Induced $W_\infty$ Gravity as a WZNW Model

TL;DR

This paper derives the explicit form of the quantum effective action for a chiral $ ext{W}_ ext{infinity}$-symmetric system coupled to a $ ext{W}_ ext{infinity}$-gravity background, revealing its geometric structure and symmetries.

Contribution

It provides a systematic derivation of the $ ext{W}_ ext{infinity}$ quantum effective action as a geometric action on a coadjoint orbit, linking it to a WZNW model and uncovering hidden symmetries.

Findings

Explicit form of the $ ext{W}_ ext{infinity}$ effective action as a geometric orbit action

Identification of hidden $SL( infty; ext{IR})$ Kac-Moody currents

Expression of the energy-momentum tensor in Sugawara form

Abstract

We derive the explicit form of the Wess-Zumino quantum effective action of chiral $\Winf$-symmetric system of matter fields coupled to a general chiral $\Winf$-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" $SL(\infty;\IR)$ Kac-Moody currents and the associated $SL(\infty;\IR)$ Sugawara form of energy-momentum tensor component $T_{++}$ as a consequence of the $SL(\infty;\IR)$ stationary subgroup of the relevant $\Winf$ coadjoint orbit.