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Smeared versus localized sources in flux compactifications Smeared - - PowerPoint PPT Presentation

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Smeared versus localized sources in flux compactifications Smeared vs. localized sources Timm Wrase Timm Wrase Flux compact. BPS case non-BPS case Localiz. effects Conclusion Based on: TW, Zagermann 1003.0029 Bl ab ack,

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SLIDE 1

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Smeared versus localized sources in flux compactifications

Timm Wrase

Based on: TW, Zagermann 1003.0029 Bl˚ ab¨ ack, Danielsson, Junghans, Van Riet, TW, Zagermann 1009.1877

String Vacuum Project meeting Fall 2010

1 / 16

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SLIDE 2

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Classical type II flux compactifications

Most constructions of dS vacua use non-perturbative effects for moduli stabilization dS after uplift which breaks explicitly SUSY

KKLT, Large Volume

2 / 16

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SLIDE 3

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Classical type II flux compactifications

Most constructions of dS vacua use non-perturbative effects for moduli stabilization dS after uplift which breaks explicitly SUSY

KKLT, Large Volume

It is in principle possible to stabilize all moduli classically

Villadoro, Zwirner hep-th/0503169 DeWolfe, Giryavets, Kachru, Taylor hep-th/0505160 C´ amara, Font, Ib´ a˜ nez hep-th/0506066

2 / 16

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SLIDE 4

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Classical type II flux compactifications

Most constructions of dS vacua use non-perturbative effects for moduli stabilization dS after uplift which breaks explicitly SUSY

KKLT, Large Volume

It is in principle possible to stabilize all moduli classically

Villadoro, Zwirner hep-th/0503169 DeWolfe, Giryavets, Kachru, Taylor hep-th/0505160 C´ amara, Font, Ib´ a˜ nez hep-th/0506066

2 / 16

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SLIDE 5

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Classical type II flux compactifications

Most constructions of dS vacua use non-perturbative effects for moduli stabilization dS after uplift which breaks explicitly SUSY

KKLT, Large Volume

It is in principle possible to stabilize all moduli classically

Villadoro, Zwirner hep-th/0503169 DeWolfe, Giryavets, Kachru, Taylor hep-th/0505160 C´ amara, Font, Ib´ a˜ nez hep-th/0506066

Can we find classical dS vacua?

Hertzberg, Tegmark, Kachru, Shelton, Ozcan 0709.0002 [astro-ph]

2 / 16

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SLIDE 6

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Type II supergravity

The classical ingredients for type II supergravity theories are RR-fluxes Fp, NSNS H-flux, R6, Oq-planes, ...

3 / 16

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SLIDE 7

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Type II supergravity

The classical ingredients for type II supergravity theories are RR-fluxes Fp, NSNS H-flux, R6, Oq-planes, ... For smeared Oq-planes we find a 4D scalar potential V (ρ, φ, . . .) =

  • p

VFp + VH + VR6 − VOq, where ρ = (vol6)1/3 and φ is the dilaton.

3 / 16

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SLIDE 8

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Type II supergravity

The classical ingredients for type II supergravity theories are RR-fluxes Fp, NSNS H-flux, R6, Oq-planes, ... For smeared Oq-planes we find a 4D scalar potential V (ρ, φ, . . .) =

  • p

VFp + VH + VR6 − VOq, where ρ = (vol6)1/3 and φ is the dilaton.

When is ∂ρV = ∂φV = 0 and V > 0 possible?

Hertzberg, Kachru, Taylor, Tegmark 0711.2512 [hep-th]

3 / 16

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SLIDE 9

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

We can evade a no-go theorem involving ρ and φ with the following minimal ingredients Curvature IIA IIB VR6 ∼ −R6 ≤ 0 O4, H, F0 O3, H, F1 VR6 ∼ −R6 > 0 O4, F0 O4, F2 O6, F0 O3, F1 O3, F3 O3, F5 O5, F1

Hertzberg, Kachru, Taylor, Tegmark 0711.2512 [hep-th] Silverstein 0712.1196 [hep-th] Haque, Shiu, Underwood, Van Riet 0810.5328 [hep-th] Caviezel, Koerber, K¨

  • rs, L¨

ust, TW, M. Zagermann 0812.3551 [hep-th] Flauger, Robbins, Paban, TW 0812.3886 [hep-th] Danielsson, Haque, Shiu, Van Riet 0907.2041 [hep-th] de Carlos, Guarino, Moreno 0907.5580, 0911.2876 [hep-th] Caviezel, TW, Zagermann 0912.3287 [hep-th] TW, Zagermann 1003.0029 [hep-th] Danielsson, Koerber, Van Riet 1003.3590 [hep-th] Danielsson, Haque, Koerber, Shiu, Van Riet, TW 1011.xxxx [hep-th]

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SLIDE 10

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Smeared versus localized sources

O-planes are localized objects Smearing was necessary to solve equations of motion

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SLIDE 11

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Smeared versus localized sources

O-planes are localized objects Smearing was necessary to solve equations of motion

When is smearing δ(Oq) ≈ 1 a valid approximation?

