An eigenvalue problem for a fermi system and lie algebras

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New perturbation theory of low-dimensional quantum liquids II: operator description of Virasoro algebras in integrable systems

We show that the recently developed {\it pseudoparticle operator algebra} which generates the low-energy Hamiltonian eigenstates of multicomponent integrable systems also provides a natural operator representation for the the Virasoro algebras associated with the conformal-invariant character of the low-energy spectrum of the these models. Studying explicitly the Hubbard chain in a non-zero chemical potential and external magnetic field, we establish that the pseudoparticle perturbation theory provides a correct starting point for the construction of a suitable critical-point Hamiltonian. We derive explicit expressions in terms of pseudoparticle operators for the generators of the Virasoro algebras and the energy-momentum tensor, describe the conformal-invariant character of the critical point from the point of view of the response to curvature of the two-dimensional space-time, and discuss the relation to Kac-Moody algebras and dynamical separation.Comment: 35 pages, RevteX, preprint UA

Integrable anyon chains: from fusion rules to face models to effective field theories

Starting from the fusion rules for the algebra $SO(5)_2$ we construct one-dimensional lattice models of interacting anyons with commuting transfer matrices of `interactions round the face' (IRF) type. The conserved topological charges of the anyon chain are recovered from the transfer matrices in the limit of large spectral parameter. The properties of the models in the thermodynamic limit and the low energy excitations are studied using Bethe ansatz methods. Two of the anyon models are critical at zero temperature. From the analysis of the finite size spectrum we find that they are effectively described by rational conformal field theories invariant under extensions of the Virasoro algebra, namely $\mathcal{W}B_2$ and $\mathcal{W}D_5$, respectively. The latter contains primaries with half and quarter spin. The modular partition function and fusion rules are derived and found to be consistent with the results for the lattice model.Comment: 43 pages, published versio

Structure and Classification of Superconformal Nets

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.Comment: 49 pages. Section 8 has been removed. More details concerning the diffeomorphism covariance are give

From Operator Algebras to Superconformal Field Theory

We make a review on the recent progress in the operator algebraic approach to (super)conformal field theory. We discuss representation theory, classification results, full and boundary conformal field theories, relations to supervertex operator algebras and Moonshine, connections to subfactor theory and noncommutative geometry

Free fermions and the classical compact groups

There is a close connection between the ground state of non-interacting fermions in a box with classical (absorbing, reflecting, and periodic) boundary conditions and the eigenvalue statistics of the classical compact groups. The associated determinantal point processes can be extended in two natural directions: i) we consider the full family of admissible quantum boundary conditions (i.e., self-adjoint extensions) for the Laplacian on a bounded interval, and the corresponding projection correlation kernels; ii) we construct the grand canonical extensions at finite temperature of the projection kernels, interpolating from Poisson to random matrix eigenvalue statistics. The scaling limits in the bulk and at the edges are studied in a unified framework, and the question of universality is addressed. Whether the finite temperature determinantal processes correspond to the eigenvalue statistics of some matrix models is, a priori, not obvious. We complete the picture by constructing a finite temperature extension of the Haar measure on the classical compact groups. The eigenvalue statistics of the resulting grand canonical matrix models (of random size) corresponds exactly to the grand canonical measure of non-interacting free fermions with classical boundary conditions.Comment: 35 pages, 5 figures. Final versio