Generalisation of Teichmueller space
The quantization of the Teichmueller spaces of Riemann surfaces has found important applications to conformal field theory and N=2 supersymmetric gauge theories. The aim of the talk is to construct generalizations of the quantum Teichmueller theory which would, in particular, describe the quantum theory of the Teichmueller spaces of Super-Riemann surfaces. The starting point of the project was the observation that the quantum Teichm\"ueller theory can be build combinatorially from a simple quantum group, a deformation of the Borel half of SL(2). The idea is to replace the (quantum) group SL(2) by a suitable quantum super-group. The result is to demonstrate that the resulting quantum theory is nothing but the quantum theory of the Teichmueller spaces of Super-Riemann surfaces."