On deformation quantizations of symplectic supervarieties

TL;DR

This work extends deformation quantization to symplectic supervarieties by constructing an injective period map from the quantization classes to de Rham cohomology and relating quantizations of a supervariety to those of its even reduction. It develops a comprehensive super-geometry toolkit—$\mathcal{D}_X$-modules, jet bundles, and Harish-Chandra torsors—and an associated period-map framework that mirrors the Bezrukavnikov–Kaledin program. A key contribution is showing admissibility and splitting for certain super nilpotent orbits, enabling a full classification of their deformation quantizations via $H^2_{\mathrm{dR}}(\mathbb{O})_{\hbar}$. The approach integrates supersymmetric generalizations of D-modules, Hochschild–Serre extensions, and torsor methods to illuminate connections with representation theory of Lie superalgebras and related structures like W-algebras. The results pave the way for a super-analogue of orbit-method-type correspondences and deepen the understanding of quantizations in the super geometric setting.

Abstract

We classify deformation quantizations of the symplectic supervarieties that are smooth and admissible. This generalizes the corresponding result of Bezrukavnikov and Kaledin to the super case. We relate the equivalence classes of quantizations of supervarieties with that of their even reduced symplectic varieties. Finally, we prove that certain nilpotent orbits of basic Lie superalgebras are admissible and splitting, and classify their deformation quantizations.

Theorems & Definitions (40)

• Example 2.1
• Proposition 3.1: K$\ddot{\text{u}}$nneth formula
• Theorem 3.2
• proof
• Proposition 3.3: Spencer resolution
• proof
• Proposition 3.4
• proof
• Theorem 3.5
• proof