arXiv:hep-th/9304011 v1 Apr 5 1993

sense if we rotate φ → it (as we may when µ = 0), and regard X as a spatial variable. We
are therefore discussing theory B of sec. 5.4. As explained there, the vertex operators are
given by (5.24) and we have energy and momentum conservation laws for the amplitude
〈 ∏ T + ∏
k i
T
−
pi
〉 given by
∑
ki + ∑ p i = 0
s = 2 − ∑ (1 + 1 2 k i) − ∑ (1 − 1 2 p i) = 0 .
(14.9)
Standard vertex operator calculations now give
〈∏ ∏ 〉 ∫
T
+
k i
T
−
pi
=
N
∏
i=4
∏
d 2 z i |z ij | −2s ij
, (14.10)
where we take the three points at 0, 1, ∞ as usual, s ij = β i β j − 1 2 k ik j , β = √ 2 + k/ √ 2
for T + k , and β = √ 2 − p/ √ 2 for T − p .
i 0, and p i + p j > 1, then the integral (14.10) is convergent and well-defined, and
results in
〈T + k
N∏
Tp − i
〉 =
i=1
N∏ π N−2
∆(m i )
(N − 2)! . (14.11)
i=1
This has been shown in [49,48] by analytic arguments and in [155] by an elegant algebraic
technique. Note in particular that:
1) p i , k i ∈ IR. We have put µ = 0 so there is no longer any rationale to impose the
Seiberg bound (3.40).
2) As in 26 dimensions, we can continue to other momenta for which the integral representation
does not converge. Then there are poles, but in this case they occur for
p i = 1, 2, . . . . These are known as the “leg poles.”
3) We already see a remarkable difference between D = 2 and D > 2 strings since in
general there is no simple closed formula for (14.10) for N > 4.
Let us now consider other combinations of chiralities. We find a new surprise. Because
of kinematic “coincidences”, one cannot define the integrals, even by analytic continuation,
since one is always sitting on top of a Γ-function pole or zero. Indeed it has been argued
in [48,49] that these amplitudes are zero, at least for generic external momenta.
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