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 A of von Neumann algebras on the real line. In the first part we have proved the uniqueness of 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 Vir_1 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 A in B and A is the fixed point of B w.r.t. a compact gauge group, then any locally normal, primary KMS state on A extends to a locally normal, primary state on 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.Comment: 36 pages, no figure. Dedicated to Rudolf Haag on the occasion of his 90th birthday. The final version is available under Open Access. This paper contains corrections to the Araki-Haag-Kaster-Takesaki theorem (and to a proof of the same theorem in the book by Bratteli-Robinson). v3: a reference correcte

Wearable wireless tactile display for virtual interactions with soft bodies.

We describe here a wearable, wireless, compact, and lightweight tactile display, able to mechanically stimulate the fingertip of users, so as to simulate contact with soft bodies in virtual environments. The device was based on dielectric elastomer actuators, as high-performance electromechanically active polymers. The actuator was arranged at the user's fingertip, integrated within a plastic case, which also hosted a compact high-voltage circuitry. A custom-made wireless control unit was arranged on the forearm and connected to the display via low-voltage leads. We present the structure of the device and a characterization of it, in terms of electromechanical response and stress relaxation. Furthermore, we present results of a psychophysical test aimed at assessing the ability of the system to generate different levels of force that can be perceived by users.The authors gratefully acknowledge financial support from COST – European Cooperation in Science and Technology, within the framework of “ESNAM – European Scientific Network for Artificial Muscles” (COST Action MP1003). Gabriele Frediani also acknowledges support from the European Commission, within the framework of the project “CEEDS: The Collective Experience of Empathic Data Systems” (FP7-ICT-2009.8.4, Grant 258749) and “Fondazione Cassa di Risparmio di Pisa,” within the framework of the project “POLOPTEL” (Grant 167/09

Representations of Conformal Nets, Universal C*-Algebras and K-Theory

We study the representation theory of a conformal net A on the circle from a K-theoretical point of view using its universal C*-algebra C*(A). We prove that if A satisfies the split property then, for every representation \pi of A with finite statistical dimension, \pi(C*(A)) is weakly closed and hence a finite direct sum of type I_\infty factors. We define the more manageable locally normal universal C*-algebra C*_ln(A) as the quotient of C*(A) by its largest ideal vanishing in all locally normal representations and we investigate its structure. In particular, if A is completely rational with n sectors, then C*_ln(A) is a direct sum of n type I_\infty factors. Its ideal K_A of compact operators has nontrivial K-theory, and we prove that the DHR endomorphisms of C*(A) with finite statistical dimension act on K_A, giving rise to an action of the fusion semiring of DHR sectors on K_0(K_A)$. Moreover, we show that this action corresponds to the regular representation of the associated fusion algebra.Comment: v2: we added some comments in the introduction and new references. v3: new authors' addresses, minor corrections. To appear in Commun. Math. Phys. v4: minor corrections, updated reference