# Dimers on surface graphs and spin structures. II

## TL;DR

This paper extends the geometric understanding of dimer models on surface graphs by generalizing spin structures to surfaces with boundary and exploring their implications in quantum field theory reformulations.

## Contribution

It generalizes the correspondence between edge orientations and spin structures to surfaces with boundary and analyzes how cutting and gluing affect the partition function.

## Key findings

- Extended spin structure correspondence to surfaces with boundary.
- Described how cutting and gluing operations modify spin structures.
- Reformulated the dimer model as a quantum field theory.

## Abstract

In a previous paper, we showed how certain orientations of the edges of a graph G embedded in a closed oriented surface S can be understood as discrete spin structures on S. We then used this correspondence to give a geometric proof of the Pfaffian formula for the partition function of the dimer model on G. In the present article, we generalize these results to the case of compact oriented surfaces with boundary. We also show how the operations of cutting and gluing act on discrete spin structures and how they change the partition function. These operations allow to reformulate the dimer model as a quantum field theory on surface graphs.