# Coadjoint orbits of the Virasoro algebra and the global Liouville equation

Date: 1998

MTMT: 1292288

WoS ID: 000071843300009

Scopus ID: 0039547318

### Abstract:

The classification of the coadjoint orbits of the Virasoro
algebra is reviewed and then applied to analyze the so-called
global Liouville equation. The review is self-contained,
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<sub>22</sub> 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 behavior 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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