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

TL;DR

This paper explores the relationship between holomorphic conformal field theories and dualities in 3D topological quantum field theories, revealing new dualities involving non-invertible anyon condensation and extending known classifications.

Contribution

It introduces a framework linking holomorphic CFTs with topological dualities, including novel sporadic dualities and generalizations involving non-invertible anyons and fusion categories.

Findings

Discovery of new dualities from $c=24$ holomorphic theories.

Identification of dualities involving non-abelian anyon condensation.

Extension of duality patterns to an infinite series of 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.