Anomalies for Conformal Nets Associated with Lattices and T-Kernels

Abstract

Let \(L\subseteq \mathbb {R}^{n}\) an even lattice and \(T_{L}=\mathbb {R}^{n}/L\) the associated torus. Associated with L we construct \(T_{L}\)-kernel on a hyperfinite factor type \(\mathcal {A}_{L}\), i.e. a monomorphism \(T_{L}\rightarrow \textsf{Out}(\mathcal {A}_{L})\), and compute Sutherland’s Sutherland, C.E.: Cohomology and extensions of von Neumann algebras. I. Publ. Res. Inst. Math. Sci. 16(1), 105–133 (1980); Sutherland, C.E.: Cohomology and extensions of von Neumann algebras. II. Publ. Res. Inst. Math. Sci. 16(1), 135–174 (1980) obstruction class in \(H^{3}_{\textrm{Borel}}(T_{L},\mathbb {T})\cong H^{4}(BT_{L},\mathbb {Z})\), which is an invariant of the \(T_{L}\)-kernel and an obstruction to the existence of a twisted crossed product by \(T_{L}\). As a Corollary, we obtain that for any n-torus T any class in \(H^{3}_{\textrm{Borel}}(T,\mathbb {T})\) arises as an obstruction for a T-kernel on the hyperfinite type III\(_{1}\) factor R.

The construction is an analogue of the construction of Vaughan Jones for finite groups on the hyperfinite type II\(_{1}\) factor but is also motivated by and has applications to conformal nets. Namely, there is an associated local extension \(\mathcal {A}_{L}\supseteq \mathcal {A}_{\mathbb {R}^n}\) of conformal nets and the \(T_{L}\)-kernel corresponds to a family of \(T_{L^*}\)-twisted sectors representations whose anomaly (obstruction) can be identified with the inner product on L seen as a class in \(H^{4}(BT_{L},\mathbb {Z})\cong \operatorname {Sym}^{2}(L,\mathbb {Z})\).