M. Zagermann - The Backreaction of Localized Sources and de Sitter Vacua

The Backreaction of Localized Sources and de Sitter Vacua

Marco Zagermann(Leibniz Universität Hannover & QUEST)

Donji Milanovac, August 29, 2011

Montag, 29. August 2011

Based on:

Blåbäck, Danielsson, Junghans, Van Riet, Wrase, MZ, w. i. p. Blåbäck, Danielsson, Junghans, Van Riet, Wrase, MZ (2011)Blåbäck, Danielsson, Junghans, Van Riet, Wrase, MZ (2010)

Wrase, MZ (2010)Caviezel, Wrase, MZ (2009)

Caviezel, Koerber, Körs, Lüst, Wrase, MZ (2008) Caviezel, Koerber, Körs, Lüst, Tsimpis, MZ (2008)

As well as


Outline

1. Smearing D-branes and O-planes

2. Classical de Sitter vacua

3. Smearing in the BPS-case I

4. Smearing in the BPS-case II

5. Smearing in the non-BPS case

6. Conclusions


1. Smearing D-branes and O-planes


D-branes & O-planes are important ingredients in phenomenologically realistic type II compactifications:



D-brane O-plane

T > 0 T < 0Tension:Montag, 29. August 2011


E.g.

• Chiral matter



E.g.

• Chiral matter

• Supersymmetry breaking



...

E.g.

• Moduli stabilization

• Chiral matter

• Supersymmetry breaking

( →Tadpole cancellation, non-pert. effects, etc.)


But: Dp-branes and Op-planes...

• ... have mass

→ Backreaction on metric (e.g. warp factor)



• ... have mass

• ... carry RR-charge


→ Source RR-potentials



• ... have mass

• ... carry RR-charge

• ... couple to the dilaton


→ Source RR-potentials

→ Nontrivial dilaton profile (except for p = 3)


x

Profile of warp factor, dilaton or RR-pot.

D-brane or O-plane


This backreaction is absent if all brane masses and charges are cancelled locally by putting the right number of D-branes on top of O-planes



O6

2 D6






In all other cases:

• take backreaction into account

or

• make sure it can be neglected


A common approach:

Take backreaction into account at most in an averaged or integrated sense


Example: Tadpole cancellation with D3/O3:

Locally: dF5 = H3 ∧ F3 − µ3δ6(O3/D3)

→ Nontrivial C4 - profile → complicated



Locally:

Globally:

dF5 = H3 ∧ F3 − µ3δ6(O3/D3)

0 =�M(6) H3 ∧ F3 − µ3

�NO3 − 1

4ND3

�


Instead:



Locally:

Globally:

dF5 = H3 ∧ F3 − µ3δ6(O3/D3)

0 =�M(6) H3 ∧ F3 − µ3

�NO3 − 1

4ND3

�

→ Global cancellation of tadpole by choosing appropriate flux & brane #‘s


F5

Instead:



Locally:

Globally:

dF5 = H3 ∧ F3 − µ3δ6(O3/D3)

0 =�M(6) H3 ∧ F3 − µ3

�NO3 − 1

4ND3

�

→ Global cancellation of tadpole by choosing appropriate flux & brane #‘s

But neglection of precise - profile


F5

C4

Instead:


At the level of the 10D eoms, this averaging is often implemented by “smearing” the local brane sources:



Localized brane source “Smeared” brane source



Localized brane source

x

ρ(x)

x

x

ρ(x)

x

“Smeared” brane source


Mathematically: δ → const.

(More generally: δ → smooth function)




→ Nice simplification: Warp factor, dilaton and certain RR-potentials (e.g. ) may often be assumed const.C4




→ Construction of many interesting flux backgrounds as explicit solutions to the 10D (smeared) eoms.

→ Nice simplification: Warp factor, dilaton and certain RR-potentials (e.g. ) may often be assumed const.

