A toy model for categorical charges

TL;DR

We address the problem of categorifying boundary charges in a $(3+1)d$ higher gauge model with a 2-group symmetry $\mathbb G(2,3)$ that couples a $0$-form $\mathbb Z_2$ and a $1$-form $\mathbb Z_3$. Our approach constructs an exactly solvable lattice Hamiltonian whose symmetry-preserving Neumann boundary hosts electric excitations organized as 2-representations of $\mathbb G(2,3)$, with domain walls and condensation defects encoding the fusion data. We provide an explicit classification of simple 2-representations and 1-intertwiners, derive their horizontal composition rules to reproduce the boundary fusion, and show that the boundary 1-charges obey a fusion algebra $\mathcal E_{\eta} \boxtimes \mathcal E_{\mu} \simeq \mathcal E_{\eta\mu} \boxplus \mathcal E_{\eta\bar\mu}$. The results establish a concrete link between higher gauge theory, condensation defects, and the categorical charges governing symmetry-preserving boundaries, with generalization to arbitrary $2$-groups discussed as future work.

Abstract

We consider a higher gauge topological model in three spatial dimensions whose input datum is a 2-group encoding the mixing of a 0-form $\mathbb Z_2$- and 1-form $\mathbb Z_3$-symmetry. We study the excitation content of the theory on the symmetry-preserving boundary. We show that boundary operators are organised into the fusion 2-category of 2-representations of the 2-group. These can be interpreted as categorical charges for an effective boundary model that inherits a global 2-group symmetry from the bulk topological order. Interestingly, we find that certain simple 2-representations are physically interpreted as composites of intrinsic excitations and condensation defects.