The Kauffman skein algebra of a surface at $\sqrt{-1}$

TL;DR

This paper investigates the Kauffman skein algebra at the complex parameter sqrt(-1), revealing its structure as an algebra of parallel transport operators on the moduli space of flat connections, and analyzes trace asymptotics in TQFT.

Contribution

It provides a novel interpretation of the Kauffman algebra at sqrt(-1) as parallel transport operators and studies trace asymptotics in non-standard TQFT regimes.

Findings

Kauffman algebra at sqrt(-1) corresponds to parallel transport operators.

Asymptotic behavior of curve-operator traces analyzed.

Interpretation of sqrt(-1) case in TQFT context.

Abstract

We study the structure of the Kauffman algebra of a surface with parameter equal to sqrt(-1). We obtain an interpretation of this algebra as an algebra of parallel transport operators acting on sections of a line bundle over the moduli space of flat connections in a trivial SU(2)-bundle over the surface. We analyse the asymptotics of traces of curve-operators in TQFT in non standard regimes where the root of unity parametrizing the TQFT accumulates to a root of unity. We interpret the case of sqrt(-1) in terms of parallel transport operators.