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Sine-Liouville gravity as a Vertex Model on Planar Graphs

Ivan Kostov

TL;DR

The paper presents a lattice regularisation of sine-Liouville gravity via a one-parameter 7-vertex model on planar graphs and derives its continuum limit through a dual large-N matrix model (7vMM). It computes sphere and disk partition functions from the spectral curve and reveals a novel boundary structure expressed through a generalized K-Bessel function, linking non-perturbative physics to Matrix Quantum Mechanics (MQM). The bulk physics matches between 7vMM and MQM, providing two complementary realizations of the same world-sheet theory, while boundary data differ and reflect distinct brane content. The work advances the holographic understanding of 2D gravity with sine-Liouville matter, clarifies gravity-induced radius renormalisation, and highlights rich connections between integrable lattice models, matrix models, and non-perturbative 2D string theory.

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 7-vertex model (7vM). The 7vM is characterised by a temperature coupling and its continuum limit is characterised by a massive, dilute and dense phases similarly to the $O(n)$ loop model. We compute the sphere and disk partition functions of the 7vM from the spectral curve of the dual matrix model, abbreviated here as 7vMM. The disk partition function for fixed length is expressed in terms of an uncharted deformation of the K-Bessel functions. We argue that 7vMM and Matrix Quantum Mechanics (MQM) provide two complementary non-perturbative realisations of sine-Liouville gravity. Specifically, we find that the continuum limit of 7vMM and the 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 the 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 extremities of the flow are described by a free boson coupled to Liouville gravity and compactified at circles with two different radii.

Sine-Liouville gravity as a Vertex Model on Planar Graphs

TL;DR

The paper presents a lattice regularisation of sine-Liouville gravity via a one-parameter 7-vertex model on planar graphs and derives its continuum limit through a dual large-N matrix model (7vMM). It computes sphere and disk partition functions from the spectral curve and reveals a novel boundary structure expressed through a generalized K-Bessel function, linking non-perturbative physics to Matrix Quantum Mechanics (MQM). The bulk physics matches between 7vMM and MQM, providing two complementary realizations of the same world-sheet theory, while boundary data differ and reflect distinct brane content. The work advances the holographic understanding of 2D gravity with sine-Liouville matter, clarifies gravity-induced radius renormalisation, and highlights rich connections between integrable lattice models, matrix models, and non-perturbative 2D string theory.

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 7-vertex model (7vM). The 7vM is characterised by a temperature coupling and its continuum limit is characterised by a massive, dilute and dense phases similarly to the loop model. We compute the sphere and disk partition functions of the 7vM from the spectral curve of the dual matrix model, abbreviated here as 7vMM. The disk partition function for fixed length is expressed in terms of an uncharted deformation of the K-Bessel functions. We argue that 7vMM and Matrix Quantum Mechanics (MQM) provide two complementary non-perturbative realisations of sine-Liouville gravity. Specifically, we find that the continuum limit of 7vMM and the 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 the 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 extremities of the flow are described by a free boson coupled to Liouville gravity and compactified at circles with two different radii.

Paper Structure

This paper contains 21 sections, 108 equations, 3 figures.

Table of Contents

  1. Introduction and summary
  2. Sine-Liouville gravity
  3. Sine-Gordon QFT
  4. Sine-Liouville gravity
  5. The 7-vertex model on planar graphs
  6. Definition and loop expansion
  7. The 7-vertex matrix model
  8. Collective field and Virasoro constraint
  9. Thermodynamical limit and spectral curve
  10. Boundary value problem
  11. Phase diagram and continuum limit
  12. Uniformisation of the boundary value problem in the continuum limit
  13. Solution at the critical point $(M=0)$
  14. Solution of the boundary value problem in the continuum limit
  15. Computation of the boundary entropy
  16. ...and 6 more sections

Figures (3)

  • Figure 1: The vertices of the 7-vertex model.
  • Figure 2: A loop configuration on a trivalent graph with the topology of the disk and on its dual triangulation. To preserve the information about the topology the lines should be thickened.
  • Figure 3: The domains ${\mathbb{C}}_+$ and ${\mathbb{C}}_-$ of the principal sheet of the Riemann surface of $Y(x)$ (left) and in the principal strip of the $s$-parametrisation plane (right)