The Moduli of Annuli in Random Conformal Geometry

We obtain exact formulae for three basic quantities in random conformal geometry that depend on the modulus of an annulus. The first is for the law of the  modulus of the Brownian annulus describing the scaling limit of uniformly sampled planar maps with annular topology, which is as  predicted from the ghost partition function in bosonic string theory. The second is for the law of the modulus of the annulus bounded by a loop of a simple conformal loop ensemble (CLE) on a disk   and the disk boundary.  The formula is as conjectured from the partition function of the $O(n)$ loop model on the annulus derived by Cardy (2006). The third is for the annulus partition function of the $\SLE_{8/3}$ loop introduced by Werner (2008). It again confirms a prediction of Cardy (2006). The physics  principle underlying  our proofs is that  2D quantum gravity coupled with conformal matters can be decomposed into three conformal field theories (CFT): the matter CFT, the Liouville CFT, and the ghost CFT. At the technical level, we rely on two types of integrability in Liouville quantum gravity, one from the scaling limit of random planar maps, the other from the  Liouville CFT. We expect our method to be applicable to a variety of questions related to the random moduli of non-simply-connected random surfaces. Joint work with Morris Ang and Guillaume Remy.

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