Conference CBMS - American Mathematical Society · 2019-02-12 · 84 Hugh L. Montgomery, Ten...

Conference Board of the Mathematical Sciences

CBMS Regional Conference Series in Mathematics

Number 29

Lectures on Syinplectic Manifolds

Alan Weinstein

American Mathematical Society with support from the

National Science Foundation

Other Titles in This Series

86 Sorin Popa, Classification of subfactors and their endomorphisms, 1995 85 Michio Jimbo and Tetsuji Miwa, Algebraic analysis of solvable lattice models, 1994 84 Hugh L. Montgomery, Ten lectures on the interface between analytic number theory and

harmonic analysis, 1994 83 Carlos E. Kenig, Harmonic analysis techniques for second order elliptic boundary value

problems, 1994 82 Susan Montgomery, Hopf algebras and their actions on rings, 1993 81 Steven G. Krantz, Geometric analysis and function spaces, 1993 80 Vaughan F. R. Jones, Subfactors and knots, 1991 79 Michael Frazier, Bjorn Jawerth, and Guido Weiss, Littlewood-Paley theory and the study

of function spaces, 1991 78 Edward Formanek, The polynomial identities and variants of n x n matrices, 1991 77 Michael Christ, Lectures on singular integral operators, 1990 76 Klaus Schmidt, Algebraic ideas in ergodic theory, 1990 75 F. Thomas Farrell and L. Edwin Jones, Classical aspherical manifolds, 1990 74 Lawrence C. Evans, Weak convergence methods for nonlinear partial differential equations,

1990 73 Walter A. Strauss, Nonlinear wave equations, 1989 72 Peter Orlik, Introduction to arrangements, 1989 71 Harry Dym, J contractive matrix functions, reproducing kernel Hilbert spaces and

interpolation, 1989 70 Richard F. Gundy, Some topics in probability and analysis, 1989 69 Frank D. Grosshans, Gian-Carlo Rota, and .Joel A. Stein, Invariant theory and

superalgebras, 1987 68 J. William Helton, .Joseph A. Ball, Charles R • .Johnson, and .John N. Palmer,

Operator theory, analytic functions, matrices, and electrical engineering, 1987 67 Harald Upmeier, Jordan algebras in analysis, operator theory, and quantum mechanics,

1987 66 G. Andrews, q-Series: Their development and application in analysis, number theory,

combinatorics, physics and computer algebra, 1986 65 Paul H. Rabinowitz, Minimax methods in critical point theory with applications to

differential equations, 1986 64 Donald S. Passman, Group rings, crossed products and Galois theory, 1986 63 Walter Rudin. New constructions of functions holomorphic in the unit ball of en, 1986 62 Bela Bolloblis, Extremal graph theory with emphasis on probabilistic methods, 1986

61 Mogens Flensted-.Jensen, Analysis on non-Riemannian symmetric spaces, 1986 60 Gilles Pisier, Factorization of linear operators and geometry of Banach spaces, 1986 59 Roger Howe and Allen Moy, Harish-Chandra homomorphisms for j:J-adic groups, 1985 58 H. Blaine Lawson, .Jr., The theory of gauge fields in four dimensions, 1985 57 .Jerry L. Kazdan, Prescribing the curvature of a Riemannian manifold, 1985 56 Hari Bercovici, Ciprian Foia§, and Carl Pearcy, Dual algebras with applications to invariant

subspaces and dilation theory, 1985 5 5 William Arveson, Ten lectures on operator algebras, 1984 54 William Fulton, Introduction to intersection theory in algebraic geometry, 1984 5 3 Wilhelm Klingenberg, Closed geodesics on Riemannian manifolds, 198 3 52 Tsit-Yuen Lam, Orderings, valuations and quadratic forms, 1983 51 Masamichi Takesaki, Structure of factors and automorphism groups, 1983

(Continued in the back of this publication)

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Conference Board of the Mathematical Sciences

CBMS Regional Conference Series in Mathematics

Number 29

Lectures on Symplectic Manifolds

Alan Weinstein

Published for the Conference Board of the Mathematical Sciences

by the American Mathematical Society

Providence, Rhode Island with support from the

National Science Foundation

http://dx.doi.org/10.1090/cbms/029

Expository Lectures

from the CBMS Regional Conference

held at the University of North Carolina

March 8-12, 1976

Prepared by the American Mathematical Society

with partial support from the National Science Foundation grant MPS-74-23180.

