A presentation for the Virasoro and super-Virasoro algebras

It is shown that the entire Virasoro, Ramond and Neveu-Schwarz algebras can each be constructed from a finite number of well-chosen generators satisfying a small number of conditions. The most economical sets consist of just two starting generators in all cases, subject to eight conditions for the Virasoro case, five conditions for the Ramond case, and nine conditions for the Neveu-Schwarz case.Work supported by the U.S. Department of Energy, Division of High Energy Physics, Contract W-31-109-ENG-38     

Unitary representations of the Virasoro and super-Virasoro algebras

It is shown that a method previously given for constructing representations of the Virasoro algebra out of representations of affine Kac-Moody algebras yields the full discrete series of highest weight irreducible representations of the Virasoro algebra. The corresponding method for the super-Virasoro algebras (i.e. the Neveu-Schwarz and Ramond algebras) is described in detail and shown to yield the full discrete series of irreducible highest weight representations.     

The quantum group structure associated with non-linearly extended Virasoro algebras

Recently, an infinite family of chiral Virasoro vertex operators, whose exchange algebra is given by the universalR-matrix ofSL(2) _q , has been constructed. In the present paper, the case of non-linearly (W-) extended Virasoro symmetries, related to the algebrasA _N,N>1, is considered along the same line. Contrary to the previous case (A ₁) the standardR-matrix ofSL(N+1)q does not come out, and a different solution of Yang and Baxter's equations is derived. The associated quantum group structure is displayed.Unité Propre du Centre National de la Recherche Scientifique, associée à l'École Normale Supérieure et à l'Université de Paris-Sud     

Localization and the Weyl algebras

Physics of Atomic Nuclei - Let W n (ℝ) be the Weyl algebra of index n. It is well known that so(p, q) Lie algebras can be viewed as quadratic polynomial (Lie) algebras in W n (ℝ) for p...     

Virasoro algebras in the theory of bosonicp-branes

It is shown that in the case of closed bosonicp-branes there existp + 1 mutually commuting Virasoro algebras which are a direct generalization of the string case. The existence of these algebras allows us to conclude that a non-Abelian string spectrum and quantum anomalies are admissible. The generalization of these results for the supersymmetric case is also discussed.     

Quantization on the Virasoro group

The quantization of the Virasoro group is carried out by means of a previously established group approach to quantization. We explicitly work out the two-cocycles on the Virasoro group as a preliminary step. In our scheme the carrier space for all the Virasoro representations is made out of polarized functions on the group manifold. It is proved that this space does not contain null vector states, even forc1, although it is not irreducible. The full reduction is achieved in a striaghtforward way by just taking a well defined invariant subspace _{(c, h)} , the orbit of the enveloping algebra through the vacuum, which is irreducible for any value ofc andh. _{(c, h)} is a proper subspace of the space of polarized functions for those values ofc andh for which the Kac determinant is zero. We give the local version of these group representations as well as the associated classical phase space structures, i.e., symplectic form and Noether invariants.Research partially supported by the Conselleria de Cultura de la Generalitat Valenciana, the Plan de formación del Personal investigador, the Comision Interministerial de Ciencia y Tecnologia (CICYT) and the British Council     

Fusion Rules for the Continuum Sectors of the Virasoro Algebra with c=1

The Virasoro algebra with c = 1 has a continuum of superselection sectors characterized by the ground state energy h 0. Only the discrete subset of sectors with h = s ², s ₀, arises by restriction of representations of the SU(2) current algebra at level k=1. The remaining continuum of sectors is obtained with the help of (localized) homomorphisms into the current algebra. The fusion product of continuum sectors with discrete sectors is computed. A new method of determining the sector of a state is used.     

SL(2,C)-gauge theory andN=1 supergravity

It is shown that as Riemannian space may be taken to give rise to a Poincaré gauge theory of gravitation, the superspace where the coordinates are given by (X, ), being a spinorial variable gives rise to anSL(2, C)-gauge theory and corresponds toN= 1 supergravity. It leads to a conserved current and the conserved quantity here corresponds to isospin, where the latter is taken to be generated from conformal reflection. Thus, supergravity plays a predominant role in the microlocal space-time.     

