Tensor network approach to electromagnetic duality in (3+1)d topological gauge models
Clement Delcamp
TL;DR
The article develops a tensor-network framework for (3+1)d topological gauge models with finite gauge group $G$, encoding gapped boundaries as module 2-categories and linking the two canonical boundary descriptions to virtual symmetry 2-categories $\mathsf{2Vec}_G$ and $\mathsf{2Rep}(G)$. It proves a Morita equivalence between these 2-categories and shows the bulk order is captured by the Drinfeld centre, thereby realising electromagnetic duality in a concrete lattice setting. Specialising to $G=\mathbb{Z}_2$ recovers known tensor-network dual representations of the 3D toric code and clarifies the connection between boundary Ising-Kramers–Wannier dualities and bulk gauge dualities. The framework also relates the 3+1d toric code to a 2-form $\mathbb{Z}_2$ gauge Walker–Wang model, illustrating the broader landscape of dualities in higher-dimensional topological phases. Overall, the work provides explicit tensor-network constructions tied to higher-categorical boundary data, enabling dual realizations of the same bulk topological order.
Abstract
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 $\mathsf{2Vec}_G$ of $G$-graded 2-vector spaces and $\mathsf{2Rep}(G)$ of 2-representations of $G$, respectively. In virtue of the Morita equivalence between $\mathsf{2Vec}_G$ and $\mathsf{2Rep}(G)$ -- 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 $G = \mathbb Z_2$, we recover tensor network representations that were recently introduced, as well as the relation between the electromagnetic duality of a pure (3+1)d $\mathbb Z_2$ gauge theory and the Kramers-Wannier duality of a boundary (2+1)d Ising model.