2000 Mathematics Subject Classification. Primary 53C15.

Library of Congress Cataloging-in-Publication Data

Weinstein, Alan, 1943-Lectures on symplectic manifolds. (Regional conference series in mathematics; no. 29) "Expository lectures from the CBMS regional conference held at the University of North Car

olina, March 8-12, 1976." Bibliography: p. 1. Symplectic manifolds-Addresses, essays, lectures. I. Conference Board of the Mathemat

ical Sciences. II. Title. III. Series. QA1.R33 no. 29 [QA649] 510'.8s [516'.362] 77-3399 ISBN 0-8218-1679-9

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©Copyright 1977 by the American Mathematical Society. All rights reserved. Second printing, with corrections, 1979.

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1098765 04 03 02 01 00

CONTENTS

Introduction ............................................................................................................................................ .

Lecture 1. Symplectic manifolds and lagrangian submanifolds, examples ................................... 3

Lecture 2. Lagrangian splittings, real and complex polarizations, Kahler manifolds............... 7

Lecture 3. Reduction, the calculus of canonical relations, intermediate polarizations ........... .11

Lecture 4. Hamiltonian systems and group actions on symplectic manifolds ........................... 15

Lecture 5. Normal forms ..................................................................................................................... 22

Lecture 6. Lagrangian submanifolds and families of functions ................................................... 25

Lecture 7. Intersection Theory of lagrangian submanifolds ......................................................... 29

Lecture 8. Quantization on cotangent bundles ............................................................................... 31

Lecture 9. Quantization and polarizations ....................................................................................... 35

Lecture IO. Quantizing lagrangian submanifolds and subspaces, construction of the Maslov

bundle .................................................................................................................................................. 39

References ................................................................................................................................................ 45

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40 ALAN WEINSTEIN

e-i(S(x)+c>a(x) =constant,

so a(x) must be a constant multiple of eiS(x). The presence of this multiplicative constant

is unavoidable, since the function S(x) is determined by L only up to an additive constant.

If we replace dS by an arbitrary closed I-form w on X, the lagrangian submanifold

L = w(X) is quantizable just when the function u(x) satisfying du = w is globally defined

modulo 2?T: this is the case when the integral of w around every closed curve in Xis an inte

ger multiple of 21T, or, equivalently, when the class [w/21T] E l/1 (X; R) is in the image of

111 ( X; Z). In this case, the function eiu(:<) is globally defined and may be associated with w(X).

We have now associated, to each quantizable lagrangian section L = w(X) ~ T* X, a

function on X. To get a I /2-density on X, we should add one piece of information - a 1/2·

density on L, which we can pull back to X and multiply by the function. This suggests that

the objects to be quantized in general should be pairs consisting of a lagrangian submanifold

/. and a I /2-density on L. Such a pair will be called a quasi-classical state (see [SL 2) ). (In

case L is the graph of a symplectomorphism f: P 1 -+ P 2 , it carries a natural 1 /2-density in

duced from the symplectic form on P1 or P2 .)

Remaining within cotangent bundles, we may ask next how to quantize a quasi-classical

state (L, cS) in T* X for which L does not project diffeomorphically onto X. How, for exam·

pie, shnuld we quantize the fibre x = 0 in T*R? Any function on T*X x S 1 of the form

.,,{x, ~. 0) = e--io a(x) is already constant along the horizontal lifts of the fibre x = 0 (as it

is along the horizontal lifts of all fibres), so none is distinguished by that condition. Since

the fibre x = 0 lies only over the origin in R, it is tempting to quantize it by a Dirac delta

"function" supported at the origin. It turns out that argument by continuity leads to the

same conclusion, as we shall now see.

Consider the line L defined by x = 0 in the (x, n-plane, equipped with the I /2-density

p = Cid~ 11 ' 2 , where C is a constant. It is the limit as m -+ 0 of the line Lm defined by

x = m~, equipped with the 1 /2-density Pm whose expression in the fcoordinate is Cid~ 11 / 2 .

