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$\mathbb{Z}_2$ Lattice Gerbe Theory

$2$-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.