Convex polytopes and quantization of symplectic manifolds

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 ⊂ ℝⁿ is a convex polytope, can we compare the number |P ∩ ℤⁿ| 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⁻ⁿ|kP ∩ ℤⁿ|. As we will recall, a link between both problems is provided by the study of Hamiltonian symmetries.

Example 1: Consider the region Δ_n of ℝⁿ consisting of all points v = (t₁, t₂, … , t_n), such that coordinates t_i of v are nonnegative and satisfy the inequation t₁ + t₂ + ⋯ +t_n ≤ 1 (Fig. 1).

Figure 1.

Standard simplex.

Let us consider the dilated simplex kΔ_n. The volume of kΔ_n is the homogeneous function of k,

Consider the number of points v = (u₁, u₂, … , u_n) with integral coordinates u_i in kΔ_n. If k is any nonnegative integer, this number is given by a polynomial function of k:

This function is not homogeneous in k, but clearly its top order term in k is equal to the volume kⁿ/n! of kΔ_n. Remark that p_n(k) is an integer for all k, while the volume of kΔ_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” kΔ_n.

Volumes of Symplectic Manifolds and of Convex Polytopes

A symplectic manifold of dimension 2n is a manifold M with a closed nondegenerate two-form Ω. The simplest example is the phase space ℝ²ⁿ = ℝⁿ × ℝⁿ, with symplectic coordinates (q₁, q₂, … , q_n; p₁, p₂, … , p_n) and symplectic form Ω = Σ_k=1ⁿ 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 (2π)⁻ⁿΩⁿ/n!, and the symplectic volume of M is the integral of the Liouville form over M.

Example 2: Consider the sphere S ⊂ ℝ³ with radius 1. We project S on ℝ via the height z. The image of S is the interval [−1, 1] (Fig. 2).

Figure 2.

The sphere.

In coordinates, (x = cos φ, y = sin φ, z), the volume form Ω of S is dφ ∧ dz. This gives a system of Darboux coordinates (outside north and south poles). In particular the symplectic volume of S is (2π)⁻¹(4π) = 2. This is also the length of [−1, 1].

Let P be a convex polytope in ℝⁿ. This means P is the convex hull of a finite set of points of ℝⁿ. 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 (t₁, t₂, … , t_n, φ₁, φ₂, … , φ_n) on an open dense subset U_P of M_P. Here the point v = (t₁, t₂, … , t_n) varies in the interior P⁰ of P, and φ_k are angles between 0 and 2π. Thus U_P is isomorphic to P⁰ × S¹ × S¹ × ⋯ × S¹, where S¹ 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 [−1, 1] is the sphere S. The inflated symplectic manifold corresponding to the simplex Δ_n of Example 1 is the projective space P_n(ℂ). We realize P_n(ℂ) as the space:

with identification of all proportional points z and e^iθz in the sphere S²ⁿ⁺¹ in ℂⁿ⁺¹.

Darboux coordinates on an open dense set are (t₁, t₂, … , t_n, φ₁, φ₂, … , φ_n) ↦ (t₁^1/2 e^iφ₁, ⋯ , t_n^1/2 e^iφ_n, ), where (t₁, t₂, … , t_n) vary in Δ_n⁰.

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 ℝⁿ or, more generally, the integral over P of any exponential function on ℝⁿ. 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 a₁^p, … , a_n^p 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 φ = (φ₁, φ₂, … , φ_n) of ℝⁿ, such that 〈a_k^p, φ〉 ≠ 0 for all vertices p and edges vectors a_k^p,

1

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

2

The formula obtained is independent of the choice of φ. 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) = {e^iφ} be the circle group. Assume that U(1) acts on our symplectic manifold M (Ω 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:

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 ∂/∂φ, the energy is the height function f = z, and the fixed points are the north and south poles.

