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On geometric quantization of \(b^m\)-symplectic manifolds

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Abstract

We study the formal geometric quantization of \(b^m\)-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.

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Notes

  1. Such a line bundle with connection is needed for the formal quantization to be well-defined, even in the b-symplectic case discussed in [4]. We would like to thank the referee for pointing out this omission in [4] and in an earlier version of this paper.

  2. We write Q(M) by abuse of notation even though the quantization as defined depends on the data \(\omega , L, \nabla .\)

  3. Some care must be taken when \(\alpha \) is a singular value of the moment map; see below.

  4. This sign convention is inspired by the results of [2] for presymplectic manifolds; in the symplectic case, these orientations always agree, and the sign is always positive.

  5. Note that since the moment map is singular on Z,  and N is compact, \((M \times N)//_0 T = ((M-Z) \times N)//_0 T \), so that \((M \times N)//_0 T \) is compact and symplectic.

  6. This sign convention is inspired by the results of [2] for presymplectic manifolds; in the symplectic case, these orientations always agree, and the sign is always positive.

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Acknowledgements

We thank Cédric Oms for carefully reading a first version of this article. We would also like to thank the referee for several helpful comments and corrections. In particular, the referee pointed out to us the delicate issue of the potential dependence of quantization on the choice of a connection with curvature given by the symplectic form, in the case where the quotients may be orbifolds. This issue was overlooked in [4], and we address it in this paper. It does not affect the statements of the main theorems and addressing it requires only some care with the proofs, not any major changes.

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Authors and Affiliations

  1. Department of Mathematics, MIT, Cambridge, MA, 02139, USA

    Victor W. Guillemin

  2. Laboratory of Geometry and Dynamical Systems, Department of Mathematics, EPSEB, Universitat Politecnica de Catalunya- BGSMath Barcelona, Graduate School of Mathematics, Barcelona, Spain

    Eva Miranda

  3. IMCCE, CNRS-UMR8028, Observatoire de Paris, PSL University, Sorbonne, Université 77 Avenue Denfert-Rochereau, 75014, Paris, France

    Eva Miranda

  4. Department of Mathematics, Northeastern University, Boston, MA, 02115, USA

    Jonathan Weitsman

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  1. Victor W. Guillemin
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  2. Eva Miranda
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Correspondence to Eva Miranda.

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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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Guillemin, V.W., Miranda, E. & Weitsman, J. On geometric quantization of \(b^m\)-symplectic manifolds. Math. Z. 298, 281–288 (2021). https://doi.org/10.1007/s00209-020-02590-w

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