Generalized Level-Rank Duality, Holomorphic Conformal Field Theory, and Non-Invertible Anyon Condensation

TL;DR

The paper develops a unified framework linking holomorphic CFTs to 3D TQFT dualities via topological interfaces and anyon condensation, highlighting how $c=24$ Schellekens theories seed both sporadic and infinite families of dualities. By identifying Lagrangian algebras and performing abelian or non-abelian condensations, it derives dualities of the form $\mathrm{G}_{k}/\mathcal{A}_{\mathrm{G}} \cong \mathrm{H}_{-\tilde{k}}/\mathcal{A}_{\mathrm{H}}$, up to gravitational CS terms, and uncovers new cases where non-invertible one-form symmetries are gauged. The work presents explicit examples including Spin$(n^{2})_{2}$ dualities with twisted Dihedral gauge theories, and demonstrates that the center of Tambara-Yamagami categories $\mathrm{TY}[\mathbb{Z}_{2n+1}]$ corresponds to $\mathrm{Spin}(2n+1)_{2} \times \mathrm{SU}(2n+1)_{-1}$, with broader implications for holomorphic CFTs at central charges $c=2(k-1)$. Overall, it advances a program to realize generalized level-rank dualities and symmetry TFTs directly from holomorphic edge theories.

Abstract

We study the interplay between holomorphic conformal field theory and dualities of 3D topological quantum field theories generalizing the paradigm of level-rank duality. A holomorphic conformal field theory with a Kac-Moody subalgebra implies a topological interface between Chern-Simons gauge theories. Upon condensing a suitable set of anyons, such an interface yields a duality between topological field theories. We illustrate this idea using the $c=24$ holomorphic theories classified by Schellekens, which leads to a list of novel sporadic dualities. Some of these dualities necessarily involve condensation of anyons with non-abelian statistics, i.e. gauging non-invertible one-form global symmetries. Several of the examples we discover generalize from $c=24$ to an infinite series. This includes the fact that Spin$(n^{2})_{2}$ is dual to a twisted dihedral group gauge theory. Finally, if $-1$ is a quadratic residue modulo $k$, we deduce the existence of a sequence of holomorphic CFTs at central charge $c=2(k-1)$ with fusion category symmetry given by $\mathrm{Spin}(k)_{2}$ or equivalently, the $\mathbb{Z}_{2}$-equivariantization of a Tambara-Yamagami fusion category.

Figures (8)

• Figure 1: The local operators of a holomorphic 2D CFT decomposed into chiral subalgebras $\mathrm{G}_{k} \times \mathrm{H}_{\tilde{k}}$ can be constructed by stretching an anyon in the $\mathrm{G}_{k} \times \mathrm{H}_{\tilde{k}}$ Chern-Simons theory from the standard chiral WZW boundary condition (orange, left) to a topological boundary condition separating $\mathrm{G}_{k} \times \mathrm{H}_{\tilde{k}}$ from an appropriate gravitational Chern-Simons term (blue, right). The index $\mu = 0,1,\ldots,N^{ij}$ specifies the pairing of the characters. The yellow background emphasizes regions with trivial anyon data.
• Figure 2: Equivalence of three different concepts for Niemeier Holomorphic CFTs. The partition function can be expressed in terms of an extension of the chiral algebra, which also corresponds to gauging an abelian symmetry of the initial CFT. In 3D TQFT terms this corresponds to a Lagrangian subgroup, or gauging of an abelian one-form symmetry of the corresponding TQFT. Meanwhile, the same partition function can also be obtained from the glue code of the lattice.
• Figure 3: A topological interface (in blue) between Chern-Simons theories $\mathrm{SU}(N)_{k} \times \mathrm{U}(k)_{N}$ and $\mathrm{U}(N k)_{1}$ follows from the existence of the conformal embedding \ref{['confembed']}. The yellow background is used to emphasize that $\mathrm{U}(Nk)_{1} \cong -2 N k \mathop{\mathrm{\mathrm{CS}_{\mathrm{grav}}}}\nolimits$ has trivial anyon data.
• Figure 4: A topological interface (in blue) between $\mathrm{SU}(N)_{k}$ and $\mathrm{U}(k)_{-N}$ (up to a gravitational Chern-Simons term) implies the existence of the Level-Rank duality $\mathrm{SU}(N)_{k} \cong \mathrm{U}(k)_{-N}$.
• Figure 5: The anyons in the $\mathrm{G}_{k} \times \mathrm{H}_{\tilde{k}}$ decomposition of the holomorphic chiral algebra define a Lagrangian algebra in the $\mathrm{G}_{k} \times \mathrm{H}_{\tilde{k}}$ Chern-Simons theory. This specifies the lines that can terminate on the topological boundary. The yellow background emphasizes the region with trivial anyon data.

Theorems & Definitions (1)

• Proposition 1