Some interesting aspects of the SU(2)-Chern-Simons theory arose from the study of its asymptotic properties in the semiclassical limit. For instance, Berezin-Toeplitz theory defines for a closed, oriented, smooth surface a family of deformations of the Poisson algebra of its moduli space of flat SU(2)-connections, parametrised by the Teichmüller space. The dependence on the parameter can be measured via the Hitchin connection, by defining a formal version of it by means of an asymptotic expansion in the mentioned limit.
This presentation will focus on the problem of finding an analogue of this formal connection for the situation of SL(2,C), together with a trivialisation thereof. Although the Hitchin-Witten connection, analogous in this situation to the Hitchin connection, admits an immediate asymptotic expansion in the full (complex!) quantum parameter t, the existence of a trivialisation for the corresponding formal connection has a non-vanishing cohomological obstruction. Said obstruction, however, disappears if one considers asymptotic expansions only in the imaginary part of the quantum parameter. Moreover, in the particular case of a surface of genus one, an explicit trivialisation can be found, which can also be related to an exact trivialisation of the Hitchin-Witten connection exhibited by Witten in this special case.
The content of the presentation is a joint work with Jørgen Ellegaard Andersen.