Discrete Dirac Operators, Critical Embeddings and Ihara-Selberg Functions

TL;DR

This paper develops a discrete analogue of fermionic partition functions on Riemann surfaces using graph embeddings, Dirac operators, and Ihara-Selberg functions, linking combinatorial and geometric aspects of statistical physics.

Contribution

It introduces a novel discrete framework connecting Dirac operators, spin structures, and Ihara-Selberg functions to compute Ising model partition functions on embedded graphs.

Findings

Partition function expressed as a linear combination of Ihara-Selberg functions.

Each Ihara-Selberg function is computable in polynomial time.

Discrete Feynman functions serve as analogues of Pfaffians of Dirac operators.

Abstract

The aim of the paper is to formulate a discrete analogue of the claim made by Alvarez-Gaume et al., realizing the partition function of the free fermion on a closed Riemann surface of genus g as a linear combination of 2^{2g} Pfaffians of Dirac operators. Let G=(V,E) be a finite graph embedded in a closed Riemann surface X of genus g, x_e the collection of independent variables associated with each edge e of G (collected in one vector variable x) and S the set of all 2^{2g} Spin-structures on X. We introduce 2^{2g} rotations rot_s and (2|E| times 2|E|) matrices D(s)(x), s in S, of the transitions between the oriented edges of G determined by rotations rot_s. We show that the generating function for the even subsets of edges of G, i.e., the Ising partition function, is a linear combination of the square roots of 2^{2g} Ihara-Selberg functions I(D(s)(x)) also called Feynman functions. By a result of Foata--Zeilberger holds I(D(s)(x))= det(I-D'(s)(x)), where D'(s)(x) is obtained from D(s)(x) by replacing some entries by 0. Thus each Feynman function is computable in polynomial time. We suggest that in the case of critical embedding of a bipartite graph G, the Feynman functions provide suitable discrete analogues for the Pfaffians of discrete Dirac operators.