GL(2,R) dualities in generalised Z(2) gauge theories and Ising models

TL;DR

This paper explores algebraic duality transformations in generalized Z(2) gauge theories and Ising models, unifying classical dualities and extending partition function relations across various lattice geometries.

Contribution

It introduces a framework using algebra and coalgebra structures to describe dualities, generalizes classical dualities, and derives explicit partition function relations for diverse lattice types.

Findings

Classical Kramers-Wannier dualities are special cases of the new transformations.

Explicit partition function relations are derived for finite triangulations.

A continuous gauge coupling symmetry transformation is identified in two dimensions.

Abstract

We study a class of duality transformations in generalised Z(2) gauge theories and Ising models on two- and three-dimensional compact lattices. The theories are interpreted algebraically in terms of the structure constants of a bidimensional vector space H with algebra and coalgebra structures, and it is shown that for any change of basis in H there is a related symmetry between such models. The classical Kramers and Wannier dualities are described as special cases of these transformations. We derive explicit expressions for the relation between partition functions on general finite triangulations for these cases, extending results known for square and cubic lattices in the thermodynamical limit. A class of symmetry transformations in which the gauge coupling changes continuously is also studied in two dimensions.