Asymptotic properties of the Hitchin-Witten connection

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 $1$.