Hypergroup Symmetry in Relative Quantum Field Theories and Chiral Algebras

Abstract

A QFT is said to be relative if it lives at the boundary of a topological QFT in one higher dimension. We develop a general framework for working with noninvertible symmetries of relative theories in two spacetime dimensions, extending several well-known results for absolute QFTs. We emphasize various new features which arise in the relative setting, including the role of topological surfaces of the bulk, and the appearance of hypergroups and certain generalizations of tube algebras known as dome algebras. Our formalism is particularly well-suited for studying rational chiral algebras, where it predicts that finite symmetries are in explicit one-to-one correspondence with conformal embeddings of finite index. We describe several implications of our framework for absolute theories. First, we explain how to "glue" together symmetries of the left- and right-moving chiral algebras of a 2D CFT to produce topological line defects of the full theory. Second, we derive a precise correspondence between boundary conditions of a 2D CFT and symmetries of its chiral algebra. This correspondence has several structural corollaries: in diagonal rational CFTs, we demonstrate that the topological line defects of the theory act transitively on its boundary conditions, and further that the identity Cardy state has the smallest $g$-function amongst all boundary conditions, including those which only preserve Virasoro symmetry. We conclude by illustrating our results in a variety of examples. For instance, we show that, if there exists a rational chiral algebra with central charge $c=8$ whose modular tensor category is the Drinfeld center of the Haagerup fusion category, then it must arise as the fixed points of a rank-2 hypergroup acting on the $SU(3)_1\otimes (E_{6})_1$ chiral algebra.