptic complex and use the Atiyah-Singer theorem (cf. for instance Eguchi, Gilkey and Hanson). 4) A hermitian I metric (p. 334) on an almost complex manifold (V2n , J) is a riemannian metric g which is invariant under J:

hermitian metric

g(u, v ) = g(Ju, Jv) .

A Kfihler metric (p. 334) on an almost complex manifold (Vz, , J) is a hermitian metric g such that the exterior 2-form ~ defined by 4~(u, v) = g(u, Jv) ,

u, v ~ TV2. ,

is closed. The form 4) is then called the Kfihler form. An almost KhMer manifold is an almost complex manifold (Vzn, J) with a Kfihler metric g. A Kiihler manifold is an almost Kfihler manifold whose almost complex structure (Vz,, J) derives from a complex structure. Give conditions on gq for the following g to be a Kiihler metric written in a basis A i, Ai of T* | g = gqA ~| AJ + gb.Ai | A j . Answer 4" The tensors A~| Aj, Ai | Xj, /~i (~/~j /~i (~) /~j form a basis of (T* | | (T~* | A hermitian metric on 112, is a section of (T* | C) | (T* | C) which is invariant under J, symmetric, and real, i.e., g(u, v ) = g(v, u) = g*(u, v) ,

The section

g=l(gqZ

|

Vu, v E T x .

+g~

|

)

or more explicitly g(u, v) = 89

l~i' U)( ~J, V) "4-g~( ~i U)( 1~], V))

is invariant under J because ( l~ i J u ) --- i( i~ i u ) and ( ,~J, v ) = - i ( ,~J, v ) ~

9

1A hermitian scalar product is, in other contexts, a complex valued bilinear form such that Such a definition could be used here, if the invariance under J were stated in the form g(Ju, v)= ig(u, v) (cf. Chern p. 9).

g(u, v)= g*(o, u).

Kfihler metric

Kfihler form almost K/ihler Kfihler manifold

300


If g contained terms in /~i~)hj or/~i~)/~j it could not be invariant. g satisfies g(u, v ) = g(v, u) and g(u, v) = g*(u, v) if and only if

gq = g]i

and

gq = g j~.

Remark: It is usual to abbreviate the expression of g, as in the case of riemannian metrics, to i-j g = gqA A . The K/ihler 2-form ~b corresponding to g and J

4,(u, v ) = g(u, Jv) = g(Jv, u ) = - g ( v , Ju)= -ok(v, u) is given by

i~i' U)( ~J, Sv) dl- gi,( XJ, u ) ( i~ i, Jv) .

~b(U, V) = 89 It is easy to show that ~b = - 89

| )~J - )tJ| A') - - 89

~ A )tJ.

The metric g is K/ihlerian if r is closed, i.e., if

~kgii A k A A i A ~j + gk gi7Xk ^ A i A ~j = 0 which is equivalent to

t~k gq - ai gk]

=

0

and

~kgi7 -

ajg~ = O.

5) Consider a Kiihler manifold. Compute the connection and curvature of

its riemannian metric (which is also the Kiihler metric), making use of the canonical 1-forms dz i, d~ ~. Show that its Ricci tensor determines a closed 2-form p. Answer 5" We have seen in 4) that a K/ihler metric, here on a complex manifold, can be written g = gq dzi d~?J = T~a 0~0~, where the (0 '~) = (dz j, d~?j) are 2n independent linear complex-valued forms on V2~. The usual computations determine the connection coefficients of the torsionless metric connection. Since go = g~7 = 0, we find that

l~jik-- 89

+ Ggfi).

If 4> is closed, we can write (cf. 4))

l~]ik = g i'[ajgk7

14. KAHLER M A N I F O L D S - CALABI-YAU SPACES

301

and ~k

=

il

i

g Oig~l = ( ~ k ) * "

All other Christoffel symbols are zero. The only nonvanishing components of the Riemann tensor are computed to be R k r~i/ = -- ~m l"]ik

and those related by symmetry and complex conjugation. The Riemann tensor has an extra symmetry R k rai] = R#~i

k

"

The Ricci tensor is i

R,~j = R;,~j = aj 8m(log det g ) . It follows from the hermiticity of g that det g is real and the Ricci tensor is hermitian, i.e.,

R#~ = R,~j ,

R#~ = Rmj.

It then follows from the expression of Rj~ (and the properties &e = that the 2-form p = iRj~

t~2 ._ 0 )

dz j A d z k ,

is a real-valued closed 2-form in V2., and therefore defines an element of the cohomology g r o u p H2(V2n). Despite appearances, p need not be exact. The quantity det g is not a scalar but a density, so the expression for Rj~ is not a globally defined differential; it will change as one moves from coordinate patch to coordinate patch. It can be shown that the cohoInology class of p is independent of the K~ihler metric, and is equal to the first Chern class C 1 of the tangent bundle to the complex manifold. It follows that a complex manifold cannot admit a Ricci-flat K/ihler metric if C 1 r Complex manifolds with a Ricci-flat K~ihler metric are called Calabi-Yau spaces. The existence of a Ricci-flat K~ihler metric on any complex manifold with C 1 = 0 which admits a K/ihler metric has been proven by Yau.

6) Let (V2,, J, g) be a Kiihler manifold. Show that its holonomy group ( p . 386) (with respect to the riemannian connection) is a subgroup of

U(n). Show that it is a subgroup of SU(n) if and only if Vzn is Ricci fiat.

Calabi-Yau spaces

302

V. RIEMANNIAN MANIFOLDS. K,g,HLERIAN MANIFOLDS

Answer 6: By the Ambrose-Singer theorem (p. 389) the Lie algebra of an holonomy group is spanned by elements S2 (u, v), where s'-2is the curvature J ~ V I V Fn are in the Lie algebra of U (n), and 2-form. It can be checked that R~t in the Lie algebra of SU(n) if Rtr~ = 0. For a more complete proof see reference [5].

Remark: This last property has attracted the interest of physicists to six dimensional Calabi-Yau spaces. In particular superstrings can consistently propagate only on a ten-dimensional manifold. For a physically realistic model, this manifold must be of the form V4 • V6, where V4 is ordinary spacetime and V6 is a six-manifold which is at least approximately Ricci flat. For four-dimensional supersymmetry to be unbroken, V6 must also be K/ahler. For more details, see reference [6]. REFERENCES [1] A. Lichnerowicz, Th&~rie globale des connexions et des groupes d'holonomie, ed. Cr6mon~se Rome, English translation. [2] S.S. Chem, Complex manifolds without potential theory (Springer, 1979). [3] T. Eguchi, P. Gilkey and A. Hanson, Gravitation, gauge theories and differential geometry, Physics Reports 66 (1980) 213-293. [4] G. Horowitz, "What is a Calabi-Yau space?', in Proceedings of Workshop on unified string theories, Santa Barbara (August 1985). [5] S. Yau, Proc. Natl. Acad. Sci. 74 (1977) 1798. [6] P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B258 (1985) 46.

V BIS. CONNECTIONS ON A PRINCIPAL FIBRE BUNDLE

1. AN EXPLICIT PROOF OF THE EXISTENCE OF INFINITELY MANY CONNECTIONS ON A PRINCIPAL BUNDLE WITH PARACOMPACT BASE (p. 363)

Let P be a principal bundle over X, with projection/7" P --+ X. Let { Ui } be a covering of X. Construct explicitly the connection form 09i on z r - l ( u i ) obtained from a given form ff)i on Ui by a choice o f a section fi over Ui. Check explicitly that c o - ~ i (Oi o Fl)wi is a connection on P, where {Oi } is a smooth partition of unity on X. Show that the space o f all connections on a principal bundle is not a vector space. Answer: Let q~i be the trivialization on 17 -1 (Ui) defined by a section 3~: over Ui. Following the construction given on (p. 364), the pull back of 09 i on Ui x G by q ~ l is (4)f-l*cOi)x,gi(V, w ) -

A d ( g ~ l ) ( & i ) x ( V ) + (OMC)gi(W),

(1)

where (v, w) E TxUi x Tgi G, 0MC is the Maurer-Cartan form on G, (0MC) g, (w) -- 6; (p. 168).

.~(x,gi)

~ i / ~ , (x,gi) P, , ~ ~ ( ! , e ) Ui

(x,e)

!

Ui

I ! I I

Uj

x 303

\\~x Ui

4,

304


This construction can be done over each patch. If x is in the intersection of two patches, x E U~ N Uj, the representative of p ~ / / - l ( u i N Uj) with I I ( p ) = x is (x, gj) in the r with A

A

gj "-- Cj,x ~ r -1 gi = g j i ( x ) g i

(2)

"

Let {0~} be a smooth partition of unity in X. Then {0~ o//} is a smooth partition of unity on the principal bundle P = H - I ( x ) . We define a 1-form on P with values in the Lie algebra ~ of G by setting

,o = E ( oU)o j.

(3)

J

The sum is meaningful because, by definition of a partition of unity, only a finite number of O/'s are different from zero for any given x ~ X. The pull back o n U i x G by r - ~ of the restriction of to to I I - ~(Ui) is -1 )(X, g i ) ( r - 1 , (0j)x,g i

(~i-l*(0)x,gi "- E ( ~ ~ 1 7 6

J

--" E Oj(x)((~i J

-1,

X E Ui ,

,

gi ~ G

O')j)x,g i 9

(4)

Each term in the sum is well defined on U i x G. It has its support in (U i N Uj) x G and is given by

oj(x)(

j

(v, w) = Oj(x)(

j- 1 , o r ,

-1,

(V, W)

, i- 1 , to/)~,g,(v ~ W) = 0j(x)(g/ith -1, t

= 0j(x)(~b/

(0/)x,gj(o, g/iw),since g / = g/i(x)gi.

Using formula (1) and the invariance of the M a u r e r - C a r t a n 1-form gives -1,

Oj(x)(ch/

(0j)x,g~(v, w ) = O j ( x ) ( A d ( g ~ ) ( ~ j ) ~ ( v )

+ (OMC)gi(W)) .

(5)

Reporting these values in formula (4) gives, since E/0i(x ) = 1, (i~) i - l ,

We can 1. r -1, 2. r 3. r

(1-

(0)x,g i(v, w)=

now check (0)(0, w) = (0 depends (0)(x,gih)(V,

~_, Oj(x)(md( g71)(~j)x(V) J

+ (OMC)gi(W))

that (0 is a connection (see p. 169) if, differentially on (x, gi). wh) -- A d ( h - 1)(r (x,g,)(v , w ) .

This calculation shows that the affine sum of two connections 0(01 + 0)o) 2 is a connection, but that toI + to2 is not a connection. Hence,

2. GAUGE TRANSFORMATIONS

305

the space of all connections on a principal bundle is not a vector space but an affine space. Note, however, that this calculation is not necessary to show that the space of connections is an affine space. Indeed, let A be the fundamental vector field (p. 360) defined by an element A of the Lie algebra c~, then ~o2(/[ ) = A . aW 1 "+" ~0.) 2 is a connection only if a A + flA = A; i.e., ~

"-" 1 - -

Og .

2. G A U G E T R A N S F O R M A T I O N S * (p. 364)

1) Use the sections s i and sj canonically associated to the trivializations (4)i, Ui) and (~j, Uj) o f a principal bundle e to prove the relationship between the connection f o r m s ~i and ~j, at a p o i n t x ~ U i f) Uj, in the local gauges c~i and Cj:

OJi(V) --

Ad(gji(x)

-1)~](v) -%OMC( g j i ( X ) V ) , t

(1)

where the transition functions are A

A

gq" U i N Uj ~ G by x ~ gq(x) = 6~,x o ckj-1 , the vector v E T x (U i f3 Uj) and OMC is the M a u r e r - C a r t a n 1-form on G. A n s w e r 1" The definition of s i is Si(X) = r

-1 (x, e)

x ~ U;, e unit of G

Therefore x

sj t I \

,j,~

ij(x )

~._.------~e

j •

S/ .~_ r j- 1 o r

si 9

*Written in collaboration with T. Jacobson and S. Carlip.

(2)

306

V BIS. C O N N E C T I O N S

ON A PRINCIPAL

FIBRE BUNDLE

we shall show that

(3)

sj(x) = G,,~x>S,(X) . Indeed in a trivialization, say ~bj, we have

4,j o sj(x) = (x, e) ,

ck i- 1 )(x, e) = (x, gz(x)e)

4,j ~ si(x) = (r

by definition of the transition functions. They act on G by left action, however, at the identity

g,(x)e = eg,(x) the left action is identical to a right action and we can write down eq. (3), where Rg,(,) i s the right a c t i o n of g,(x) on P. It is such that Rgji(x ) o Rgkj(X ) = Rgkj(x)gji(x ) = Rgki(X). At x, the connection 1-forms in the local gauges corresponding to a given connection to on the G-principal bundle are by definition toi =

s i

to

and

toj =

s *j

tO ,

sj(x) = R~,~>si(x) . Let and

x(t) be a curve on the base manifold X such that x(O)= x E U i N Uj dx(t)/dtl,__o = v. Then, at x

s.~ ,o(v)

= ~o -dr (sj(x(t)))l,-o

)

d = o)(--~l (l~gq(x(t))si(x(l)))lt=o)

d_

d

= ,o(Tt ( ~ , (.)si(x(t)))l,=o + -d] (Rg,(.(,))si(x))l,=o

)

9

The first term on the right-hand side is equal to

a)(1~'sq(x)s'i(x)o) = Ad(gq(x)-l)w(s~(x)v)

(p. 361).

The~ argument in the second term is a vertical vector. By inserting Rg,(x)-~Rso(x ) in the argument, one obtains d (~gij(X)-,g,(,(,))sj(x))l,=o ) = -dt d tg, (x)-'g'(x(t))l'=~ ~o -~ (p. 361, property 1 of definition c)

= L~,~x>-,g,(x)(o) t

t

= oM~(g;j(x)(o)) by definition of the M a u r e r - C a r t a n form

= g, (x)OM~(v) and the desired relation is proved.

3. H O P F

FIBERING

S 3~

S1

307

2) Using the_ sections_ s i and sj, prove the relationship between the representatives ~ and g2j of the curvature in the local gauges"

g2j = A d ( gij(x)-l )g2i .

(4)

Answer 2" The calculation is similar to the derivation of (1), but here the term giving rise to the M a u r e r - C a r t a n form will not contribute since g2 vanishes on vertical vectors:

a ( u , o) = a(s (u), (o)) = a(R (u), = Ad(gg )s ' a ( u , v). 3. H O P F F I B E R I N G

$3---~ S 2

1) a) S 3 is the submanifold of ~ 4 with equation E i=1 4 (xi) 2 = 1. Show that it can be represented in the space C 2 o f two complex variables by Z1Z1 -'b Z2Z 2 ---~1.

b) Show that S z can be represented as the set o f equivalence classes o f pairs of complex numbers [z 1, z2], not both zero, with the equivalence relation (z 1, z : ) = (Az 1, Az2), h ~ C, h ~ 0 (i.e., S 2 is identified with the complex projective line CP~). 2) Define a projection p" S 3~ S 2 such that p - l ( x ) , x E S 2, is diffeomorphic to S 1. Show that it endows S 3 with a principal fibre bundle structure over S 2.

Answer la: S 3 is the submanifold of ~4 with equation Ei4~ (xi) 2-" 1. It can be represented as the set of pairs (z 1, z2) with z~Z 1 + z2z2 = 1 by setting z~ = x I + ix 2, z 2 = x 3 + ix 4. Answer lb" The complex projective line C P 1, set of equivalence classes [z I , z2], is made of two sets" one in which z 2 ~ 0 and one with z 2 = 0 and thus z I ~ 0. This last set contains only 1 point since (z 1, 0) = (z'1, 0) for all z I , z'1 E C. The first set can be identified with the complex line C since, if z z ~ 0 , (z 1, z : ) = ( Z l / Z 2, 1). Thus C P 1 is identified with C (equivalently R E) plus one point, that is the sphere S 2, for instance by a stereographic projection from the north pole of S 2 onto R 2, and the additional point of C P 1 identified with this north pole. A n analytic identification of S 2 defined as the submanifold E~_-i (xi) 2 = 1 of ~3 with the set of equivalence classes [z 1, z2] can be checked to be

308

V BIS. CONNECTIONS ON A PRINCIPAL FIBRE BUNDLE 1 X =

2~:~/~Z1772

2 X =

Z1Z71 + Z2Z72

25r Z I Z 1 + Z2Z72

3 Z I / 1 - Z222 X = Z l Z 1 + Z2Z 2

(1)

Answer 2: We define a map p" 5 3 " - - > 5 2 by (Zl, Z2)---~[zl, z2] , when ZlZ71 + z2~?2 = 1. This mapping is surjective since for any pair with z I and z 2 not both zero (Z 1, Z 2 ) =

( Z I ~ 1 -tS Z2Z2) 1/2 ' (Z1Z 1 + Z2Z2) 1/2

'

and the last pair satisfies the condition for it to represent a point of S 3. The reciprocal image p-l(x) of a point x E S 2 represented by the equivalence class [z 1, z2] is the set of points of S 3 represented by a pair (z'1, z~) with (z~, z2) "" ( Z l , z 2 ) thus Z1

P

Z1= ~

p

( Z I Z 1 "+" Z2Z2) 1/2 '

Z2 = ~

Z2 (Z1Z 1 + Z2Z2) 1/2 '

!

--

Z l Z ~ + Z2Z 2'-' = 1

(2)

with IAI- 1. The set IXl = 1 in C is identified with S 1, or U(1), and is in bijective correspondence with p - ~(x). Formula (2) gives the local trivialization p - I(U) = qb(U x S 1). The fibering $3---~ S 2 is called "Hopf fibering" because it was discovered by H. Hopf. Other fiberings of spheres, also called Hopf fiberings, into principal bundles are S 7~

S4

group S 3

sn---. ~Pn ,

real projective space,

s2n+l----~ C P n ,

group U(1) quaternion projective space, group SU(2).

54n+ 3 ----~ ~ - ~ n ,

group Z 2

REFERENCE N. Steenrod, The Topology of Fibre Bundles (Princeton University Press, sixth printing 1967).

4. SUBBUNDLES AND R E D U C I B L E BUNDLES*

1) Let H be a Lie subgroup o f a Lie group G. Show that the two following statements are equivalent. *For a physical application, see for instance Problem V bis 5, Broken symmetry.

309

4. SUBBUNDLES AND REDUCIBLE BUNDLES

a) The G-principal bundle P = ( P, M, zr, G) admits an H-principal subbundle PH -- (PH, M, JrH, H) with PH C P, and JrH = 7rlpH. b) The G-principal bundle P admits a family of local trivializations with H-valued transition functions. Answer: If P satisfies a), it is reducible to PH by the definition of reducibility (p. 131). The equivalence of a) and b) results therefore from the theorem (p. 382): A principal fibre bundle (P, M, Jr, G) is reducible to (PH, M, 7rH, H) with H a subgroup of G if and only if it admits a family of local trivializations whose transition functions take their values in H. We recall the proof that a :=> b. Let (PH, M, zrn, H) be a subbundle of (P, M, Jr, G). We shall show that the trivialization (~OHi, Ui C M) on P/4 defines an H-valued set of transition functions for P. Indeed, let p ~ PH C P and YrH(p) = x ; the fibre of P at x is

where p g -

Jr -1 (x) - {pg, g ~ G}

R~,(p);

let ((Di, Ui) be the trivialization of P defined by qPi (pg) -- ~oi(p)g - (x, A qgHi,x(p)g ),

p E PH,

Jr(p)--x

E Ui

with the standard notations (p. 125). The corresponding transition functions A

gi.j (X) ~ qgi,x A

A -1

o

qV.j,x

A --

qaHi,x g

o

g

--IA--I

qgH.j,x

A-1

-- (flHi,x o (flHj,x E H

take their values in H. 2) Show that a G-principal bundle (P, M, yr, G) is reducible to an Hprincipal bundle (PH, M, JrH, H) if and only if the bundle P \ H with typical fibre G \ H (the space of left cosets g H) associated to P by the canonical left action of G on G \ H admits a cross section. Answer 2: (p. 385)

a) Assume there exists a cross section a" M - + P \ H ;

let P . = (#-1 o a ) ( M ) where # is the canonical map #" P -+ P \ H,

by p w+ [p], p ~ p' if p ' = Rhp,

h E H.

Then P . C P and the injection identity j" P . --+ P is a bundle morphism which commutes with the right action of H on Pc~. According to the definition (p. 131) of reducible principal bundles, P is reducible to the H-principal bundle P~.

310

V BIS. C O N N E C T I O N S ON A PRINCIPAL FIBRE B U N D L E

b) Conversely, let PH C P be a reduced bundle of P, then/z o i is constant on the fibres of Pt-/, and -1

o" - - / z o i oTz"H

defines a section M ~ P \ H. G

P P

IG\H

'~

x I I i

x

(x)

I- _ ol •

A point IPl in P \ H is the equivalence class of points p E P such that Pl "~ P2 if Pl =

p2h.

Remark: As a particular case of the previous study, we find the result that a principal bundle p is trivial (reducible to a bundle with group reduced to unity) if and only if it admits a cross section. Remark: For a study of the existence of cross sections on the associated bundle P \ H we refer the reader to C.J. Isham, "Space-time topology and spontaneous symmetry breaking", J. Phys. A. Math. Gen. 14 (1981) 29432956.

5. BROKEN SYMMETRY AND BUNDLE REDUCTION, HIGGS MECHANISM INTRODUCTION

vacuum

The action functional of a system is often invariant under a group G. We have shown (Problem II 5, Invariance) that the Euler-Lagrange equations of the system have the same symmetries as the action; hence the family of solutions of the Euler-Lagrange equations is invariant under G but a given solution is usually not invariant under G. A solution which minimizes the energy of the system is called vacuum. One expects the vacuum to be maximally symmetric and to be unique, but even the vacuum may not be invariant under G but under subgroup

5. BROKEN SYMMETRY AND BUNDLE REDUCTION

311

H of G and possibly not unique. One says that this vacuum breaks the symmetry or that the G-symmetry of the system is spontaneously broken to n .

Example 1: Einstein equations are invariant under isometrics but a given space-time has in general no nontrivial isometry. The Einstein vacuum is Minkowski space, invariant under the maximum possible group of isometries of a 4-dimensional pseudo-riemannian manifold. Example 2" Let f -- ( f a ) , a - 1 . . . . . N be a scalar multiplet on the Minkowski space time M 4, that is a mapping M 4 --+ R N by x w-~ f ( x ) ( f a ( x ) ) , a - 1 , . . . , N. Consider the lagrangian L(f)-

~

~x o

-

t Oxi ]

a=l

Einstein

vacuum scalar multiplet

(1)

i=1

where the potential V" ]~N ~ for instance V given by V ( v ) --

-- V o f,

spontaneously broken symmetry

I~

is invariant under rotations in ]1~N . Take

(va) 2 - k 2

,

v - - (v a) E R N.

(2)

a=l

The energy of f (p. 513) and [Problem II 7, Stress energy] is the integral, which is independent of x ~ - t if f is a solution of the Euler equation of L" Ox o ,] + ~

-~ xO=t

a=l

Ox----T

+ V (f

(x))

dx 1 dx 2 dx 3.

(3)

i=1

This integral attains an absolute minimum when f (x) - v0, some constant vector in R N, and v0 is an absolute minimum of V. If V is given by (2) the vector v 6 I[~u is an absolute minimum of V if its extremity is on the sphere S u - 1 of radius k" ~(va)

2 -k2--0.

(4)

a=l

The lagrangian (1) is invariant under the transformations f (x) --+ S f (x),

S ~ O(N),

orthogonal transformations of R u

while a solution v0 of (4), a point on S N-1 C ]~N is invariant only under the subgroup of O(N), isomorphic to O ( N - 1), which leaves this point fixed.

energy

312


Remark 1" The point v invariant under O(N).

0 is, in this example, a maximum of V. It is

Remark 2: The sphere S N - 1 where the vacua take their values is the homogeneous space O ( N ) / O ( N - 1) [Problem V 10, Invariant]. The purpose of this problem is to show how, in some cases, one exploits the H-symmetry of the vacuum. 1.

EXISTENCE AND PROPERTIES OF H-INVARIANT VACUA

We consider the general case of a physical system on a (pseudo)riemannian manifold M whose unknown quantities are a connection co on a principal G-bundle P over M (p. 364) and a Higgs field q~ of type (p, E) (p. 404), i.e., a mapping from P into a given complex - or real - vector space E, equivariant under a representation p of G by a group/-' of linear transformations of E:

~(pg) _ p(g-l)~(p),

minimal coupling

p ~ P,

g ~ G,

p ( g - l ) ~ 17

We denote by A, F, 4~ the gauge potential, gauge field, scalar-multiplet pull backs of co, $2 (the curvature of co) and q~ by a local section s" U ~ P, U C M. We denote by V4~ the covariant derivative of 4~ in the connection co, by Vq~ its pull back. If the Higgs field 4~ is coupled to the gauge field F only by the covariant derivative Vq~ in the connection o9, the coupling is called minimal. We consider a potential V: E ~ R. We suppose E endowed with a hermitian - or real - scalar product, invariant under F'. The following lagrangian defined on U by

L ( A , ~) -

88gUC~gVPFuv . Fcrp + 89gUVVu~ . V v ~ - V(d?)

(5)

is a well-defined function on M because its values in U are independent of the local section s of P, since it is invariant under the transformations F w-~ A d ( g - I ) F , q~ w-~ p ( g - l ) ~ , V4~ w-~ p ( g - i ) v 4 ~ . The notations are

{eu}, # 1 . . . . . d is a basis for Tx M, g#v(x) is the metric tensor on M. -

F#v -- Filzvei, F/z v -

0 Ai

_

{ei }, 0 Ai +

i -- 1 . . . . . n is a basis for the Lie algebra ~Ct(G).

ci

A.j Ak

F#~ . F,~ -- gij Fiu~,FD%, - cbaea,

{ea},

a-I

c i are the structure of constants G

o

g - (gij) Ad-invariant scalar product on S ( G ) , ..... NbasisofE.

5. B R O K E N S Y M M E T R Y A N D B U N D L E R E D U C T I O N

(V~,~b)a = au6

r

r =

a

t

a

i

a

313

b

+ (Pe(A'~,e,))abdP b - 0~,r + At, T ~ bdp ,

(V. r 1 7 6

h = (hab) is a scalar product in E invariant under p(g), g E G . The field equations are [Problem II 5, Invariance] a

i

V g F i~'v + dp T a b ~

V/zVP'r a -- O V / O r

vr b

=

O,

(6a) (6b)

a ~- O .

If the metric on M is properly riemannian the energy of the system is (for the hyperbolic case see Problem II 7, Stress energy) U(A, r

= f ( 88g*'~g~~

F~o + 89 g~'~V~,r .V,~b + V(r

d/x.

(7)

We suppose the scalar products in ~ ( G ) are positive definite and there exists a subset Z C E such that V attains its absolute minimum on 2. 1) a) Show that ,~ is a union o f orbits o f F, representation o f G by p. b) Show that each orbit can be identified with the homogeneous space F \ F H where Fn is the isotropy subgroup o f a point o f this orbit (FH is the representative by p o f a subgroup H o f G). Answer la: v E ~ if and only if V ( v ) = m, the absolute minimum of V. Since V is invariant under F we have V ( p ( g)v) = V(v) ,

Vp(g) E F .

Answer lb: Choose a point v 0 ~ ,~, denote by FHo C F its isotropy group, i.e., the set of elements p ( h ) E F such that p(h)v o = vo ,

h C Ho .

Let v I - p ( g ) v o be in the orbit of v0 under F. Its isotropy group FH1 is the set of elements p ( g ) p ( h ) p ( g -1) ~ F since v I = p(g)p(h)v o = p(g)p(h)p(g-1)Vl . if p is a left action H 1 = gHog -1", if p is a fight action/-/1 = g - l H o g ; in both cases H I = H0 and we can identify each orbit of F with the homogeneous space F / F H where FH is the isotropy group of an arbitrary point of the orbit.

314


2) Show that if P is a trivial bundle the vacua exist and are such that

F =0 Vud? - 0

i.e.,

S-2=OonP

(8a)

i.e.,

V(b--Oon P

(8b)

d~ i.e., c~ takes its values in ~ .

(8c)

Answer 2: If P is trivial it admits a flat c o n n e c t i o n - i.e., with zero curvature. It can be determined by choosing in a given cross section so of P, the representative A0 = 0. Relative to this cross section (b) reduces to Oug)o(x ) = 0; its solution is q~0(x) = vo, constant vector in E. The condition (c) says that v0 ~ Z:. It is clear that the energy attains its absolute minimum for the considered fields, and that they satisfy the field equations. 3) We shall now examine under which conditions on the possibly nontrivial bundle P the system admits a vacuum of the type given in question 2). a) Give necessary topological conditions for the existence of a flat connection on P. b) Show that if there exists an equivariant mapping 49" P ~ 2?1, where E1 is an orbit of F in E, then the group G of the bundle P is reducible to the group H whose representative by p is the isotropy group of a point of El. c) Suppose conversely that P is reducible to a principal bundle with the group H, whose image by p is the isotropy group of a given point vo ~ICE.

Denote by PH (cfi Problem V bis 4, Subbundles) the corresponding subprincipal bundle. Given a covering of M by open sets Ui over which PH is trivial there exist sections si: Ui --~ PH C P, and sj(x) ----si(x)hji(x) with x E Ui N U j, hji (x) E H. Show that one can define an equivariant mapping: ~" P -+ ,~l by setting ~(S(X))-

DO,

(b(p) -- p(g-1)vo

X E Ui,

if

1)o ~ El

(9)

p -- si(x)g.

d) Show that if PH admits a flat connection the same is true of P. Show that the mapping 4)" P --+ ~ l defined above has a vanishing covariant derivative in this connection.

Answer 3a: If P is nontrivial it may not admit a connection with zero curvature: a necessary condition for the existence of such a connection is the


315

vanishing of the characteristic classes of P ( p ) . A nonzero characteristic class is also called a topological charge. It can be proved [Kobayashi-Nomizu, Vol. I, p. 92] that M is simply connected P admits a fiat connection if and only if P is a trivial bundle. Remark: if A ~ 0 but F 0 in U C M the equations (8b) are a completely integrable system, thus admit solutions at least locally. We shall retum to the solution of (8b) after the next question. A n s w e r 3b (See also answer a p. 309): Suppose there exists q~" P --+ 2? E1 U . . . U ~ , such that r - p ( g - l ) ~ ( p ) . Denote by v0 some point of E1 attained by r and denote by Q the subset of P such that

r

q 9

The projection Jr" P --+ M restricted to Q still covers M" indeed let p Jr -1 (x) since r 6 271 there exists g 6 G such that ~(pg) _ p(g-1)~(p)

_ vo,

p 9 re - l ( x ) ,

thus p g 9 rCQ l (x).

The action of G on P induces an action of H on Q because q h ~ Q if q 6 Q, h 6 H, since if r

[k(qh) - p ( h - 1 ) ( b ( q ) - vo

- v0.

Thus Q is a principal subbundle PH of P. A n s w e r 3c (See also answer b p. 310)" The problem is to prove that r does not depend on the choice of the local sections si of P/4. Consider indeed the two sections si and sj over Ui N Uj. We have r

-- p ( h j i 1( X ) ) ~ ( S i ( X ) )

- - YO

since v0 is invariant under p ( h ) . On the other hand if p - - s i ( x ) g we have p -- s j ( x ) h . ~ 1(x)g. Applying the previous definition to the section sj gives r

- p(g-lhji(x))vo

- p(g-l)vo,

thus the same value r A n s w e r 3d: Suppose that PH admits a flat connection & H - We define the horizontal subspace of a connection & on P by the general method (cf. p. 380): the horizontal subspace of & at q ~ PH is the same as the horizontal subspace of cbn; at p -- qg, the horizontal subspace horp P will be (cf. p. 380)

horp P - R'g horq

PH.

topological charge

316

V BIS. CONNECTIONS ON A PRINCIPAL F I B R E B U N D L E

It is easy to check that the definition does not depend on the choice of q E PH" The connection 03 is flat if the connection o3H (cf. p. 388) is fiat. Let i: PI.I-.--~P be the identity injection. The field 4,H = thoi has a vanishing covariant derivative in any H-connection. The vanishing of V~b in the connection 03 is a particular case (with Y = 0) of results which will be proved in section 2. 2. HIGGS MECHANISM

We consider a lagrangian of type (5) and suppose that the system admits a vacuum 4, such that (see question 2) F=O,

Higgs mechanism

V~b = 0 ,

4, takes its values in .~.

The lagrangian (5) is expressed in terms of quantities defined on the G-principal bundle P. We shall exploit the fact that P is reducible to an H-principal bundle PH and reexpress the lagrangian in terms of quantities defined on PH. Some of the new terms can be interpreted as massive fields and this procedure is called the (generating mass) Higgs mechanism. 4) Let .~(G) and .~(H) be the Lie algebras of G and H, set

.~ ( G )

=

.~ ( H ) ~3 Y[ .

See [Problem III 7, Homogeneous spaces] ]'or properties of Ys Let to be a connection on P and i'to its pull back on Pn by the identity injection i: PH--~ P. Set i ' t o = toll + T

where ton is a 1-form with values in ~ ( H ) and Y is a 1-form with values in 9g. Show that ton is a connection on PH and Y a tensorial form (p. 373) of type (Ad, Y{). Answer" Let h E H C G, and R h the right action on Pn. We have Rh(i* to) = A d ( h - 1)(i * to), i.e.,

RhtoH "~"R h T = Ad(h-1)toH + Ad(h-1)Y 9 It follows that ton is a connection on PH. To prove that Y is horizontal, i.e., vanishes on vertical vectors VH vr ~ TPH we compute

i* to(VH vert) = to(i'VH vert) =- ~rH vert (~ ~ ( H ) ==~toH(VH vert) "

5. B R O K E N S Y M M E T R Y A N D B U N D L E R E D U C T I O N

317

Hence

y (VH vert)

- - O.

m

5) Let Y ~ Tp P be the w-horizontal lift of Y e Tx M and YH E Tp PH be its wH-horizontal lift. Show that

(

d

)

Y - - i f YH + -~ pexp(--ty(YH))lt=O

(10)

.

