TFT construction of RCFT correlators II: unoriented world sheets

Abstract

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 [I]. Here we show that the construction of [I] 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. We study the classification of reversions, work out the construction of the annulus, Möbius strip and Klein bottle partition functions, and discuss properties of defect lines on non-orientable world sheets. As an illustration, the Ising model is treated in detail. manche meinen lechts und rinks kann man nicht velwechsern. werch ein illtum! Ernst Jandl [ J ]

Nonsemisimple Fusion Algebras and the Verlinde Formula

“…For instance, the state spaces of the three-dimensional TFT are the spaces of chiral blocks of the CFT, and the modular S matrix (or, to be precise, the symmetric matrix that diagonalizes the fusion rules) is, up to normalization, the invariant of the Hopf link in the three-dimensional TFT. Also, a full (nonchiral) CFT based on a given chiral CFT corresponds to a certain Frobenius algebra in the category C, and the correlation functions of the full CFT can be determined by combining methods from three-dimensional TFT and from noncommutative algebra in monoidal categories [27,28]. In the nonrational case, C is no longer modular, in particular not semisimple, but in any case it should still be an additive braided monoidal category.…”

Uniqueness of open/closed rational CFT with given algebra of open states

“…Composing the left hand side of (5.10) from the right with e γ × r ′ δ one finds 12) while the same manipulation of the right hand side results in 13) where the second equality uses the definition of composition in C. Thus the left and right sides of formula (5.10) are equal when composed with e γ × r ′ δ , for any choice of γ and δ. Since the latter morphisms form a basis of Hom C⊠C (U k × U k , V × V ′ ), we have indeed equality already in the form (5.10).…”

“…For more details on the invariants resulting from glueing tori see appendix A.3 of [13]; the factor Dim(C) −1 appears as a consequence of tft C (S 3 ) = Dim(C) −1/2 . Comparing the two results we get h jl = Dim(C) −1 (shs) jl .…”

Non-compact string backgrounds and non-rational CFT

“…At least for rational models (i.e. for compact targets) the program can be made much more precise [91,92] (see also [93,94,95,96] The chiral algebra in question is the so-called coset algebra SL 2 (R)/U(1). Its description is a bit indirect through the SL 2 (R) current algebra.…”