Chiral Tube Algebras I: Topological Defect Lines, Twisted Modules, and Finite Gauging

Abstract

Chiral algebras and topological defect lines (TDLs) represent two complementary notions of symmetry in 2d conformal field theories. In this paper, we introduce chiral tube algebras to unify and extend these two notions. Chiral tube algebras generalize chiral algebras in two ways. First, they extend the action of chiral algebras beyond the local Hilbert space to include defect Hilbert spaces twisted by TDLs. Second, they allow for non-local chiral currents attached by TDLs and thus can map between different defect Hilbert spaces, analogous to the tube algebras of TDLs. Since local chiral currents can become non-local after finite gauging, chiral tube algebras provide a natural framework for describing the image of chiral algebras under such gauging. We illustrate this framework through a variety of examples that generalize familiar chiral algebras, including Kac-Moody algebras, $\mathcal{W}$ algebras, superconformal algebras, and their orbifolds/bosonizations. We construct their irreducible modules, which are isomorphic to twisted modules of the corresponding chiral algebras, and use them to organize local and defect Hilbert spaces. In a subsequent paper, we will study chiral tube algebras generated by non-local chiral currents with fractional spins, which have no counterparts in chiral algebras.