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From combinatorial maps to correlation functions in loop models
by Linnea GransSamuelsson, Jesper Lykke Jacobsen, Rongvoram Nivesvivat, Sylvain Ribault, Hubert Saleur
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Submission summary
Authors (as registered SciPost users):  Linnea GransSamuelsson · Jesper Lykke Jacobsen · Rongvoram Nivesvivat · Sylvain Ribault 
Submission information  

Preprint Link:  https://arxiv.org/abs/2302.08168v2 (pdf) 
Date accepted:  20230906 
Date submitted:  20230825 11:00 
Submitted by:  Ribault, Sylvain 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
In twodimensional statistical physics, correlation functions of the O(N) and Potts models may be written as sums over configurations of nonintersecting 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 n 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 singlevalued, and we call it an npoint 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) 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 fourpoint functions on the sphere.
Author comments upon resubmission
List of changes
 On page 5, in order to clarify the origin of combinatorial maps, we have rewritten the first full paragraph after Figure (1.4). ("The main idea...")
 On page 5, we have rewritten the last paragraph before the Highlights, in order to improve the discussion of the interpretation of correlation functions in terms of local fields.
 On page 18, we have enumerated the parameters of our sum over loop configurations, in order to clarify which parameters are discrete or continuous. In particular, the angles are parameters of the loops themselves and not of the sum. The angular momentums are discrete, and the only continuous parameters of the sum are the weights of closed loops.
 On page 21, we have rewritten the last paragraph of Section 3 in order to better conclude the comparison with the Coulomb gas approach.
Published as SciPost Phys. 15, 147 (2023)