The correlators of two-dimensional rational conformal field theories that are obtained in the TFT construction of [FRS I, FRS II, FRS IV] are shown to be invariant under the action of the relative modular group and to obey bulk and boundary factorisation constraints.We present results both for conformal field theories defined on oriented surfaces and for theories defined on unoriented surfaces.In the latter case, in particular the so-called cross cap constraint is included.Contents 1 Introduction 342 2 Consistency conditions for CFT correlators 347 3 Proof of modular invariance 361 4 Proof of boundary factorisation 370 5 Proof of bulk factorisation 380 A Algebras in modular tensor categories 403 B The assignment X C(X) 409 C Two-point correlators 416 D Some properties of (bimodule) morphisms 423
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Fjelstad et al. (2006) studied this question.