Higher - spin Realisations of the Bosonic String

It has been shown that certain W algebras can be linearised by the inclusion of a spin–1 current. This provides a way of obtaining new realisations of the W algebras. Recently such new realisations of W3 were used in order to embed the bosonic string in the critical and non-critical W3 strings. In this paper, we consider similar embeddings in W2,4 and W2,6 strings. The linearisation of W2,4 is already known, and can be achieved for all values of central charge. We use this to embed the bosonic string in critical and non-critical W2,4 strings. We then derive the linearisation of W2,6 using a spin–1 current, which turns out to be possible only at central charge c = 390. We use this to embed the bosonic string in a non-critical W2,6 string. Supported in part by the U.S. Department of Energy, under grant DE-FG05-91-ER40633 With the discovery of the property of the W3 algebra that it can be linearised by the inclusion of a spin–1 current [1], new realisations were constructed for the purpose of building the corresponding W3 strings [2, 3]. An unusual feature of these realisations is that the spin–3 current contains a term linear in a ghost-like field. The realisations also close when this term is omitted, under which circumstance the corresponding string theory is equivalent to the one that is based on the Romans’ free-scalar realisation [4]. However, when this term is included, the corresponding BRST operator is equivalent to that of the bosonic string, which can be shown by making a local canonical field redefinition [2]. Thus the new realisations provide embeddings of the bosonic string in the W3 string. It is interesting to generalise the above consideration to the embedding of the bosonic string inW2,s strings, whereW2,s denotes the conformal algebra generated by a spin–s current together with the energy-momentum tensor. The W2,s strings based on free-scalar realisations were extensively discussed in Ref. [5], where it was shown that when s ≥ 3 the cohomologies describe Virasoro strings coupled to certain minimal models. TheW2,s algebras exist at the classical level for all positive integer values of s. However, at the quantum level, for generic values of s, a W2,s algebra exists only for a finite set of special values of central charge [6, 7], which in particular does not include the critical value. The exceptions are the W2,s algebras for s = 1, 2, 3, 4 and 6, for which the central charge can be arbitrary. Although the W2,s algebra does not close at the critical central charge for generic values of s, it is nevertheless possible to buildW2,s strings with free-scalar realisations. It was shown in Ref. [8] that one can first use the free-scalar realisation to write down the classical BRST operator, and then quantise the theory by renormalising the transformation rules and adding necessary quantum counter-terms. The new realisations that were constructed in Ref. [2], which provide embeddings of the bosonic string in W3 = W2,3 strings, do not generate the W3 algebra at the classical level. The W3 symmetry arises only as a consequence of quantisation. Thus it seems that if we are to use such new realisations for values of s other than 3, we must restrict our attention to the cases s = 1, 2, 4 and 6, for which the quantum algebras exist. The embeddings of the bosonic string in the W2,s string for s = 1, 2 and 3 were discussed in Ref. [2]. In this paper, we shall focus our attention on the remaining cases s = 4 and 6. It is instructive to begin by studying the form of the linearisation of the W3 and W2,4 algebras, for which the results were obtained in Refs. [1, 9]. The associated linearised W1,2,3 and W1,2,4 algebras take the form: T0(z)T0(0) ∼ c 2z4 + 2T0 z2 + ∂T0 z , T0(z)W0(0) ∼ sW0 z2 + ∂W0 z , T0(z) J0(0) ∼ c1 z3 + J0 z2 + ∂J0 z , J0(z) J0(w) ∼ − 1 z2 , (1) J0(z)W0(w) ∼ hW0 z , W0(z)W0(w) ∼ 0 , where s = 3 and 4 respectively. The coefficients c, c1 and h are given by c = 50 + 24t + 24 t2 , c1 = − √ 6(t+ 1 t ) , h = √ 3 2 t , (s = 3) c = 86 + 30t + 60 t2 , c1 = −3t− 4 t , h = t . (s = 4) (2) The currents of the W3 and W2,4 algebras are then given by T = T0 , W = W0 +WR , (3) where WR is the Romans type realisation constructed from T0 and J0. For the cases where s = 3, 4 and 6, WR takes the form [8]