COADJOINT ORBITS OF THE VIRASORO ALGEBRA AND THE GLOBAL LIOUVILLE EQUATION

The classification of the coadjoint orbits of the Virasoro algebra is reviewed and then applied to analyze the sc-called global Liouville eq uation. The review is self-contained, elementary and is tailor-made fo r the application. It is well known that the Liouville equation for a smooth, real field phi under periodic boundary condition is a reductio n of the SL(2,R) WZNW model on the cylinder, where the WZNW field g is an element of SL(2, R) is restricted to be Gauss decomposable. if one drops this restriction, the Hamiltonian reduction yields, for the fie ld Q = kappa g(22) where kappa not equal 0 is a constant, what we call the global Liouville equation. Corresponding to the winding number of the SL(2,R) WZNW model, there is a topological invariant in the reduc ed theory, given by the number of zeros of Q over a period. By the sub stitution Q = +/- exp(-phi/2), the Liouville theory for a smooth phi i s recovered in the trivial topological sector. The nontrivial topologi cal sectors can be viewed as singular sectors of the Liouville theory that contain blowing-up solutions in terms of phi. Since the global Li ouville equation is conformally invariant, its solutions can be descri bed by explicitly listing those solutions for which the stress-energy tensor belongs to a set of representatives of the Virasoro coadjoint o rbits chosen by convention. This direct method permits to study the '' coadjoint orbit content'' of the topological sectors as well as the be havior of the energy in the sectors. The analysis confirms that the tr ivial topological sector contains special orbits with hyperbolic monod romy and shows that the energy is bounded from below in this sector on ly.