Warped Conifolds and Cosmology

Dariush KavianiIPM school of Particles and Accelerators

Talk prepared for the 3rd IPM School and Workshopon Applied AdS/CFT February 22-28, 2014

Table of Contents

Warped string compactifications from AdS/CFT

Supergravity perturbations and moduli stabilization

Application to D-brane inflation

From D-branes to AdS/CFT

As our starting point, we consider a stack of N D3-branes.

A stack of N D3-branes realizes N = 4 supersymmetricSU(N) gauge theory in 4D.

It also generates a curved 10D background of closed type IIBsuperstring theory with the metric given as follows:

ds2 =

1+

L4

r4

−1/2(−(dx0)2+(dx i )2)+

1+

L4

r4

1/2(dr2+r

2dΩ2

5).

For small r the second part in the metric becomes moreimportant and the geometry approaches AdS5 × S

5.

The AdS/CFT correspondence

The AdS/CFT duality relates 4D conformal gauge theory tostring theory on AdS5 space times a 5D compact space. In thecase of N = 4 SYM theory this compact space is a 5D sphere[Maldacena; Gubster, Klebanov, Polyakov; Witten (1998)].

The geometrical symmetry of the AdS5 space realizes theconformal symmetry of the gauge theory.

The AdSd space is a hyperboloid given by

(X 0)2 + (X d)2 −d−1

i=1

(X i )2 = L2.

with the metric of the form

ds2 =

L2

z2

dz

2 − (dx0)2 +d−2

i=2

(dx i )2.

Remarks on symmetries

The low energy physics of D3-branes on a Calabi-Yau manifoldis conformally invariant and N = 4 supersymmetric.

Calabi-Yau compactifications give rise to many moduli. Inorder to fix the moduli it is necessary to break conformalinvariance and most of the supersymmetry.

Exactly the same issue arises in AdS/CFT. String theory onAdS5 × S

5 is dual to N = 4 SYM theory. To find string dualsof gauge theories with confinment and chiral symmetrybreaking one must reduce the symmetry; in the SUGRAcontext this generates potential which can fix some moduliand stabilize a hierarchy.

A simple way of reducing the symmetry is to place theD3-branes not at a smooth point of the transverse spacebut at a Calabi-Yau singularity.

Cone-Brane Dualities

To reduce the number of supersymmetries, we put the parallelD3-branes at a conical Calabi-Yau singularity.

Conical singularities

By a conical singularity on an n-dimensional manifold Yn, wemean a point (which we will label as r = 0) near which themetric can locally be put in the form

gmndxmdx

n = dr2 + r

2gijdx

idx

j (i , j = 1, ..., n − 1).

Here gij is a metric on an n − 1-dim. base manifold Xn−1.

The basic property of this metric, which makes it “conelike”,is that there is a group of diffeomorphisms of Yn that rescalethe metric.

This group is r → λr with λ > 0, and thus isomorphic to R∗+.

One calls Yn a cone over the base manifold Xn−1.

For n > 2, the condition that Yn is Ricci-flat is that that Xn−1

is an Einstein manifold of positive curvature.

D3-branes at conical singularities

We are interested in n = 6. We take spacetime to beR3,1 × Y6 , with R3,1 being 4D Minkowski and Y6 as above.

Thus to reduce the number of supersymmetries in AdS/CFT,we may place the stack of N D3-branes at the tip of a 6DRicci-flat cone Y6 whose base is a 5D Einstein manifold X5.

Taking the near-horizon (r → 0) limit of the backgroundcreated by the N D3-branes, one finds the space AdS5 × X5,where X5 is the 5D Einstein manifold bearing N units of R–R5-form flux and the length is now given by

L4 =

√πκN

2Vol(X )= 4πgsNα π3

Vol(X ).

This type IIB background is conjectured to be dual to the IRlimit of the gauge theory on N D3-branes at the tip of thecone Y6 [Kachru, Silverstein; Lawrence, Nekrasov, Vafa; ...].

