Classification of local conformal nets. Case \(c < 1\). (English) Zbl 1083.46038

Kawahigashi, Yasuyuki; Longo, Roberto

Ann. Math. (2) 160, No. 2, 493-522 (2004).

Mathematical and physical motivations (e.g., 2-dimensional conformal field theory) have inspired the study of local nets \({\mathcal A}\) (obeying a number of axioms) of von Neumann algebras \({\mathcal A}(I)\), indexed by intervals \(I\) of the circle \(S^1\). By the diffeomorphism covariance, implemented by the projective unitary representation \(U\) of the Lie group \(\mathop{\text{Diff}}(S^1)\) on the Hilbert space of \({\mathcal A}\), an irreducible net \({\mathcal A}\) canonically contains the subnet of factors \({\mathcal A}_{\text{ Vir}}(I)=U(\mathop{\text{Diff}}(I))''\subset{\mathcal A}(I)\), related to representations of the Virasoro algebra with fixed scalar central charge \(c\). The possible values are \(c=1-{6\over m(m+1)}\), \(m=2\), 3, …, or \(c\geq 1\).
Viewing irreducible \({\mathcal A}\) with \(c<1\) as a finite-index extension of the Virasoro subnet, the paper derives a first general classification, labelling such nets bijectively with pairs of \(A-D_{2n}-E_{6,8}\) Dynkin diagrams whose Coxeter numbers difference equals 1. The authors use \(\alpha \)-induction, cf.[Commun. Math. Phys. 208, 429–487 (1999; Zbl 0948.46048); ibid. 210, 733–784 (2000; Zbl 0988.46047)], producing modular invariants of representations of \(SL(2,{\mathbb Z})\) from extensions of \({\mathcal A}_{\text{ Vir}}\). They analyse the full list of these invariants given (for “minimal models”) for \(c<1\) by A. Cappelli, C. Itzykson and J. B. Zuber [Commun. Math. Phys. 113, 1–26 (1987; Zbl 0639.17008)] in terms of diagrams. Complete rationality of \({\mathcal A}\) in the sense of Kawahigashi–Longo–Müger follows from the isomorphy of \({\mathcal A}_{\text{ Vir}}\), \(c<1\), with the coset net studied by F. Xu [Commun. Math. Phys. 211, 1–43 (2000; Zbl 1040.81085)]. Various known cosets models fall naturally into the classification.

Reviewer: Andrej V. Bulinski (Moskva)

MSC:

46L60	Applications of selfadjoint operator algebras to physics
46L37	Subfactors and their classification
81T40	Two-dimensional field theories, conformal field theories, etc. in quantum mechanics

Keywords:

von Neumann factors; local net; subnet of subfactors; conformal invariance; Virasoro algebra; central charge; complete rationality; braidings; extension; DHR-sectors; \(\alpha \)-induction; \(SL(2; \mathbb Z)\) modular invariants; CIZ-classification; Dynkin diagrams; \(Q\)-systems; coset models

Citations:

Zbl 0948.46048; Zbl 0988.46047; Zbl 0639.17008; Zbl 1040.81085