Convex polytopes and quantization of symplectic manifolds - PNAS

Proc. Natl. Acad. Sci. USA Vol. 93, pp. 14238–14242, December 1996 Colloquium Paper

This paper was presented at a colloquium entitled ‘‘Symmetries Throughout the Sciences,’’ organized by Ernest M. Henley, held May 11–12, 1996, at the National Academy of Sciences in Irvine, CA.

Convex polytopes and quantization of symplectic manifolds `LE VERGNE MICHE Ecole Normale Supe´rieure et Unite ´ Associeé 762 du Centre National de la Recherche Scientifique, Departement de Mathematiques et d’Informatique, Ecole Normale Supe´rieure, 45 rue d’Ulm, 75005 Paris, France

ABSTRACT Quantum mechanics associate to some symplectic manifolds M a quantum model Q(M), which is a Hilbert space. The space Q(M) is the quantum mechanical analogue of the classical phase space M. We discuss here relations between the volume of M and the dimension of the vector space Q(M). Analogues for convex polyhedra are considered. Quantum mechanics enables us to associate discrete quantities to some geometric objects. For example, the volume of a compact symplectic manifold has a quantum analogue that is the dimension of the quantum model Q(M) for M. It is important to understand the relation between both quantities, as the volume of the manifold M is just a ‘‘limit’’ of the dimension of Q(M). A similar comparison problem is the following: if P , R n is a convex polytope, can we compare the number uP ù Z nu of points in P with integral coordinates and the volume of P? It is clear that the volume of P is obtained as the limit when k tends to ` of k 2nukP ù Z nu. As we will recall, a link between both problems is provided by the study of Hamiltonian symmetries. Example 1: Consider the region D n of R n consisting of all points v 5 (t 1, t 2, . . . , t n), such that coordinates t i of v are nonnegative and satisfy the inequation t 1 1 t 2 1 z z z 1t n # 1 (Fig. 1). Let us consider the dilated simplex kD n. The volume of kD n is the homogeneous function of k,

vol~ kD n! 5

FIG. 1.

plectic volume of M is the integral of the Liouville form over M. Example 2: Consider the sphere S , R 3 with radius 1. We project S on R via the height z. The image of S is the interval [21, 1] (Fig. 2). In coordinates, (x 5 =1 2 z 2 cos f , y 5 =1 2 z 2 sin f , z), the volume form V of S is d f ` dz. This gives a system of Darboux coordinates (outside north and south poles). In particular the symplectic volume of S is (2p)21(4p) 5 2. This is also the length of [21, 1]. Let P be a convex polytope in R n. This means P is the convex hull of a finite set of points of R n. Under some conditions, which will be stated in the next section, there exists a compact symplectic manifold M P of dimension 2n, with Darboux coordinates (t 1, t 2, . . . , t n, f 1, f 2, . . . , f n) on an open dense subset U P of M P. Here the point v 5 (t 1, t 2, . . . , t n) varies in the interior P 0 of P, and f k are angles between 0 and 2p. Thus U P is isomorphic to P 0 3 S 1 3 S 1 3 z z z 3 S 1, where S 1 is the unit circle. The symplectic volume of M P will then be equal to the Euclidean volume of P. We can think of M P as an inflated version of P. The inflated symplectic manifold corresponding to the interval [21, 1] is the sphere S. The inflated symplectic manifold corresponding to the simplex D n of Example 1 is the projective space P n(C). We realize P n(C) as the space:

kn . n!

Consider the number of points v 5 (u 1, u 2, . . . , u n) with integral coordinates u i in kD n. If k is any nonnegative integer, this number is given by a polynomial function of k:

p n~ k ! 5 ukD n ù Z nu 5

Standard simplex.

~ k 1 1 !~ k 1 2 ! · · · ~ k 1 n ! . n!

This function is not homogeneous in k, but clearly its top order term in k is equal to the volume k nyn! of kD n. Remark that p n(k) is an integer for all k, while the volume of kD n is only a rational number. We will see that the number p n(k) arises naturally as the dimension of the quantum space associated to a symplectic manifold M n(k) of dimension 2n constructed by ‘‘inflating’’ kD n.