In the spirit of Maldacena’s conjecture [Maldacena (1998)],we may identify the field theory on the D3-branes at a conicalsingularity as the dual of Type IIB string theory on AdS5 ×X5.

We now argue that the field theory is N = 1 superconformal,instead of N = 4 supersymmetric.

Remarks on the supersymmetries

The AdS5 × X5 differs from the N = 4 SU(N) gauge theory.

This is clear from the fact that only S5 preserves the maximal

number of supersymmetries (namely 32) whereas otherEinstein manifolds lead to reduced supersymmetries.

Counting supersymmetries

The condition for an unbroken supersymmetry is that thereshould exist a nontrivial solution to the Killing spinor equation

∇Mε = 0, ε(x , y) = ξ(x)⊗ η(y).

Here ε is four-component Majorana spinor.

The Killing spinor equation in the conical metric hmn is

(∂m +1

4ωmabΓ

ab)η = 0.

Evaluating this in the metric we find [Kehagias (1998)]

(∂i +1

4ωijkΓ

jk +1

2Γri )η = 0.

This is equivalent to the X5 part of the Killing spinor equationin Type IIB compactification on AdS5 × X5, including theeffects of the 5-form field strength (encoded in Γri = Γisns).

Therefore the number of unbroken supersymmetries on X5 isthe same as the number of unbroken supersymmetries on the6D cone.

If the cone is a manifold of SU(3) holonomy (a CY 3-fold),then there are 8 unbroken supersymmetries–4 for left-moversand 4 for right-movers.

These cases correspond to N = 1 superconformal fieldtheories (SCFT) in 4D, i.e. we may construct such fieldtheories as the infrared limits of the theories on D3-branesplaced at Calabi-Yau singularities.

The Klebanov-Witten background

A prototypical example of a generic Calabi-Yau singularity isthe conifold point!

So we put a stack of N D3-branes on the conifold, and obtaina supergravity solution dual to the N = 1 SCFT in 4D[Klebanov, Witten (1998)].

The Conifold

The conifold is a Calabi-Yau 3-fold cone Y6 defined as ahypersurface in C4 by the simple equation[Candelas, de la Ossa (1990)]

4

i=1

z2i = 0.

The base is a coset space T1,1 =

SU(2)× SU(2)

/U(1),

which has SU(2)× SU(2)× U(1) isometry.

S 3

S 3

S 3

S 3

S 2

S 2

S 2

S 2

small resolutiondeformation

Figure: The singular conifold (middle) and its blow-ups, the resolvedconifold (right) and the deformed conifold (left).

The toplology of T 1,1 is S3 × S2, and the Sasaki-Einstein

metric on T1,1 is [Candelas, de la Ossa (1990)]!

ds2T 1,1 =

1

9

dψ+

2

i=1

cos θidϕi

2+

1

6

2

i=1

dθ2i +sin2 θidϕ

2i

2,

where θi ∈ [0,π], ϕi ∈ [0, 2π], ψ ∈ [0, 4π].

The gravity side of KW

The gravity side (of the dual pair) is then given by theAdS5 × T

1,1 geometry.

According to our counting of supersymmetries, thiscompactification should be dual to an N = 1 SCFT in 4D.

The dual gauge theory of KW

The dual gauge theory is given by the worldvolume theory onN D3-branes at the conifold geometry.

The N = 1 SCFT on N D3-branes at the apex of the conifoldhas gauge group SU(N)× SU(N) coupled to bifundamentalchiral superfields A1, A2 in (N, N), and B1, B2 in (N,N).

The bifundamental fields interact though a uniqueSU(2)AxSU(2)B invariant superpotential given as:

W = ijkl trAiBkAjBl .

This theory also has a baryonic U(1): Ak → eiaAk ; Bl → e

−iaBl .

Checking the duality

To check the duality, match the moduli.