The “exact stationary phase formula” (4) of Duistermaat–Heckman for ∫_M e^iβf(x)dx (where dx denotes the Liouville measure) compute exactly this function of β in terms of the fixed points of the symmetry group U(1) on M. If the set of fixed points is finite,

3

Here p runs through all fixed points and a_k^p are integers such that X_f is, near each p, equal to a product of infinitesimal rotations with speed a_k^p. 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 e^iβf when f generates a periodic flow. Allowing manifolds M_P 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 = U(1) × U(1) × ⋯ × U(1). Assume we have d commuting Hamiltonian functions (f₁, f₂, … , f_d) generating the action. Let f : M → ℝ^d be the map with components f_i. 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 → 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 hyperplanes. Thus, P is the union of convex polytopes C̄, where C is a connected component of the set of regular values of f (Fig. 3).

Figure 3.

Regular values of the moment map.

Let t be a point in P. For φ = (φ₁, φ₂, … , φ_d), we denote by 〈φ, f〉 the function Σ_k=1^d φ_kf_k. Integrating the function e^i〈φ,f〉 first on {f(x) = t}, then on P, we have:

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 − 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) = 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) = {f(x) = t}/(x ↦ g·x). 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 = (f₁, … , f_d) 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:

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

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 = 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 push-forward measure by the moment map?

We need to recall some of the basic constructions of quantum mechanics. The quantum model Q(M,Ω) of the classical model M is searched as a vector space of functions on M. For the phase space ℝ²ⁿ = ℝⁿ × ℝⁿ, then Q(M,Ω) 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,Ω) to be the space of functions of p_k or of the variables q_k, in the Schrödinger model, or as well functions of the complex variables z_k = p_k + iq_k in the Fock–Bargmann model. Furthermore, ideally, Q(M,Ω) 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,Ω). The construction of Q(M, Ω) with all possible Hamiltonian symmetries of M lifted to the space Q(M,Ω) is not possible, but we will see that a quantum model Q(M,Ω) 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 Ω is integral. Then M is “quantizable” in the sense of Kostant and Souriau (26, 27); we have Ω = iF, where F is the curvature of ℒ, the Kostant–Souriau line bundle on M. We will construct Q(M,Ω) [denoted also by Q(M,ℒ)] via a positive almost-complex structure (see refs. 13 and 14). Roughly speaking, locally M is modeled on ℝ²ⁿ with symplectic coordinates (q_k,p_k), and this means, intuitively, that we will construct Q(M,Ω) as functions on M of the n complex variables z_k = p_k + 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^iφ_kz_k. To be precise, if M is a compact complex manifold and ℒ → M is an holomorphic line bundle on M with positive curvature form given by the formula:

where s is a nonvanishing holomorphic section of ℒ over a chart, then Q(M,mΩ) coincide with the space H⁰(M,𝒪(ℒ^m)) of holomorphic sections of the line bundle ℒ^m = ⊗^mℒ when m is sufficiently large. Unfortunately, it is necessary to allow virtual vector spaces in the construction of Q(M,Ω). We define the space Q(M,Ω) to be [Ker D⁺] − [Ker D⁻], where D is the ∂̄-operator (associated to the almost-complex structure) on ℒ-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,mΩ) = Σ_k=0ⁿ (−1)^kH^k(M,𝒪(ℒ^m)). When m is sufficiently large, the positivity assumption on ℒ implies that all cohomology spaces H^k(M,𝒪(ℒ^m)) vanishes for k > 0. We write dim Q(M,Ω) for the integer (maybe negative) dim[Ker D⁺] − dim[Ker D⁻]. This integer is the quantum analogue of the symplectic volume of M.

Let P be a convex polytope in ℝⁿ 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 ℒ_P. The space Q(M_P,ℒ_P) has a basis indexed by points with integral coordinates contained in P.

Example: Points v = (u₁, u₂, … , u_n) with integral coordinates in the simplex kΔ_n label the monomial basis z₁^u₁z₂^u₂ … z_n^u_nz_n+1^(k−Σui) of the space Q(P_n(ℂ),ℒ_k) of homogeneous polynomials of degree k in n + 1 variables.

There is a formula to compute the sum over integers contained in P of any exponential function on ℝⁿ (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 a₁^p, … , a_n^p 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 φ = (φ₁, φ₂, … , φ_n) of ℝⁿ, such that 〈a_k^p, φ〉 ≠ 0 for all vertices p:

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 φ 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 = U(1) be acting on a quantizable manifold M. If the action of U(1) lifts to ℒ, 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,Ω). The operator F on the vector space Q(M,Ω), such that the one parameter group (e^iφ) of symmetries of M lifts in the action of e^iφF, is a self-adjoint operator on Q(M,Ω) 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:

This formula is similar to Eq. 3 in the “classical” case for ∫_M e^iβfdx.

