Projectively Equivariant Quantization and Symbol calculus in dimension 1|2

TL;DR

This paper develops an explicit formula for projectively equivariant quantization and symbol calculus in dimension 1|2, revealing isomorphisms between differential operators and symbols under osp(2|2) symmetry.

Contribution

It extends the explicit construction of osp(2|2)-equivariant quantization to higher-order differential operators in dimension 1|2, including non-resonant cases.

Findings

Spaces of differential operators are isomorphic to symbol spaces as orthosymplectic modules.

Explicit formulas for the equivariant quantization are derived.

The quantization extends previous second-order results to higher orders.

Abstract

The spaces of higher-order differential operators (in Dimension 1|2), which are modules over the stringy Lie superalgebra K(2), are isomorphic to the corresponding spaces of symbols as orthosymplectic modules in non resonant cases. Such an osp (2|2)-equivariant quantization, which has been given in second-order differential operators case, keeps existing and unique. We calculate its explicit formula that provides extension in particular order cases.