Dimers on surface graphs and spin structures. II

In a previous paper [3], we showed how certain orientations of the edges of a graph Gamma embedded in a closed oriented surface Sigma can be understood as discrete spin structures on Sigma. We then used this correspondence to give a geometric proof of the Pfaffian formula for the partition function of the dimer model on Gamma. 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.