Quantisation of super Teichmueller theory

TL;DR

This paper develops a quantum version of the Teichmueller spaces for super Riemann surfaces, incorporating spin structure dependence and demonstrating invariance under triangulation changes.

Contribution

It introduces a novel quantisation method for super Teichmueller spaces that accounts for spin structures and proves invariance under triangulation refinements.

Findings

Constructed a quantum Teichmueller space for super Riemann surfaces.

Encoded spin structure dependence via combinatorial refinement.

Established a projective unitary representation ensuring triangulation independence.

Abstract

We construct a quantisation of the Teichmueller spaces of super Riemann surfaces using coordinates associated to ideal triangulations of super Riemann surfaces. A new feature is the non-trivial dependence on the choice of a spin structure which can be encoded combinatorially in a certain refinement of the ideal triangulation. By constructing a projective unitary representation of the groupoid of changes of refined ideal triangulations we demonstrate that the dependence of the resulting quantum theory on the choice of a triangulation is inessential.