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SLIDE 12

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Smeared versus localized sources

O-planes are localized objects Smearing was necessary to solve equations of motion

When is smearing δ(Oq) ≈ 1 a valid approximation?

Negative curvature R6 < 0 requires (in the localized case) large warping or large stringy corrections

Douglas, Kallosh 1001.4008 [hep-th]

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SLIDE 13

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

An example with BPS sources

Giddings, Kachru and Polchinski found localized no-scale Minkowski solutions with O3-planes

Giddings, Kachru, Polchinski hep-th/0105097

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SLIDE 14

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

An example with BPS sources

Giddings, Kachru and Polchinski found localized no-scale Minkowski solutions with O3-planes

Giddings, Kachru, Polchinski hep-th/0105097

smeared case H, F3, O3 ds2 = ds2

4 + ds2 6

0 = dF5 = H ∧ F3 − ˜ µ3

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SLIDE 15

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

An example with BPS sources

Giddings, Kachru and Polchinski found localized no-scale Minkowski solutions with O3-planes

Giddings, Kachru, Polchinski hep-th/0105097

smeared case H, F3, O3 ds2 = ds2

4 + ds2 6

0 = dF5 = H ∧ F3 − ˜ µ3 localized case H, F3, O3, F5, A ds2 = e2Ads2

4 + e−2Ads2 6

dF5 = H ∧F3 − ˜ µ3δ(O3)

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SLIDE 16

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

An example with BPS sources

Can solve the 10D equations of motions in both cases Find no-scale Minkowski vacua Internal space is (conformally) Ricci-flat

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SLIDE 17

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

An example with BPS sources

Can solve the 10D equations of motions in both cases Find no-scale Minkowski vacua Internal space is (conformally) Ricci-flat But localization effects are large ∇2e−4A = −e−φ|H|2 + ˜ µ3δ(O3)

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SLIDE 18

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

An example with BPS sources

Can solve the 10D equations of motions in both cases Find no-scale Minkowski vacua Internal space is (conformally) Ricci-flat Complex structure moduli and φ are stabilized

smeared case F3 = −e−φ ⋆6 H localized case F3 = −e−φ⋆6H

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SLIDE 19

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

An example with BPS sources

BUT F3 = −e−φ⋆6H = −e−φ ⋆6 H since warp factor cancels: ⋆6H ≈

  • det (e2Ag6)
  • e−2Ag−1

6

3 H

8 / 16

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SLIDE 20

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

An example with BPS sources

BUT F3 = −e−φ⋆6H = −e−φ ⋆6 H since warp factor cancels: ⋆6H ≈

  • det (e2Ag6)
  • e−2Ag−1

6

3 H

Moduli values at minimum unchanged! Approximation δ(O3) ≈ 1 is “ok”

8 / 16

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SLIDE 21

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

An example with BPS sources

BUT F3 = −e−φ⋆6H = −e−φ ⋆6 H since warp factor cancels: ⋆6H ≈

  • det (e2Ag6)
  • e−2Ag−1

6

3 H

Moduli values at minimum unchanged! Approximation δ(O3) ≈ 1 is “ok”

smeared: H and F3 stabilize moduli localized: ˜ µ3δ(O3), F5, A give corrections of equal size ⇒ corrections from ˜ µ3δ(O3), F5, A cancel each other

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SLIDE 22

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

A T-dual example with BPS sources

T-duality along one H-flux direction ↔ Douglas, Kallosh H → R6 < 0 F3 → F4 O3 → O4 F5 → F4 A → A

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SLIDE 23

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

A T-dual example with BPS sources

T-duality along one H-flux direction ↔ Douglas, Kallosh H → R6 < 0 F3 → F4 O3 → O4 F5 → F4 A → A Conclusions remain unchanged:

Douglas, Kallosh ⇒ warping effects are large

But again smeared moduli values are unaffected Note: √g10R6 < 0 ⇒ VR6 > 0 (no ‘uplift’ to dS, solutions are Minkowski)

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SLIDE 24

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

An example with non-BPS sources

Replace O3-plane by D3-brane:

smeared case H, F3, D3 ds2 = ds2

4 + ds2 6

0 = dF5 = H ∧ F3 − µ3 localized case H, F3, D3, F5, A, . . . ds2 = e2Ads2

4 + ds2 6

dF5 = H ∧F3 −µ3δ(O3)

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SLIDE 25

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

An example with non-BPS sources

smeared case H, F3, D3 ds2 = ds2

4 + ds2 6

0 = dF5 = H ∧ F3 − µ3

Can solve the 10D equations of motions

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SLIDE 26

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

An example with non-BPS sources

smeared case H, F3, D3 ds2 = ds2

4 + ds2 6

0 = dF5 = H ∧ F3 − µ3

Can solve the 10D equations of motions Find AdS solutions V = VF3 + VH − VR6 + VD3 < 0 Internal space is positively curved: e.g. S3 × S3

