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 and quantization of symplectic

MICHE`LE VERGNE

Ecole Normale Supe´rieureet Unite´Associe´e762 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 sym- plectic 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 of M and the dimension of the 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 has a quantum analogue that is the dimension of the quantum model Q(M) for M.Itis important to understand the relation between both quantities, as the volume of the manifold M is just a ‘‘limit’’ of the FIG. 1. Standard . dimension of Q(M). A similar comparison problem is the n following: if P ʚ ޒ is a convex , can we compare the plectic volume of M is the integral of the Liouville form over n number ͉P പ ޚ ͉ of points in P with integral coordinates and M. the volume of P? It is clear that the volume of P is obtained Example 2: Consider the sphere S ʚ ޒ3 with radius 1. We Ϫn n as the limit when k tends to ϱ of k ͉kP പ ޚ ͉. As we will recall, project S on ޒ via the height z. The image of S is the interval a link between both problems is provided by the study of [Ϫ1, 1] (Fig. 2). Hamiltonian symmetries. In coordinates, (x ϭ ͌1 Ϫ z2 cos ␾, y ϭ ͌1 Ϫ z2 sin ␾, z), n consisting of all the volume form ⍀ of S is d␾ ٙ dz. This gives a system of ޒ Example 1: Consider the region ⌬n of points v ϭ (t1, t2,..., tn), such that coordinates ti of v are Darboux coordinates (outside north and south poles). In nonnegative and satisfy the inequation t1 ϩ t2 ϩ ⅐⅐⅐ϩtn Յ1 particular the symplectic volume of S is (2␲)Ϫ1(4␲) ϭ 2. This (Fig. 1). is also the length of [Ϫ1, 1]. Let us consider the dilated simplex k⌬n. The volume of k⌬n Let P be a convex polytope in ޒn. This means P is the convex is the homogeneous function of k, hull of a finite set of points of ޒn. Under some conditions, which will be stated in the next section, there exists a compact kn symplectic manifold M of dimension 2n, with Darboux co- vol͑k⌬n͒ ϭ . P n! ordinates (t1, t2,..., tn, ␾1, ␾2,..., ␾n) on an open dense subset UP of MP. Here the point v ϭ (t1, t2,...,tn) varies in Consider the number of points v ϭ (u1, u2,..., un) with 0 the interior P of P, and ␾k are between 0 and 2␲. Thus integral coordinates ui in k⌬n.Ifkis any nonnegative integer, 0 1 1 1 1 UP is isomorphic to P ϫ S ϫ S ϫ ⅐⅐⅐ϫS , where S is the this number is given by a polynomial function of k: unit circle. The symplectic volume of MP will then be equal to P M k ϩ 1 k ϩ 2 ··· kϩn the Euclidean volume of . We can think of P as an inflated n ͑ ͒͑ ͒ ͑ ͒ pn͑k͒ ϭ ͉k⌬n പ ޚ ͉ ϭ . version of P. The inflated symplectic manifold corresponding n! to the interval [Ϫ1, 1] is the sphere S. The inflated symplectic manifold corresponding to the simplex ⌬ of Example 1 is the This function is not homogeneous in k, but clearly its top order n n projective space Pn(ރ). We realize Pn(ރ) as the space: term in k is equal to the volume k ͞n!ofk⌬n. Remark that 2 2 2 i␪ pn(k) is an integer for all k, while the volume of k⌬n is only a ͕͉z1͉ ϩ ···͉zn͉ ϩ͉znϩ1͉ ϭ1͖/͑z3e z͒, rational number. i␪ We will see that the number pn(k) arises naturally as the with identification of all proportional points z and e z in the dimension of the quantum space associated to a symplectic sphere S2nϩ1 in ރnϩ1. manifold Mn(k) of dimension 2n constructed by ‘‘inflating’’ Darboux coordinates on an open dense set are (t1, t2,..., 1͞2 i␾1 1͞2 i␾n ,(t1 e ,⅐⅐⅐, tn e , ͌1Ϫ•tk) ۋ (k⌬n. tn, ␾1, ␾2,..., ␾n 0 where (t1, t2,...,tn) vary in ⌬n. of Symplectic Manifolds and of Although not every polytope P can be inflated to a smooth symplectic manifold MP, it may be worthwhile to give imme- A symplectic manifold of