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

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

1993-01-30

arxiv.org/abs/hep-th/9301128v2

Mod. Phys. Lett. A8 (1993) 1139

Physics

High Energy Physics

High Energy Physics - Theory

19 pages, LATEX, with notation $w_N$ changed to $\bar{w}_N$, with 3 more references and with note added in proof

Scientific paper

10.1142/S0217732393002622

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.

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