QUANTUM TEICHMULLER THEORY AND TQFT

By using quantum Teichmuller theory, a new type of three-dimensional TQFT has been constructed with the following distinguishing features: it uses the combinatorial framework of shaped triangulations; it takes its values in the space of tempered distributions; the fundamental building block of the theory is given by Faddeev's quantum dilogarithm. The semi-classical behavior and the geometrical ingredients suggest that the constructed TQFT is related to exact solution of quantum Chern-Simons theory with gauge group SL(2, C). We also remark that quantum Teichmuller theory itself admits an additional real parameter which preserves unitarity but affects the projective factor in the corresponding mapping class group representation.