  Hyperbolic lattices underlie a new form of quantum matter with potential
applications to quantum computing and simulation and which, to date, have been
engineered artificially. A corresponding hyperbolic band theory has emerged,
extending 2-dimensional Euclidean band theory in a natural way to higher-genus
configuration spaces. Attempts to develop the hyperbolic analogue of Bloch's
theorem have revealed an intrinsic role for algebro-geometric moduli spaces,
notably those of stable bundles on a curve. We expand this picture to include
Higgs bundles, which enjoy natural interpretations in the context of band
theory. First, their spectral data encodes a crystal lattice and momentum,
providing a framework for symmetric hyperbolic crystals. Second, they act as a
complex analogue of crystal momentum. As an application, we elicit a new
perspective on Euclidean band theory. Finally, we speculate on potential
interactions of hyperbolic band theory, facilitated by Higgs bundles, with
other themes in mathematics and physics.
