The logarithmic Cardy case: Boundary states and annuli

TL;DR

This paper develops a universal, model-independent framework for boundary states in a broad class of conformal field theories, including logarithmic theories, extending the well-known rational case.

Contribution

It introduces a universal description of boundary states in finite CFTs using categorical methods, generalizing the Cardy case beyond rational theories.

Findings

Boundary states are described by objects in the category of chiral data.

Annulus amplitudes are computed and interpreted as partition functions.

The framework applies to all finite CFTs, including logarithmic theories.

Abstract

We present a model-independent study of boundary states in the Cardy case that covers all conformal field theories for which the representation category of the chiral algebra is a - not necessarily semisimple - modular tensor category. This class, which we call finite CFTs, includes all rational theories, but goes much beyond these, and in particular comprises many logarithmic conformal field theories. We show that the following two postulates for a Cardy case are compatible beyond rational CFT and lead to a universal description of boundary states that realizes a standard mathematical setup: First, for bulk fields, the pairing of left and right movers is given by (a coend involving) charge conjugation; and second, the boundary conditions are given by the objects of the category of chiral data. For rational theories our proposal reproduces the familiar result for the boundary states of the Cardy case. Further, with the help of sewing we compute annulus amplitudes. Our results show in particular that these possess an interpretation as partition functions, a constraint that for generic finite CFTs is much more restrictive than for rational ones.