Non-Invertible Anyon Condensation and Level-Rank Dualities

TL;DR

The paper develops a unified framework for dualities among 3D topological quantum field theories by employing non-invertible anyon condensation, extending classic level-rank dualities to include Maverick cosets and conformal embeddings. It casts coset and bulk-boundary relations into a Frobenius-algebra/gauging language, demonstrating that non-invertible one-form symmetries must be gauged to realize dualities with a single boundary vacuum. Through explicit non-abelian condensation calculations, it unveils new dualities, including parafermion realizations, orbifold branches, and a novel SU(2)_{N} presentation as (USp(2N)_{1} × SO(N)_{-4}) / \\mathcal{A}_{N}, and it extends the coset inversion philosophy to Maverick cosets and beyond. The results provide a cohesive mechanism linking bulk MTCs, boundary RCFTs, and a broad spectrum of dualities across unitary, Spin, and exceptional groups, with potential implications for topological phases and boundary CFT classifications. Overall, the work offers a versatile toolbox for deriving and checking 3D dualities via non-invertible symmetry gauging and clarifies the role of non-abelian anyons in conformal embeddings and Maverick cosets.

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.

Figures (5)

• Figure 1: On the left: the theory $(G_{k} \times H_{-\tilde{k}}) / Z$, where the common center $Z$ of $G_{k}$ and $H_{-\tilde{k}}$ has been gauged, in the presence of the CFT coset boundary condition which is denoted $(G_{k}/H_{\tilde{k}})_{Z}$. On the right: a topological interface connecting the product $G_{k} \times H_{-\tilde{k}}$ without the common center gauged and $(G_{k} \times H_{-\tilde{k}})/Z$.
• Figure 2: Coset boundary condition with single vacuum $(G_{k}/H_{\tilde{k}})_{Z}$ in the presence of the topological interface joining $G_{k} \times H_{-\tilde{k}}$ and $(G_{k} \times H_{-\tilde{k}}) / Z$.
• Figure 3: Pushing the topological interface towards the boundary and fusing it with the coset boundary condition $(G_{k}/H_{\tilde{k}})_{Z}$ generates a gapless boundary for the $G_{k} \times H_{-\tilde{k}}$ Chern-Simons theory (with no common center symmetry gauged). By the nature of the construction, the result is generically a gapless boundary intertwined with a topological sector (i.e., there are now many vacua/topological local operators at the boundary) that is denoted as $(G_{k}/H_{\tilde{k}})$ (with no $Z$ subindex). The gapless boundary condition so generated corresponds to (the chiral version of) the associated gauged WZW model.
• Figure 4: In the canonical CFT boundary condition (in orange), all lines of the bulk CS theory end perpendicularly at the boundary on a (non-topological) local operator of the WZW theory based on the same group and level. Relatedly, pushing a line to the boundary in parallel to it gives rise to a Verlinde line of the corresponding WZW theory.
• Figure 5: In a topological boundary condition not all lines of the bulk theory end perpendicularly to the boundary. Only those generating a Lagrangian algebra can end (see Appendix \ref{['FrobeniusAlgebras']} for a precise mathematical definition), resulting in a topological local operator at the junction. When $Z$ is abelian as it is assumed in this section, the Lagrangian algebra is moreover a Lagrangian subgroup. Under parallel fusion, those lines in the Lagrangian subgroup become invisible at the boundary.