QM Research Seminar: Condensable Defects in 4D Dijkgraaf-Witten Models

Speaker: Hao Xu
(Georg-August University of Göttingen) 


Abstract: Given a finite symmetry group $G$ with an anomaly $\pi \in \mathrm{H}^4(G,U(1))$, one can construct a twisted 4D gauge theory called a Dijkgraaf-Witten model. These models are exactly solvable and play important roles in both higher energy physics and topological orders of quantum phases of matter. People have previously studied (codim > 1) topological defects (such as generalizations of Wilson lines) within them, which together form a **braided fusion 2-category**, $\mathscr{Z}(\mathbf{2Vect}^\pi_G)$, the Drinfeld center of $\pi$-twisted $G$-crossed finite semisimple linear categories.
Anyons (i.e. topological line defects) can condense in 3D quantum systems to produce new ones, with gapped 2D domain walls sitting between these 3D systems. It has been known for a decade that such a physical process can be described mathematically considering connected étale algebras (which in physical terms correspond to a condensable set of anyons), their ordinary and local modules within modular tensor categories. In particular, the after-condensation phase is trivial if and only if the condensable set of anyons is maximal, and hence the gapped 2D domain wall becomes a 2D topological boundary of the original phase. 
My doctoral project is to develop a mathematical framework for the same picture but in one higher dimension. This goal has been achieved with Décoppet in a series of past papers, where we define **étale algebras** and their **local modules** in braided fusion 2-categories and study their properties. In a recent independent work, I applied the general machinery to $\mathscr{Z}(\mathbf{2Vect}^\pi_G)$, where étale algebras mean $\pi$-twisted $G$-crossed braided multifusion 1-categories. Built upon essential tools developed by Drinfeld et al., I obtain a **classification of connected étale algebras** in $\mathscr{Z}(\mathbf{2Vect}^\pi_G)$. In particular, Lagrangian algebras in $\mathscr{Z}(\mathbf{2Vect}^\pi_G)$ correspond to 3D topological boundaries of 4D Dijkgraaf-Witten Models. 
Décoppet has shown that the Drinfeld center of an arbitrary fusion 2-category is either equivalent to a 4D Dijkgraaf-Witten model $\mathscr{Z}(\mathbf{2Vect}^\pi_G)$ or a more complicated fermionic version. If time allows, I would introduce my classification result for all **fusion 2-categories** based on studying Lagrangian algebras in 4D Dijkgraaf-Witten models.