arXiv:hep-th/9304011 v1 Apr 5 1993

More generally, we may calculate the one macroscopic loop amplitude for general
perturbed (2, 2m − 1) models coupled to gravity along the following lines. The genus zero
limit of the string equation (7.43) can be written
∑
t j u j = 0 . (10.8)
j≥0
(Note µ = −t 0 . Recall that the t j describe the coupling to the various scaling operators
and if the largest nonvanishing t j has j = m, we are describing 2D gravity coupled to the
(2, 2m − 1) minimal model.) Using (10.8), can explicitly evaluate the loop amplitude as
〈
W (l)
〉h=0 = 1 ∫ ∞
dx e −lu(x;ti) = 1 ∫ ∞
dy ( ∑
∞ j t
κl 1/2 −t 0
κl 1/2 j y j−1) e −ly
u j=1
= 1 ∞∑
j t j u j−1/2 ψ
κl
j−1 (˜l) ,
j=1
(10.9)
where
ψ j (x) ≡ j! x −j−1/2 (1 + x + x 2 /2! + · · · x j /j!) e −x , (10.10)
and ˜l ≡ ul. (We assume here that all but finitely many t j are nonzero.) The relation of
these amplitudes to the Bessel functions of the continuum theory is less evident than in
(10.7) and will be explained in sec. 10.3 .
Similarly, we can study the genus zero approximation to the propagator as
〈
W (l1 )W (l 2 ) 〉 h=0 = e−(l 1+l 2 )u
√
l1 l 2 κ 2
∫ ∞ ∫ −t0
dx
−t 0 −∞
= 2 √ l 1 l 2
e −u(µ)(l 1+l 2 )
l 1 + l 2
.
dy e −(x−y)2 (l 1 +l 2 )/(4κ 2 l 1 l 2 )
(10.11)
Exercise.
Prove (10.11) by inserting (10.3) into (10.2). Similarly, try to prove
〈 W (l1 ) W (l 2 ) W (l 3 ) 〉 = 2 ∂u
∂t 0
√
l 1 l 2 l 3 e −u(l 1+l 2 +l 3 ) . (10.12)
(We will prove this more efficiently below.)
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