Sine-Liouville gravity as a Vertex Model on Planar Graphs

Abstract

We investigate the universal behaviour of a one-parameter generalisation of the six-vertex model on planar graphs, which we refer to as the seven-vertex model, or 7vM for quick reference. The 7vM is characterised by a temperature coupling and its continuum limit exhibits massive, dilute and dense phases similarly to the $O(n)$ loop model. However, there is an important distinction: the loop weights are no longer topological and the dynamics of the loops is now entangled with the local geometry of the lattice. From the dual matrix model we derive explicit expressions for the sphere and disk partition functions in the continuum limit. The disk partition function for fixed length is a deformation of the Bessel integral known as the Kr\"atzel function. We argue that the 7v matrix model (7vMM) and Matrix Quantum Mechanics (MQM) provide two complementary non-perturbative realisations of sine-Liouville gravity. Specifically, we find that the continuum limits of 7vMM and MQM share the same classical spectral curve but describe two different types of branes in sine-Liouville gravity. The 7vMM precisely covers the range of parameters where the Minkowskian MQM lacks a simple interpretation in terms of multiple tachyon scattering. We investigate the flow relating the dilute and dense phases and argue that this flow is the gravitational analogue of the massless flow in the sine-Gordon model with imaginary mass coupling. The two endpoints of the flow are described by a free boson coupled to Liouville gravity and compactified on circles with two different radii.