Second-Order Conformally Equivariant Quantization in Dimension 1|2

TL;DR

This paper advances the theory of conformally equivariant quantization on supermanifolds, specifically for the supercircle with two odd variables, establishing isomorphisms of differential operator spaces and deriving explicit quantization formulas.

Contribution

It extends conformally equivariant quantization to the supercircle with multiple odd variables, proving module isomorphisms and providing explicit formulas for the quantization map.

Findings

Second order differential operators are isomorphic to their symbols as osp(2|2)-modules.

The conformal equivariant quantization map is unique.

Explicit formula for the quantization map is derived.

Abstract

This paper is the next step of an ambitious program to develop conformally equivariant quantization on supermanifolds. This problem was considered so far in (super)dimensions 1 and 1|1. We will show that the case of several odd variables is much more difficult. We consider the supercircle $S^{1|2}$ equipped with the standard contact structure. The conformal Lie superalgebra $\mathcal{K}(2)$ of contact vector fields on $S^{1|2}$ contains the Lie superalgebra osp(2|2). We study the spaces of linear differential operators on the spaces of weighted densities as modules over osp(2|2). We prove that, in the non-resonant case, the spaces of second order differential operators are isomorphic to the corresponding spaces of symbols as osp(2|2)-modules. We also prove that the conformal equivariant quantization map is unique and calculate its explicit formula.