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

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The SEENET-MTP Workshop JW2011
Scientific and Human Legacy of Julius Wess
27-28 August 2011, Donji Milanovac, Serbia

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

  1. 1. The Backreaction of Localized Sources and de Sitter Vacua Marco Zagermann (Leibniz Universität Hannover & QUEST) Donji Milanovac, August 29, 2011Montag, 29. August 2011
  2. 2. 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) As well as Wrase, MZ (2010) Caviezel, Wrase, MZ (2009) Caviezel, Koerber, Körs, Lüst, Wrase, MZ (2008) Caviezel, Koerber, Körs, Lüst, Tsimpis, MZ (2008)Montag, 29. August 2011
  3. 3. 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. ConclusionsMontag, 29. August 2011
  4. 4. 1. Smearing D-branes and O-planesMontag, 29. August 2011
  5. 5. D-branes & O-planes are important ingredients in phenomenologically realistic type II compactifications:Montag, 29. August 2011
  6. 6. D-branes & O-planes are important ingredients in phenomenologically realistic type II compactifications: D-brane O-plane Tension: T>0 T<0Montag, 29. August 2011
  7. 7. D-branes & O-planes are important ingredients in phenomenologically realistic type II compactifications: E.g. • Chiral matterMontag, 29. August 2011
  8. 8. D-branes & O-planes are important ingredients in phenomenologically realistic type II compactifications: E.g. • Chiral matter • Supersymmetry breakingMontag, 29. August 2011
  9. 9. D-branes & O-planes are important ingredients in phenomenologically realistic type II compactifications: E.g. • Chiral matter • Supersymmetry breaking • Moduli stabilization ( →Tadpole cancellation, non-pert. effects, etc.) ...Montag, 29. August 2011
  10. 10. But: Dp-branes and Op-planes... • ... have mass → Backreaction on metric (e.g. warp factor)Montag, 29. August 2011
  11. 11. But: Dp-branes and Op-planes... • ... have mass → Backreaction on metric (e.g. warp factor) • ... carry RR-charge → Source RR-potentialsMontag, 29. August 2011
  12. 12. But: Dp-branes and Op-planes... • ... have mass → Backreaction on metric (e.g. warp factor) • ... carry RR-charge → Source RR-potentials • ... couple to the dilaton → Nontrivial dilaton profile (except for p = 3)Montag, 29. August 2011
  13. 13. Profile of warp factor, dilaton or RR-pot. x D-brane or O-planeMontag, 29. August 2011
  14. 14. 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-planesMontag, 29. August 2011
  15. 15. 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 D6Montag, 29. August 2011
  16. 16. 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-planesMontag, 29. August 2011
  17. 17. 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-planesMontag, 29. August 2011
  18. 18. In all other cases: • take backreaction into account or • make sure it can be neglectedMontag, 29. August 2011
  19. 19. A common approach: Take backreaction into account at most in an averaged or integrated senseMontag, 29. August 2011
  20. 20. Example: Tadpole cancellation with D3/O3: Locally: dF5 = H3 ∧ F3 − µ3 δ6 (O3/D3) → Nontrivial C4 - profile → complicatedMontag, 29. August 2011
  21. 21. Example: Tadpole cancellation with D3/O3: Locally: dF5 = H3 ∧ F3 − µ3 δ6 (O3/D3) → Nontrivial C4 - profile → complicated Instead: Globally: 0= M(6) H3 ∧ F3 − µ3 NO3 − 1 ND3 4Montag, 29. August 2011
  22. 22. Example: Tadpole cancellation with D3/O3: Locally: dF5 = H3 ∧ F3 − µ3 δ6 (O3/D3) → Nontrivial C4 - profile → complicated Instead: Globally: 0= M(6) H3 ∧ F3 − µ3 NO3 − 1 ND3 4 → Global cancellation of F5 tadpole by choosing appropriate flux brane #‘sMontag, 29. August 2011
