Sphere and disk partition functions in Liouville and in matrix integrals

TL;DR

This work computes sphere and disk partition functions in the semiclassical Liouville theory and in the corresponding double-scaled matrix-integral duals, focusing on the unambiguous ratio $Z_{\text{sphere}}/Z_{\text{disk}}^2$ and demonstrating a precise numerical match between the two frameworks. The authors carefully treat saddle points, one-loop determinants, and the division by conformal-group volumes (PSL$\left(2,\mathbb{C}\right)$ and PSL$\left(2,\mathbb{R}\right)$) using a Fadeev–Popov construction, then apply the matching to the $(2,p)$ minimal string and its large-$p$ limit that approaches JT gravity, where the sphere partition function diverges. On the matrix-model side, they construct analytic, analytic-in-$\epsilon^2$ potentials yielding a universal edge density, compute the genus-zero free energy and FZZT disk observable, and extract a universal nonanalytic term that matches the Liouville result in the appropriate limit. Overall, the paper provides a nontrivial cross-check between Liouville theory and matrix-integral duals and clarifies the JT gravity divergence via the Liouville/matrix-model correspondence.

Abstract

We compute the sphere and disk partition functions in semiclassical Liouville and analogous quantities in double-scaled matrix integrals. The quantity sphere/disk^2 is unambiguous and we find a precise numerical match between the Liouville answer and the matrix integral answer. An application is to show that the sphere partition function in JT gravity is infinite.