$u z 1u 2 1 · · · u z nu 2 1 u z n11u 2 5 1 % / ~ z 3 e iuz ! , with identification of all proportional points z and e iuz in the sphere S 2n11 in C n11. Darboux coordinates on an open dense set are (t 1, t 2, . . . , t n, f 1, f 2, . . . , f n) ° (t 1y2 e if1, z z z , t 1y2 e ifn, =1 2 • t k), 1 n where (t 1, t 2, . . . , t n) vary in D 0n. Although not every polytope P can be inflated to a smooth symplectic manifold M P, it may be worthwhile to give immediately a formula to compute the volume of any convex polytope P in R n or, more generally, the integral over P of any exponential function on R n. We state it for a generic polytope P with n edges through each vertex. At each vertex p of P, let us draw n vectors a p1, . . . , a pn on the edges through p. Let us normalize these vectors such that the parallelepiped constructed on these n vectors has volume 1. Then for a vector f

Volumes of Symplectic Manifolds and of Convex Polytopes A symplectic manifold of dimension 2n is a manifold M with a closed nondegenerate two-form V. The simplest example is the phase space R 2n 5 R n 3 R n, with symplectic coordinates (q 1, q 2, . . . , q n; p 1, p 2, . . . , p n) and symplectic form V 5 • nk51 dq k ` dp k. Around each point of a symplectic manifold M, it is possible to find such Darboux coordinates. The Liouville form on M is the volume form (2 p ) 2nV nyn!, and the sym14238

Proc. Natl. Acad. Sci. USA 93 (1996)

Colloquium Paper: Vergne

FIG. 2.

The sphere.

5 ( f 1, f 2, . . . , f n) of R n, such that â pk, f & Þ 0 for all vertices p and edges vectors a pk,

E

e ^f,t&dt 5 ~ 21 ! n

P

OP p

e ^f,p& n p k51^ a k,

f&

.

[1]

Here the sum runs over all vertices p of P. The volume of P is then obtained as a limit when f tends to 0. This leads to a formula for the volume of P in terms of vertices and edges:

vol~ P ! 5 ~ 21 ! n

O P

^f, p&n

p

n!

f&

n p 1 ^ a k,

.

[2]

The formula obtained is independent of the choice of f. As we will see, this formula, which can be proved in an elementary way (see refs. 1–3), has a beautiful generalization in symplectic geometry, the Duistermaat–Heckman formula. Let U(1) 5 {e if} be the circle group. Assume that U(1) acts on our symplectic manifold M (V being invariant). It is important to try to find an energy function for this action—that is, a real valued function f on M, such that the vector field X generated by the action of U(1) is the Hamiltonian vector field associated to f. This means that X is the vector field given in a system of Darboux coordinates by:

O

f f 2 . q j p j j51p j q j n

Xf 5

In particular, f is constant on the trajectories of the group U(1), by Noether’s theorem. The critical points of f are exactly the fixed points of the action of U(1) on M. In Example 2, for the rotation around the z axis, X is the vector field yf, the energy is the height function f 5 z, and the fixed points are the north and south poles. The ‘‘exact stationary phase formula’’ (4) of Duistermaat– Heckman for * M e ibf(x)dx (where dx denotes the Liouville measure) compute exactly this function of b in terms of the fixed points of the symmetry group U(1) on M. If the set of fixed points is finite,

E

M

e ibfdx 5

O p

~ 21 ! n

e ibf~p! i bn n

P

a pk

.

