Cardy Algebras, Sewing Constraints and String-Nets

TL;DR

This paper extends the string-net construction for Cardy algebras to open-closed world sheets, establishing a correspondence between string-net solutions to sewing constraints and $( ext{C}| ext{Z}( ext{C}))$-Cardy algebras, thus providing an alternative proof of prior results.

Contribution

It generalizes the string-net approach to include open-closed surfaces and more general field algebras, linking solutions of sewing constraints to Cardy algebras.

Findings

String-net spaces for $( ext{C}| ext{Z}( ext{C}))$-Cardy algebras produce solutions to sewing constraints.

Solutions to sewing constraints determine $( ext{C}| ext{Z}( ext{C}))$-Cardy algebras up to isomorphism.

Provides an alternative proof of previous results using string-net methods.

Abstract

In \cite{Schweigert:2019zwt} it was shown how string-net spaces for the Cardy bulk algebra in the Drinfeld center $\mathsf{Z}(\mathsf{C})$ of a modular tensor category $\mathsf{C}$ give rise to a consistent set of correlators. We extend their results to include open-closed world sheets and allow for more general field algebras, which come in the form of $(\mathsf{C}|\mathsf{Z}(\mathsf{C}))$-Cardy algebras. To be more precise, we show that a set of fundamental string-nets with input data from a $(\mathsf{C}|\mathsf{Z}(\mathsf{C}))$-Cardy algebra gives rise to a solution of the sewing constraints formulated in \cite{Kong_2014} and that any set of fundamental string-nets solving the sewing constraints determine a $(\mathsf{C}|\mathsf{Z}(\mathsf{C}))$-Cardy algebra up to isomorphism. Hence we give an alternative proof of the results in \cite{Kong_2014} in terms of string-nets.