In order to match the two couplings to the moduli of the typeIIB theory on AdS5 × T

1,1, one notes that the integrals overthe S

2 of T 1,1 of the NS-NS and R-R 2-form potentials, B2

and C2 , are moduli. In particular, the two gauge couplings aredetermined as follows [KW (1998); Morrison, Plesser (1999)]

g−21 + g

−22 ∼ e

−φ, g−21 − g

−22 ∼ e

−φ

S2B2

− 1/2

.

The two gauge couplings do not flow, and indeed can bevaried continuously without ruining conformal invariance!

The Klebanov-Strassler background

Indeed, placing the D3-branes at the conifold point reduces thesupersymmetry to N = 1. But this does not break conformalinvariance, so it is necessary to add ‘fractional’ branes localized atthe conifold singularity [Gubster, Klebanov (1998)].

Breaking conformal invariance

To break conformal invariance, add to the N D3-branes MD5-branes wrapped over the (collapsed) S2 at the tip of theconifold; these D5-branes are known as fractional D3-branes[Klebanov, Nekrasov (1999)].

The SUGRA dual of this field theory has then M units of the3-form flux, in addition to N units of the 5-form flux:

S3F3 = M,

T 1,1F5 = N.

In the SUGRA description the 3-form flux is the source ofconformal symmetry breaking. Indeed, now the 2-form fieldB2 cannot be kept constant and acquires a radial dependence[Klebanov, Nekrasov (1999)].

S2B2 ∼ Me

φ ln(r/r0).

This results a new nonsingular SURGRA solution [Klebanov,Strassler (2000)] whose dual gauge theory (discussed below)has a relative gauge coupling which runs logarithmically[Klebanov, Nekrasov (1999)].

The gauge theory of KS

The addition of M fractional branes at the singular pointchanges the gauge group to SU(N +M)× SU(N).

The four chiral superfields A1,A2,B1,B2 remain, now in therepresentation (N+M,N) and its conjugate (N+M,N), asdoes the superpotential W = ijkl trAiBkAjBl .

Since the AdS5 radial coordinate r is dual to the RG scale[Maldacena; Gubser, Klebanov, Polyakov; Witten (1998)],the relation

g−21 −g

−22 ∼ e

φ

S2B2

−1/2

with

S2B2 ∼ Me

φ ln(r/r0)

implies a logarithmic running of the relative gauge coupling,g−21 − g

−22 , in the SU(N +M)× SU(N) gauge theory.

It has been shown (by Klebanov-Strassler) that such gaugetheories have an exact anomaly-free Z2M R-symmetry, whichis broken dynamically, as in pure N = 1 YM theory, to Z2.

In the SUGRA, this occurs through the deformation of thesingular conifold.

The 10D geometry of KS

The 10D geometry dual to the gauge theory on these branesis the warped deformed conifold, or the KS solution.

The deformed conifold is defined by the constraint:

4

i=1

z2i = 2.

The warping in this background is induced by type IIBbackground fluxes and the warped metric takes the form

ds210 = h(y)1/2 gµνdx

µdx

ν

4D

+h(y)−1/2gmndy

ndy

m

6D CY

.

The 6D metric on the deformed conifold takes the form[Candelas and de la Ossa (1990)]:

gmndymdy

n =1

24/3K (η)

1

3K (η)3dη2 + (g5)2+ cosh2

η

2(g3)2

+(g4)2+ sinh2η

2(g1)2 + (g2)2

.

The gi ’s are forms representing the angular directions, given by

g1,3 =

e1 ∓ e

3

√2

, g2,4 =

e2 ∓ e

4

√2

, g5 = e

5

e1 = − sin θ1dϕ1, e

2 = dθ1, e3 = cosψ sin θ2dϕ2 − sinψdθ2,

e4 = sinψ sin θ2dϕ2 + cosψdθ2, e

5 = dψ + cos θ1dϕ1 + cos θ2dϕ2.