when f generates a periodic f low. Allowing manifolds MP with singularities, one may recover Eq. 1 as a particular case of Duistermaat–Heckman formula (Eq. 3). If M is compact connected, the image of the manifold M by f is an interval [a, b], and all points above the end points a or b of the interval, being critical points of f, are fixed by the action of U(1). This simple observation for the case of an Hamiltonian action of the circle group has a deep generalization for any torus action. Let M be a compact symplectic manifold with an action of a d-dimensional torus G 5 U(1) 3 U(1) 3 z z z 3 U(1). Assume we have d commuting Hamiltonian functions (f1, f2, . . . , fd) generating the action. Let f : M 3 Rd be the map with components fi. Then Atiyah–Guillemin–Sternberg theorem (5–7) asserts that the image of M by f is a convex polytope P. This implies a strong link between convex polytopes and Hamiltonian actions of compact abelian groups. More generally, Kirwan’s theorem (8) associates a convex polytope in a Weyl chamber to any Hamiltonian action of a compact Lie group on a compact symplectic manifold M. Let us here consider for simplicity only a Hamiltonian torus action. The map f : M 3 P will be called the moment map. Clearly, all points of M above vertices of P are fixed points of G. Singular values of f lies on hyper¯ , where C planes. Thus, P is the union of convex polytopes C is a connected component of the set of regular values of f (Fig. 3). Let t be a point in P. For f 5 ( f 1, f 2, . . . , f d), we denote by ^ f , f& the function • dk51 f kf k. Integrating the function e i^f,f& first on {f(x) 5 t}, then on P, we have:

E

e i^f,f&dx 5

M

E

e i^t,f&h ~ t ! dt.

P

Duistermaat–Heckman formula implies that h(t) is a continuous function of t [ P. It is given by a polynomial function h C of degree at most (n 2 d) on each connected component C of the set of regular values. What is the meaning of this function h(t)? Assume t a regular value of f. Each fiber {f(x) 5 t} is connected and stable by the action of G. The reduced fiber is the Marsden– Weinstein symplectic quotient (9) obtained by ‘‘reduction of degrees of freedom’’ M red(t) 5 {f(x) 5 t}y(x ° gzx). The space M red(t) is a symplectic space that may have some singularities. In the next theorem, we summarize these results of Atiyah, Guillemin–Sternberg, and Duistermaat–Heckman on Hamiltonian torus actions. THEOREM 1. Let f 5 (f1, . . . , fd) be the moment map for a Hamiltonian torus action on a compact connected symplectic manifold M. Then, (i) the image of M by f is a convex polytope P, and (ii) we have:

[3]

Here p runs through all fixed points and apk are integers such that Xf is, near each p, equal to a product of infinitesimal rotations with speed apk. Clearly this formula can be used to compute the volume of M, in the case where M has a circular Hamiltonian symmetry. Note the similarity between integral over a convex polytope P in Rn of exponential functions and integral over a symplectic manifold M of the function eibf

14239

FIG. 3.

Regular values of the moment map.

14240



E

M

e

i^f,f&

dx5

E

e

i^t,f&

h(t)dt.

P

The continuous function h(t), supported on the convex polytope P 5 f(M), is locally polynomial of degree at most (n 2 d). It is given in function of the fiber of the moment map by the formula:

h(t)5volMred(t). In particular, the symplectic volume of M is calculated by integrating over P the locally polynomial function h(t). The simplest case of this theorem is the case of a completely integrable action, where d 5 n. Then M is the manifold M P constructed by inflating P. There are only a finite set F of fixed points under the action of the group G, and each of the point f(p) for p [ F is a vertex of P. Each fiber of the map f is a single orbit for the action of G; consequently M red(t) is just a point for all t [ P. Thus the image of the Liouville measure is the characteristic function of P, identically 1 on P. In particular, as we have already noted, the volume of M P is equal to the volume of P. Using Duistermaat–Heckman formula (Eq. 3), we can compute h(t) alternatively either in function of the fixed points of the action of G or in function of the volume of the reduced fiber M red(t). A formula, similar to (Eq. 3), exists to compute the integral on M of any equivariant cohomology class. This is the ‘‘abelian’’ localization formula in equivariant cohomology (10, 11). In ref. 12, Witten remarked that any integral of De Rham cohomology classes over the reduced fiber can be computed in function of fixed points for the action of G in M. This will be referred to as the ‘‘nonabelian’’ localization formula. We will see the fundamental implications of this observation for quantum mechanics. Dimensions and Number of Points in Convex Polytopes Is there a ‘‘quantum analogue’’ of Duistermaat–Heckman theorem on the piecewise polynomial behavior of the pushforward measure by the moment map? We need to recall some of the basic constructions of quantum mechanics. The quantum model Q(M,V) of the classical model M is searched as a vector space of functions on M. For the phase space R 2n 5 R n 3 R n, then Q(M,V) is a space of functions on M depending only on n ‘‘commuting’’ variables among the 2n variables (q k, p k). We thus can choose Q(M,V) to be the space of functions of p k or of the variables q k, in the Schro ¨dinger model, or as well functions of the complex variables z k 5 p k 1 iq k in the Fock–Bargmann model. Furthermore, ideally, Q(M,V) should still carry the Hamiltonian symmetries of M. Although there is no invariant way in general to select n commuting variables, we can sometimes still lift some symmetries of M to symmetries of Q(M,V). The construction of Q(M, V) with all possible Hamiltonian symmetries of M lifted to the space Q(M,V) is not possible, but we will see that a quantum model Q(M,V) can be constructed in a satisfactory way, in the case of a compact symplectic manifold M with integral symplectic form and with symmetries coming from an action of a compact symmetry group G. Assume the symplectic form V is integral. Then M is ‘‘quantizable’’ in the sense of Kostant and Souriau (26, 27); we have V 5 iF, where F is the curvature of +, the Kostant– Souriau line bundle on M. We will construct Q(M,V) [denoted also by Q(M,+)] via a positive almost-complex structure (see refs. 13 and 14). Roughly speaking, locally M is modeled on R 2n with symplectic coordinates (q k,p k), and this means, intuitively, that we will construct Q(M,V) as functions on M of the n complex variables z k 5 p k 1 iq k. This procedure is well adapted to the action of the circular symmetry in the (q k, p k)

plane, which transforms z k to e ifkz k. To be precise, if M is a compact complex manifold and + 3 M is an holomorphic line bundle on M with positive curvature form given by the formula:

F 5 ¯ logusu 2, where s is a nonvanishing holomorphic section of + over a chart, then Q(M,mV) coincide with the space H 0(M,2(+ m)) of holomorphic sections of the line bundle + m 5 R m+ when m is sufficiently large. Unfortunately, it is necessary to allow virtual vector spaces in the construction of Q(M,V). We define the space Q(M,V) to be [Ker D 1] 2 [Ker D 2], where D is the ¯-operator (associated to the almost-complex structure) on +-valued forms on M of type (0, q). In the case of a holomorphic line bundle over a compact complex manifold, we take Q(M,mV) 5 • nk50 (21) kH k(M,2(+ m)). When m is sufficiently large, the positivity assumption on + implies that all cohomology spaces H k(M,2(+ m)) vanishes for k . 0. We write dim Q(M,V) for the integer (maybe negative) dim[Ker D 1] 2 dim[Ker D 2]. This integer is the quantum analogue of the symplectic volume of M. Let P be a convex polytope in R n such that the vertices of P have integral coordinates. Then we can inflate P to an algebraic manifold M P with a Kostant–Souriau line bundle + P. The space Q(M P,+ P) has a basis indexed by points with integral coordinates contained in P. Example: Points v 5 (u 1, u 2, . . . , u n) with integral coordinates in the simplex kD n label the monomial basis i) of the space Q(P n(C),+ k) of homogez 1u1z 2u2 . . . z nunz (k2•u n11 neous polynomials of degree k in n 1 1 variables. There is a formula to compute the sum over integers contained in P of any exponential function on R n (1–3). We state it here only for a Delzant polytope P with n edges through each vertex. A polytope P is a Delzant polytope, if each vertex p of P have integral coordinates and if, furthermore, we can draw n vectors a p1, . . . , a pn on the edges through p, with integral coordinates, and such that the parallelepiped constructed on these n vectors has volume 1. Then for a small vector f 5 ( f 1, f 2, . . . , f n) of R n, such that â pk, f & Þ 0 for all vertices p:

O

e ^f,u& 5

u[PùZn

OP p

e ^f,p& n 1~1

p

2 e â k, f &!