Answer: By construction we have TpP -- TpPH + ~ p ,

where ~gp is the subset of the vertical space at p canonically isomorphic to Jc/ in the decomposition ~CP(G) -- .L~(H) 9 ~ . Set - - i ' ( Y n + Z).

Since zr'(I3) -- zr'(I?H), it follows that Z is a vertical vector in TpP. It follows from i ' c o - (_1)n -i- y that 0 - o ( Y ) = oo(i'(YH + Z)) -- (OH 4- Y)(YH + Z) = O)H(t'H) + Y(I'H) + i*w(Z) = Y(YH) + i*w(Z),

i.e., Z e ~ is the vertical vector defined by - y (YH) ~ ~ isomorphism JU ~ ~ .

in the canonical

6) Let WH be a flat connection on PH ; let

C H - - ~ oi" PH -'-> Z C E. CH is called a v a c u u m solution if its covariant derivative with respect to WH vanishes,

VH ;M(p) =0. Show that the covariant derivative Vr defined by a)is given by

Vr

V) - P'e(Y (V))r

with

i*w -- WH + y,

V e Tp PH.

(11)

Answer 6: Let hor~o label the horizontal component of a vector in the connection w.

vacuum solution


318

7q~(i'V) -- q~'(hor~o(i'V))

(d

by definition,

= ~'(i'hor~oH(V)) -t-~' i ' - ~ p e x p ( - t g ( V ) ) l t = o

)

according to (10) and since y(VH vert) -- O, d = v H ~ H + -~ p(exp(ty(V)))~H(p)lt=O by definition of q~ --

pte(Y(V))~H

since

VH~H

- - O.

7) Show that i'S2 -- 12H + xTHy + 89 y],

(12)

where I2 and I2H are the curvatures defined by o9 and OOH and V H the covariant derivative defined by WH. With the notation of paragraph 5, let Yj and YHj be the horizontal lifts of Yj ~ Tx M at p 6 P defined respectively by co and coil; let

~'.j - i' (~'Hj + Z j) ~ Tp P,

j -- 1,2

where Zj(p)

d

-- -~

pexp(--ty(~'Hj))lt=O,

V -- i*oa --

OaH,

~r(p) - x . (13)

Let Vi ~ Tp PH be such that

horoo(i'Vj ) - Y.j; then (i ~VI, i ~V2) -- i* dco(l?l, 1~2) -- di* (]~l,)52) = dcoH(l~Hl, ]?H2) + dY(l?Hl, I?H2) + i* d~(YH1, Z2) + i* dco(Zl, I~H2) + i* do3(Zl, Z2).

The first two terms

dCoH(YH1, YH2)

-- ~2H(VI,

g2),

d?' (]~H 1, ]~H2) -- V H y (Vl ' g2).

To compute the remaining three terms, we write (p. 207) i* dog(l~Hl, Z2) --

i*~,nlOO(Z2)

-- i * ~ Z z O g ( ~ ' H 1 )

--

i*co([YH1, Z2])

and two similar equations. Note first that

co(YH1) -- --co(Z1)

(14)

319


since I71 is co-horizontal. Next, extend Z.j(p) defined by (13) to a Killing vector field Z./in a neighborhood of p. By definition ^

co(Zj) -- Zj --constant 6 j U

(15)

and its Lie derivatives vanish. We can write --i*co([YH1,

Z2])

--

i*co(..~Zz~'H1) 1

,~,

Z2 -

--)/(YH2).

= i* lim-(co I?nl) - co(I?H1)) t=0 t ( g (t)-I with d dt g(t)lt=o

-

Hence d -i*w([I?nl, Z2]) -- i* ~(Ad(g(t))co(~'H1))lt=O _

D

by (14) and (15)

= --[Y(YH2), Y(YH1)] -- --[y(g2),

y(gl)]

since Y is tensorial with respect to oH. Finally, i*S2(V1,

V2) - -

S'2H(V1, V2)

+ V H

•

v2) + [•

•

Hence (p. 374) i'S2

- - if2 H --[- V H

z + 89 z].

8) The lagrangian (5) is defined on the base M of the G-principal bundle P, i.e., in terms of the pullbacks by a section s.j : Uj C M --+ P o f co, S-2, and (5. Choose a section s.j such that s.j: U.j --+ P H C P. Use equations (11) and (12) to reexpress the lagrangian in terms of quantities defined on PH and interpret the lagrangian. Answer 8" The pullbacks of (11) and (12) give respectively

V~b(Y)

-

-

pte(y(Y)) ~,

!

!

s.iY -- V -- i V,

F - - FH + VH~, + 89

~'].

~ -- s.~y.

320


In the lagrangian evaluated at a vacuum solution 4' and written schematically L = + one notes i) the following quadratic terms in

lv"

massive gauge potential

71)

+

+

which can be interpreted as the lagrangian of a massive field ~/. Note that ~/is not a gauge potential because it is not the pullback of a connection. It is sometimes called a massive gauge potential because it traces its origin to the gauge potential A. ii) The field ~ is minimally coupled to the gauge potentials a3H pullbacks of connections on PH-

(Nambu) Goldstone boson

iii) The terms P'e(~'bt (x))~b are called (Nambu) Goidstone bosons relative to the vacuum ~b. Since i = 1 , . . . , dim X, there are (dim G - d i m H ) Goldstone bosons. There are as many Goldstone bosons as there are massive fields ~. Given the lagrangian (5) we have established necessary conditions for the existence of a vacuum, namely a fiat connection, and a mapping r shown to be equivalent to the existence of a principal subbundle Pn of P. In the previous problem we have shown that the existence of a principal subbundle P,v of P is equivalent to the existence of a section rt,'. M---, P/ H ,

where P / H is the associated bundle to P by the action of G on G / H . If the G-principal bundle P necessary to define the lagrangian (5) is reducible to an H-principal bundle PH, the lagrangian (5) can be expressed in terms of massive fields ~ and gauge potentials o3n pullbacks of connections on PH.

REFERENCES Y. Kerbrat and H. Kerbrat-Lunc, "Spontaneous symmetry breaking and principal fibre bundles", J. Geometry and Physics Vol. III, No. 2 (1986). C.J. Isham, "Space-time topology and spontaneous symmetry breaking", J. Phys. A: Math. Gen. 14 (1981) 2943-2956.

6. EULER-POINCARI~ CHARACTERISTIC

321

L.D. Faddeev and A.A. Slavnov, Gauge Fields Introduction to Quantum Theory (Benjamin Cummings, Reading, MA, 1980). R.O. Fulp and L.K. Norris, "Splitting of the connection in gauge theories with broken symmetry", J. Math. Phys. 24 (1983) 1871-1887. M.E. Mayer, "Symmetry breaking in gauge theories", Hadronic Journal 4 (1981) 108-152.

6. THE EULER-POINCARt~ CHARACTERISTIC A supplement to this problem entitled "The Euler class" can be found near the end of the book. INTRODUCTION

Chern (1979) called the Euler characteristic "the source and common cause of a large number of geometrical disciplines" and illustrated this relationship as follows. Hodge theorem, p. 400 Atiyah-Singer index theorem, p. 401

Total curvature (Gauss-Bonnet-Chern-Avez theorem, p. 395, Gaussian curvature, p. 396)

f

Euler number Euler-Poincar~ characteristic (p. 293)

/

Combinatorial topology (Simplex decomposition p. 216, Poincar6-Hopf theorem, p. 396)

Homology (Betti numbers, p. 224, de Rham theorem, p. 226) Characteristic classes (Euler class, p. 394)

This problem touches upon these various aspects of the Euler characteristic and in particular provides key elements of proofs and examples of the Poincar6-Hopf theorem and of the Gauss-Bonnet-Chem-Avez theorem. Several of the examples are given in two-dimensional spaces, thereby reproducing some classic results of curves and surfaces of euclidean spaces. 1. POINCARI3-HOPF THEOREM

The E u l e r n u m b e r X (M) of a compact manifold M is equal to the sum of the indices (p. 396) of the zeros of any smooth vector fieid v on M which

Euler number

322


has only isolated zeros. If M has a boundary, the vector field v is required to point outward at all boundary points. a i) Compute the indices i at xo of the vectorfields drawn on figure 1 on a 2-dimensional manifold. a ii) Construct a vectorfield with a zero of arbitrary index. a iii) Show that the group of transformations generated by any of these vector fields does not act freely on M. a iv) Observe that the sum ofthe indices ~ i of the zeros ofany vectorfield with isolated zeros which can be drawn on the 2 sphere is 2. Check that i = 0 on the 2 torus.

-...1.f /',, 9XO

. 1f /\ ~

9x o

y

Fig. 1.

Answer 1a i: The index of a vector field v (p. 396) at an isolated zero x0 is the degree (pp. 221,557, 558) of the map: x --+ fi(x) (unit vector in the direction of v) from a small sphere E(x0) of center x0 into the unit sphere. The sphere e(xo) is oriented as the boundary of the corresponding disk. For instance the index of the sink can be read off the following picture where the map: x --+ fi(x) maps points from the left sphere into points of the right sphere: the map is bijective and orientation preserving, thus the index is 1.

6. EULER-POINCARIE CHARACTERISTIC

xz

^

x51

323

~(x.)

~(x z) Fig. 2.

Similarly we find for the indices of the various cases drawn on fig. 1. Vector field Index

a 1

/3 ~/ t~ ~" rl 1 -1 -2 1 2

Answer l a ii: Let z = x + iy; the map v" z ~ z n, with n a positive integer, is of degree n, hence the vector field o(x, y ) = ( R e ( x + iy) n, Im(x + iy) n) has a zero at the origin of index n.

Answer l a iii: T h e local g r o u p of transformations {o-g~s~} generated by any of the vector fields drawn on fig. 1 leaves x 0 fixed" trg~s~(Xo)= Xo, therefore the group does not act freely (p. 153) on M. b) We can relate the topological properties of a 2-dimensional compact manifold M to the nature of the vector fields on M by triangulating M and constructing a vector field with the following zeros

a source in each triangle a saddle at each edge Fig. 3.

a sink at each vertex

i) Show that the sum of the indices of this vector field on M is

~i=F-E+V, M

where F is the number of triangles, E the number of edges, V the number of vertices.

324


ii) Show that F - E + V is a topological invariant equal to the alternate sum of the Betti numbers of M (p. 224)" 2

F-

E + V-

~-~(--)kb k. k=0

Show that for any n-dimensional compact triangulable manifold M n

n

k=0

k=0

where Otk is the dimension of the vector space Ck(M, R) of real valued k chains on M. Hence Z i -- X (M) Poincar6-Hopf theorem (quoted p. 396). M iii) Give the Euler numbers of S 2 and T 2.

Answer lb i" The indices of sources, saddles and sinks are, respectively, + 1 , - 1 , + l , hence y~ i - F - E + V. Inserting a segment A D in the A BC triangle does not change the alternate sum F - E + V. Answer l b ii: We shall now prove that F - E + V - b 2 - bl + b0 or in general (see Singer and Thorpe, p. 142) for an n-dimensional compact triangulable manifold M n

n

k=0

k=0

where C~k is the dimension of the vector space Ck(M, I~) of real valued k chains. For each 0 < k < n, the boundary operator 0 maps [see Problem IV 1, Cohomology] the vector space Ck (M, I~) of k-chains into Ck-l (M, ]~)"

O" Ck(M, I~) --+ Ck-I (M, R), C_ 1 is by definition the empty space. C~k -- dim Ck (M, It~)

by definition of Ck (M,/t~);

by the rank and nullity theorem of linear algebra we have oek -- dim Zk (M, R) + dim Bk-1 (M, I1~), where Zk is the kernel of O" Ck -+ Ck-1 and Bk-1 is the image of O" Ck --+ Ck-1.

325

6. EULER-POINCARt~ CHARACTERISTIC On the other hand bk = dim Hk(M, [~) = dim Zk(M , R) -- dim Bk(M, R) . Hence n

(--l)kbk = ~ (--1) k d i m Z k Jr ~ (--1) k+l dim B k .

x(M)-~ k =0

k =0

k =0

Now B,,(M n) = 0 because there is no n + 1 chain, hence

( - 1 ) k§ dim Bk = ~ ( - 1 ) ~dim B~_a k=0

1=1 n

- ~

( - 1 ) t dim

BI_ 1 ,

l=0

since B_ 1 = O. Finally, n

x ( M ) = ~ ( - 1 ) k (dim Z k + dim B k _ , ) k=O

= ~ (-1)kay. k=0

Remark: In the case of a connected c o m p a c t orientable 2-dimensional manifold M, b 2 = 1 because B 2 = 0, and H 2 has only one g e n e r a t o r K h o m e o m o r p h i c to M. T h e Betti n u m b e r b0 = 1 for the following reasons: Let A and B be 2 vertices of K, t h e n there is a sequence of vertices A ~ = A, A 2, 9 9 9 A r+l B such that A~, A~+I are the vertices of a 1-simplex tr~ -

-

"

a f~, utr: = u ~2~ (A i i=l

Ai+l)=

uA-

uB ,

u~R.

i--1

T h e r e f o r e uA is h o m o l o g o u s to uB. A n y 0-cycle is of the form E uitr ~ and is homologous to (E ui)A. Two 0-cycles uA and vB with u ~ v, are not h o m o l o g o u s , because if they were u A - oB would be a bounding cycle, but all bounding cycles are of the form u A - uB. H e n c e Ho(M, I~) is isomorphic to R. Note that the Poincar6 duality t h e o r e m , i.e., bp = bn_ p (quoted p. 228), yields b 0 = bn.

Answer l b iii: Examples: T h e triangulations of S 2 and T 2 are as follows:

326


The eight curvilinear triangles ABE, AED, ADF, AFB, BCE, CED, DCF, BFC determine a triangulation of the sphere.

For the torus, the representation by means of a rectangle with opposite sides identified can be used. A

F,

,E

c

B

C

A

D

D

E

E

~

~

E

.~

Fig. 4.

On S 2, F - E + V -- 2; on T 2, F - E + V --0. One can also compute X ($2) and X (T2) from the Betti numbers given on (p. 224). Remark" One can check that inserting segments in a triangulation creates only bounding cycles or cycles homologous to existing cycles; for instance, if we join FB in the torus triangulation by inserting 3 segments FH, HI, and IB, we create several bounding cycles and one 1-cycle, namely, BF + FH + HI + IB, homologous to the existing 1-cycle, AD + DE + EA.

Gauss map

c) We can prove that the sum of the indices Y-~Mi of a vector field v on M does not depend on the choice of v by using the properties of the Gauss map. The Gauss m a p G" OM - ~ S n - I assigns to each point x e OM the outward unit normal n (x). Let M C I~n be a compact n-manifold with smooth boundary OM (see paragraph d f o r manifolds without boundary). Show that

deg(G, 8 M ) - Z

i,

M

where )-~M i is the sum of the indices o f any vector field v on M with isolated zeros, pointing outward at the boundary. Answer 1c" Remove an s-ball around each zero of the vector field v in M; let/~/be the new manifold. The boundary aA?/has several components: 8 M and the boundary of each e-ball. We shall show that the sum of the degrees of the map ~: x --~ ~(x) - v(x)/Iv(x)l on the various components of OM vanishes. Indeed ^

~IOM OM ~ S~-~ is the restriction to 0 A?/of a smooth map

~. ~ _ ~ sn-I which implies that the degree of ~103~/is zero (see Milnor 1965)"


327

Let y be a regular value of ~1011~/. By definition (p. 557), degree (~, O/l), y) -- ~-:sign J~(x) for all x ~ ~ - l ( y ) . The set {~-l(y)} is an arc (or a finite union of arcs) whose boundary points a, b are on OM, if ~(a) points inward, ~3(b) points outward. b

Fig. 5. M is the annulus between the 2 circles.

Thus for each pair of bounding points sign J~ (a) - - sign J~ (b). Now ~ J]0 M is homotopic to G, since, by hypothesis, v points outward at the boundary of M. Hence deg(~, OM) --deg(G, OM). The degree of ~ on Oe(xo) with e(xo) the e ball centered at a zero x0 of v is equal by definition to minus the index of v at x0. The minus sign is introduced by the orientation of 0e which is not the boundary of the ball ex0 but one component of the oriented (p. 218) boundary of the oriented manifold M ^

deg(~, Oe(xo)) - - i . ^

And since the sum of the degrees of ~31[0 M on all its components vanishes deg(G, 0M) - ~ i

-- 0.

M

Gauss map

Gauss map L

v

Fig. 6. G preserves the orientation of a locally convex surface and reverses the orientation of a locally saddle-like one.

328

V BIS. CONNECTIONS ON A PRINCIPAL FIBRE B U N D L E

d) Let M C I~ N be a compact n-manifold without boundary. Let Ne be a smooth manifold with boundary which is a closed e-neighborhood 1 o f M. Show that the index sum o f any vector field v on M is i -- deg(G, ONe).

Z M

Hence Y]~M i is independent of v. Answer 1d" We shall only give the proof for the case where the zeros of v are nondegenerate. See [Milnor 1965] for zeros which are degenerate and more details on the proof. If the vector field v on M can be extended to a vector field w on Ne such that w points outward on the boundary, and such that w and v vanish at the same points and have the same indices, then the result follows from the result of the previous section. Let x 6 Ns and r(x) ~ M such that [Ix - r(x)[I is minimum. To construct w we set w ( x ) - (x - r(x)) + v(r(x)); for e sufficiently small r(x) is well defined and smooth, x onal to Tr(x)M, w points outward along the boundary ON~ -- e} and vanishes only at the zeros of v. The derivatives nents of w at a zero x0 of v are such that det(Ow~/Ox ~) In another notation

dwxo (u) = dvxo (u) dwxo (u) - u

r(x) is orthog=_ {llx - r(x)ll of the compodet(Ov~/Ox~).

for all u ~ Txo M for u in the orthogonal complement of Txo M.

The determinant of dwxo and the determinant of dvxo are equal and so are the indices. 2.

VAN HOVE'S SINGULARITIES 2

The thermodynamical properties of a crystal are to a large extent determined by its frequency distribution function g(v) where the frequency v is defined as follows: Let s(a) be the basic vectors defining a crystal cell. Let h (c~) be the reciprocal lattice of the crystal defined by the duality in IRn, usually IR3,

(h(~),s(/j))- 6~. 1Set of points of the euclidean space I~N which are at a distance _< e from M. For the existence of such an Ne see [Milnor 1965 problem 12, w 2See also the Reeb theorem, p. 592.


329

The elastic vibrations of a crystal are superpositions of normal modes, each of which is a plane wave vibration with wave vector 27rq. The frequency is a multivalued periodic function of the wave vector (i.e., the dispersion relation v (q))

v(q + ~-~ nah (~)) --v(q),

naEZ,

i.e., a function defined on a toms. The analytic singularities of the frequency distribution g(v) originate from the critical points of v(q) on each of its branches, (i.e., the zeros of its gradient, namely the zeros of the group velocity of the q-wave). Van Hove used the fact that the number and the nature of the critical points of a function are constrained by the topology of its domain (Morse theorem) to constrain the occurrence and the nature of the singularities of the frequency distribution - known nowadays as the Van Hove singularities.

Let f be a differentiable function on a compact manifold M and Cx the number of its critical points, assumed nondegenerate, of Morse index )~ (p. 571); show that ~ ( - 1)~ Cx is equal to the Euler-Poincard number. Therefore the number of critical points of a function is constrained by the topology of its domain.

Answer 2: Let f : M --+ R. The critical points of f are the points X(i ) such that

af

Ox~ (x(j)) = O,

for every ~ = 1 . . . . . dim M.

The index of a critical point x(./) of f is the number of negative eigenvalues of the hessian 02f/Ox~Ox/3 at x(j). To assume that a critical point

x(j) of f is nondegenerate is to assume that det(Ozf/ox(.~)Ox~(j)) r O. The gradient of f defines a covariant vector field v of components

va(x) = 3f /Ox ~. The zeros of v are the critical points of f . The index of v at a critical point x(j) is the degree of the map

~: x --+ f ' ( x ) / l f ' ( x ) l on a small sphere S.i =_~(x(.j)) centered at x(.i): deg(b, S.j, y) - ~

sign J~(x) for all x 6 ~-I (y) C S/by definition (p. 557).

A nondegenerate critical point of f is a zero of v where dv is nonsingular; hence ~ is bijective on S j, and x is unique"

index of a critical point

330

V BIS. CONNECTIONS ON A PRINCIPAL FIBRE B U N D L E

deg(~, Sj, y) - sign J~ (x) = signdet(O2f/Ox ~ Ox l~)

= ( - 1)x

by continuity from x to x(j),

where ,k is the number of negative eigenvalues of the hessian of f at x(j). Finally, summing over all the critical points of f , ~(-1)XCx--~i-x(M). M

3.

GAUSS-BONNET-CHERN-AVEZ

THEOREM(quoted p. 395)

Let M be an even dimensional oriented compact riemannian or pseudoriemannian manifold with metric g; let X (M) be its Euler number (p. 224) and y its Euler class (p. 394), then x(m)--

f, M

A p r o o f f o r some embedded manifold. Let M C R n be a compact nmanifold with boundary OM. Let G be the Gauss map defined in section 1 c), and let J (x) be the jacobian o f G at x. a) Show that f

J(x)r~M = deg(G) volume S n-I ,

~JM

where r~M is the volume element on 0 M induced by the euclidean volume element in R n. b) Assume OM is even dimensional. Let a ~ S n - I be such that a and - a are regular values o f G. Let v be the vector field on 0 M defined by v(x) = ( a l G ( x ) ) G ( x ) - a, where ( a l G ( x ) ) is the euclidean scalar product. Show that the index o f v at its zeros is 1 if G preserves the orientation and - 1 otherwise. Show that the sum o f the indices o f v at all x where G (x) = a is equal to deg(G). Show that the sum o f the indices of v at all x where G(x) = - a is equal to deg(G). Show that if 0 M is even dimensional

2 deg(G) - X (0 M). A n s w e r 3a: [See Guillemin and Pollack.]

331


f J(X)~'aM-- f a*~s~-,-aM OM

deg(G) f

sn-1

"/'Sn_l

= deg(G) vol S n- 1.

Answer 3b" The vector field v vanishes at the points x 0 such that G(x o) = +-a. The index of v is the degree of the map 6 " x ~ 6(x) of a small sphere ae(Xo) around x o and deg(6, ae(Xo), y) = ~

sign Jacobian of t3(x) for all x E t3-1(y).

Since __a are regular values of G, 6'(x 0) is an isomorphism and index of v at x 0 = sign (Jacobian of t3)(x 0) v'(x) = (ala'(x))a(x)

+ (ala(x))a'(x)

.

It follows from (G(x){G(x))= 1 that (G(x){G'(x))=0. In particular 0 = ( G ( x o ) l G ' ( x o ) ) = ( +--alG'(xo) a n d v'(Xo) = (a[ ___ a ) G ' ( x o ) = +- G ' ( x o ) .

Thus two cases arise: (Jacobian of 6)(Xo)= (Jacobian of G)(xo) or

(Jacobian of t3)(x0)= (Jacobian of ( - G))(x0) = ( - 1 ) ~ - l ( J a c o b i a n of G)(xo) = (Jacobian of G)(xo)

since n -

1 is even.

In both cases the index of v is 1 [respectively - 1 ] if G preserves [reverses] the orientation. In summary, (index of v)(x)= deg v on ae(x)

by definition

= sign (Jacobian of v)(x)

by definition and by isomorphism of v'(x)

= sign (Jacobian of G)(x)

by calculation.

It follows that

E

{x0; G(xo)=a}

indices of v at x 0 = ~

sign Jacobian of G at x o = deg G .

The same sum over the points x 0 such that G(xo) = - a is also equal to the

332


degree of G. Hence by the Poincar6-Hopf theorem

i = x(OM) = f y = 2 deg(G, aM) aM

curvature integral

if aM is even-dimensional.

aM

Because of the relation between the Euler-Poincar6 characteristic and the curvature (Gauss-Bonnet theorem), the degree of the Gauss map G on aM is also called the curvature integral of aM. Example" Let aM = S 2 and v be a vector field on aM with zeros of index 1 at the North and South Poles. ~.V

. G

i

S2

Fig. 7. A Picasso view of aM, its Gauss map, and the vector field v vanishing at N and S. 4. SUPERTRACEOF e x p ( - A ) a) Show that on a compact Riemannian manifold M, the Euler number is

x(M)=de-do,

(1)

where d e [respectively, do]/s the dimension of the space of harmonic forms to of even [odd] degree, /too = O. Answer 4a: According to Hodge's theorem (p. 400) the space of harmonic p-forms on a compact manifold M is isomorphic to HP(M), hence d e is the sum of the even Betti number and d o the sum of the odd ones. The Euler-Poincar6 characteristic (alternate sum of Betti number) is equal to the analytical index of the elliptic complex {dp, Ep } where dp is the exterior derivative defined on p-forms and Ep the vector bundle of p-forms on M. The laplacian A on a compact manifold M is spectrum )t0 = 0 , h i , . . . , A , , . . . and A - h , kernel. The spectrum of the operator e x p ( - h i ) , . . . , e x p ( - A , ) , . . . . Its trace in the functions spanned by its eigenvectors is

an operator with discrete has a finite dimensional exp(-za) is e x p ( - h 0 ) , Hilbert space of normed

oo

Tr exp(-za) = ~

u. e x p ( - A . )

n----O

v, = dim. of the space spanned by the k, eigenvectors.


333

b) Show that the supertrace of exp(-A) defined by Str e x p ( - a ) = Tr exp(-zl)lg+ - Tr e x p ( - A ) ] g - , where ~+ and Yg- are the Hilbert spaces of even and odd forms on a compact manifold M, is a topological invariant [see McKean and Singer]. Answer 4b: [See Getzler.] Let Q=d+8,

where 6 is the metric transpose of d defined on (p. 297). Then Q. ye ~-~ ~ is a self-adjoint operator such that zi = Q2. Let ~ be the eigenspaces of the (positive) operator .4 corresponding to the eigenvalues h >__0. We have +

Q. ~ ; ~ ~

-T-

.

...

(2)

+

Indeed, let f ~ ~ A , then AQ f = Q af= a QL hence Q f E ~-;. The operator Q Z restricted to Y(~ is equal to the operator multiplication by h" Q2Ix; = x.

(3)

It follows from (2) and (3) that, if ,~ ~ 0 , ~ - l / 2 Q [ ~ .7.;t-~/2Ql~ ~ and, for h r dim ~ f = dim ~ - . Now Str exp(- A) = ~ e x p ( - h)(dim ~

- dim ~ - )

is the inverse of

(by definition)

A

= dim ~ o - dim ~ o = x(M)

since dim ~

= dim ~ - for h ~ 0

(by eq. (1))

since a harmonic form is an eigenstate of A with 0 eigenvalue. The Laplace-Beltrami operator is an example of "supersymmetric quantum mechanics" and displays cancellations analogous to the cancellations of supersymmetric quantum field theory (SUSY). For an explicit calculation of x ( M ) by means of a path integral calculation of Str exp(-A) see [Witten, DeWitt]. REFERENCES J.W. Milnor, Topology from the Differentiable Viewpoint (The University Press of Virginia, Charlottesville, 1965) pp. 32-37. J.W. Milnor, Morse Theory (Princeton University Press, 1963). S.S. Chern, Selected Papers (Springer-Verlag, New York, 1978) pp. 83-136, i.e., "A simple intrinsic proof of the Gauss-Bonnet formula for closed riemannian manifolds", Ann.

334


Math. 46 (1945) 674-684; "Characteristic classes of hermitian manifolds", Ann. Math. 47 (1946) 85-121. S.S. Chern, "Curves and Surfaces in Euclidean Space", in Studies in Global Geometry and Analysis, ed. S.S. Chern, in Vol. 4, Studies in Mathematics (Prentice-Hall, Englewood Cliffs, NJ, 1967). S.S. Chern, "From Triangles to Manifolds", Amer. Math. Monthly 86 (1979) 339-349. V. Guillemin and A. Pollack, Differential Topology (Prentice-Hall, Englewood Cliffs, NJ, 1974) pp. 194-199. L. Van Hove, "The Occurrence of Singularities in the Elastic Frequency Distribution of a Crystal", Phys. Rev. 89 (1952) 1189-1193. For a pedagogical introduction see C. Kittel, Introduction to Solid State Physics, Fourth edition (Wiley, New York 1971), p. 210 and p. 703. See also S.I. Goldberg, Curvature and Homology (Academic Press, New York, 1962). I.M. Singer and J.A. Thorpe, Lecture Notes on Elementary Topology and Geometry (Scott, Foresman, Glenview II., 1967). E.M. Patterson, Topology (Oliver and Boyd, Edinburgh, 1959). E. Getzler, "Atiyah-Singer Index Theorem", in Critical Phenomena, Random Systems, Gauge Theories, eds. K. Osterwalter and R. Stora (North-Holland, Amsterdam, 1985). H.P. McKean and I.M. Singer, "Curvature and the eigenvalues of the laplacian", Journal Diff. Geom. 1 (1967)43-69. E. Witten, "Supersymmetry and Morse theory", J. Diff. Geom. 17 (1982) 661. E. Witten, "Holomorphic Morse inequalities", in Algebraic and Differential Topology, Global Differential Geometry, ed. G. Rassias pp. 318-333 (Teubner Texte z. Mat. 1984). B.S. DeWitt, Supermanifolds (Cambridge University Press, 1984).

7. E Q U I V A L E N T

equivalent

BUNDLES

Let ~ and ~ ' be two principal b u n d l e s with the s a m e space B, base M, and g r o u p G, defined respectively by the local trivializations (U~, the) and (U~, ~b~), a n d the transition functions y~j, y~;. T h e b u n d l e s are said to be equivalent if on e a c h U~ there exists a c o n t i n u o u s m a p p i n g A i ' U ~~ G such that the h o m o m o r p h i s m G---~ G d e d u c e d for each x E U~ from the mapping ~b, o ~bi-l" U~ x G---~ U~ • G is the left p r o d u c t by A~(x). 1) S h o w that the transition functions o f two equivalent bundles are such that

"y~j(x) = A 71(x)Ti;(x)A;(x) .

(1)

2) Show that if ~ and ~ ' are equivalent bundles there are h o m o m o r p h isms ~ ~ ~ ' and ~'---~ ~ which induce the identity map on M.

8. U N I V E R S A L B U N D L E S ; B U N D L E C L A S S I F I C A T I O N

335

R e m a r k : Principal bundles with the same group and base such that there exist such homomorphisms are also called equivalent. A n s w e r 1" By definition of Ai we have A

A

t

((~i,x o (~i ,x - 1 ) .__ ~ i ( X ) )

X ~ Ui 9

A transition function 3'//is defined by A

A

--1

ch,.x ~ ch j.x = y,j(x),

x ~ U, N U j ,

g E G.

We have therefore (

)

A

t y;j x = 4~,x

A o ~ , - 1:,x

A

._

A

A

A

t o ~ i , -x1 o ~ i , x O t ~ j , -x1O 4~,x

= ~l(x)'Yi](X)t~j(X

A ~j

,x o ( ~ '/,x

1

) .

l(ui) in ~ into p-l(u~) in ~ ' by its representative in local trivializations, given by

A n s w e r 2" Define a map from p -

(X, g ) E c~,(p-~(U~))

h i" ( x , g)~---)(x, Ai(x)g),

(x, a,(x)g) We check, using (1), that the maps so defined in different trivializations, coincide in p - ~(U~) N p-~(Uj), and thus define a bundle map h. It obviously induces the identity on M. It is easy to show, by expressing the right actions in local trivializations, that the mapping h is an homomorphism.

8. UNIVERSAL BUNDLES. BUNDLE CLASSIFICATION

Let ~1 and ~2 be two bundles with fibres F 1 and F2, groups G 1 and G 2. A mapping h" ~ i - ~ ~2 is said to be a bundle homomorphism (p. 380) if there exists a homomorphism ~ : G 1--~ G 2 and if h commutes with the actions of G 1 and G 2, namely: hRg 1 = RgEh

on

~1 ,

g2 = ~i~gl 9

The map h then induces a map h between the base spaces" h =pE~176 and the following diagram is commutative.

,

bundle homomorphism

336

V BIS. C O N N E C T I O N S

ON A PRINCIPAL FIBRE BUNDLE

~1

) ~2

Pl ~ M1

h ~'p2 ) ME

In question 1 we shall study the converse. 1. PULLBACK OF A BUNDLE

a) Let f: M---~ M ' be a continuous map, and let gd' be a bundle over M ' with group G and fibre F. Construct a bundle ~ over M, with the same group G and fibre F, such that there exists a h o m o m o r p h i s m h" ~ ~ ~ ' and f = h. A n s w e r la: i) We can construct ~ by a pedestrian pull back of the bundle

structure of ~ ', namely consider a cover { U'i } of M', the corresponding local trivializations p'(U'i)---~ U'i x F with the transition functions y~j, and define ~ by the cover { Ui}, pull back by f of the open sets U I, the transition functions y~j by pull back of the y~j, i.e., y~j = y~jof, the fibre F x as the equivalence class of copies F~ of F defined by (x, y~)= (x, yj) if x E U~ f"l Uj, yj = yij(x)yi, the trivialization over U~ being p - l ( u i ) = U i • F~ and the projection 7r, independent of the trivialization "rr(Fx)= x. The homomorphism h: ~--~ ~ ' is defined in the considered trivializations, by (x, y g ) ~ ( f ( x ) , yg); it is independent of the trivialization since y~j(x)y/T~i(f(x))yj

9

ii) We can define ~ as the subspace of M x ~ ' of these pairs (x, b') such that f ( x ) = 7r'(b'). It is given a bundle structure with base M by the natural projection zr" M x ~'---~ M, restricted to ~. A right action of G -a on F x = 7r (x), fibre at x in ~ , which is identified to the fibre at f ( x ) in ~J ', is given by It completes the definition of the bundle structure of ~. We have then obviously hRg = Rgh ,

pull back

g E G

if the mapping h" ~ ~ ~ ' is defined to be the natural projection M x ~'---~ ~ ' by (x, b ' ) ~ b', restricted to ~. The two constructions (i) and (ii) give isomorphic bundles. The bundle is called the pull back of ~ ' by f, and denoted ~ = f* ~ '. It can be shown [Steenrod, p. 53] that if M is a paracompact manifold and if f0 and fa are two homotopic maps M ~ M' the two pulled back bundles

8. UNIVERSAL BUNDLES" BUNDLE CLASSIFICATION

337

~1 - - Jo*~' and ~ 2 - - fl * ~ ' of the same bundle ~ ' are equivalent bundles, in the sense that there exists a bundle homomorphism ~ 1 --+ ~ 2 which induces the identity map M --+ M, and conversely. b) Show that if M is a manifold diffeomorphic to 1Rn, then any bundle over M is equivalent to a trivial bundle.

