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

TL;DR

This paper constructs and analyzes anomalies (obstructions) for conformal nets associated with lattices, linking them to cohomological invariants and twisted sectors, with implications for understanding symmetries in conformal field theory.

Contribution

It introduces a novel construction of $T$-kernels for conformal nets linked to lattices and computes their obstruction classes, extending Jones' finite group approach to infinite-dimensional settings.

Findings

Computed Sutherland's obstruction class in cohomology.

Showed any class in $H^{3}_{ ext{Borel}}(T,T)$ arises as an obstruction.

Connected lattice inner products to cohomological invariants.

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 $T_{L}\to\mathsf{Out}(\mathcal{A}_{L})$, 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 $H^{4}(BT_{L},\mathbb{Z})\cong \operatorname{Sym}^{2}(L,\mathbb{Z})$.