.

[4]

This formula, which can be proved in an elementary way, is a particular case of Atiyah–Bott formula (15) for the manifold M P (with our assumption on P, the manifold M P is indeed smooth). The number of integral points in P is then obtained from the above formula as a limit when f tends to 0. Comparing with Eq. 2, we see that this leads to a formula (see refs. 2 and 3) for this number in terms of the volume of P, volumes of faces of P, and Bernoulli numbers. Let G 5 U(1) be acting on a quantizable manifold M. If the action of U(1) lifts to +, this provides an energy function f. This function f takes integral values on fixed points. The action of G lifts to an action on Q(M,V). The operator F on the vector space Q(M,V), such that the one parameter group (e if) of symmetries of M lifts in the action of e ifF, is a self-adjoint operator on Q(M,V) with integral eigenvalues. If the set of fixed points of G on M is finite, then Atiyah–Bott fixed point formula, with notations as in Eq. 3, gives:

TrQ~M,V!~ e ibF! 5

OP p

e ibf~p! n k51~ 1

p

2 e i b a k!

.

This formula is similar to Eq. 3 in the ‘‘classical’’ case for * M e ibfdx.


Colloquium Paper: Vergne Let G 5 U(1) 3 U(1) z z z 3 U(1) be a group of d commuting circular symmetries of M. Assume that this action lifts to an action on the line bundle +. Thus, the image f(M) of M is a convex polytope P with vertices that are points of R d with integral coordinates. Let F k be the operator on Q(M,V) associated to the one parameter group e ifk. We think of F k as the quantum analogue of the observable f k. Eigenvalues of F k are the ‘‘quantum’’ levels of the energy function f k. As we will see, the multiplicity of the eigenvalue u k is related to the classical level of energy, where the energy function takes the value u k. Consider the common eigenspace decomposition for the commuting self-adjoint operators F k:

Q ~ M,+ ! 5 % t[ZdQ t, where for t 5 (u 1, u 2, . . . , u d) a point in R d with integral coordinates,

Q t 5 $ v [ Q ~ M,V !u F kv 5 u kv,k 5 1, 2, . . . , d % . We denote by q(+,t) the dimension of Q t. If + is fixed, we denote it simply by q(t). Let f [ R d and let ^ f , F& 5 • k f kF k. Then the matrix of the action of exp i^ f , F& in Q(M,V) is diagonal with the diagonal term e iu1f1e iu2f2 z z z e iudfd appearing q(t) times. Thus the trace of exp i^ f , F& (or, more exactly, the super trace) is:

TrQ~M,V!e i^f,F& 5

O

q ~ t ! e i^t,f&.

t[Zd

The function f ° TrQ(M,V)e i^f,F& is the analogue in the quantum case of the function f ° * M e i^f,f& in the classical case. Its Fourier decomposition • t[Zd q(t)e i^t,f& should be related to the Fourier decomposition * M e i^f,f&dx 5 * P e i^t,f&h(t)dt. However, h(t)dt is a continuous measure on P, while q(t) is a discrete measure supported on P ù Z d. How is q(t) related to the reduced fiber M red(t) above t? The following theorem was stated by Guillemin and Sternberg in (16) as a conjecture. It is the generalization of Kirwan’s theorem (17) on geometric invariant theory for actions of complex reductive groups on projective varieties: the geometric quotient can be realized as a ‘‘symplectic quotient’’ (I will describe the theorem here only in the torus case). This theorem was proved recently using different approaches and in various degrees of generality by a number of mathematicians (13, 14, 18 –23). An excellent survey is given by Sjamaar in ref. 24. The impetus was probably given by the nonabelian localization formula of Witten (12) and further work by Jeffrey and Kirwan (25). It gives a fundamental justification for choosing Q(M,V) as ‘‘the’’ quantum model for the classical model M. THEOREM 2. Let M be a quantizable compact connected symplectic manifold. Let f 5 (f1, . . . , fd) be the moment map for a Hamiltonian torus action on M lifting to Kostant–Souriau line bundle +. Then: (i) The multiplicity function q(t) is supported on Zd ù P. Thus we have:

