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This is the part II of a two-part work started in arXiv:0807.3356 [math.QA].In part I, Cardy algebras were studied, a notion which arises from theclassification of genus-0,1 open-closed rational conformal field theories. Inthis part, we prove that a Cardy algebra also satisfies the higher genusfactorisation and modular-invariance properties formulated inarXiv:hep-th/0612306 in terms of the notion of a solution to the sewingconstraints. We present the proof by showing that the latter notion, which isdefined as a monoidal natural transformation, can be expressed in terms ofgenerators and relations, which correspond exactly to the defining data andaxioms of a Cardy algebra.

Cardy algebras and sewing constraints, II