Cardy algebras and sewing constraints, I

TL;DR

This paper introduces Cardy algebras as a framework for understanding the consistency conditions of two-dimensional open-closed rational conformal field theories at genus 0 and 1, establishing existence and uniqueness results.

Contribution

It defines Cardy algebras, proves their properties, and shows that rational closed CFTs can be extended to open-closed CFTs under certain conditions.

Findings

Proved existence and uniqueness of Cardy algebras.

Demonstrated extension of rational closed CFTs to open-closed CFTs.

Connected Cardy algebras to sewing constraints solutions.

Abstract

This is part one of a two-part work that relates two different approaches to two-dimensional open-closed rational conformal field theory. In part one we review the definition of a Cardy algebra, which captures the necessary consistency conditions of the theory at genus 0 and 1. We investigate the properties of these algebras and prove uniqueness and existence theorems. One implication is that under certain natural assumptions, every rational closed CFT is extendable to an open-closed CFT. The relation of Cardy algebras to the solutions of the sewing constraints is the topic of part two.