dimension 2n is a manifold M with diately a formula to compute the volume of any convex a closed nondegenerate two-form ⍀. The simplest example is polytope P in ޒn or, more generally, the integral over P of any the phase space ޒ2n ϭ ޒn ϫ ޒn, with symplectic coordinates exponential function on ޒn. We state it for a generic polytope n (q1, q2,...,qn;p1,p2,...,pn) and symplectic form ⍀ϭ•kϭ1 P with n edges through each . At each vertex p of P, let p p dqk ٙ dpk. Around each point of a symplectic manifold M,it us draw n vectors a1,...,an on the edges through p. Let us is possible to find such Darboux coordinates. The Liouville normalize these vectors such that the parallelepiped con- form on M is the volume form (2␲)Ϫn⍀n͞n!, and the sym- structed on these n vectors has volume 1. Then for a vector ␾

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when f generates a periodic flow. 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 generali- zation for any torus action. Let M be a compact symplectic manifold with an action of a d-dimensional torus G ϭ U(1) ϫ U(1) ϫ ⅐⅐⅐ϫ U(1). Assume we have d commuting Hamiltonian functions (f1, d f2,...,fd) generating the action. Let f : M 3 ޒ be the map with components fi. Then Atiyah–Guillemin–Sternberg the- orem (5–7) asserts that the image of M by f is a convex polytope P. This implies a strong link between convex FIG. 2. The sphere. polytopes and Hamiltonian actions of compact abelian n p groups. More generally, Kirwan’s theorem (8) associates a ϭ (␾1, ␾2,...,␾n)ofޒ , such that ͗ak, ␾͘Þ0 for all vertices p and edges vectors ap, convex polytope in a Weyl chamber to any Hamiltonian k action of a compact Lie group on a compact symplectic e͗␾,p͘ manifold M. Let us here consider for simplicity only a e͗␾,t͘dt ϭ ͑Ϫ1͒n . [1] Hamiltonian torus action. The map f : M 3 P will be called ͸ n p ͵ p kϭ1͗ak, ␾͘ P ͹ the moment map. Clearly, all points of M above vertices of P are fixed points of G. Singular values of f lies on hyper- Here the sum runs over all vertices p of P. The volume of P is planes. Thus, P is the union of convex polytopes C¯, where C then obtained as a limit when ␾ tends to 0. This leads to a is a connected component of the set of regular values of f formula for the volume of P in terms of vertices and edges: (Fig. 3). Let t be a point in P. For ␾ ϭ (␾1, ␾2,...,␾d), we denote n ͗␾, p͘ by ͗␾, f͘ the function •d ␾ f . Integrating the function ei͗␾,f͘ vol͑P͒ ϭ ͑Ϫ1͒n . [2] kϭ1 k k ͸ n p ␾ first on {f(x) ϭ t}, then on P, we have: p n!͹1͗ak, ͘ ␾ The formula obtained is independent of the choice of .Aswe i͗␾,f͘ i͗t,␾͘ will see, this formula, which can be proved in an elementary ͵ e dx ϭ ͵ e h͑t͒dt. way (see refs. 1–3), has a beautiful generalization in symplectic M P , the Duistermaat–Heckman formula. Let U(1) ϭ {ei␾} be the circle group. Assume that U(1) acts Duistermaat–Heckman formula implies that h(t) is a contin- on our symplectic manifold M (⍀ being invariant). It is uous function of t ʦ P. It is given by a polynomial function hC important to try to find an energy function for this action—that of degree at most (n Ϫ d) on each connected component C of is, a real valued function f on M, such that the vector field X the set of regular values. generated by the action of U(1) is the Hamiltonian vector field What is the meaning of this function h(t)? Assume t a associated to f. This means that X is the vector field given in regular value of f. Each fiber {f(x) ϭ t} is connected and stable a system of Darboux coordinates by: by the action of G. The reduced fiber is the Marsden– Weinstein symplectic quotient (9) obtained by ‘‘reduction of n g⅐x). The space ۋ Ѩf Ѩ Ѩf Ѩ degrees of