Now /, 111 is also defined by the equation ~ = x/m, so it is dS(R), where S(x) = x 2 /2m; in

the x-coordinate, our I /2-density becomes

cld(:...)11'2 = c ldxl112. 1 m lml1/2

By our quantization rule for sections of T* X, we should associate with (Lm, Pm) the 1 /2·

density (C/iml 1 ' 2 )eix 2 / 2 m ldxl 1 ' 2 on X. (This is not in L 2 ; recall, however, that the ob

jects associated with lagrangian submanifolds of (P, il) are generally contained in some

exte11sio11 of the Hilbert space which quantizes (P, il).) Since (L, p) = limm_ 0 (Lm, Pm).

it is natural to try to quantize (L, p) by

(JO.I) lim C eix2/2mldxlt/2 m-+O lmll/2 .

We cannot make sense out of the last limit in the space of c- 1 /2-densities on R, but the

limit does exist in the following "weak" sense. If u = u(x) ldx 1112 is any c- 1/2-density

on R with compact support and 11 = u(x) ldx 1112 is any C 00 1 /2· density on R, we can

form their product by

http://dx.doi.org/10.1090/cbms/029/11

SYMPLECTlC MANIFOLDS 41

<u, u) = JR u(x)u(x) dx.

In this way, the space E(IRl 1 ' 2) of C"° 1/2-densities on R is identified with a space of

linear functionals on the space V(IR1 1' 2) of compactly supported C"° 1/2-densities. The

full space of linear functionals on V(IR 11 ' 2), continuous with respect to a certain C 00 topol

ogy (see [S]), is denoted by V'(IRl 1 ' 2). Since V(IRl 1 ' 2) contains E(IRl 112) (as a dense

subset), its elements are called generalized 1 /2-densities, or I /2-density-valued distributions,

on R.

Now we may try to find the limit {IO.I) in V'(l"R 1112). To do this, we must evaluate

(10.2) lim f--c- eix2/2m v(x) dx m-o lmjl/2

where u(x) is a compactly supported C 00 function on R. In fact, the principle of stationary

phase tells us that the limit (10.2) exists if m approaches 0 with a fixed sign; it is equal to

C(27r)-112 e(i11/4)sgn m u(O).

The functional u(x) ldx 1112 t-+ u(O) belongs to V'( IR 11 ' 2); it is called a Dirac delta func

tional at the origin and will be denoted by 6 0 1dxl 1' 2. Then we have, in V'(IRl 112),

lim _c __ e;x2/2m ldxll/2 = C(27r)-1/2eH11/46 ldxl'/2. m->±0 lmjl/2 o

Thus, we are lead to quantize the quasi-classical state (x = 0, C Id~ 11' 2) by the distribution

(determined up to a factor of i) C(27r)- 112e±in/4 60(1dx1)112 on X.

We shall now describe a general construction which is motivated by this example. Let

V be a real vector space of dimension n. Any lagrangian subspace l of T* V = V x V*

which is transversal to the "vertical" space {O} x V* (which we will denote simply by V*)

is the graph of a symmetric mapping AL: v -+ v•; equivalently' l = dS L ( V), where s L

is the quadratic function SL(x) = ~AL(x)(x). (We remove the indeterminacy in SL by re

quiring SL (O) = 0.) If p is any 1 /2-density on l, we may regard it as well as a 1 /2-density

on V by pullback, and we quantize the pair (L, p) by the 1/2-density iSL(x) p on V. -

If l is the vertical space, its quantization should be a delta functional supported at

0 E V. Again, the principle of stationary phase justifies this choice, but instead of working

out the details of this we will pass immediately to the general problem of quantizing a pair

(L, p ), where l is an arbitrary lagrangian subspace of T* V, and p is a translation-invariant

1 /2-density on L.

