Geometric quantizations of mixed polarizations on K\"ahler manifolds with T-symmetry

TL;DR

This paper studies the geometric quantization of a specific mixed polarization on compact K"ahler manifolds with T-symmetry, showing how the quantum space decomposes and relates to symplectic reduction.

Contribution

It establishes the structure of the quantum space associated with the mixed polarization and proves that geometric quantization commutes with symplectic reduction in this setting.

Findings

Quantum space consists of distributional sections supported on specific fibers.

Decomposition of quantum space matches the weight decomposition for T-symmetry.

Quantum spaces are isomorphic to sections over symplectic quotients, confirming quantization commutes with reduction.

Abstract

Let $M$ be a compact K\"ahler manifold equipped with a pre-quantum line bundle $L$. In [9], using $T$-symmetry, we constructed a polarization $\mathcal{P}_{\mathrm{mix}}$ on $M$, which generalizes real polarizations on toric manifolds. In this paper, we obtain the following results for the quantum space $\mathcal{H}_{\mathrm{mix}}$ associated to $\mathcal{P}_{\mathrm{mix}}$. First, $\mathcal{H}_{\mathrm{mix}}$ consists of distributional sections of $L$ with supports inside $\mu^{-1}(\mathfrak{t}^{*}_{\mathbb{Z}})$. This gives $\mathcal{H}_{\mathrm{mix}}=\bigoplus_{\lambda \in \mathfrak{t}^{*}_{\mathbb{Z}} } \mathcal{H}_{\mathrm{mix}, \lambda}$. Second, the above decomposition of $\mathcal{H}_{\mathrm{mix}}$ coincides with the weight decomposition for the $T$-symmetry. Third, an isomorphism $\mathcal{H}_{\mathrm{mix}, \lambda} \cong H^{0}( M//_{\lambda}T, L//_{\lambda}T)$, for regular $\lambda$. Namely, geometric quantization commutes with symplectic reduction.