Answer l b: If M is diffeomorphic to R n then the identity map R n --+ R n is homotopic to a constant map R n -+ x0; one says that R n is contractible to a point. The pull back of a given bundle ~.~' with base M by the identity map M -+ M gives the bundle ~.~, while the pull back of ~.~' by a constant map is a trivial bundle.

contractible

a point

2. UNIVERSAL BUNDLE The statement in 1b shows that there is a way to classify the bundles with given group G, base M and fibre F if they can all be obtained as pull back of some "universal" bundle, with group G, fibre F and base some space X, by using the homotopy classes of maps M --+ X. Show that if a principal bundle @'G -- (U, G, X), with group G and base X is arcwise connected and such that its first n homotopy groups vanish" 7r 1 ( U ) - - y r 2 ( U ) . . . . .

y r n ( U ) -- 0

(1)

then every principal bundle with group G and base a paracompact manifold o f dimension < n can be obtained as the pull back of @'G by a mapping f" M - - + X . Such a bundle ~r is called n-universal. Since the pulled back bundles are equivalent for homotopic maps M --+ X these maps are called classifying maps. A n s w e r 2: We shall give an idea of the construction of the homomorphism h" ~.~ --+ ~'c using (1). The mapping/7t, which will have to be continuous, can always be defined on fibres over isolated points of M, basis of ~ , by setting where ~: G --+ G x and ~: G --+ G y are respectively mappings from G onto the fibre Gx over x 6 M and onto the fibre G y o v e r some point y ~ X, which commute with the right action of G on the considered fibres. Such an h can be considered as a homomorphism from the restriction of to fibres over the 0-cells of a complex decomposition (triangulation) of M (cf. Problem IV 1, Cohomology) into ~'c. We shall first show that h can be extended to the restriction of ~ over the 1-cells. N

n-universal classifying maps

to

338


Let D 1 be a 1-cell in M; since Da is contractible (homeomorphic to [0, 1]) the portion of ~ over Da is equivalent to the trivial bundle D 1 x G, and the restriction of ~ over OD1 is equivalent to aD1 x G. T h e b o u n d a r y a Da consists of two points x0, x a E M, above which the map h has been defined, i.e., we know hlaoxxG" aD 1 x G---~_9/a. We define f to be the mapping aDa ~ q/a defined by f(x) = h(x, e), x E OD1. Since ~ a is arcwise connected, f extends to a (continuous) map still denoted f: D1 ~ ~ We define the map h" D a x G--~ ~ a by

h(x, g)= P( f(x), g), where P is the mapping q/cxG--~q/a

by

(y,g)~Rgy.

It is straightforward to check that h takes the preassigned values on o~D1 x G and is a homeomorphism D 1 x G---> q/a. Analogous reasoning will give the conclusion by induction for k cells, k-< n. The boundary a D k of a k-cell D k is homeomorphic to a k - 1 sphere Sk_ 1 sphere, and maps from a D k into ~ can be extended to D k because 7rk_1 = 0.

3.

EXISTENCE OF UNIVERSAL BUNDLES

It can be proved that every compact Lie group G is isomorphic to a subgroup of an orthogonal group O k [cf. Chevalley, Theory of Lie groups p. 211]. It can also be proved [cf. Steenrod, p. 103] that the Stiefel manifolds Vn+k, k --On+k\Ok have zero homotopy groups 7ri(Vn+k,k)--" 0 for l n the (horizontal (p. 373)) 2k form f(~2) vanishes, therefore T f (co) is closed. If 2 k - 1 > n the 2 k - 1 form f(~o(t), ~ ( t ) , . . . , ~2(t)) is zero (it is horizontal because, unlike co(t), ~0(t) is horizontal); the cohomology class [Tf(co(t)] is independent of t; it follows that [Tf(co)] is independent of co, indeed two arbitrary connections coo and col can always be imbedded in a 1-parameter family [Problem V bis 1, Explicit proof] co(t) - (1 - t)o~0 + tool, and [Tf(co(t))] -- [Tf(co0)] -- [Tf(col)].

4) We consider the case of riemannian connections on a manifold X of dimension n. a) Show that if f is an Ad O(n) invariant polynomial of odd degree and S2 a riemannian curvature 2-form then f (~2) =_O.

b) Show that if co(t) is a 1 -parameter family of connections of conformal metrics one has

344

V BIS. CONNECTIONS ON A PRINCIPAL FIBRE BUNDLE f ( d t o ( t ) ~ ( t ) .. ~(t))=0 dt ' ' "'

for any Ad ~l(n, R) invariant polynomial of even degree 2s. Deduce that, if f(I2) = 0, the cohomology class of Tf(to) is then a conformal invariant. c) Show that if f is an Ad ~l(n, R) invariant polynomial then f(s itself is a conformal invariant, expressible in terms of the Weyl tensor. Answer 4a: The elements of the Lie algebra ~(n) are the real, antisymmetric n x n matrices. Thus the curvature of a riemannian connexion satisfies the transposition law

tO---O; therefore if f has degree k f(to)

= (-1)kf(o)

;

but since f is a symmetric multilinear mapping from qd into R we have

f(V,...,

V)- t(f(V,..., V))--f(tv,...,

tv)

thus f(a)

= f('a)

comparing the two relations gives f ( O ) = 0 if k is odd.

Answer 4b" We consider the family of conformal metrics on X g(t) = exp(2a(t)g) , where g is a given metric and a a scalar function on X, depending on the parameter t. We make the computation in a local trivialization: if it is defined by a local section s: X D U---~ P we have, since the pull back is an homomorphism of the exterior algebra,

s*f --~ ' O(t), " " '

12(0 = f s* -d-[ dto , s, O ( t ) , . . . .

Let A be the matrix valued 1-form representative of a riemannian connexion in a coordinate chart, given in terms of Christoffel symbols by

A kj = Ai k

dx i

We have computed (p. 351) the difference of the representative A(t) and A of the connections to(t) and to defined by g(t) and g. It is given by the 3-tensor: Fii~(t) - Fi~ = 8 f 9aja(t) + 8jkaia(t) - gug kt~ta(t)

10. C H E R N - - S I M O N S CLASSES

345

thus the derivative d A ( t ) / d t is of the form qS(t)--=

dA(t) = I d b + a + fl dt '

da b = -dt '

where I is the n • n unit matrix and a , / 3 the n • n matrices of 1-forms with elements da = aj y dx ,

=-g

kt gij

da u dxi .

Using the multilinearity of f, and since db is scalar valued, we have

f ( ~ ( t ) , ~2(t), . . . , ~ 2 ( t ) - f ( I db + a + [3, ~2(t), . . . , ~2(t)) -db

f(l,

O(t),

. . . , O(t))

+ f([3, a ( t ) , . . . ,

+ f(a,

O(t),

. . . , O(t))

a(t)).

The first term is zero because it is now a polynomial of odd degree in 12. The two other terms can also be shown to be zero using the algebraic Bianchi identities (cf. Chern, p. 59). This result together with the results of paragraph 3 imply that [Tf(to(t)] is independent of t, i.e., is a conformal invariant.

Answer 4c: The Weyl tensor C is expressible in terms of the Riemann, Ricci and scalar curvature tensor (cf. p. 351); one can show that f ( C ) and f ( O ) differ by an exact differential (cf. Avez). 5) On a subset U C X above which H - I ( u ) gives, by choosing A 0 = 0:

is trivializable formula (5)

1

g.

f ( F ) = kd J f ( A , Ft, . . . , Ft) dt . o

The 2 k - 1 form Q ( A ) = - f l f ( A , F t , . . . , F,) dt is called a local ChernSimons form. It can be expressed in terms o f A . Give this expression when f is a second order Ad Gl(n) invariant polynomial. Answer 5" We have A , = tA, Ft = tdA + 89

A] = tF + 89 z - t)[A, A] ,

thus f ( A , F , , . . . , F,) can be expressed as a polynomial in t with coefficients functions of A and F, or d A and [A, A]. If f has degree two then

f ( A , Ft)= t f ( A , d A ) + 89 and the local C h e r n - S i m o n s form is

[A, Z ] )

local Chern-Simons form


346

[.

1

Q ( A ) - J f ( A , Ft) dt = 89 f ( A , dA) + ~f ( A , [A, A]).

(9)

0

A second order Ad GL(n) invariant polynomial on (gl(n) is given by

f(X1, X2) = tr(X1, X2),

X 1

and X 2 being n x n matrices.

If A is a ~l(n) valued 1-form, (9) reads

Q(A) = 89{tr(Z A dA) + ~ tr(A A [m, A ] ) } . Chern-Simons form

6) More generally, call Chern-Simons form a 2 k - 1 form which is the projection on X of the horizontal form on P, given in terms of two connections ~176and too by 1

Q(1)(to,, too)-- k j f(to, - too, at, 9 9 9, a t ) dt. 0

Show that if too, to1, to2 are three connections on P then (cf. Guo et al.) Q(1)(to,, to2) + Q(1)(to2, too) + Q(')(too, to1) = dQ(2)(to2, to1, too) (10)

where Q(2) is the 2 k - 2 horizontal form" Q(2)(to2, tol, t o O ) - " k ( k - 1)

f

f(~bl, ~ 2 ,

~'~tlt2,''',

~'~tlt 2)

dtl dt2

t 1+ t2-< 1

with ~1 ~ 0)1 ~ toO ~

~2 ~ 0)2 ~ toO

and 12,~,2 the curvature of the connection totlt 2 -- toO + t l ( t o l -- toO) + t 2 ( ~ 2 -

tOO).

For a generalization of this relation to forms Q(2r) of degree 2 k - r , depending on r + 1 connections [cf. Guo et al.].

Answer6" If 0 (2)= k ( k - 1)

f

f(m I -- (.00, (.O2 -- s

atlt2,...

, atlt2 )

dtx dt2,

t 1+t 2-1

we have by the properties of exterior derivation, since %, too, to2 are 1-forms and 12t~t2 a 2-form on which f depends in a symmetric way:

10. CHERNmSIMONS CLASSES dQ (2)= k(k-

1)

f

{f(d(tol- t~

347

to2- too, f2,,,2,..., n,,, 2)

tl +t2 S 1 , where ~g acts on ~t. Eq. (6a) is the additive version of the cocycle *In general physical situations to takes its values in the space of local functionals on ~t. [See, for instance, Dubois Violette.]

352


condition derived in [Problem IV 4, Cohomology, eq. (8)]. We rewrite (6a) as follows: dto = 0 .

(6b)

Remark: We refer the reader to [Jackiw] for an example in quantum mechanics of representations U satisfying (4) and ( 5 ) - namely a representation of Galilean boosts. 4. PROJECTIVE REPRESENTATIONS OF c~ projective (ray) representation

A mapping U of ~ginto L(gt, ~ ) is called a projective (ray) representation, if it is associative

U(g3)(U( g2)U(g,)) = (U( g3)U(g2)) U(g,)

(7)

U(gE)U(gl) : exp(ia(A, gl, gE))U(gEgl),

(8)

and if

phase of representation

where a: M • ~ x ~g-->S 1 is called the phase of the representation. a) ~Show that, if U satisfies (8), it is associative if and only if a is a cocycle. b) Show that if the mapping U is given by (4) it is always associative: U defined by (4) is always a projective representation, with zero phase if to is a cocycle. c) Show that if U is a projective representation and if a = dfl, one can find tB(A g) a representation ~1 = e " U which satisfies (5). Answer 4a: We shall compute the action of U(ga)(U(gE)U(gl)) on and the action of (U(g3)U(gE))U(gl) on 1/' and equate them: U( ga)(U( g2)U( gl))~'(A) = U( ga) e x p ( i a ( A , gl, gE))U( gl g2)~(A) = exp(ia( Aga, gl, g2) + i a ( A , glgE, ga)) x U(gagEgl)~(A), (U( ga)U(g2)) U(gl)O(A) = exp(ia(A, gE, g3)) U( g3gE)U(gl)O(A) = exp(ia(A, g2, g3)) + i a ( A , gl, gEg3)) x U(gagEgl)~(A). The associativity holds if and only if a( Aga, g~, g 2 ) - a ( A , g2, g3) + a ( A , gEgl, g 3 ) - a ( A , gx, gagE) = 0 mod 2 ~r.

(9a)

A calculation similar to the one leading to (8) in [Problem IV 4,

353

11. COCYCLES ON THE LIE ALGEBRA

Cohomology] shows that (9a) can be called a 2-cocycle condition for the mapping a" ~r x ~d x ~

S1

with ~3 acting on ~r

and we rewrite (9a) as follows" dc~ = 0 .

(9b)

Remark: See [Problem IV 6, Short exact] for an example of a projective representation (4) when c~ is not a trivial cocycle- namely the quantum mechanical representation of translations in phase space.

Answer 4b: We have shown in Answer 3) that if U acts on ~ by (4) then U( g2)U( g~) = exp(i doo)U( g2g~) . U satisfies the relation (8) with a phase c~ = dw; the cocycle condition da = 0 is always satisfied and U is associative.

Remark: An action of U(g) on ~p(A) more general than (4), U( g)$(A) = e(A, g)$(A g) with 16(A, g)l not necessarily equal to 1, can define an associative projective representation (8) with a nontrivial cocycle. For properties of 6 and for applications to Physics see for instance [Faddeev, Zumino].

Answer 4c: Given a projective representation U the problem is to find /3(A, g) such that U =exp(ifl)U

with (J(g2)(J(g~)=/](g2gl),

i.e.

exp(i/3(A, gz))U(g2) exp(i/3(A, ga))V(gz)qJ(A) = exp(i/3(a, gzga)U(gzgx)qJ(A)

(10)

with U(g) satisfying (8) and (7). Rewriting (10) and, using (8) gives /3(A, g2) + fl( Ag2, ga) + a(A, ga, g 2 ) - fl(A, gzgl)= O.

(11)

This equation says that the cocycle a isw a coboundary, a = d/3. Hence if a = d/3 one can find a representation U satisfying (5).

Remark" We can weaken the associative requirement and replace it by (U( gl)U(g2)) U(g3) = exp(i/3(A, gl, g2, g3)) U( gl)(U(g2)U(g3)),

354

V BIS. CONNECTIONS ON A PRINCIPAL F I B R E B U N D L E

where/3 is obtained by requiring associativity of four-fold products. We refer the reader to Jackiw for a study and example of representations of this type. 5. CENTRAL EXTENSIONS OF .o~(~)

Let U be a smooth* mapping from ~ = c=(M, G) into the space of linear maps L(~, rF) =_L; let

We denote by ~b = U'(e) r the element of L image of ~b by the linear map U'(e)" T,~---~ L, and by [q~,, ~2] the usual bracket of linear maps, i.e.,

[q~l, ~2] = t~lq~2 - q~2q~l.

(12)

a) Show that if U satisfies (8) and U(e) = 1 (e unit of ~, 1 unit of L, i.e., the identity map) then: [q~, ~2] = [~bx, ~b2]+ C(A, 4h, ~b2)1,

(13)

where [~b~, ~b2] is the element of T~ ~ corresponding to the Lie bracket of the elements of .~(~) defined by 4~ and ~b2, [ ~ 2 ] its image by U'(e) and where C is a numerical valued 2-cochain on ~ ( ~ ) . (To simplify the proof, take the group G to be a linear group.)

b) Show that if U is associative C satisfies the cocycle condition [Problem IV 7, Cohomology]. Answer 5a: We denote by exp t$ the 1-parameter subgroup of ~3 whose tangent at unity is ~b. This subgroup exists when @ = C~(M, G) and is given by

(exp t~)(x) = exp t~(x) E G ,

x E M,

dp(x) E ~(G)

We recall that, for an ordinary Lie group G, we have the Campbell Hausdorff expansion formula exp tu 1 exp su 2 = exp(tu I + su 2 + 89

u2] + . . . )

u 1, u 2 E ~ ( G ) . (14)

(exp t~bI exp s~b2)(x) = exp(t~bl(X)) exp(s~2(x))

(15)

which gives, by the Campbell-Hausdorff expansion formula for small s *In the sense of Gateaux derivative [Problem II 1, Supersmooth], ~3 being a manifold modelled on the locally convex vector space .2'(@).


355

and t (for a linear Lie group G the exponentials on the right are usual exponentials of matrices).

(exp tt~l exp St#2)(X ) = exp(/thl(X ) + s(~2(X ) + 89

(~2(X)] + ' " ") ;

in other words (exp t$1 exp s4h) = exp(t4h + s$2 + 89

4h] + ' " ").

(16)

We now define exp t6 to be the image by U of the 1-parameter group exp t$, i.e., exp tt~ = U(exp t $ ) .

(17)

We define d 8 = ~-~ U exp

t4~[,=0

(18)

9

We have, then, by derivation of (17) that ~ is deduced from 4~ by the linear map, U'(e)" 8 = U'(e)$. By the hypothesis (8) on U we have exp t61 exp s62 = exp ia(A, exp t~b1, exp s~b2)U(exp t~b1 exp

sck2). (19)

Using (16), eq. (19) reads

exp tt~l exp S~2 = exp ia(A, exp t~l, exp

s~2)U(exp(s~2 + tt~x + lst[t~2, t~2] +..-))

(20)

and we have immediately, using (18) ~

(exp t ~ exp s~2) ,-~--o = ~ 2

(product of linear maps).

Therefore d2 [~1, ~2] = dt ds (exp t~l exp s~2 - exp s~2 exp t~ 1)1,_~.o 9 Computing the right-hand side of (20) we find an expression of the form

[~1, t])2]--[~P

2] -[- C(A, t#l , ~2)1

where

[~2]

-" O'(e)[(~l, t#2]

(21)

356


and C(A, 4~1, t~2) is a bilinear and antisymmetric mapping from ~ ( ~ ) into L depending (20) on OgxCt,Og2a and also U'(e), and U"(e) if U is not linear.

Answer 5b" The brackets [~1, ~2]

a n d [~1, ~2] both satisfy the Jacobi identity since they are brackets in the Lie algebra ~ ( ~ ) and in the Lie algebra of linear maps, respectively. One shows that the image [~b~, ~ ] = U'(e)[4h, ~b2] satisfies the Jacobi identity if and only if U is associative by generalizing classical methods [cf. for instance Pontryagin].

[[~,~'~21, ~31 + [[4~3, 4'11, ~21 + [ [ ~ ~ 3 1 , ~,l = 0. Using these results and the definition of C we obtain

C(A, [61, 62], 63) + C(A, [63, 61], q~2) + C(A, [62, 63], 6 1 ) = 0 which expresses [Problem IV 7, Cohomology] that C is a cocycle on dC = 0 . 6. LIE ALGEBRA OF

C~1761, G)

Cocycles for the Lie algebra of the space C| 1, G) of smooth maps from the circle S ~ into a finite dimensional Lie group G, that is of C = mappings from R into G, q~~--~g(q~) which are periodic of period 2~r. The group structure of G defines a group structure on C=(S ~, G) by pointwise multiplication (glg2)(~o) := gl(~o)g2(~o).

(22)

a) Construct a basis for the Lie algebra .~(C=(S 1, G)). b) Identify the space of constant maps in C| 1, G). c) Consider a projective representation of .~(C=(S 1, iT)) on a Hilbert space

~(. Derive the general expression for the complex valued 2-cocycles on ~(C=(S 1, G)), and compute the dimensionality of its second cohomology group n2(,.~P(C~ 1, G)); C) when G is simple. Answer 6a: The Lie algebra of the group C| 1,G) is the space C| 1, ~(G)). A basis for this infinite dimensional vector space, endowed with the Fr6chet topology of uniform convergence of maps and each of their derivatives is the denumerable set (we suppress the notation | since the meaning is clear) {0a cos n~o, Oa sin n~o;

n ~ [~, {0a } a basis of *~(G)}.


357

We shall use for simplicity in writing (expansions will then have complex conjugate coefficients)

G~a = Oazn ,

with z = e ir n E 7/.

The general formula of section 1 reads now [Gn,

GT]

= [Oa, Ob]Z m+n .

Thus, if fab denotes the structure constants of G,

[G n,,

n +mr Oc ' Jab

am]__z

b

[Gna a mb l - - f cab G m+n c

(23) "

A Lie algebra with Lie bracket (23) is called an (untwisted) affine Kac-Moody algebra. For applications of the Kac-Moody algebra to physics see for instance [Dolan, Goddard et al.]. Answer 6b" A constant map in cg

=

Kac-Moody algebra

C~(S 1, G) is a map

q9 ~-->c , c a fixed element of G. The space of constant maps in C~ a, G) is isomorphic to G. The quotient space C=(S a, G ) / G is called the loop g r o u p of G, and is usually denoted

OG

-~ C ~ ( S

1,

G)/G.

Answer 6c: Let G ~"be the representative of G~ in a projective representation of ~ ( ~ ) on a Hilbert space ~ : according to Answer 5) and formula (23) we have [(~m, 0~1 =fCbffr7 +n + C(G ~a ' G"b )

(24)

1] a

where C is a cocycle on ~(~3), i.e., a bilinear antisymmetric map from ~(q3) • ~(q3) into R, which satisfies the equation

C([G m , G"a], G ck) + C([Gb,,, Go], k G T ) + C ( [ G rk G a ] , G b ) = 0 , ioe. d faab C( G m+n a , Gck ) + f bc C( G n+k d ,

G"

a ) -]"

d f ca C( G dk+m

,

Gn)=0 b "

(25) (26)

It is easy to show that when G is an orthogonal group* (then fdab = fabd =

C ( G t., G-b t) = Al6,b

for A ~ C ,

is a solution of (26) when m + n + k = 0. *For other cases see [Goddard and Olive, paragraph 1.3].

1~ Z

(27)

loop group

358

V BIS. C O N N E C T I O N S O N A P R I N C I P A L F I B R E B U N D L E

Eq. (27) suggests that, under the same hypothesis on fc b,

C(G k, a m b )=

1~kt~a,bt~m,_k ,

A E C ,

k, m ~ 7/

(28)

satisfies (26), and it is straightforward to check that, indeed, it is so. One can check that the cocycles (28) on ~(C~(S ~, G)) are nontrivial if h ~ 0 (i.e., C is not a coboundary) by constructing another representation ~ " = (~" + K:ll a

a

~

One finds that there is no choice of K~" for all n which will make vanish. Hence eq. (28) provides the set of all nontrivial cocycles for ~(C| 1, G)) and H2(.~(C~(S 1, G))) is a space with 1 complex dimension.

[~"o,~b]--fa~C'7 A

m

c

-+-n

Remark: All cocycles on the Lie algebra of semi-simple Lie groups are trivial [see for instance Guillemin et al., or Jacobson]. 7.

LIE ALGEBRA OF DIFF S 1

Let Diff S ~, be the group of diffeomorphisms of the circle, where the group structure is defined by map composition. a) Derive the Lie bracket of the Lie algebra ~(Diff S 1) (not a current algebra). b) Consider a projective representation (6) of .~(Diff S 1) on a Hilbert space ~. Spell out the relation (13). c) Derive the general expression for the complex valued 2-cocycles on ~(Diff $1). Show that the second cohomology group HE(.~(Diff $1); C) is isomorphic to C. d) Find subalgebras of .~(Diff S 1) for which the cocycles vanish. Remark: These two examples are instructive in many ways, but they are not generic: their cocycles take their values in C; in general cocycles for an infinite dimensional group ~dtake their values in a space of functionals. See examples for instance in [Bonora et al.] and [Dubois-Violettel'2]. Answer 7a: The group Diff S 1 is the set of smooth diffeomorphisms ~b: $1---~ S 1 with the group law defined by map composition, ~b1, ~b2 Diff S 1 implies ~bI o ~b2 E Diff S ~. The Lie algebra ~ ( D i f f S 1) is the space of smooth vector fields on S 1, identified with the set of derivations of smooth functions r ~ f ( r on S ~. The Lie bracket of two smooth vector fields is d d] fl(~O) ~-~, f2(~p) ~ = (fl(~o)f~(~O)- f~(r

d ~--~.

(29)

11. COCYCLES ON THE LIE A L G E B R A

359

Using the previous notation z = e i~' and d/dq~ = iz d / d z , we consider the set of vector fields on S 1 Li(z ) = z_i+ , d

dz '

iEZ.

It is a basis for ~ ( D i f f S ~) and its Lie algebra is [L,, Lj] = (i - j ) L , + j .

(30)

A n s w e r 7b: Let L i be the representative of L i in a projective representation of ~(Diff S 1) on ~ : it satisfies

[L~, Lj] = ( i - j)Li+ j + C(L~, Lj)I] ,

L~ E L ( ~ , ~ ) ,

(31a)

where C is a 2-cocycle on the Lie algebra ~ ( D i f f S ' ) , i.e., a bilinear, antisymmetric function of its argument satisfying the cocycle condition: C([L i, Lj], Lk) + C([Lj, Lk] , Li) + C([Lk, Li], L j ) = 0.

(32)

A n s w e r 7c: Complex-valued 2-cocycle on ~ ( D i f f S 1) are obtained by solving eq. (32) algebraically. First set k = - i - j; then (32) together with (30) gives (i - j)C(Li+j, L_i_j) + (2j + i ) C ( L _ i , Li) - (2i + j ) C ( L _ j , Lj) = 0.

(33)

It is easy to show by induction that the solution of (33) is C ( L k, L _ k ) = Ak 3 q- ~ k ,

A,/x E C ,

k~ Z

and the solution of (32) is C ( L k, L j ) = (Ak 3 + txk)6j,_k,

A, /x E C .

(34)

R e m a r k : Using (34), eq. (31a) reads [Lk, Lj] = (k - j ) L k §

+ ( Xk 3 + fzk)Sj,_k .

(316)

This algebra is a central extension [see Problem IV 6, Short exact] of ~(Diff S 1). To identify the second cohomology group H 2 ( ~ ( D i f f $1); C) we need to identify which of the cocycles (34) are trivial. We know from Answer 4c) that if the cocycle in (31b) is trivial, then there exists a representation L i such that A

[L, L,]- (k-j)L,.

(35)

Set Lk "= L k + V6k, o ,

vE C

(36)

360


and determine v so that (Lk} satisfies (35). An easy calculation gives [L k, Lj] = (k - j)Lk+ j + ( h k 3 + (/x - 2v)k)Sj._ k .

(37)

Eq. (37) says that if h = 0 one can find a representation {Zk} satisfying (35) by choosing v = / x / 2 . The trivial cocycles are

C ( L k, L j ) = tzkSj _k ,

The nontrivial cocycles are

C ( L k, L j ) = (Ak3+ t.,k)Sj._k ,

/x~C. A,/~EC.

The second cohomology group is n 2 ( ~ ( D i f f $1); C ) ~ C . A n s w e r 7d: Consider the infinite dimensional subalgebras W 0 and W 0 of .~(Diff S 1), +

W o =(Lk, k>-O},

Wo = {Lk;k- L(~*(q3), *(~)) such that (L2, ~Id*(L~)L*) = ( ~Id(L,)L2, L* ) , L,, L 2 ~ s

L* E ~ * ( ~ ) .

(6)

~ d * is called the coadjoint representation. Show that if fg is the group Diff S ~ of C ~ diffeomorphisms of S 1, and { L i } , i E 2z, the basis o f ~(cg) considered in [Problem V bis 11, Cocycles], then ~Id*(Lj)L , k = (2]"- k ) L ,k-y ,

(7)

where L*k denotes the element o f ~ * ( ~ ) defined by (v, L ' k ) = vk, V E+oo i = _~ V L i E ~ ( ~ ) . Show that ~ d * is an antirepresentation of ~ ( ~ ) , namely [~d*(Lj), ~d*(Li) ] = (i-j)~d*(Li+j)

[Lj, Li] = ( j -

whereas

i)Li+ j .

(8)

coadjoint representation


366

Remark: ~ ' * (~) is not a Lie algebra (the bracket of two one-forms is not a one-form but a two-form). coadjoint representation

Remark: There is another definition for the eoadjoint representation of S(~)

namely (L2, ~ ' d * ( L 1 ) L * ) - -(~ld(L1)L2, L*).

(8a)

With this definition the coadjoint representation is a representation- not an antirepresentation, and

~r

-- (k - 2 j ) L *k-s .

(9)

The group representations corresponding to these Lie algebra representations are such that (,s~/'d(g)L2, g d * ( g - 1 ) L *) --(L2, L*) corresponding to (6), (,~d(g)L2, ,~"d*(g)L*) - (L2, L*) corresponding to (8a). b) Denote by A* the vector subspace of S * (f~) A. . . A S * (f~), n products,

generated by the exterior products A L'In ],

ordered indices. Show that the linear mapping p*" 0~(~) --+ L(A*, A n) determined by {L*li A

9

9

9

I1 . . . . . In

rl , . . . L,ln Pn ( L)(L*II A A ) -- Z L ,I! A ... A , ~ d * ( L ) L

*#k A ... A L *In

k=l

(10)

with the definition ( 8 a ) f o r the coadjoint representation, is a homomorphism of Lie algebras. Answer 2a" Let L* 6 ~ ' * ( ~ ) and Ll 6 c ~ ( ~ ) . Formula (6) gives the action of s~'d* (L l)L* on any L2 E 9o(~) as a linear continuous map into R, i.e., determines s~e'd*(L i ) L* as an element of S * (~), and ,~d* (L 1) as an element of L ( t * (~), oL,f* (~)). A straightforward calculation, using the definitions and Jacobi identity shows that ,~d* is an antirepresentation, namely

[,r162

1), ,.dd*(L2)] "-- ,r

- ,~d*(Lz),.~td*(L1)

-- - ~ ' d * ([L l, L2]). Using the fact that, for the given basis of S ( D i f f S l ) we have [L j,

we find (V i Li, ~ d * ( t . j ) L

Li] -- (j - i ) L j + i

*k) -- 1)i([L.j, Li], L ' k ) -

l)i (j - - / ) ( L j + I ,

= ( j - - 1 ) 6 ~ + 1p i --(2j--1)6~l vk-l,

L *k)

12. V I R A S O R O

REPRESENTATION

O F ~ ( D i f f S 1)

367

i.e., *k = (2j

~r

-

k)L*~'-t

A n s w e r 2b: We need to check that

p2(L,)] = ( k We can check it in a few lines for n = 2 using the defining equation (9). The calculation generalizes in a straightforward manner for arbitrary n. 3.

SEMI-INFINITE FORMS

The space of semi-infinite forms introduced by [Feigin], applied to physics by [Frenkel] is the_ space A*, which, together with the following restrictions, is labelled A*. An element of A* is a semi-infinite product (infinite only on one side) L *11 A L *12 A ' - -

A

L *z~-a A L *Ik

A L *I~:+1 A . . -

that from an arbitrary index k on to infinity the indices decrease 1 by 1. For example,

such

L .3 A

L .1A L *(-s)

Ij+ = / j - 1

A L * ( - 6 ) A L * ( - 7 ) A " - " A L *lj

j--3

is an element o f / i * . a) Define i ( t m ) and

~.(t *m) by oo

i(Lm)( L "11 A ' ' "

A L *'p A ' " ") "~ E

p>--I

(--I)P-I(

t m , L *,p ) L '1 A . . .

A...

A L*lp

(11a)

and E ( L * m ) ( L ,I, A " ' " A L ,lp A "" ") = L * M A L *I1 ^ . . . A L ,Ip

(llb) C o m p u t e their anticommutators and c o m p a r e them with (3). A n s w e r 3a: It follows readily from the antisymmetry of the A-product

that

[i(Lk), i(Lm)]+ = 0 ,

(12a)

[E(L*~'), E(L*"n)]+ = 0.

(12b)

An explicit calculation of [i(Lk) , E(L*m)]+ operating on a generic ele-

semi-infinite forms


368

ment of A * shows that

[i(Lk),

~(L*m)]nt- --(Lk,

L *m ) -- ~t~.

II

(12c)

b) Consider the following mapping" 9 . ~r Pcx~

(13)

._~ L ( A ~

by poo(Lk)(L

*il

A - - . A L .I/' A . . - )

OO

9--- ~

L *ll A . "

A

se/d*(Lk)(L *0) A

...

for k =fi 0.

(13a)

p=l

We note that for k # 0 this series terminates since for Ip smaller than a fixed number s~d*(Lk)(L *0 ) ,~ L *O-k can be found in the term on which p ~ (L k) operates. For k -- 0

p~(Lo)(L

A ... A L *0 A ...) "-- ho(L *!1 A ... A L'11' ...), (13b)

where h0 is a real finite number which depends on L *~ A .-. A L*O . . . .

Show that the mapping (13) defines a projective representation of S ( f f f ) on A* Compute the 2-cocycle C(Li L j) of this projective representation. [See Problem V bis 11, Cocycles and IV 6, Short exact. ] O0

9

Remark" The reason for defining p~o(Lo) by (13b) is related to the definition, given below, of the quantum B RST operator in terms of normal ordered products.

Answer 3b: To prove that (13) defines a projective representation, one computes [p~ (L j), p ~ (Lk)]. It is easier to do so by considering first the case when j and k are both positive or both negative. One finds that in these cases there are no cocycles. Thus, the cocycles, if any, are proportional to 6.i,-k. A direct straightforward calculation shows that p * defines a Virasoro representation. We have shown in the previous problem [Problem V bis 11, Cocycles, eq. (38)] that if Li is the representative of Li in a Virasoro representation

[Lk, L i] -- (k - j ) L k + j + X(k 3 - k)6j,-k. So far, X is arbitrary because we have not specified the representation beyond eirninating the trivial cocycles. We want to compute X for the representation p ~ defined by (13), narne]y for L / - - P~c (L j). Note that [El, L - I ] - - 2L0

and

[Le, L - e ] - - 4 L 0 + 6Z;

12. VIRASORO REPRESENTATION OF ~(DiffS 1)

369

hence,

[L2, L _ 2 ] - 2 [ L , , L _ , ] = 6 A . Applying both sides of this equation to a generic element in A* gives = -26/12 ; i.e., the representation p* carries a nontrivial cocycle

C(tk ,

tj)=

-

~26 ( k ~ -

k)a;

(14)

_~.

