Liouville conformal field theory on Riemann surface with boundaries

TL;DR

This paper rigorously constructs Liouville conformal field theory on Riemann surfaces with boundaries, establishing a Markov property and fusion estimates, advancing the mathematical foundation for boundary conformal bootstrap.

Contribution

It provides a unified rigorous construction of boundary Liouville CFT and proves key properties, serving as a foundation for future conformal bootstrap research.

Findings

Established a Markov property for boundary Liouville CFT

Proved fusion-type estimates in Boundary LCFT

Serves as a reference for conformal bootstrap on Riemann surfaces 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.