arXiv:hep-th/9304011 v1 Apr 5 1993

in [156] it is shown that the “leg poles” in (14.11) may be interpreted in terms of on-shell
intermediate discrete states. The vanishing of the mixed chirality amplitudes is important
in their discussion. While this makes sense from the worldsheet point of view, the existence
in spacetime of (normalizable!) modes which are only physical at discrete momenta is quite
peculiar and has not been adequately interpreted.
14.3. Wall vs. Bulk Scattering
We finally discuss the relation of the B-matrix to the W -matrix, that is, we compare
amplitudes at µ = 0 with amplitudes at µ > 0. Since we cannot expand in µ we have no
right to expect a simple relation. Moreover, the perturbative W -matrix does not have a
good µ → 0 limit. Nevertheless, there is an interesting series of conjectures explored in
[47,48,49] on the relation between these S-matrices. We describe these here.
To compare, we must continue back to Euclidean space and impose the Seiberg bound
(3.40). Thus we only consider processes with T + k , k > 0, and T − k
, k < 0. The chirality
rule thus becomes the rule that amplitudes are generically zero unless all but one of the
momenta k i have the same sign. Without loss of generality, we take k 1 , . . . k N < 0, hence
s = 0 implies that k N+1 = N − 1. We now try to relate the µ = 0 and µ > 0 theories by
integrating over the Liouville zero mode as in sec. 3.9, splitting φ = φ 0 + ˜φ. This gives:
〈V · · · V 〉 = µ s Γ(−s) 〈V · · · V 〉 ˜φ,µ=0
. (14.12)
Since s = 0, the RHS is ill-defined, but the pole of the Γ-function has a nice physical
interpretation. Returning to fixed area correlators we see that it arises from an ultraviolet
A → 0 divergence. That is, in spacetime terms, a divergence from an integration over the
volume of the φ-coordinate. We may therefore regulate the theory and consider log µ the
regularized volume of the world,
µ s Γ(−s)| s=0 →
∫ ∞
ɛ
dA
A e−µA → log(µ ɛ) . (14.13)
To extract the residue of the s = 0 pole, we divide by the volume of φ-space. As mentioned
in sec. 3.9, the Liouville interaction is effectively zero in “most” of φ-space so we should be
able to treat φ as a free field in this regime and calculate the residue of the s = 0 pole with
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