Completeness Conditions for Boundary Operators in 2D Conformal Field Theory

TL;DR

This paper investigates boundary operators in 2D conformal field theories, revealing new phenomena like multiplicities and fixed-point ambiguities, and derives polynomial equations for open spectrum determination.

Contribution

It extends previous work on sewing constraints by including additional open sectors and introduces a new tensor governing the open spectrum in non-diagonal models.

Findings

Identification of multiplicities and fixed-point ambiguities in boundary algebras.

Derivation of polynomial equations for the tensor determining open spectra.

Extension of boundary sewing constraints to include new open sectors.

Abstract

In non-diagonal conformal models, the boundary fields are not directly related to the bulk spectrum. We illustrate some of their features by completing previous work of Lewellen on sewing constraints for conformal theories in the presence of boundaries. As a result, we include additional open sectors in the descendants of $D_{odd}$ $SU(2)$ WZW models. A new phenomenon emerges, the appearance of multiplicities and fixed-point ambiguities in the boundary algebra not inherited from the closed sector. We conclude by deriving a set of polynomial equations, similar to those satisfied by the fusion-rule coefficients $N_{ij}^k$, for a new tensor $A_{a b}^i$ that determines the open spectrum.