$\mathbb{Z}_2$ Lattice Gerbe Theory

TL;DR

This paper studies the phase transitions of a $ abla_2$ lattice gerbe theory with $ abla_2$ gauge fields on hypercubic lattices, revealing dimension-dependent behaviors including phase transitions and dualities, extending understanding of higher-form gauge theories.

Contribution

It demonstrates that $ abla_2$ lattice gerbe theories exhibit dimension-dependent phase transitions and dualities, connecting them to generalized Ising models and extending the theoretical framework of higher-form gauge theories.

Findings

Phase transition for Wilson surfaces occurs in dimensions greater than 3.

In 3D, the model is dual to an infinite coupling model with no transition.

In 4D, the model is dual to the 4D Ising model with a continuous transition.

Abstract

$2$-form abelian and non-abelian gauge fields on $d$-dimensional hypercubic lattices have been discussed in the past by various authors and most recently by Lipstein and Reid-Edwards. In this note we recall that the Hamiltonian of a $\mathbb{Z}_2$ variant of such theories is one of the family of generalized Ising models originally considered by Wegner. For such "$\mathbb{Z}_2$ lattice gerbe theories" general arguments can be used to show that a phase transition for Wilson surfaces will occur for $d>3$ between volume and area scaling behaviour. In $3d$ the model is equivalent under duality to an infinite coupling model and no transition is seen, whereas in $4d$ the model is dual to the $4d$ Ising model and displays a continuous transition. In $5d$ the $\mathbb{Z}_2$ lattice gerbe theory is self-dual in the presence of an external field and in $6d$ it is self-dual in zero external field.