Asymptotic properties of the Hitchin–Witten connection

We explore extensions to SL(n, C)-Chern–Simons theory of s ome results obtained forSU(n)-Chern–Simons theory via the asymptotic properties of the Hitchin connectionand 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 quantumparameter t = k + is and investigate the existence of a formal trivialisation. Afterreducing the problem to a recursive system of differential equations, we identify acohomological obstruction to the existence of a solution. We explicitly provide onefor the first step in the specific case of an operator of order zero, and show in generalthe vanishing of a weakened version of the obstruction. We also provide a solution forthe whole recursion in the case of a surface of genus one.