The lagrangian subspace L rnojects onto a subspace WL ~ v. We may guess that the

quantization of (l, p) should consist of distributions which are supported on W L, but which

distributions should they be? The subspace F = W L E0 V* is coisotropil. in V E0 V*, and F 1 = 0 E0 Wi, where Wt ~ V* is the usual annihilator of W L. Since V* /WL ~ Wt, the reduced sym

plectic space F/F1 is naturally isomorphic to W L E0 Wt. The reduction (L n F)/(L n F 1)

of l to Ff F1 ~ W L E0 w; projects onto W L, so it is the graph of a symmetric mapping AL * wL -wL.

This suggests that we quantize (l, p) by the function e< 1t2>iAL(x){x) on WL times


42 ALAN WEINSTEIN

some "delta functional" along WL. Jn fact, some juggling of 1/2-densities and exact sequences

(see (GU] or (W 7)) shows that there is a natural isomorphism h between the space IL 1112

of I /2-densities on the vector space L and the space Hom(IVI' ' 2 , I WL I) of homomorphisms

from 1/2-densities on V to I-densities on WL. (In case WL = V, his just the pull back iso

morphism.) The latter space may be identified with a subspace of V'(I VI' ' 2): Given a: I Vl 112

~ I WL I and a "test-density" u E V(I Vl 1 ' 2 ), we may 11.pply a to it and restrict to WL• obtain·

ing an element of V(l W LI) which may be integrated over W L to give a complex number.

Thus we are led to quantize (L, p) by the functional

u ,___._ I (I /2)iA L(.x)(.x)((j" ) ] .-~ W e LP o u '

L

which we will denote by o(L, p).

The I -dimensional case suggests that we should modify o(L, p) by constant factors of

(211') 1 / 2 and ein/4 . We shall now show that, with such modifications, the quantization of

lagrangian subspaces becomes a continuous process.

We may study the process (l, p) - o(L, p) and, incidentally, define a differentiable

structure on the lagrangian grassmannian L(T* JI), by using a special covering of L(T* JI). For each K E L(T* JI) which is transversal to {O} x v• (which we denote simply by

V*), we define UK to be the set of L E L(T* V) which are transversal to K. Given

any /, E L(T* V), there is a K which is transversal to both L and v•' so the UK's

cover L(T* V).

Now we may identify UK with the space of quadratic forms on V*. In fact, the la

grangian splitting T* V = V* $ K induces an isomorphism of K with V** (it is the negative

of the isomorphism of K with V given by projection along V*) as in Lecture 2, and hence

an isomorphism of T* V with V* Ea V** = T*( V*). Now each L in UK is identified with

a lagrangian subspace aK (L) of T* V* which is transverse to V**. By the earlier construc-A a k(L ).

lion in this section, WC then have the symmetric mapping V* V** and the quadratic

function Sux<Ll<n = ~AaK(L) (~)(0 on V*. We will write A(V*IL IK) for AaKl(L)·

The mappings UK~ Sym(V*, V**) defined by i;,.· (L) = A(V* IL IK), which are bi·

jective, will be taken as the charts for L(T* JI). To show that the charts give a differentiable

structure, we must study the transition map on Ux 1 n Ux 2 . WritingA(VIKIV*) for the

symmetric map from V to v• of which K is the graph, and identifying V with v•• in the

usual way. we have the following lemma, whose proof we omit.

LEMMA. (a) If LE UK 2• then L lies in llx 1 if and only if the operator 1-A(V*IL IK2 )[A( VIK2 IV*) -A(VIK1 IV*)] is invertible.

(b) If LE UK, n LJK2• then

.pK 1(L) ={I - .pK 2(L)[A(VIK2 IV*)-A(VIK1 IV*))}- 11Px2(L).

This lemma implies immediately that the charts (U x, .Px ) define a differentiable struc·

lure on L(T* V).

We return now to the quantization of lagrangian subspaces. The correspondence


SYMPLECTIC MANIFOLDS 43

(L, p) 1--+ 6(L, p) is clearly continuous for those L which are transversal to the vertical. We

look next at those L which are transversal to the "horizontal'', i.e. L E U v where we write

V for Ve {O}. The trick is to reduce this to the first case by using the Fourier transform,

which is the analytic analogue of interchanging V and V*.

Suppose for the moment that L is transversal to both the horizontal and the vertical.

