The Landscape of String theory

Transcription

1 The Landscape of String theory Dieter Lüst, LMU (ASC) and MPI München

2 The Landscape of String theory Dieter Lüst, LMU (ASC) and MPI München in collaboration with Riccardo Appreda, Ralph Blumenhagen, Gabriel L. Cardoso, Mirjam Cvetic, Johanna Erdmenger, Florian Gmeiner, Viviane Grass, Michael Haack, Daniel Krefl, Gabriele Honecker, Costas Kounnas, Jan Perz, Marios Petropoulos, Susanne Reffert, Robert Richter, Christoph Sieg, Maren Stein, Stephan Stieberger Antoine van Proeyen, Dimitri Tsimpis, Timo Weigand and Marco Zagermann

3 I) Introduction Geometry: Calabi-Yau spaces, mirror symmetry, generalized spaces, D-branes, K-theory,... Black Hole Entropies String Theory SUSY field theories Particle physics, astrophysics, cosmolgy

4 String theory: Describes the interactions and the spectrum of 1-dimensional extended (closed & open) strings.

5 String theory: Describes the interactions and the spectrum of 1-dimensional extended (closed & open) strings. High energies (E O(M Planck ), D = 10) : Theory of Quantum Gravity (quantized space-time?)

6 String theory: Describes the interactions and the spectrum of 1-dimensional extended (closed & open) strings. High energies (E O(M Planck ), D = 10) : Theory of Quantum Gravity (quantized space-time?) Intermediate energies (E O(TeV), D = 4(6?)) : Perturbative (supersymmetric?) Standard Model (by compactification) with 3 quark families.

7 String theory: Describes the interactions and the spectrum of 1-dimensional extended (closed & open) strings. High energies (E O(M Planck ), D = 10) : Theory of Quantum Gravity (quantized space-time?) Intermediate energies (E O(TeV), D = 4(6?)) : Perturbative (supersymmetric?) Standard Model (by compactification) with 3 quark families. Low energies (E O(GeV), D = 4(5?)) Can string theory provide some non-perturbative informations for QCD?

8 Count the number of consistent string solutions Vast landscape with N sol = discrete vacua! (Lerche, Lüst, Schellekens (1986), Douglas (2003))

9 Count the number of consistent string solutions Vast landscape with N sol = discrete vacua! (Lerche, Lüst, Schellekens (1986), Douglas (2003)) Two strategies to find something interesting:

10 Count the number of consistent string solutions Vast landscape with N sol = discrete vacua! (Lerche, Lüst, Schellekens (1986), Douglas (2003)) Two strategies to find something interesting: Explore all mathematically consistent possibilities: top down approach (quite hard), string statistics (perhaps some anthropic point of view is necessary?)

11 Count the number of consistent string solutions Vast landscape with N sol = discrete vacua! (Lerche, Lüst, Schellekens (1986), Douglas (2003)) Two strategies to find something interesting: Explore all mathematically consistent possibilities: top down approach (quite hard), string statistics (perhaps some anthropic point of view is necessary?) Do not look randomly - look for green (promising) spots in the landscape model building, bottom up approach.

12 Outline Heterotic string compactifications Type II orientifolds models Intersecting brane models and their statistics D-instantons: non-perturbative couplings The string landscape: fluxes and branes AdS 2 AdS 4 flux vacua and black holes flux vacua and domain walls

13 Outline Heterotic string compactifications Type II orientifolds models Intersecting brane models and their statistics D-instantons: non-perturbative couplings The string landscape: fluxes and branes AdS 2 AdS 4 flux vacua and black holes flux vacua and domain walls (Review: D. Lüst, arxiv:0707:2305)

14 Outline Heterotic string compactifications Type II orientifolds models Intersecting brane models and their statistics D-instantons: non-perturbative couplings The string landscape: fluxes and branes AdS 2 AdS 4 flux vacua and black holes flux vacua and domain walls (Review: D. Lüst, arxiv:0707:2305)

15 III) Heterotic String Compactifications: 10 dimensions: gauge group G 10 = E 8 E 8, (SO(32))

16 III) Heterotic String Compactifications: 10 dimensions: gauge group G 10 = E 8 E 8, (SO(32)) N = 1 supersymmetric compactification to 4 dimensions:

17 III) Heterotic String Compactifications: 10 dimensions: gauge group G 10 = E 8 E 8, (SO(32)) N = 1 supersymmetric compactification to 4 dimensions: (i) Choice of internal 6-dimensial manifold M 6 : Calabi-Yau space, orbifold

18 III) Heterotic String Compactifications: 10 dimensions: gauge group G 10 = E 8 E 8, (SO(32)) N = 1 supersymmetric compactification to 4 dimensions: (i) Choice of internal 6-dimensial manifold M 6 : Calabi-Yau space, orbifold (ii) Choice of stable vector bundle V = H over M 6 : SUSY: F ab = F tadpoles: c 2 (V ) = 0 ab = g ab F ab = 0;

19 III) Heterotic String Compactifications: 10 dimensions: gauge group G 10 = E 8 E 8, (SO(32)) N = 1 supersymmetric compactification to 4 dimensions: (i) Choice of internal 6-dimensial manifold M 6 : Calabi-Yau space, orbifold (ii) Choice of stable vector bundle V = H over M 6 : SUSY: F ab = F tadpoles: c 2 (V ) = 0 ab = g ab F ab = 0; 4D gauge group G 4 : G 10 H G 4

20 III) Heterotic String Compactifications: 10 dimensions: gauge group G 10 = E 8 E 8, (SO(32)) N = 1 supersymmetric compactification to 4 dimensions: (i) Choice of internal 6-dimensial manifold M 6 : Calabi-Yau space, orbifold (ii) Choice of stable vector bundle V = H over M 6 : SUSY: F ab = F tadpoles: c 2 (V ) = 0 ab = g ab F ab = 0; 4D gauge group G 4 : G 10 H G 4 4D chiral matter fields: N F = c 3 (V )

21 Heterotic CY Compactifications

22 Heterotic CY Compactifications (i) (elliptically fibred) CY with H being a simple group: (Friedman, Morgan, Witten (1997); Andreas, Curio, Klemm (1999); Donagi, Ovrut, Pantev, Waldram (2000);...)

23 Heterotic CY Compactifications (i) (elliptically fibred) CY with H being a simple group: (Friedman, Morgan, Witten (1997); Andreas, Curio, Klemm (1999); Donagi, Ovrut, Pantev, Waldram (2000);...) GUT Models: H = SU(4) = G 4 = SO(10) H = SU(5) = G 4 = SU(5) No adjoint Higgs: need discrete Wilson line to break G 4 to SU(3) SU(2) U(1)

24 Heterotic CY Compactifications (i) (elliptically fibred) CY with H being a simple group: (Friedman, Morgan, Witten (1997); Andreas, Curio, Klemm (1999); Donagi, Ovrut, Pantev, Waldram (2000);...) GUT Models: H = SU(4) = G 4 = SO(10) H = SU(5) = G 4 = SU(5) No adjoint Higgs: need discrete Wilson line to break G 4 to SU(3) SU(2) U(1) (ii) (elliptically fibred) CY with H being a non-simple group: (Blumenhagen, Honecker, Weigand (2005);...)

25 Heterotic CY Compactifications (i) (elliptically fibred) CY with H being a simple group: (Friedman, Morgan, Witten (1997); Andreas, Curio, Klemm (1999); Donagi, Ovrut, Pantev, Waldram (2000);...) GUT Models: No adjoint Higgs: need discrete Wilson line to break G 4 to Flipped SU(5): H = SU(4) = G 4 = SO(10) H = SU(5) = G 4 = SU(5) SU(3) SU(2) U(1) (ii) (elliptically fibred) CY with H being a non-simple group: (Blumenhagen, Honecker, Weigand (2005);...) H = SU(4) U(1) = G 4 = SU(5) U(1) SM: H = SU(5) U(1) = G 4 = SU(3) SU(2) U(1)

26 Resume: heterotic strings:

27 Resume: heterotic strings: Heterotic CY compactifications are mathematically interesting, but also complicated.

28 Resume: heterotic strings: Heterotic CY compactifications are mathematically interesting, but also complicated. 3 generation CY models without exotic partices can be constructed. (Braun, He, Ovrut, Pantev (2005); Bouchard, Donagi (2005))

29 Resume: heterotic strings: Heterotic CY compactifications are mathematically interesting, but also complicated. 3 generation CY models without exotic partices can be constructed. (Braun, He, Ovrut, Pantev (2005); Bouchard, Donagi (2005)) Nice SO(10) GUT models from orbifolds. (Kobayashi, Raby, Zhang (2004);,... Buchmüller, Hamaguchi, Lebedev, Ratz (2005); Lebedev, Nilles, Raby, Ramos-Sanchez, Ratz, Vaudrevange (2006);...)