TrQ(M,V)ei^f,F&5

O

q(t)ei^t,f&.

t[ZdùP

(ii) The value q(t) at an integral value t of f is related to the reduced fiber Mred(t) by the formula:

q(t)5dim(Q(Mred(t),+red(t)). We need to explain the last formula. At an integral value t of the moment map f, the reduction + red(t) of the Kostant– Souriau line bundle is defined to be the line bundle +u f21(t)yG. This is a Kostant–Souriau line bundle for M red(t). Thus we can

14241

define the quantum ‘‘volume’’ dim Q(M red(t), + red(t)) of the reduced fiber, which is indeed an integer. We thus see that this theorem is the quantum analogue of the classical decomposition of Theorem 1. Let us give an idea of a proof of this theorem. From the Atiyah–Segal–Singer formula for the index of twisted Dirac operator, the number Q(M red(t), + red(t)) is the integral over the reduced fiber M red(t) of a de Rham cohomology class, involving the Todd class of M red(t). Jeffrey–Kirwan–Witten nonabelian localization formula shows that it possible to compute this number in function of fixed points for the action of G on the ambient manifold M. On the other hand, TrQ(M,V)e ifF is itself given in function of the fixed points for the G-action by Atiyah–Bott–Segal–Singer formula. Careful examination of both formulas leads to the comparison result. Consider again the simplest case of this theorem for the case of a completely integrable action on a quantizable manifold M. Then f(M) is a convex polytope with integral vertices. Each point t [ P ù Z n labels an eigenvector for the quantum representation of G. In particular, we have dim Q(M,V) 5 card(P ù Z n). Let us compare the continuous measure h(t) and the discrete measure q(t) for the case of the circle action z ° e iuz on C n11 with energy function f 5 izi 2. Although this example is not compact, level surfaces of f are compact and the same theorem still holds. The quantum space Q(C n) is the Bargmann space of holomorphic functions in (n 1 1) variables, and we have the eigenspace decomposition:

Q ~ C n! 5 % kQ k, where Q k is the space of homogeneous polynomials of degree k in (n 1 1) variables. In this case, we have:

h ~ t ! 5 t n/n!, q~t! 5

~ t 1 1 !~ t 1 2 ! · · · ~ t 1 n ! . n!

It is also possible to prove, as announced by Meinrenken and Sjamaar, a remarkable ‘‘continuity property’’ for the function q(+,t). As the function t 3 q(+,t) is a priori defined only on the finite set P ù Z n, we need to enlarge its domain of definition for stating a meaningful continuity property. It would be worth to investigate further its continuity properties in terms of both variables + and t. Let m be any positive integer. Define q(m,t) to be q(+ m,t). Then t ° q(m,t) is supported on mP ù Z d. Let C be a connected component of the set of regular values of f. Consider the open convex cone # in R d11 with base C.

# 5 $~ t,v ! , t . 0, v [ tC % . The function q(m,t) is defined on # ù Z d11. A quasipolynomial function on R d11 is a function in the algebra generated by polynomials and periodic functions (with sufficiently large period). Then, there exists a unique quasipolynomial function ¯ ù q C on R d11 such that q(m,t) 5 q C(m,t) for all (m,t) [ # Z d11. The next example shows that, inexorably, quasipolynomial functions appear in the subject. Example 3. Consider the case of an action of U(1) on M 5 P1(C) 3 P1(C) by eiu on the first factor and e2iu in the second factor. Then the function f takes values in [0, 3] with singular

14242



values 0, 1, 2, 3. We have for m $ 0, 0 # t # 3m, t, m integers,

q ~ m,t ! 5

t 3 1 1 1 ~ 21 ! t , 2 4 4

0 # t # m,

1 m 3 q ~ m,t ! 5 1 1 ~ 21 ! m , 2 4 4 q~m,2m 1 t! 5 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 13.

1 m2t 3 1 1 ~21!m2t , 2 4 4

14.

m # t # 2m, 2m # 2m 1 t # 3m.

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