Z2 lattice Gerbe Theory

Discretized formulations of 2-form Abelian and non-Abelian gauge fields on d-dimensional hypercubiclattices have been discussed in the past by various authors and most recently by Lipstein and Reid-Edwards [J. High Energy Phys. 09 (2014) 034]. In this paper we recall that the Hamiltonian of a Z2 variant ofsuch theories is one of the family of generalized Ising models originally considered by Wegner. For such“Z2 lattice Gerbe theories” general arguments can be used to show that a phase transition for Wilsonsurfaces will occur for d > 3 between volume and area scaling behavior. In 3d the model is equivalentunder 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 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.