From coadjoint orbits to scale invariant WZNW type actions and 2D quantum gravity action

By using a geometrical procedure outlined by Balachandran we are able to show that the Wess-Zumino-Novikov-Witten action arises naturally from the coadjoint orbit of a purely central extended coadjoint vector of the Kac-Moody group. Consideration of other orbits leads to an action that motivates us to write a possibly new class of scale invariant models which are 'interacting' WZNW models. We are able to construct several scale invariant theories that appear to be interacting WZNW models. We then apply these techniques to the Virasoro group and show explicitly that the coadjoint orbit corresponding to Diff S1/S1 leads precisely to Polyakov's 2D quantum gravity which uses the light-cone gauge. From here it is possible to consider any arbitrary coadjoint orbit and construct an action. In particular, we are able to write actions for the orbits Diff S1/SL(2, R)n invariant under Ln, L0, L-n of the Virasoro algebra. We make several comments with regard to extending these models. (orig.)