A mathematical theory of topological invariants of quantum lattice systems

TL;DR

This paper develops a mathematical framework linking topological invariants like Hall conductance to gauge symmetries in quantum lattice systems, applicable to complex geometries beyond traditional field theory.

Contribution

It introduces a local Lie algebra approach over Grothendieck sites to define topological invariants of gapped quantum states, extending analysis to arbitrary lattice geometries.

Findings

Hall conductance acts as an obstruction to gauge symmetry promotion.

Constructs topological invariants from local Lie algebra symmetries.

Applicable to complex lattice geometries beyond standard field theory.

Abstract

We show that Hall conductance and its non-abelian 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 algebra 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 spin system form a local Lie algebra 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.