String Theory and M-Theory

10.2 Flux compactifications of the type IIB theory 497
near the tip of the cone, the flux solution is similar to the one described in
the previous section. Globally, however, the background solution must be
changed, since we are interested in a compact solution. The conifold solution
presented in the previous section is noncompact with r unbounded. This
can be interpreted as a singular limit of a compact manifold in which one of
the cycles degenerates to infinite size.
Let us assume that there are M units of F3 flux through an A-cycle and
−K units of H3 flux through a B-cycle, that is,
1
2πα ′
F3 = 2πM and
A
1
2πα ′
B
H3 = −2πK. (10.160)
Using Poincaré duality, the superpotential can then be written as
W = G3 ∧ Ω = (2π) 2 α ′
M Ω − Kτ Ω , (10.161)
The complex coordinate describing the cycle collapsing at the tip of the
conifold is
z = Ω. (10.162)
The discussion of special geometry in Section 9.6 explained that the dual
coordinates, that is, the coordinates defining periods of the B-cycles, are
functions of the periods of the A-cycles. More concretely, since we are describing
a conifold singularity, we can invoke the result derived in Section 9.8
that
Ω = z
log z + holomorphic. (10.163)
2πi
B
Using these results, the Kähler covariant derivative of the superpotential
can be rewritten in the form16 DzW (2π) 2 α ′
M
log z − iK + . . .
(10.164)
2πi gs
in the limit in which K/gs is large. The equation DzW = 0 is solved by
A
B
A
z e −2πK/Mgs . (10.165)
Thus, one obtains a large hierarchy of scales if, for example, M = 1 and
K/gs = 5. It is assumed that the dilaton is frozen in this solution.
The solution for the warp factor can be estimated in the following way. As
16 This assumes z ≪ 1, which is the case of interest.