arXiv:hep-th/9304011 v1 Apr 5 1993

Those who look for special states in the singularities of the c = 1 S-matrix are like the
man who settled in Casablanca for the waters. They were misinformed.
The situation is rather more confused for the bulk–scattering matrix described in
chapt. 14.
5.6. Bosonic String Amplitudes and the “c > 1 problem”
In this section we consider some of the “tachyonic” divergences that occur in bosonic
string theories.
First Description
Let us return to the operator formalism description of string amplitudes. In general,
the amplitudes A h,n are meaningless because of the singularities of the string density Ω
on the boundaries of moduli space. A traditional way of avoiding this problem has been
the introduction of supersymmetry. An alternative way around the problem is provided
by low dimensional string theory[4], since in low dimensions the tachyon (which causes the
divergences) becomes massless or massive as we have seen in (5.16). We can see how this
comes about by considering the one-loop partition function in the example of a general
linear dilaton background (i.e. non-zero Q in (5.15)) coupled to some matter conformal
field theory C,
Ω 1,0 ∼ dτ η 2 ∧ d¯τ ¯η 2 Z Liouville⊗C (q, ¯q) . (5.28)
(Note the leading η 2¯η 2 is from the ghosts.) The behavior of the partition function as
q → 0, which accounts for the tachyon divergences of the theory, is obtained by writing
the partition function as a sum over eigenstates of L 0 , L 0 :
Z Liouville⊗C = ∑ i
∫ ∞
0
dE f i (E) (q¯q) 1 2 E2 + 1 8 Q2 +∆ i −26/24 , (5.29)
where f i (E) represents the density of Liouville states in (3.31). Including also the leading
(q¯q) 2/24 from the ghosts in (5.28), we arrive at the condition [7,73] for no tachyonic
divergences:
min ∆i ∈C
{
}
1
2 E2 + 1 8 Q2 + ∆ i − 1 ≥ 0 =⇒ c eff (C) ≡ c − 24 min∆ i ≤ 1 . (5.30)
From this point of view, we see that the problem is not necessarily that c > 1 per se, but
is rather an issue involving the value of c together with the spectrum of the theory.
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