F.A. Bais

Extensions of the Virasoro algebra constructed from Kac-Moody algebras using higher order Casimir invariants

Abstract We consider bosonic extensions of the Virasoro algebra that can be obtained from Kac-Moody algebras g by generalizing the Sugawara construction to the higher order Casimirs of g. In this paper we explicitly construct the algebra of a primary field of dimension 3 constructed from the 3rd order Casimir of AN−1. For N = 3 we compare our results to the Z3-extended Virasoro algebra proposed by Fateev and Zamolodchikov.

Coset construction for extended Virasoro algebras

We discuss extensions of the Virasoro algebra obtained by generalizing the Sugawara construction to the higher order Casimir invariants of a Lie algebra g. We generalize the GKO coset construction to the dimension-3 operator for g =A N 1 and recover results of Fateev and Zamolodchikov if N = 3. Branching rules and generalizations to all simple, simply-laced g are discussed.

Covariantly coupled chiral algebras

Abstract New extended conformal algebras are constructed by conformal reductions of sl N WZWN models. These are associated with the inequivalent sl 2 embeddings into sl N . Among other things, the conformal weights of the generators and the occurrence of Kac-Moody and W n subalgebras are determined by the branching rules of the adjoint representation for the particular embedding. For some representative classes the algebras are constructed explicitly. In general they are coupled chiral algebras suggesting that they correspond to the symmetries of certain interacting conformal field theories. Moreover we find that a (minimal) covariant coupling is present which is related to a generalized Gelfand-Dickii structure. Some aspects of the quantization are addressed, in particular the c -values are determined. We introduce a new hybrid realization of KM algebras which interpolates between a realization of currents and of free fields, in which the constraints can be imposed in a very natural way.

Condensate-induced transitions between topologically ordered phases

We investigate transitions between topologically ordered phases in two spatial dimensions induced by the condensation of a bosonic quasiparticle. To this end, we formulate an extension of the theory of symmetry-breaking phase transitions which applies to phases with topological excitations described by quantum groups or modular tensor categories. This enables us to deal with phases whose quasiparticles have noninteger quantum dimensions and obey braid statistics. Many examples of such phases can be constructed from two-dimensional rational conformal field theories, and we find that there is a beautiful connection between quantum group symmetry breaking and certain well-known constructions in conformal field theory, notably the coset construction, the construction of orbifold models, and more general conformal extensions. Besides the general framework, many representative examples are worked out in detail.

TOPOLOGICAL FIELD THEORY AND THE QUANTUM DOUBLE OF SU(2)

We study the quantum mechanics of a system of topologically interacting particles in 2 + 1 dimensions, which is described by coupling the particles to a Chern-Simons gauge field of an inhomogeneous group. Analysis of the phase space shows that for the particular case of ISO(3) Chern-Simons theory the underlying symmetry is that of the quantum double D(SU(2)), based on the homogeneous part of the gauge group. This in contrast to the usual q-deformed gauge group itself, which occurs in the case of a homogeneous gauge group. Subsequently, we describe the structure of the quantum double of a continuous group and the classification of its unitary irreducible representations. The comultiplication and the R-element of the quantum double allow for a natural description of the fusion properties and the non-abelian braid statistics of the particles. These typically manifest themselves in generalised Aharonov-Bohm scattering processes, for which we compute the differential cross sections. Finally, we briefly describe the structure of D(SO(2,1)), the underlying quantum double symmetry of (2+1)-dimensional quantum gravity.

Dimensional reduction of gauge theories yielding unified models spontaneously broken to SU3×U1

By dimensional reduction of pure gauge theories (with gauge groupG) over a compact coset spaceS/R, one obtains four-dimensional theories where scalar fields and a symmetry breaking potential appear naturally. We present an analysis of all such unified models with simpleG which are spontaneously broken toSU3×SU2×U1, and which can be obtained by this technique with the added restriction thatS⊂G. Although the bosonic sectors appear promising, no cases are found with the correct quantum numbers for the surviving fermions.

Tensor product representations of the quantum double of a compact group

Abstract:We consider the quantum double of a compact group G, following an earlier paper. We use the explicit comultiplication on in order to build tensor products of irreducible *-representations. Then we study their behaviour under the action of the R-matrix, and their decomposition into irreducible *-representations. The example of is treated in detail, with explicit formulas for direct integral decomposition (“Clebsch–Gordan series”) and Clebsch-Gordan coefficients. We point out possible physical applications.

Geometry of Coset Spaces and Massless Modes of the Squashed Seven Sphere in Supergravity

Abstract We derive some general results for Killing vectors on arbitrary coset manifolds and explicitly exhibit the squashed seven-sphere as the coset space Sp 4 ×Sp 2 /Sp 2 ×Sp 2 . Using these results, we then analyze the zero-mass sector of supergravity of the squashed S 7 and argue that it is not interpretable as a spontaneously broken version of N =8 supergravity. We also point out the existence of a new solution which combines squashing and torsion.

Torus compactification for non-simply laced groups

Abstract Non-simply laced gauge groups are obtained from torus compactification of the interacting 26-dimensional bosonic string.

Broken quantum symmetry and confinement phases in planar physics

Many two-dimensional physical systems have symmetries which are mathematically described by quantum groups (quasitriangular Hopf algebras). In this Letter we introduce the concept of a spontaneously broken Hopf symmetry and show that it provides an effective tool for analyzing a wide variety of phases exhibiting many distinct confinement phenomena.