, Volume 315, Issue 3, pp 771-802,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 15 Jul 2012

Thermal States in Conformal QFT. II


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