Super pentagon relation and mapping class group

Teichmueller space (TS) is a fundamental space that is important in many areas of mathematics and physics.The generalizations of this space on supermanifolds called super-Teichmueller spaces (STS). Super means that the structure sheaf is Z/2Z graded and contains odd or anti-commuting coordinates. The STS arise naturally as higher Teichmueller spaces corresponding to supergroups, which play an important role in mathematical physics. We construct a quantisation of the TS 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. Yamazak