Induced $W_\infty$ Gravity as a WZNW Model

We derive the explicit form of the Wess-Zumino quantum effective action of chiral W ∞ symmetric system of matter fields coupled to a general chiral W ∞ -gravity background. It is expressed as a geometric action on a coadjoint orbit of the deformed group of area-preserving diffeomorphisms on cylinder whose underlying Lie algebra is the centrally-extended algebra of symbols of differential operators on the circle. Also, we present a systematic derivation , in terms of symbols, of the “hidden” SL ( ∞ ; IR) Kac-Moody currents and the associated SL ( ∞ ; IR) Sugawara form of energy-momentum tensor component T ++ as a consequence of the SL ( ∞ ; IR) stationary subgroup of the relevant W ∞ coadjoint orbit.