Abstract
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 {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 Kong et al. (Adv Math 262:604–681, 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 Kong et al. (2014) in terms of string-nets.
Highlights
String-net spaces were originally introduced by Levin and Wen in [LW05] in order to describe phenomena of topological phases of matter on surfaces
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
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
Summary
String-net spaces were originally introduced by Levin and Wen in [LW05] in order to describe phenomena of topological phases of matter on surfaces. In terms of string-nets and the first main result, whose precise formulation in the main text is Theorem 6.4, is Theorem I The set of correlators corr gives a solution to the sewing constraints for the conformal block functor B with boundary coloring determined by the (C|Z(C))-Cardy algebra. Theorem II An assignment of fundamental string-net correlators based on boundary colorings (Hcl , Hop) in C, and Z(C), solving the sewing contraints, determines a (C|Z(C))-Cardy algebra (Hcl , Hop, ιcl−op), which is unique up to isomorphism. The graphical representation of consistency relations for Cardy algebras appear directly as string-nets on surfaces and can be manipulated .
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