Title: Classical Liouville Action and Uniformization of Orbifold Riemann Surfaces: A Geometric Approach to Classical Correlation Functions of Branch Point Vertex Operators
Presenter: Behrad Taghavi (Institute for Research in Fundamental Sciences)
Time & Date: 14:00 IRST - 04 & 11 December 2023
Location: Ferdowsi University, Faculty of Science, Physics Department, Administration Room
Abstract: In this talk, based on 2310.17536, we study the classical Liouville field theory on Riemann surfaces of genus g>1 in the presence of vertex operators associated with branch points of orders mi>1. In particular, classical correlation functions of branch point vertex operators on a closed Riemann surface are related to the on-shell value of Liouville action functional on the same Riemann surface but with the insertion of conical points (of angles 2pi/mi) at the location of these operators. With this motivation, and using the results of 1508.02102 and 1701.00771, we will study the appropriate classical Liouville action on a Riemann orbisurface using the Schottky global coordinates. We will also study the first and second variations of this action on the Schottky deformation space of Riemann orbisurfaces and show that the classical Liouville action is a Kähler potential for a special combination of Weil-Petersson and Takhtajan-Zograf metrics which appear in the local index theorem for Riemann orbisurfaces (see 1701.00771). The obtained results can then be interpreted in terms of the complex geometry of Hodge line bundle equipped with Quillen’s metric over the moduli space of Riemann orbisurfaces.