Remark: Working with the Lie algebra ~ ( ~ = Diff S 1) is simpler than working with the diffeomorphism group of S 1 for 2 reasons: 9 In a group of diffeomorphisms there are elements of the group arbitrarily close to the identity which cannot be reached by exponentiating its Lie algebra. 9 Diff S 1 is multiply connected, rr l(Diff S 1) = Z. See for instance [Milnor]. See [Segal] for nontrivial projective representations of Diff S 1 which induce representations of ~ ( D i f f S 1) with trivial cocycles. 4. NORMAL ORDERING. QUANTUM BRST OPERATORS

Normal ordering is a prescription indicated by a pair of double dots and defined as follows: Choose a fixed arbitrary integer k 0, then

{i(Lm)~.(L *m) . i(Lm)E(L ,m ). =__ __~.(L *m)i(Lm) "e(L*m)i(tm) : -

_i(Lm)e(L ,m)

e(g,m)i(gm)

normal ordering

if m k0 ifm k 0

The vector w 0 = L ' k ~ A L'k~

A ' ' " with all the slots filled in

is called the v a c u u m . One can interpret ~(L *m) a s a creation operator and i(tm) a s an annihilation operator. For m_< k 0 normal ordering means creation operator on the right, annihilation operators on the left, and the reverse situation for m > k 0. Normal ordering eliminates the infinity which would occur from --I- o o

E m=

[i(Zm), ~ ( L * m ) ] + -oo

= E m

(Lm, L *m) = E ~ . m

vacuum

370


Indeed,

"( Z -I- Z ) [i(Lm)'E(L*m)]'-O" mko ghost field operator conjugate ghost field operator

In physics, is called a ghost field operator, ~a ~--i(La) is called a conjugate ghost field operator.

77~ = E ( L *" ) ^

Let La be the representative of La be the representative of La in a possibly projective representation Jr to be chosen later. The operator

(15) ^

quantum BRST charge operator

is called the quantum BRST charge operator in the representation La = zr(La). This equation has a formal resemblance with (5), which gives the coboundary operator d in terms in e and i. But, in contrast with d, the quantum BRST is not necessarily nilpotent. We know [Problem V bis 11, Cocycles, Answer 7] that, if one defines the space of physical states as the kernel of the quantum B RST operator, consistency can be achieved only if the quantum BRST is nilpotent. It can be nilpotent only if the cocycle introduced by the choice of representation La --+ La precisely cancels the terms introduced by - 89fat), 9~crl a Oh'. This is the case in particular when La are represented by operators on the Fock module of a bosonic string in 26 dimensions. Interesting relations are obtained by computing the commutators of p ~ ( L n ) with i ( L m ) and E(L*") and expressing p ~ in terms of i's and 6'S.

We have presented the version of the B RST operator which comes naturally in an hamiltonian theory where the first class constraints together with the Poisson bracket are a representation of the Lie algebra _SP(~'). Its relationship with the lagrangian BRST operators [see for instance Beaulieu] can be found in [Fradkin, Vilkovisky]. For the construction of the classical BRST operator, its relationship with the constraint analysis of dynamical systems, the Lie algebra cohomology and the geometric interpretation of ghosts and conjugate ghosts, see [Kostant et al., McMullan]. REFERENCES See also references at the end of [Problem V bis 11, Cocycles]. Becchi, C., A. Rouet and R. Stora, "Renormalization of the abelian Higgs-Kibble model", Comm. Math. Phys. 42 (1975) 127; Ann. Phys. (N.Y.) 98 (1976) 287.

12. VIRASORO REPRESENTATION OF ~(Diff S 1)

371

I.V. Tyutin, "Gauge invariance in field theory and in statistical mechanics", Lebedev preprint FIAN 39 (1975) (unpublished). Baulieu, L., "Perturbative Gauge Theories", Physics Reports 129 (1985) 1-74. Dubois-Violette, M., M. Talon and C.M. Viallet, "BRS algebras. Analysis of the consistency equations in gauge theory", Commun. Math. Phys. 102 (1985) 105-122. Feigin, B.L., "The semi-infinite homology of the Kac-Moody and Virasoro Lie algebras", Russian Math. Surveys 39 (1984) 155-156. Frenkel, I.B., H. Garland and G.J. Zuckerman, "Semi-infinite cohomology and string theory", Proc. Natl. Acad. Sci. USA 83 (1986) 8442-8446. Milnor, J., "Remarks on infinite dimensional Lie groups", in RelativitY, Groupes et Topologie II, eds. B.S. DeWitt and R. Stora, North-Holland, Amsterdam, 1984) pp. 1010-1057. Segal, G., Comm. Math. Phys. 80 (1981) 301-342 (in particular p. 331). Batalin, I.A. and G.A. Vilkovisky, "Relativistic S-Matrix of dynamical systems with boson and fermion constraints", Phys. Lett. B69 (1977) 309-312. Batalin, I.A. and G.A. Vilkovisky, "Closure of the gauge algebra, generalized Lie equations and Feynman rules", Nucl. Phys. B234 (1984) 106-124. Fradkin, E.S. and G.A. Vilkovisky, "Quantization of relativistic systems with constraints", Phys. Lett. 55B (1975) 224-226; Equivalence of canonical and covariant formalisms in quantum theory of the gravitational field (CERN Preprint TH2332-CERN, June 1977). Kostant, B. and S. Sternberg, Ann. Phys., 176 (1987) 49-113. McMullan, D., J. Math. Phys. 28 (1987) 428.


VI. DISTRIBUTIONS

1. E L E M E N T A R Y SOLUTION OF THE WAVE E Q U A T I O N IN d-DIMENSIONAL SPACETIME

C denotes the cone in R d defined by x 0 >- 0 ,

R ~- 891762 - r 2) = 0,

r 2 --=

d-1

(x' )2

i-1

8n is the Dirac measure (pp. 438, 512) on C, defined by

< ~R ~0) = f ~(X i ) Rd-1

dx1...

d x d-1

where q3 denotes the function induced by ~ ~ ~ ( R a) on C (p. 421) ~(x i) = q~(x~ xi)lxO=, .

1) Determine values of d is 2) Determine 3) Show that

0o6n, ai6 g and compare their values when d > 4, For which 6~ = (1/x~ defined? D6 R when d = 4. when d is even one can define a distribution by setting E=

10o

6R"

Show that D E has support 0 and & proportional to 6. Answer 1" By definition of the derivative of a distribution (p. 446) =

Hence (0~

~) = -

f

d x 1 . . . . d x d--1

Rd-1

~176176

and, since 373

r

(1)

374

VI. DISTRIBUTIONS

r i ~0 - -

xi

r i~

r 0o ~O ,

f ( xi

- - 7 0~

< Oi~R' ~ ) ~- --

+ 0,~

) dxl'''dxd-1 r

.

(2)

Rd-~

If d -> 4 and q~ ~ ~ we have the equality of convergent integrals:

f R,/- ~

j

9 9 9

r

-"T

19

9 9 9d x d -

dxl

r

Ra- l

(3)

Therefore (2) can be written, if d-> 4 i

aiSR = X r

i

ao 8R + x 8r "

(3a)

We have

<x1 o908R' ~ > = - - < t S R ,

f(Oo

00(p

r

r2

r3

Rd-1

if the integral is convergent for all ~0 ~ ~ , that is if d-> 5. In this case (3a) can be written xi

a~Sr = - ~ aoSr.

(3a)'

Note: If instead of the measure 8r we were derivating a Lorentz invariant C 1 function f ( R ) we would have obtained e. f ( R ) = % o x ~ f ' ( R ) . Formula (3a)' is analogous if we set 1

x

I

a08r = 8 r .

This definition is invariant under a Lorentz transformation yO= L O x ~ ,

~.oy.yO=__ % o x . x o

since by properties of change of variables on distributions (p. 456) 1

o

1

o

yO OyO 8r = L O x ~ Lo ~ ~

8R

and using (3a)' 1

yO

0

Oy~

8R =

xi

L~ ~

( L o o _ __0

x

)

Loi

0_~ 8R

Ox ~

"

1. WAVE EQUATION IN d-DIMENSIONS

375

(L~ t3) is the inverse of the Lorentz transformation (Lt3~), and w e know that L~ o = Lo ~ ,

L~ i = - L i o

hence, as announced 1 d 1 O y0 0y~ 6 R = p 0X ~ 6R" 6~ is not defined if d - < 4 , since (1/x~

is not defined in this case.

A n s w e r 2" If / ~ is defined (i.e., on R d - {0}, or on R d if d > 4 ) 6~-(1/x~ is also defined we shall have, as in 1)

1

2

O06R -- XO~ R ,

?

O 00~R "-- 6 R "4- ( X 0 ) 2 6 R

and

.

d-1 ~__

?

2

t

.

i-1

thus [--16R = d6 R' + 2 R 6 "R= r

(d-

4 ) 6 'R ,

(4)

is defined if d > 6 by

= ((I/xO)Oo)2C~R

< (1

=

x

f(02

ooq~

30 o q~ ~. 3-~ d x ~ . . d. x . d r

r

[~d-1

1

r

Thus (4) is valid on R d if d > 6. It is always valid on ~ a _ {0}. If d = 4, r-16R = 0 on R 4 - {0}. To compute [--16R on R 4 we go back to the definition ([--16R, qo ) = (6R, [--1r ) = f

d x l dxzr dx3 [Zlc#.

~3

For arbitrary d a straightforward computation gives, since x ~

(x,)

E] ~ = - A ~ + 2x~ 0 i 0 0 ~ At- 0 i - r r

Since O i ( x i / r ) - - ( d [--]q9 rp = -

0'0~

~

~ = ZO 2

2 ) / r we have for any d ~ r rp

~

_

2X i

F ~

0i~1 -[-

(d - 2)qJ rp+ 1 '

A~ 2 _q r--F-+ r p + i ~i r --r qj '

~ll

--

00~

q = 89

"

r on c , (4a)

376

VI. DISTRIBUTIONS

hence i f p = l

andd=4,

i.e.,q=l

D~

A~

. . . .

F

I- 2 a i ( r - 2 x i q , )

.

l"

Since q~ has compact support, we have if d = 4

t- Aq~ dxl dx2 dx3 ; .1 F IR3

(l--16r, q~) = - /

we c o m p u t e this integral on R 3 - B, (ball of radius e centered at 0) and let e tend to zero and find (cf. p. 512) (F16r, q~) = 4~rq~(O) ; that is, when d = 4 0 6 n = 4r

.

(1/47r)6 r is the elementary solution with support in C of E3 when d = 4. Answer

3" The distribution E=

(1

~

090)(d-4)/2

(d-4)/2

6r=Br

is defined for d-> 4 and even, by

because the function 1

F2+(d-4)/2

1

F

d/2

is integrable on R a-1 if d-> 4. We c o m p u t e l--1B(np - 2 ) , p = d / 2 on R a - {0}"

[-'](~(Rp-2) "-" d(~ (p-l) + 2RB R(p) from the induction formula q = 0, 1 , . . . , d(~ ( p - l )

=

p

2 R B (np ) = ( d - 2 q ) B (np - ' ) + 2 ( R S (P - q ) ) ( q ) .

Since R 6 r - 0 we deduce

F](~ (p-2) -- 0

on

R a - {0}

(the formula is not valid on R e because 6(nq) is not defined there if q>d/2).

2. SOBOLEV EMBEDDING T H E O R E M

377

On R d we have

([--]~(p-2) ~O) (~(p-2) [---]~) (__l)P-2(~R, (O~01) (p-2) ,

=

,

=

U]q~

/

.

Computations analogous to the ones done previously show that, if q~ has compact support (divergences integrate to zero on R ~-1)

> = K (O) with some constant K, thus

['-]6(p-2) __ 1(6

.

By explicit computation K can be found to be (or see Leray p. 103 which uses another method and other notations)" K = 1" 3 . . . . . (d - 3) vol

S d-2

,

d p=~->2.

The volume of the sphere S 2n is 1) vol

S 2n

-"

2(2zr)" 1.3 . . . . . (2n - 1) '

therefore K = 2(27r) p-1 . REFERENCES J. Leray, Hyperbolic differential equations, I.A.S. Princeton (1953). Y. Choquet-Bruhat, Bull. Soc. Math. de France, 81, (1953) 225-288.

2. S O B O L E V E M B E D D I N G T H E O R E M

I. DEFINITIONS

m

We have defined (p. 486) the Sobolev space Wp ( U ) , m a nonnegative integer, 1 -~ p _~ ~, U open set of R", as the space of distributions on U which, together with their derivatives of order -<m can be identified with functions in L P ( U ) . An alternative, equivalent definition (Meyers and Serrin 1964, cf. Adams p. 52) when p < ~, is that Wpm( U ) is the closure in 1)The volume of S 2"+a is 2"rr"+I/n!.

378

VI. DISTRIBUTIONS

the norm

Ilfllw; =

mu Io'f(x)l

dx

of the space of C m functions on U for which this norm is finite" in other m words this last space is dense in the space W p (U) as previously defined, if 1-< p < oo. 1) Show that the result does not extend to p = oo.

Answer 1" The norm in W 7 ( U ) is

[[fllw= = E

SuplOy(x)l.

Ilium x~U

C~'(U) space

segment property

Functions in c m ( u ) for which this norm is bounded constitute the complete (Banach) space C'~(U), not equal - even for m = 0 - to w m ( u ) . m W T ( U ) is however, like all Wp ( U ) , a Banach space. H, (U) -= W 2( U ) is a Hilbert space. It is known that Co(R" ) is dense in W p ( R " ) , 1 - < p < o o . In general m C o(U) is not dense in Wp (U) (cf. p. 468). The set of restrictions to U of functions in Co(R" ), denoted C~176 is m dense in W p ( U ) , 1 -< p < oo if U has the segment property. An open set U C R" is said to have the segment property if for every x E OU, boundary of U, there exists an open set Ux ~ x and a nonzero vector ox such that if y E 0 f] U~ then y + tv~ ~ U for 0 < t < 1. 2) Show that the open set of R 2 U = { 0 < [xl[ < 1, O < x 2 < 1)

does not have the segment property. Give a necessary condition for an open set U C R" with ( n - 1) dimensional boundary to have the segment property. Show that the function on the previous domain U C R 2 given by u(xl' X2) =

1

0

if if

x~>O x I 0 and h > 0 such that at each point x ~ U one can draw an axially symmetric cone of vertex x, angle a, and height h. A fundamental theorem is the following. II.

SOBOLEV EMBEDDING THEOREM

Let U be an open set in R" with the cone property, then the following continuous imbeddings hold for 1 -< p < ~. i) If m > n/p, then W p ( U ) ~ C~ m (hence Wp (U) ~ Lq(u) Vq >-p). ii) If m < n/p, then Wp (U) ~ L m

q(u),

l ~ p n/2

since (Schwartz inequality)

(1 + lyl~)ml~f(y)i ~ dy

f ( ~ f ) ( y ) dy < Rn

and

(1 + lYl~) -~

Rn

f

(1

+

dyj

Rn

lyl=) -m

d y = K < oo

if

rn >

n/2.

Rn

If ~ f E L2(R"), f can be identified with a continuous and bounded function (p. 474) 1 f (~;f)(y) eix.r dy , f(x) = (27r)n Rn

K

Ilfllco(.~)-< (2r

I1(1 + lyl~)m'~fll,~(..)

9

Using the Plancherel theorem (p. 490) we find that,

II fll~o(..)-< cII/11,,.(..),

m > n/2

here C is a constant depending only on n and m. 5) Proof of part ii); to simplify the writing we first take n = 3, p = 2. a) Suppose first that f ~ Co(R3). Obtain an inequality for If(x)[ 4 by writing it as an integral over a coordinate line. Multiply all possible inequalities and integrate successively over x 1, x 2, x 3. Use the Schwartz inequality to prove

Ilfll~-< cllvfll,~,

if

f e Co(JR3).

2. SOBOLEV EMBEDDING THEOREM b) Extend the result to f E n 1. Prove that then f ~ L

q,

381 2 and f appear in different integrals. Answer: From H61der's inequality we get

f dxxn-l f yn-l (x, y)>

0

0

co

_
2,p

1/q Jq,ot,

'lf(x)l

0

oo

J q ~ = f dx x"-l f dy y,-1 [f(x)[q~lf(Y)[ q~ ' (x, y)q 0

0

In order to separate f from (x, y)> we introduce (x/y) ~ times its inverse and obtain from Schwarz' inequality

Jq ~ '6>6' "--"

S

~

---

9

Answer 2" We show that if f ~ H~,~ we have also tr ~+kDkf ~ H~_g for 0 -- k _-_s. We have 03ior "- X/O "-1

thus IDa[ < 1 on R" 9

more generally, for each k there is a constant C such that

Io%1-< Ccr -~§

9

(1)

We deduce therefore from the Leibnitz formula, with Cm,~ some constants >0:

IDt(crsf)l-C8/~

s E N, 6 E [~

ii) H,.a - . H,,(8 +n/Z)/e-n/2

,

S ~ N, 6 ~ ~ .

We deduce from (3) that trY(x) -< tr(y) -< 21/2o'e(x)

ore(

and

=

ax

8ij

)

e) x'xJ

(1 -

,Ofi OX

(x) =

(

~-1

-

and thus, for any function f on [~n

)

X'XI

6u - (1 - e) or(x)2

cr

e-1

(4)

(x)

(x)

O~(Lf)

Oy i

(y).

(5)

The quadratic form between parelathesis is uniformly bounded if 0-< e < 1, as well as its inverse 9 .

8u + ( 1 - e)

XtX l

1 + ~lxl ~ "

Therefore there exist positive constants C1 and C 2 such that

C, [g(x)Df(x)[ 1 the space W p is the dual of a Banach space WP'~, with 1 / p ' = 1 - 1/p (p. 489). The ball ,

-

is compact in the weak-star topology [Problem 1 13, Compactness]; therefore there exists a subsequence of { f,}, still denoted (fn}, such that it converges in this topology to h E W~. The convergence is 'de, Vg E WP's, :IN such that [(h - f~, g)! < e if n > N and implies a fortiori the convergence of (f~} to h in ~ ' , thus f = h. 9. DISTRIBUTION WITH L A P L A C I A N IN L2([~ n)

1) Show that if a tempered distribution u on R ~ ( p . 476) is such that its laplacian 32U AU = n

i=1 ( ~ 2 i 2

is a function in L 2(En) then the same is true o f each o f its second partial derivatives.

2) Show that if u is a distribution on R ~ such that its first derivatives Ou and its laplacian Au are locally (i.e., on each compact set) square integrable functions, then the same is true o f each o f its second derivatives. Answer 1" If u is a tempered distribution it admits a Fourier transform o = ofu, and so does Au, with (cf. p. 477)

~Au = - j y j 2 v ,

lyl2=~

(yi)2 ,

i=1

If zau E L2(~n), the same is true of o~Au, (p. 490)"

-JyJZv

~ L2(R ")

from which we deduce y'yJv ~ L2(Rn), .

since

lyl-2y~y j

.

Vi, ] = 1 , . . . ,

n

is in L~176 - and f ~ L 2, g E L = implies fg ~ L z. There-

400

vI. DISTRIBUTIONS

fore, by the inverse Fourier transform 092 u 09X i 09X j

E L2(R").

Moreover

Ily,y,ol[=,2

[llyl o11,2,

thus

1

8Zu

3 X i ~ X j L2 -- Alul p

F"(t) >- f NAIu] p d/z(g)>- C f l u [ p d/z(m) St

St

with C = inf N2,4 = a2A. V,,+I

On the other hand, if V~ is a compact manifold IF,(t)[-
- C l f u ( t ) l

~ .

We first deduce from this inequality that, for all t

F"(t) >_O . Therefore if

f

St o

OoU d/z(m) : F'(to) > O,

(1)

10. N O N L I N E A R WAVE E Q U A T I O N IN CURVED SPACETIME

403

we have ! P Fu(t ) >- Fu(to) > O,

V t >- to, o

thus Fu(t ) is a strictly increasing function for t -> t 0, and there exists t I >__t o such that

Fu(t)>O,

t>-t 1 .

Therefore (1) can now be written

ru(t ) >_ C ( F , ( t ) ) p

t> t1

and we have d d t (F')2 = 2 F ' F " >- 2 C F ' ( F u ) p ,

t >-- tl .

This inequality gives by integration t 2 2C (F~(t)) >_ p + 1 {(Fu(t))p+l - (Fu(tl))P+l} + (F'u(/1))2 '

that is, since the right-hand side is positive when t-> t 1 an inequality of the form

F~,(t)>_ { C l ( F u ( t ) ) p+I + C2} 1/2 ,

t>_ t I

which implies

F~(t) >- y ( t ) ,

t>t I

where y is the solution of the p r o b l e m

y ' = ( f l y p+I + C2) 1/2 ,

Y(/1) = F u ( l l ) ;

that is y such that Y /.

t = t~ + 1 (C~z p+l + C2)-1/2 d z ,

Yl = F u ( t l ) .

Yl

Since C 1 - - 2 ( p + 1 ) - 1 C > 0 and ClYlp+I + C 2 > 0 , the solution y(t) tends to infinity, if p > 1, when t tends to the finite value oo

T=

f (c z,

+

C2) -1/2 d z .

Yl

REFERENCES Y. Choquet-Bruhat, Comptes Rendus Ac. Sc. Paris (1987).

404

VI. DISTRIBUTIONS

We give below a summary and references for results about the existence and nonexistence of global solutions of nonlinear wave equations on Minkowski spacetime M,+ 1 obtained in recent years. 1) Case n = 3 F. John, Ann. Maths. 28 (1979) 235-261. This article proves the blow up of solutions of

Flu >_ Aiul p ,

A > 0,

1 < p < 1 + X/2

for Cauchy data in Co(R 3) which satisfy either fSo aoU d3x > 0 or Uiso - Alsl p , f(0) = 0,

Vs~R, A >0 lim sup f(s)/Isl s--0

3 and equations

Flu = f(u, au, a2u) with f a smooth function such that f(0, 0, 0) = 0,

f ' ( 0 , 0, 0) = 0.

(2)

S. Klainerman, "Global existence for nonlinear wave equations", Comm. pure and app. Maths 33 (1980) 43-101. D. Christodoulou, "Global solution of nonlinear hyperbolic equations for small initial data", Comm. pure and app. Maths XXXIX (1986) 267-282. This last article also contains a proof of the same theorem in the case n - 3 when f satisfies the further condition that it vanishes when its arguments 8u and a2u are replaced respectively by a null (for the Minkowski metric) covector y and its tensor product y | y. 3) The global existence on R n§ 1, n-> 3, for small data in Co, under the general hypothesis

11. HARMONIC COORDINATES IN GENERAL RELATIVITY

405

(2) on f, for the equation

Du + u = f(u, Ou, 02u) has been given in: S. Klainerman, "Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space time dimensions", Comm. pure and app. Maths. XXXVIII (1985) 631-641. See also L. Hormander, Institute Mittag-Lefler reports Lund (1985) 86, 87.

11. H A R M O N I C

COORDINATES

IN GENERAL

RELATIVITY

1) Let g be a riemannian metric, of arbitrary signature on a d-dimensional manifold M. Show that in a coordinate system where the metric satisfies the h a r m o n i c i t y conditions _a~ r, h

FX--g

~,~ t3 = 0 ,

h=O,...,d-1

(1)

the Einstein equations in vacuum, R i c c ( g ) = O, reduce to a quasi-diagonal, 2 quasi-linear system, with principal operator g~r 2) Show that the conditions (1) express that the mapping f" U---~ f ( U ) by x ~ (x ~) (U domain o f the coordinate chart) is a harmonic map f r o m the riemannian manifold (U, g) onto ( f ( U ) , e) where e is the standard euclidean metric o f Nd. 3) Generalize to the case where (1) is replaced by an harmonicity condi-

tion with respect to an arbitrary given metric e on M. Answer 1" By the definition of the R i e m a n n and Ricci t e n s o r w e h a v e , in an arbitrary c o o r d i n a t e system (cf. p. 306) V~,Vt3g(,~) _ Vt3V~'g(~) _= R~'t3 ~g(') = R~t3 , = ~ ~, are c o n s i d e r e d as the c o m p o n e n t s of a c o v a r i a n t vector, w h e r e ..(a) for fixed a ( d u m b index). With this notation

-g

~ ~

-g~'~F~ ~.

Thus R ~ ~_V~'V~g(~)+ g~O~F'~; since R ~ is s y m m e t r i c we have also R ~t3= 89

(~) + V"V'~g (t3) + g~OAF ~ + g ~ 0 hF r )

(2)

harmonicity conditions

406

VI. DISTRIBUTIONS

We remark that

= _(gl~XgaV + gaXg/SV)[v, X/z] with [v, X/z] =_ gv~ Fx~u the Christoffel symbols (p. 308), hence V/~g(~) + VUg(u/~) --= _ •2 (g[SXgOtV + gUXgl~V)Ougxv _ Oug~ . We deduce from (2), after computation

~l~) + 89 R al~ =-- R(h

F ~ + g~X Vx F [3),

where R(h ) is the second-order, quasidiagonal operator et~

R(h)

_

n2 6,,~/3 _ gXU gpV Fx~pFu/Sv. I gXU,XU

The system is hyperbolic in the sense of Leray (p. 520) if g is hyperbolic, since it is quasi-diagonal with principal part I gXU OZug~l~" By lowering the indices we find that R,~/~ is also of the form Ru/~ ---- ~(h) "'u/~ - 1 (gux V/~ F x + g/~xVu F x ) p(h) with "'a/~ a quasi-diagonal second-order hyperbolic operator with principal part - 89gXU 02u ga/J .

Answer 2" The mapping f is represented in local coordinates (x x) on U and f (U) by the identity map

f x (x u) = x x" The equations for harmonic maps [Problem V 1 1, Harmonic] reduce to (1) if f ( U ) is endowed with a flat metric for which the x x are canonical coord i n a t e s - where its Christoffel symbols are zero.

Answer 3" Suppose the identity map M ~ M is a harmonic map from (M, g) onto (M, e). Then, in local coordinates

[:X_ g ~ # ( _ F X + l~ Z ) _ 0 , where the/~'ax# are the Christoffel symbols of e. The previous computations show that we have identities of the type R~# = gXUCTxfugal~ + 89

+ gaXOxFl~ ) + H~

Vg),

(3)

where ~' is the covariant derivative in e. Decomposition (3) is tensorial since F is a vector and is valid on the whole manifold. ^

12. LERAY THEORY OF HYPERBOLIC SYSTEMS

407

12. L E R A Y T H E O R Y OF H Y P E R B O L I C SYSTEMS. T E M P O R A L G A U G E IN G E N E R A L R E L A T I V I T Y

Let (M, g) be a smooth riemannian manifold of hyperbolic signature with M = S x ff~. The lines {x} x ff~, x ~ S are supposed time-like and orthogonal to the space-like submanifolds S t = S x {t}. In adapted coordinates, (x i) coordinates in S and x~ E R, the metric g reads ds 2

= -a2(dx~

2

+ gij dxi dxJ

i,j=l,...,d-1,

(1)

= (gq) is the metric induced on S, by g, when x ~ t. a is a function called the lapse. Denote by K - ( K q ) the second fundamental form (p. 315) of a submanifold S,. 1) Express the Ricci tensor of M, at each point (x, x ~ t) in terms of the metric ~,, second fundamental form K and lapse a of S t, their covariant derivatives in the metric ~,, and "time" derivatives (i.e., partial derivatives with respect to x~

2) Show that if S ~ ---R~ - ~ g~t~R is the Einstein tensor then OL

2

--~-(R _ K ij K ij + (KI)2),

Soo

S,o=- a(-fTjK[ + ViKt) . -

l

(2a) (2b)

Express these identities in an intrinsic (i.e., coordinate-free) manner.

3) Show that the quantity OoRij- a2(V, Sjo + VjS/o) contains no third derivatives of g or a. Show that it contains second derivatives of K only through the operator 2 _ ~i~i F - l = ~12 aoo t~

if a is chosen such that

a 2 trK+a

,

=0,

af=

Oo

a.

Give a relation between a and d e t ( ~ ) equivalent to the above equation.

4) Using the previous results, deduce from the Einstein equations in vacuum R ~ = O, in an appropriate gauge, a Leray hyperbolic system.

lapse

408

VI. DISTRIBUTIONS

5) E x a m i n e w h e t h e r a s o l u t i o n o f the L e r a y s y s t e m thus obtained satisfies the original Einstein equations.

St, whose equation is x ~ t.

A n s w e r 1" Let n be the unit normal t o We have g ( n , n ) = - 1 , n~ = 0, thus

no

( gOO)-1/2

and (cf. p. 315) Kij=aFo

=

1 2 a '9~176

All the Christoffel symbols of g are then easily computed (overlined quantities are relative to the space metric g)

Foij

=

-oLK

10ia I -'00i ~ Ol

,j

i~i j =

,

Foio

a1

a o ia

Ki, ,

Fo~

1 ~

~ ~00t , Ot

We deduce from these formulas and the definition of Ricc(g), after some simplifications" -

h

Rq =-- R# - a-lOoKi/ - 2 K i h K /

+

h

K i / K h - ot

-1

~it~jol ,

Ro,---a(-VhK, h + 0,Khh),

Roo-- Ot2(Og

-

1vitriol 4;" Ol

-1

h

Oog h -

1

gigj

i

) ,

and using aog '~ = 2 a K ~i ,

the scalar curvature R = g~176 + g 'JR 6 =- R + ( K h ) 2 + K / K / -

2 a -1 OoKhh -- 2 a

- 1 Vit~ ia .

A n s w e r 2: We deduce from the identities in answer 1) the following ones 1

So ~ = Ro ~ - 2 g~176 R_

2

-2- ( ~ - K ij Kj, + ( K h) 2

So,-= R o , - a ( - - ~ h K i h + a, Khh)

(2a')

(2b')

which depends only on ~ and K on S t, and a. We remark t h a t o~-25o0 is the expression in the chosen coordinates 2 0 - 1 i (g0;=0, g00=-a thusn =a , n = 0 ) of the scalar function : S j_l -- S" (n | n) - S,~t3n"n t3 ,

12. L E R A Y T H E O R Y

OF HYPERBOLIC

SYSTEMS

409

while a - l S o i are the components in the chosen coordinates of the covariant vector S~ on St, S• = i*(S . (n | 7r)) ,

where 7r is the projection operator on S t, i the inclusion map of S t into M. In arbitrary coordinates (S . (n | 7r))~ = S,,13n~n~Tr~ ,

7r~ = g~ + n~n~ .

R e m a r k : In arbitrary coordinates (x i, x~ we have

S• L -'(S_l_)

with x ~

t the equation of S t,

( n 0 ) 2 S 00 '

_ n o S~ "

i ---

The identities (2a'), (2b') read, intrinsically 2S•177= / ~ -

K - K + (tr K) 2 ,

S• = - V . K + Vtr K . The right-hand side depends only, for x ~ forms of S t as a submanifold of (M, g).

t, on the two fundamental

A n s w e r 3" From the definitions of the connection and the Ricci tensor one deduces the formulae [Lichnerowicz], setting V0~ij = if;j, l

0o ['k = 89

--

--I

--t

(3)

gjh + Vjgih --Vhgij) ,

(4)

0o~i j = 89~kh ( ~ k ( ~ i ~ ; h + Vjgi'h) -- V~Vjghk) "

Using these relations, the Ricci identity and the identities (2) we obtain

~

--

~

01~ r

~-~

t

+ --~ K i j - ~ ot

--

-

h

1

+ f i j - - ViVj~ ot

t

a ' ViVja - F~j - k ' ~ k a + KijK~ , h + K ij K h'h -- 2K~mKj , m - 2K~mKj,m + --~ Ol

(5)

with f/j

-

+

-

-

-(Vi~r h a)K/h + (~r(/a)Vj)K h + ( V i % a ) K h k

h

h

(6)

+ a(Rih j K k - Rh(iKj) ) , r

t

-2

2

-

-h

a =Ooa, K i j = O o K i j ; [-]=-a 0 o - V h V . 2-9 We see on (5) that OoR~j- a V(iSj)o contains no third derivatives of the

410

vI. DISTRIBUTIONS

~'s. The expression (5) will contain second derivatives of the K's only through the operator El, and no third derivatives of a, if we choose a such that a 2 trK+a since we have o l -V-i ~ j K hh = ot

!

=0

(7)

-l~i~](ot2Khh) + m i j

with mij

-- 2(Via)VjK

~ - 2 ( V j a)ViK - hh - 2 a - x ( V i a ) ( ~ j a ) K

hh _ 2 ( ~ , ~ ) a ) K hh 9

With the choice (7) expression (5) reduces to OoR,j- ~

~-~

fiy + hi.i,

(8)

where f~j is given by (6) and n~j by 7/~ij = - - m i i + g i j g h h - K h V i v j O l -- FiiktVkOl t

m

tm

- 2 K i m K j - 2KimK j

9

We remark that we have (det g)' = -2aK I det g the relation (7) is therefore (det g)' - - 2 a ' det

=0

and its general solution for the scalar a is a = (det

g)l/2a-1/2

(9)

with a an arbitrary, positive, scalar density on S (independent of x~ Remark: The equation a t q-

a 2 tr K = 0

expresses that the submanifolds S t satisfy the harmonicity condition V~V~x~

~ ' r-'~o,

= 0.

A n s w e r 4: Identity (8) shows that, with the choice (9) for a, and goi = 0 , the equations in vacuum R ~ = 0 imply aoRii- a2V(iSi)o--alqKii

+ fii + nii = O,

(10a)

12. LERAY THEORY OF HYPERBOLIC SYSTEMS aog i]

(10b)

2 a K ij

=

411

These equations are equivalent if a ~ 0 to a quasidiagonal third-order system for if, with principal operator D&0. The operator I--l&0 is hyperbolic if a ~ 0 and ~ is properly riemannian: the dual characteristic cone in the tangent space is

X~ 12 (X~ 2 g i l X X ) = 0 ot __

-

i

j

it is cut in three distinct points by every straight line passing through a point where 1

--~ (X~

a

-

i

gijX X

2-

j

> O.