Quantum affine algebras and deformations of the Virasoro and 237-1237-1237-1

Using the Wakimoto realization of quantum affine algebras we define new Poisson algebras, which areq-deformations of the classicalW. We also define their free field realizations, i.e. homomorphisms into some Heisenberg-Poisson algebras. The formulas for these homomorphisms coincide with formulas for spectra of transfer-matrices in the corresponding quantum integrable models derived by the Bethe-Ansatz method.Partially supported by NSF grants DMS-9205303 and DMS-9296120.     

Higher-order polarizations on the Virasoro group and anomalies

In a previous paper the authors showed that the space of (first order) polarized functions on the Virasoro group is not, in general, irreducible. The full reduction was explicitly achieved by taking the orbit of the enveloping algebra through the vacuum. This additional step provided the proper quantization in the strong-coupling domain 0<c1. In this paper we introduce the concept of higher order polarization as a generalization of that of polarization. We prove that the imposing of the additional (higher-order) polarization conditions is equivalent to the taking of the above-mentioned orbit. This demonstrates that the generalized (higher-order) polarization conditions suffice to obtain the irreducible Hilbert spaces. We also discuss the need for higher order polarizations in terms of anomalies.Research partially supported by the Comision Interministerial de Ciencia y Tecnologia (CICYT)     

Vertex representations forN-toroidal Lie algebras and a generalization of the Virasoro algebra

Vertex representations are obtained for toroidal Lie algebras for any number of variables. These representations afford representations of certainn-variable generalizations of the Virasoro algebra that are abelian extensions of the Lie algebra of vector fields on a torus.Work supported in part by the Natural Sciences and Engineering Research Council of Canada.     

A Coset-Type Construction for the Deformed Virasoro Algebra

An analog of the minimal unitary series representations for the deformed Virasoro algebra is constructed using vertex operators of the quantum affine algebra U_q(sl₂). A similar construction is proposed for the elliptic algebra A_q,p(sl₂).     

Covariant Mellin transforms and the Weyl group quantization

The outline of the formalism with diagonalized dilation generator is presented. The covariant Mellin transform defining quanta with definite dimensionality and SL(2, C) quantum numbers is proposed.     

Dynamics on the Virasoro group, 2D gravity and hidden symmetries

We derive the action of 2D gravity from the two-cocycle of the Virasoro group by using a previously introduced method for constructing a dynamical system from a Lie group. A natural construction and explanation of the hidden SL(2, )-Kac-Moody symmetry, of general applicability, is provided. We also show how the proper quantization goes beyond the ordinary co-adjoint orbit method, and the possible connection with the Poisson-Lie groups.     

Kac-Moody group representations and generalization of the Sugawara construction of the Virasoro algebra

We discuss the dynamical structure of the semidirect product of the Virasoro and affine Kac-Moody groups within the framework of a group quantization formalism. This formalism provides a realization of the Virasoro algebra acting on Kac-Moody Fock states which generalizes the Sugawara construction. We also give an explicit construction of the standard Kac-Moody group representations associated with strings on SU(2) and recover, in particular, the renormalization factor of L(z) Research partially supported by the Conselleria de Cultura de la Generalitat Valenciana, the Plan de Formacion del Personal Investigador, the Comision Asesora de Investigacion Cientifica y Tecnica (CAICYT), and The British Council.     

Macdonald polynomials from Sklyanin algebras: A conceptual basis for thep-adics-quantum group connection

We establish a previously conjectured connection betweenp-adics and quantum groups. We find in Sklyanin's two parameter elliptic quantum algebra and its generalizations, the conceptual basis for the Macdonald polynomials, which interpolate between the zonal spherical functions of related real andp-adic symmetric spaces. The elliptic quantum algebras underlie theZ _n -Baxter models. We show that in then limit, the Jost function for the scattering offirst level excitations in the 1+1 dimensional field theory model associated to theZ _n -Baxter model coincides with the Harish-Chandra-likec-function constructed from the Macdonald polynomials associated to the root systemA ₁. The partition function of theZ ₂-Baxter model itself is also expressed in terms of this Macdonald-Harish-Chandrac-function, albeit in a less simple way. We relate the two parametersq andt of the Macdonald polynomials to the anisotropy and modular parameters of the Baxter model. In particular thep-adic regimes in the Macdonald polynomials correspond to a discrete sequence of XXZ models. We also discuss the possibility of q-deforming Euler products.Work supported in part by the NSF: PHY-9000386