Open Descendants in Conformal Field Theory

TL;DR

Open Descendants extends Rational Conformal Field Theory to unoriented surfaces with boundaries by introducing two generalized fusion structures: a signed-integer tensor that preserves the fusion algebra for diagonal models and a positive-integer tensor that classifies boundary operators in non-diagonal models via polynomial sewing constraints. The framework integrates torus modular invariants with Klein bottle, annulus, and Möbius-strip amplitudes, deriving boundary reflection coefficients, simple-current extensions, and Y-algebras that connect bulk data to the open sector and Chan-Paton factors. Through explicit SU(2) WZW examples ($k=2$, $k=6$) and their $Z_T$ and crosscap data, the work demonstrates how the open spectrum is determined by the closed-sector data via sewing constraints and modular invariance, while tadpole conditions further constrain the Chan-Paton content. Collectively, the results provide a systematic method to classify and construct open-descendant theories on unoriented surfaces, with potential applications to string theory and boundary phenomena in condensed matter systems.

Abstract

Open descendants extend Conformal Field Theory to unoriented surfaces with boundaries. The construction rests on two types of generalizations of the fusion algebra. The first is needed even in the relatively simple case of diagonal models. It leads to a new tensor that satisfies the fusion algebra, but whose entries are signed integers. The second is needed when dealing with non-diagonal models, where Cardy's ansatz does not apply. It leads to a new tensor with positive integer entries, that satisfies a set of polynomial equations and encodes the classification of the allowed boundary operators.