Dimers on graphs in non-orientable surfaces

TL;DR

This paper presents a Pfaffian formula for calculating the dimer model's partition function on graphs embedded in non-orientable surfaces, using geometrical methods and a correspondence with pin^- structures.

Contribution

It introduces a new Pfaffian formula for non-orientable surfaces, generalizing previous results with a purely geometrical approach.

Findings

Provides a computationally suitable Pfaffian formula

Establishes a correspondence between orientations and pin^- structures

Simplifies previous theoretical results

Abstract

The main result of this paper is a Pfaffian formula for the partition function of the dimer model on a graph G embedded in a closed, possibly non-orientable surface S. This formula is suitable for computational purposes, and it is obtained using purely geometrical methods. The key step in the proof consists of a correspondence between some orientations on G and the set of pin^- structures on S. This generalizes (and simplifies) the results of a previous paper [2].