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Chern-Simons theory in 3-dimensions has found its application in string theory as well as in pure mathematics. However, while the study of those theories for non-graded gauge groups is very advanced, the Chern-Simons theories for super-groups are much less studied.

In the first part of the talk we show how to construct a quantisation of the Teichmüller Space of super Riemann surfaces using coordinates associated to ideal triangulations of super Riemann surfaces. A new feature is the non-trivial dependence on the choice of a spin structure which can be encoded combinatorially in a certain refinement of the ideal triangulation. By constructing a projective unitary representation of the groupoid of changes of refined ideal triangulations we demonstrate that the dependence of the resulting quantum theory on the choice of a triangulation is inessential. Super pentagon relations is the main equation in the super-groupoid relations (arxiv:1512.02617). We also find the super generalisation of the generator of mapping class group acts on the Hilbert space of the once-puncture torus. We will show how these result will help to find CS invariant of mapping torus which may be the limit of the partition function of particular type of 3d, N=2 theory. This is based on the ongoing project with M. Pawelkiewicz and M. Yamazaki.

In the second part I will present the Hamiltonian quantisation of Chern-Simons theory for a super-group G on a 3-dimensional manifold M that locally admits a
foliation Sigma x R, where Sigma is a Riemann
surface. The quantisation is intimately connected to the representation
theory of the quantum group U_q(g) with q being a root of
unity. As an end result, one obtains a representation of the mapping
class group on the ``constant time'' slice Sigma, from which one can
calculate a super-symmetric manifold invariants. The general construction will be illustrated by the prototypical example of the Chern-Simons theory with GL(1|1)-gauge group and Sigma being a torus. The mapping class group in that
case is just SL(2,Z). This is based on arxiv:1811.09123.