form is primitive. On a six-torus h1,0 = 0 and there are harmonic (2, 1)-forms

that are not primitive. If h1,0 = 0 supersymmetry is unbroken if

(2,1)

G3 ∈ Hprimitive . (10.115)

Note that besides being primitive, the χ± are also imaginary self-dual. The

behavior of three-forms under the Hodge-star operation is displayed in the

table. Expressing the Levi“Civita tensor in the form

µabc¯qr = ’i (ga¯gb¯gc¯ ± permutations) (10.116)

pqr

p ¯¯

allows us to check these rules by the reasoning of Exercise 10.4. Then

Eq. (10.110) agrees with the condition that G3 is imaginary self-dual.

„¦ = ’i„¦

(3, 0) „¦

(2, 1) χ± χ± = iχ±

χ± = ’iχ±

(1, 2) χ±

¯ ¯ ¯

¯ ¯ ¯

(0, 3) „¦ „¦ = i„¦

An example: ¬‚ux background on the conifold

As discussed in Chapter 9, di¬erent Calabi“Yau manifolds are connected

by conifold transitions. At the connection points the Calabi“Yau manifolds

degenerate. This section explores further the behavior of a Calabi“Yau

manifold near a conifold singularity of its moduli space. By including these

singular points it is possible to describe many, and possibly all, Calabi“Yau

manifolds as part of a single connected web. In order to be able to include

these singular points, it is necessary to understand how to smooth out the

singularities. This can be done in two distinct ways, called deformation and

resolution.

Conifold singularities occur commonly in the moduli spaces of compact

Calabi“Yau spaces, but they are most conveniently analyzed in terms of

488 Flux compacti¬cations

S2

2

S

3

S3 S

S2

S3

Fig. 10.4. The deformation and the resolution of the singular conifold near the

singularity at the tip of the cone.

the noncompact Calabi“Yau space obtained by magnifying the region in

the vicinity of a singularity of the three-fold. This noncompact Calabi“

Yau space is called the conifold, and its geometry is given by a cone. This

section describes the space-time geometry of the conifold, together with its

smoothed out cousins, the deformed conifold and the resolved conifold. Type

IIB superstring theory compacti¬ed on a deformed conifold is an interesting

example of a ¬‚ux compacti¬cation. It is the superstring dual of a con¬n-

ing gauge theory, which is described in Chapter 12. Here we settle for a

supergravity analysis.

The conifold

At a conifold point a Calabi“Yau three-fold develops a conical singularity,

which can be described as a hypersurface in 4 given by the quadratic

£

equation

4

(wA )2 = 0 wA ∈ 4

for . (10.117)

£

A=1

This equation describes a surface that is smooth except at w A = 0. It

describes a cone with an S 2 — S 3 base. To see that it is a cone note that if

wA solves Eq. (10.117) then so does »w A , where » is a complex constant.

Letting w A = xA + iy A , and introducing a new coordinate ρ, Eq. (10.117)

can be recast as three real equations

1 1

x · x ’ ρ2 = 0, y · y ’ ρ2 = 0, x · y = 0. (10.118)

2 2

10.2 Flux compacti¬cations of the type IIB theory 489

√

S3

The ¬rst equation describes an with radius ρ/ 2. Then the last two

equations can be interpreted as describing an S 2 ¬bered over the S 3 . It can

be shown that a Ricci ¬‚at and K¨hler metric on this space is given by a cone

a

ds2 = dr2 + r2 dΣ2 , (10.119)

where r = 3/2ρ2/3 and dΣ2 is the metric on the ¬ve-dimensional base,

which has the topology S 2 — S 3 . Explicitly, the metric on the base can be

written in terms of angular variables

2 2

1 1

2

2

dθi + sin2 θi dφ2 .

2

dΣ = 2dβ + cos θi dφi + (10.120)

i

9 6

i=1 i=1

The range of the angular variables is

0 ¤ β ¤ 2π, 0 ¤ θi ¤ π 0 ¤ φi ¤ 2π,

and (10.121)

for i = 1, 2, while 0 ¤ r < ∞. This space has the isometry group SU (2) —

SU (2) — U (1).13

In order to describe this background in more detail, it is convenient to

introduce the basis of one-forms

1 1

g1 = √ (e1 ’ e3 ), g2 = √ (e2 ’ e4 ),

2 2

1 1

g3 = √ (e1 + e3 ), g4 = √ (e2 + e4 ), (10.122)

2 2

g 5 = e5 ,

with

e1 = ’ sin θ1 dφ1 , e2 = dθ1 ,

e3 = cos 2β sin θ2 dφ2 ’ sin 2βdθ2 ,

(10.123)

e4 = sin 2β sin θ2 dφ2 + cos 2βdθ2 ,

e5 = 2dβ + cos θ1 dφ1 + cos θ2 dφ2 .

In terms of this basis the metric takes the form

4

1 1

dΣ = (g 5 )2 +