The dimer model and the 2D Dirac fermion are two important examples of critical systems: the former being a stat mech model and the latter being simple quantum field theory. We show that these theories are broadly speaking the same. We show that the partition functions of both in the presence of a background gauge field gives the same quantity in the continuum limit, known mathematically as the isomonodromic tau function. The computation on the dimer side involves lattice-level identities of discrete holomorphic functions and leads to an interesting expansion of the tau function in terms of a series of holomorphic integrals.
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