Natural and Projectively Invariant Quantizations on Supermanifolds

TL;DR

This paper extends the concept of natural, projectively invariant quantizations to supermanifolds, generalizing previous work and providing a new globalization method for supergeometry quantizations.

Contribution

It adapts Bordemann's method to supermanifolds, establishing a natural globalization of the equivariant quantization in supergeometry.

Findings

Constructed a supermanifold quantization invariant under projective transformations.

Extended the quantization to arbitrary degree symbols.

Connected the supermanifold quantization with previous degree-two symbol quantizations.

Abstract

The existence of a natural and projectively invariant quantization in the sense of P. Lecomte [Progr. Theoret. Phys. Suppl. (2001), no. 144, 125-132] was proved by M. Bordemann [math.DG/0208171], using the framework of Thomas-Whitehead connections. We extend the problem to the context of supermanifolds and adapt M. Bordemann's method in order to solve it. The obtained quantization appears as the natural globalization of the $\mathfrak{pgl}({n+1|m})$-equivariant quantization on ${\mathbb{R}}^{n|m}$ constructed by P. Mathonet and F. Radoux in [arXiv:1003.3320]. Our quantization is also a prolongation to arbitrary degree symbols of the projectively invariant quantization constructed by J. George in [arXiv:0909.5419] for symbols of degree two.