Conformal Bootstrap for surfaces with boundary in Liouville CFT. Part 1: Segal axioms

TL;DR

This paper establishes the foundational axioms and gluing properties for Liouville conformal field theory on surfaces with boundaries, paving the way for a rigorous conformal bootstrap approach in this setting.

Contribution

It introduces Segal's axioms for surfaces with corners in Liouville CFT and proves the gluing property, enabling the conformal bootstrap formula with boundary conditions.

Findings

Proved Segal's axioms for surfaces with boundaries.

Established the gluing property for boundary amplitudes.

Developed spectral decomposition for the semi-group of half-annuli.

Abstract

This paper is the first part of the proof of the conformal bootstrap for Liouville conformal field theory on surfaces with a boundary, devoted to Segal's axioms in this context. We introduce the notion of Segal's amplitudes on surfaces with corners and prove the gluing property for such amplitudes. The semi-group of half-annuli and its generator are studied and we develop the necessary material for proving its spectral decomposition using scattering theory in the companion paper \cite{GRW2}. The Segal gluing properties and the spectral decomposition allows us to prove the conformal bootstrap formula for correlation functions of Liouville conformal field theory with a boundary. This has several important applications to the study of conformal blocks (analyticity and convergence) in \cite{remypreprint}, in the construction of a unitary representation of mapping class group in the space of conformal blocks, and the study of random moduli \cite{ARSmoduliRPM} in Liouville quantum gravity.