Topological low-temperature limit of Z(2) spin-gauge theory in three dimensions

TL;DR

This paper analyzes the low-temperature behavior of Z(2) lattice gauge theory in three dimensions, revealing its topological nature and algebraic structure, and introduces duality relations among classical spin models.

Contribution

It reformulates the theory algebraically, shows the ground state degeneracy is topologically invariant, and connects the low-temperature limit to a topological field theory.

Findings

Ground state degeneracy is a topological invariant.

In the low-temperature limit, the algebra reduces to a Hopf algebra.

New duality relations among classical spin models are derived.

Abstract

We study Z(2) lattice gauge theory on triangulations of a compact 3-manifold. We reformulate the theory algebraically, describing it in terms of the structure constants of a bidimensional vector space H equipped with algebra and coalgebra structures, and prove that in the low-temperature limit H reduces to a Hopf Algebra, in which case the theory becomes equivalent to a topological field theory. The degeneracy of the ground state is shown to be a topological invariant. This fact is used to compute the zeroth- and first-order terms in the low-temperature expansion of Z for arbitrary triangulations. In finite temperatures, the algebraic reformulation gives rise to new duality relations among classical spin models, related to changes of basis of H.