Topology and Duality in Abelian Lattice Theories

TL;DR

This paper develops a general method to find dual models of Abelian lattice theories with inhomogeneous interactions, revealing the role of cohomology and enabling the construction of self-dual theories, with applications to Z_N and U(1) gauge theories.

Contribution

It provides an explicit construction for duality in Abelian lattice models with arbitrary topology and inhomogeneity, including self-duality conditions and applications to specific gauge theories.

Findings

Dual models incorporate disorder loops related to cohomology generators.

Explicit construction of self-dual theories when obstructions are present.

Calculation of n-point functions for U(1) gauge theory on tori.

Abstract

We show how to obtain the dual of any lattice model with inhomogeneous local interactions based on an arbitrary Abelian group in any dimension and on lattices with arbitrary topology. It is shown that in general the dual theory contains disorder loops on the generators of the cohomology group of a particular dimension. An explicit construction for altering the statistical sum to obtain a self-dual theory, when these obstructions exist, is also given. We discuss some applications of these results, particularly the existence of non-trivial self-dual 2-dimensional Z_N theories on the torus. In addition we explicitly construct the n-point functions of plaquette variables for the U(1) gauge theory on the 2-dimensional g-tori.