From combinatorial maps to correlation functions in loop models

Abstract

In two-dimensional statistical physics, correlation functions of the O(N)O(N) and Potts models may be written as sums over configurations of non-intersecting loops. We define sums associated to a large class of combinatorial maps (also known as ribbon graphs). We allow disconnected maps, but not maps that include monogons. Given a map with nn vertices, we obtain a function of the moduli of the corresponding punctured Riemann surface. Due to the map’s combinatorial (rather than topological) nature, that function is single-valued, and we call it an nn-point correlation function. We conjecture that in the critical limit, such functions form a basis of solutions of certain conformal bootstrap equations. They include all correlation functions of the O(N)O(N) and Potts models, and correlation functions that do not belong to any known model. We test the conjecture by counting solutions of crossing symmetry for four-point functions on the sphere.