Two-dimensional gravity and its W3-extension: Strongly coupled unitary theories

For strongly coupled 2D-gravity, with central charge Cgrav=1+6(s+2), s=0, ±1, a chiral (2,2)-operator Φ satisfies a closed exchange algebra on the unit circle, with a consistent restriction to a unitary subspace of the Virasoro representation. In this paper, this result of Gervais and Neveu is first extended to a larger unitary space, with characters equal to those of a free boson compactified on a circle with radius √2(2-s). Second, Φ is shown to take a simple form when expressed in terms of the operators whose exchange algebra coincides with the universal R-matrix of the quantum group SL(2)q. Third, in the case of strongly coupled A2 Toda theories, i.e. 'W3-extended 2D-gravity', the generalization of the Φ-field is obtained for CT=2+24(s+2). Going from A1 (Liovilletriple bond2D-gravity) to A2 brings in interesting novel features, such as an intriguing U(1) gauge field configuration defined on the A2 weight lattice. (orig.)