The dual of the above cone, that is the light cone of the metric, determines the dependence domain of the solution, in a g r e e m e n t w i t h physical expectation. A n s w e r 5: We shall prove: L e m m a Let ~ and K verify the hyperbolic system (10a,b), and a be given by (9), then the Einstein tensor S ~ corresponding to the metric

-aZ(dx~ 2 + gi] dXi dxJ,

a = (det g,)1/Za-1/2

(11)

verifies a linear, homogeneous hyperbolic system. P r o o f : By the Bianchi identities we have

V.S ~ - 0 which can be written, modulo linear terms in S "a -

iO

= O,

(12a)

&0S j0 + ~,S,j _- 0.

(12b)

&oS~176 + ViS

Eqs. (10) say that the metric (11) verifies the equations 8 o R i j - aZ(viSjo + VjSio ) = O.

We have, in the case of gio = 0 ROo _ 2S00

1

2 ghk R Ol

hk

thus Sq- R q -

g'(ghkRhk

--

ot2S00)

412

VI. DISTRIBUTIONS

and eqs. (10) imply, modulo linear terms in S "a

aOSij ,~ _ot2(~isi~ + vJs iO) + 20t2gqVhShO + c~2g'&0S~176 from which we deduce, modulo linear terms in S ~a, by the Bianchi identity (12b)

OoSij = -a2(~7'S j~ + ~rJS'~ + cz2gij VhshO 9

(13)

We deduce from which eq. (13) modulo linear terms in S ~, and Vi S ~

ViaoS' = -a2V~V~S j~ 9 The Bianchi identities (12b) imply therefore, modulo linear terms in S "~ and V~S~ lqS j~ = 0 .

(14)

The system (12a, 13, 14) is a linear homogeneous system for the S ~a which can be shown to be hyperbolic by derivating equation (14) with respect to x ~ We then obtain a third-order equation 7"l&oS/~ = 0 , where the symbol =0 means modulo linear terms in S ~, their first derivatives and the second derivatives of only S j~ (we use (12) and (13) to eliminate second derivatives of S ~176 and S~J). By the uniqueness theorem for a solution of the Cauchy problem for hyperbolic systems we shall have S ~a = 0 (in appropriate functional spaces of tensor fields on M) if S "a is zero on S O together with its derivatives of order - O} c 5r 2. Show that for Im z > 0 , the function W(z) . - ~ W ( L )

is analytic. 3. For y > 0 let Wy(x)= W(x + iy) and show that there are integers n and m and a polynomial C(y) so that

y.(a 9y t(l The polynomial bound on Wy(x) for each y ensures that for fixed y this function defines a distribution Wy E 5e'. 4. Show that lim Wy ~ 6e' y~O

and sup ~ W r ~ [0, oo). 5. Show that lim Wy = W . y~0

This establishes that W is the "boundary value" of an analytic function. To proceed: 1. F o r y = I m z > 0 , f~(k) = (2~r)-lx(k) e ikx-kr is C ~176 supported in [ - 1 , oo) and decays exponentially as k ~ +oo. To show the fz ~- ~, choose integers m and n. Then sup (1 + k2) m d" k -d~ L(k) = (2"tr) -1 sup ( 1 +

k2)m i-0 ~ (n)i X (i)(k)(iz)n-'

eikZ

_< (2~r) -1 sup [(1 + k2) m e-kY[ k>-I

..[_(2,,lr)-Ii=1 ~ (n)2m y n i e Izl-*

s u p IX (k)[ . ~'~ - l 0. To establish analyticity it is sufficient to show that W(z) is once (and thus infinitely) differentiable in this complex sense for Im z > 0 . This means that we must show that

W(z))

lim ~ - X ( w ( z q- ~ ) I~l--,0 exists. By linearity, ~ - I ( W ( z + ~z)_

W(z))-- ~:-I(~'W(L+~r ) -- ~ ; W ( L ) ) = ~ W ( ~ : - l ( f z + e - fz)).

Hence the limit exists if ~: - l(fz +e - fz) converges in 5e(R) as I r ---~0. Since we expect that

dfz(k) dz =ikfz(k) ' we will show that ow - lim ~:-l(fz+~(k ) - f~(k)) = ikfz(k ) . I~l--,0 (Note that ikfz(k ) E 5t'(R) by a proof like that in 1. above). To begin, we first r e m a r k that le ik' -- 11 = ~ 1 (ik~:)n 0 Wr(g ) is given by an absolutely convergent integral, we may differentiate with respect to y under the integral. Recalling that Wr(x ) = W(z) is analytic so that Cauchy's theorem applies, we find dl

f

dt

dY ~Wy(g) = dx ~dy~W(x + iy)g(x) dl

= f dx[(i)l-d-2xtW(x+ iy)] g(x)= (-i)' f dx W(x+ iy)g(t)(x). As y+0, the derivations of Wr(g) are bounded by the as Wr (g): dl

dyt Wy(g) - y -1 so we may take a ~ ~ and conclude that Wy(g)=O. 5. We are now ready to show that W is the boundary value of Wy. By this we mean that for any g ~ 9 o lim Wy(g) = W ( y ) . y$0

Since

Wy( g) = (~Wy)(~g)

and

W(g) = ( ~ W ) ( ~ g )

and ~Wy and ~ W are supported on [0, ~)_ we may without loss of generality consider test functions g with sup ~ ( g ) C [0, ~). The regularity theorem for tempered distributions (Schwarz, 1957, 1959, Reed and Simon, 1972) ensures that ~ W has the representation f

N

dn

~W(h) = dk ~, a,(x) dx" h(x) n=0

for some finite N and continuous, polynomially bounded functions a,(x). Hence, Wy (g)

f

= dx g(x) ~ W(fx +iy ) = f dx g(x)

f

d n fx+iy(k) dk ~N an(k ) -d~ n=O

= j=o

] (21r)-1

= .~--o •

dx g(x) (-Y)J-'

fax ~

dk a,,(k)x(n-J)(k)(ix-y)J e ik~-k" dk a.(k)x(U-;)(k) e

(ix) t eikxg(x),

where the reversal of the order of integrations is justified by the absolute convergence of the integrals. Thus we find Wy(g)= ~, ~ n ,,=0 i=O j

dk an(k)x (,,_j)(k)

(e_ky(~g)(x))

N an(k ) - d" = f dk ~, ~ (e-ky(~g)(k)) n=O

since ~ g and all its derivatives are supported on [0, oo) while X ("-j) is supported on [ - 1 , O] for j ~ n. The conclusion is

424

VI. DISTRIBUTIONS

Wy( g) =

~ W ( e - ( ) Y ( ~ g ) ( 9) ) =

W(~(e-()Y(~g)( 9))).

It thus only remains to be shown that 6e-lim ~ ( e - ( ) Y ( ~ g ) ( 9) ) = g , y~0

or equivalently, since the inverse Fourier transform is continuous on 5e, that ~-lim

y&O

e-ky(~g)(k) = (~g)(k) .

Toward this end, choose integers m and n. Then sup k (1 +

k2)m ~d~

O

1+

k2

+ ~ yi(in )

s pl(1 + k2) m+l (n Uk ('~g))(k)l supI( 1 +

k2)'(~g)~n'(k)l.

i-1

The sum above is a polynomial in y with no constant coefficient-it clearly vanishes as y $0. The first term may be rewritten -ky

1-e = sup sup k>0 1 + k2 k>0

f

k

d k ' ~--~7

1 + k '2

0

sup{yf k>0

(1 eke)

0

k

k

(1 + k '2)

- 2 f dk' k'(1.(.1+-k;2-~e -k'y) } 0

oo

~Y

f

dk' >0. 1 + k '2 y,o

0

REFERENCES Streater, R.F. and A.S. Wightman, PCT, Spin and Statistics, and All that, (Benjamin Cummins, Reading, Mass., 1964). Reed, M. and B. Simon, Methods of Modern Mathematical Physics (Academic Press, New York) 1972 (Vol. 1), 1975 (Vol. 2). Glimm, J. and A. Jaffe, Quantum Physics (Springer-Verlag, New York, 1981). Schwartz, L., Th~orie des distributions (Hermann, Pads) 1957 (Part I), 1959 (Part II).

15. BOUNDS ON THE NUMBER OF BOUND STATES

425

15. B O U N D S ON T H E N U M B E R OF B O U N D STATES OF T H E S C H R O D I N G E R O P E R A T O R *

This problem concerns the spectrum of Schrbdinger-type operators H = - A + V(x) acting on L2(R"). We assume that lim V(x) = 0 Ixl~ tends to zero at infinity. Simple modifications allow us to treat confining potentials. 1) Start from the energy functional for n >-3, defined by (integrals are taken over R ~) E(CJ) = T(,/I) + U(~),

T(qJ) = f dnx IV (x)l z

U( ~,) = f dnx V(x) l $(x) l2

f dnx I~r

2 - 1,

for q, E H I(R") and suitable potentials V(x). Bind the kinetic energy T(q,) using Sobolev' s inequality, use H6lder' s inequality for the potential energy contribution U(~b). Combine both to obtain a condition excluding negative energy bound states. Observe that - A + V(x)>-0 implies the absence o f such states [1]. 2) Consider the reduced wave equation for a radial symmetric Schr6dinger problem in R 3 with angular momentum l

dz l(l + 1) ) -~ + 2 + V(r) ut,E(r ) = Eut,e(r ) , dr z r

Ut,e(O ) = 0

on L2(R +, dr). Assume that there exist ut bound states. The zero energy solution splits the half line into ut disjoint intervals, according to the nodal theorem (see for instance [2]). Take first l = 0; integrate the Schr6dinger equation between successive nodes to obtain rp+l

f

dr[(-~)2

+ V(r)u2(r)} = 0 ,

where u(r)= Uo,o(r) .

r,

Apply the reasoning o f part 1) to each interval, observe that Sobolev's inequality applies to finite intervals as well. Sum up the contributions to *Contributed by H. Grosse.

426

VI. DISTRIBUTIONS

obtain a bound on the number of bound states of the form

IV]

v0_< S3 f dr r21Vl 3/2

89

=

v)

0

where Iv I_ denotes the absolute value of the negative part of V. Relate S 3 to the Sobolev constant [11. 3) Generalize 2) to angular momentum l: Transform from r and u(r) to z = In r ~ ( - o o , oo) and ~b(z)= u(r)/V7 and apply a scaling argument [1,3].

Answer 1" From Sobolev's inequality we get T(qj)> s~ll~ll =p with p = 2n/(n - 2). Let V = If+ - V_ be the decomposition of V into positive and

negative parts; IVl_ = V_. H61der's inequality yields V(0) >-- l i E _ Ilqll~,ll ~, for 1/q + 2/p = 1. Combining both bounds gives a lower bound on the energy functional 2 2 -+=1, p n

E(~) >m (S~ -IIV_ I1~,=)11~011~p

from which we conclude that there is no negative energy bound state if F

s \ - J d"xlVl "'~ < -

1

'

S~ = [Trn(n - 2)1 -n/2

F(n) r(n/2)

"

For a one-parameter family of similar conditions, see [6].

Answer 2" From the zero-energy solution of the reduced radial wave equation

--~r 2 + V(r) u(r) = 0 ,

u(0) = 0 ,

we obtain by integrating between successive zeros rp+ 1

0=

f

dr{(du(r)) 2+V(r)

rp rp+l

rp+ l

2/3

u 6 ( r ) ~ 1/3

:> (,if3 2/3- ( f dr r2]V] 3-/2) )( f d r r 4 "] rp

rp

where we used the fact that from Sobolev's lemma

15. B O U N D S ON T H E N U M B E R OF B O U N D STATES oo

oo

f dr r2(d@/dr) z inf o 4,ec~,(R§

427

(f) oo

dr rZ~6(r)

f dr(du(r)/dr) 2

a/3

=

inf

u~C~(R+)

o

6 (f)1,3 dr u (r) /r 4

0

oo

0 = ( 4 7 r ) m2/3S3 = S3- 1 / 3

,

@(r) = u(r) /r , and the same constant appears if one restricts integration to a finite interval. Adding up the inequalities rp+l

1 1, of sections of the bundle of r-tensors on X is the space of r-tensor distributions on X which, together with their covariant derivatives of order -<m can be identified with tensors in L P(X, h). This space is a Banach space, with the norm Iltllw

m =

_ f [V~t(x)[ v {k~<m h} 1/p d/~(

)

X m In the case of a compact manifold X the space W p (X, h) does not depend on h (though the norm of a particular section depends on h).

Remark" It is not necessary to endow X with a metric to define W Pm

spaces of sections of a vector bundle over X. One can instead work through the charts of an atlas. The definition will be independent of the choice of the atlas if X is compact. In the case of spinor fields on (X, h) it may be convenient to use the covariant derivative in h, and an appropriate, noncanonical, norm in the fibre. The injectivity radius of a riemannian manifold (X, h) is by definition the greatest number 6 such that

where ~x > 0 is the maximal radius of a geodesic sphere in X centered at x for it to be diffeomorphic by the exponential mapping to an open neighborhood of 0 in the tangent space TxX.

injectivity radius

430

VI. D I S T R I B U T I O N S

a) Show that on a compact manifold one has always 6 > O, but only 6 > 0

if X is noncompact and h arbitrary. b) Show that a riemannian manifold with nonzero injectivity radius is

complete. euclidean at infinity

c) A riemannian manifold (X, h) is called euclidean at infinity if it is the

union of a compact set K and an open set U and if (U, h lu) is isometric to the exterior of a ball of R ~ with its euclidean metric. Show that if (X, h) is euclidean at infinity it has a nonzero injectivity radius. Answer a: ~x is a continuous function of x E X which attains its minimum at a point x 0 ~ X if X is compact. Therefore 8x -_- 8x0 = 6 > 0.

Answer b: Let (xn} be a Cauchy sequence in X: for each E > 0 there exists n such that d(x,,Xm)<e

if

n,m>_N.

We choose e < 6 and we consider the geodesic ball B~ of center xN and radius 6. The sequence {Xm}, m >- N lies in B, C B~ and B~ is diffeomorphic to a ball of R ~. The image of the closure of B, is complete in R ~, therefore the sequence {xn} converges in X.

Answer c: We define a compact K~ D K which is the set of points where distance to K is less or equal to some given number d > 0 . A point x E U - K 1 is at a distance > d from K, and therefore is the center of a geodesic ball in the euclidean metric of radius at least d. On the other hand 8x, x E K 1, attains on the compact K 1 its minimum, which is therefore positive. II 2. DENSITY THEOREM denotes C o ( X ) the space of C | tensors on X of some given order r, compact support. It can be proved (cf. Aubin [3], Wamon [6]) Co(X) is dense in W ~ h ) = LP(X) Pl C o ( X ) is dense in Wp(X, h) if ( X , h ) has a nonzero injectivity radius, and uniformly bounded riemann curvature. (iii) Co(X) is dense in W mp (X, h), m > 1, if (X, h) has nonzero injectivity radius and if the riemann curvature is uniformly bounded on X as well as its derivatives of order -<m"

One with (i) (ii)

suplV ~ Riem(h)[ 0, v m v = Vmu, a n d

IlvllLq

-
TeG--g

is a vector spaces isomorphism. The equivalence (a) r

(c) can be proved in a similar way.

II

Remark. If A 6 A2(G) has one of these properties, then A(e) = 0, l(e) = O, m(e) = 0, where e is the neutral element of G. In particular, this is the case when (G; A) is a PLG. From 1.4 and 1.5, we can say that (G; A) is a PLG if and only if: (a') [A; A] = 0, i.e., A is a cocycle in the Poisson cohomology on G defined by the Schouten bracket. (b') l(g . h) = l(g) + (Adg)| where l(g) -- (Tgpg-l)| E IJ | {t, for all g, h ~ G, i.e., the mapping g ~ G --+ (Tgpg-l)| E g | ~1 is a 1-cocycle on ~1relative to the adjoint representation of g on ~t | {t. The notion of a Lie algebra can be seen as the infinitesimal definition of the notion of a Lie group. What then is the infinitesimal definition of a PEG? This will be the notion of a Lie bialgebra, which can be reached by looking more closely at the definition (a'), (b') of a PLG.

1.6. L e t A C A 2(G) and suppose the mapping

l'g6G Now let E =_ Tel " g e6G.

> (Tgpg-,)|

E g | 1~.

> g | fj be the tangent mapping to l at the point

3. POISSON-LIE GROUPS

447

Prove that if A has the Drinfeld property, or equivalently, if l is a 1-cocycle relative to the adjoint representation o f G on 1~ | g, then E is a 1-cocycle on g with values on g | g relative to the adjoint action o f g on g | g, i.e.,

~([x; y ] ) -

(adx @1 + 1 | a d x ) ~ ( y ) - (ady |

+ 1 @ ady)~(x),

f o r all x, y ~ ~. Answer. From the Definition (b) in 1.5, we easily obtain the following properties of l"

(i) l(e)-O

(ii) l(g -1) = - Adg-1 9 for all g 6 G. From the same definition and these two properties, we easily obtain, for all g l, g2 E G , t h e e q u a l i t y " -1

)-/(gl)-I-(Adg,)N21(g2)-

I ( g l " g2" g 1

(Adgl.g2.gl ~)|

In this equality, we now substitute g2 - - e x p ty, y ~ g, t ~ R"

l(gl "expty'g 1-1 ) --/(gl) + (Adgl )|

(expty)

--

(Adgl.expty.g _1 1 )N2/ (gl) .

We then calculate the derivative of this mapping at the point t -- 0, i.e., d

-1

Tel " - ~ ( g l " e x p t y " g 1 )It=0

= (Adg I ) |

. -d~ l ( e x p t y ) l t = o

d ~-; (Adg 1 9Adexp ty .Adgl 1 ) |

"/(gl).

Then: Tel" (Adg 1(y)) -- Ad |gl "Tel(y)

- - ( A d g 1 9ady- Adg~-i |

q- I @ Adg 1 9ady- Adgll ) . / ( g l ) .

In this equality, we then substitute g l - e x p s x, x ~ g, s ~ fiR, and compute the derivative of this mapping at s - 0: d d --ds (Tel" Adexp sx (y))Is=0 = --ds(Ad| expsx

"rel(y)_,s= )1 0

d

(Adexpsx 9ady 9Adexp(-sx) | ds Jr- I | Adexpsx" ady. Adexp(-sx) ) 9

sx)Is=O"

SUPPLEMENTS AND ADDITIONAL PROBLEMS

448

Likewise" __

d

d (Tel" Adexpsx(Y))Is=O- Tel'-~s(Adexpsx(Y))ls= o -- Tl" [x; y],

ds d d ) ) ls=0 ---- ds (" Ad e| x p s x --ds(Ad e| x p s x " Tel(.Y-,, -- d((exp(adsx))| ds

s

)1- , s = 0

"

Tel(y)

=o" Tel(y) -- (adx | I + I | adx)- Tel(y),

d (Adexpsx 9ady. Adexp(-sx) | d-~ + I | Adexpsx "ady. Adexp(-sx) )" l (exp sx)[s=O = (ady | -t- I | ady). Tel(x). The required derivative at s - 0 is then" Tel([x; y]) -- (adx|

I + I | adx). T e l ( y ) - ( a d y |

I + I | ady). Tel(x). m

Given a Poisson-Lie group (G; A) the essential property of cocycle E : 0 --+ 0 | 0 is that its transposed mapping Et: 0* | 0* --+ 0* will define a Lie algebra structure on 0". This property can be proved by solving the following problem:

1.7,

Let A ~ A2(G) (note that we need not assume that A verifies the Drinfeld property). Let us consider the mapping l 9G --+ 0 | 0 as in 1.6 (which need not be a 1 -cocycle). Let us also consider the following skewsymmetric mapping on C o~ (G)" {~0; ~ } G

--

A(d~" d~)

f o r all ~p, gr ~ C ~176 (G). Prove the following relation"

(d{go; ~}G(e); x ) - (dgo(e)/x d~(e); E(x))

( -= (Et(dgo(e)/x d~p(e)); x))

f o r all x ~ O. Equivalently, prove the relation"

d({~; ~ r } c ) ( e ) - 6t(d~(e) A d~(e)).


449

Answer.

(d({~; ~})(e); x} _ d ({99 7r }(exp tx))It=0 dt d _

-- dt ((TePexptx)|

A d~(exptx)))It=0

-- d ( l ( e x p t x ) ( d q ) ( e x p t x ) o TePexptx A d ~ ( e x p t x ) o TePexptx))]t= 0 dt d = --l(exptx)lt=o(dgo(e) A d~r(e)) - E(x) (dgo(e) A dO(e))

dt = (ago(e) A dl/r (e); e ( x ) ) - (Et(dgo(e) A dgr(e)); x).

The above equality is obviously verified for any P L G (G; A); furthermore {; }6 is the Lie bracket of the Poisson structure defined by A on the manifold G. F o r P L G , we can also prove: 1.8. Let (G; A) be a PLG. Let 1(g) -- (Tg pg-1) A (g) be the corresponding 1cocycle on G, with values on 1~ A 1~, relative to the adjoint representation o f G on g A g. Let ~ = Tel : g --~ g /X 13 be the corresponding 1-cocycle on g with values on g A g relative to the adjoint representation o f g on 1~ A g. Let us define the following bilinear mapping on g*: [~1; ~2]1~* ~--- Et(~l A ~2)

f o r all ~1, ~2 E 1~*. Prove that the pair (13"; [; ]~*) is a Lie algebra. Answer. It is clear that [; ]9, is bilinear and skewsymmetric. It remains to

be proved that it also verifies the Jacobi identity. Let q91, q)2 E C ~176 ( G ) and be such that ~1 = dqgl ( e ) , ~2 - - d q ) z ( e ) , ~3 - - d~03(e). Then: [~1; s~219* -- et(d~l (e) A drp2(e)) -- d({rpl ; rP2}G) (e). Also: [~1; [~2; ~3]g*]9, -- [dgol (e); d({g02; (Tgpg-i )|

E g| ... p |

(12)

for all g, h 6 G,

(13)

verifies the relation: l ( g . h) - l(g) + (Adg) |

l(h);

i.e., if and only if I is a 1-cocycle on G, with values in ~1| -p-" | to the adjoint representation of G on g|

relative

|

Let us now consider the notion (11). For any given h 6 G, the term P(h) defines a left-invariant tensor field on G; and for any given g 6 G, the term (Tgph)| P(g) defines a right-invariant tensor field on G. This observation leads us to reformulate condition (11) using the result in 3.1.

(Thkg) |

3.4. Let G be a connected Lie group, and g its Lie algebra. Prove that P ~ AP(G) verifies the Drinfeld property if and only if'. (i) P is zero at e ~ G, i.e., P (e) = O. (ii) L x P is a left-invariant tensor field for any left-invariant vector field X ~ A I (G). Equivalently, 3.1, L y L x P - - 0 f o r any right-invariant vector

field Y ~ A pI(G).

Answer. P verifies the Drir~feld property if it verifies (11) or, equivalently, if it is verified (13). Let us suppose that P verifies (11). Setting g = h = e in (11), we get P (e) = 0. (i) is thereby proved. We now prove (ii). Let x ~ 0 and x x ~ A~(G). The flow of x x is Ft -Pexptx, t ~ ~. Thus,

d (Lx~ P ) (g) -- -~(T~,.exptx Pexp(_tx) )|

( P (g . exp tx) ) ]t=o

d )| -- __d-'-~((TgexptxPexp(-tx) d -t- -~(Tg.exptx(Pexp(-tx)) |

o (Texptx Xg)| p " P ( e x p t x ) ) [ t = 0 o (Tgloexptx) |

" P(g))lt=o.


461

(In the second equality, we use (11) for P ( g 9exptx).) The second of the last two terms is obviously zero. The first is: d

dt ((TgexptxPexp(_tx))|

d

-- d t - - ( ( r e ; k g ) |

o (Texptx)~g) |

o (ZexptxPexp(_tx))

d

= (TeXg)|

.

P(exptx))[t=o

.

P(exptx))lt=o

|

)|

9 P(exptx))lt=

o

= (Te)~g)OP(LxxP)(e). We thus get the equality:

( L x x P ) ( g ) -- (Te,kg)OP(LxxP)(e), (ii) is thereby proved. Proving that, if P verifies (i) and (ii), then P verifies the Drinfeld condition will be equivalent to proving that the mapping (12) verifies (13); or equivalently, given that G is connected, the relation:

l ( g . exptx) -- l(g) + (Adg)|

for all x e g.

First, let us compute the derivative of each side of this equality (14)" d - - l (g exp

d - -~ ((Tg.exp tx 9p ( g . e x p t x ) - l ) |

d

--(TgPg-,)| = (T~,.e•

" Pexp(-tx)) |

(g . exp tx)) . . . . .

" P(Pexptx(g)))

(L~x P ) ( g . exp tx) . . . . .

9P(g.e•

-- (Ze(P(g.exptx)-I

9P

0 Xg.exptx)) |

-- (Adg.exptx )|

.

(LxxP)(e)

).

For the derivative of the right side of (14), we get"

d (l(g) + (Adg) | dt d -- ( A d g ) |

-- (Adg) |

(Texptx

= (Adg.exp tx ) |

l(exptx)) ..... " Pexp(-tx) " l)exp-tx

)|

(Lxx P) (e).

)or

P(exptx)) tx

)|

(14)

462


We thereby prove that the derivatives of the terms on both sides of (14) are equal, i.e., for all t 6 N: d(l(g, dt

exp tx) - l ( g ) - (Adg) |

l ( e x p t x ) ) --O.

Thus, l(g . exp tx) - l(g) - (Adg) |

9l ( e x p t x ) - l(g) - l(g) - 0 - O,

where we make use of the fact l (e) = 0 since P (e) = 0. The proof is now complete. II 3.5. Prove that the space o f contravariant skewsymmetric tensors on G verifying the Drinfeld property has the structure of a graded Lie algebra relative to the Schouten bracket. Answer. We must prove that, if P and Q verify the Drinfeld property, so does [P; Q]. From 3.4 this is equivalent to proving that [P; Q](e) = 0 and Lx~ [P; Q] is a left-invariant tensor field for any x 6 ~. (a) From the local expresion of [P; Q], it is clear that [P; Q](e) = 0 when P(e) -- Q(e) -- O. (b) For R 6 Ar(G), the relation that defines the graduated structure of skewsymmetric tensors is:

R]; P] + ( - l ) q ' r [ [ R ; P]; Q] + (--1)P'r[[P; a]; R ] - 0 . (15) Whenever R = X is a vector field, the above relation is: (-l)Pq[[o;

[P; ix; o]] + [ix; P]; o ] - [x; [P; el], that is: L x [ P ; Q] = [LxP; Q] + [P; L x Q ] . Let Y be any vector field on G. As above, we get: L y L x [ P ; Q] = [ L y L x P ; Q] + [LxP; L y Q ]

+ [ L y P ; L x Q ] + [P; L r L x Q ] .

(16)

Let AI(G), Apl(G) be respectively the spaces of left and right-invariant

vector fields on G. Let us suppose that X ~ A l(G), Y 6 Af,l(a) . From 3.4 tensors Lx P, Lx Q are left-invariant. Therefore, LyLx P - - 0 , and LyLx Q = 0. Also, [X; Y] = 0; wherefore LxLy P = LyLx P = 0 and LxLy Q = LyLx Q = 0; and Ly P, Ly Q are right-invariant tensors. From these relations, question 3.2, and (16), we get Lr, L x [ P ; Q] = 0 .


463

From this relation, the equality [P; Q](e) = 0 and from 3.4, [P; Q] verifies the Drinfeld property. The graded character of the algebra is a consequence of (15). II 3.6, Let G be the simply-connected Lie group with Lie algebra ~. Let (~; [; ]; e) be a Lie bialgebra (2.1). Prove that there is only one Poisson-Lie structure on G with this Lie bialgebra. Answer. (a) From the classical Lie group and Lie algebra cohomology, the

1-cocycle e :9 ---> ~ | g relative to the adjoint representation of g on g | determines a unique l- cocycle on G, 1 : G --+ g | ~, such that Tel -- E. Let us therefore define the following skewsymmetric contravariant 2-tensor: A ( g ) -- (TePg)|

. l(g).

(b) From question 1.5, the tensor A verifies the Drinfeld property. From question 3.5, the 3-tensor [A; A] also verifies this property. Therefore, from 3.4, [A; A ] ( e ) - 0, and Lxz[A; A], x z 6 A~(G), x 6 9 is a leftinvariant tensor field, i.e., Lxz[A; A ] ( g ) - (Te)~g) |

(Lxz[A; A])(e).

To prove that (G A) is a Poisson-Lie group, it remains to be proved that Lxz[A; A](e) = 0;

for all x 6 9,

in which case Lxz[A; A](g) = 0, and therefore (3.1), the tensor [A; A] is right-invariant. Thus: [A; A](g) ----0, as required. (c) From the definition of the Lie derivative, we have: (Lxx[A; A])(g). (d~pl (g); d~02(g); d~o3(g)) = Lxx. ([A; A](g)(d~01 (g); d~02(g); d~03(g))) - [ A ; A]. ((Lxxd991)(g); d992(g); d993(g)) - [ A ; A]. (d991(g); (Lxzdq92)(g); d993(g)) - [A; A]. (d~01(g); d~p2(g); (Lxxdq93)(g)). Whence: (Lx~.[A; A])(e)(dq)l (e); dcp2(e); dq)3(e)) = (Lx~ ([A; Al(d~01 ; d~02; d~o3)))(e).

(17)

464


From question 1.7, we get: (d{~pl ; ~p2})(e) - ~t(dqgl (e) | dq92(e)). Using for example the local expression of the Schouten bracket [A; A], we can easily get the following equality: [A; Al(dqgl ; dqg2; dq93) - 2({{q)l ; 992}; q)3} + p.c.). Expression (1 7) can then be written: (Lx~tA; Al)(e)(d~ol (e); drpz(e); d993(e)) = 2((Cxx {{991; 992}; 993} ) ( e ) + p.c.) = 2(((d{ {(/91; q)2}; 993 })(e); x} + p.c.) = 2((~t@t(dq)l (e) | dgo2(e)) | dgo3(e)); x)-I- p.c.) = 2({[[d~ol (e); d~o2(e)]o, ; d~o3(e)],, ; x ) + p.c.) = 0, where [; ]~, is the Lie bracket on g* defined by the mapping ~t.g, | g, _+ g*. We thus obtain (Lxx[A; A])(e) --0, and consequently [A; A] --0. II 4.

EXACT POISSON-LIE GROUPS AND THE GENERALIZED CLASSICAL YANG-BAXTER EQUATION

This problem consider the particular case of Poisson-Lie groups where cocycles I and E are exact: l - Or; e - gr, r ~ g | g. These are basic to the understanding of the mathematical structure of the integrable dynamical systems connected with the Inverse Scattering Method. The generalized classical Yang-Baxter equation and the notions of quasi-triangular exact Lie bialgebras are brought into play. (Ref. [1 ], [2], [3], [7].) 4.1.

We begin by introducing a few notations. (a) Let M be a differentiable manifold. Let A2(M) be the spaces of contravariant skewsymmetric 2-tensors on M. Let A 6 AZ(M). We define the vector spaces homomorphism" #'c~ E AI(M)

> #c~ E AI(M)

by the relation" (/3; #c~)- A(/3; c~);

for all o~,/~ ~ AI(M).

(b) Let G be a connected Lie group with Lie algebra g. Let r 6 g | g. The canonical isomorphism: r ~g|

> 7 ~ Hom(g*; g)


465

is defined by the relation: (r/; 7 ( ~ ) ) - (~ | ~;r);

for all ~, r/~ 0*-

Let A ( g ) -- (Telg)| A(e) -- r be the left-invariant 2-tensor defined by the element r ~ 0 | 0- Then" #g" ~ (g) 6 T* g G

> #g~ (g) 6 Tg G

and (q(g); # g ~ ( g ) ) - A(g)(rl(g); ~(g)). Let us suppose that ~ (g) and r/(g) are left-invariant 1-forms: ~(g) - - ( T g X g _ l ) t . ~(e), 9

rl(g) - - (Tg)~g-1

)t

9q(e),

where ~ (e), r/(e) e Te*G ~- 0. We then get: ((TgXg-,)t.

r/(e);

#g(Tg),..g-, ) t .

~(e))

=

= r (r/(e); ~(e)) --(o(e); ?(~(e))), for all g 6 G. Therefore (r/(e); #e~(e)) = (o(e); ?(~(e))), whereby we get #e-r. (c) The expressions of the function [A; Aq(otl; ol2; or3) given in the following three questions will be needed subsequently. 4.2.

Let M be a differentiable manifold and A , A 1 ~ A2(M). Prove that the Schouten bracket can be written as:

[A; A'](ol 1; oe2 ; oe3) --

_

((L#~loe2 ; #'u 3) + (L#,~l or2 ; #c~3) + p.c.),

(18)

where # and #I are the homomorphisms defined at the begining o f this section. Answer. In the natural coordinates corresponding to a local chart on M, the

components of the tensor [A A] are" [A; A t ] i ' j k -

1 , ijk A t n . a t ( A ~) rs q_ emns ijk Amn t)ts _~. [6nrs . . at (A ).

From the definition abc

1 2 3

(Or 1 A ot 2 A ot B ) i.j k - - 6 i.j k Ota Otb Otc ,

466


the left-hand side of the expression we need to prove is then:

The right-hand side of the expression we want to prove is then:

All the terms in this expression containing a derivative &a:: cancel out, because A and A' are skewsymmetric tensors. We then get:

) + (L#r,ra*;#a3)+ p.c.

(L#,lU 2 ; ##a 1 3

= A f ' . ( a r ( A ' ) i k ) . ~.ia " : + ( a , A " ) ) ( A ' ) ' ' . a :

.a?.a:+p.c.