30 Resume: heterotic strings: Heterotic CY compactifications are mathematically interesting, but also complicated. 3 generation CY models without exotic partices can be constructed. (Braun, He, Ovrut, Pantev (2005); Bouchard, Donagi (2005)) Nice SO(10) GUT models from orbifolds. (Kobayashi, Raby, Zhang (2004);,... Buchmüller, Hamaguchi, Lebedev, Ratz (2005); Lebedev, Nilles, Raby, Ramos-Sanchez, Ratz, Vaudrevange (2006);...) Moduli stabilization (bundle moduli!) with H-flux is difficult. (Strominger (1985), Becker, Becker, Dasguta, Green (2003); Curio, Cardoso, Dall Agata, Lüst, Krause (2003/04/05); Braun, He, Ovrut (2006))

31 Outline Heterotic string compactifications Type II orientifolds models Intersecting brane models and their statistics D-instantons: non-perturbative couplings The string landscape: fluxes and branes AdS 2 AdS 4 flux vacua and black holes flux vacua and domain walls

32 Outline Heterotic string compactifications Type II orientifolds models Intersecting brane models and their statistics D-instantons: non-perturbative couplings The string landscape: fluxes and branes AdS 2 AdS 4 flux vacua and black holes flux vacua and domain walls

33 IV) (Intersecting) D-brane models: (Bachas (1995); Blumenhagen, Görlich, Körs, Lüst (2000); Angelantonj, Antoniadis, Dudas Sagnotti (2000); Ibanez, Marchesano, Rabadan (2001); Cvetic, Shiu, Uranga (2001);...)

34 IV) (Intersecting) D-brane models: (Bachas (1995); Blumenhagen, Görlich, Körs, Lüst (2000); Angelantonj, Antoniadis, Dudas Sagnotti (2000); Ibanez, Marchesano, Rabadan (2001); Cvetic, Shiu, Uranga (2001);...) Now consider open string compactifications with intersecting D-branes Type IIA/B orientifolds:

35 IV) (Intersecting) D-brane models: (Bachas (1995); Blumenhagen, Görlich, Körs, Lüst (2000); Angelantonj, Antoniadis, Dudas Sagnotti (2000); Ibanez, Marchesano, Rabadan (2001); Cvetic, Shiu, Uranga (2001);...) Now consider open string compactifications with intersecting D-branes Type IIA/B orientifolds: They exhibit several new features:

36 IV) (Intersecting) D-brane models: (Bachas (1995); Blumenhagen, Görlich, Körs, Lüst (2000); Angelantonj, Antoniadis, Dudas Sagnotti (2000); Ibanez, Marchesano, Rabadan (2001); Cvetic, Shiu, Uranga (2001);...) Now consider open string compactifications with intersecting D-branes Type IIA/B orientifolds: They exhibit several new features: Non-Abelian gauge bosons live as open strings on lower dimensional world volumes π of D-branes.

37 IV) (Intersecting) D-brane models: (Bachas (1995); Blumenhagen, Görlich, Körs, Lüst (2000); Angelantonj, Antoniadis, Dudas Sagnotti (2000); Ibanez, Marchesano, Rabadan (2001); Cvetic, Shiu, Uranga (2001);...) Now consider open string compactifications with intersecting D-branes Type IIA/B orientifolds: They exhibit several new features: Non-Abelian gauge bosons live as open strings on lower dimensional world volumes π of D-branes. Chiral fermions are open strings on the intersection locus of two D-branes: N F = I ab #(π a π b ) π a π b

