  Hyperbolic lattices are a new type of synthetic quantum matter emulated in
circuit quantum electrodynamics and electric-circuit networks, where particles
coherently hop on a discrete tessellation of two-dimensional negatively curved
space. While real-space methods and a reciprocal-space hyperbolic band theory
have been recently proposed to analyze the energy spectra of those systems,
discrepancies between the two sets of approaches remain. In this work, we
reconcile those approaches by first establishing an equivalence between
hyperbolic band theory and $U(N)$ topological Yang-Mills theory on higher-genus
Riemann surfaces. We then show that moments of the density of states of
hyperbolic tight-binding models correspond to expectation values of Wilson
loops in the quantum gauge theory and become exact in the large-$N$ limit.