Equality (18) follows from this relation in conjunction with (19) and (20).

rn


467

4.3. Let A, A' E A 2 (G) be left-invariant tensor fields on G defined as" A ( g ) -- (Te~g)|

A ' ( g ) -- (Te~.g)|

',

where r, r I ~ g A g. Let 9.1(g) be the enveloping algebra of g, and let p23 be the permutation of the second and third factors in the tensor product 9.1(g) | 9.1(g) | 9.1(g). Let us define the following elements ofg.l(g) |

r12-- r @ 1;

rl3

p23

_

9r12;

r23 -- 1 @ r

and also the bracket: 9

I

I

!

[rij, rmn] -- rijrmn -- rmnri j,

.

i, j, m, n -- 1,2, 3,

using the product of 9.1(g). Let us also define the element

(

Jr; r'] -- -- [r12; rfl3]-+- [r12; r~3] -F [r13; r23] -[-[ri12 ", r13] -t-[rtl2; r23] -+-[r'13," r23])" Prove that Jr; r ~] is an element of g | g | g, and following a similar argument as in 3.2, prove that

[A; A'](g) -- (TeXg)a3[r; r']. 4.4. Using the notations and the result of the preceding question, by taking into account together the result in question 4.1, prove the following equality"

[r; r'](ot' : oe2; ot3) - -((oe3; [7(oe'): 7'(oe2)] + [~'(otl): ~(ol2)]] + p.c.), (21) w h e r e 0/1 o12 0/3 G 1~*

Answer. When no confusion results, we will identify the element 0/i E ~t* T~'(G), i - 1,2, 3, with the invariant 1-form (Tglkg_l)tol i ~ 0/i (g) as re-

quired in the following proof. From relations (20) and (18) in 4.1, we can write:

[A; All(o/l; 0/2.0/3) _ ((0/2; L#ffl#f0/3)+ (0/2; L#totl#0/3)+

-(L#c~, (ot2; #'oe 3) +

p.c.)

L#,c~l(cr2; #or 3) + p.c.).

(22)

The terms in the second line of this expression are zero because the functions (oti; #crJ)(g) are constant: every oe' (g) is a left-invariant 1-form, and


468

#, #~ are respectively defined through the left-invariant tensor fields A (g), A' (g). We thus get: [A; At](otl; c~2; ot3)(e)

--(ot2(e); [7(c~1 (e)); 7'(c~3(e))])q-(ot2(e); [7'(c~1 (e)); 7((ot3)(e))]) +p.c. This expresion is precisely (21), since [A; A~](e) = [r; r~], 4.2, and since, at this point, we make the identification Oti ~ Ott (e), i -- 1,2, 3. I (d) One of the results from question 4.3 is that the 2-tensor A;~(g)

--

(Te~.g)|

r ~ g / x g,

verifies the equivalence [A;~; A ; ~ ] - 0 ~

[r 12" r 13] + [r 12" r 23] + [r 13" r 23] - - 0

i.e., (G; A~) is a left-invariant Poisson structure if and only if the element r = A (e) 6 g/x g is a solution of the classical Yang-Baxter equation [r; r] = 0. It is obvious that this Poisson structure can not be a Lie-Poisson structure unless r = 0. A Lie-Poisson structure on G is defined, 1.4, by a 2-tensor A such that [A; A] = 0, and A verifies the Drinfeld property: A(g.

h)=

ThUg"

A ( h ) + Tgph " A ( g ) .

Even if the 2-tensor A x (g) does not verify this property, it may be that the 2- tensor A ( g ) -- A ~ ( g ) - A o ( g ) - - ( T e ) ~ g )| r - ( T e p g )| (23) does. The sing - in this definition is due to the fact that A (e) = 0, from the Drinfeld condition. From the definition of A, we get: A(g . h)

--

(Te)~g.h) |

"r

-

(Tepg.h) |

r.

Also: Th)~g " A ( h ) -- T h ; k g ( A z ( h )

- Ap(h))

-- ( T e Z g . h ) " r -- Te(Zg " p h ) " r

and TgPh

" A(g)

-- TgPh(Az(g)

-

Ap(g))

-- Te(Ph

" )~g) . r - - T e P g . h

" r.

From the last three expresions, we can see that the tensor A defined in (23) verifies the Drinfeld condition. Note that, in this computation, we have not had to suppose that A~, and consequently Ap, defines a Poisson structure on G; i.e., the classical Yang-Baxter equation [r; r] -- 0 does not


469

hold. Were we to suppose that this equation does hold, then [Ax; Ak] = 0 and [Ap; A p] - - 0 . F r o m question 3.2, we then get [ A ; A ] -- [ A k ; A k ] -- [ A k ; Ap] -- [Ap; A k ] -t- [Ap; Ap] = O.

Therefore, the pair (G; A = Ak - A p ) , where Ak(g) = Tekg .r, A p ( g ) = TePg 9r and [r; r] - - 0 , is a P o i s s o n - L i e group. We no longer suppose with respect to expression (23) that [r; r] = 0, or equivalently A k and A p defines a Poisson structure on G. F r o m the question 1.5, the mapping: l" g E G ----+ ( T g p g - ~ ) | is therefore a 1-cocycle on G, with values in g | g, relative to the adjoint action of G on g | g. We can easily compute this cocycle as determined by r ~g|

l(g)

-- (rgpg-l)|174

r - (TePg) |

r).

Thus, l(g) -- (Adg) |

r - r.

In terms of the cohomology on G, with values on O | g, relative to the adjoint representation, this last expression tells us that the 1-cocycle I is the cobondary of the zero cochain r 6 g | g: 1 =0r. The corresponding 1-cocycle on g with values in g | g corresponding to the adjoint representation of g on g | g is e = Tel. Therefore, for all x ~ g, r

-- T ~ l ( x ) -

d ~l(exptx)lt=o

_-- __d ((Adexptx)| dt A

~,

9r -- r ) l t = 0 = (adx | A

+ 1 Q adx) 9r.

Then" E(x) -- adx 9r - 6r, where ad stands for the adjoint representation of g on g | g. (e) The tensor A defined by the expression (23) verifies the Drinfeld property. Now we will look for a necesary and sufficient condition for r g | g whereby the tensor A in (23) also verifies [A; A] = 0, i.e., whereby the condition on r for (G; A = A x - A p) is a P o i s s o n - L i e group. F r o m questions 1.8 and 3.6, we can see that this is equivalent to finding the condition on r such that e t . g , | g, __~ g,, where e - ~r, defines a Lie algebra structure on ~*.


470

First, let us introduce some additional notations. F r o m the definition of isomorphism: ~ ' g | g > Hom(g*; g), 4.1 (b) we have, for all ~, r/6 g*, xEg, (~; ~r(x)o) - (~ | ~; ~r(x)) = - ( a d * x ~ | r / + ~ | ad*x rl; r) .~,

-

-

(~; adx 97(r/) - F(ad*x 9r/)).

Therefore: 6 r ( x ) -- adx o ~ + F o (adx) t.

(24)

We simply define ~ 6 Hom(g*; 9) as" g~ (x) -= ~r (x) - adx o ~ - ~ o ad*x. 4.5. Let r ~ g | g. Let

[; 1~, "g* x $*

> 9"

(25)

be the bilinear mapping defined by the exact l-cocycle E - 7 "g --+ g x g. Prove that: F [s~; ~]~, -- (6r)t(s~ | r/) -- ad*;(,)

9~ + ad*,~,(~) 9rl.

(26)

Answer. We have, for all x ~ g,

([~; r/]~. ;x) . . . . . = ....

(~; a d x . 7(r/) - F(ad*x 9r/)) (ad*~(0) ~ ; x ) - (ad~,(~)x; rl).

Then" [~; r/]~. - ad*~(,) 9~ + ad*~t(~) 9r/. Obviously, the definition of ~t. g. _4 (9*)* - g is as follows"

=

|

|

p,2(r))_

p,:(r),)

Thus ~t _ p 12(r ) where the components of r 6 g | g permute under the action of P 12. We now determine a necesary and sufficient condition on making the bilinear form, [; ]g,, on g*, defined by the exact l-cocycle E -- 3r, skewsymmetric.


471

4.6. Let r -- s + a ~ g | g where s and a are respectively the symmetric and skewsymmetric components o f r. Prove that the mapping (25) is skewsymmetric if and only if s ~ g | g is a zero cocycle: 6s - O, i.e.,

adx 9~ - ~. ad*x;

f o r all x 6 g,

(27)

or equivalently, if and only if

ad*~(~) 977 + ad*~(,7) 9~ - 0;

f o r all ~, rl ~ g*.

(28)

Moreover, supposing 6s --O, then 6r -- 6a, and

[~:; r/]~, -- [~; r/]~, -- ad*~(,7 ) ~ - ad*~(~) r/.

(29)

Answer. Clearly, f i t _ _fi and ~t _ ~.

(a) The equality (26) is therefore" [~; ~]~, -- ad*~(,7) 9~ + ad*Ft(~) .r/ = (ad*~(,7) 9~ + ad*i(~). ~7) + (ad*a(rj) 9~ - ad*a(~). 77). Therefore, [; ]~, is skewsymmetric if and only if: ad*~(~) 9r / + ad*~(,7) 9~ - - 0 . Expression (29) thereby is proved. (b) On the other hand: 0 -- (ad*~(~) -~ + ad*~(~) .r i; x} . . . . . = (r/; a d x . ~ ( ~ ) ) + (~; a d x . ~ ( ~ ) ) - (r/; a d x . ~ ( ~ ) } - (r/; ~. ad*x ~}. Therefore adx o g - g o ad*x, for all x ~ g. The conditions (27), (28), (29) are then equivalent. (c) Also, by definition, 6s(x) - adx 9s. By (24) this is equivalent to: 6g - 6(x) - adx o g - g o ad*x. The relations (27), (28), (29) are then equivalent to 6s - - 0 , i.e., to the invariance of s by the adjoint action of g on g | g" adx 9s - (adx | 1 + 1 | adx) 9s - 0. F These results are equivalent to the following one: [; ]g. is skewsymmetric if and only if it is defined by the skewsymmetric part of r. II

472

SUPPLEMENTSANDADDITIONALPROBLEMS

In order to get a necessary and sufficient condition on r for the skewsymmetric mapping [; ]g. to verify the Jacobi identity, we shall write an equivalent of expression (21) in 4.4 for the tensor [a; a], making use of the form F

4.7.

Let a ~ 0/x g. Let [a; a] 6 g/x 0/x 0 be the element defined in 4.2 and 4.3. Let [; ]~, be the skewsymmetric bilinear form in 4.5. Prove that the expression (21 ) in 4.4 f o r r = r I = a can be written as: _l -adx[ ~ a.

~

a ~ 3]0" a . x)+ p.c. a](~l. ~2. ~3) _ ([[~1. ~ 2]0,. ~

~

~

~

(30)

where: ad-'~"0 |

--+ 0 | is the adjoint representation of 0 on 0 ~ ad--"~| 1 + 1 | adx | + 1 | 1 | ~ O. In consequence, [; ]g. verifies the Jacobi identity if and only if the tensor [a; a] is invariant by the adjoint representation of a~d o f 0 on 0 | equivalently, if and only if [a;a] is a zero cocycle in the corresponding cohomoladx |

ogy. Answer. We develop both sides of the expression (30); by the result in 4.4

and the Jacobi identity for 0, we find that both sides coincide.

From the expression (29), the definition of ad, ad* and the Jacobi identity for 0, we get: 9

2

a

3]a

+ p.c. =

.

.

.

.

-- - ( ad*x ~3 ; 3(ad*~(~,) ~2 _ ad*a (;2) ~ I )> + (ad*x .~3; [fi(~,); ~(~2)]) q_ p.c.

(31)

Furthermore ad-'~[a; a](~l; ~2; ~3) _ - [ a ; a](ad*x ~1; sea; ~3)

-[a; a](~l; ad*x ~2; ~3) _ [a; a](~l; ~2; ad*x~3) = - I a ; al(ad*x ~l; b~2;~:3) _~_p.c. (in the last equality, we make use of the fact that [a; a] E 0/x g/x g). We now develop the last term by referring to expression (21) in 4.3. We thus have: - [ a ; a](~l ; ~2; ~3) _+_p . c . - 2(~3; [fi(ad*x ~'); t1(~2)]) -k-Z(ad*x ~3; [a(~l); fi(~:2)])-t-2(~3; [fi(~l); a(ad*x ~2)])-t- p.c.


From the equality 3

--

473

_fit, we also get:

(~3; [a(ad*x ~ 1); 3(~2)]} = -{a(ad*a{~2 ) ~3); ad*x ~ 1), and {~3; [3(~ 1); fi(ad,x

~2)])_

{a( ad*a(~ 1) ~3); ad*x ~2).

Finally, we obtain: ad-"~[a; a](~ 1; se2; se3) _ -[a; a](se 1; ~2; se3) + p.c. = 2{ad, x ~3; [3(~1); 3(~2)]}_ 2{3(ad, a(~, ) ~2); ad,x ~3) + 2(a (ad.a(~2) se,); ad. x se3)+ p.c. If we compare this expression with expression (31), we obtain expression (30). m We can summarize the results in 4.4, 4.5 and 4.6 as follows: 4.8. Let g be a Lie algebra and r - s + a ~ g | g where s and a are respectively the symmetric and skewsymmetric parts o f r. The set (g; [; ]; ~ -- 6r) is a (exact) Lie bialgebra if and only if s ~ g | g and [a; a] 6 g | g | g are O-cocycles relative to the adjoint representations o f g o n g| and g| respectively.

4.9. Definitions (a) We will say that the skewsymmetric tensor a ~ g/x g is a solution o f the generalized classical Yang-Baxter equation if the tensor [a; a] ~ g A g A g is a O-cocycle in the Chevalley cohomology o f g with values in g | g | g, equivalently with values in g/x g/x g, and relative to the adjoint representation, ad, o f g on g| equivalently with values in gA3. Also equivalently, if the tensor [a; a] is ad invariant. Whenever a ~ g/x g is a solution of the generalized classical Yang-Baxter equation, we will say that the (exact) Lie bialgebra (g; [; ]; ~ - 6a) is quasitriangular. (b) Whenever [a; a] = 0, we will say that a is a solution of the classical Yang-Baxter equation, and the Lie bialgebra (g; [; ]; ~ - 6a) will be called triangular

474


4.10. Let G be a simply-connected Lie group. Let r be any element in g | g. Let e -- 6r "g --+ g | g be the exact 1-cocycle on g, with values on g | g, relative to the adjoint representation o f t3 on 0 | g. Prove that the 1-cocycle l " G ~ g | g on G with values on g | g relative to the adjoint representation o f G on g | g such that Tel -- 6r = ~ is the exact 1-cocycle" l - Or"

l(g)-Ad

|

f o r all g E G.

Answer. From the action of O on the 0-cochains, it is clear that l consequently, l is an exact 1-cocycle. In particular, l ( g . h) - l(g) + Ad |

9

Or;

for all g, h E G.

We need only prove that Tel = 8r, by reason of the bijective correspondence between the 1-cocycles for the Chevalley cohomology of 0 and for the cohomology of the Lie group G. But this is the same proof as the one at the end of section 4.4. II 4.11. Let r = s + a E g | g. Let us suppose that (g; [; ]; e -- 8 r ( = 6a)) is a (quas i t r i a ~ u l a r ) Lie bialgebra. (Equivalently, let us suppose that adx 9s - 0 and adx[a; a] - 0 . ) Let G be the simply connected Lie group whose Lie algebra is g. Let (G; A) be the Poisson-Lie group determined by the above Lie bialgebra. Prove that: A ( g ) - AXr(g) - A P t ( g ) - AXa(g) - A~(g);

f o r all g E G,

where AXr (g) -- (Te~.g)|

APr(g)--(TePg)|

Answer. By 4.10, the l-cocycle on G, l, corresponding to the 1-cocycle - 6 r on g is" l(g) - A d g ~

- r -- Adg~2.a - a.

(ad-"~s - 0 for all x E g =, Ad~ 2 s -- s for all g E G). From 3.6, we know that tl~e tensor field A ( g ) -- (TePg)|

3. P O I S S O N - L I E GROUPS

475

defines the Lie-Poisson structure on G corresponding to the Lie bialgebra in the question. So: A ( g ) -- (TePg)|

= (Te)~g)|

r - r) . . . . .

- (TePg)|

-- (Te~,g)|

- (TePg)|

= A~r (g) - APr (g) -- A~a (g) - APa (g).

This tensor field is precisely the one that could be conjectured after the reasoning concerning expression (23). m As a consequence of this result, we obtain: 4.12. Let {eu; # -- 1 . . . . . n} be any basis on g, and r - rUVeu | ev ~ g | g, any element such that (g; [; ]; e - 6r) is a quasitriangular Lie bialgebra. The corresponding Poisson bracket on the Poisson-Lie group (G; A ) determined by the above Lie bialgebra is:

{~o; 7r} - rtZV(Lx~o . Lx~Tr - Lx~cp 9Lxp~);

f o r all ~o, 7r e C ~ ( G )

where xtz~ (g) -- Te)~g " e u, x p (g) - TePg " ev.

Note, 4.6, that this bracket does not depend on the symmetric part of r since Adg s -- s, for all g 6 G. 4.13. Let a, a ~ ~ g A g and Aa,, APa be respectively left and right-invariant 2tensors as in 4.11. Let us suppose that a and a ~ are solutions o f the generalized Y a n g - B a x t e r equation. p Prove that the 2-tensor A~a - A a, is a Poisson tensor on the simply connected Lie group G with Lie algebra g, if a n d only if [a; a] -- [a~; a~]. In the p a r i c u l a r case in which a and a ~ are solutions o f the classical YangBaxter equation, the tensors A~a, Aa,, p A a~ + A a" p Aa~ - Aa'p are Poisson tensors. A

Answer. From the equality adx 9[a; a] - 0 for all x e g, we get

(Adg) |

9[a; a] -- [a; a];

for all g E G.

I f [ A a~ - A a , ,aAp " Z - A P a , ] - O , w e h a v e " [Aa~',Aa~]+[Aap, " , A a p , ] - 0 g i v e n p that, by 3.2, [Aa~; AaP,]- 0. The tensors [Aa~; Aa~] and [AaP,; Aa,] are respectively left-invariant and right-invariant. Moreover: [Aa~; A a ~ ] ( g ) - (Te)~g)|

a]


476

and [Aa,;P AaP,](g) -- -(TePg)|

a'].

By adding these two expressions, we get: (TePg) | (Adg[a; a] - [a'; a']) -- 0;

for all g E G.

Thus, Adg~3[a; a] - [a'; a'], and ultimately, [a; a] - [a'; a']. The converse now becomes obvious.

II

REFERENCES [1] V.G. Drinfeld, "Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations", Sov. Math. Dokl. 27 (1983) 68-71. [2] M.A. Semenov-Tian-Shansky, "What is a classical R-matrix?", Funct. Anal. Appl. 17(4) (1983) 259-272. [3] Y. Kosmann-Schwarzbach, Lie bialgebras, "Poisson Lie groups and dressing transformations", Preprint Centre de Math6matiques. Ecole Polytechnique, No. 97-22. Palaiseau (France), 1997. [4] A. Lichnerowicz, "Les vari6t6s de Poisson et leurs alg6bres de Lie associ6es", J. Diff. Geom. 18 (1983) 523. [5] C. Moreno, J.A. Pereira da Silva, "Algebres de Lie-Semenov et groupes de Lie-Poisson", in: Geometria, Ffsica-Matemdtica e outros Ensaios. Homenagem a Ant6nio Ribeiro Gomes. Departamento de Matemfitica Universidade de Coimbra, Portugal, 1998. [6] J.H. Lu, A. Weinstein, "Poisson Lie groups, dressing transformations and the Bruhat decomposition", J. Diff. Geom. 31 (1990) 501-526. [7] I. Vaisman, Lectures on the Geometry of Poisson Manifolds (Birkhtiuser-Verlag, Berlin, 1994). [8] A. Guichardet, Cohomologie des grupes topologiques et des algebres de Lie (Ferdinand-Nathan, Paris, 1980). [9] P. Liberman, Ch. Marie, Symplectic Geometry and Analytical Mechanics (Reidel, Dordrech, 1987). [10] Y. Choquet-Bruhat, C. DeWitt-Morette, M. Dillard-Bleick, Analysis, Manifolds and Physics (North-Holland, Amsterdam, 1987). [11] R. Abraham, J. Marsden, T. Ratiu, Manifolds, Tensor Analysis, and Applications (AddisonWesley, London, 1983). [12] W.T. Van Est, "Group cohomology and Lie algebra cohomology in Lie groups" I, II, Indagationes Mathematicae 15 (1953) 484-492, 493-504.

4. V O L U M E OF THE S P H E R E S n

A Supplement to Problem V.4 (pp. 240-243) Use the expression for the metric of S n in polar coordinates to compute its volume COn by induction on n when n is even and when n is odd. Show that in each case 27rn /2 09n_ 1 ---

F(n/2)

4. VOLUME OF THE SPHERE S n Answer.

The volume COn of

477

is given by the integral

Sn

with rn - (sin o n )

rn-1

n-1

dO n

"El - -

sin01 d01

Y2

Therefore 7r

- f(sin0)

COn - - I n C O n - 1 ,

dO.

0

Integration by parts gives if n > 1 n-1 n

~ / n - 2 n

- -

hence, since I1 - 7r and I2 - 2 I2p ~

I2p+l

(2p-1)

x(2p-3)...3xl

2p x ( 2 p - 2 ) .... 4 x 2 2p x ( 2 p - 2 ) . . . 4 x 2 --

(2p-1)

x(2p-3)...3x

1

and one finds using induction CO2p--

2(2zr) p (2p . 1)(2p . . . 3).

2zrp +1 3

X 1'

CO2p+I-

~

p!

Suppose that n - 1 -- 2p -t- 1. Then by the definition of the g a m m a function 2yen~ 2 COn-1 "-- ~ .

r(n/2)

Suppose that n - 1 - 2p. We have 27rP COep = (~ _ 1)(~ - 2 ) . . - 3

• 1"

By the property F ( x + 1) = x F ( x ) of the g a m m a function and F ( 1 ) -zr l/Z, we have 1 --

2

F(3/2) rrl/2

n '

9

~

~

2

F(n/2) 1 -- F ( n / 2 -

1)

from which we deduce that the given expression of COn-1 holds also when n is odd.

478


5. TEICHMULLER SPACES INTRODUCTION AND DEFINITIONS Teichmuller spaces

A diffeomorphism q9 of a riemannian manifold (M, g) is a confeomorphism if there exists on M a positive function f such that:

qg*g-- f g . It is known that the confeomorphisms of an n-dimensional (connected) riemannian manifold form a Lie group of dimension at most 89 (n + 1) (n + 2) if n > 2. It has been proved (cf. J. Ferrand 1995 and references therein) that the group of confeomorphisms of (M, g) is the isometry group of some conformal space (M, f g ) except if (M, g) is a metric sphere S n or an euclidean space E n. Analogous properties do not necessarily hold for pseudo riemannian metrics. The isometry group of a compact riemannian manifold is compact. The conformal group of S n is not compact. Two riemannian metrics g and g~ on a manifold M are called eonformally equivalent if there exists a diffeomorphism 99 homotopic to the identity and a positive function f such that on M ~o* g - f g ' .

Teichmuller space

genus

When n = 2 the space of conformally inequivalent metrics on a compact manifold M has the remarquable property to be isomorphic to a finite dimensional vector space. We will study this space. The Teichmuller space T ( M ) of a compact (connected, without boundary) smooth 2-dimensional manifold M is the space of conformally inequivalent smooth riemannian metrics which can be put on M. It has come to play an important role in geometry and physics, in particular in string theory. We shall give, in the case where M is oriented, the main steps of a proof that T ( M ) can be given the structure of a finite dimensional smooth manifold diffeomorphic to [R6 G - 6 with G > 1 the genus of M The genus of a 2-dimensional connected compact (with or without boundary) manifold M is the maximum number G of cuts one can inflict to M along a simple curve (continuous injection of a circle into the interior of M) so that the resulting manifold remains connected (in some definitions the genus is G + 1). It can be proved that two compact 2-dimensional manifolds without boundary are homeomorphic if and only if they have the same genus and are either both orientable or both non orientable. The sphere S 2 has genus zero, the toms T 2 has genus 1, an orientable "n holes surface" has genus n. The Betti number bl of an orientable connected compact

5. TEICHMULLER SPACES

479

2-dimensional manifold (without boundary) of genus G is bl = 2G. Its Euler-Poincar6 n u m b e r (cf. Problem 6 chapter Vbis "Euler P o i n c a r 6 . . . " ) is X = b0 - bl + b2 = 2 - 2G. Recall that if R is the scalar curvature of a 1 riemannian metric on M , then X -- 7Y fM R dr.

1. Denote by A/l [resp. M s ] the space of smooth [resp. in the Sobolev space Hs ] riemannian metrics on the 2-dimensional connected compact without boundary smooth manifold M. Denote by T) [resp. 79s] the space o f smooth [resp. Hs] strictly positive functions on M. a) Show that the relation 7E given in Ads x A/Is by (g', g) ~ 7-4.if there exists f ~ T)s such that g' = f g is an equivalence relation (p. 5) if s > 1. Denote by A d s / ~ . the set of these equivalent classes. b) Admit the fact (Kazdan and Warner) that the equation on (M, g) for the function oAgo-

~

e a -- R

has one and only one solution o- ~ Hs for each given if g ~ Hs, s > 2, and X < O. Give a bijection between space of Hs metrics with scalar curvature equal to - 1 M with G > 1. c) Show that A d s I is a C ~ submanifold of A/ls. Give a point g.

function R ~ H s _ 2 Ads/7)s and AAs 1, on a given oriented its tangent space at

Answer 1. a) Tr is defined if s > 1 because Hs is an algebra if s > gn - 1. It is reflexive, symmetric, and transitive. b) If M is 2-dimensional the scalar curvatures of two conformal metrics g and f g with f = e c~ are linked by (cf. p. 351) R (g) = A g o - --i--e c~R (e ~ g) hence if g is given we shall have R (e" g) = - 1 if and only if a satisfies the equation Ago-

--

e ~ = R(g)

this equation has one and only one solution a on the oriented manifold M if G > 1, since then X < O. If g and g~ are conformal, g~ -- e)'g, we have, by uniqueness of the solution of the above equation, R (e~ ~) -- - 1 if and only if a - a ' -t- )v, therefore to g and g' are associated the same metric g E . M s 1. The mapping A d s / ~ s -+ A d s 1 thus defined is bijective.


480

c) M s is an open set in the vector space of Hs symmetric 2-tensors, if s > 2. The subset . M s 1 is defined by the equation R (g) -- - 1 . Multiplication properties of Hs spaces show that g ~ R(g) is a differentiable mapping from M s into Hs. Its derivative at g is the linear mapping from covariant 2-tensors into scalars given by, since R = g i j R i j h w-~

DR(g).h --

- h ij Ri.j -Jr-gij

DRij.h,

hiJ _ gik g j l h k l .

We find using the expression of the Ricci tensor for a riemannian manifold of arbitrary dimension n

DRij .h = Vk (D F~j .h) - Vi (D

F.~k .h),

V covariant derivative in g,

DF~..h - l gkl (7.jhil d-- Vihi.j - 71hi.j) hence

DR(g).h - - - h i.j Ri.j - Ag tr h + V.V.h,

tr h - - g i . J h i j ,

V.V.h

-- 7i

V j h ij.

One shows that R(g) is a submersion when R(g) < 0, h e n c e J ~ s 1 a submanifold of Ads (cf. p. 239, 242), by proving that DR(g) is then a mapping 1 onto Hs-2. Indeed if hi.j - - n ' r g i j then

1

DR(g).h -- - - r R n

-

( 1l-)-

n

Agr = k

has a solution r 6 Hs for any k ~ Hs-2.

Note. When M is 2-dimensional we have R i j - - I g i j R, therefore for all h

1 DR (g).h - - - R tr h - A g tr h + 7. V .h. 2 2) Show that the group ~ ) s + l of Hs+l diffeomorphisms of M acts on A d s 1. Split the tangent space at g to 3/l~. 1 into a tangent space to the orbit of g through the action of 7) s+l and an L2 orthogonal complement. Show that this orthogonal complement is finite-dimensional.

Answer 2. The metrics g and r

have scalar curvatures related by

R(cp* g) - r R(g) =_ (R(g)) o ~o-1 . Therefore if one belongs t o . / ~ 1 SO does the other. The orbit Og of g by 79 is the space of metrics isometric to g. The generator of a one parameter group of isometries of g is a Killing vector field.

5. TEICHMULLER SPACES

481

The tangent space at g t o (.9g is therefore the set of symmetric 2-covariant tensors on M of the form L xg, L x the Lie derivative with respect to a vector field X on M. Every 2-covariant Ha. tensor field admits a unique L2 orthogonal decomposition h--h+Lxg

whereV.h--0

because the operators X ~ L x g and h ~-+ V.h are L2 adjoint, the first one has injective principal symbol (p. 397), and the operator X ~-~ V . L x g is elliptic with kernel {Y, L y g = 0}. Suppose that h ~ rg.A,4s 1 that is D R ( g ) . h - O. By the previous argument or direct verification, D R ( g ) . L x g = 0, hence DR(g).h = 0, which reduces to Ag tr h - tr h -- 0

which implies tr h - 0.

The L2 orthogonal complement of the tangent space to Og in TgAAL71 is therefore the space Z ~ T of transverse (V.h --0), traceless, symmetric covariant, Ha., 2 tensors on M. On a 2-dimensional manifold the operator V. on traceless symmetric 2tensors has injective symbol, i.e., is elliptic, its kernel is finite-dimensional. To determine its dimension we shall use results of analysis on complex manifolds. 3. a) Show that an almost complex (cf p. 331) structure on a 2-dimensional manifold M is always a complex structure. b) Show that the linear mapping J : TxM --+ TsM defined by the mixed t e n s o r --g$btg where g$ is the contravariant tensor associated to g and #g the volume 2-form (i.e. J{.! - - _ g i k ~k.j ) defines a complex structure on M, which is the same for two conformal metrics.

Answer 3. a) When n -- 2 the integrability condition for an almost complex structure to be complex is identically satisfied. b) J is the same for g and f g because ( f g ) $ - f - l g and ttt.g - - f l Z g , if /l--2.

J defines an almost complex (hence here complex) structure if j 2 _ - I d . This property is frame independent. Take an orthonormal frame in Tx M. Then g$i.j _ ~i.j and IJ, i.j -- ~1 ~ i.j12 ' hence j_(0

-1) 1

0

j2 '

_Id

482


4. Show that the space ~?~T of transverse traceless 2-covariant symmetric tensors on (M, g) is isomorphic to the space of 2-covariant tensors on (M, J) of type (0, 2) and holomorphic (cf. Chapter V, Problem 294). Answer 4. Take coordinates where gij - f6ij (always possible locally) then J has the expression given above, z - x 1 + ix 2 is a local coordinate for the complex structure (M, J). A 2-covariant tensor of type (0, 2) reads H -- F (z, z) dz | dz it is holomorphic if F does not depend on z. Let h 6 ZTza~.A straightforward computation, using also tr h - 0 (cf. Fisher and Tromba), gives, with gij - f3i.i (V.h)i-

f-I

O ij. Ox./h

In the chosen coordinates set h l l - - h 2 2 u, h i 2 V.h -- 0 reads as the Cauchy-Riemann equations

Ou OX 2

8v OX 1 '

Ou

8v

OX 1

OX 2 '

h21 - - v ,

then

conditions for u + iv to be an analytic function of z - x + iy. We have h -- u ( d x I t~) dx I - dx 2 | dx 2) -- u ( d x I (~) dx 2 + dx 2 (~) dx 1)

-- ~ e {(u -k- iv) dz | dz }. The (0, 2) analytic tensor H defined locally by H -- (u + iv)dz | dz extends to a holomorphic tensor defined on (M, J) by

H-

h + iJh,

where

(Jh)ij

-

Ji~hu,

J h is a symmetric 2-tensor, as can be checked in the chosen frame. CONCLUSION Riemann-Roch theorem

Teichmuller space Bochner

The R i e m a n n - R o c h t h e o r e m says that on a 2-dimensional compact orientable manifold the vector space of (0, 2) holomorphic 2-tensor is 6G - 6 dimensional, G > 1 the genus of M. Therefore Z'zTT is also 6G - 6 dimensional. The Teichmuiler space 7" is the set of equivalence classes .A/[sl/~)~+1, 7~ +l the subset of D s+l of diffeomorphisms homotopic to the identity. It has been proved by Bochner that 7)0 acts freely on a riemannian manifold (M, g) with negative definite Ricci curvature (i.e. ~0*g--g only if

6. YAMABE PROPERTY ON COMPACT MANIFOLDS

483

~0 is the identity map). This property is also a consequence of the theorem of Eells and Sampson saying that there is a harmonic mapping in each homotopy class of maps from (M, g) into (M ~, g~) if the Riemann curvature of g~ is non positive. A 2-dimensional manifold with R(g) -- - 1 has Ricci(g) = _ l g , negative definite. Hence Do acts freely on .Ads 1. It is possible to endow T with the structure of a C ~ manifold of dimension 6G - 6 (dimension of the L2 orthogonal complement of the tangent to the orbit Og) diffeomorphic t o [R6 G - 6 .

Eells and Sampson

REFERENCES Ferrand, J., Math. Ann. 1995. Fisher, A. and Tromba, A., Math. Ann. 267 (1984) 311-345. Earle, C. and Eells, J., J. Diff. Geom. 3 (1969) 19-43. Kazdan, J. and Warner, R., Ann. of Maths. 99 (1974) 14-47.


INTRODUCTION

It is an old result that every 2-dimensional compact riemannian manifold admits a conformal metric with constant scalar (gaussian) curvature. A riemannian manifold (M, g) is said to have the Yamabe p r o p e r t y if there exixts on M a metric g~ conformal to g with constant scalar curvature R ~. This property is important in geometry for the classification of compact manifolds, and in General Relativity in relation with the solution of the constraints (cf. Y. Choquet-Bmhat and J. York, 1979). The original proof of Yamabe contained a flaw and it has required a considerable effort from the mathematicians (Tmdinger, Aubin, Schoen) to obtain results valid in all cases and for every dimension. In the following (M, g) denotes a smooth, compact, n-dimensional riemannian manifold with n > 3. We shall see that all such manifolds possess the Yamabe property.

