Geometry and Physics on $w_{\infty}$ Orbits

TL;DR

This paper explores the geometric and physical properties of $w_{}$ orbits using coadjoint orbit techniques, linking them to $w$ gravity, 2D turbulence, and string theory, and identifying critical central charges.

Contribution

It applies the coadjoint orbit method to $w_{}$, derives geometric actions, and connects these to $w$ gravity, turbulence, and string theory, revealing new insights into orbit structures and central charges.

Findings

Derived geometric actions for $w_{}$ and $ar{w}_N$ orbits.

Connected $w$ gravity and turbulence through the Hamiltonian structure.

Identified the critical central charge of the $ar{w}_N$ string.

Abstract

We apply the coadjoint orbit technique to the group of area preserving diffeomorphisms (APD) of a 2D manifold, particularly to the APD of the semi-infinite cylinder which is identified with $w_{\infty}$. The geometrical action obtained is relevant to both $w$ gravity and 2D turbulence. For the latter we describe the hamiltonian, which appears to be given by the Schwinger mass term, and discuss some possible developments within our approach. Next we show that the set of highest weight orbits of $w_{\infty}$ splits into subsets, each of which consists of highest weight orbits of $\bar{w}_N$ for a given N. We specify the general APD geometric action to an orbit of $\bar{w}_N$ and describe an appropriate set of observables, thus getting an action and observables for $\bar{w}_N$ gravity. We compute also the Ricci form on the $\bar{w}_N$ orbits, what gives us the critical central charge of the $\bar{w}_N$ string, which appears to be the same as the one of the $W_N$ string.