  23. 23. Example: Tadpole cancellation with D3/O3: Locally: dF5 = H3 ∧ F3 − µ3 δ6 (O3/D3) → Nontrivial C4 - profile → complicated Instead: Globally: 0= M(6) H3 ∧ F3 − µ3 NO3 − 1 ND3 4 → Global cancellation of F5 tadpole by choosing appropriate flux brane #‘s But neglection of precise C4 - profileMontag, 29. August 2011
  24. 24. At the level of the 10D eoms, this averaging is often implemented by “smearing” the local brane sources:Montag, 29. August 2011
  25. 25. At the level of the 10D eoms, this averaging is often implemented by “smearing” the local brane sources: Localized brane source “Smeared” brane sourceMontag, 29. August 2011
  26. 26. At the level of the 10D eoms, this averaging is often implemented by “smearing” the local brane sources: x x Localized brane source “Smeared” brane source ρ(x) ρ(x) x xMontag, 29. August 2011
  27. 27. Mathematically: δ → const. (More generally: δ → smooth function)Montag, 29. August 2011
  28. 28. Mathematically: δ → const. (More generally: δ → smooth function) → Nice simplification: Warp factor, dilaton and certain RR-potentials (e.g. C4 ) may often be assumed const.Montag, 29. August 2011
  29. 29. Mathematically: δ → const. (More generally: δ → smooth function) → Nice simplification: Warp factor, dilaton and certain RR-potentials (e.g. C4 ) may often be assumed const. → Construction of many interesting flux backgrounds as explicit solutions to the 10D (smeared) eoms. Early work, e.g.: Acharya, Benini,Valandro (2006) Grana, Minasian, Petrini, Tomasiello (2006) Koerber, Lüst, Tsimpis (2008)Montag, 29. August 2011
  30. 30. Smearing also brings a welcome simplification to dimensional reduction:Montag, 29. August 2011
  31. 31. 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.Montag, 29. August 2011
  32. 32. 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)Montag, 29. August 2011
  33. 33. 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. 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)Montag, 29. August 2011
  34. 34. In particular: Gauged SUGRA theories obtained from (twisted) torus orientifolds make implicit use of such a smearing E.g. Angelantonj, Ferrara, Trigiante (2003) Derendinger, Kounnas, Petropoulos, Zwirner (2004) Roest (2004) + ...Montag, 29. August 2011
  35. 35. Summary: Smearing D-branes and O-planes is a commonly employed simplification to obtain explicit 10D flux compactifications or consistently truncated 4D LeffMontag, 29. August 2011
  36. 36. Summary: Smearing D-branes and O-planes is a commonly employed simplification to obtain explicit 10D flux compactifications or consistently truncated 4D Leff It takes into account some brane backreaction in an averaged sense, but ignores local backreaction on warp factor, dilaton or certain RR-potentialsMontag, 29. August 2011
  37. 37. Summary: Smearing D-branes and O-planes is a commonly employed simplification to obtain explicit 10D flux compactifications or consistently truncated 4D 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?Montag, 29. August 2011
  38. 38. Question seems particularly important for...Montag, 29. August 2011
  39. 39. 2. Classical de Sitter vacuaMontag, 29. August 2011
  40. 40. de Sitter compactifications are hard to build at leading order in gs and α (No comparable problems for Minkowski or AdS)Montag, 29. August 2011
  41. 41. de Sitter compactifications are hard to build at leading order in gs and 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)Montag, 29. August 2011
  42. 42. de Sitter compactifications are hard to build at leading order in gs and α (No comparable problems for Minkowski or AdS) “No-go” theorems: E.g.: Gibbons (1984); Fluxes + D-branes de Wit, Smit, Hari Dass (1987) but no O-planes Maldacena, Nuñez (2000) Steinhardt, Wesley (2008) Hertzberg, Kachru, Taylor, Tegmark (2007) Danielsson, Haque,Shiu,Van Riet (2009) Wrase, MZ (2010)Montag, 29. August 2011