freedom’’ Mred(t) ϭ {f(x) ϭ t}͞(x X ϭ Ϫ . f ͸ Mred(t) is a symplectic space that may have some singularities. jϭ1Ѩpj Ѩqj Ѩqj Ѩpj In the next theorem, we summarize these results of Atiyah, In particular, f is constant on the trajectories of the group U(1), Guillemin–Sternberg, and Duistermaat–Heckman on Hamil- by Noether’s theorem. The critical points of f are exactly the tonian torus actions. fixed points of the action of U(1) on M. THEOREM 1. Let f ϭ (f1,...,fd) be the moment map for a In Example 2, for the rotation around the z axis, X is the Hamiltonian torus action on a compact connected symplectic vector field Ѩ͞Ѩ␾, the energy is the height function f ϭ z, and manifold M. Then, (i) the image of M by f is a convex polytope the fixed points are the north and south poles. P, and (ii) we have: The ‘‘exact stationary phase formula’’ (4) of Duistermaat– i␤f(x) Heckman for ͐M e dx (where dx denotes the Liouville measure) compute exactly this function of ␤ in terms of the fixed points of the symmetry group U(1) on M. If the set of fixed points is finite,

ei␤f͑p͒ ei␤fdx ϭ ͑Ϫ1͒n . [3] ͸ n n p ͵ p i ␤ ak M ͹

p Here p runs through all fixed points and ak are integers such that Xf is, near each p, equal to a product of infinitesimal p rotations with speed ak. 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 ޒn of exponential functions and i␤f integral over a symplectic manifold M of the function e FIG. 3. Regular values of the moment map. Downloaded by guest on September 26, 2021 14240 Colloquium Paper: Vergne Proc. Natl. Acad. Sci. USA 93 (1996)

i␾k plane, which transforms zk to e zk. To be precise, if M is a ͵ ei͗␾,f͘dxϭ͵ ei͗t,␾͘h(t)dt. compact complex manifold and ᏸ 3 M is an holomorphic M P bundle on M with positive curvature form given by the formula: The continuous function h(t), supported on the convex poly- ¯ 2 tope P ϭ f(M), is locally polynomial of degree at most (n Ϫ d). F ϭѨѨlog͉s͉ , It is given in function of the fiber of the moment map by the formula: where s is a nonvanishing holomorphic section of ᏸ over a chart, then Q(M,m⍀) coincide with the space H0(M,ᏻ(ᏸm)) of m m h(t)ϭvolMred(t). holomorphic sections of the line bundle ᏸ ϭ R ᏸ when m is sufficiently large. Unfortunately, it is necessary to allow In particular, the symplectic volume of M is calculated by virtual vector spaces in the construction of Q(M,⍀). We define integrating over P the locally polynomial function h(t). The the space Q(M,⍀) to be [Ker Dϩ] Ϫ [Ker DϪ], where D is the simplest case of this theorem is the case of a completely Ѩ¯-operator (associated to the almost-complex structure) on integrable action, where d ϭ n. Then M is the manifold MP ᏸ-valued forms on M of type (0, q). In the case of a constructed by inflating P. There are only a finite set F of fixed holomorphic line bundle over a compact complex manifold, we points under the action of the group G, and each of the point n k k m take Q(M,m⍀) ϭ •kϭ0 (Ϫ1) H (M,ᏻ(ᏸ )). When m is f(p) for p ʦ F is a vertex of P. Each fiber of the map f is a single sufficiently large, the positivity assumption on ᏸ implies that orbit for the action of G; consequently Mred(t) is just a point all cohomology spaces Hk(M,ᏻ(ᏸm)) vanishes for k Ͼ 0. We for all t ʦ P. Thus the image of the Liouville measure is the write dim Q(M,⍀) for the integer (maybe negative) dim[Ker characteristic function of P, identically 1 on P. In particular, as Dϩ] Ϫ dim[Ker DϪ]. This integer is the quantum analogue of we have already noted, the volume of MP is equal to the volume the symplectic volume of M. of P. Let P be a convex polytope in ޒn such that the vertices of P 3 Using Duistermaat–Heckman formula (Eq. ), we can