arXiv:hep-th/9304011 v1 Apr 5 1993

Finally, we must relate these nonperturbative S-matrices to string perturbation theory.
By double-scaling, the string perturbation series can be extracted by restoring the string
coupling µ → µ/κ and taking κ → 0 asymptotics. This is the same as taking µ → ∞
asymptotics holding p i or ω i fixed. Thus we need the asymptotic behavior of the bounce
factors. To all orders of perturbation theory, we can replace the expression (13.32) by the
simpler expression for the Euclidean bounce factor at p > 0:
Here the Q k are polynomials in p.
R p = (−iµ) −p Γ( 1 2
− iµ + p)
Γ( 1 2 − iµ) ∼ 1 +
∞∑
k=1
Q k (p)
µ k . (13.36)
13.6. Properties of S-Matrix Elements
From the above algorithm one can calculate any S-matrix element. Some general
properties of the S-matrix elements following from the above construction are the following.
First let us define some notation. By KPZ scaling, the Euclidean S-matrix elements
〈 k∏ 〉
T q + i
i=1
h
= µ 2h+k−1− 1 2
∑
|qi | F h (q 1 , . . . q k ) (13.37)
define certain functions F h (q 1 , . . . q k ) associated to the moduli spaces M h,k of curves with
h handles and k punctures. Defining different kinematic regions C α as in eq. (13.29), one
can then show:
Some Properties of perturbative amplitudes:
i) F h is parity-invariant: F h (q i ) = F h (−q i ).
ii) F h is continuous on { ∑ q i = 0} ∩ IR k
iii) In C α , F h is a polynomial in the momenta with rational coefficients. In general the
polynomial is different in different regions. That is, the expressions are continuous
but not continuously differentiable.
iv) The degree of the polynomial is 2k + 4h − 3.
v) As any momentum goes to zero, we have
〈 k∏
i=1
T + q i
〉
qi →0
∼
|q i | ∂ 〈∏
∂µ
j≠i
T + q i
〉
. (13.38)
vi) If q i ∈ Z, then F h = 0 for sufficiently large genus, specifically, for 2h − 2 + k > ∑ |q i |.
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