$\mathbb{Z}_2$ Lattice Gerbe Theory

Understand

$2$-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.