com- have integral coordinates. Then we can inflate P to an alge- pute h(t) alternatively either in function of the fixed points of braic manifold M with a Kostant–Souriau line bundle ᏸ . The the action of G or in function of the volume of the reduced P P space Q(M ,ᏸ ) has a basis indexed by points with integral fiber M (t). A formula, similar to (Eq. 3), exists to compute P P red coordinates contained in P. the integral on M of any equivariant cohomology class. This is Example: v u u u the ‘‘abelian’’ localization formula in equivariant cohomology Points ϭ ( 1, 2,..., n) with integral coordi- (10, 11). In ref. 12, Witten remarked that any integral of De nates in the simplex k⌬n label the monomial basis u1 u2 un (kϪ•ui) Rham cohomology classes over the reduced fiber can be z1 z2 ...zn znϩ1 of the space Q(Pn(ރ),ᏸk) of homoge- computed in function of fixed points for the action of G in M. neous polynomials of degree k in n ϩ 1 variables. This will be referred to as the ‘‘nonabelian’’ localization There is a formula to compute the sum over integers n formula. We will see the fundamental implications of this contained in P of any exponential function on ޒ (1–3). We observation for quantum mechanics. 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 Dimensions and Number of Points in Convex Polytopes p of P have integral coordinates and if, furthermore, we can p p draw n vectors a1,...,anon the edges through p, with integral Is there a ‘‘quantum analogue’’ of Duistermaat–Heckman coordinates, and such that the parallelepiped constructed on theorem on the piecewise polynomial behavior of the push- these n vectors has volume 1. Then for a small vector ␾ ϭ (␾1, n p forward measure by the moment map? ␾2,...,␾n)ofޒ, such that ͗ak, ␾͘Þ0 for all vertices p: We need to recall some of the basic constructions of e͗␾,p͘ quantum mechanics. The quantum model Q(M,⍀) of the ͗␾,u͘ e ϭ n p . [4] classical model M is searched as a vector space of functions on ͸ ͸ ͗ak,␾͘ n p 1͑1 Ϫ e ͒ M. For the phase space ޒ2n ϭ ޒn ϫ ޒn, then Q(M,⍀) is a space uʦPപޚ ͹ of functions on M depending only on n ‘‘commuting’’ variables This formula, which can be proved in an elementary way, is a among the 2n variables (qk, pk). We thus can choose Q(M,⍀) particular case of Atiyah–Bott formula (15) for the manifold to be the space of functions of pk or of the variables qk, in the MP (with our assumption on P, the manifold MP is indeed Schro¨dinger model, or as well functions of the complex vari- smooth). ables zk ϭ pk ϩ iqk in the Fock–Bargmann model. Further- The number of integral points in P is then obtained from the more, ideally, Q(M,⍀) should still carry the Hamiltonian above formula as a limit when ␾ tends to 0. Comparing with symmetries of M. Although there is no invariant way in general Eq. 2, we see that this leads to a formula (see refs. 2 and 3) for to select n commuting variables, we can sometimes still lift this number in terms of the volume of P, volumes of faces of some symmetries of M to symmetries of Q(M,⍀). The con- P, and Bernoulli numbers. struction of Q(M, ⍀) with all possible Hamiltonian symmetries G U M of M lifted to the space Q(M,⍀) is not possible, but we will see Let ϭ (1) be acting on a quantizable manifold . If the that a quantum model Q(M,⍀) can be constructed in a action of U(1) lifts to ᏸ, this provides an energy function f. satisfactory way, in the case of a compact symplectic manifold This function f takes integral values on fixed points. The action M with integral symplectic form and with symmetries coming of G lifts to an action on Q(M,⍀). The operator F on the vector i␾ from an action of a compact symmetry group G. space