Anomalies for conformal nets associated with lattices and T-kernels

Let L⊆ Rⁿ an even lattice and TL=Rⁿ/L the associated torus. Associated with L we construct TL--kernel on a hyperfinite factor type AL, i.e. a monomorphism TL(AL), and compute Sutherland's obstruction class in H³Borel(TL,T) H⁴(BTL ,Z), which is an invariant of the TL--kernel and an obstruction to the existence of a twisted crossed product by TL. As a Corollary, we obtain that for any n-torus T any class in H³Borel(T,T) arises as an obstruction for a T-kernel on the hyperfinite type III₁ factor R. The construction is an analogue of the construction of Vaughan Jones for finite groups on the hyperfinite type II₁ factor but is also motivated by and has applications to conformal nets. Namely, there is an associated local extension AL⊇ ARⁿ of conformal nets and the TL--kernel corresponds to a family of TL^--twisted sectors representations whose anomaly (obstruction) can be identified with the inner product on L seen as a class in H⁴(BTL,Z) Sym²(L,Z).