arXiv subjects
Publications and source records attributed to Sebastiano Carpi.
Given a completely rational conformal net A on the circle, its fusion ring acts faithfully on the K_0-group of a certain universal C*-algebra associated to A, as shown in a previous paper. We prove here that this action can actually be identified with a Kasparov product, thus paving the way for a fruitful interplay between conformal field theory and KK-theory.
We consider the smallest values taken by the Jones index for an inclusion of local conformal nets of von Neumann algebras on S^1 and show that these values are quite more restricted than for an arbitrary inclusion of factors. Below 4, the only non-integer admissible value is 4\cos^2 π/10, which is known to be attained by a certain coset model. Then no index value is possible in the interval between 4 and 3 +\sqrt{3}. The proof of this result based on α-induction arguments. In the case of values below 4 we also give a second proof of the result. In the course of the latter proof we classify all possible unitary braiding symmetries on the A D E tensor categories, namely the ones associated with the even vertices of the A_n, D_{2n}, E_6, E_8 Dynkin diagrams.
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.
We study the general structure of Fermi conformal nets of von Neumann algebras on the circle, consider a class of topological representations, the general representations, that we characterize as Neveu-Schwarz or Ramond representations, in particular a Jones index can be associated with each of them. We then consider a supersymmetric general representation associated with a Fermi modular net and give a formula involving the Fredholm index of the supercharge operator and the Jones index. We then consider the net associated with the super-Virasoro algebra and discuss its structure. If the central charge c belongs to the discrete series, this net is modular by the work of F. Xu and we get an example where our setting is verified by considering the Ramond irreducible representation with lowest weight c/24. We classify all the irreducible Fermi extensions of any super-Virasoro net in the discrete series, thus providing a classification of all superconformal nets with central charge less than 3/2.
Let M be a von Neumann algebra on a Hilbert space H with a cyclic and separating unit vector Ωand let ωbe the faithful normal state on M given by ω(\cdot)=(Ω,\cdotΩ). 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 ω=ω\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_iΩ} and \bar{NΩ} respectively satisfy e=\bigwedge_{i\in I}e_i.This proves a conjecture of V.F.R. Jones and F. Xu in \cite{JonesXu04}.
A Moebius covariant net of von Neumann algebras on S^1 is diffeomorphism covariant if its Moebius symmetry extends to diffeomorphism symmetry. We prove that in case the net is either a Virasoro net or any at least 4-regular net such an extension is unique: the local algebras together with the Moebius symmetry (equivalently: the local algebras together with the vacuum vector) completely determine it. We draw the two following conclusions for such theories. (1) The value of the central charge c is an invariant and hence the Virasoro nets for different values of c are not isomorphic as Moebius covariant nets. (2) A vacuum preserving internal symmetry always commutes with the diffeomorphism symmetries. We further use our result to give a large class of new examples of nets (even strongly additive ones), which are not diffeomorphism covariant; i.e. which do not admit an extension of the symmetry to Diff^+(S^1).
We classify Haag-dual Poincaré covariant subsystems $\B\subset \F$ of a graded-local net $\F$ on 4D Minkowski spacetime which satisfies standard assumptions and has trivial superselection structure. The result applies to the canonical field net $\F_\A$ of a net $\A$ of local observables satisfying natural assumptions. As a consequence, provided that it has no nontrivial internal symmetries, such an observable net $\A$ is generated by (the abstract versions of) the local energy-momentum tensor density and the observable local gauge currents which appear in the algebraic formulation of the quantum Noether theorem. Moreover, for a net $\A$ of local observables as above, we also classify the Poincaré covariant local extensions $\B \supset \A$ which preserve the dynamics.
We discuss various aspects of the representation theory of the local nets of von Neumann algebras on the circle associated with positive energy representations of the Virasoro algebra (Virasoro nets). In particular we classify the local extensions of the $c=1$ Virasoro net for which the restriction of the vacuum representation to the Virasoro subnet is a direct sum of irreducible subrepresentations with finite statistical dimension (local extensions of compact type). Moreover we prove that if the central charge $c$ is in a certain subset of $(1,\infty)$, including $[2,\infty)$, and $h \geq (c-1)/24$, the irreducible representation with lowest weight $h$ of the corresponding Virasoro net has infinite statistical dimension. As a consequence we show that if the central charge $c$ is in the above set and satisfies $c\leq 25$ then the corresponding Virasoro net has no proper local extensions of compact type.
We show that the sectors with lowest weight $h\geq 0$, $h\neq j^2$, $j\in {1/2}\ZZ$ of the local net of von Neumann algebras on the circle generated by the Virasoro algebra with central charge c=1 have infinite statistical dimension.
Let F be a local net of von Neumann algebras in four spacetime dimensions satisfying certain natural structural assumptions. We prove that if F has trivial superselection structure then every covariant, Haag-dual subsystem B is the fixed point net under a compact group action on one component in a suitable tensor product decomposition of F. Then we discuss some application of our result, including free field models and certain theories with at most countably many sectors.