Q(M,⍀), such that the one parameter group (e )of i␾F Assume the symplectic form ⍀ is integral. Then M is symmetries of M lifts in the action of e , is a self-adjoint ‘‘quantizable’’ in the sense of Kostant and Souriau (26, 27); we operator on Q(M,⍀) with integral eigenvalues. If the set of have ⍀ϭiF, where F is the curvature of ᏸ, the Kostant– fixed points of G on M is finite, then Atiyah–Bott fixed point Souriau line bundle on M. We will construct Q(M,⍀) [denoted formula, with notations as in Eq. 3, gives: also by Q(M,ᏸ)] via a positive almost-complex structure (see ei␤f͑p͒ refs. 13 and 14). Roughly speaking, locally M is modeled on ޒ2n i␤F TrQ͑M,⍀͒͑e ͒ ϭ n p . ͸ i␤ak with symplectic coordinates (qk,pk), and this means, intu- p kϭ1͑1 Ϫ e ͒ itively, that we will construct Q(M,⍀) as functions on M of the ͹ n complex variables zk ϭ pk ϩ iqk. This procedure is well This formula is similar to Eq. 3 in the ‘‘classical’’ case for ͐M i␤f adapted to the action of the circular symmetry in the (qk, pk) e dx. Downloaded by guest on September 26, 2021 Colloquium Paper: Vergne Proc. Natl. Acad. Sci. USA 93 (1996) 14241

Let G ϭ U(1) ϫ U(1) ⅐⅐⅐ϫ U(1) be a group of d define the quantum ‘‘volume’’ dim Q(Mred(t), ᏸred(t)) of the commuting circular symmetries of M. Assume that this action reduced fiber, which is indeed an integer. We thus see that this lifts to an action on the line bundle ᏸ. Thus, the image f(M) theorem is the quantum analogue of the classical decomposi- of M is a convex polytope P with vertices that are points of ޒd tion of Theorem 1. with integral coordinates. Let Fk be the operator on Q(M,⍀) Let us give an idea of a proof of this theorem. From the i␾k associated to the one parameter group e . We think of Fk as Atiyah–Segal–Singer formula for the index of twisted Dirac the quantum analogue of the observable fk. Eigenvalues of Fk operator, the number Q(Mred(t), ᏸred(t)) is the integral over are the ‘‘quantum’’ levels of the energy function fk.Aswewill the reduced fiber Mred(t) of a de Rham cohomology class, see, the multiplicity of the eigenvalue u is related to the k involving the Todd class of Mred(t). Jeffrey–Kirwan–Witten classical level of energy, where the energy function takes the nonabelian localization formula shows that it possible to value uk. Consider the common eigenspace decomposition for compute this number in function of fixed points for the action the commuting self-adjoint operators Fk: of G on the ambient manifold M. On the other hand, i␾F TrQ(M,⍀)e is itself given in function of the fixed points for the Q͑M,ᏸ͒ ϭ  tʦޚdQt, G-action by Atiyah–Bott–Segal–Singer formula. Careful ex- d where for t ϭ (u1, u2,..., ud) a point in ޒ with integral amination of both formulas leads to the comparison result. coordinates, Consider again the simplest case of this theorem for the case of a completely integrable action on a quantizable manifold M. Qt ϭ ͕v ʦ Q͑M,⍀͉͒Fkv ϭ ukv,k ϭ 1, 2, . . . , d͖. Then f(M) is a convex polytope with integral vertices. Each point t ʦ P പ ޚn labels an eigenvector for the quantum We denote by q(ᏸ,t) the dimension of Q .Ifᏸis fixed, we t representation of G. In particular, we have dim Q(M,⍀) ϭ denote it simply by q(t). Let ␾ ʦ ޒd and let ͗␾, F͘ϭ• ␾ F . k k k card(P പ ޚn). Then the of the action of exp i͗␾, F͘ in Q(M,⍀)is Let us compare the continuous measure h(t) and the diagonal with the diagonal term eiu1␾1eiu2␾2 ⅐⅐⅐eiud␾d appearing ei␪z ۋ discrete measure q(t) for the case of the circle action z q(t) times. Thus the trace of exp i͗␾, F͘ (or, more exactly, the ރnϩ1 2 super trace) is: on with energy function f ϭ ʈzʈ . Although this example is not compact, level surfaces of f are compact