Non-Invertible Anyon Condensation and Level-Rank Dualities

Abstract

We derive new dualities of topological quantum field theories in three spacetime dimensions that generalize the familiar level-rank dualities of Chern-Simons gauge theories. The key ingredient in these dualities is non-abelian anyon condensation, which is a gauging operation for topological lines with non-group-like i.e. non-invertible fusion rules. We find that, generically, dualities involve such non-invertible anyon condensation and that this unifies a variety of exceptional phenomena in topological field theories and their associated boundary rational conformal field theories, including conformal embeddings, and Maverick cosets (those where standard algorithms for constructing a coset model fail.) We illustrate our discussion in a variety of isolated examples as well as new infinite series of dualities involving non-abelian anyon condensation including: i) a new description of the parafermion theory as $(SU(N)_{2} \times Spin(N)_{-4})/\mathcal{A}_{N},$ ii) a new presentation of a series of points on the orbifold branch of $c=1$ conformal field theories as $(Spin(2N)_{2} \times Spin(N)_{-2} \times Spin(N)_{-2})/\mathcal{B}_{N}$, and iii) a new dual form of $SU(2)_{N}$ as $(USp(2N)_{1} \times SO(N)_{-4})/\mathcal{C}_{N}$ arising from conformal embeddings, where $\mathcal{A}_{N}, \mathcal{B}_{N},$ and $\mathcal{C}_{N}$ are appropriate collections of gauged non-invertible bosons.