Boundary Loop Models and 2D Quantum Gravity

TL;DR

This paper analyzes the O(n) loop model on a dynamically triangulated disk with new boundary conditions, connecting it to Liouville theory and confirming conjectured boundary exponents via KPZ correspondence.

Contribution

It introduces and studies the Jacobsen-Saleur boundary conditions in the dense phase of the loop gas, linking boundary correlators to Liouville theory and validating conjectured exponents.

Findings

Boundary two-point functions match Liouville theory results.

The boundary exponents agree with JS conjectures via KPZ correspondence.

The model generalizes known boundary conditions like Neumann and Dirichlet.

Abstract

We study the O(n) loop model on a dynamically triangulated disk, with a new type of boundary conditions, discovered recently by Jacobsen and Saleur. The partition function of the model is that of a gas of self and mutually avoiding loops covering the disk. The Jacobsen-Saleur (JS) boundary condition prescribes that the loops that do not touch the boundary have fugacity n in [-2,2], while the loops touching at least once the boundary are given different fugacity y. The class of JS boundary conditions, labeled by the real number y, contains the Neumann (y=n) and Dirichlet (y=1) boundary conditions as particular cases. Here we consider the dense phase of the loop gas, where we compute the two-point boundary correlators of the L-leg operators with mixed Neumann-JS boundary condition. The result coincides with the boundary two-point function in Liouville theory, derived by Fateev, Zamolodchikov and Zamolodchikov. The Liouville charge of the boundary operators match, by the KPZ correspondence, with the L-leg boundary exponents conjectured by JS.