Matrix Cartan superdomains, super Toeplitz operators, and quantization

TL;DR

This paper develops a non-perturbative quantization framework for hermitian symmetric supermanifolds using super Toeplitz operators, linking classical super Poisson structures with quantum algebras.

Contribution

It introduces a novel quantization scheme based on super Toeplitz operators on graded Hilbert spaces, extending non-perturbative quantization to supermanifolds.

Findings

Reproduces the invariant super Poisson structure in the classical limit

Defines a C*-algebra generated by super Toeplitz operators

Establishes a link between classical and quantum supergeometry

Abstract

We present a general theory of non-perturbative quantization of a class of hermitian symmetric supermanifolds. The quantization scheme is based on the notion of a super Toeplitz operator on a suitable Z_2 -graded Hilbert space of superholomorphic functions. The quantized supermanifold arises as the C^* -algebra generated by all such operators. We prove that our quantization framework reproduces the invariant super Poisson structure on the classical supermanifold as Planck's constant tends to zero.