Condensable Topological Defects in 4D Dijkgraaf-Witten models—Classification of Fusion 2-Categorical Symmetries

Given a finite group $G$ a 4-cocycle $\pi \in \mathrm{H}^4(G,U(1))$, the 4D Dijkgraaf–Witten model provides an exactly solvable gauge theory with applications in high-energy physics and topological phases of matter. The topological defects in this model form a braided fusion 2-category $\mathscr{Z}(\mathbf{2Vect}^\pi_G)$, the Drinfeld center of $\pi$-twisted $G$-crossed finite semisimple linear categories. Extending the theory of anyon condensation in 3D, my thesis develops a higher-dimensional framework using étale algebras and their local modules in braided fusion 2-categories, in collaboration with Décoppet. In particular, I classify connected étale algebras in $\mathscr{Z}(\mathbf{2Vect}^\pi_G)$, in terms of homotopy data generalizing the group cohomology and Brauer-Picard groups. Additionally, Décoppet has shown that the Drinfeld center of any fusion 2-category is either a 4D Dijkgraaf–Witten model or a fermionic analogue. Time permitting, I will also discuss classification results for fusion 2-categories via the my classification results for Lagrangian algebras.