1. Show that (M, g) has the Yamabe property with gl _- q)4/(n-2) g

if and only if 99 satisfies the equation, with R the scalar curvature of g, k n A g q ) -- Rq) + Rtq9 (n+2)/(n-2) ---- O,

k n = 4(n - 1)/(n - 2).

(1)

Yamabe property

484

elliptic operators


A n s w e r 1. Straightforward calculation (cf. p. 351 and problem 6, chapter V, p. 247). 2. It can be proved (Palais, 1968) that a linear operator of order m on a compact smooth riemannian manifold (M, g), with smooth coefficients a~ m (the hypotheses on smoothness can be weakened), L =_ Y-~=0 a~ v k , which is elliptic (p. 397), is a Fredholm (p. 401) operator from the Sobolev space (p. 486) W s+mH p into WpS if s >_ 0, 1 < p < cx~, in particular it satisfies" I s o m o r p h i s m t h e o r e m for elliptic operators. If the linear elliptic operator L with smooth coefficients is injective, then it is an isomorphism from Wp +m onto Wp for any (integer) s > 0, 1 < p < ee (the inverse of L is the Volterra convolution (p. 471) with the Green function of L on M). Use this theorem and the L e r a y - S c h a u d e r degree theory (p. 563-565 and the application on p. 591) to prove that if R < 0 then (M, g) has the Yamabe property with R I < O. A n s w e r 2. Suppose R < 0 and let R ~ be some negative constant. We show the existence of a solution q) > 0 of (1) on (M, g) as follows. Consider the linear equation, with t 6 [0, 1] and c some number A g u -- u -- --c + t f (v),

f (v) =-- --v + c + R v - R ' v ( n + z ) / ( n - 2 ) .

(2t)

Suppose v > 0 and v 6 Wp, s > n / p , then (problem 3, chapter VI) f ( v ) Wp. Therefore (2t) has one and only one solution u 6 Wp +2, and there exists a constant C such that for all these v Ilu IIw/,;+2 _< f i l e + t f (v) ll wi~, . Consider a bounded subset s C Wps defined by the two inequalities

Ilvll wi; < K,

0 < a < v ( x ) < b, Vx ~ M .

s is an open subset of W~p if C O C W~, i.e. if s > n / p . Each mapping .T't 9I2 --+ W~ defined by v ~-+ u is a compact mapping (p. 563) because the injection W~ +2 --+ Wp is compact (extension of the Kondrakov theorem quoted in Problem 2, chapter VI, p. 383). The fixed points of ~ l are the functions u such that 1 + u is solution of (1); .To has one fixed point in S2, u - c, if Ilcllwi,; < K, i.e. c ( v o l M ) 1/p < K , and a < c < b. The existence of a solution of (1) (non necessarily unique) will follow from the Leray-Schauder theory if we prove that no mapping ~-t 9S2 - s U 0s t 6 [0, 1], has a fixed point on the boundary 0s of s we must prove that a solution of (2t) with v - u and satisfying Ilullw,,.,. _< K,

O < a < _v ( x ) < b ,_

x6M,


485

verifies the corresponding strict inequalities. A C 2 function u such that u (x) > a on M can attain its m i n i m u m a at a point x of M only if at that point Au > 0 ( m a x i m u m principle, p. 500). A fixed point u of Ut is C 2 since u 6 Wp +2 C C 2 if s > n / p ; it will verify Au < 0 when u -- a if

P (a) =_ Ra

Rta (n+2)/(n-2) < O.

-

If R < 0 on M and R ~ is a negative constant this inequality is satisfied as soon as a < (IR'linfM I R I ) ( n - 2 ) / 4 . W e shall have P(b) > 0, hence u cannot attain its upper value b, if b > (IRtl suPM IRI). We choose a > 0 and b > a satisfying these inequalities. Then 0 < a < u < b on M . We choose c b e t w e e n a and b. On the other hand, if u is a fixed point of f't in ~ we have f (u) ~ W ~ - - L p with an L p n o r m depending only on the data and the choice of b. The inequality recalled for elliptic equations shows that there exists a constant K ~ such that for all fixed points of the m a p p i n g s Ut satisfy

Ilullw __ O.

c) Show that if q < n(n - 2) there exists a subsequence of {qg(N)} which

converges to a function q)q in D satisfying a Yamabe equation with )~q = lZq.

d) Show that q)q > O.

Answer 4a. For any q, Jq is invariant by homothetic rescaling of qg. Answer 4b. The non existence of q)(N), given N, would lead to a contradiction, since/Zq is the infimum of Jq. We can choose q)(U) >_ 0 because/Zq


487

is also the infimum of Jq restricted to ~p > 0. Indeed if 99 6 H1 then also I~ol ~ H1 since Iris011- IX7~01, and Jq(~O) - Jq(l~01). Therefore l Z q - inf Jq(Igol) ~ 99eD

inf

qg~D, ~p>0

Jq(~O)> lZq. --

Answer 4c. S i n c e II(tg(N)I]L2q -- 1, q > 1, we have IIq)(N)IIL2 uniformly bounded when M is compact since (Holder inequality, p. 53) II~ollt2 _< II~ollt2qII111t2q,,

q1 + ~1 - 1 , II111L2qt -- (Vol(M , g))1/2q' .

-

On the other hand Jq (~P(N)) [/Zqo 1. c) Prove that if lz < 0 there exists a sequence {99qN}, which converges in HI to a strictly positive function q9 when qN converges to n / ( n -- 2). Conclude that (M, g) has then the Yamabe property. Hint. Show that the 99q, q ~ [q0, n / ( n - 2)[, are uniformly bounded. d) Show that a sufficient condition f o r lz < 0 is f M R dv < O. A n s w e r 6b. If ~ 6 D A C ~ it is straightforward to prove that Jq (~) is a

continuous function of q 6 [1, n / ( n - 2)] and to deduce from this property the continuity in q of/~q, infimum of Jq (cf. Aubin, 1982, p. 128). On the otherhand ifVol(M, g) - 1 andq0 < q we have ]l!/*]]tzq0 -< II 7~ II t2q by the Holder inequality, therefore

IJq(~)l ~ IJq0(~)[

and consequently

ICZql~ I/zqol ifq0 < q .

It results that/Zq, q 6 (1, n / ( n - 2)], is either everywhere positive, everywhere zero or everywhere negative. A n s w e r 6c. Suppose/z < 0. Then/Zq < 0, q 6 [1, n / ( n - 2)].

We have seen that ~pq, 1 _< q < n / ( n - 2), is continuous on M hence attains its maximum m q at a point Xq ~ M . At such a point we have (A99q)(Xq) < 0. We deduce then from the Yamabe equation satisfied by qgq, with #q < 0

mq5

-R(xq)

2q - 1, then by the Yamabe equation satisfied by ~0q and the Sobolev imbedding theorem there exist other constants C such that

II~OqIIw;/~2q_~,z

-< C,

]]Lr/(Zq-l)}

II~OqIIL r I - r. By induction one sees that q)q is uniformly in H2, and hence admits a subsequence which converges strongly in Hi to q) satisfying the Yamabe equation and that q) is smooth and strictly positive (cf. the proof in answer 3d). 8) It can be proved (cf. Aubin, 1982, p. 45) that on a riemannian manifold of dimension n > 3, with bounded curvature and injectivity radius (Problem 16, Chapter 6, p. 429) 3 > 0, the Sobolev imbedding theorem (problem 2, Chapter 6, p. 379) L p C H I , p -- 2 n / ( n - 2), can be written

(IlqglIL,,

2

~ (gn +~)(IlVqglIL2) 2 -+- A(E) (llcP IIL2) 2

where Kn is the best Sobolev constant obtained on OR" (cf. Problem 4, Chapter 6, p. 389), hence independent of (M, g), E is an arbitrary strictly positive number, and A (e) > 0 depends on (M, g) and ~.


491

8a) Show that the best Sobolev constant (Kn) -1 =

~o

IlVq:,ll 2L2

inf

'

Sobolev constant

cc~o(~ .) I1~oll2L2n/(n_2)

obtained on p. 389 can be written Kn -- 4(OOn)-2/n{n(n - 2)} -1,

~On volume of the sphere S n.

Answer 8a. We have found on p. 389, with the notation Sn - K n 1 (Kn) -1 --n(n-2)7rlP(n/2----~)} 2/n'F(n) A straightforward calculation gives (cf. an analogous one in Problem 4, Chapter V, Sphere S n, p. 240)

F(n)--2"-la'-l/2F Kn --

~ F "2

hence

4 { F((n + l)/2) } 2/n n(n - 2) 2~(n+l)/2

which, due to the value recalled in problem 4, chapter V of the volume of the sphere S n, is the indicated expression.

(_On

8b) Show that on S n, # - n (n - 1)co21n.

Hint: use the conformal invariance o f / , and the function on [Rn given by u (x) -- (1 q- r 2) ( 2 - n ) / 2

r --

]x l

o

Answer 8b. The given function u is such that (cf. Problem 4, Chapter VI, The b e s t . . . ) inf

~cg~

11V~~]122(~ ~)

") II~pll2L,,(R,)

=

Kn 1

=

[]Vu J122(~~)

Ilull 2L P ( R n)

On the other hand we know that S n minus one point is conforma| to [~n which has zero scalar curvature. We deduce therefore from the definition of the Yamabe invariant I ~ ( S n) - - k n K n

1 --

n(n

-

1)(Wn) 2/n.

492

SUPPLEMENTS AND ADDITIONALPROBLEMS

8c) Prove that f o r all manifolds (M, g) with bounded curvature and positive injectivity radius the Yamabe invariant Ix satisfies the inequality . 2/n

tx < n(n--1)COn

.

Hint: use the family of functions q)~,e -- rl~ uc~, where tie is a cut off function with support in a geodesic ball of radius e and u~ is a function given in normal coordinates (p. 326) by uc~ -- { c~2nt-r2 }(2-n)/2 O/

A n s w e r 8c (inspired from the survey article of Lee and Parker). The radial functions uc~ on ~n have the same Sobolev quotients as u for any positive c~ because they read, with r - c~p -

OrU~ = (2 -- n)ol(2-nl/2p(p 2 + 1) -n/2.

+ 1)}

The m a x i m u m of uc~ is attained for r = 0 and becomes more and more sharp as ot decreases: the compactly supported functions qgE,~, defined on a manifold via normal coordinates will have a Sobolev quotient which approximates K n I when E and ot are small enough. Indeed let ~0~,~ - r/~u~, where tie is a C 1 function on the manifold M with support in a geodesic ball Be of radius e, depending only on the radial normal coordinate r, with tie < 1 in Be, tie = 1 in BE~2 and IVrlE I < CE -I (choice possible cf. p. 529), while u~ is the function expressed in normal coordinates by the previous formula. We have then V~oE,~ = (8r~O~,~, 0 . . . . ,0). Recall that in normal coordinates there exists a constant C, depending on the curvature of (M, g), such that in BE dv -- f dr0,

dvo -- r n- 1 dr d o n -

l,

If - 11 CPN is a C* principal fiber bundle, because it can be defined by the family of trivializations. UA/- zr -1 (Ui) -+ (Ui, C*) by z w, (p, zi). The transition function Ui t~ Uj -+ C is p ~-+ z i / z j. The commutative group C* acts on the fiber F/-l (p) by z~-+ cz. c) The intersections of CS N with the open sets Ui are an open covering of CS N, which project onto the covering Ui, i - 1, . . . , N + 1, b y / 7 since in every f i b e r / 7 - l ( p ) of C N+I - {0} there is an element with norm 1. Zi The trivializations are now Ui I~s -~ (Ui, U (l)) by z w-> (p, ~ ) , since the z--~-~take their values in U ( l ) . The same holds for the transition functions Iz'l z~ IzYl which give therefore to CS N the structure of a U ( I ) bundle over z-J zi ' CPN. Let z i -- z i + iy i, with x i and yi real numbers, CS N is given by N+I i=1

N+I [zi12 ~ Z ( Ixi 12 + lyi 12) -- 1, i=1

therefore c a N can be identified with S 2N+I d) The homotopy groups zrk, k < 2N, of S 2/~+1 are zero (II, p. 43). Problem 9" Consider the 1 -form defined on c a N by the pull back through the embedding c a u ~ C u + l o f the l-form coz -- (zl'), z 6 C N+I. Show that this 1-form, also denoted ~o, is a connection 1-form on the U (1) principal bundle CS N --> C p N. A n s w e r 9" The analytic l-form on C S N given by coz at the point z ~ C S N is a connection l-form because (cf. p. 361)" 1. It takes its values in the Lie algebra i[R of U ( I ) because v E TzCS N implies (z, v) imaginary. 2. It is invariant under the action of U (1) (an abelian group) since COcz(CV) - (cz, cv) -- (z, v)

if I c l - 1.

9. THE BERRY AND AHARONOV-ANANDAN PHASES

509

3. If u ~ TzCS N is tangent to the fiber Vz, there is a curve t w-~ C(t) e i~~ such that dC dt

-= igb(O)z t=0

therefore COz(U) -- (z, u) -- (z, igb(O)z) -- igb(O)(z, z) -- igb(O) ~ i~.

igb(0) is the Lie algebra element t~ canonically associated with the vertical vector u. The horizontal subspace of co at z is orthogonal to z.

~ CS N

is defined by

OOz(UH) - - O,

i.e.,

Problem 10" Let V'[0, T] w-~ C PN by t w-> p(t) be a closed loop in C PN determined by the projection on C PN of a cyclic mapping t w-> go(t) ~ CS u. a) Determine the element o f the holonomy group o f the connection co on the U (1) bundle CS u --+ C PN which is determined by this loop. b) Show that it coincides with the A - A phase o f the mapping 7/. Answer 10" a) The parallel transport of a point q ~ ~ S N along y is defined by the solution of the differential equation

dq dv dt -- O-q--dT'

q (0) - ~ (0),

(16)

where aq is the linear mapping Tp(CPN) ---->Tq(CSN) determined by the connection 1-form. In answer 3b, we have rewritten (16) in terms of the pull back of the connection 1-form, and obtained the corresponding equation in a trivia|ization. Suppose the loop V lies in a coordinate patch of CPN, for instance Ux+l, that is the mapping ~ defining that loop takes its values in U N + I [-/-I(UN+I) , i.e., ~N+l(t) 7~0, t ~ [0, T]. Let ~b be the trivialization of CS u over UN+I given by 4~" UAN+I --+ zN+I

(UN+I, U(1)) by z w-~ (p, iziU+~). Consider the canonical section s ' p w+ (p, 1). The mapping az, z ~ CS N, is given by TpCPN --+ TzCS u by u (u, -(s*rO)p(U)). Equation (1) reads then (cf. 8b), with q~(z) - (p, e k~

i d--~- -t- (s*co)•

~

- O.

510


Therefore we have by integration a curve C ' [ t l , t2] --+ C(t) 6 UN+ 1 r (t2) - r

+ i

f s,~ C

Since s is a diffeomorphism between UN+ 1 and S(UN+I) C C S N the above formula can be written r

- r

+ i

f

(17)

6O.

t/

s(c)

b) If C is the closed loop y in C PN defined by the projection of the orcyclic mapping 7r, then s ( y ) is a closed loop in CS u defined by a mapping ]r with the same projection and such that ~r u + l / ] ~ l N+l - 1, hence -

-

(,))-'

(,)1.

The formula (17) reads therefore

f(d6 ) T

r

-- r

+i

-d-t- ~r(t)

,

0 T

r

- r

+i

f(d

i --~ 0 ( t )

0

)

dt - ~.(T) + ,k(0)

if 7r is ot cyclic we have k(T) - k(O) -- -or. The element ei'p(0)e -i~~ of U (1) (i.e., of the holonomy group) associated to the parallel transport along 9/is therefore the number/3 associated to 7r defined in problem 8. If the closed loop y is not included in one coordinate patch of C PN, we split it into arcs included into such patches and use the transition function to obtain the formula (17) for the element of the holonomy group associated to y. Problem 1 l" Indicate how the results of paragraphs 9 and l0 extend to the case where 7-[ is a separable Hilbert space.

9. THE BERRY AND AHARONOV-ANANDAN PHASES

511

Answer 11" A separable Hilbert space is isomorphic to the space l 2 of sequences of complex numbers {zi}, i 6 N, such t h a t E i ~ l ]zil 2 is bounded. The scalar product of z l, Z2 E ~ is (Zl IZ2) - - E i =. Ic x ~~IZ2 i i The sphere S in the subspace of 7-/of sequences {z i} such t h a t ZiC~=l Izil 2 - 1. The previous reasonings apply, with a covering of 7-[ being defined by the open sets Z i =~O,i E N . Problem 12: Let M be a smooth manifold as in the beginning of this problem, but we are only given for each x E M a 1-dimensional equivalence class p ( x ) of vectors in S, namely we are given a smooth mapping Y : M ~ 79. Denote by B := Y*S the bundle obtained by pulling back S by Y (p. 336). (B is a U ( 1 ) ) b u n d l e with base M. Y is its "classifying map".) Show that the Berry-Simon connection on B is the pull back of the A.A. connection on S. Answer 12: L e t / 7 : S --+ 79 be the projection of S onto its base 79. The bundle B is obtained by identification with the subset of 79 {17-1(y(x)), x E M}. Its base is M, its fiber at x is I T - l ( y ( x ) ) , isomorphic to U(1), its transition functions are obtained from the transition functions of S --+ 79. An equivalent definition is that the right action of U (1) on the fibers is the same in S --+ 79 and B --+ M. If Ui is an open set of 7-9 over which S is trivial, then Ui - - y - 1 ( U i ) is an open set of M over which B is trivial. The mapping ~'/-/M 1(ui) --+ 17 -1 (Ui) is given in these trivializations by Y - (Y, Id), its derivatives is (Y', Id). A section of B an open set U i C M is determined by a mapping s : x ~-+ X(i)(x) E 1 7 - 1 ( Y ( x ) ) C S. Straightforward computations show that the pull back by s of the pull back by Y of the A.A. form on S is the 1-form on u i given by (a(i))x(V) -- (s*~'* CO)x(v) -- W(~os)(x) (~"s'v) -- (dX(i)(x)vIX(i ) (x)). Therefore

A(i)

is the representative of the B.S. 1-form in ui.

REFERENCES [ 1] T. Kato, "On the adiabatic theorem of quantum Mechanics", J. Phys. Soc. Japan 5 (1950) 435-439. [2] M.V. Berry, "Quantal phase factors accompanying adiabatic changes", Proc. Roy. Soc. London, Ser A 392 (1984) 45. [3] B. Simon, "Holonomy, the quantum adiabatic theorem, and Berry's phase", Phys. Rev. Lett. 24 (1983) 2167-2170. [4] Y. Aharonov and J. Anandan, "Phase change during a cyclic quantum evolution", Phys. Rev. Lett. 58 (1987) 1593-1596.

Berry-Simon connection


512

[5] A. Bohm, L.J. Boya, A. Mostafazadeh and J. Rudolph, "Classification theorem for principal fibre bundles, Berry's phase, and exact cyclic evolution", J. Geometry and Phys. 12 (1993) 13-28; A. Mostafazadeh and A. Bohm, "Topological aspects of the non-adiabatic Berry phase", J. Phys. A: Math. Gen. 26 (1993) 5473-5479. [6] For a review article and applications to chemistry see: C.A. Mead, "The geometric phase in molecular systems", Rev. Mod. Phys. 64 (1992) 51-87.

10. A DENSITY THEOREM A Suplement to Problem VI.6 (pp. 393-396) Show that 79(~ n) =- C ~ ( ~ n) is dense in Hs,~([Rn). Hs.S(~ n)

Answer: We use the notations of Vol. II, problem VI 6, p. 393. We consider a truncating sequence "it"u ( g o l . I, p. 434) defined by the composition of a function rl of one variable and a sequence YN of functions on urn; namely we set Z"N - - Z"1 o Y N ,

where rl ~ C ~ ( ~ ) , i.e. rl is a C ~ function on [R with compact support. We choose it such that rl(y)-I

ify2.

We take YN

--

N -I logo',

o'(x) --= (1 + Ix12) 1/2,

x ~ urn.

Z"N is then in C ~ ( ~ n) with rU(X) -- 1 if log~r(x) < N, r U ( X ) - 0 if logcr(x) > 2N. The family of functions rN is uniformly bounded on ~n. We have OrN

0rl

Ox i

Oy Ox i

1 X i 0r 1 Nor a Oy

OyN

i~rN Therefore the family cr Ii~x--7] is uniformly bounded on urn. An analogous proof shows that each cr ~ ID ~ rNI is uniformly bounded on [Rn. Let f 6 Hs,~. Then r u f -- fN ~ Hs,s and has compact support. We have

I I / - fu IIH,,,- IIs('-

Ef S-2+

o,o -f 0 S-2+

0 S2_

The result follows from the multiplication law of C ~ functions and distributions. A n s w e r 4b: Let r 6 791, then

(V~z, r ) - - ( ~ , V . r ) - - f

V~r~co. S

Choose local coordinates such that f ( x ) =_ x ~ Then X 1, . . . , X n are local coordinates on ~' and, if r has compact support in the chart,

f v~r~co-f O~(pr~)dxl...dx'-f Oo(pr~ Z7

xO=0

dx n .

x0=0

In the chosen coordinates we have therefore V i 6 Z" - - 0 and we set 8'Z - --V06~. Hence in these coordinates VaE - ~6~. This tensorial expression is valid in all coordinates, with 8 IZ the distribution of order 1 defined above in particular coordinates. 5. A tensor T is said to be regularly C k d i s c o n t i n u o u s across ,V if 1) T is C ~ in $2+ and S2_. 2) T, VT, . . . , V ~T converge uniformly to tensors denoted T+, ( V T ) + , . . . . (V~T)+ when x ~ $2+ tends to a point of Z7 [resp. T_, ( V T ) _ , . . . . (V~T)_ when x ~ S2_ tends to Z]. To T, . . . , V ~ T, locally integrable tensors in I2+ U S2_ are associated tensor distributions in $2. On the other hand the following ordinary tensors are defined and continuous on ZT: [T] =_ T+ - T_ . . . . .

[V ~ T] =-- (V k T)+ - (V k T ) _ .

and such that [T]al...a p --- [Totl...ap], with an analogous property for the derivatives. a) Define [T]3z: . . . . . [V~T]SE as g e n e r a l i z e d tensors o f order 0 (measures) with s u p p o r t Z .

discontinuous tensor

518

sUPPLEMENTS AND ADDITIONAL PROBLEMS

b) S h o w there exists a t e n s o r distribution t o f o r d e r 1 with s u p p o r t Z such that -[VT]

z

--

et.

c) S h o w there exists a t e n s o r distribution v o f o r d e r 2 such that V(s | t) + s | Vt + s | 1 6 3| v_= [VVT]Sz. A n s w e r 5a: These measures are defined by

([V ~ T]8,r,, q g ) - i [Vk T].~o~o,

q9 e Dp+~ if T is p-covariant.

E

A n s w e r 5b: By the definitions the components of V([T]6x) are such that

E

Hence in adapted coordinates, using integration by parts a;

=

from which results V i ( [ T ] 6 z ) u i . . . u F = [Vi Tui...u,,]6 z

and the announced tensorial relation where t is the tensor with components in adapted coordinates such that (t~,~...~p, r

-

f

{[T~,...~p]Oo(p~o) - [VoT~,...~,,,]p~o} dx I ... d x " .

x0--0

A n s w e r 5c" Analogous to the previous one. R e m a r k : If T is continuous across Z and VT regularly discontinuous then Hadamard relation

(Hadamard relation)

[VT]6x -- s | t, where t is a tensor valued measure with support Z' and components in adapted coordinates given by t~ ~...~ p = - [Vo T~ ~.. .~ ,, ] S z .

shock equations

6. The shock equations in mechanics can be obtained by looking for dis-

11. TENSOR DISTRIBUTIONS ON SUBMANIFOLDS

519

tribution solutions which are regularly discontinuous tensors across a 3 dimensional submanifold 27 of the 4 dimensional space time. Consider for example the case of a perfect relativistic fluid. The dynamical equations are conservation laws Vc~T at; -- 0 for the stress energy tensor

T ~ =- (r + p)u~ul~ + p g ~

stress energy tensor

r, p specific energy density and pressure, u unit 4-velocity, g space time metric. We say that the fluid undergoes a shock across 27 if T and V T are regularly discontinuous across I7.

Determine the equations satisfied by the stress energy tensor of a fluid which undergoes a shock across ~ .

shock

Answer 6: If T and VT are regularly discontinuous across Z: the derivative of the tensor distribution T is given by: =

+

where Vc,{T ~ } denotes the locally integrable function in 1"2 equal to V~ T ~/~ in S2+ and in S2_. The sum of a locally integrable tensor and a measure with support I7 is zero if and only if both are zero. The equation V~ T c'/~ = 0 is therefore equivalent to the equations V~{T ~/~} and

g~[T~t~]--O,

they are the R a n k i n e Hugoniot equations.

7. Study gravitational shocks as a consequence of Einstein equations in vacuum (cf Problem 11, chapter 6), supposing that the physical metric g is continuous while its partial derivatives are C 2 regularly discontinuous across Z . Answer 7: In order to have covariant formulas we endow the space time V with a given C ~ metric e (which can be the euclidean metric if permitted by the topology of V) and denote by 0 the covariant differentiation in the metric e. The equations satisfied in a generalized sense are Einstein equations in vacuo which read, with H depending smoothly on its arguments 2 R ~ =_ 1 i g ~ O~g~ u + OkF~ + Ou Fx + H~, (g, Og) --O,

F ~ = gC,~(V~/~ - E~r

E~/~ Christoffel symbols of e.

Rankine Hugoniot equations gravitational shocks


520

( F ~ -- 0 are the generalized harmonicity conditions.) The results of 4 show on, the one hand, that there exists a symmetric space time 2-tensor y defined and continuous on Z' such that

[O~g)~u] = g.I~V)~u; on the other hand they give the formulae 2

2

Ooq3g)~u -- Oo4~{g)~u } + g.ot,e..l~y x u 6 z , 0zF u = 0 x { F u} + e)~[Fu]6z, + e)~[Fu]-I-

Rx. -- {R)~.)--I-

eu[Fxl) x.

The sum of a locally integrable function on V and a measure with support Z' is zero if and only if both are zero. The measure with support Z' in the above equation is zero if and only if the coefficient of 6z is zero. We consider two cases: 1) o on Z , then gxu - (ex[Fu] + gu[Fxl)( 89176 -1 such a discontinuity in Og across Z' is not considered as significant because it can be destroyed by a C ~ regularly C l discontinuous across Z', change of coordinates. 2) gC~/~g~g/~ = 0, then ~ is a null vector in the physical metric. In this case the Einstein equations reduce to {Rxu}-0 Rankine Hugoniot equations

and

[Fx]--0.

The second set of conditions, which are independent of the choice of the background metric e, are the gravitational Rankine Hugoniot equations. They read, using the expressions of F and [0g], setting e~ = g~t,gr e ~0c~/~ -- 0,

with

O~l~ - - Y~l~ -

I

got~ gZU Y x # .

An extension of the given method permits a study of the propagation of gravitational shocks (see Lichnerowicz, 1973) analogous to those obtained for high frequency waves (Choquet-Bruhat, 1969). REFERENCES Choquet(Four~s)-Bruhat, Y., "Distributions sur les multiplicit6s", C.R.A.S. 236 (1953). De Rham, G., Varidtds Differentiables (Hermann, 1955). Lichnerowicz, A., "Propagateurs et commutateurs en relativit6 g6n6rale", Publications de I'I.H.E.S. n ~ 10 (1961). Pichon, G., Etude des fluides relativistes visqueux et chargds, Th~se (Paris, 1964). Choquet-Bruhat, Y., "Espaces temps g6n6raux, chocs gravitationnels", Ann. Inst. Poincar6 8 (4) (1968) 327-338. Choquet-Bruhat, Y., "Construction de solutions radiatives approach6es des 6quations d'Einstein", Comm. Math. Phys. 12 (1969) 16-35.

12. DISCRETE BOLTZMANN EQUATION

521

Lichnerowicz, A., "Ondes de choc gravitationnelles et 61ectromagnEtiques", Comptes Rend. Acad. Sci. 276 (1973) 1385. Choquet-Bruhat, Y., "Distributions on manifolds, applications to shocks and discrete models", in: GeneralizedFunctionsand Applications,Ed. R.S. Patak (Plenum Press, 1993).

12. DISCRETE BOLTZMANN EQUATION We recall that the usual B o l t z m a n n equation on a pseudo riemannian man-

ifold (V, g) reads

Boltzmann equation

E f =_ p C~ Of pa Of Ox----S + Op~ = Z ( f ) where x a, pC~ are local coordinates in the tangent bundle T(V) to V, representing position and 4-momentum of particles; f is the distribution function of particles on T(V); (p, P) is the tangent vector to the trajectories of particles in the phase space T (V) between collisions: the operator 12 is the Lie derivative along such a free trajectory, E f = 0 expresses the conservation of f in the absence of collisions. The term Z ( f ) expresses the loss and gain of particles with momentum p undergoing collisions at the point x. If no other field is present the particles follow between collisions geodesics of the metric g. In the general case we have:

Pa =_-F~.p,~p. + Qa where Q represents the non gravitational force fields. If Q results from an electromagnetic field represented by an exterior 2-form F, then Q~ = eF~ p~,

e the charge of the particles.

In a collisionless model Z ( f ) = O, the equation is called the Vlasov equation. Suppose that the momenta of the particles can take only a finite number of values, depending on the point of V where the particle is located, take a distribution generalized function of the form

N f (x, p) = ~

aI(X)6Bt(x)(P).

I=l

Where ai is a smooth function and BI a smooth vector field on V while 6B~(x)(P) is the measure on T(V) defined by (Vol. I. p. 430)

(~BI(x)(P), q)(X, p ) ) - f (tg(X, BI(X)) r], v

q) E "D(T(V)),

Vlasov equation


522

with in local coordinates r / = p dx ~ . . . dx n, with p -- [det g[1/2 w h e n V is endowed with a pseudo-riemannian metric g. Note that the volume element of T ( V ) is then p 2 d x O . . , dx ndpo ... dpn. 1) Compute the partial derivatives of the generalized functions (~BI on

T(V). ~Qa

2) Suppose that Q is such that ~

- 0 (show that such is the case if Q is the Lorentz force of an electromagnetic field). Write the system of partial differential equations satisfied by a l and B I when f satisfies the Vlasov equation in a generalized sense: the system is called a discrete Vlasov equation. 3) Propose a model for a discrete Boltzmann equation.

Answer 1" Let q9 be a C cr function with compact support on T(V). The partial derivatives of 6B~ are such that (we suppress the explicit dependence of 6B~ on x and p to shorten the writing, but keep it in ~0 to make the proof more transparent)

0

-- --(SBt, P

-2 0(,02~ (x' P)) \.

/

Op~

Hence, since p does not depend on p"

( OP~ 6B,, (x , p))__ f

V

We use the property p-1 0

{ a (x, p) OP~

o,

~ p - F~x, together with the identity

ax0 (x, B1(x))

P)}

p=Bt(x)

OXa

-~p-[3

p=BI

the definition of 6B~ and the compactness of the support of q9 to obtain

Ox~

Ox~ Opl~"

12. DISCRETE BOLTZMANN EQUATION

523

Answer 2: Straightforward computation leads then to the following formula

0 pa ~ 6 e i

=

(

+ (Qa

V B; +

3p ~

)

pX ptZ 3p ~

If Q is an electromagnetic force then QC~ _ eFffp~ , hence ;~ QO~= 0. The Op~ Vlasov equation is satisfied in a generalized sense if and only if all independent measures and doublets on the left hand side vanish. This fact leads to the equations, written when ~ - 0 " B~I V/3B~ - {QU}p=BI - - 0 , V 3(aIB~l) -- O,

I -- 1 . . . . , N,

I -- 1 , . . . , U.

3) A natural model for a discrete Boltzmann equation is obtained by replacing in the usual Boltzmann equation the collision term I ( f ) by a sum of measures" N

Z ( f ) =- ~

ZI 8B,

I=1

where ZI represents the balance of loss and gain of particles with BI momentum at the point x. For binary collisions ZI is of the form It=

~

{ o K L a K a L - - O ' KI JL a , a j }

J,K,L

where the a KL are given positive functions on V. The equations are B~IV3B,ot - {Qa

V/~ ( a / B ~ ) - - Z I ,

} p=BI

-- O,

I -- 1,

" " "

, N,

I - - 1 . . . . . U.

4a) Consider the metric and the external force Q as given. Show that the system of equations is a nonlinear, causal, Leray hyperbolic first order system, when the B I are time like or null. b) Study the case where the metric is considered as a field variable satisfying Einstein equations whose source is the stress energy tensor generated by the distribution generalized function f , while Q is the Lorentz force

Leray hyperbolic first order

524


corresponding to an electromagnetic field generated by f when particles have an electric charge e.

Leray weights

Answer 4" a) Give to the equations (1), (2) and to the unknown BI, ai the Leray weights (problem 7, Chapter V, p. 255)

m(1)-l,

Leray criteria

m(2)-0

a l Blot B I [~, I

causal system Gevrey class

n(Bi)--2, n(ai)--l.

The characteristic matrix is then a diagonal matrix with elements the hyperbolic polynomials (of degree 1) B~ Xc~. If the vectors B I are time like or null, the cones (half spaces) B~ X~ < 0 have a non empty intersection, which contains the interior of the future light cone. The Leray criteria for a strictly and causal hyperbolic system are satisfied. b) The stress energy tensor and electric current generated by f are respectively the tensor and the vector on V given by Totl~ -- Z

Leray weights

and

Jot - - ~

e a l Blot. l

Consider the Einstein equations in harmonic gauge (problem 11, chapter VI, p. 405) and Maxwell equations in Lorentz gauge (p. 337). There are no Leray weights which make the characteristic matrix diagonal with elements hyperbolic polynomials, except when the space-time is 2-dimensional. The characteristic determinants is a product of terms B~ Xot and got/~XotXr the system is causal and non strictly hyperbolic in the sense of Leray-Ohya, the Cauchy problem for such systems is well posed in a Gevrey class (cf. Leray-Ohya, 1967). REFERENCES Pichon, G. et Huyn-Servet, M., "Distribution de Boltzmann discr6te sur un fibr6 tangent", J. Math. pures et appliqu6es (1992). Choquet-Bruhat, Y. and Pi,chon, G., "Plasmas with discrete velocities", in: Discrete Models in Fluid Dynamics, Ed. A. Alves (World Scientific, 1991). Leray, J. et Ohya, Y., "Syst6mes non lin6aires hyperboliques non stricts", Math. Ann. (70) (1967) 167205.

