arXiv:hep-th/9304011 v1 Apr 5 1993

To determine ∆, we employ the same scaling argument that led to (2.22). We shift
ϕ → ϕ + ρ/γ with e ρ = A on the right hand side of (2.24), to find
F Φ (A) = A−Qχ/2γ−1+α/γ
A −Qχ/2γ−1 F Φ (1) = A α/γ F Φ (1) ,
where the additional factor of e ρα/γ = A α/γ comes from the e αφ gravitational dressing of
Φ 0 . The gravitational scaling dimension ∆ defined in (2.24) thus satisfies
∆ = 1 − α/γ . (2.25)
Solving (2.23) for α with the same branch used in (2.19),
α = 1 2 Q − √
1
4 Q2 − 2 + 2∆ 0 = 1 √
12
(√
25 − D −
√
1 − D + 24∆0
)
(2.26)
(for which α ≤ Q/2, and α → 0 as D → −∞). Finally we substitute the above result for
α and the value (2.19) for γ into (2.25), and find 7
∆ =
√ 1 − D + 24∆0 − √ 1 − D
√
25 − D −
√
1 − D
. (2.27)
We can apply these results to the (p, q) minimal models [21] mentioned after (2.22).
These have a set of operators labelled by two integers r, s (satisfying 1 ≤ r ≤ q − 1, 1 ≤
s ≤ p − 1) with bare conformal weights ∆ 0 = ( (pr − qs) 2 − (p − q) 2) /4pq (we take p > q).
Coupled to gravity, these operators have dressed Liouville exponents
1 − ∆ r,s = α r,s
γ
p + q − |pr − qs|
=
2q
1 ≤ r ≤ q − 1, 1 ≤ s ≤ p − 1 (2.28)
(p − q)2
(note also that c = 1 − 6
pq
=⇒ γ =
√
2q
p ,
√ √
2p 2q
Q = q + p
)
,
in agreement with the weights determined from the (p, q) formalism (to be discussed in
sections 7.4 and 7.7 ) for the generalized KdV hierarchy (see e.g. [34,35]).
7 We can also substitute α = γ(1 − ∆) from (2.25) into (2.23) and use − 1 γ(γ − Q) = 1 (from
2
before (2.19)) to rederive the result ∆−∆ 0 = ∆(1−∆)γ 2 /2 for the difference between the “dressed
weight” ∆ and the bare weight ∆ 0 [33].
22