(g i )2 . (10.124)

9 6

i=1

The conifold has a conical singularity at r = 0. In fact, this would also

13 Compact Calabi“Yau three-folds do not have continuous isometry groups.

490 Flux compacti¬cations

be true for any choice of the ¬ve-dimensional base space other than a ¬ve-

sphere of unit radius. As was already mentioned, in the case of the conifold

there are two ways of smoothing out the singularity at the tip of the cone,

called deformation and resolution.

The deformed conifold

The deformation consists in replacing Eq. (10.117) by

4

(wA )2 = z, (10.125)

A=1

where z is a nonzero complex constant. Since w A ∈ 4 we can rescale

£

these coordinates and assume that z is real and nonnegative. This de¬nes

a Calabi“Yau three-fold for any value of z. As a result, z spans a one-

dimensional moduli space. At the singularity of the moduli space (z = 0)

the manifold becomes singular (at ρ = 0).

For large r the deformed conifold geometry reduces to the singular conifold

with z = 0, that is, it is a cone with an S 2 — S 3 base. Moving from ∞

towards the origin, the S 2 and S 3 both shrink. Decomposing w A into real

and imaginary parts, as before, yields

z = x · x ’ y · y, (10.126)

and using the de¬nition

ρ2 = x · x + y · y, (10.127)

shows that the range of r is

z ¤ ρ2 < ∞. (10.128)

As a result, the singularity at the origin is avoided for z > 0. This shows

that as ρ2 gets close to z the S 2 disappears leaving just an S 3 with ¬nite

radius.

The resolved conifold

The second way of smoothing out the conifold singularity is called resolution.

In this case as the apex of the cone is approached, it is the S 3 which shrinks

to zero size, while the size of the S 2 remains nonvanishing. This is also

called a small resolution, and the nonsingular space is called the resolved

conifold.

In order to describe how this works, let us make a linear change of variables

10.2 Flux compacti¬cations of the type IIB theory 491

to recast the singular conifold in the form

XU

det = 0. (10.129)

VY

Away from (X, Y, U, V ) = 0 this space is equivalently described as the space

XU »1

= 0, (10.130)

VY »2

in which »1 and »2 don™t both vanish. The solutions for »i are determined

up to an overall multiplicative constant, that is,

»∈

(»1 , »2 ) »(»1 , »2 ) with . (10.131)

£

As a result, the variables (X, Y, U, V ) and (»1 , »2 ) lie in 4 — P 1 and

£ £

satisfy the condition (10.130). This describes the resolved conifold, which

is nonsingular. Why is the singularity removed? In order to answer this

question note that for (X, Y, U, V ) = (0, 0, 0, 0) this space is the same as

the singular conifold. However, at the point (X, Y, U, V ) = (0, 0, 0, 0) any

solution for (»1 , »2 ) is allowed. This space is P 1 , which is a two-sphere.

£

Fluxes on the conifold

Let us now consider a ¬‚ux background of the conifold geometry given by

N space-time-¬lling D3-branes located at the tip of the conifold, as well as

M D5-branes wrapped on the S 2 in the base of the deformed conifold and

¬lling the four-dimensional space-time. These D5-branes are usually called

fractional D3-branes.

This background can be constructed by starting with a set of M D5-

branes, which give

F3 = 4π 2 ± M. (10.132)

S3

This can also be written as

M±

ω 3 = g 5 § ω2 ,

F3 = ω3 where (10.133)

2

and

1

ω2 = (sin θ1 dθ1 § dφ1 ’ sin θ2 dθ2 § dφ2 ) . (10.134)

2

In order to describe a supersymmetric background, the complex three-form

G3 should be an imaginary self-dual (2, 1)-form. This implies that an H3

¬‚ux has to be included. Imaginary self-duality determines the H3 ¬‚ux to be

3

gs M ± dr § ω2 ,

H3 = (10.135)

2r

492 Flux compacti¬cations

where gs = e¦ is the string coupling constant, which is assumed to be

constant, while the axion C0 has been set to zero. Once H3 and F3 are both

present, F5 is determined by the Bianchi identity

˜

dF5 = H3 § F3 + 2κ2 T3 ρ3 , (10.136)

to be

˜

F5 = (1 + 10 )F, (10.137)

where

1

F = π(± )2 Ne¬ (r)ω2 § ω3 (10.138)

2

and

3 r

gs M 2 log

Ne¬ (r) = N + . (10.139)

2π r0

Note that the total ¬ve-form ¬‚ux is now radially dependent, with

1

˜

F5 = (± )2 πNe¬ (r). (10.140)

2

Σ

The geometry in this case is a warped conifold, where the metric has the

form

ds2 = e2A(r) ·µν dxµ dxν + e’2A(r) (dr2 + r2 dΣ2 ). (10.141)

10

The metric of the base, dΣ2 , is given in Eq. (10.120). The volume form for

the metric in these coordinates is given by

√ 1