Early work, e.g.: Acharya, Benini, Valandro (2006)Grana, Minasian, Petrini, Tomasiello (2006) Koerber, Lüst, Tsimpis (2008)

C4


Smearing also brings a welcome simplification to dimensional reduction:



For compactifications on group or coset manifolds(incl. (twisted) tori, spheres) without brane sources, the restriction to the left-invariant modes yields a consistent truncation.




On a torus this corresponds to keeping only the constant Fourier modes:

φ(x, y) =�∞

n=0 φn(x)einy −→ φ0(x)




Remains valid in presence of brane-like sources if these are suitably smeared. E.g. Grana, Minasian, Petrini, Tomasiello (2006)

Caviezel, Koerber, Körs, Lüst, Tsimpis, MZ (2008)Cassani, Kashani-Poor (2009)


Gauged SUGRA theories obtained from (twisted) torus orientifolds make implicit use of such a smearing

In particular:

E.g. Angelantonj, Ferrara, Trigiante (2003)Derendinger, Kounnas, Petropoulos, Zwirner (2004)

Roest (2004) + ...


Smearing D-branes and O-planes is a commonly employed simplification to obtain explicit 10D flux compactifications or consistently truncated 4D

Summary:

Leff



It takes into account some brane backreaction in an averaged sense, but ignores local backreaction on warp factor, dilaton or certain RR-potentials

Summary:

Leff



It takes into account some brane backreaction in an averaged sense, but ignores local backreaction on warp factor, dilaton or certain RR-potentials

Question: Is this always a good approximation?

Summary:

Leff


Question seems particularly important for...


2. Classical de Sitter vacua


de Sitter compactifications are hard to buildat leading order in and gs α�

(No comparable problems for Minkowski or AdS)




“No-go” theorems:

E.g.: Gibbons (1984); de Wit, Smit, Hari Dass (1987) Maldacena, Nuñez (2000) Steinhardt, Wesley (2008)

Hertzberg, Kachru, Taylor, Tegmark (2007)Danielsson, Haque,Shiu,Van Riet (2009)Wrase, MZ (2010)







Fluxes + D-branesbut no O-planes







Fluxes + D-branesbut no O-planes

Fluxes + D-branes + O-planes with

�d6y

�g(6)R(6) ≥ 0


Two approaches:

(i) Go beyond leading order

E.g. non-perturbative quantum corrections → KKLT


Two approaches:



Hard to build explicit and controlled examples


Two approaches:




(ii) Work harder at leading order

→ “Classical” de Sitter vacua?


Two approaches:




(ii) Work harder at leading order

→ “Classical” de Sitter vacua?

Simplest way to evade no-go‘s: O-planes +neg. curvature


Has met with partial success:

4D de Sitter extrema found for certain group/coset spaces that allow for an SU(3)-structure with

Caviezel, Koerber, Körs, Lüst, Wrase, MZ (2008)Flauger, Paban, Robbins, Wrase (2008)

Caviezel, Wrase, MZ (2009)

See also: Haque, Shiu, Underwood, Van Riet (2008)Danielsson, Haque, Shiu, Van Riet (2009)

Andriot, Goi, Minasian, Petrini (2010)Dong, Horn, Silverstein, Torroba (2010)

R(6) < 0


Has met with partial success:

4D de Sitter extrema found for certain group/coset spaces that allow for an SU(3)-structure with

Caviezel, Koerber, Körs, Lüst, Wrase, MZ (2008)Flauger, Paban, Robbins, Wrase (2008)

Caviezel, Wrase, MZ (2009)

See also: Haque, Shiu, Underwood, Van Riet (2008)Danielsson, Haque, Shiu, Van Riet (2009)

Andriot, Goi, Minasian, Petrini (2010)Dong, Horn, Silverstein, Torroba (2010)

R(6) < 0

Explicit uplift to 10D knownDanielsson, Koerber, Van Riet (2010)

Danielsson, Haque, Koerber, Shiu, Van Riet, Wrase (2011)


Examles found so far not yet fully satisfactory:

• All contain at least one tachyon




• Possible issues with flux quantizationDanielsson, Haque, Koerber, Shiu, Van Riet, Wrase (2011)




• Possible issues with flux quantization

• Validity of smearing ? → “Douglas-Kallosh problem”

Danielsson, Haque, Koerber, Shiu, Van Riet, Wrase (2011)


The Douglas-Kallosh problem:Douglas, Kallosh (2010)


The Douglas-Kallosh problem:

Spaces of constant negative curvature require an everywhere negative energy density

In the absence of warping and higher curvature terms:

Douglas, Kallosh (2010)






R < 0






R < 0

ρ < 0





But smeared O-planes can provide precisely that!So where is the problem?






