TFT construction of RCFT correlators II: unoriented world sheets Original Research Article

A full rational CFT, consistent on all orientable world sheets, can be constructed from the underlying chiral CFT, i.e., a vertex algebra, its representation category C, and the system of chiral blocks, once we select a symmetric special Frobenius algebra A in the category C [Nucl. Phys. B 646 (2002) 353]. Here we show that the construction of [Nucl. Phys. B 646 (2002) 353] can be extended to unoriented world sheets by specifying one additional datum: a reversion σ on A—an isomorphism from the opposed algebra of A to A that squares to the twist. A given full CFT on oriented surfaces can admit inequivalent reversions, which give rise to different amplitudes on unoriented surfaces, in particular to different Klein bottle amplitudes.