Rice University Department of Mathematics Colloquium

Determinants, Liouville action and analytic geometry of moduli spaces of Riemann surfaces

4:00 pm Thursday, November 12, 2015
Leon Takhtajan(UW Madison)

Abstract: Regularized determinants of elliptic operators play fundamental role in geometry and physics. I will start reviewing classical Kronecker limit formula and its generalization to compact Riemann surfaces of genus $g > 1$. It gives a holomorphic factorization of determinants of Laplacians on Riemann surfaces and is based on the Atiyah-Singer index theorem for families; the latter is understood as explicit computation of the Chern form of the determinant line bundle with the Quillen’s metric.

In this context I will elucidate the role of the Liouville action and will highlight the interplay between the quantum Liouville theory and analytic geometry of moduli spaces. I will conclude by discussing recent computation of the Chern forms of certain Hermitian line bundles over the moduli spaces of punctured Riemann surfaces.

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