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This paper is the first of a series of works on the conformal bootstrap inLiouville conformal field theory (CFT) with boundaries. We focus here on thecase of the annulus with two boundary insertions, each of which lies on thedifferent connected components of the boundary. In the course of proving thebootstrap formula, we established several properties on the correspondingannulus conformal blocks:  1) we show that they converge everywhere on the spectral line and they arecontinuous with respect to the spectrum and the primary weights.  2) we relate them to their torus counterparts by rigorously implementingCardy's doubling trick for boundary CFT,  3) we solve a conjecture of Martinec on the annulus partition function,  4) we also extend the bootstrap formula to the one-point case.  As an application of our bootstrap result, we give an exact formula for thebosonic LQG partition function of the annulus when $\gamma\in (0,2)$. Our paperserves as a key ingredient in the recent derivation of the random moduli forthe Brownian annulus by Ang, Remy, and Sun (2022). We also solve several otherconjectures relate to torus conformal blocks which arise from physicsliterature.

Conformal Bootstrap on the Annulus in Liouville CFT