Mathematics Colloquia and Seminars

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In this talk I shall survey Witten path integral derivation of the TQFT axioms and his derivation of the gauge theory construction. The resulting gauge theory construction of representations of mapping class groups of surfaces is completly rigorous. I shall then explain the asymptotic expansion conjecture, which is concerned with the semi-classical limit of these partition functions. This is a mathematically well formulated conjecture, but it is motivated by stationary phase approximation of the path integral expression of the partition function in a parameter called the level, which plays the role of one over Planks constant. From its formulation, it is clear that this conjecture implies strong relations between the quantum invariants and the classical algebraic topology of 3-manifolds such as Chern-Simons values on flat connections and the first fundamental group. Finally I shall present a proof of the conjecture for the class of 3-manifolds which are mapping cylinders of finite order diffeomorphims of 2-dimensional surfaces by using the gauge theory constructed. This proof uses the Lefschetz-Riemann-Roch theorem on the singular algebraic variety, the moduli space semi-stable bundles on a Riemann Surface, of which the difeomorphism is an automorphism of.