Asymptotic properties of the Hitchin-Witten connection

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We explore extensions to SL(n, C)-Chern-Simons theory of some results obtained for 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.
Original languageEnglish
PublisherarXiv.org
Publication statusPublished - 13 May 2018

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