Let G = U(1) × U(1) ⋯ × U(1) be a group of d commuting circular symmetries of M. Assume that this action lifts to an action on the line bundle ℒ. 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,Ω) associated to the one parameter group e^iφ_k. 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:

where for t = (u₁, u₂, … , u_d) a point in ℝ^d with integral coordinates,

We denote by q(ℒ,t) the dimension of Q_t. If ℒ is fixed, we denote it simply by q(t). Let φ ∈ ℝ^d and let 〈φ, F〉 = Σ_k φ_kF_k. Then the matrix of the action of exp i〈φ, F〉 in Q(M,Ω) is diagonal with the diagonal term e^iu₁φ₁e^iu₂φ₂ ⋯ e^iu_dφ_d appearing q(t) times. Thus the trace of exp i〈φ, F〉 (or, more exactly, the super trace) is:

The function φ ↦ Tr_Q(M,Ω)e^i〈φ,F〉 is the analogue in the quantum case of the function φ ↦ ∫_M e^i〈φ,f〉 in the classical case. Its Fourier decomposition Σ_t∈ℤ^d q(t)e^i〈t,φ〉 should be related to the Fourier decomposition ∫_M e^i〈φ,f〉dx = ∫_P e^i〈t,φ〉h(t)dt. However, h(t)dt is a continuous measure on P, while q(t) is a discrete measure supported on P ∩ 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,Ω) as “the” quantum model for the classical model M.

Theorem 2. Let M be a quantizable compact connected symplectic manifold. Let f = (f₁, … , f_d) be the moment map for a Hamiltonian torus action on M lifting to Kostant–Souriau line bundle ℒ.

Then:

(i) The multiplicity function q(t) is supported on ℤ^d ∩ P. Thus we have:

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

We need to explain the last formula. At an integral value t of the moment map f, the reduction ℒ_red(t) of the Kostant–Souriau line bundle is defined to be the line bundle ℒ|_f⁻¹(t)/G. This is a Kostant–Souriau line bundle for M_red(t). Thus we can define the quantum “volume” dim Q(M_red(t), ℒ_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), ℒ_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, Tr_Q(M,Ω)e^iφF 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 ∩ ℤⁿ labels an eigenvector for the quantum representation of G. In particular, we have dim Q(M,Ω) = card(P ∩ ℤⁿ).

Let us compare the continuous measure h(t) and the discrete measure q(t) for the case of the circle action z ↦ e^iθz on ℂⁿ⁺¹ with energy function f = ∥z∥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) variables, and we have the eigenspace decomposition:

where Q_k is the space of homogeneous polynomials of degree k in (n + 1) variables. In this case, we have:

It is also possible to prove, as announced by Meinrenken and Sjamaar, a remarkable “continuity property” for the function q(ℒ,t). As the function t → q(ℒ,t) is a priori defined only on the finite set P ∩ ℤⁿ, 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 ℒ and t.

Let m be any positive integer. Define q(m,t) to be q(ℒ^m,t). Then t ↦ q(m,t) is supported on mP ∩ ℤ^d. Let C be a connected component of the set of regular values of f. Consider the open convex cone 𝒞 in ℝ^d+1 with base C.

The function q(m,t) is defined on 𝒞 ∩ ℤ^d+1. A quasipolynomial function on ℝ^d+1 is a function in the algebra generated by polynomials and periodic functions (with sufficiently large period). Then, there exists a unique quasipolynomial function q_C on ℝ^d+1 such that q(m,t) = q_C(m,t) for all (m,t) ∈ 𝒞̄ ∩ ℤ^d+1.

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 = P₁(ℂ) × P₁(ℂ) by e^iθ on the first factor and e^2iθ in the second factor. Then the function f takes values in [0, 3] with singular values 0, 1, 2, 3. We have for m ≥ 0, 0 ≤ t ≤ 3m, t, m integers,