, Volume 315, Issue 3, pp 771-802,
Open Access This content is freely available online to anyone, anywhere at any time.

Thermal States in Conformal QFT. II

Abstract

We continue the analysis of the set of locally normal KMS states w.r.t. the translation group for a local conformal net ${\mathcal{A}}$ of von Neumann algebras on ${\mathbb{R}}$ . In the first part we have proved the uniqueness of the KMS state on every completely rational net. In this second part, we exhibit several (non-rational) conformal nets which admit continuously many primary KMS states. We give a complete classification of the KMS states on the U(1)-current net and on the Virasoro net Vir1 with the central charge c = 1, whilst for the Virasoro net Vir c with c > 1 we exhibit a (possibly incomplete) list of continuously many primary KMS states. To this end, we provide a variation of the Araki-Haag-Kastler-Takesaki theorem within the locally normal system framework: if there is an inclusion of split nets ${\mathcal{A}\subset \mathcal{B}}$ and ${\mathcal{A}}$ is the fixed point of ${\mathcal{B}}$ w.r.t. a compact gauge group, then any locally normal, primary KMS state on ${\mathcal{A}}$ extends to a locally normal, primary state on ${\mathcal{B}}$ , KMS w.r.t. a perturbed translation. Concerning the non-local case, we show that the free Fermi model admits a unique KMS state.

Communicated by Y. Kawahigashi
Research supported in part by the ERC Advanced Grant 227458 OACFT “Operator Algebras and Conformal Field Theory”, PRIN-MIUR, GNAMPA-INDAM and EU network “Noncommutative Geometry” MRTN-CT-2006-0031962.
Dedicated to Rudolf Haag on the occasion of his 90th birthday