A mathematical theory of topological invariants of quantum lattice systems
Abstract
We show that Hall conductance and its nonabelian and higher-dimensional analogs are obstructions to promoting a symmetry of a state to a gauge symmetry. To do this, we define a local Lie system over a Grothendieck site as a pre-cosheaf of Lie algebras with additional properties and propose that a gauge symmetry should be described by such an object. We show that infinitesimal symmetries of a gapped state of a quantum lattice system form a local Lie system over a site of semilinear sets and use it to construct topological invariants of the state. Our construction applies to lattice systems on arbitrary asymptotically conical subsets of a Euclidean space including those which cannot be studied using field theory.
Fields of Study & Subjects
Topological and Geometric Data Analysis
Quantum chaos and dynamical systems
Publication details
- Journal / source
- Communications of the American Mathematical Society
- Volume
- 6
- Issue
- 15
- Pages
- 793–843
- Publication date
- 30 June 2026
- Language
- EN