Conformal Bootstrap on the Annulus in Liouville CFT

Baojun Wu

Published 2022-03-22Version 1

This paper is the first of a series of work on boundary bootstrap in Liouville conformal field theory (CFT), which focuses on the case of the annulus with two boundary insertions lying on different boundaries. In the course of proving the bootstrap formula, we established several properties on the corresponding annulus conformal blocks: 1) we show that they converge everywhere on the spectral line and they are continuous with respect to the spectrum; 2) we relate them to their torus counterparts by rigorously implementing Cardy's doubling trick for boundary CFT; 3) we extend the definition of these conformal blocks to the whole Seiberg bound; 4) when there is one insertion and its weight is $\gamma$, we show that the block degenerates and the bootstrap formula settles a conjecture of Martinec, which was properly written in Ang, Remy and Sun (2021). As an intermediate step, we obtain robust estimates on the boundary-bulk correlator around the spectrum line. As an application of our bootstrap result, we give an exact formula for the bosonic LQG partition function of the annulus when $\gamma\in (0,2)$. Our paper serves as a key ingredient in the recent derivation of the random moduli for the Brownian annulus by Ang, Remy and Sun (2022).