11 / 16

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SLIDE 27

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

An example with non-BPS sources

smeared case H, F3, D3 ds2 = ds2

4 + ds2 6

0 = dF5 = H ∧ F3 − µ3

Can solve the 10D equations of motions Find AdS solutions V = VF3 + VH − VR6 + VD3 < 0 Internal space is positively curved: e.g. S3 × S3 Complex structure and φ stabilized (F3 = −e−φ ⋆6 H) volume moduli stabilized: e.g. RS3

ij = 1 2e−φ|H|2gS3 ij

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SLIDE 28

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

An example with non-BPS sources

smeared case H, F3, D3 ds2 = ds2

4 + ds2 6

0 = dF5 = H ∧ F3 − µ3

Can solve the 10D equations of motions Find AdS solutions V = VF3 + VH − VR6 + VD3 < 0 Internal space is positively curved: e.g. S3 × S3 Complex structure and φ stabilized (F3 = −e−φ ⋆6 H) volume moduli stabilized: e.g. RS3

ij = 1 2e−φ|H|2gS3 ij

no SUSY but volume and dilaton masses above BF bound

11 / 16

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SLIDE 29

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

An example with non-BPS sources

localized case H, F3, D3, F5, A, . . . ds2 = e2Ads2

4 + ds2 6

dF5 = H ∧F3 −µ3δ(O3)

(Assume) F1 = 0, F3 = −e−φ⋆6H for arbitrary g6 Combine eoms for F3, H, F5 and external Einstein: e−2AR4 = −(1+1)µ3δ(D3)

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SLIDE 30

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

An example with non-BPS sources

localized case H, F3, D3, F5, A, . . . ds2 = e2Ads2

4 + ds2 6

dF5 = H ∧F3 −µ3δ(O3)

(Assume) F1 = 0, F3 = −e−φ⋆6H for arbitrary g6 Combine eoms for F3, H, F5 and external Einstein: e−2AR4 = −(1+1)µ3δ(D3)

The smeared solution cannot be localized?

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SLIDE 31

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

BPS versus non-BPS sources

BPS condition in GKP

1 4(T m m − T µ µ )loc ≥ µ3ρloc 3

For O3, D3 and D3 we have T m

m = 0.

O3 D3 D3 ρloc

3

− 1

4

  • 1

1 − 1

4T µ µ

µ3ρloc

3

−µ3ρloc

3

µ3ρloc

3

BPS

  • ×
  • Force between D3 and fluxes H, F3

⇒ no static localized solution

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SLIDE 32

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

non-BPS sources smeared case localized case

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SLIDE 33

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Localization effects

Localization effects are generically large in flux compactifications ∇2eA = |flux|2

p −δ(source)

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SLIDE 34

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Localization effects

Localization effects are generically large in flux compactifications ∇2eA = |flux|2

p −δ(source)

Warping suppressed in the large volume limit g6 → λ2g6?

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SLIDE 35

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Localization effects

Localization effects are generically large in flux compactifications ∇2eA = |flux|2

p −δ(source)

Warping suppressed in the large volume limit g6 → λ2g6? Yes!

1 λ2 ∇2eA = 1 λ2p |flux|2 p

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SLIDE 36

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Localization effects

Localization effects are generically large in flux compactifications ∇2eA = |flux|2

p −δ(source)

Warping suppressed in the large volume limit g6 → λ2g6? Yes! But so are the fluxes!

1 λ2 ∇2eA = 1 λ2p |flux|2 p

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SLIDE 37

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Localization effects

Localization effects are generically large in flux compactifications ∇2eA = |flux|2

p −δ(source)

Are there regions of small warping eA ≈ 1 and (∇A)2 ≪ 1?

15 / 16

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SLIDE 38

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Localization effects

Localization effects are generically large in flux compactifications ∇2eA = |flux|2

p −δ(source)

Are there regions of small warping eA ≈ 1 and (∇A)2 ≪ 1? Not really: ∇2eA = eA∇2A + eA(∇A)2 ≈ ∇2A = |flux|2

p

15 / 16

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SLIDE 39

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Conclusion:

Explicit examples:

smeared BPS sources are ok non-BPS solutions problematic

Localization effects comparable to fluxes

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SLIDE 40

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Conclusion:

Explicit examples:

smeared BPS sources are ok non-BPS solutions problematic

Localization effects comparable to fluxes

Outlook:

Study solutions close to BPS point Construct localized, non-BPS examples Generalize findings to intersecting branes

16 / 16

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SLIDE 41

Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case

  • Localiz. effects

Conclusion

Conclusion:

Explicit examples:

smeared BPS sources are ok non-BPS solutions problematic

Localization effects comparable to fluxes

Outlook:

Study solutions close to BPS point Construct localized, non-BPS examples Generalize findings to intersecting branes

THANK YOU!

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