Classical Liouville Action and Uniformization of Orbifold Riemann Surfaces

In this talk, based on 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 $m_i > 1$. 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 $2\pi/m_i$) 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. 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 arXiv:1701.00771).