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

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 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 remypreprint, in the construction of a unitary representation of mapping class group in the space of conformal blocks, and the study of random moduli ARSmoduliRPM in Liouville quantum gravity.