The warp factor takes the form [Klebanov, Strassler (2000)]:

hKS = 2(gsMα)2 −8/3I (η),

I (η) ≡ ∞

ηdx

x cosh x − 1

sinh2 x(sinh x cosh x − x)1/3.

The string tension, is proportional to h(y)−1/2 and isminimized at y = 0. It blows up at large y (near theboundary) where space is ‘near-AdS’.

Flux compactification of type IIB theory

The KS SUGRA solution is an as example of a Calabi-Yau fluxcompactification of IIB theory [GKP (2002)].

The type IIB action takes the form:

S IIB =1

2κ210

d10x

|g |

R− | ∂τ |2

2(Imτ)2− |G3|2

12 Imτ− | F5|2

4 · 5 !

+1

8iκ210

C4 ∧ G3 ∧ G

∗3

Im(τ)+ S loc, with G3 = F3 − τH3.

The metric and self-dual 5-form read as

ds210 = h(y)1/2 gµνdx

µdx

ν

4D FRW

+h(y)−1/2gmndy

ndy

m

6D CY

,

F5 = (1 + 10)dα(y) ∧ dx

0 ∧ dx1 ∧ dx

2 ∧ dx3.

The supergravity equations of motion [GKP (2002)]

The Einstein equations and 5-form Bianchi identity imply

∆(0)Φ− =h2 .eΦ

24|G−|2 + h

−3/2|∇Φ−|2

+2κ2h1/21

4(Tm

m − Tµµ )

local − T3ρloc3

,

G± ≡ (i ± 6)G3, Φ± ≡ h ± α.

The equation of motion for the three-form flux is

dΛ+i

2

dτ

Im(τ)∧ (Λ+ Λ) = 0,

where by definition

Λ ≡ Φ+G− + Φ−G+.

The imaginary self-dual (ISD) conditions [GKP (2002)]

The LHS of the above eq. integrates to zero, so the globalconstrains for the supergravity solution are:• The three-form flux is imaginary self-dual,

6G3 = iG3.

• The warp factor and four-form potential are related,

h = α.

• The localized sources saturate a ‘BPS-like’ bound

1

4(Tm

m − Tµµ )

local = T3ρloc3 .

A compactification satisfying these conditions is called ISD!

Application to D-brane inflation

The ISD compactification In an ISD compactification fluxes generate a warped throat

which smoothly closes off in the IR and is glued to thecompact CY space at the UV-end.

D3-branes sit at the IR location due to attractive forces, andmoduli stabilizing D7-branes enter the throat from the UV endwrapping certain four-cycles bearing nonperturbative effects.

Embed a mobile D3-brane in the ISD flux compactificationand study its motion in the deep throat region.

D-brane inflation in the ISD compactification

The effective action for the D3-brane takes the form:

SD3 = −g−1s

d4x√−g

T3h(y)

−1(γ−1DBI − 1) + V (φm)

,

γ−1DBI =

1− h(y)gmng

µν∂mφm∂nφn/T3.

Inflation occurs as a result of D3-brane motion in a stronglywarped region of the throat background [Silverstein, Tong(2004); Alishahiha, Silverstein, Tong (2004)]:

In a strongly warped region h(y) 1 the kinetic energy’spre-factor h(y)−1 term suppresses it relative to V (φm) evenwhen the motion is relativistic and the potential is steep.

Multifields arise naturally in DBI inflation, but it is NOTexplained where do they come from. Only few models explainthis and compute their effects [Gregory, Kaviani (2012);Kaviani (2013)]!

D-brane inflation and moduli stabilization

Moduli stabilization

CY compactifications include complex and Kahler moduli[Candelas, de la Ossa (1991)].

In CY flux compactifications of type IIB theory ISD fluxesstabilize the complex moduli but not the Kahler moduli[Giddings, Kachru, Polchinski (2002)].

