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In the limit of the lattice spacing going to zero, we consider the dimermodel on isoradial graphs in the presence of singular $SL(N,\mathbb{C})$ gaugefields flat away from a set of punctures. We consider the cluster expansion ofthis twisted dimer partition function show it matches an analogous clusterexpansion of the 2D Dirac partition function in the presence of this gaugefield. The latter is often referred to as a tau function. This reproduces andgeneralizes various computations of Dub\'edat (J. Eur. Math. Soc. 21 (2019),no. 1, pp. 1-54). In particular, both sides' cluster expansion are matched upterm-by-term and each term is shown to equal a sum of a particular holomorphicintegral and its conjugate. On the dimer side, we evaluate the terms in theexpansion using various exact lattice-level identities of discrete exponentialfunctions and the inverse Kasteleyn matrix. On the fermion side, the clusterexpansion leads us to two novel series expansions of tau functions, oneinvolving the Fuschian representation and one involving the monodromyrepresentation.

2D Fermions and Statistical Mechanics: Critical Dimers and Dirac Fermions in a background gauge field