Description of W ∞ Symmetry in Coset Models

We derive the transformation laws in the target space of the SL(2, R)/U(1) coset model which generate the classical W∞ symmetry. Presently serving in the Greek Armed Forces. Present address: Theory Division, CERN, CH-1211, Geneva 23, Switzerland, email: [email protected] It was realized recently that the concept of W -algebras [1], their decompactification limits (algebras of the W∞ type) [2, 3, 4] and their non-compact analogues [5], play an important role in such varied subjects as CFTs, their integrable off-critical siblings, integrable hierarchies, string theory both perturbative and non-perturbative, etc. In this context, W -algebras are described as abstract operator algebras that organize the operator content of 2-d theories, they are local and generically contain higher spin operators, which are responsible for their non-linearity. They reflect the underlying parafermionic symmetry of CFTs, which is generically non-local, and can be simply described as a local subalgebra of the enveloping parafermionic algebra. In the σ-model description of such CFTs, the presence of theW -algebras is associated with the existence of chiral symmetries. However, in the σ-model context one usually works in perturbation theory around the decompactification limit (α → 0). Thus, classically, the symmetry will be of the W∞ type, and α ′ corrections will renormalize it accordingly; for example, in the compact case the renormalized W -algebra should be finitely generated. Such a target space description of W -symmetries has not attracted any attention so far. We feel that it is interesting to explore it, in order to get a better grasp of W -symmetries from a (more conventional) lagrangian point of view, which usually provides an easier link to geometric concepts and their generalizations. The purpose of this letter is to use the parafermionic description of W -algebras together with the σ-model description of the parafermionic currents [6] in order to obtain a systematic formulation of W -symmetries in target space. The CFT models we will be considering are of the coset (G/H) type for which a σ-model description is available as gauged WZW models [7]. Our method is applicable to all such models, both compact and non-compact, but for notational simplicity we will only consider the SL(2, R)/U(1) model. At the clasical level theW -symmetry is identical for various non-compact versions of a compact target space (that is the manifolds obtained by analytic continuation of the compact manifold), for example the N → ∞ limit of the WN algebra associated to the SU(2)N/U(1) model and the k → ∞ limit of the Ŵ∞(k) algebra of the SL(2, R)k/U(1) model coincide [5]. The SL(2, R)k/U(1) coset model has received considerable attention recently since it admits a geometric interpretation (for k large) as a 2-d black hole [8]. Writing down the gauged WZW action and integrating out the gauge fields we obtain to leading order in k the following action (in isothermal coordinates) S = k 4π ∫ ∂u∂̄ū+ ∂ū∂̄u 1− uū . (1) It differs from the conventional σ-model on S (or its compact analogue) in that the target space metric is (1 − uū) instead of (1 − uū). Also, the difference between SU(2)/U(1) and SL(2, R)/U(1) is reflected in the range of the coordinates u, ū. The 1 classical equations of motion can be derived from the variation of S δS ∼ ∫ (δuEū + δūEu). (2) They read Eu = Eū = 0 (3) with Eu = ∂∂̄u 1− uū + ū∂u∂̄u (1− uū)2 , (4a) Eū = ∂∂̄ū 1− uū + u∂ū∂̄ū (1− uū)2 . (4b) The action (1) has a classical U(1) symmetry u→ ue , ū→ ūe (5) reflecting the killing symmetry of the the target manifold. The associated current is J = ū∂u − u∂ū 1− uū , J̄ = ū∂̄u− u∂̄ū 1− uū (6) which is conserved but not chiral ∂J̄ + ∂̄J = 0 (7) using the equations of motion. We can define the generating parafermion (non-local) currents in the standard way [6] by dressing the gauge currents with Wilson lines ∗ ψ+ = ∂u √ 1− uū V+ , ψ− = ∂ū √ 1− uū V− (8) where V± = exp [