arXiv:hep-th/9304011 v1 Apr 5 1993

where |Σ〉 is a state created by the rest of the surface (as is standard in discussions of
the “operator formalism” [21–23]). If we interpret the integral in (3.59) as an OPE over
macroscopic vertex operators V E , then since we sum over operators with weights ∆ ≥
Q 2 /8, we see that the OPE is much softer than in ordinary CFT. This discussion can be
generalized.
The essential message is that while we may insert microscopic operators on a surface we
should only do so “externally.” We must factorize on macroscopic states. The factorization
on macroscopic states also ameliorates the disastrous sum noted above for α + β < 1 2 Qχ.
The essentially non-free field nature of the operator product expansion in Liouville
theory accounts for some unusual properties of the theory.
As one example, note that
since the Liouville theory is conformal for all µ > 0, the cosmological constant e γφ is
an exactly marginal operator.
This appears to conflict with the fact that its n-point
correlation functions are nonvanishing, since the standard obstruction to exact marginality
is the existence of a “potential” for such couplings. However, the standard discussion of
the obstruction to exact marginality does not apply because of the strange nature of the
operator product expansion. In later sections on string theory (sec. 5.6 ) we will see that
the unusual OPE of Liouville also has important consequences for the finiteness of the
theory and for the existence of an infinite dimensional space of background deformations.
3.9. Liouville Correlators from Analytic Continuation
In the past two years there has been very interesting progress in understanding Liouville
correlation functions via “analytic continuation in the number of operators.” The first
step in the calculation of continuum correlators was provided in [45], where the free field
formulation by zero mode integration of the Liouville field was established. The essential
idea is to treat the Liouville path integral measure as a free field measure and separate out
a zero-mode φ 0 via φ = φ 0 + ˆφ, so that [dφ] = [d ˆφ] dφ 0 . The integral over the zero mode is
∫ ∞
−∞
s ≡ 1 γ
∑
dφ 0 e
αi φ 0 e
−Qχφ 0
/2−Be γφ 0
(
1
2 Qχ − ∑ α i
)
= 1 Γ(−s) Bs
γ
B ≡ µ ∫ √ĝ
e
γ ˆφ .
8πφ 2
(3.60)
In references [46–49], it is proposed that when s ∈ Z + (so there is no negative curvature
solution) the ˆφ integral can be done using free field techniques. One then obtains a class
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