Quantum mechanics and non-abelian theta functions for the gauge group $SU(2)$

TL;DR

This paper explores the non-abelian theta functions in $SU(2)$-Chern-Simons theory using classical theta function methods, quantum group quantization, and skein algebra, establishing foundational results for quantum topology.

Contribution

It introduces a quantum mechanical framework for non-abelian theta functions and proves a Stone-von Neumann theorem for the moduli space of flat $SU(2)$-connections.

Findings

Derived the formula for the Reshetikhin-Turaev representation on the torus.

Established a quantum mechanical derivation of skein relations for handle slides.

Proved a Stone-von Neumann theorem for the moduli space of flat $SU(2)$-connections.

Abstract

This paper outlines an approach to the non-abelian theta functions of the $SU(2)$-Chern-Simons theory with the methods used by A. Weil for studying classical theta functions. First we translate in knot theoretic language classical theta functions, the action of the finite Heisenberg group, and the discrete Fourier transform. Then we explain how the non-abelian counterparts of these arise in the framework of the quantum group quantization of the moduli space of flat $SU(2)$-connections on a surface, in the guise of the non-abelian theta functions, the action of a skein algebra, and the Reshetikhin-Turaev representation of the mapping class group. We prove a Stone-von Neumann theorem on the moduli space of flat $SU(2)$-connections on the torus, and using it we deduce the existence and the formula for the Reshetikhin-Turaev representation on the torus from quantum mechanical considerations. We show how one can derive in a quantum mechanical setting the skein that allows handle slides, which is the main ingredient in the construction of quantum $3$-manifold invariants.