  43. 43. de Sitter compactifications are hard to build at leading order in gs and α (No comparable problems for Minkowski or AdS) “No-go” theorems: E.g.: Gibbons (1984); Fluxes + D-branes de Wit, Smit, Hari Dass (1987) but no O-planes Maldacena, Nuñez (2000) Steinhardt, Wesley (2008) Fluxes + D-branes + O-planes Hertzberg, Kachru, Taylor, Tegmark (2007) 6 Danielsson, Haque,Shiu,Van Riet (2009) with d y g(6) R(6) ≥ 0 Wrase, MZ (2010)Montag, 29. August 2011
  44. 44. Two approaches: (i) Go beyond leading order E.g. non-perturbative quantum corrections → KKLTMontag, 29. August 2011
  45. 45. Two approaches: (i) Go beyond leading order E.g. non-perturbative quantum corrections → KKLT Hard to build explicit and controlled examplesMontag, 29. August 2011
  46. 46. Two approaches: (i) Go beyond leading order E.g. non-perturbative quantum corrections → KKLT Hard to build explicit and controlled examples (ii) Work harder at leading order → “Classical” de Sitter vacua?Montag, 29. August 2011
  47. 47. Two approaches: (i) Go beyond leading order E.g. non-perturbative quantum corrections → KKLT Hard to build explicit and controlled examples (ii) Work harder at leading order → “Classical” de Sitter vacua? Simplest way to evade no-go‘s: O-planes + neg. curvatureMontag, 29. August 2011
  48. 48. Has met with partial success: 4D de Sitter extrema found for certain group/coset spaces that allow for an SU(3)-structure with R(6) 0 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)Montag, 29. August 2011
  49. 49. Has met with partial success: 4D de Sitter extrema found for certain group/coset spaces that allow for an SU(3)-structure with R(6) 0 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) Explicit uplift to 10D known Danielsson, Koerber,Van Riet (2010) Danielsson, Haque, Koerber, Shiu,Van Riet, Wrase (2011)Montag, 29. August 2011
  50. 50. Examles found so far not yet fully satisfactory: • All contain at least one tachyonMontag, 29. August 2011
  51. 51. Examles found so far not yet fully satisfactory: • All contain at least one tachyon • Possible issues with flux quantization Danielsson, Haque, Koerber, Shiu,Van Riet, Wrase (2011)Montag, 29. August 2011
  52. 52. Examles found so far not yet fully satisfactory: • All contain at least one tachyon • Possible issues with flux quantization Danielsson, Haque, Koerber, Shiu,Van Riet, Wrase (2011) • Validity of smearing ? → “Douglas-Kallosh problem”Montag, 29. August 2011
  53. 53. The Douglas-Kallosh problem: Douglas, Kallosh (2010)Montag, 29. August 2011
  54. 54. The Douglas-Kallosh problem: Douglas, Kallosh (2010) In the absence of warping and higher curvature terms: Spaces of constant negative curvature require an everywhere negative energy densityMontag, 29. August 2011
  55. 55. The Douglas-Kallosh problem: Douglas, Kallosh (2010) In the absence of warping and higher curvature terms: Spaces of constant negative curvature require an everywhere negative energy density R0Montag, 29. August 2011
  56. 56. The Douglas-Kallosh problem: Douglas, Kallosh (2010) In the absence of warping and higher curvature terms: Spaces of constant negative curvature require an everywhere negative energy density R0 ρ0Montag, 29. August 2011
  57. 57. The Douglas-Kallosh problem: Douglas, Kallosh (2010) In the absence of warping and higher curvature terms: Spaces of constant negative curvature require an everywhere negative energy density But smeared O-planes can provide precisely that! So where is the problem?Montag, 29. August 2011
  58. 58. The Douglas-Kallosh problem: Douglas, Kallosh (2010) In the absence of warping and higher curvature terms: Spaces of constant negative curvature require an everywhere negative energy density But smeared O-planes can provide precisely that! So where is the problem? True O-planes are not smeared!Montag, 29. August 2011
