von Neumann Subfactors and Non-invertible Symmetries

TL;DR

The work establishes a subfactor-based framework to analyze non-invertible 2D symmetries, encoding fusion categories, module categories, and gauging data in principal and dual principal graphs. It provides a practical criterion for gauging via non-negative integer solutions to fusion-matrix equations and uses quiver representations to parcel vacuum structures and particle-soliton degeneracies in both gapless CFTs and gapped TQFTs. Through detailed case studies of Rep$(D_4)$, Rep$(A_4)$ (including non-invertible triality CT), and Rep$(A_5)$, the paper constructs generalized orbifold groupoids, exposes self-dualities in $SU(2)_1/A_5$, and links subfactor data to global manipulations on conformal manifolds. The results yield a concrete, diagrammatic route to classify gapped phases and boundary conditions for $ olinebreak olinebreak ext{C}$-symmetric theories, with a clear path to applications in conformal manifolds and SymTFT frameworks.

Abstract

We use the language of von Neumann subfactors to investigate non-invertible symmetries in two dimensions. A fusion categorical symmetry $\mathcal{C}$, its module category $\mathcal{M}$, and a gauging labeled by an algebra object $\mathcal{A}$ are encoded in the bipartite principal graph of a subfactor. The dual principal graph captures the quantum symmetry $\mathcal{C}'$ obtained by gauging $\mathcal{A}$ in $\mathcal{C}$, as well as a reverse gauging back to $\mathcal{C}$. From a given subfactor $N \subset M$, we derive a quiver diagram that encodes the representations of the associated non-invertible symmetry. We show how this framework provides necessary conditions for admissible gaugings, enabling the construction of generalized orbifold groupoids. To illustrate this strategy, we present three examples: Rep$(D_4)$ as a warm-up, the higher-multiplicity case Rep$(A_4)$ with its associated generalized orbifold groupoid and triality symmetry, and Rep$(A_5)$, where $A_5$ is the smallest non-solvable finite group. For applications to gapless systems, we embed these generalized gaugings as global manipulations on the conformal manifolds of $c=1$ CFTs and uncover new self-dualities in the exceptional $SU(2)_1/A_5$ theory. For $\mathcal{C}$-symmetric TQFTs, we use the subfactor-derived quiver diagrams to characterize gapped phases, describe their vacuum structure, and classify the recently proposed particle-soliton degeneracies.

Figures (27)

• Figure 1: Properties of a gaugeable algebra object: unit, associativity, Frobenius property (one row for each property).
• Figure 2: Left: resolving an endpoint into a semi-infinite strip of $M$ region (grey) among $N$ region (white); Right: resolving a trivalent junction $x$ into a Y-shaped $M$ region (grey) separating three $N$ regions (white). Adapted from a figure in Bischoff_2015.
• Figure 3: Resolution of the unit, symmetric, and Frobenius properties from Figure \ref{['fig:Frobenius']}. All lines are replaced by a thin strip of $M$ region inside $N$ regions, and all the conditions are satisfied provided that the $M-N$ boundary (described by the map $\iota: N \rightarrow M$) is deformable. Adapted from a figure in Bischoff_2015.
• Figure 4: Principal diagram and dual Ppincipal diagram for the subfactor corresponding to gauging $1 + \eta$ in the Ising fusion category.
• Figure 5: The quiver diagram of actions of elements in Ising fusion category on all its module categories. These three dots are the module categories for the Ising category under the algebra object $A = 1 + \eta$, which also appeared in Figure \ref{['fig:ising_principal']} and (\ref{['eqn:Ising_modules']}).