and the same i͗␾,F͘ i͗t,␾͘ theorem still holds. The quantum space Q(ރ ) is the Barg- TrQ͑M,⍀͒e ϭ ͸ q͑t͒e . n tʦޚd mann space of holomorphic functions in (n ϩ 1) variables, and we have the eigenspace decomposition: i͗␾,F͘ TrQ(M,⍀)e is the analogue in the ۋ The function ␾ i͗␾,f͘ .M e in the classical case͐ ۋ quantum case of the function ␾ i͗t,␾͘ Q͑ރn͒ ϭ  kQk, Its Fourier decomposition •tʦޚd q(t)e should be related to i͗␾,f͘ i͗t,␾͘ the Fourier decomposition ͐M e dx ϭ͐P e h(t)dt. However, h(t)dt is a continuous measure on P, while q(t)isa where Qk is the space of homogeneous polynomials of degree discrete measure supported on P പ ޚd. How is q(t) related to k in (n ϩ 1) variables. In this case, we have: the reduced fiber Mred(t) above t? The following theorem was stated by Guillemin and h͑t͒ ϭ tn/n!, Sternberg in (16) as a conjecture. It is the generalization of Kirwan’s theorem (17) on geometric invariant theory for ͑t ϩ 1͒͑t ϩ 2͒···͑tϩn͒ actions of complex reductive groups on projective varieties: q͑t͒ ϭ . the geometric quotient can be realized as a ‘‘symplectic n! quotient’’ (I will describe the theorem here only in the torus case). This theorem was proved recently using different It is also possible to prove, as announced by Meinrenken and approaches and in various degrees of generality by a number Sjamaar, a remarkable ‘‘continuity property’’ for the function of mathematicians (13, 14, 18–23). An excellent survey is q(ᏸ,t). As the function t 3 q(ᏸ,t)isa priori defined only on given by Sjamaar in ref. 24. The impetus was probably given ޚn by the nonabelian localization formula of Witten (12) and the finite set P പ , we need to enlarge its domain of further work by Jeffrey and Kirwan (25). It gives a funda- definition for stating a meaningful continuity property. It mental justification for choosing Q(M,⍀) as ‘‘the’’ quantum would be worth to investigate further its continuity properties model for the classical model M. in terms of both variables ᏸ and t. m THEOREM 2. Let M be a quantizable compact connected Let m be any positive integer. Define q(m,t)tobeq(ᏸ ,t). d. Let C be aޚ q(m,t) is supported on mP പ ۋ Then t symplectic manifold. Let f ϭ (f1,...,fd) be the moment map for a Hamiltonian torus action on M lifting to Kostant–Souriau line connected component of the set of regular values of f. Con- bundle ᏸ. sider the open convex Ꮿ in ޒdϩ1 with base C. Then: (i) The multiplicity function q(t) is supported on ޚd പ P. Thus Ꮿ ϭ ͕͑t,v͒, t Ͼ 0, v ʦ tC͖. we have:

i͗␾,F͘ i͗t,␾͘ The function q(m,t) is defined on Ꮿ പ ޚdϩ1. A quasipolyno- TrQ(M,⍀)e ϭ q(t)e . ͸ dϩ1 tʦޚdപP mial function on ޒ is a function in the algebra generated by polynomials and periodic functions (with sufficiently large (ii) The value q(t) at an integral value t of f is related to the period). Then, there exists a unique quasipolynomial function dϩ1 reduced fiber Mred(t) by the formula: qC on ޒ such that q(m,t) ϭ qC(m,t) for all (m,t) ʦ Ꮿ¯ പ ޚdϩ1. q(t)ϭdim(Q(M (t),ᏸ (t)). red red The next example shows that, inexorably, quasipolynomial We need to explain the last formula. At an integral value t functions appear in the subject. Example 3. Consider the case of an action of U(1) on M ϭ of the moment map f, the reduction ᏸred(t) of the Kostant– P (ރ) ϫ P (ރ)byei␪on the first factor and e2i␪ in the second Souriau line bundle is defined to be the line bundle ᏸ͉fϪ1(t)͞G. 1 1 This is a Kostant–Souriau line bundle for Mred(t). Thus we can factor. Then the function f takes values in [0, 3] with singular Downloaded by guest on September 26, 2021 14242 Colloquium Paper: Vergne Proc. Natl. Acad. Sci. USA 93 (1996)

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