General WZW Research Articles

PRESENTATIONS OF WESS–ZUMINO–WITTEN FUSION RINGS

The fusion rings of the Wess–Zumino–Witten models are re-examined. Attention is drawn to the difference between fusion rings over ℤ (which are often of greater importance in applications) and fusion algebras over ℂ. Complete proofs are given by characterizing the fusion algebras (over ℂ) of the SU (r+1) and Sp (2r) models in terms of the fusion potentials, and it is shown that the analagous potentials cannot describe the fusion algebras of the other models. This explains why no other representation-theoretic fusion potentials have been found. Instead, explicit generators are then constructed for general WZW fusion rings (over ℤ). The Jacobi–Trudy identity and its Sp (2r) analogue are used to derive the known fusion potentials. This formalism is then extended to the WZW models over the spin groups of odd rank, and explicit presentations of the corresponding fusion rings are given. The analogues of the Jacobi–Trudy identity for the spinor representations (for all ranks) are derived for this purpose, and may be of independent interest.

THE GENERAL TWISTED OPEN WZW STRING

We recently studied two large but disjoint classes of twisted open WZW strings: the open-string sectors of the WZW orientation orbifolds and the so-called basic class of twisted open WZW strings. In this paper, we discuss all T-dualizations of the basic class to construct the general twisted open WZW string — which includes the disjoint classes above as special cases. For the general case, we give the branes and twisted noncommutative geometry at the classical level and the twisted open-string KZ equations at the operator level. Many examples of the general construction are discussed, including in particular the simple case of twisted free-bosonic open strings. We also revisit the open-string sectors of the general WZW orientation orbifold in further detail. For completeness, we finally review the general twisted boundary state equation which provides a complementary description of the general twisted open WZW string.

TWISTED OPEN STRINGS FROM CLOSED STRINGS: THE WZW ORIENTATION ORBIFOLDS

Including world-sheet orientation-reversing automorphismsĥσ∈H-in the orbifold program, we construct the operator algebras and twisted KZ systems of the general WZW orientation orbifoldAg(H-)/H-. We find that the orientation-orbifold sectors corresponding to each ĥσ∈H-are twisted open WZW strings, whose properties are quite distinct from conventional open-string orientifold sectors. As simple illustrations, we also discuss the classical (high-level) limit of our construction and free-boson examples on Abelian g.

Twisted Einstein Tensors and Orbifold Geometry

Following recent advances in the local theory of current-algebraic orbifolds, we study various geometric properties of the general WZW orbifold, the general coset orbifold and a large class of (nonlinear) sigma model orbifolds. Phase-space geometry is emphasized for the WZW orbifolds — while for the sigma model orbifolds we construct the corresponding sigma model orbifold action, which includes the previously-known general WZW orbifold action and general coset orbifold action as special cases. We focus throughout on the twisted Einstein tensors with diagonal monodromy, including the twisted Einstein metric, the twisted B field and the twisted torsion field of each orbifold sector. Finally, we present strong evidence for a conjectured set of twisted Einstein equations which should describe those sigma model orbifolds in this class which are also one-loop conformal.

EXTENDED OPERATOR ALGEBRA AND REDUCIBILITY IN THE WZW PERMUTATION ORBIFOLDS

Recently the operator algebra, including the twisted affine primary fields, and a set of twisted KZ equations were given for the WZW permutation orbifolds. In the first part of this paper we extend this operator algebra to include the so-called orbifold Virasoro algebra of each WZW permutation orbifold. These algebras generalize the orbifold Virasoro algebras (twisted Virasoro operators) found some years ago in the cyclic permutation orbifolds. In the second part, we discuss the reducibility of the twisted affine primary fields of the WZW permutation orbifolds, obtaining a simpler set of single-cycle twisted KZ equations. Finally we combine the orbifold Virasoro algebra and the single-cycle twisted KZ equations to investigate the spectrum of each orbifold, identifying the analogues of the principal primary states and fields also seen earlier in cyclic permutation orbifolds. Some remarks about general WZW orbifolds are also included.

THE GENERAL COSET ORBIFOLD ACTION

Recently an action formulation, called the general WZW orbifold action, was given for each sector of every WZW orbifold. In this paper we gauge this action by general twisted gauge groups to find the action formulation of each sector of every coset orbifold. Connection with the known current-algebraic formulation of coset orbifolds is discussed as needed, and some large examples are worked out in further detail.

On Orientifolds of WZW Models and their Relation to Geometry

We investigate D-branes in orientifolds of WZW models. A connection between the conformal field theory approach to orientifolds and the target space motivated analysis is established. In particular, we associate previously constructed crosscap states to involutions of the group manifold and their fixed point sets. Whereas our analysis of D-branes in orientifolds of general WZW models is restricted to special D0-branes, we investigate all symmetry preserving branes of SU(2)-orientifolds in detail. For that case, the location of the orientifold fixed point set is independently determined by scattering localized graviton wave packets.

MORE ABOUT ALL CURRENT-ALGEBRAIC ORBIFOLDS

Recently a construction was given for the stress tensors of all sectors of the general current-algebraic orbifold A(H)/H, where A(H) is any current-algebraic conformal field theory with a finite symmetry group H. Here we extend and further analyze this construction to obtain the mode formulation of each sector of each orbifold A(H)/H, including the twisted current algebra, the Virasoro generators, the orbifold adjoint operation and the commutator of the Virasoro generators with the modes of the twisted currents. As applications, general expressions are obtained for the twisted current–current correlator and ground state conformal weight of each twisted sector of any permutation orbifold A(H)/H, H⊂SN. Systematics are also outlined for the orbifolds A(Lie h(H))/H of the (H and Lie h)-invariant conformal field theories, which include the general WZW orbifold and the general coset orbifold. Finally, two new large examples are worked out in further detail: the general SNpermutation orbifold A(SN)/SNand the general inner-automorphic orbifold A(H(d))/H(d).

Universal aspects of string propagation on curved backgrounds.

String propagation on $D$-dimensional curved backgrounds with Lorentzian signature is formulated as a geometrical problem of embedding surfaces. When the spatial part of the background corresponds to a general WZW model for a compact group, the classical dynamics of the physical degrees of freedom is governed by the coset conformal field theory $\frac{\mathrm{SO}(D\ensuremath{-}1)}{\mathrm{SO}(D\ensuremath{-}2)}$, which is universal irrespective of the particular WZW model. The same holds for string propagation on $D$-dimensional flat space. The integration of the corresponding Gauss-Codazzi equations requires the introduction of (non-Abelian) parafermions in differential geometry.

Semiclassical versus exact solutions of charged black hole in four dimensions and exact O( d, d) duality

We derive a charged black hole solution in four dimensions described by SL(2, R ) × SU(2) × U(1)/U(1) 2 WZW coset model. Using the algebraic hamiltonian method we calculate the corresponding solution that is exact to all orders in 1 k . It is shown that unlike the 2D black hole, the singularity remains also in the exact solution, and moreover, in some range of the gauge parameter the space-time does not fulfil the cosmic censor conjecture, i.e. we find a naked singularity outside the black hole. Exact dual models are derived as well, one of them describes a 4D space-time with a naked singularity. Using the algebraic hamiltonian approach we also find the exact to all orders O( d, d) transformation of the metric and the dilaton field for general WZW coset models and show the correction with respect to the transformations in one-loop order.