W 3 constructions on affine Lie algebras

We use an argument of Romans showing that every Virasoro construction leads to realizations of W3, to construct W3 realizations on arbitrary affine Lie algebras. Solutions are presented for generic values of the level as well as for specific values of the level but with arbitrary parameters. We give a detailed discussion of the ŝu(2)l-case. Finally, we discuss possible applications of these realizations to the construction of W -strings. Aspirant N.F.W.O. Belgium, e-mail: [email protected] Onderzoeker, I.I.K.W. Belgium, e-mail: [email protected] 1. It has become clear in the last few years that W -algebras—nonlinear extensions of the Virasoro algebra by additional primary fields of dimension greater than two—play an important rôle in two-dimensional conformal field theory as well as in integrable systems. These algebras have recently been gauged, giving the so-called W -gravity theories. In order to study the representation theory of these algebras, it is convenient to realize them in terms of simpler structures, such as free fields or affine currents. The interest in realizations of conformal field theories on affine algebras has increased considerably since it has been noted that the Sugawara and coset energy-momentum tensors are but specific examples of far more general “Virasoro constructions”. The most general realization of a Virasoro algebra as a quadratic combination of affine currents (including possible background charges) leads to a set of coupled algebraic equations [1, 2]. Though only quadratic, these “master equations” have so far resisted any attempt at a general solution. However, the use of specific ansätze has led to numerous new solutions, lifting a tip of the veil on this space of constructions. See e.g. [3, 4] for some of these new solutions and early developments. For W -algebras, and in particular for its simplest example, W3, an analogous approach has been unsuccessful up to now. In fact, the straightforward albeit tedious task of writing down the corresponding “master equations” for W3—and thus generalizing the coset solutions of [5]—still remains to be done. So far, the most general results have been obtained in [6] for W3, using N free scalar fields—i.e. an abelian affine algebra (û(1)) . In this letter, we construct realizations of W3 for arbitrary affine Lie (super)algebras. The starting point is an argument by Romans in [6] which reduces the problem ofW3 constructions to the one of Virasoro constructions. Although this certainly does not give the most general W3 construction, it allows us to associate many different such constructions with every affine algebra. We illustrate this method with various affine algebras, yielding constructions for generic values of the level as well as for specific values of the level but with free parameters. In particular, we write down explicitly a realization of W3 in terms of ŝu(2) currents for generic value of the level. Also, we show how the c = 2 free field realization of W3 from [7] can be recast as an affine one-parameter construction on ŝu(2)4 × û(1). Finally, we discuss possible applications of these realizations to the constructions of W -strings. 2. The W3 algebra [8] is an extension of the Virasoro algebra, which is