${\mathbb{Z}}_2$ Lattice Gerbe Theory

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.

v2: references added, abstract tweaked, (at least some of the) timeline clarified

References in corpus (11)

• Covariant Action for a D=11 Five-Brane with the Chiral Field
• Interacting Chiral Gauge Fields in Six Dimensions and Born-Infeld Theory
• Confining Strings
• Phases of Antisymmetric Tensor Field Theories
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Cited by in corpus (5)

• Fracton Topological Order, Generalized Lattice Gauge Theory and Duality
• $(k,n)$-fractonic Maxwell theory
• Universality Classes of Stabilizer Code Hamiltonians
• Model of Fractionalization of Faraday Lines in Compact Electrodynamics
• String confinement in 2-form lattice gauge theory