Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 4

Special Issue in honor of Victor Guillemin

Guest Editors: Yael Karshon, Richard Melrose, Gunther Uhlmann, and Alejandro Uribe

Bohr–Sommerfeld quantization of $b$-symplectic toric manifolds

Pages: 2169 – 2194

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n4.a15


Pau Mir (Laboratory of Geometry and Dynamical Systems, Universitat Politècnica de Catalunya, Barcelona, Spain)

Eva Miranda (Laboratory of Geometry and Dynamical Systems & Institut de Matemàtiques de la UPC-BarcelonaTech (IMTech), Universitat Politècnica de Catalunya, Barcelona, Spain; and CRM Centre de Recerca Matemàtica, Bellaterra, Barcelona, Spain)

Jonathan Weitsman (Department of Mathematics, Northeastern University, Boston, Massachusetts, U.S.A.)


We introduce a Bohr–Sommerfeld quantization for bsymplectic toric manifolds and show that it coincides with the formal geometric quantization of $\href{ https://mathscinet.ams.org/mathscinet/relay-station?mr=3804693}{[\textrm{GMW18b}]}$. In particular, we prove that its dimension is given by a signed count of the integral points in the moment polytope of the torus action on the manifold.

The first-named author is supported by the Doctoral INPhINIT-RETAINING grant ID 100010434 LCF/BQ/DR21/11880025 of “la Caixa” Foundation; by the AEI grant PID2019-103849GB-I00 of MCIN/ AEI /10.13039/501100011033; by the AGAUR project 2021 SGR 00603 Geometry of Manifolds and Applications, GEOMVAP; and by the AGRUPS 2023 grant by UPC.

The second-named author is partially supported by the Spanish State Research Agency AEI/10.13039/501100011033, through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (project CEX2020-001084-M); and by the grant PID2019-103849GB-I00. The author is also partially supported by the AGAUR project 2021 SGR 00603 Geometry of Manifolds and Applications, GEOMVAP; and by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2021.

The third-named author is supported by a Simons collaboration grant.

Received 10 March 2022

Received revised 19 September 2022

Accepted 13 October 2022

Published 20 November 2023