## Three-manifold invariants derived from conformal field theory and projective representations of modular groups

The purpose of this paper is to give a brief review on the author's approach to define 3-manifold invariants using representations of modular groups appearing in conformal field theory on Riemann surfaces. After the Witten's discovery of 3-manifold invariants based on Chern-Simons gauge theory. Reshetikhin and Turaev gave a Dehn surgery formula for Witten invariant, which was studied extensively by Kirby and Melvin and others. In this paper the authors describe the nature of these representations. The representations of the mapping class groups discussed in this paper are projectively linear and we write down the associated 2-cocyle in a combinatorialmore »