# Tensor network approach to electromagnetic duality in (3+1)d topological gauge models

Given the Hamiltonian realisation of a topological (3+1)d gauge theory with finite group G, we consider a family of tensor network representations of its ground state subspace. This family is indexed by gapped boundary conditions encoded into module 2- categories over the input spherical fusion 2-category. Individual tensors are characterised by symmetry conditions with respect to non-local operators acting on entanglement degrees of freedom.

In the case of Dirichlet and Neumann boundary conditions, we show that the symmetry operators form the fusion 2-categories 2Vec_{G} of G-graded 2-vector spaces and 2Rep(G) of 2-representations of G, respectively. In virtue of the Morita equivalence between 2Vec_{G }and 2Rep(G) — which we explicitly establish — the topological order can be realised as the Drinfel’d centre of either 2-category of operators; this is a realisation of the electromagnetic duality of the theory. Specialising to the case G = Z_{2},we recover tensor network representations that were recently introduced, as well as the relation between the electromagnetic duality of a pure Z_{2 }gauge theory and the Kramers-Wannier duality of a boundary Ising model.

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