Kahler moduli can be stabilized by nonperturbative effects(e.g. on wrapped D7-branes) [Kachru, Kallosh, Linde, Trivedi(2003)], the ‘KKLT’ scenario.

Kahler moduli stabilization breaks the no-scale structure whichinduces pertubations around the ISD SUGRA sol.[Baumann,Dymarsky, Kachru, Klebanov, McAllister (2010)].

Corrections to the inflaton potential

The pertubations around the ISD SUGRA sol. are describedby the SUGRA eq. of motion [Baumann, Dymarsky, Kachru,Klebanov, McAllister (2010)]

∆(0)Φ− =gs

96|Λ|2 + R4.

The potential of the inflationary D3-brane moving in this inthis background receives corrections ∆V of the form[Baumann, Dymarsky, Kachru, Klebanov, McAllister (2010)]:

∆V = T3(h(y)− α) ≡ T3Φ−,

Φ−(r ,Ψ) =

LM

ΦLM fLM(r)YLM(Ψ).

⇒ Compactification effects induce harmonic dependentcorrections to the inflaton action!

Corrections to the D3-brane potential: The UV case

In the UV the supergravity perturbations take the form[Baumann, Dymarsky, Kachru, Klebanov, McAllister (2010)]:

Φ−(r , θ) =

M

ΦLMYLM(Ψ)

r

rUV

∆(L)

,

∆(L) ≡ −2 +6[l1(l1 + 1) + l2(l2 + 1) + R2/8] + 4.

The radial scaling dimensions computed from AdS/CFT read[Baumann, Dymarsky, Kachru, Klebanov, McAllister (2010)]:

∆ = 1,3

2, 2s , 2,

5

2, 3, ...

Corrections to the D3-brane potential: The UV/IR case

At leading order, the corrected UV/IR consisted D3-branepotential reads [Gregory, Kaviani (2012); Kaviani (2013)]:

T3V = T3

1

2m

20 [r(η)2 + c2K (η) sinh η cos θ] + V0

+VF + VD ,

VF + VD =2κ2a2n|A0|2e−2anσ∗(η,θ)

U[η,σ∗(η, θ)]2|g(η, θ)|2/n

×U[η,σ∗(η, θ)]

6+

1

an

1− |W0|

|A0|eanσ∗(η,θ)

g(η, θ)1/n

+F (η, θ)

+

Duplift

U[η,σ∗(η, θ)]2.

The D3-brane equations of motion

The simplest multifield (DBI) D3-brane equations of motiontake the form [Gregory, Kaviani (2012); Kaviani (2013)]:

H2 =

E

3M2pl

, H = −(E + P)

2M2pl

,

η = − 3H

γ2DBI

η +h

γDBIhη2(1− γDBI) +

h

2h2A(γ−1

DBI − 1)2

− 1

2A(Aη2 − B

θ2) + hθηVθ

γDBIT3− (1− hAη2)

Vη

γDBIAT3

−ηθ(1− γ−1DBI)

hθ

h,

θ = −3H θ

γ2DBI

+ (1− γDBI)θηh

γDBIh− θη

B

B+ hθη

Vη

γDBIT3

−(1− hB θ2)Vθ

γDBIBT3− (1− γ−1

DBI)

θ2 −

(1− γ−1DBI)

2hB

hθ

h.

0

20

40

60

80

t

0

1 109

2 109

3 109

4 109

t

ΓDBI

Figure: The number of e-foldings and the γDBI-factor with and withoutharmonic dep. corrections [Gregory, Kaviani (2012); Kaviani (2013)].

Summary and outlook

We constructed warped throat geometries form AdS/CFT.

These warped string compactifications provided us withsuitable inflationary backgrounds.

We have discussed how inflation can occur in such warpedstring solutions.

We saw that moduli stabilization induces harmonicdependence which can increase the inflationary capacity.

However, we also saw that the level of nongaussianity in DBIbrane inflation remains too large.

The next step would be to see if certain corrections maydecrease the level of nongaussianity.

Thank you!