W∞ gravity

Abstract

We construct a gauge theory of woo gravity coupled to scalars. For scalar fields valued in some-finite-dimensional Lie algebra, a complete realisation of gauged w~ can be given using a number of gauge fields equal to the rank of the Lie algebra. We show how the theory can be truncated to describe a gauge theory of Ws gravity, and we give the explicit result for W3. We show that the transformations of the scalar fields can be interpreted as a non-linear realisation on the coset w~ +oo/w~.

N=2 W Supergravity

“…Indeed, in the one-scalar realization of the bosonic woo-gravity theory the lowestspin s = 1 transformation is not gauged. 30 The reason for this difference is that we are working here with a two-scalar realization where the lowest-spin s = 1/2 current is given by w(1/2) = Dφ¯and, since <φ¯(1)φ¯(2» = 0, a single contraction between two s = 1/2 currents does not give a central term. Therefore, one can treat the s = 1/2 transformation on the same footing as the higher-spin transformations.…”

“…& M. de Roo (26) The delta function Δ(Zl -Z2) is defined by (27) We have furthermore defined a regularization where (28) The inverse operator 1/ D¯is defined by the relations (29) By taking repeated derivatives one can derive the general identities (30) Under the leading order inhomogeneous terms in the gauge transformations δA (1) …”

“…The aim here is to extend the ordinary string to a so-called "W string." Recently, several steps in this programme have been undertaken, including the gauging of the W symmetries, [28][29][30][31] an investigation of the anomaly-structure in W gravity,28,32-39 and a study of the spectrum of W strings. 40,41 It seems natural to investigate the role of supersymmetry in the above examples.…”

W-Strings on Curved Backgrounds

“…Since ordinary string theory can be considered as a gauge theory based on the Virasoro algebra, one can analogously define a W-string theory as a gauge theory based on a W-algebra [2] (or any other higher spin conformally extended algebra [1]). Actions for a large class of W-string theories have been constructed so far [3][4][5][6][7][8][9][10]. These actions essentially describe a W-string propagating on a flat background spacetime metric.…”

Covariant w∞ gravity and its reduction to WN gravity

“…Another approach to classical W gravity was advocated in ref. [ 7 ], where it was proposed to view the W~v gravities as reductions of a woo gravity. The latter corresponds to the woo algebra which is, however, a linear algebra.…”