Anomalies for conformal nets associated with lattices and $T$-kernels

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 $T_{L}\to\mathsf{Out}(\mathcal{A}_{L})$, and compute Sutherland's obstruction class in $H^{3}_{\mathrm{Borel}}(T_{L},\mathbb{T})\cong H^{4}(BT_{L} ,\mathbb{Z})$, 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 $H^{3}_{\mathrm{Borel}}(T,\mathbb{T})$ 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 $T_{L^\ast}$--twisted sectors representations whose anomaly (obstruction) can be identified with the inner product on $L$ seen as a class in $H^{4}(BT_{L},\mathbb{Z})\cong \operatorname{Sym}^{2}(L,\mathbb{Z})$.