Liouville conformal field theory on Riemann surface with boundaries

Abstract

In this note, we give a unified rigorous construction for the Liouville conformal field theory on compact Riemann surface with boundaries for $\gamma \in (0,2]$ and prove a certain type of Markov property. We also prove some fusion-type estimates in the Boundary LCFT. This note will serve as a reference for a program leading to the conformal bootstrap for the Riemann surface with boundaries and several related projects.

21 pages, 2 figures. In this version we have added some fusion-type estimate in Section 6.2. arXiv admin note: text overlap with arXiv:1607.08467 by other authors