It will help to use coordinates in what follows, so let (x 1 , ... , xn) be coordinates on V and

(~I' ... , ~n) the dual coordinates on V*. We write ldx 1112 for ldx 1 /\ • · • I\ dXn 1112 and

the same for the rs. L is described by the equation ~ =Ax, where A =A( VIL IV*) is sym·

metric and invertible (since L is transversal to the horizontal) and p = c ldx 1112 for some con· stant c. 6(L, p) is then ce(l 12)i.A (x)(x) ldx 1112, and its Fourier transform is

F6(L, p) = { J e-i<t ,x> ce(i/2).A (x)(x) dx} Id~ 11 /2.

(The "natural" Fourier transform on l /2-densities involves the canonical l /2-density

ldx 1 1 1 2 Id~1 1 t2 on T* V.) This Fourier transform is computed, for instance, on pp. 144-145

of [HO] . It is

Fo(L, p) = c(27r)n/2 t<wf4)sgn A e-W2).A -1 mm ldet A 1-112Id~1112,

where sgn A js the signature of the symmetric form A. Now we may interpret the formula for

F6(L, p) in terms of the coordinate ..Py on U v· In fact, B = A(V*IL IV) is the negative of

the inverse of A = A(VIL IV*), and ldet A 1- 1 12 Id~1 1 1 2 = ldx 1112 on L, so

c I det A 1- 112 Id~ 1112 is just the pullback p of p to v• by the projection along V, so we may

write

(10.3)

If L is not transversal to the vertical, we get a similar result. For instance, if/, = v• and p = c Id~ 1112, then 6(V*, p) = c6 0 ldx1 1 ' 2 , and

F6(V*, p) = cld~l 1 12 = p, which we can write in the form (10.3) if we remove the factor (211')"'2 , since B = A(V*IV*IJ')

is zero in this ·case.

A similar computation shows that, for general L E U v•

(10.4)

where B = A(V*IL IV) and pis the pullback of p to V*.

We can see now how to "correct" 6(L, p). The signature of Bis cc,ngruent mod 2 to

the rank of B, which is in turn equal to the dimension of W L. It follows from ( 10.4) that,

except for "multiplicative jumps" of powers of ei71'/2 = i when thr: signature of B changes, the map

(L, p) I-+ (27r)-(l/2)dim WLei(w/4)dim WLS(L, p)

is continuous for L E U v· (The jumps occur when L fails to be transversal to v• .) To take into account the jumps, we may associate to (L, p) the 4-tuple

e(L, P) ={(27r)-(l/2)dim WLei(w/4)dim WL;kS(L, p)lk = O, l, 2 , 3}


44 ALAN WEINSTEIN

of distributions, which does depend continuously on (L, p).

Finally, we must see what happens on UK for K other than V. It turns out that the map

(L, p) 1-+ e(_L, p) is still continuous -instead of just the Fourier transform, one must con

sider F (e-(i/2)A ( v• 1L IK)-1 (.x)(.x) f,(l, p)).

Now consider the set M (T* V) = {(l, 6(l, p)) Ip E IL 1112 } ~ L (T* V) x V'(IVl 1 ' 2 ),

with the projection M(T* V)-+ L(T* V) given by (L, 6(L, p)) I--+ L. Our calculations show

that M(T* V) is a complex line bundle over L(T* V) whose structure group reduces to the

discrete group of multiplications by {I, i, - 1, - i}. Analysis of the jumps when L is not

transversal to V shows that this bundle is exactly the Maslov line bundle used in the theory

of Fourier integral operators. We have seen, therefore, an "analytic" realization of the

Maslov bundle arising from quantization theory.

What structures on T* V did we use to construct the Maslov bundle? Besides the sym

plectic structure and linear structure, we used the polarization given by the vertical space

V*, but the horizontal space V played no essential role. In general, if we have a symplectic

vector space E with a distinguished linear real polarization 7r, we may prequantize E as a

symplectic manifold and consider the distribution space V'(E, 7r) in which the Hilbert space

obtained by geometric quantization sits as a dense subspace. (The choice of a horizontal

space enables one to identify V'(E, 7r) with V'(IE/7r 1112 ).) In the bundle L(E) x V'(E, 7r) over L(E), there is then a distinguished "Maslov bundle" M(E, 11'). This construction can be applied fibre by fibre, if we have a symplectic vector bundle E-+ n with two real polariza

tions to produce a line bundle over B. If E = T L(T* X), where L ~ T* X is a lagrangian sub

manifold and the polarizations are given by TL and TL (fibres), one recovers the Maslov

bundle over L used in [HO]. Its holonomy is the mod 4 reduction of the Maslov class

which we defined in Lecture 6.