38 IV) (Intersecting) D-brane models: (Bachas (1995); Blumenhagen, Görlich, Körs, Lüst (2000); Angelantonj, Antoniadis, Dudas Sagnotti (2000); Ibanez, Marchesano, Rabadan (2001); Cvetic, Shiu, Uranga (2001);...) Now consider open string compactifications with intersecting D-branes Type IIA/B orientifolds: They exhibit several new features: Non-Abelian gauge bosons live as open strings on lower dimensional world volumes π of D-branes. Chiral fermions are open strings on the intersection locus of two D-branes: N F = I ab #(π a π b ) π a π b Can be combined with background fluxes moduli stabilization (GKP) and ds-vacua (KKLT)

39 Perturbative type II orientifolds contain: (Review: Blumenhagen, Körs, Lüst, Stieberger, hep-th/ )

40 Perturbative type II orientifolds contain: (Review: Blumenhagen, Körs, Lüst, Stieberger, hep-th/ ) Closed string 6-dimensional background geometry: -Torus, orbifold, Calabi-Yau space, generalized spaces with torsion,... - Background fluxes: H NS 3, F R p, ω geom

41 Perturbative type II orientifolds contain: (Review: Blumenhagen, Körs, Lüst, Stieberger, hep-th/ ) Closed string 6-dimensional background geometry: -Torus, orbifold, Calabi-Yau space, generalized spaces with torsion,... - Background fluxes: H NS 3, F R p, ω geom Space-time filling D(4+p)-branes wrapped around internal p-cycles: - Open string matter fields.

42 Perturbative type II orientifolds contain: (Review: Blumenhagen, Körs, Lüst, Stieberger, hep-th/ ) Closed string 6-dimensional background geometry: -Torus, orbifold, Calabi-Yau space, generalized spaces with torsion,... - Background fluxes: H NS 3, F R p, ω geom Space-time filling D(4+p)-branes wrapped around internal p-cycles: - Open string matter fields. Strong consistency conditions: - tadpole cancellation with orientifold planes. - space-time supersymmetry (brane stability)

43 Perturbative type II orientifolds contain: (Review: Blumenhagen, Körs, Lüst, Stieberger, hep-th/ ) Closed string 6-dimensional background geometry: -Torus, orbifold, Calabi-Yau space, generalized spaces with torsion,... - Background fluxes: H NS 3, F R p, ω geom Space-time filling D(4+p)-branes wrapped around internal p-cycles: - Open string matter fields. (diophantic Strong consistency conditions: equations with - tadpole cancellation with orientifold planes. finite no. of - space-time supersymmetry (brane stability) solutions!)

44 Space-time filling D-branes:

45 Space-time filling D-branes: Geometrical, large radius regime: IIA: special lagrangian submanifolds: D6 on 3-cycles at angles Mirror symmetry (SYZ) IIB: points, (complex lines), divisors, (CY) with gauge bundles: D3 (D5) D7 (D9)

46 Space-time filling D-branes: Geometrical, large radius regime: IIA: special lagrangian submanifolds: D6 on 3-cycles at angles Mirror symmetry (SYZ) IIB: points, (complex lines), divisors, (CY) with gauge bundles: D3 (D5) D7 (D9)! M IR (3,1) D6 O6 D6

47 Space-time filling D-branes: Geometrical, large radius regime: IIA: special lagrangian submanifolds: D6 on 3-cycles at angles Mirror symmetry (SYZ) IIB: points, (complex lines), divisors, (CY) with gauge bundles: D3 (D5) D7 (D9) SM Soft SUSY breaking HS

48 (Intersecting) D-brane statistics How many orientifold models exist which come close to the (spectrum of the) MSSM? (Blumenhagen, Gmeiner, Honecker, Lüst, Stein, Weigand, hep-th/ , hep-th/ , hep-th/ ; related work: Dijkstra, Huiszoon, Schellekens, hep-th/ ; Anastasopoulos, Dijkstra, Kiritsis, Schellekens, hep-th/ ; Douglas, Taylor, hep-th/ ; Dienes, Lennek, hep-th/ )

49 (Intersecting) D-brane statistics How many orientifold models exist which come close to the (spectrum of the) MSSM? (Blumenhagen, Gmeiner, Honecker, Lüst, Stein, Weigand, hep-th/ , hep-th/ , hep-th/ ; related work: Dijkstra, Huiszoon, Schellekens, hep-th/ ; Anastasopoulos, Dijkstra, Kiritsis, Schellekens, hep-th/ ; Douglas, Taylor, hep-th/ ; Dienes, Lennek, hep-th/ ) Example: M 6 = T 6 /(Z 2 Z 2 ) IIA orientifold:

