Intersecting Jones projections

Let M be a von Neumann algebra on a Hilbert space H with a cyclic and separating unit vector \Omega and let \omega be the faithful normal state on M given by \omega(\cdot)=(\Omega,\cdot\Omega). Moreover, let {N_i :i\in I} be a family of von Neumann subalgebras of M with faithful normal conditional expectations E_i of M onto N_i satisfying \omega=\omega\circ E_i for all i\in I and let N=\bigcap_{i\in I} N_i. We show that the projections e_i, e of H onto the closed subspaces \bar{N_i\Omega} and \bar{N\Omega} respectively satisfy e=\bigwedge_{i\in I}e_i.This proves a conjecture of V.F.R. Jones and F. Xu in \cite{JonesXu04}

Energy bounds for vertex operator algebra extensions

Let V be a simple unitary vertex operator algebra and U be a (polynomially) energy-bounded unitary subalgebra containing the conformal vector of V. We give two sufficient conditions implying that V is energy-bounded. The first condition is that U is a compact orbifold for some compact group G of unitary automorphisms of V. The second condition is that V is exponentially energy-bounded and it is a finite direct sum of simple U-modules. As consequence of the second condition, we prove that if U is a regular energy-bounded unitary subalgebra of a simple unitary vertex operator V, then V is energy-bounded. In particular, every simple unitary extension (with the same conformal vector) of a simple unitary affine vertex operator algebra associated with a semisimple Lie algebra is energy-bounded

La resistenza culturale nel Libano contemporaneo. Le sfide di artiste locali e profughe = Cultural resistance in contemporary Lebanon: The challenges of local and refugee artists

Sulla base di interviste condotte nel 2018, questo articolo analizza le somiglianze e le differenze che intercorrono tra le sfide che i “fautori della cultura” – artiste in primis – cittadine libanesi e rifugiate palestinesi e siriane devono affrontare nel contesto libanese. Dopo un’illustrazione dello scenario storico-politico libanese e di come in esso la “resistenza culturale” emerge in modo poliedrico, gli autori individuano aree d’incontro e di potenziale solidarietà tra gruppi. L’articolo discute la cosiddetta “umanitarizzazione” dei finanziamenti, attraverso la quale vengono sostenuti e potenziati soprattutto i progetti artistici che possono fungere da strumento di neutralità politica e di “medicalizzazione” dei traumi post-guerra. Tale fenomeno genera in parte una depoliticizzazione ed esteticizzazione dell’arte, “demobilitando” quindi la verve politica dietro al lavoro culturale e, allo stesso tempo, lega la sopravvivenza materiale di tali spazi culturali a cicliche crisi umanitarie

Classification of subsystems, local symmetry generators and intrinsic definition of local observables

We give a general overview of results about subsystems of local nets of von Neumann algebras in close connection with the problem of characterizing the abstract algebra of observables through the existence of Wightman currents

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

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

Super-KMS functionals for graded-local conformal nets

Motivated by a few preceding papers and a question of R. Longo, we introduce super-KMS functionals for graded translation-covariant nets over R with superderivations, roughly speaking as a certain supersymmetric modification of classical KMS states on translation-covariant nets over R, fundamental objects in chiral algebraic quantum field theory. Although we are able to make a few statements concerning their general structure, most properties will be studied in the setting of specific graded-local (super-) conformal models. In particular, we provide a constructive existence and partial uniqueness proof of super-KMS functionals for the supersymmetric free field, for certain subnets, and for the super-Virasoro net with central charge c>= 3/2. Moreover, as a separate result, we classify bounded super-KMS functionals for graded-local conformal nets over S^1 with respect to rotations.Comment: 30 pages, revised version (to appear in Ann. H. Poincare

Unitary representations of the W3-algebra with c ≥ 2

We prove unitarity of the vacuum representation of the W_3-algebra for all values of the central charge c ≥ 2.We do it by modifying the free field realization of Fateev and Zamolodchikov resulting in a representation which, by a nontrivial argument, can be shown to be unitary on a certain invariant subspace, although it is not unitary on the full space of the two currents needed for the construction. These vacuum representations give rise to simple unitary vertex operator algebras. We also construct explicitly unitary representations for many positive lowest weight values. Taking into account the known form of the Kac determinants, we then completely clarify the question of unitarity of the irreducible lowest weight representations of the W_3-algebra in the 2 ≤ c ≤ 98 region

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

Spectral triples and the super-Virasoro algebra

We construct infinite dimensional spectral triples associated with representations of the super-Virasoro algebra. In particular the irreducible, unitary positive energy representation of the Ramond algebra with central charge c and minimal lowest weight h=c/24 is graded and gives rise to a net of even theta-summable spectral triples with non-zero Fredholm index. The irreducible unitary positive energy representations of the Neveu-Schwarz algebra give rise to nets of even theta-summable generalised spectral triples where there is no Dirac operator but only a superderivation.Comment: 27 pages; v2: a comment concerning the difficulty in defining cyclic cocycles in the NS case have been adde