Asymptotic properties of the Hitchin-Witten connection

TL;DR

This paper studies the asymptotic behavior of the Hitchin-Witten connection in complex Chern-Simons theory, identifying obstructions and solutions for its trivialisation, especially for genus one surfaces.

Contribution

It extends previous results on the Hitchin connection to SL(n,C) theories, analyzing formal trivializations and cohomological obstructions in the context of complex Chern-Simons theory.

Findings

Identified a cohomological obstruction to trivializing the formal Hitchin-Witten connection.

Explicitly found a solution for the first recursive step in the case of an order 0 operator.

Proved the vanishing of a weakened obstruction and solved the recursion for genus one surfaces.

Abstract

We explore extensions to $\operatorname{SL}(n,\mathbb{C})$-Chern-Simons theory of some results obtained for $\operatorname{SU}(n)$-Chern-Simons theory via the asymptotic properties of the Hitchin connection and its relation to Toeplitz operators developed previously by the first named author. We define a formal Hitchin-Witten connection for the imaginary part $s$ of the quantum parameter $t = k+is$ and investigate the existence of a formal trivialisation. After reducing the problem to a recursive system of differential equations, we identify a cohomological obstruction to the existence of a solution. We explicitly find one for the first step, in the specific case of an operator of order $0$, and show in general the vanishing of a weakened version of the obstruction. We also find a solution of the whole recursion in the case of a surface of genus $1$.