Gauging Categorical Symmetries in 3d Topological Orders and Bulk Reconstruction

TL;DR

The work develops a rigorous framework for gauging categorical symmetries in 3d topological orders via anyon condensation, treating condensing lines as Frobenius algebra objects within a modular tensor category to obtain a child theory and a corresponding ungauging procedure that reconstructs the parent from wall data and the Drinfeld center. It extends the condensation program from abelian to nonabelian anyons, introducing algorithms for module decomposition, sequential condensing, and spin-structure couplings, and links these procedures to modular invariants, including extension, permutation, and super modular variants tied to congruence subgroups \Gamma(n). Through explicit examples—Ising⊗Ising to Toric Code, SU(3)_3 from Spin(8)_1, and tensorized theories—the paper demonstrates how consistency relations and S-matrix data can be used to deduce or constrain the S-matrix of the parent theory and to reveal bulk-boundary structure and symmetry actions. The results advance understanding of bulk reconstruction, the classification of modular invariants beyond Lagrangian algebras, and the interplay between higher-form categorical symmetries, spin structures, and modular data, with concrete computational methods for 3d topological orders and their interfaces.

Abstract

We use the language of categorical condensation to give a procedure for gauging nonabelian anyons, which are the manifestations of categorical symmetries in three spacetime dimensions. We also describe how the condensation procedure can be used in other contexts such as for topological cosets and constructing modular invariants. By studying a generalization of which anyons are condensable, we arrive at representations of congruence subgroups of the modular group. We finally present an analysis for ungauging anyons, which is related to the problem of constructing a Drinfeld center for a fusion category; this procedure we refer to as bulk reconstruction. We introduce a set of consistency relations regarding lines in the parent theory and wall category. Through use of these relations along with the $S$-matrix elements of the child theory, we construct $S$-matrix elements of a parent theory in a number of examples.

Figures (8)

• Figure 1: The diagram on the left is the axiom that multiplication and comultiplication can be composed into $\varphi$, i.e. all bubbles can be closed. The diagram on the right shows that the composition of comultiplcation and multiplication can be decomposed as a composition of ($\text{id}_\varphi\times$multiplcation) and (comultiplcation$\times\text{id}_{\varphi}$) or (multiplication$\times \text{id}_\varphi$) and ($\text{id}_\varphi\times$comultiplication)
• Figure 2: The left diagram shows that multiplication is associatve. The right diagram shows that comultiplication is coassociative
• Figure 3: The physical picture of condensation looks like inserting a fine mesh of the algebra that takes the form of a surface when zoomed out. The dark line at the boundary represents a module for the algebra.
• Figure 4: Each of the black tori represents the spatial dimensions of the 3d theory, with time running horizontally. The blue torus indicates a defect that can be placed in this quantum mechanics model at an instant in time. The black tori are both acted on by the modular group, so the defect $M$ intertwines the two actions. The 2d theory on the black tori can in particular be the chiral or anti-chiral half of a WZW model.
• Figure 5: We give a top down view of the interface, which is represented by the solid line, that separates theories $\mathcal{C}$ and $\mathcal{D}$. Suppose that $\alpha$ is a line that exists in the parent theory, but lifts off to the child theory. Then it can pass by the totally confined object in two equivalent ways.

Theorems & Definitions (1)

• Theorem 1.1