Light cone W _n geometry and its symmetries and projective field theory

Abstract

I show that the generalized Beltrami differentials and projective connections which appear naturally in induced light cone W n gravity are geometrical fields parametrizing in one-to-one fashion generalized projective structures on a fixed base Riemann surface. I also show that W n symmetries are nothing but gauge transformations of the flat SL(n, C) vector bundles canonically associated to the generalized projective structures. This provides an original formulation of classical light cone W n geometry. From the knowledge of the symmetries, the full BRS algebra is derived. Inspired by the results of recent literature, I argue that quantum W n gravity may be formulated as an induced gauge theory of generalized projective connections. This leads to projective field theory. The possible anomalies arising at the quantum level are analyzed by solving Wess-Zumino consistency conditions. The implications for induced covariant W n gravity are briefly discussed. The results presented, valid for arbitrary n, reproduce those obtained for n = 2, 3 by different methods.

Higher dimensional uniformisation and W-geometry

“…It is possibly a Kahler manifold but we do not know how to establish this. The W -manifold described here does not seem to have any relation to the manifold introduced by Zucchini as a possible candidate for W -space [25].…”

Primary currents and Riemannian geometry in algebras

“…This type of equations has been previously obtained in a systematic way [70] through vanishing curvature conditions (a useful trick already used in [12,41]). Even if there exist few Lagrangian models giving rise to field equations containing Bol's operators [41], a general construction of such Lagrangians is still missing.…”

Large chiral diffeomorphisms on Riemann surfaces and W-algebras

“…, (3.9) a transformation law which shows that the local scalar fields Z have to be transformed in a homographic way in accordance with the Zucchini's point of view [7] on W-algebras.…”

“…Particular interest has been devoted to the so-called W-algebras [3] which come out from different principles [4,5,6] and the question of their geometric origin remains still unclear or unsatisfactory despite the various attempts given by [5,7,8,9,10,11].…”