arXiv:hep-th/9304011 v1 Apr 5 1993

The condition (5.30) is of course only a necessary condition. We should also worry
about the existence of divergences when operators approach each other. In this case the
softening of the Liouville operator product expansion discussed above explains the lack of
divergences on the boundaries of punctured moduli space. In particular, if we look at the
operator product of two dressed matter primaries Φ 1 e αφ and Φ 2 e βφ , then from (3.59) we
have
Φ 1 e αφ (z, ¯z) Φ 2 e βφ (w, ¯w)
∼ ∑ ∫ ∞
dE c 1,2,(X,E) |z − w| 2( 1 2 E2 + 1 8 Q2 +∆ X −2) Φ ∆X V E (w, ¯w)
∆ X
0
(5.31)
(where c 1,2,(X,E) is the coefficient of the field Φ ∆X
and its gravitational dressing V E (w, ¯w)
in the operator product expansion of the two above operators). The worst singularity at
z = w comes from the contribution near E = 0,
1
|z − w| 2 |z − w| 1 12 (1−c eff (C)) ,
and is integrable when the condition (5.30) is satisfied. (The case c eff (C) = 1 is a borderline
case. In the c = 1 model, it turns out that c 1,2,E → 0 as E → 0.)
Based on these two examples, we may guess that all bosonic string amplitudes in fact
do exist when (5.30) is satisfied.
The matrix model approach to 2D string theory has
the great virtue of confirming this, and moreover gives an infinite dimensional space of
background perturbations.
Second Description
We can also describe these divergences from the point of view of the spacetime theory
by interpreting the norm of the plumbing fixture coordinate q as |q| = e −s , where s is a
proper time coordinate such as introduced following (5.3) for the field theory propagator.
From this point of view, we see that the divergences are due to on-shell tachyons and
massless particles. When (5.30) is satisfied as a strict inequality, we see that the amplitudes
are finite because only zero-momentum massive particles flow.
particles present a special case at c = 1, but they are derivatively coupled.
73
As usual, the massless