Von Neumann subfactors and non-invertible symmetries

We use the language of von Neumann subfactors to investigate non-invertible symmetries in two dimensions. A fusion categorical symmetry \mathcal{C} 𝒞 , its module category \mathcal{M} ℳ , and a gauging labeled by an algebra object \mathcal{A} 𝒜 are encoded in the bipartite principal graph of a subfactor. The dual principal graph captures the quantum symmetry \mathcal{C}' 𝒞 ′ obtained by gauging \mathcal{A} 𝒜 in \mathcal{C} 𝒞 , as well as a reverse gauging back to \mathcal{C} 𝒞 . From a given subfactor N \subset M N ⊂ 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) ( D 4 ) as a warm-up, the higher-multiplicity case Rep (A_4) ( A 4 ) with its associated generalized orbifold groupoid and triality symmetry, and Rep (A_5) ( A 5 ) , where A_5 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 c = 1 CFTs and uncover new self-dualities in the exceptional SU(2)_1/A_5 S U ( 2 ) 1 / A</mml:mi