HIGHER-SPIN EXTENDED CONFORMAL ALGEBRAS AND W-GRAVITIES

The construction of classical W3 gravity is reviewed. It is suggested that the hidden symmetry for quantum W3 gravity in the chiral gauge is not SL(3,R) but a group contraction of this, ISL(2,R). This is extended to W(N) gravity, and the case of W4 gravity is presented in detail. The gauge transformations are realized on D free bosons, with the spin-n conserved current (2 less-than-or-equal-to n less-than-or-equal-to N) taking the form d(i(i)...i(n)) partial + phi-i(l)...partial + phi-i(n) for some constant tensor d(i(i)...i(n)). The d-tensors must satisfy N - 2 non-linear algebraic constraints and these constraints are shown to be satisfied if the d-tensors are taken to be the structure-tensors of an Nth degree Jordan algebra. The relation with Jordan algebras is used to give solutions of the d-tensor constraints for any value of D, N. The free-boson construction of the W(N) algebras is generalized to give a Sugawara-type construction of a large class of classical extended conformal algebras. The chiral gauging of any classical extended conformal algebra is shown to require only a linear Noether coupling to world-sheet gauge-fields, while gauging a non-chiral algebra in general leads to a non-polynomial action. A number of examples are examined, including W(x), W-supergravity, Knizhnik-Berschadsky supergravity and "W(N/M" algebras. Theories of higher-spin W-gravity of the type described are only possible in one and two space-time dimensions, and the one-dimensional case is briefly discussed. The covariant formulation of W-gravity is briefly discussed and the relation between classical and quantum extended conformal algebras is analyzed.