Discrete Dirac Operators, Critical Embeddings and Ihara-Selberg Functions

The aim of the paper is to formulate a discrete analogue of the claim made by Alvarez-Gaume et al., ([1]), realizing the partition function of the free fermion on a closed Riemann surface of genus g as a linear combination of 22g Pfaffians of Dirac operators. Let G = (V,E) be a finite graph embedded in a closed Riemann surface X of genus g, xe the collection of independent variables associated with each edge e of G (collected in one vector variable x) and Σ the set of all 22g spin-structures on X. We introduce 22g rotations rots and (2|E| × 2|E|) matrices ∆(s)(x), s ∈ Σ, of the transitions between the oriented edges of G determined by rotations rots. 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 22g IharaSelberg functions I(∆(s)(x)) also called Feynman functions. By a result of Foata– Zeilberger holds I(∆(s)(x)) = det(I − ∆′(s)(x)), where ∆′(s)(x) is obtained from ∆(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.