Geometric Quantization on the Super-Disc

TL;DR

This paper explores the geometric quantization of an infinite-dimensional super-disc, a super-homogeneous space relevant for coupled boson-fermion systems, establishing a framework for classical and quantum analysis.

Contribution

It introduces a super-disc as a super-homogeneous manifold with a symplectic form, demonstrating geometric quantization in this infinite-dimensional setting.

Findings

Defined a super-disc with a natural symplectic form

Established classical dynamics on the super-disc

Quantized the system using geometric quantization methods

Abstract

In this article we discuss the geometric quantization on a certain type of infinite dimensional super-disc. Such systems are quite natural when we analyze coupled bosons and fermions. The large-N limit of a system like that corresponds to a certain super-homogeneous space. First, we define an example of a super-homogeneous manifold: a super-disc. We show that it has a natural symplectic form, it can be used to introduce classical dynamics once a Hamiltonian is chosen. Existence of moment maps provide a Poisson realization of the underlying symmetry super-group. These are the natural operators to quantize via methods of geometric quantization, and we show that this can be done.