arXiv:hep-th/9304011 v1 Apr 5 1993

3.8. Operator Products in Liouville Theory
The existence of the Seiberg bound (3.40) dictates that the operator product expansion
in Liouville theory will be rather different from that in free field theory. Indeed it is not
clear that the notion of the operator product expansion is the correct language to use when
discussing Liouville correlators. Already in the semiclassical theory, we shall see that the
putative short distance behavior of correlators depends on global properties of the surface
and the “operator product expansion” appears to be nonlocal [7,8].
Fig. 4: Left: Insertion of operators e αφ and e βφ in a surface with boundary. Right:
the same operation on a higher genus surface.
Consider the state arising from the insertion of two vertex operators e αφ and e βφ on
a surface with boundary C as in the l.h.s. diagram of fig. 4, or, more generally, on a higher
genus surface as in the r.h.s. diagram of fig. 4. In the semiclassical approximation, the
state created by the surface on the boundary C has a wavefunction that depends on the
zero-mode φ 0 of φ as
ψ(φ 0 ) ∼ e αφ 0 +βφ 0 − 1 2 Qχφ 0 , (3.58)
where χ = 1 − 2h is the Euler character for the surface (with a single boundary).
α + β < 1 2 Qχ, then the wavefunction (3.58) is a real exponential diverging at φ 0 → −∞
(short distance). Then we may expect to replace the holes in the diagrams in fig. 4 each
by a sum of local (“microscopic”) operators, as in ordinary conformal field theory. Note
that since the three-point functions are generically nonzero, we would naively expect a
disastrous sum over operators with conformal dimensions unbounded from below.
If α + β > 1 2Qχ, on the other hand, then the state is normalizable and we certainly
cannot expand it in an operator product expansion of local operators. Instead the state
must be expanded in the normalizable macroscopic state operators. Thus, by sewing, the
surface amplitude must have the form
〈
e αφ(z,¯z) e βφ(z,¯z) · · ·〉 ∼
∫ ∞
0
dE c αβE |z − w| 2(E2 /2+Q 2 /8−∆ α −∆ β ) 〈E|Σ〉 , (3.59)
39
If