Cardy algebras and sewing constraints, I
Liang Kong, Ingo Runkel
TL;DR
This work develops Cardy algebras as a categorical framework for open-closed rational CFT by connecting a 3D TFT/ modular tensor category approach with a vertex operator algebra perspective. It introduces the ambidextrous pair of functors $T$ and $R$ to transport Frobenius and algebra structures between $\mathcal{C}$ and $\mathcal{C}^2_{\,\pm}$, and defines modular invariance for the closed sector. The main contributions are reconstruction and uniqueness results: any simple modular invariant commutative symmetric Frobenius closed algebra $A_{cl}$ in $\mathcal{C}^2_{\,\pm}$ admits a simple open part $A_{op}$ making a Cardy algebra, and the open-closed data are classified up to Morita equivalence by the full centre $Z(A)$. These results provide a rigorous categorical bridge between genus-0/1 consistency and sewing constraints, enabling reconstruction of open-closed CFTs from boundary data and linking two complementary formulations of the theory.
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.