Projectively equivariant quantizations over the superspace $\R^{p|q}$

TL;DR

This paper extends the theory of projectively equivariant quantization to supermanifolds, establishing existence, uniqueness, and explicit formulas, with special cases where a family of quantizations arises due to algebraic structure.

Contribution

It generalizes classical quantization results to supergeometry, providing new existence, uniqueness, and explicit formula results for super projective spaces.

Findings

Existence and uniqueness of quantization when the superalgebra is simple.

Presence of a one-parameter family of quantizations when the algebra is not simple.

Explicit formulas involving a generalized divergence operator.

Abstract

We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra pgl(p+1|q) is simple, our result is similar to the classical one in the purely even case: we prove the existence and uniqueness of the quantization except in some critical situations. When the projective superalgebra is not simple (i.e. in the case of pgl(n|n)\not\cong sl(n|n)), we show the existence of a one-parameter family of equivariant quantizations. We also provide explicit formulas in terms of a generalized divergence operator acting on supersymmetric tensor fields.