The classification of the coadjoint orbits of the Virasoro algebra is reviewed and is then applied to analyze the so-called global Liouville equation. The review is selfcontained, elementary and is tailor-made for the application. It is well-known that the Liouville equation for a smooth, real field ϕ under periodic boundary condition is a reduction of the SL(2, R) WZNW model on the cylinder, where the WZNW field g ∈ SL(2, R) is restricted to be Gauss decomposable. If one drops this restriction, the Hamiltonian reduction yields, for the field Q = κg 22 where κ = 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 reduced theory, given by the number of zeros of Q over a period. By the substitution Q = ± exp(−ϕ/2), the Liouville theory for a smooth ϕ is recovered in the trivial topological sector. The nontrivial topological sectors can be viewed as singular sectors of the Liouville theory that contain blowing-up solutions in terms of ϕ. Since the global Liouville equation is conformally invariant, its solutions can be described by explicitly listing those solutions for which the stress-energy tensor belongs to a set of representatives of the Virasoro coadjoint orbits chosen by convention. This direct method permits to study the 'coadjoint orbit content' of the topological sectors as well as the behaviour of the energy in the sectors. The analysis confirms that the trivial topological sector contains special orbits with hyperbolic monodromy and shows that the energy is bounded from below in this sector only.

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The construction and the symmetries of Chern-Simons vortices in harmonic and uniform magnetic force backgrounds found by Ezawa, Hotta and Iwazaki, and by Jackiw and Pi are generalized using the non-relativistic Kaluza-Klein-type framework presented in our previous paper. All Schrödinger-symmetric backgrounds are determined.

The chiral WZNW symplectic form Ω ρ chir is inverted in the general case. Thereby a precise relationship between the arbitrary monodromy dependent 2-form appearing in Ω ρ chir and the exchange r-matrix that governs the Poisson brackets of the group valued chiral fields is established. The exchange r-matrices are shown to satisfy a new dynamical generalization of the classical modified Yang-Baxter (YB) equation and Poisson-Lie (PL) groupoids are constructed that encode this equation analogously as PL groups encode the classical YB equation. For an arbitrary simple Lie group G, exchange r-matrices are found that are in one-to-one correspondence with the possible PL structures on G and admit them as PL symmetries.