Cardy Algebras, Sewing Constraints and String-Nets

In Schweigert and Yang (Methods Appl Symmetry Integr Geom, 2021) it was shown how string-net spaces for the Cardy bulk algebra in the Drinfeld center Z ( C ) of a modular tensor category 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 ( C | Z ( C ) ) -Cardy algebras. To be more precise, we show that a set of fundamental string-nets with input data from a ( C | Z ( C ) ) -Cardy algebra gives rise to a solution of the sewing constraints formulated in Kong et al. (Adv Math 262:604–681, 2014) and that any set of fundamental string-nets solving the sewing constraints determine a ( C | Z ( C ) ) -Cardy algebra up to isomorphism. Hence we give an alternative proof of the results in Kong et al. (2014) in terms of string-nets.