  The Berry connection plays a central role in our description of the geometric
phase and topological phenomena. In condensed matter, it describes the parallel
transport of Bloch states and acts as an effective "electromagnetic" vector
potential defined in momentum space. Inspired by developments in mathematical
physics, where higher-form (Kalb-Ramond) gauge fields were introduced, we
hereby explore the existence of "tensor Berry connections" in quantum matter.
Our approach consists in a general construction of effective gauge fields,
which we ultimately relate to the components of Bloch states. We apply this
formalism to various models of topological matter, and we investigate the
topological invariants that result from generalized Berry connections. For
instance, we introduce the 2D Zak phase of a tensor Berry connection, which we
then relate to the more conventional first Chern number; we also reinterpret
the winding number characterizing 3D topological insulators to a Dixmier-Douady
invariant, which is associated with the curvature of a tensor connection.
Besides, our approach identifies the Berry connection of tensor monopoles,
which are found in 4D Weyl-type systems [Palumbo and Goldman, Phys. Rev. Lett.
121, 170401 (2018)]. Our work sheds light on the emergence of gauge fields in
condensed-matter physics, with direct consequences on the search for novel
topological states in solid-state and quantum-engineered systems.