But smeared O-planes can provide precisely that!So where is the problem?

True O-planes are not smeared!







R < 0

ρ < 0


So how can negative curvature be sustained if O-planes are localized (as they should be)?


So how can negative curvature be sustained if O-planes are localized (as they should be)?

Note: Is a general issue of negative internal curvature, not necessarily related to dS


Possible ways out:

- Everywhere strongly varying warping

(- Or higher curvature terms relevant)



Possible ways out:



Varying warping is automatically induced by localized O-planes and D-branes


`


Possible ways out:



Varying warping is automatically induced by localized O-planes and D-branes

But if it varies strongly everywhere, it is unclear whether this is still well-approximated by the smeared solution with constant warp factor.


`


x

e2A(x)

x

e2A(x)

Localized O-plane with everywhere strongly varying warp factor

Smeared O-plane with constant warp factor


Our question:

How reliable is the smearing procedure in general?


Our question:


1) Do smeared solutions always have a localized counterpart?

2) If yes, how much do their physical properties differ? (e.g. w.r.t. moduli values, cosmological constant,...)


Our question:


1) Do smeared solutions always have a localized counterpart?

2) If yes, how much do their physical properties differ? (e.g. w.r.t. moduli values, cosmological constant,...)

For 2), cf. also “warped effective field theory” E.g DeWolfe, Giddings (2002)

Giddings, Maharana (2005)Frey, Maharana (2006)

Koerber, Martucci (2007)Douglas, Torroba (2008)

Shiu, Torroba, Underwood, Douglas (2008)+later papers


3. Smearing in the BPS case I


Need simple toy models where a localized solution is accessible → compare to the smeared solution

Blåbäck, Danielsson, Junghans, Van Riet, Wrase, MZ (2010,2011)


Prime candidate: Flux compactifications à la GKP

Giddings, Kachru, Polchinski (2001)

= best understood type of flux compactification with backreacting localized sources

Need simple toy models where a localized solution is accessible → compare to the smeared solution

Blåbäck, Danielsson, Junghans, Van Riet, Wrase, MZ (2010,2011)


Simplest version:

• M(10) = M(4) ×w M(6)


Simplest version:

•

• O3-planes filling M(4) and pointlike on M(6)

M(10) = M(4) ×w M(6)


Simplest version:

•


• F3 and H3 Flux on M(6)

M(10) = M(4) ×w M(6)


Simplest version:

•


• F3 and H3 Flux on M(6)

M(10) = M(4) ×w M(6)

dF5 = H3 ∧ F3 − µ3δ6(O3/D3)•


•

•O3

O3

F3

H3

M(6)

F5

and e sourced by fluxes and O3-planes 2A(x)+


Localized case:

ds210 = e2Ads̃24 + e−2Ads̃26


Localized case:

F5 = −(1 + ∗10)e−4A ∗6 dα

ds210 = e2Ads̃24 + e−2Ads̃26


Localized case:

F5 = −(1 + ∗10)e−4A ∗6 dα

ds210 = e2Ads̃24 + e−2Ads̃26

∇̃2(e4A − α) = R̃(4) + e

−6A

��∂(e4A − α)��2

+ 1

2e2A+φ

��F3 + e−φ∗̃6H3

��2


Localized case:

F5 = −(1 + ∗10)e−4A ∗6 dα

⇒ R̃(4) ≤ 0 (AdS or Minkowski)

ds210 = e2Ads̃24 + e−2Ads̃26

∇̃2(e4A − α) = R̃(4) + e

−6A

��∂(e4A − α)��2

+ 1

2e2A+φ

��F3 + e−φ∗̃6H3

��2


Localized case:

F5 = −(1 + ∗10)e−4A ∗6 dα


Minkowski ⇔ e4A − α = const.