  59. 59. The Douglas-Kallosh problem: Douglas, Kallosh (2010) In the absence of warping and higher curvature terms: Spaces of constant negative curvature require an everywhere negative energy density R0 ρ0Montag, 29. August 2011
  60. 60. So how can negative curvature be sustained if O-planes are localized (as they should be)?Montag, 29. August 2011
  61. 61. 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 dSMontag, 29. August 2011
  62. 62. Possible ways out: Douglas, Kallosh (2010) - Everywhere strongly varying warping (- Or higher curvature terms relevant)Montag, 29. August 2011
  63. 63. Possible ways out: Douglas, Kallosh (2010) - Everywhere strongly varying warping ` (- Or higher curvature terms relevant) Varying warping is automatically induced by localized O-planes and D-branesMontag, 29. August 2011
  64. 64. Possible ways out: Douglas, Kallosh (2010) - Everywhere strongly varying warping ` (- Or higher curvature terms relevant) 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.Montag, 29. August 2011
  65. 65. 2A(x) 2A(x) e e x x Localized O-plane with Smeared O-plane with everywhere strongly constant warp factor varying warp factorMontag, 29. August 2011
  66. 66. Our question: How reliable is the smearing procedure in general?Montag, 29. August 2011
  67. 67. Our question: How reliable is the smearing procedure in general? 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,...)Montag, 29. August 2011
  68. 68. Our question: How reliable is the smearing procedure in general? 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 papersMontag, 29. August 2011
  69. 69. 3. Smearing in the BPS case IMontag, 29. August 2011
  70. 70. Need simple toy models where a localized solution is accessible → compare to the smeared solution Blåbäck, Danielsson, Junghans,Van Riet, Wrase, MZ (2010,2011)Montag, 29. August 2011
  71. 71. 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 sourcesMontag, 29. August 2011
  72. 72. Simplest version: • M(10) = M(4) ×w M(6)Montag, 29. August 2011
  73. 73. Simplest version: • M(10) = M(4) ×w M(6) • O3-planes filling M(4) and pointlike on M(6)Montag, 29. August 2011
  74. 74. Simplest version: • M(10) = M(4) ×w M(6) • O3-planes filling M(4) and pointlike on M(6) • F3 and H3 Flux on M(6)Montag, 29. August 2011
  75. 75. Simplest version: • M(10) = M(4) ×w M(6) • O3-planes filling M(4) and pointlike on M(6) • F3 and H3 Flux on M(6) • dF5 = H3 ∧ F3 − µ3 δ6 (O3/D3)Montag, 29. August 2011
  76. 76. H3 • O3 M(6) F3 • O3 2A(x) + F5 and e sourced by fluxes and O3-planesMontag, 29. August 2011
  77. 77. Localized case: ds2 = e2A d˜2 + e−2A d˜2 10 s4 s6Montag, 29. August 2011
  78. 78. Localized case: ds2 = e2A d˜2 + e−2A d˜2 10 s4 s6 F5 = −(1 + ∗10 )e−4A ∗6 dαMontag, 29. August 2011
  79. 79. Localized case: ds2 = e2A d˜2 + e−2A d˜2 10 s4 s6 F5 = −(1 + ∗10 )e−4A ∗6 dα 2 2 ˜ ˜ −6A 1 2A+φ −φ ˜ ∇2 (e4A − α) = R(4) +e 4A ∂(e − α) + 2 e F3 + e ∗6 H3 Montag, 29. August 2011
  80. 80. Localized case: ds2 = e2A d˜2 + e−2A d˜2 10 s4 s6 F5 = −(1 + ∗10 )e−4A ∗6 dα 2 2 ˜ ˜ −6A 1 2A+φ −φ ˜ ∇2 (e4A − α) = R(4) +e 4A ∂(e − α) + 2 e F3 + e ∗6 H3 ˜ ⇒ R(4) ≤ 0 (AdS or Minkowski)Montag, 29. August 2011
  81. 81. Localized case: ds2 = e2A d˜2 + e−2A d˜2 10 s4 s6 F5 = −(1 + ∗10 )e−4A ∗6 dα 2 2 ˜ ˜ −6A 1 2A+φ −φ ˜ ∇2 (e4A − α) = R(4) +e 4A ∂(e − α) + 2 e F3 + e ∗6 H3 ˜ ⇒ R(4) ≤ 0 (AdS or Minkowski) e4A − α = const. Minkowski (Λ=0) F3 + e−φ ∗6 H3 = 0 ˜ (BPS-like cond.‘s)Montag, 29. August 2011
