Strict Quantization for Compact Pseudo-K\"ahler Manifolds and Group Actions

TL;DR

This paper extends strict quantization techniques to pseudo-Kähler manifolds with group actions, introducing a Berezin transform with asymptotic expansion to establish strict quantization in this broader setting.

Contribution

It generalizes strict quantization from Kähler to pseudo-Kähler manifolds with group actions, introducing a Berezin transform with asymptotic properties.

Findings

Established a Berezin transform with a complete asymptotic expansion.

Proved certain quantization maps are strict in the pseudo-Kähler setting.

Extended the framework of Berezin-Toeplitz quantization to new geometric contexts.

Abstract

The asymptotic results for Berezin-Toeplitz operators yield a strict quantization for the algebra of smooth functions on a given Hodge manifold. It seems natural to generalize this picture for quantizable pseudo-K\"ahler manifolds in presence of a group action. Thus, in this setting we introduce a Berezin transform which has a complete asymptotic expansion on the preimage of the zero set of the moment map. It leads in a natural way to prove that certain quantization maps are strict.