On geometric quantization of $$b^m$$ b m -symplectic manifolds
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Mathematische Zeitschrift
On geometric quantization of bm -symplectic manifolds Victor W. Guillemin1 · Eva Miranda2,3 · Jonathan Weitsman4 Received: 23 September 2019 / Accepted: 24 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We study the formal geometric quantization of bm -symplectic manifolds equipped with Hamiltonian actions of a torus T with nonzero leading modular weight. The resulting virtual T −modules are finite dimensional when m is odd, as in [4]; when m is even, these virtual modules are not finite dimensional, and we compute the asymptotics of the representations for large weight.
1 Introduction The purpose of this paper is to continue our study of formal geometric quantization for a class of Poisson manifolds, the bm -Poisson manifolds which appeared first in the thesis of Geoffrey Scott [14], in the case where these manifolds are equipped with Hamiltonian torus actions. In the case m = 1, those are the b-symplectic manifolds of [5,6] whose quantizations
V. Guillemin is supported in part by a Simons collaboration grant. E. Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016 and partially supported by the grants reference number MTM2015-69135-P (MINECO/FEDER) and reference number 2017SGR932 (AGAUR). Eva Miranda was supported by a Chaire d’Excellence of the Fondation Sciences Mathématiques de Paris when this project started and this work has been supported by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-10-LABX-0098). J. Weitsman was supported in part by NSF grant DMS 12/11819 and by the Simons collaboration grant 579801.
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Eva Miranda [email protected] Victor W. Guillemin [email protected] Jonathan Weitsman [email protected]
1
Department of Mathematics, MIT, Cambridge, MA 02139, USA
2
Laboratory of Geometry and Dynamical Systems, Department of Mathematics, EPSEB, Universitat Politecnica de Catalunya- BGSMath Barcelona, Graduate School of Mathematics, Barcelona, Spain
3
IMCCE, CNRS-UMR8028, Observatoire de Paris, PSL University, Sorbonne, Université 77 Avenue Denfert-Rochereau, 75014 Paris, France
4
Department of Mathematics, Northeastern University, Boston, MA 02115, USA
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are constructed in [4]. The methods we use here are very similar to those of [4], and as in the case of b-symplectic manifolds, the quantizations turn out to have remarkable properties: We obtain finite dimensional virtual modules for m odd and infinite dimensional virtual modules with remarkable asymptotic properties for m even. We hope this repertory of examples will give further intuition about what might be a geometric quantization for more general Poisson manifolds. Let m be a positive integer. A bm -symplectic manifold is a smooth manifold M, along with a smooth hypersurface Z ⊂ M, and a choice of an m-germ of a C ∞ -function at Z along with a closed, nondegenerate bm -form ω of degree 2 on M (see
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