Geometric quantizations of mixed polarizations on Kähler manifolds with T-symmetry

2023/01/03 by Leung, Naichung Conan, Wang, Dan · 2 citations
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2301.01011

Abstract

Let M be a compact Kähler manifold equipped with a pre-quantum line bundle L. In [9], using T-symmetry, we constructed a polarization P_mix on M, which generalizes real polarizations on toric manifolds. In this paper, we obtain the following results for the quantum space H_mix associated to P_mix. First, H_mix consists of distributional sections of L with supports inside μ^-1(\mathfrakt^*_ℤ). This gives H_mix=\bigoplus_λ∈ \mathfrakt^*_ℤ H_{mix, λ}. Second, the above decomposition of H_mix coincides with the weight decomposition for the T-symmetry. Third, an isomorphism H_{mix, λ} ≅ H⁰( M//_λT, L//_λT), for regular λ. Namely, geometric quantization commutes with symplectic reduction.

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• Quantization in mixed polarization via transverse Poincaré-Birkhoff-Witt theorem
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