(BPS-like cond.‘s)

(Λ=0)

ds210 = e2Ads̃24 + e−2Ads̃26

∇̃2(e4A − α) = R̃(4) + e

−6A

��∂(e4A − α)��2

+ 1

2e2A+φ

��F3 + e−φ∗̃6H3

��2

F3 + e−φ∗̃6H3 = 0


Localized case:

F5 = −(1 + ∗10)e−4A ∗6 dα




(Λ=0)

Moreover: R̃(6)ij = 0, φ = φ0 = const.

ds210 = e2Ads̃24 + e−2Ads̃26

∇̃2(e4A − α) = R̃(4) + e

−6A

��∂(e4A − α)��2

+ 1

2e2A+φ

��F3 + e−φ∗̃6H3

��2

F3 + e−φ∗̃6H3 = 0


Smeared case:

F5 = −(1 + ∗10)e−4A ∗6 dα




(Λ=0)


ds210 = e2Ads̃24 + e−2Ads̃26

∇̃2(e4A − α) = R̃(4) + e

−6A

��∂(e4A − α)��2

+ 1

2e2A+φ

��F3 + e−φ∗̃6H3

��2

F3 + e−φ∗̃6H3 = 0





(Λ=0)


ds210 = ds̃24 + ds̃26

F5 = −(1 + ∗10)e−4A ∗6 dα

∇̃2(e4A − α) = R̃(4) + e

−6A

��∂(e4A − α)��2

+ 1

2e2A+φ

��F3 + e−φ∗̃6H3

��2

F3 + e−φ∗̃6H3 = 0

Smeared case:





(Λ=0)


ds210 = ds̃24 + ds̃26

F5 = 0

∇̃2(e4A − α) = R̃(4) + e

−6A

��∂(e4A − α)��2

+ 1

2e2A+φ

��F3 + e−φ∗̃6H3

��2

F3 + e−φ∗̃6H3 = 0

Smeared case:





(Λ=0)


ds210 = ds̃24 + ds̃26

F5 = 0

0 = R̃(4) + 1

2eφ��F3 + e

−φ∗̃6H3

��2

F3 + e−φ∗̃6H3 = 0

Smeared case:



Minkowski ⇔

(BPS-like cond.)

(Λ=0)


ds210 = ds̃24 + ds̃26

F5 = 0

0 = R̃(4) + 1

2eφ��F3 + e

−φ∗̃6H3

��2

F3 + e−φ∗̃6H3 = 0

Smeared case:


have a localized Minkowski counterpart with

F3 + e−φ∗̃6H3 = 0

⇒ Smeared Minkowski vacua with BPS-type fluxes,

F3 + e−φ∗̃6H3 = 0



F3 + e−φ∗̃6H3 = 0


F3 + e−φ∗̃6H3 = 0

ISD flux: fixes complex structure moduli and dilaton



F3 + e−φ∗̃6H3 = 0


F3 + e−φ∗̃6H3 = 0

ISD flux: fixes complex structure moduli and dilaton

The smeared and localized BPS-solution have these moduli fixed at the same value and have the same

cosmological constant (zero)


At least for these physical quantities the localization effects (warping etc.) cancel out.

The BPS-nature ensures that the smearing is quite harmless.


Intuitive understanding:

O-planes and fluxes are BPS w.r.t. one another

⇒ O-plane charge and mass can be freely distributed without affecting the flux


4. Smearing in the BPS case II


Take smeared GKP-solution with M(6) = T6

and BPS-flux F3 + e−φ∗̃6H3 = 0




T-dualize along a circle with H-flux

⇒ IIA compactification on a twisted torus with

F4 -fluxwrapped (and smeared) O4-planes and




T-dualize along a circle with H-flux

⇒ IIA compactification on a twisted torus with

F4 -fluxwrapped (and smeared) O4-planes and

⇒ Twisted torus has constant negative curvature !