This analytic construction of the Maslov bundle puts the theory· of Fourier integral

operators in a new perspective and suggests an extension of Hormander's symbol construction to arbitrary distributions (see [WE 6) and [WE 7] ). On the other hand, the theory of

Fourier integral operators itself provides a means for quantizing certain lagrangian submanifolds. We refer the reader to [DJ and (HO] for expositions of this theory, mentioning only

that the phase functions of Lecture 6 play an important role.


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Other Titles in This Series (Continued from the front of this publication)

50 James Eells and Luc Lemaire, Selected topics in harmonic maps, 1983 49 John M. Franks, Homology and dynamical systems, 1982 48 W. Stephen Wilson, Brown-Peterson homology: an introduction and sampler, 1982 47 Jack K. Hale, Topics in dynamic bifurcation theory, 1981 46 Edward G. Effros, Dimensions and C*-algebras, 1981 45 Ronald L. Graham, Rudiments of Ramsey theory, 1981 44 Phillip A. Griffiths, An introduction to the theory of special divisors on algebraic curves,

1980 43 William Jaco, Lectures on three-manifold topology, 1980 42 Jean Dieudonne, Special functions and linear representations of Lie groups, 1980 41 D. J. Newman, Approximation with rational functions, 1979 40 Jean Mawhin, Topological degree methods in nonlinear boundary value problems, 1979 39 George Lusztig, Representations of finite Chevalley groups, 1978 38 Charles Conley, Isolated invariant sets and the Morse index, 1978 3 7 Masayoshi Nagata, Polynomial rings and affine spaces, 1978 36 Carl M. Pearcy, Some recent developments in operator theory, 1978 35 R. Bowen, On Axiom A diffeomorphisms, 1978 34 L. Auslander, Lecture notes on nil-theta functions, 1977 33 G. Glauberman, Factorizations in local subgroups of finite groups, 1977 32 W. M. Schmidt, Small fractional parts of polynomials, 1977 31 R. R. Coifman and G. Weiss, Transference methods in analysis, 1977 30 A. Pelczyriski, Banach spaces of analytic functions and absolutely summing operators,

1977 29 A. Weinstein, Lectures on symplectic manifolds, 1 977 28 T. A. Chapman, Lectures on Hilbert cube manifolds, 1976 27 H. Blaine Lawson, Jr., The quantitative theory of foliations, 1977 26 I. Reiner, Class groups and Picard groups of group rings and orders, 1976 25 K. W. Gruenberg, Relation modules of finite groups, 1976 24 M. Hochster, Topics in the homological theory of modules over commutative rings, 197 5

23 M. E. Rudin, Lectures on set theoretic topology, 1975 22 O. T. O'Meara, Lectures on linear groups, 1974 21 W. Stoll, Holomorphic functions of finite order in several complex variables, 197 4 20 H. Bass, Introduction to some methods of algebraic K-theory, 1974 19 B. Sz.-Nagy, Unitary dilations of Hilbert space operators and related topics, 1974 18 A. Friedman, Differential games, 1974 17 L. Nirenberg, Lectures on linear partial differential equations, 1973 16 J. L. Taylor, Measure algebras, 1973 15 R. G. Douglas, Banach algebra techniques in the theory of Toeplitz operators, 1973 14 S. Helgason, Analysis on Lie groups and homogeneous spaces, 1972 13 M. Rabin, Automata on infinite objects and Church's problem, 1972 12 B. Osofsky, Homological dimensions of modules, 197 3 11 I. Glicksberg, Recent results on function algebras, 1 972 I 0 B. Griinbaum, Arrangements and spreads, 1972 9 I. N. Herstein, Notes from a ring theory conference, 1971 8 P. Hilton, Lectures in homological algebra, 1971

(See the AMS catalog for earlier titles)


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