Asymptotic properties of the Hitchin–Witten connection
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Bibliographic record
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.
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