arXiv:hep-th/9304011 v1 Apr 5 1993

3. Brief Review of the Liouville Theory
In this chapter, we touch briefly on some of the highlights of Liouville theory from
the viewpoint advocated in [7,17,8,36]. For other points of view on the Liouville theory,
see [37,38], and the sequence of works [39]. The classical Liouville theory was extensively
studied at the end of the nineteenth century in connection with the uniformization problem
for Riemann surfaces. We will sketch some of this in sections 3.2, 3.10 .
3.1. Classical Liouville Theory
We choose some reference metric ĝ on a surface Σ. The Liouville theory is the theory
of metrics g on Σ, and the Liouville field φ is defined by
The action is
∫
S Liouville =
g = e γφ ĝ . (3.1)
d 2 z √ ( 1 ĝ
8π ( ˆ∇φ) 2 + Q )
8π φR(ĝ) + µ ∫
8πγ 2
d 2 z √ ĝ e γφ , (3.2)
very similar to the background–charge theory (1.14) with a pure imaginary background
charge Q = 2iα 0 . The interaction given by µ (the “cosmological constant” term), while
soft, will be seen to have profound effects on the theory. For the particular choice
Q = 2/γ , (3.3)
the action (3.2) defines a classical conformal field theory, invariant under the Weyl transformations
ĝ → e 2ρ ĝ γφ → γφ − 2ρ . (3.4)
Remark: The linear shift in φ under a conformal transformation shows that φ can be
interpreted as a Goldstone boson for broken Weyl invariance (broken by the choice of ĝ).
Exercise. Classical Liouville theory
a) Using the transformation properties of the Ricci scalar in two dimensions,
R[e 2ρ ĝ] = e −2ρ( R[ĝ] − ˆ∇ 2 2ρ ) , (3.5)
compute the change in the action (3.2) under (3.4), and show that for Q = 2/γ the
change is independent of φ (so doesn’t affect the classical equations of motion).
b) Show that the classical equations of motion for (3.2) may be expressed as
i.e. they describe a surface with constant negative curvature.
R[g] = − 1 2 µ , (3.6)
(We take µ positive.)
Using again (3.5) (in the form R[g] = e −γφ( R[ĝ] − ˆ∇ 2 γφ ) , with φ as in (3.1)), note that
(3.6) is explicitly invariant under (3.4).
23