b

Using the previous expressions for the K¨hler potential and the superpoten-

a

tial the potential takes the form

|‚ρ W |2

1 3¯ ¯

¯

2

|‚ρ W | ’

V= (W ‚ρ W + W ‚ ρ W ) + φφ. (10.314)

¯

12L2

6L 2L

As explained in Chapter 10, including the e¬ects of the anti-D3-brane gives

an additional term in the potential

|‚ρ W |2

1 3¯ D

¯

¯

2

|‚ρ W | ’

V= (W ‚ρ W + W ‚ ρ W ) + φφ + ,

¯

12L2 (2L)2

6L 2L

(10.315)

where D is a positive constant. This potential can be expanded about a

minimum in which ρ = ρc , and φ = 0. After transforming to a canonically

normalized ¬eld • = φ/ 3/(ρ + ρ), the potential can be written in the form

¯

V0 (ρc ) 2

≈ V0 (ρc ) 1 + •• .

V= ¯ (10.316)

(1 ’ ••/3)2

¯ 3

This potential leads to a slow-roll parameter · = 2/3, which again indicates

that no slow-roll in¬‚ation can be described in this scenario, at least not in

an obvious manner. Allowing a certain amount of ¬ne tuning of the inter-

brane distance would obviously solve this problem. As previously mentioned,

other alternatives based on in¬‚ation are currently explored in the literature.

Other approaches aim to propose an alternative to in¬‚ation such as brane

gases, time-dependent warped geometries, models based on Matrix theory

or models that make a connection to the dS/CFT correspondence. It is fair

to say that, even though it is an exciting prospect, the application of string

theory to cosmology is still at its early stages.

Appendix: Dirac matrix identities 543

Hybrid in¬‚ation and exit from in¬‚ation

One very attractive aspect of D3/anti-D3-brane in¬‚ation is that it provides a

natural mechanism to end in¬‚ation based on the hybrid in¬‚ation mechanism

previously discussed. The potential for the interbrane distance discussed so

far is valid for distances that are large compared to the string scale. Since the

force between the D3-brane and the anti-D3-brane is attractive, the branes

collide and annihilate with one another. This process is described in terms of

an additional ¬eld T , which corresponds to the tachyon of hybrid in¬‚ation.

For large brane separation, this ¬eld is massive. It becomes massless once

the branes come su¬ciently close to one another and tachyonic when they

annihilate. The form of the potential describing this process is the same as

the potential for hybrid in¬‚ation previously discussed:

V (φ, T ) = a (φ/ s )2 ’ b T 2 + cT 4 + V (φ), (10.317)

where a, b and c are positive constants. The collision of branes results in

the production of strings of cosmic size, which are called cosmic strings.

Even though they are not an inevitable prediction, the discovery of such

objects would be a spectacular way to verify string theory. Further progress

in string cosmology, together with more observational data, may someday

provide direct evidence of string theory.

Appendix: Dirac matrix identities

This appendix lists various identities satis¬ed by Dirac matrices. These have

been used in this chapter to analyze the conditions for unbroken supersym-

metry of ¬‚ux compacti¬cations.

[γm , γ r ] = 2γm r {γm , γ r } = 2δm r

[γmn , γ r ] = ’4δ[m r γn] {γmn , γ r } = 2γmn r

[γmnp , γ r ] = 2γmnp r {γmnp , γ r } = 6δ[m r γnp]

[γmnpq , γ r ] = ’8δ[m r γnpq] {γmnpq , γ r } = 2γmnpq r

[γmnpqk , γ r ] = 2γmnpqk r {γmnpqk , γ r } = 10δ[m r γnpqk ]

544 Flux compacti¬cations

[γmn , γ rs ] = ’8δ[m [r γn] s] {γmn , γ rs } = 2γmn rs ’ 4δ[mn] rs

[γmnp , γ rs ] = 12δ[m [r γnp] s] {γmnp , γ rs } = 2γmnp rs ’ 12δ[mn rs γp]

[γmnpq , γ rs ] = ’16δ[m [r γnpq] s] {γmnpq , γ rs } = 2γmnpq rs ’ 24δ[mn rs γpq]

[γmnpqk , γ rs ] = 20δ[m [r γnpqk] s] {γmnpqk , γ rs } = 2γmnpqk rs ’ 40δ[mn rs γpqk]

[γmnp , γ rst ] = 2γmnp rst ’ 36δ[mn [rs γp] t]

[γmnpq , γ rst ] = ’24δ[m [r γnpq] st] + 48δ[mnp rst γq]

[γmnpqk , γ rst ] = 2γmnpqk rst ’ 120δ[mn [rs γpqk] t]

{γmnp , γ rst } = 18δ[m [r γnp] st] ’ 12δ[mnp] rst

{γmnpq , γ rst } = 2γmnpq rst ’ 72δ[mn [rs γpq] t]

{γmnpqk , γ rst } = 30δ[m [r γnpqk] st] ’ 120δ[mnp rst γqk]

[γmnpq , γ rstu ] = ’32δ[m [r γnpq] stu] + 192δ[mnp [rst γq] u]

[γmnpqk , γ rstu ] = 40δ[m [r γnpqk] stu] ’ 480δ[mnp [rst γqk] u]

{γmnpq , γ rstu } = 2γmnpq rstu ’ 144δ[mn [rs γpq] tu] + 48δ[mnpq] rstu

{γmnpqk , γ rstu } = 2γmnpqk rstu ’ 240δ[mn [rs γpqk] tu] + 240δ[mnpq rstu γk]

[γmnpqk , γ rstuv ] = 2γmnpqk rstuv ’ 400δ[mn [rs γpqk] tuv] + 1200δ[mnpq [rstu γk] v]

{γmnpqk , γ rstuv } = 50δ[m [r γnpqk] stuv] ’ 1200δ[mnp [rst γqk] uv] + 240δ[mnpqk] rstuv

Homework Problems 545

In general,

[γm1 ...mp , γ n1 ...nq ] pq odd

= 2γm1 ...mp n1 ...nq

{γm1 ...mp , γ n1 ...nq } pq even

2p!q!

’ 2!(p’2)!(q’2)! δ[m1 m2 [n1 n2 γm3 ...mp ] n3 ...nq ]

2p!q!

+ 4!(p’4)!(q’4)! δ[m1 ...m4 [n1 ...n4 γm5 ...mp ] n5 ...nq ]

’...

and

[γm1 ...mp , γ n1 ...nq ] pq even (’1)p’1 2p!q! [n1 γ n2 ...nq ]

= 1!(p’1)!(q’1)! δ[m1 m2 ...mp ]

{γm1 ...mp , γ n1 ...nq } pq odd