Abstract
The aim of the paper is to formulate a discrete analogue of the claim made by Alvarez-Gaume et al., (\cite{a}), 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.