Brownian loops and the central charge of a Liouville random surface

Abstract

We explore the geometric meaning of the so-called zeta-regularized determinant of the Laplace–Beltrami operator on a compact surface, with or without boundary. We relate the $(-\mathit{c}/2)$th power of the determinant of the Laplacian to the appropriately regularized partition function of a Brownian loop soup of intensity c on the surface. This means that, in a certain sense, decorating a random surface by a Brownian loop soup of intensity c corresponds to weighting the law of the surface by the $(-\mathit{c}/2)$th power of the determinant of the Laplacian.

Next, we introduce a method of regularizing a Liouville quantum gravity (LQG) surface (with some matter central charge parameter c) to produce a smooth surface. And we show that weighting the law of this random surface by the $(-{\mathbf{c}}^{\prime }/2)$th power of the Laplacian determinant has precisely the effect of changing the matter central charge from c to $\mathbf{c}+{\mathbf{c}}^{\prime }$. Taken together with the earlier results, this provides a way of interpreting an LQG surface of matter central charge c as a pure LQG surface decorated by a Brownian loop soup of intensity c.

Building on this idea, we present several open problems about random planar maps and their continuum analogs. Although the original construction of LQG is well defined only for $\mathbf{c}\le 1$, some of the constructions and questions also make sense when $\mathbf{c}>1$.