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

A bstract 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.