INDEX A A-valued connections, 122 A-valued covariant vectors, 122 A-valued metrics, 122, 125 A-valued spinors, 122 A-valued tangent vectors, 124 abelian representations of nonabelian groups, 110 abelianization, 111 abstract simplex, 138 abstract simplicial complex, 138 adiabatic evolution, 501,504 adjoint, 47 adjoint representation, 365 Adler-Kostant-Symes theorem, 219, 229-231 Aharonov-Anandan phase, 500, 506 almost complex, 294 almost K~ihler, 299 analytic diffeomorphism, 241 anomalies, 349, 361 antihomomorphisms of a Lie algebra, 102 Arnold's theorem, 219 associated polyhedron, 128 Atiyah-Singer index theorem, 321 (auto-) B~cklund transformation, 179

bundle homomorphism, 335 bundle reduction, 310 C Calabi-Yau spaces, 294 Campbell-Hausdorff expansion formula, 354 canonical commutation relations, 167 canonical Poisson structure, 217 canonical spin structure, 154 Cartan-Killing form, 93 Casimir functions, 185, 186, 213 Cauchy problem, 275 causal system, 524 Cayley inner product, 265 Cayley norm, 265 Cayley numbers, 264 (~ech homology, 138 central extension of a group, 166 central extension of a Lie algebra, 354 characteristic classes, 321,339 characteristic matrix, 254 characteristic space, 192 characters, 114 charge conjugation, 27, 30 Chern-Simons classes, 340, 342 Chern-Simons form, 346 classifying maps, 337 Clifford algebra as a coset of the tensor algebra, 14 Clifford algebras, 6 Clifford group, 18, 22 closed operator, 48 coadjoint orbits, 218 coadjoint orbits in G, 207 coadjoint representation, 365 coboundary, 131, 159 coboundary operator, 159, 168,364 cochain, 158 cochains on G, 168 cocycle, 132, 159, 187 cocycle condition, 161, 162 cocycles on the Lie algebra of a gauge group, 349 coframe, 289 cohomology, 127, 132 cohomology group of G with values in an abelian group, 159 cohomology of groups, 158 cohomology of Lie algebras, 167 commutant, 114 commutator subgroup, 110 compact spaces, 39 compactness in weak star topology, 40 completely integrable, 192, 288 completely integrable systems, 219

B

B-valued covariant vector, 125 B-valued metric g, 125 back face, 132, 134 B~icklund transformations, 178, 183 B~icklund transformations for evolution equations, 181 ball of Wst' in W~(~_I,398 Berezin integral, 57 Berezin integration, 57 Berezinian, 4 Berry phase, 500, 501,504 Berry-Simon connection, 511 Betti numbers, 321,324-326 Bianchi identity, 340 Bianchi-Lie transformation, 183 Bochner, 482 Bockstein map, 149 body, 2 Boltzman equation, 521 Bonnet equation, 183 bound states, 425 boundary, 129, 138 boundary of a chain, 129 boundary of a simplex, 128 broken symmetry, 310 BRST operator, 363 bundle classification, 335 525

526 completeness, 398 complex, 128 complex holomorphic manifold, 507 complex manifold, 295 composition of maps, 281 composition theorem, 387, 431 cone property, 379 conformal system for Einstein equations, 249 conformal transformation, 244 conformal transformation of nonlinear wave equations, 256 conjugate ghost field operator, 370 connections on a Spin bundle, 156 conserved current, 69 constant cocycle, 133 continuous spectrum, 48 contractible maps, 44 contractible to a point, 337 cosimplex cochain, 131 cospinor, 36, 135 cotangent vector, 124 covariant codifferential, 253 covariant derivative, 515 covariant distribution-valued tensor, 514 C-. P N , 507 critical point, 278 cup product, 132 current algebra, 350 curvature integral, 332 C.W. complexes, 158 cycles, 129 cyclic, 506 D

de Rham theorem, 321 degree of a Pin group element, 146 density theorem, 430 DeWitt algebra, 1, 119 differentiable submanifolds, 91 differential, 275 differential equations as exterior differential system, 173 dimension of a complex, 128 dimension of a simplex, 127 dimension of chains, 128 Dirac adjoint, 36 Dirac measure, 516 Dirac operator, 244 Dirac representation, xii, 31 direct products, 95 discontinuous tensor, 517 distribution-valued tensors, 514 distribution with laplacian in L2(Rn), 399 Dolbeault differential, 298 Dolbeault operators, 297 Drinfeld property, 445,460 Drinfeld compatibility condition, 451 dual Lie-bialgebras, 452 dual of a Lie algebra, 187 dual pair, 213 E edge, 127

INDEX Eells and Sampson, 483 eigenspace, 48 eigenvalue, 48 Einstein cylinder, 244 Einstein equations, 249, 31 I, 408 Einstein equations with sources, 413 Einstein vacuum, 311 Einstein-Yang-Mills equations, 286 elementary solution of the wave equation, 373 elliptic operator, 484 embedding, 92, 283 energy, 311 energy density, 276 energy density of the Yang-Mills field, 89 energy functional, 277 energy of a field, 72 energy with respect to a timelike vector field, 83 equivalent bundles, 334 equivalent spin structures, 151 equivariant momentum mapping, 214 essential spectrum, 49 euclidean at infinity, 430 Euler class, 321,330, 495 Euler equations, 76 Euler-Lagrange equations for a Dirac particle, 80 Euler numbers, 321,324 Euler-Poincar6 characteristic, 321 Euler-Poincar6 number, 329 even matrix, 3 even vector, 124 exact Poisson-Lie group, 464 exact sequence, 163 extension of a group, 164 extension of SO, 136 exterio differential system, 171,173 extremal, 275 F

face, 127, 138 field of hyperplanes, 191 Fierz identity, 15 first fundamental form, 238 first integral, 220 flow of a vector field, 219 foliation, 192 foliation of Poisson manifolds by symplectic leaves, 191 Fredholm, 49 front face, 132, 134 full pair, 213 fundamental bundle, 121 G G-module, 167 G-module with zero operators, 169 Galileo group, 96 Galileo transformation, 97 gamma matrices, 135

INDEX gamma matrices for C (n, m), 7 gamma matrices of O(n, m), 7 y+(n,m), 21 y_(n, m), 21 (Gateaux) derivative, 51,354 gauge covariant derivative, 248,293 gauge group, 349 gauge transformation, 305 Gauss-Bonnet-Chern-Avez theorem, 321,330 Gauss-Bonnet theorem, 330, 495 Gauss map, 326, 330 Gaussian curvature, 321 Gaussian integrals, 57, 61 generalized first integral, 220 genus, 478 geodesics, 275,279 geometric simplex, 127 geometrical phase, 501 (geometrical simplicial) complex, 128 Gevrey class, 524 ghost field operator, 370 ghosts, 363 global function group, 211 Goldstone bosons, 320 graded, 124 graded affine bundles, 122 graded algebra, 1 graded bundles, 118,122 graded chart, 121 graded commutative, 1 graded connections, 125 graded function, 121 graded manifold, 121 graded matrix, 3 graded spinor fields, 125 graded tangent, 124 graded tensor product, 6 graded vector bundles, 122 grading of a Clifford algebra, 12 Grassmann algebra, 119 Grassmann manifolds, 108, 109 gravitational shocks, 519 H Hadamard relation, 518 half-spinor representation, 29 hamiltonian function, 184 hamiltonian vector fields, 164, 184, 186, 188,212 hamiltonian vector fields and automorphisms of a Poisson manifold, 190 Hardy-Littlewood-S obolev inequality, 391 harmonic, 276, 277,285 harmonic gauges, 275 harmonic maps, 274, 281,406 harmonicity conditions, 405 heat equation, 175 Heisenberg group, 163-165

527 helicity, 27 helicity operator, 27, 32 hermitian metric, 299 hermitian scalar product, 299 Higgs mechanism, 310, 316 Higgs operators, 244 Hodge theorem, 321 holonomy, 504 holonomy group, 301,502 homogeneous cochain, 160 homogeneous spaces, 103, 104, 263 homogeneous symplectic spaces, 218 homology, 321 homology group, 129 homomorphism central, 141 homomorphisms of a Lie algebra, 102 homothetic space-time, 262 homotopic, 41 homotopy class, 275 homotopy groups, 41, 43 homotopy of topological groups, 46 Hopf fibering, 307 Hs.a (IR"), 512 H,. a(IRn) spaces, 396 Ha.(S n) spaces, 396 hyperbolic metric, 87 hyperdifferentiable, 52 I

ideal, 14 mdecomposable, 112 index of a Fredholm operator, 49 index of a vector field, 322 mequivalent spin structures, 150 infinite dimensional Lie algebras, 364 infinitesimal invariance, 70 mjectivity radius of a riemannian manifold, 429 integral manifold, 171 integral manifold of a field of hyperplanes, 192 invariance group, 175 invariance of the equation of motion, 79 invariant connection, 270 invariant geometries on the squashed seven spheres, 263 mvariant lagrangian, 71 mvariant metric, 288 mvariant polynomial, 343 mvolutive automorphism, 103 lsogroup, 175 lsotropy group of, 106 J

Jacobi manifold, 218 Jacobi operator, 79 Jacobi structure, 194 K

K-graded tangent bundle, 122 Kac-Moody algebras, 177, 232, 357 Kadomtsev-Petviashvili equation, 232 K~ihler form, 299 K~J.hler metric, 299

528 K~alerian manifolds, 235,294, 299 Kaluza-Klein theories, 286 Killing vector fields, 239 Kirillov-Kostant-Souriau theorem, 218 Kirillov local Lie algebras, 193 Kirillov orbits, 194 Kondrakov theorem, 383,384, 431,487 Kostant graded bundle, 118, 119, 122 L lagrangian for a free Dirac particle, 80 lagrangian for scalar functions, 76 lagrangian of a nonscalar field, 85 lagrangian of the Yang-Mills field, 85 Laplacian, 496 lapse function, 407 Lax equation, 224, 226 Lax form, 224 Lax pair, 226 left ideal, 14 Leray criteria, 524 Leray form, 516 Leray hyperbolic first order system, 523 Leray indices, 255 Leray theory of hyperbolic systems, 405, 407 Leray weights, 255,524 Lie algebra, infinite dimensions, 364 Lie algebra of observables, 196 Lie algebra of Pin(n, m), 37 Lie algebra of SO(n), 116 Lie algebra of Spin(n, m), 37 Lie algebras of linear groups, 115 Lie bialgebras, 450 Lie derivative of spinor fields, 437 linear endomorphisms, 115 little group, 106 local Chern-Simons form, 345 local function group, 211 local Lie algebras, 194 local Poisson map, 200 locally finite covering, 138 locally finite simplicial complex, 138 locally hamiltonian, 190 loop, 44 loop group, 357 Lorentz group, 96 lorentzian signature, 235 M

Majorana pinors, 27, 30 Majorana representation, 10 Manin triple, 456 mass of a metric, 262 massive gauge potential, 320 Maurer/Cartan, 305 maximum dimension, 192 minimal coupling, 312 momentum mapping, 65, 67, 69, 198, 214,217 Morse theorem, 329 multiple layers, 516

INDEX multiplication properties of Sobolev spaces, 386 multiplication theorem, 431 N

n-connected group, 151 Nambu Goldstone bosons, 320 nerve, 138 Noether's theorems, 64 non-linear-scalar field, 88 nonlinear wave equation in curved spacetime, 400 norm on y (n, m), 24 normal neighborhood, 107 normal ordering, 369 null, 238 O obstruction to the construction of spin and pin bundles; Stiefel-Whitney classes, 134 octonions, 264 odd matrix, 3 odd vector, 124 off shell, 80 on shell, 80 order (p, q) of a graded matrix, 3 orientation of a simplex, 127 orthochronous proper Lorentz, 17 orthogonal group, 17 P

parallelizable, 264 parity operator, 18, 27 partial derivatives of super differentiable mapping, 56 Pauli matrices, 8 perfect group, 111 periodicity modulo 8, 11 Pfaffian manifold, 218 phase of representation, 352 pin bundle, 137 pin frame, 137 pin group, 17, 20, 24 Pin (n, m), 20 pin structure, 137 Pin + (n), 25 Pin- (n), 25 pinors, 137 Poincar6 group, 96, 115 Poincar6-Hopf theorem, 321,330, 332 Poincar6 transformation, 97 point spectrum, 48 Poisson bracket, 165, 185, 188 Poisson-Lie groups, 443 Poisson-Lie morphisms, 443 Poisson manifolds, 184, 190, 200, 205 Poisson structure, 184, 187, 206, 207 Poisson structure defined by a contravariant tensor, 188 Poisson submanifold, 201 Poisson submersion, 215 Poisson tensor, 189

INDEX polar, 212 polar function group, 215 principal fibre bundle, 507 product Poisson manifold, 205 projective n-space, 109 protective representation, 352, 356 projective space C PN, 507 prolongation of SO, 136 prolongation structure, 177 pseudopotentials, 176 pullback of a bundle, 336

O quantum BRST operators, 368-370 quasi-linear first-order partial differential equation, 171 quasitriangular classical Yang-Baxter equation, 473 quaternions, 9, 11,264 R

rank of a Poisson manifold, 190 Rankine Hugoniot equations, 519, 520 ray representation, 352 rays, 506 realization of a Poisson manifold, 210 reducible bundles, 308 reductive, 227 reductive homogeneous spaces, 271 reductive pair, 103 representations of Spin(n, m), n + m odd, 33 residual spectrum, 48 Ricci tensor, 249 Riemann-Roch theorem, 482 Riemann tensor, 240 riemannian manifolds, 235 riemannian symmetric space, 104 S scalar curvature, 249,292 scalar curvature of S n , 241 scalar multiplet, 248,290, 311 Schouten-Nijenhuis bracket, 189 Schur's lemma, 113 Schwartz topology, 514 Schwinger functions, 414 second derivative in the directions (vl, v2), 52 second fundamental form, 279 segment property, 378 self-adjoint, 47 semi-infinite forms, 367 semidirect products, 95, 96 semilinear, 255 seven sphere, 263 sheaf, 91 shell, 80 shock equations, 519 shocks, 519 short exact sequences, 163

529 siebenbeins, 264 a-models, 274 simple cover, 138 simple (k-simple), 45 simplex, 127 simplex decomposition, 321 sine-Gordon equation, 174, 177, 179, 180, 183 singular k-simplex, 130 Sobolev constant, 491 Sobolev embedding theorem, 377, 379, 431 Sobolev inequalities, 379 Sobolev inequality on R n, 389 Sobolev spaces, 386 Sobolev spaces on riemannian manifolds, 428 solvable Lie groups, 114 soul, 2 (space time) parity operator, 18 space of distributions, 513 spacelike, 238 spaces Hs,~ (IR"), 393, 512 special orthogonal group, 17 spectrum, 48 spectrum of closed and self-adjoint linear operators, 47 sphere S n , 240 spin frame, 135 spin group, 17, 21 Spin (n, m), 21 spin structure, 136, 151 spinor, 135 spinor-cospinor, 135 spontaneously broken symmetry, 311 squashed seven sphere, 263 stability subgroup, 106 standard Manin triple, 458 standard simplex, 129 stationary metric, 72 stationary subgroup, 106 stereographic coordinates, 241 Stiefel manifold, 108 Stiefel-Whitney classes, 141 stress-energy tensor, 83 stress energy tensor of a fluid, 519 string, 275 string action, 82 structure constants of a compact Lie group, 117 subbundles, 308 subgroups of Lie groups, 92 submanifolds, 92, 275 subrepresentation, 111 superdifferentiable, 53 supersmooth, 55 supertrace, 332, 333 symmetric riemannian manifold, 107 symmetric spaces, 103, 104 symplectic leaves, 193 symplectic manifold, 185, 190, 221

530 T Taylor formula, 52 Teichmuller space, 478,482 tempered distribution, 418 tension field, 277 tensor distribution, 513 tensor product of algebras, 5 timelike, 238 topological charge, 315 torsion tensors on a symmetric space, 270 totally geodesic mapping, 282 trajectory, 65 transitive G-space, 104 transpose, 47 triangular classical Yang-Baxter equation, 473 triangulated manifold, 128 triangulations, 128,326 twisted adjoint representation, 22 Tychonoff theorem, 39 U uniformly equivalent matrices, 428 unitary representation of the Heisenberg group, 166 universal bundle, 507 universal bundles, 335,337 V vacuum, 310, 369

INDEX Van Hove's singularities, 328 vector, 124 vertex, 127, 138 Virasoro algebra, 360 Virasoro representation, 363 Vlasov equation, 521 W

Weinstein theorem, 193 Weyl Heisenberg group, 163, 164 Weyl representation, 167 Weyl spinors, 27 Weyl tensor, 249, 344, 345 Whitney product theorem, 142, 144 Whitney sum product, 145 Wightman distributions, 414 Y Yamabe equation, 486 Yamabe functional, 485 Yamabe invariant, 488 Yamabe property, 483 Yang-Baxter equation, 232, 468 Yang-Mills operator, 244 Z zero cochains, 168 zero simplex, 127

ERRATA TO A N A L Y S I S , M A N I F O L D S A N D PHYSICS. P A R T I Page* xi

Line

Errata 8

Front Matter III. Differentiable Manifolds, Chapter I

6 11 12 13 14

15 20 21 23 24 27 28 31 39 46 49 55 61 62 65 66

12 16 8 -2 3 12 5 -9 -10 12 13 4 caption -10 3 1 -16 9 -2 1 17 19 4 22 3

Example 1" Let P be the set N of all N has no maximal element. (p. 17) is open iff it is a neighborhood x" (N (x) - {x }) 71 a --/= in A. The set A is dense For each x 6 X and U 6 @" with x 6 U there exists B 6 ~ such that x 6 B and B C U. is a filter 5 ( x ) . Uj such that Vi C Uj. is simply connected when n > 2. Exercise 2, p. 68. vector space (p. 27) is bounded d l (P, 0) < a, d2 (P, 0) < a, d3 (P, 0) < a continuous (p. 21) into the neighborhood [[olxp. 56 Haar measure (p. 180). a a-field of subsets, ser Im(a)] < [ml(a) +lx2lp)l/p Let X and Y be two metric spaces, operator T on L2 ( y ) by v 2 - (vlv) add C p in the margin Let P - C ( V 3 (3))

*Refers to the revised edition. 531

532

Page*

ERRATA TO PART I

Line

Errata

-8 67 69

Vv, w E Vn(s) So we have for k odd

-12 3-10

Answer: We shall construct a covering of the product space x ~ , N X ~ which has no countable subcovering, let alone a finite one. Let A C N and VA -- x a eN Ua be an element of the topology on the product space defined as follows:

G

--J[0,2/3)

ifc~6A

(1/3,1]

ifot~A

/

"

Then {VA; A C N} is an open covering of the product space; it has no countable subcovering. Indeed let {VA~}i be any countable subset of {VA; A C N}. One can always find a point x = {x l, x2 . . . . } in the product space such that x ~ {VA~ }i. For example, let

Xi

--

1/6 5/6

if i r Ai if i ~_ Ai

then X 6 { VAi }i.

Contributed by J. Labelle.

Chapter H 72

9 -4

77 85

11 5 6

mapping: D/Ix0 = l, D Idlx0 = Id, Vx0. of f . A point where the rank is not maximal is called critical. If n = p the determinant of D f is called the Jacobian for the function q A sufficient condition theorem A sufficient condition for f

Chapter HI 111 127 129 131 135

-12 -15 8 12 -5 footnote -6

x~UcX. admissible atlas (see p. 543) replace "identical" by "isomorphic" bundle in particular (p. 376) El a subspace of E or [Osborn II 4] if follows that f * o v = v o f * .

ERRATA TO PART I

Page*

Line

151

--5

152

footnote 17 -7

153 154 155 156 158 162 164 165 165 166

176 177 186 188 190

footnote 2O -1 12, 13 -7 -3 7 7 -3 2 3 5 -11 9,12 -15

det[ x 0]

Errata

only identity transformations Svg The tensor S v g is the strain tensor generated by the vector field v ag o ah = agh (left action of G on X) or Og ocr h --- Crhg (right action of G on X) the group which defines isometries effectively, transitively, and freely on G.

[v~, v~](g)

and the n-toms 72n = R n / Z n (p. 209) (orah Oag) L and v(c,) R of v~,,) as well as the generators v(~) inserted in (5) 158 and p. 353 - [ D , u , D,/~]~ ( e ) x k is defined by matrix multiplication, antisymmetrized for every v ~ C1 X-

1 V ~ e~e[3

-8

i.e., all rotations be nonsingular, k~/~ defines a metric called

-6

jacobian matrix of the embedding mapping (p. 240)

Cartan-Killing.

(i~) - ((g 1)2 _t_ ( Z 2 ) 2 ) - 2 ( (Z2)2 -- (zl) 2 --2zlz 2 ) bl --2zlz 2 (zl) 2 --(Z2) 2 (b 2)

192

Chapter IV

196 200 203 210 211 216 218

533

footnote -2 5 9 11 16 -12 9

Then if oe and fl are 1-forms By property 2 are vectors and W a pseudo-vector div f W = grad f . W + f div W Cx : ( X ) p

ffsurface

e3

534

Page*

ERRATA TO PART I

Line

223

18 20

224

1, 2

225 226 250 253

6 -5 23, 24 17

254 271 276

11 7 -9

Errata group [See Analysis, Manifolds and Physics, Part II] Move the sentence " H p is often called the de Rham group" and the marginal note "de Rham cohomology" back to line 12. Delete and replace by: "For the cases of 0-chains see for instance [Patterson]." Insert the definition of the Euler-Poincar6 characteristic, which can be found on p. 293. co depends only on x 1 xp f o r C Cz Hp, o) cz H p. delete "A differential.., system." vector field. By the theorem on p. 248, C is completely integrable. result in a set {0(k)} 9 equations (see example p. 263) = -dy~ 9

9

. ~

~

Chapter V

285 287

14 -3

302

16

306 313 314

5 18 -5 -4 -8 -4 6 -12 -13 -11

316 317 331

-8 354

-7 12-20

(p. 134) manifold. The metric is called iorentzian. It is du i bl ~

C t

dC i --

dt dt ' In a moving frame the ( V v u ) l l - (v,,,v' a lorentzian metric it is a maximal is a minimal hypersurface; in qs* (D f ) - D(qS*/) pp. 482-486. d(*~o) (Vl . . . . . Vn/2, J Vl . . . . . J vn/2) ~CFvr/- 2q:,r/. (1) such that q~ - constant is the group of isometries and dilations - r/)~ Oo~v )" + r/xc~3/~v )~ - 2,;15r/~/~ where q~ is obtained by contraction 2q~ -- 893a va Replace lines 12 to 20 by" This transformation is defined in the case of an euclidean metric if one adds to the space a point at infinity, whose image in an inversion is the origin of this inversion. In the

ERRATA TO PART I

Page*

356

Line

13

535

Errata case of the Lorentz metric the inversion with origin point x0, for instance x0 = 0, is defined only when x 2 - O~fx~x~ # O, that is when x is not a null (i.e. lightlike) vector. The conformal group is defined in Segal cosmos (S 3 x R, ref p. 356) universal cover of the compactified Minkowski spacetime (cf. problem V.8). For more on the compactification of Minkowski space see for instance R. Penrose "Conformal Treatment of Infinity" pp. 563-584 in Relativity, Groups, and Topology (Les Houches 1963) Eds C. DeWitt and B. DeWitt (Gordon and Breach New York 1964) in empty space.

Chapter V bis 359 360 362 363 365

15 16 -14 7 2 3 9,10

369 371 372 373 375 377 379 380 381

14 8 10 -13 -13 -14 4 4 14 -8,-7 -6 -13

subspace of Tp (P). Due to Add: We also write fi -- v• (p). a 1-form on U with values - - A d ( g -1)

- (4).71.o9 kg.j i (x)

( v) 110

(

Ox/Og OLgij(x)g/Ox

15 20 -4

) ( v) w

OMc(g~'i(x)v) delete the composition sign then (p. 367), corresponding to ~ and V u ~P of a covariant vector. Add" "h -= hor" _....+1...

~veru0

differential 2-form ~2 on X Delete: "It i s . . . p. 359" connection as defined on p. 359 of the connection, as we have already seen on an

382

OLgi.j(x)g/Og

the injection f in P of G1; thus if

gi -- gij (x)gj

536

Page* 383

ERRATA TO PART I

Line

384

13 -4 -3 -12

385

17, 18

385

-9, -1

Errata the realization o of G Tg2g 1 --

PXGF

action of G1, where (P, X, Jr, G) is a principal fibre bundle Delete and insert: 2) Suppose G is reducible to G, and let P1 denote a reduced bundle with injection f" P1 -+ P. Let/z be the projection P -+ P / G 1. The mapping/x o f Delete and insert the following

Theorem. Every fibre bundle (E, X, Jr, F, G) such that the base manifold X is paracompact and the fibre F is diffeomorphic to R n admits infinitely many cross sections. Proof: It is easy to show that when E is a vector bundle, i.e. when Fx is a vector space, it admits infinitely many cross sections. Indeed, let {Oi} be a partition of unity on X subordinates to a locally finite coveting by open sets Vi such that Vi C Uj some open set of an atlas of X over which E is trivializable. Let O"i be an arbitrary cross section over Vi. Then the element of Fx given by the finite sum ff (X ) -- Z

Oi (X )O"i (X ) i

is a cross section over X. The theorem is stronger because it does not require a canonical identification of a point of Fx with the origin of R m, nor the group G to be linear. For the proof I one uses the property that a differentiable function defined on a closed set o f I~ m can be extended to the whole of It{m together with Zorn's lemma, to show that every cross section defined over a closed set Y C X can be extended to a cross section over X. I cf. R. Godement,

Theorie des Faisceaux (Hermann, Paris, 1958) p. 151

or Kobayashi and Namizu, loc. cit. Vol. I, p. 58.

386

21 -15

and p x 1 the linear product in R n and p x I the linear

ERRATA TO PART I

Page*

Line

387

-9 -12, -13

537

Errata in part A (pp. 380-381) delete i.e . . . . and insert: [ ~ 0 ( p ' ) = g~'~o(p)g-1].

389

390

18 22 -7 -5 -10 -1

391

1

9

18

p. 388 and by the subspace horizontal field is horizontal (p. 374), and that -

( u , v) 1

"''--

(2k)!

"''(Vet(I),

Vet(Z)),...,

Tx X, x -- H (p)

and be unique Add" df(X2)(Vl,..., V n ) - rr* d f ( Q ) ( V l . . . . . Vn) -- df(S2) (rr' hor Vl , . . . , JT ! hor Vn) d f ( ~ ) ( h o r v l , . . . , hor Vn) -- Df(X2)(Vl . . . . . Vn)

+ 89

20 392

394 395

397 398 399

401 402 404 405 406

4 5 7 ll 8 11 -1 -8 5 11 17,18

-7 4 ll,13 l0 figure 5 ll --13

This, together with the fact q~ being invariant on a vertical vector projects to a form q5 on X, [f(V)]

2

curvature 2-form complexification (p. 224) via a scalar product in the fibres, possibly deduced to the scalar product in Decomposition theorem. Proof: It follows formally the same lines as the proof given above for the finite dimensional space E p,x. Its validity in the new context rests therefore Zip is an elliptic ))Ap* (A is exterior product) {Dp, Ep} d x v -- ~ i , Replace 7r i by :rl D~u (v) - + A ~i,xO

ERRATA TO PART I

538

Errata

Page*

Line

407

7 -12 -10, -11 -5 2O

408 410

-13

(~)(Si (X) ) ) -1 when the connections A is flat. bold face dq~ p: g ( 1 ) --+ L(C, C) , )/1 - - --(2:rri) -1 tr Cl

. . . . . 9

'f

2zr

nd~

Sl

411 415 419 420

-10 -9 -6 18 12 -1 6 6,7,14,17

iA_in d~.

iA+ =

define the following electromagnetic field define the following Do F+ and F_ Atiyah change signs before 1/4 change signs before 1/8

421 7 -3, -4 422

References

if there is no torsion.

xRxR T. Regge, "The group manifold approach to unified gravity", in Rela-

tivitY, Groupes et Topologie II, eds.

B.S. DeWitt and R. Stora (North-

Holland, Amsterdam, 1984) pp. 933-1006.

C h a p t e r VI

435 441

455 468 472 477 478 490 494

13 -9 -8 -7 -10 16 -2 10 5 -6 13

exists C ~ on U, but cannot in general vanish for it vanishes if ~0'(a) - 0. Remark. ~06~ = 1 d

T,, = i dx Proof: X is solution in sO' since, D X - B h a s at m o s t o n e . . . i f D * h a s a n

- ( i y ) ~. p. 272 9" = I1(1 + I x 1 2 ) m / 2 5 f l 1 2 9.. Flu -- (- - - ) u -- 0

E R R A T A T O PART I

Page*

Line

499

-2

512

539

Errata D O

0

f

~

3q9 3u dx Y~,__.Ox----iOx---7

1

7

= 4----~6c. 1

= - dx l A dx Z A dx 3

t

513 522 523

13 -9 6

532 from 538 to 539

8 -2 23

sin 0 cos 4~, 0 / 2 - - sin 0 sin 4~, or3 -- cos 0 For a system with an infinite 9.. = (T, L* U) when U is C ~ with compact support 0/1 __

Y - ( x ) exp(ax) Replace by"

Answer: a) The elementary kernels E (t, s) are solutions of ( - d 2 / d t 2 - p2)E(t, s) -- ~(t, s).

(4)

The Fourier transform of this equation is ~((-dZ/dt

2 - p2)6 9 E)(r/) -- l,

(r] 2 -- p 2 ) 3 E -

1 pv( 1 ( S E ) (tl) = ~2p ~- P

1,

1 ) 0 -k- p + K l g ( r / - p) + K2a(r/-+- p). (5)

We can choose K1 and K2 such that the elementary kernel is in the convolution algebra ~ ' + or ~ ' - . Recall (Problem VI 7) that ( S Y + ) ( r / ) -- q : i ( P v 1 :k iJr6o~ / 77 k

q::i(ri T i0) -1

(6)

Hence if K + -- - K + -- irr/2p and

E + 9 ~'+ 5E+=l( __ 2p E- E ~'~E---2p

1 rl-p-i0

l(,

_

1 r/+p-i0

,)

) '

(7a)

if K1- -- - K 2 -- izr/2p and

r/-p+i0

rl+p-i0

(7b)

540

Page*

ERRATA TO PART I

Line

Errata

To compute E + from (5) using (6) one can translate (see p. 458)

Pv

(1) r/-p

+ i6 ( r / - p)

by p and the other two terms by - p "

E+(t, s) -- qzY+(t The propagator E -

-

sin(pt -

s)p -1

ps).

(8)

E + - E - is

E(t, s)

-

sin(pt -

_p-I

ps).

We could, of course have obtained (8) by solving (4) according to the method developed on p. 469, which says

E+ (t, s) -- Y• (t - s)h + (t - s) where the C ~ functions h + satisfy the homogeneous equation and the following boundary conditions: h + (0) -- 0, h +'(0) - - 1

and

h - (0) - 0, h - ' ( 0 ) - 1.

Equation (7) suggests the following integral representation for the ele-

541

-3

equation (9)

Chapter VII 549 571 590

6

footnote -7

Hilbert space of norm The Morse index is the negative of the index defined on p. 287

------ d ( . . . + I m2u 2) dx)(U, V). B

593

-4

f exp(-~t2/2)h(t) dt A

_ - 1 (h (t) -- )~t

- -h -(t)- t )~t 2

h'(t)) B -~ exp(--)~t2/2)lA

ERRATA TO PART I

Page*

Line

541

Errata

if

B

+ ~

exp(-'kt2/2)

A

h(t)

37 -

603 606 608

- 3 h'(t)t + h"(t)) dt

References

CHERN, S.S., Selected Papers (Springer-Verlag, New York, 1978). ITZYKSON, C. and J.B. ZUBER, Quantum Field Theory (McGraw-Hill, New York, 1980). SCHUTZ, B., Geometrical methods of mathematical physics (Cambridge University Press, Cambridge, 1980).

New and~or Corrected Entries for Index Page

Entries

617

Adjoint, 59 Adjoint representation, 166, 184 Automorphism, 157, 174 Cartan-Killing metric, 188 Chain, 217 Compactified Minkowski space, 354 Critical point, 72, 119,546, 547, 567 Critical value, 546, 547, 567 Decomposition theorem, 399 Degenerate, 285 Dense, 13 Dual, star, 295 Duality, 226 Dirichlet problem, 502 Einstein equation, 342, 356 Energy function, 344 Energy integral, 324, 513 Faithful, 163 Gaussian curvature, 396 Haar measure, 39, 180, 189 Hodge decomposition, 399

618

619 602

620 621

622 623 623

Page

624

624 625 626

627 628

629

Entries Hodge theorem, 400 Index of a vector field, 396 Inductive limit topology, 427 Laplacian, 318, 398, 421,449, 495, 527 Leray form, 449 Lorentzian metric, 287, 292 Lusin theorem, 41 Maxwell equations, 271,336 Normal (hypersurface), 292, 315 O(n), 175 Poincar6-Hopf theorem, 396 Principal value, 439, 532 Reflexive, 59 Shock wave, 535 Simplex, 216 Singular field, 339 Spin bundle, 417 Symplectic form, 268, 282, 552 Trace, 430, 487 Trajectory, 144, 274


Analysis, Manifolds and Physics Part II

Analysis, Manifolds and Physics, Part II - Revised and Enlarged Edition

Analysis, manifolds and physics