  82. 82. Localized case: Moreover: ˜ (6) Rij = 0, φ = φ0 = const. ds2 = e2A d˜2 + e−2A d˜2 10 s4 s6 F5 = −(1 + ∗10 )e−4A ∗6 dα 2 2 ˜ ˜ −6A 1 2A+φ −φ ˜ ∇2 (e4A − α) = R(4) +e 4A ∂(e − α) + 2 e F3 + e ∗6 H3 ˜ ⇒ R(4) ≤ 0 (AdS or Minkowski) e4A − α = const. Minkowski (Λ=0) F3 + e−φ ∗6 H3 = 0 ˜ (BPS-like cond.‘s)Montag, 29. August 2011
  83. 83. Smeared case: Moreover: ˜ (6) Rij = 0, φ = φ0 = const. ds2 = e2A d˜2 + e−2A d˜2 10 s4 s6 F5 = −(1 + ∗10 )e−4A ∗6 dα 2 2 ˜ ˜ −6A 1 2A+φ −φ ˜ ∇2 (e4A − α) = R(4) +e 4A ∂(e − α) + 2 e F3 + e ∗6 H3 ˜ ⇒ R(4) ≤ 0 (AdS or Minkowski) e4A − α = const. Minkowski (Λ=0) F3 + e−φ ∗6 H3 = 0 ˜ (BPS-like cond.‘s)Montag, 29. August 2011
  84. 84. Smeared case: Moreover: ˜ (6) Rij = 0, φ = φ0 = const. ds2 = d˜2 + d˜2 10 s4 s6 F5 = −(1 + ∗10 )e−4A ∗6 dα 2 2 ˜ ˜ −6A 1 2A+φ −φ ˜ ∇2 (e4A − α) = R(4) +e 4A ∂(e − α) + 2 e F3 + e ∗6 H3 ˜ ⇒ R(4) ≤ 0 (AdS or Minkowski) e4A − α = const. Minkowski (Λ=0) F3 + e−φ ∗6 H3 = 0 ˜ (BPS-like cond.‘s)Montag, 29. August 2011
  85. 85. Smeared case: Moreover: ˜ (6) Rij = 0, φ = φ0 = const. ds2 = d˜2 + d˜2 10 s4 s6 F5 = 0 2 2 ˜ ˜ −6A 1 2A+φ −φ ˜ ∇2 (e4A − α) = R(4) +e 4A ∂(e − α) + 2 e F3 + e ∗6 H3 ˜ ⇒ R(4) ≤ 0 (AdS or Minkowski) e4A − α = const. Minkowski (Λ=0) F3 + e−φ ∗6 H3 = 0 ˜ (BPS-like cond.‘s)Montag, 29. August 2011
  86. 86. Smeared case: Moreover: ˜ (6) Rij = 0, φ = φ0 = const. ds2 = d˜2 + d˜2 10 s4 s6 F5 = 0 2 ˜ 1 φ −φ ˜ 0 = R(4) + 2 e F3 + e ∗6 H3 ˜ ⇒ R(4) ≤ 0 (AdS or Minkowski) e4A − α = const. Minkowski (Λ=0) F3 + e−φ ∗6 H3 = 0 ˜ (BPS-like cond.‘s)Montag, 29. August 2011
  87. 87. Smeared case: Moreover: ˜ (6) Rij = 0, φ = φ0 = const. ds2 = d˜2 + d˜2 10 s4 s6 F5 = 0 2 ˜ 1 φ −φ ˜ 0 = R(4) + 2 e F3 + e ∗6 H3 ˜ ⇒ R(4) ≤ 0 (AdS or Minkowski) Minkowski (Λ=0) F3 + e−φ ∗6 H3 = 0 ˜ (BPS-like cond.)Montag, 29. August 2011
  88. 88. Smeared Minkowski vacua with BPS-type fluxes, F3 + e−φ ∗6 H3 = 0 ˜ have a localized Minkowski counterpart with F3 + e−φ ∗6 H3 = 0 ˜Montag, 29. August 2011
  89. 89. Smeared Minkowski vacua with BPS-type fluxes, F3 + e−φ ∗6 H3 = 0 ˜ have a localized Minkowski counterpart with F3 + e−φ ∗6 H3 = 0 ˜ ISD flux: fixes complex structure moduli and dilatonMontag, 29. August 2011
  90. 90. Smeared Minkowski vacua with BPS-type fluxes, F3 + e−φ ∗6 H3 = 0 ˜ have a localized Minkowski counterpart with F3 + e−φ ∗6 H3 = 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)Montag, 29. August 2011
  91. 91. At least for these physical quantities the localization effects (warping etc.) cancel out. The BPS-nature ensures that the smearing is quite harmless.Montag, 29. August 2011
  92. 92. 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 fluxMontag, 29. August 2011
  93. 93. 4. Smearing in the BPS case IIMontag, 29. August 2011
  94. 94. Take smeared GKP-solution with M(6) = T6 and BPS-flux F3 + e−φ ∗6 H3 = 0 ˜Montag, 29. August 2011
  95. 95. Take smeared GKP-solution with M(6) = T6 and BPS-flux F3 + e−φ ∗6 H3 = 0 ˜ T-dualize along a circle with H-flux IIA compactification on a twisted torus with wrapped (and smeared) O4-planes and F4 -fluxMontag, 29. August 2011
  96. 96. Take smeared GKP-solution with M(6) = T6 and BPS-flux F3 + e−φ ∗6 H3 = 0 ˜ T-dualize along a circle with H-flux IIA compactification on a twisted torus with wrapped (and smeared) O4-planes and F4 -flux Twisted torus has constant negative curvature ! Localization of O4 directly addresses DK problemMontag, 29. August 2011
  97. 97. Constructed the localized solutionMontag, 29. August 2011
  98. 98. Constructed the localized solution Warping indeed takes care of DK problemMontag, 29. August 2011
  99. 99. Constructed the localized solution Warping indeed takes care of DK problem Integrated internal curvature remains negative: 6 6 d y g (10) R(6) = g˜ (4) d y g˜ (6) − 40 (∇A)2 + 1 e 16 A R(6) 0 ˜ 3 ˜ 3 4Montag, 29. August 2011
