Classical Liouville Action and Uniformization of Orbifold Riemann Surfaces: A Geometric Approach to Classical Correlation Functions of Branch Point Vertex Operators

• Behrad Taghavi (School of Particles and Accelerators, IPM)

In this talk, based on arXiv:2310.17536, we will 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 is 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 arXiv:1508.02102 and arXiv:1701.00771, we will study the appropriate classical Liouville action on a Riemann orbisurface using the Schottky global coordinates. Additionally, classical limit of conformal Ward identities motivates us to study the first and second variations of this action on the Schottky deformation space of Riemann orbisurfaces. In this way, we are able to show that the appropriately defined 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 arXiv: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.