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A) We first describe the action of a flat line bundle N over M with equivariant structure ν on a Jandl structure J = (k, A, ϕ). According to diagram (19), π Z∗ ν : π Z∗ N → k˜ ∗ π Z∗ N is a K -equivariant structure on π Z∗ N . A. A, ϕ ⊗ π Z∗ ν). Since Z [2] π2 π1∗ π Z∗ ν Z π1 /Z πZ πZ π2∗ π Z∗ ν. J . The arguments in the proof of Lemma 2 (a) apply here too and show that this defines an action on equivalence classes. (b) Let two equivalence classes of Jandl structures be represented by J1 and J2 .

This determines a unique line bundle N → M with connection together with an isomorphism ν : π Z∗ N → A1 ⊗ A∗2 . Because (12) requires the curvatures of both A1 and A2 to be the same, N is flat. A2 . We denote the group of isomorphism classes of flat line bundles over M by Pic0 (M). It is a subgroup of the Picard group Pic(M) of isomorphism classes of hermitian line bundles with connection over M. Lemma 2. The set Hom(G, G ) of equivalence classes of stable isomorphisms is a torsor over the flat Picard group Pic0 (M).

Already the abelian case [BPS92] shows that not every rational conformal field theory that is well-defined on oriented surfaces can be considered on unoriented surfaces. A necessary condition is that the bulk partition function is symmetric under exchange of left and right movers. This restricts, for example, the values of the Kalb-Ramond field in toroidal compactifications [BPS92]. Moreover, if the theory can be extended to unoriented surfaces, there can be different extensions that yield inequivalent correlation functions.