Darboux, Moser and Weinstein theorems for prequantum systems and applications to geometric quantization

Journal of Geometry and Physics 206 (2024) 105298 Contents lists available at ScienceDirect Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/geomphys Darboux, Moser and Weinstein theorems for prequantum systems and applications to geometric quantization Eva Miranda a,∗,1,2, Jonathan Weitsman b,1,3 aDepartment of Mathematics at Universitat Politècnica de Catalunya and Centre de Recerca Matemàtica-CRM, Spain bDepartment of Mathematics, Northeastern University, Boston, MA 02115, United States of America a r t i c l e i n f o a b s t r a c t Article history: Received 19 May 2024 Received in revised form 26 July 2024 Accepted 12 August 2024 Available online 22 August 2024 Keywords: Prequantum systems Quantization Real polarization Darboux-Weinstein-Moser theorem We establish analogs of the Darboux, Moser and Weinstein theorems for prequantum systems. We show that two prequantum systems on a manifold with vanishing first cohomology, with symplectic forms defining the same cohomology class and homotopic to each other within that class, differ only by a symplectomorphism and a gauge transformation. As an application, we show that the Bohr-Sommerfeld quantization of a prequantum system on a manifold with trivial first cohomology is independent of the choice of the connection. ©2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by-nc -nd /4 .0/). 1. Introduction The Darboux theorem establishes that there are no local invariants in symplectic geometry. Namely, let Mbe a smooth manifold and let ω, ωbe two symplectic forms on M, then: Theorem 1 (Darboux [1,2]). For every point p ∈Mthere exists a neighborhood U of pand an embedding  :U→M isotopic to the inclusion and fixing psuch that ∗ω=ω|U. The global invariants of a symplectic manifold include the cohomology class defined by its symplectic form. This gives a complete classification of symplectic 2-manifolds. In general, there is the following theorem due to Moser. *Corresponding author. E-mail addresses: ev[email protected] (E. Miranda), [email protected] (J. Weitsman). 1Both authors are partially supported by Spanish State Research Agency grants PID2019-103849GB-I00 and PID2023-146936NB-I00 of MICIU/AEI / 10.13039/501100011033. 2E. Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2021 and by the Alexander von Humboldt Foundation via a Friedrich Wilhelm Bessel Research Award. E. Miranda is also supported by the Spanish State Research Agency, through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (project CEX2020-001084-M) and partially supported by and by the AGAUR project 2021 SGR 00603. 3J. Weitsman was supported in part by a Simons Collaboration Grant # 579801. https://doi.org/10.1016/j.geomphys.2024.105298 0393-0440/©2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons .org /licenses /by-nc -nd /4 .0/). E. Miranda and J. Weitsman Journal of Geometry and Physics 206 (2024) 105298 Theorem 2 (Moser [5]). Let Mbe a compact manifold endowed with two symplectic forms ωand ω. Assume that [ω] =ω, and that there exists a path ωtof symplectic forms such that [ωt] =[ω0]for all t, and with ω0=ωand ω1=ω. Then, there exists a diffeomorphism isotopic to the identity  :M→Msuch that ∗ω=ω. This proof based on Moser’s method can be adapted when symmetries are present. Theorem 3 (Weinstein [6]). Let Gbe a compact Lie group, and let Mbe a compact G-manifold endowed with two invariant symplectic forms ω, ω. Suppose there exists a path ωtof invariant symplectic forms such that [ωt] =[ω0]for all t, and with ω0=ωand ω1=ω. Then there exists a G−equivariant diffeomorphism  :M→Msuch that ∗ω=ω. Recall that if (M, ω)is a symplectic manifold, π:L →Mis a (complex) line bundle, ∇is a connection on L, and curv(∇) =ω, the quadruple (M, ω, L, ∇)is a prequantum system. The main purpose of this article is to generalize these three theorems to prequantum systems. Theorem 4. Suppose (M, ω, L, ∇)and (M, ω, L, ∇)are prequantum systems. Then for every point p ∈Mthere exists a neighborhood Uof p, an embedding  :U→M isotopic to the inclusion and fixing p, and a function φ∈C∞(U, C)such that4 ω|U=∗ω(∗L)=L|U ∇s=(∗∇)s+dφ⊗s,for any s ∈(L|U). If Lis a hermitian line bundle and ∇is a hermitian connection, we may take φ∈C∞(M, iR). For compact manifolds, we have Theorem 5. Suppose (M, ω, L, ∇)and (M, ω, L, ∇)are prequantum systems with Ma compact manifold. Assume that [ω] =ω, and that there exists a path ωtof symplectic forms with [ωt] =[ω0]for all t, such that ω0=ωand ω1=ω. Assume also that H1(M, R) =0. Then there exists a diffeomorphism  :M→M isotopic to the identity and a function φ∈C∞(M, C)such that ω=∗ω∗L=L ∇s=(∗∇)s+dφ⊗s,for any s ∈(L). If Lis a hermitian line bundle and ∇is a hermitian connection, we may take φ∈C∞(M, iR). When there are additional symmetries the following result holds. Theorem 6. Suppose Gis a compact Lie group, and suppose Mis a compact G-space. Suppose that (M, ω, L, ∇)and (M, ω, L, ∇) are G-invariant prequantum systems. Assume that [ω] =ω, and that there exists a path ωtof G-invariant symplectic forms with [ωt] =[ω0]for all t, such that ω0=ωand ω1=ω. Assume also that H1(M, R) =0. Then there exists a G-equivariant diffeomorphism  :M→M isotopic to the identity and a G-invariant function φ∈C∞(M, C) such that ω=∗ω∗L=L ∇s=(∗∇)s+dφ⊗s,for any s ∈(L). If Lis a hermitian line bundle and ∇is a hermitian connection, we may take φ∈C∞(M, iR). Remark 1.1. The function φis called a gauge transformation. We have thus proven that two prequantum systems, with associated symplectic forms lying in the same cohomology class and homotopic to each other, differ only by a symplectomorphism and a gauge transformation. Observe that it is not possible to dispense with the gauge transformation: Suppose ∇is the trivial connection on the trivialized bundle L =M×C, and ∇is some other connection on L. Then for any section s ∈(M, L), we have ∇s =∇s +dφ⊗s, where φis somewhere non-constant. And no symplectomorphism will transform φ into an everywhere constant function (since diffeomorphisms take constant functions to constant functions). As an application of this theorem, we prove that for manifolds with trivial first cohomology, the Bohr-Sommerfeld quantization of a prequantum system (M, ω, L, ∇)does not depend on the choice of the connection ∇; in other words if (M, ω, L, ∇)is another prequantum system associated with the same manifold, line bundle, and symplectic form, its 4For the definition of the pullback of a connection under a smooth map, see e.g. [4], p. 292, Lemma 3. 2 E. Miranda and J. Weitsman Journal of Geometry and Physics 206 (2024) 105298 Bohr-Sommerfeld quantization coincides with that of (M, ω, L, ∇). If H1(M) = 0a similar result holds if the holonomy representations of ∇and ∇coincide. Acknowledgment We would like to thank Peter Crooks for helpful discussions on this paper. 2. Proof of the main theorems We first prove Theorem 4. Proof. By the Darboux theorem, we know that there exists a neighborhood Uof p, which we may take to be diffeomorphic to an open ball, and an embedding φ:U→Misotopic to the inclusion preserving psuch that ∗ω=ω|U. Since is isotopic to the inclusion we also have ∗L=L|U It remains to compare the two connections and consider the difference ∇s−(∗∇)sfor any s∈(L|U). Since the space of connections on a line bundle on Uis an affine space modeled on 1U,C, we know that ∇s−(∗∇)s=ξ⊗s for some ξ∈1U,C. Since curv∇=curv ∗∇, it follows that dξ=0. Since H1(U, R) =0, ξ=dφfor some φ∈C∞(U, C). If the line bundle Lis hermitian and the connection is unitary, we may repeat this argument to show that we may take φ∈C∞(U, iR). We next prove Theorem 5. Proof. The existence of the diffeomorphism is guaranteed by Moser’s theorem. We again have ∗L =L, and again there exists ξ∈1M,Cso that ∇s−(∗∇)s=ξ⊗sfor any s∈(L). Again dξ=0, so since H1(M, R) =0, ξ=dφfor some φ, as needed. If the line bundle Lis hermitian and the connection is unitary, we may repeat the argument to show that we may take φ∈C∞(M, iR). We now proceed to the proof of Theorem 6. Proof. The existence of a G-equivariant diffeomorphism is given by Weinstein’s theorem. Since H1(M, R) =0, again there exists a function φsuch that ∇s−(∗∇)s=dφ⊗sfor any s∈(L). (2.1) Let ¯ φbe the average of φover Gusing the Haar measure μ, i.e., ¯ φ:=  G α∗ gφdμ(g), where we have denoted by αgthe diffeomorphism giving the action of g∈Gon M. Then since ∇and ∇are G-invariant, we obtain ∇s−(∗∇)s=d¯ φ⊗s by averaging both sides of equation (2.1). Again, if the line bundle Lis hermitian and the connection is unitary, we may repeat this argument to show that we may take φ∈C∞(M, iR)G. 3 E. Miranda and J. Weitsman Journal of Geometry and Physics 206 (2024) 105298 Remark 2.2. (The case where H1(M, R) = 0.) Suppose we are given two connections ∇and ∇on Lwith the same curvature. Then for any section s ∈(M, L), we have ∇s =∇ s +α⊗sfor some α∈1(M, C)with dα=0. If H1(M, C) = 0, we cannot conclude that α=dφfor some function φ. However, if ∇and ∇also have the same holonomy representation, it follows that the holonomy representation of α, thought as a connection on the trivialized bundle L ⊗L∗=C, is trivial. That means that for every oriented curve Cin M, Cα=2πn, where n ∈Z. This does not imply that αis exact; but there exists a map t:M→S1with α=t∗(dθ), where dθis the translation invariant one form on S1.5 3. Application to quantization in a real polarization Let (M, ω, L, ∇)be a prequantum system. Recall that a Lagrangian submanifold is integral if (L,∇)|is trivial as a bundle with connection. Corollary 1. Let Mbe a compact manifold with H1(M, R) =0and let (M, ω, L, ∇)and (M, ω, L, ∇)be two prequantum systems. A Lagrangian submanifold  ⊂Mis integral for (M, ω, L, ∇)if and only if it is integral for (M, ω, L, ∇). In particular, suppose we are given a real polarization of M, that is, a foliation by Lagrangian submanifolds. Consider from now on prequantum systems (M, ω, L, ∇)where Lis hermitian and the connection ∇is unitary. The Bohr-Sommerfeld quantization of (M, ω, L, ∇)associated to this polarization is given by the vector space generated by the integral leaves of the foliation, or equivalently by the covariant constant sections of (L, ∇)on those leaves. In [3] Guillemin and Sternberg consider the case where the foliation is given by a map π:M→B, where Bis a simply connected subset of Rnand the map F:M−→ Bis the moment map for an integrable system away from the singularities of the foliation. The Bohr-Sommerfeld quantization is then determined by the integral points of the image of the moment map F. Then Corollary 1implies Corollary 2. Let (M, ω)be a symplectic manifold with H1(M, R) =0. Let Lbe a complex line bundle on Mwith c1(L) =[ω]. Choose a connection ∇on Lso that (M, ω, L, ∇)is a prequantum system. Suppose we are given a real polarization of M. Then the quantization of (M, ω, L, ∇)with this real polarization is independent of the choice of ∇. Remark 3.1. Note that in the case where the foliation is given as in [3]by an integrable system, the generic leaves are tori, on which (L, ∇)may have complicated holonomy representations. We have shown that the global condition H1(M, R) =0 guarantees these holonomy representations are independent of ∇. Remark 3.2. For complex polarizations, an analog of Corollary 2holds, since the index of the Dolbeault ¯ ∂operator on (M, L) depends only on Mand c1(L) =[ω], not on the connection; this is due to the Riemann-Roch theorem ind(¯ ∂)= M Td(TM)e[ω].(3.3) Note that the right-hand side of equation (3.3) depends only on the class [ω]and not on ∇. Remark 3.4. In the case where H1(M, R) = 0, any two connections ∇and ∇with the same curvature still give the same quantization if, in addition, the trivial bundle L ⊗L∗, equipped with the flat connection arising from ∇and ∇, is trivial as a bundle with connection. Data availability No data was used for the research described in the article. References [1] G. Darboux, Sur le problème de Pfaff, Bull. Sci. Math. Astron. (2) 6 (1882) 14–36. [2] G. Darboux, Sur le problème de Pfaff, Bull. Sci. Math. Astron. (2) 6 (1882) 49–62. [3] V. Guillemin, S. Sternberg, The Gelfand-Cetlin system and quantization of the complex flag manifolds, J. Funct. Anal. 52 (1) (1983) 106–128. [4] J. Milnor, J. Stasheff, Characteristic Classes, Princeton University Press, 1974. [5] J. Moser, On the volume elements on a manifold, Trans. Am. Math. Soc. 120 (1965) 286–294. [6] A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Adv. Math. 6(3) (1971) 329–346. 5Morally α=t−1dt, where t:M→S1is a (possibly homotopically nontrivial) element of the gauge group Map(M, S1). 4