  100. 100. Constructed the localized solution Warping indeed takes care of DK problem Integrated internal curvature remains negative: 6 6 d y g (10) R(6) = g˜ (4) d y g˜ (6) − 40 (∇A)2 + 1 e 16 A R(6) 0 ˜ 3 ˜ 3 4 Despite the large warping effects, the moduli are stabilized at the same point and with the same cosmological constant as in the smeared caseMontag, 29. August 2011
  101. 101. Constructed the localized solution Warping indeed takes care of DK problem Integrated internal curvature remains negative: 6 6 d y g (10) R(6) = g˜ (4) d y g˜ (6) − 40 (∇A)2 + 1 e 16 A R(6) 0 ˜ 3 ˜ 3 4 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 natureMontag, 29. August 2011
  102. 102. 5. Smearing in the non-BPS caseMontag, 29. August 2011
  103. 103. Recall the smeared GKP solutions: 2 ˜ 1 φ −φ ˜ 0 = R(4) + 2 e F3 + e ∗6 H3 ˜ (4) ≤ 0 ⇒R (AdS or Minkowski)Montag, 29. August 2011
  104. 104. Recall the smeared GKP solutions: 2 ˜ 1 φ −φ ˜ 0 = R(4) + 2 e F3 + e ∗6 H3 ˜ (4) ≤ 0 ⇒R (AdS or Minkowski) Violating the BPS condition, i.e., assuming F3 + e−φ ∗6 H3 = 0 ˜ allows for (stable) AdS-solutions, e.g. AdS4 × S3 × S3Montag, 29. August 2011
  105. 105. Recall the smeared GKP solutions: 2 ˜ 1 φ −φ ˜ 0 = R(4) + 2 e F3 + e ∗6 H3 ˜ (4) ≤ 0 ⇒R (AdS or Minkowski) Violating the BPS condition, i.e., assuming F3 + e−φ ∗6 H3 = 0 ˜ allows for (stable) AdS-solutions, e.g. AdS4 × S3 × S3 Need D3-branes instead of O3-planesMontag, 29. August 2011
  106. 106. 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)Montag, 29. August 2011
  107. 107. 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.Montag, 29. August 2011
  108. 108. 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. For the analogous smeared non-BPS solution on AdS7 × S3 one can show that there is no continuous interpolation between the smeared solution and a fully localized counterpart (if it exists at all). Blåbäck, Danielsson, Junghans,Van Riet, Wrase, MZ (2011)Montag, 29. August 2011
  109. 109. ρ(x) xMontag, 29. August 2011
  110. 110. ρ(x) xMontag, 29. August 2011
  111. 111. ρ(x) xMontag, 29. August 2011
  112. 112. ρ(x) xMontag, 29. August 2011
  113. 113. ρ(x) x Works for BPSMontag, 29. August 2011
  114. 114. But: Only smooth non-BPS solution is the smeared one: ρ(x) xMontag, 29. August 2011
  115. 115. Moreover: If a localized solution disconnected from the smeared one exists, it must involve non-standard boundary conditions at the D6-brane (divergent H3 ). Blåbäck, Danielsson, Junghans,Van Riet, Wrase, MZ (2011)Montag, 29. August 2011
  116. 116. Moreover: If a localized solution disconnected from the smeared one exists, it must involve non-standard boundary conditions at the D6-brane (divergent H3 ). Blåbäck, Danielsson, Junghans,Van Riet, Wrase, MZ (2011) Whether this makes sense is still unclear Cf. also Bena, Grana, Halmagyi (2009)Montag, 29. August 2011
  117. 117. 6. ConclusionsMontag, 29. August 2011
  118. 118. Smearing D-branes and O-planes is a common and helpful simplificationMontag, 29. August 2011
  119. 119. Smearing D-branes and O-planes is a common and helpful simplification For BPS configurations we found this to be a quite robust approximationMontag, 29. August 2011
  120. 120. 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 problem of negatively curved spaces for BPS solutionsMontag, 29. August 2011
  121. 121. 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 problem of 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.Montag, 29. August 2011
  122. 122. Unfortunately, de Sitter vacua should be non-BPS, so it is still unclear whether smearing makes sense here.Montag, 29. August 2011
  123. 123. 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)Montag, 29. August 2011

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