# Asymptotic properties of the Hitchin–Witten connection

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## Abstract

We explore extensions to \({{\,\mathrm{SL}\,}}(n,{\mathbb {C}})\)-Chern–Simons theory of some results obtained for \({{\,\mathrm{SU}\,}}(n)\)-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 provide one for the first step in the specific case of an operator of order zero, and show in general the vanishing of a weakened version of the obstruction. We also provide a solution for the whole recursion in the case of a surface of genus one.

## Keywords

Complex Chern-Simons theory Geometric quantisation Formal Hitchin-Witten connection## Mathematics Subject Classification

53D50 57R56 81S10## References

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