⇒ Localization of O4 directly addresses DK problem


Constructed the localized solution



Warping indeed takes care of DK problem




Integrated internal curvature remains negative:�d6y

�g(10)R(6) =

�g̃(4)

�d6y

�g̃(6)

�− 40

3 (∇̃A)2 + 14e

163 AR̃(6)

�< 0




Integrated internal curvature remains negative:

Despite the large warping effects, the moduli are stabilized at the same point and with the same cosmological constant as in the smeared case

�d6y

�g(10)R(6) =

�g̃(4)

�d6y

�g̃(6)

�− 40

3 (∇̃A)2 + 14e

163 AR̃(6)

�< 0




Integrated internal curvature remains negative:

Despite the large warping effects, the moduli are stabilized at the same point and with the same cosmological constant as in the smeared case

→ Consequence of BPS nature

�d6y

�g(10)R(6) =

�g̃(4)

�d6y

�g̃(6)

�− 40

3 (∇̃A)2 + 14e

163 AR̃(6)

�< 0


5. Smearing in the non-BPS case


Recall the smeared GKP solutions:


0 = R̃(4) + 1

2eφ��F3 + e

−φ∗̃6H3

��2




0 = R̃(4) + 1

2eφ��F3 + e

−φ∗̃6H3

��2

Violating the BPS condition, i.e., assuming

F3 + e−φ∗̃6H3 �= 0

allows for (stable) AdS-solutions, e.g.

AdS4 × S3 × S3




0 = R̃(4) + 1

2eφ��F3 + e

−φ∗̃6H3

��2

Violating the BPS condition, i.e., assuming

F3 + e−φ∗̃6H3 �= 0

allows for (stable) AdS-solutions, e.g.

AdS4 × S3 × S3

Need D3-branes instead of O3-planesMontag, 29. August 2011

One can prove: A localized solution does not exist, if the fluxes satisfy the same relation as in the smeared case.

Blåbäck, Danielsson, Junghans, Van Riet, Wrase, MZ (2010)



So, if a localized solution exists, it will probably fix the moduli at different values.




So, if a localized solution exists, it will probably fix the moduli at different values.

For the analogous smeared non-BPS solution on

one can show that there is no continuous interpolation between the smeared solution and a fully localized counterpart (if it exists at all).

AdS7 × S3


Blåbäck, Danielsson, Junghans, Van Riet, Wrase, MZ (2011)Montag, 29. August 2011

ρ(x)

x


ρ(x)

x


ρ(x)

x


ρ(x)

x


Works for BPS

ρ(x)

x


Only smooth non-BPS solution is the smeared one:

ρ(x)

x

But:


If a localized solution disconnected from the smeared one exists, it must involve non-standard boundary conditions at the D6-brane (divergent H ).3

Moreover:



If a localized solution disconnected from the smeared one exists, it must involve non-standard boundary conditions at the D6-brane (divergent H ).

Whether this makes sense is still unclear

3

Cf. also Bena, Grana, Halmagyi (2009)

Moreover:



6. Conclusions


Smearing D-branes and O-planes is a common and helpful simplification



For BPS configurations we found this to be a quite robust approximation




Warp factor resolves the Douglas-Kallosh problemof negatively curved spaces for BPS solutions




Warp factor resolves the Douglas-Kallosh problemof negatively curved spaces for BPS solutions

For non-BPS configuration, the general validity of smearing could not yet (?) be confirmed and raised instead many questions/concerns.


Unfortunately, de Sitter vacua should be non-BPS,so it is still unclear whether smearing makes sense here.


Unfortunately, de Sitter vacua should be non-BPS,so it is still unclear whether smearing makes sense here.

Can we also learn something about brane backreaction in warped throats from this?

Cf. also Bena, Grana, Halmagyi (2009)


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M. Zagermann - The Backreaction of Localized Sources and de Sitter Vacua

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