50 (Intersecting) D-brane statistics How many orientifold models exist which come close to the (spectrum of the) MSSM? (Blumenhagen, Gmeiner, Honecker, Lüst, Stein, Weigand, hep-th/ , hep-th/ , hep-th/ ; related work: Dijkstra, Huiszoon, Schellekens, hep-th/ ; Anastasopoulos, Dijkstra, Kiritsis, Schellekens, hep-th/ ; Douglas, Taylor, hep-th/ ; Dienes, Lennek, hep-th/ ) Example: M 6 = T 6 /(Z 2 Z 2 ) IIA orientifold: Systematic computer search (NP complete problem): Look for solutions of a set of diophantic equations:

51 (Intersecting) D-brane statistics How many orientifold models exist which come close to the (spectrum of the) MSSM? (Blumenhagen, Gmeiner, Honecker, Lüst, Stein, Weigand, hep-th/ , hep-th/ , hep-th/ ; related work: Dijkstra, Huiszoon, Schellekens, hep-th/ ; Anastasopoulos, Dijkstra, Kiritsis, Schellekens, hep-th/ ; Douglas, Taylor, hep-th/ ; Dienes, Lennek, hep-th/ ) Example: M 6 = T 6 /(Z 2 Z 2 ) IIA orientifold: Systematic computer search (NP complete problem): Look for solutions of a set of diophantic equations: There exist about susy D-brane models on this orbifold (with restricted complex structure)! (Finiteness of models was recently proven by D.T.)

52 (Intersecting) D-brane model building Require additional further phenomenological restrictions:

53 (Intersecting) D-brane model building Require additional further phenomenological restrictions: Restriction Factor gauge factor U(3) gauge factor U(2)/Sp(2) No symmetric representations Massless U(1) Y Three generations of quarks Three generations of leptons Total

54 (Intersecting) D-brane model building Require additional further phenomenological restrictions: Restriction Factor gauge factor U(3) gauge factor U(2)/Sp(2) No symmetric representations Massless U(1) Y Three generations of quarks Three generations of leptons Total Only one in a billion models gives rise to a MSSM like vacuum!

55 More results: D-brane statistics: (Gmeiner, Stein, hep-th/ ; Gmeiner, Lüst, Stein, ; Gmeiner, Honecker, arxiv: )

56 More results: D-brane statistics: (Gmeiner, Stein, hep-th/ ; Gmeiner, Lüst, Stein, ; Gmeiner, Honecker, arxiv: ) (i) Z6-orientifold: (exceptional, blowing-up 3-cycles!) In total susy D-brane models of them possess MSSM like spectra!

57 More results: D-brane statistics: (Gmeiner, Stein, hep-th/ ; Gmeiner, Lüst, Stein, ; Gmeiner, Honecker, arxiv: ) (i) Z6-orientifold: (exceptional, blowing-up 3-cycles!) In total susy D-brane models of them possess MSSM like spectra! (ii) Similar results are obtained for SU(5) GUT models.

58 More results: D-brane statistics: (Gmeiner, Stein, hep-th/ ; Gmeiner, Lüst, Stein, ; Gmeiner, Honecker, arxiv: ) (i) Z6-orientifold: (exceptional, blowing-up 3-cycles!) In total susy D-brane models of them possess MSSM like spectra! (ii) Similar results are obtained for SU(5) GUT models. Explicit D-brane constructions: there exist many models that come close to the MSSM. Problem of exotic particles! (Chen, Li, Mayes, Nanopoulos, hep-th/ ; Chen, Li, Nanopoulos, hep-th/ ; Blumenhagen, Plauschinn, hep-th/ ; Bailin, Love, hep-th/ ; Blumenhagen, Cvetic, Marchesano, Shiu, hepth/ ; Marchesano, Shiu, hep-th/ ; Honecker, Ott, hep-th/ ;...)

59 Outline Heterotic string compactifications Type II orientifolds models Intersecting brane models and their statistics D-instantons: non-perturbative couplings The string landscape: fluxes and branes AdS 2 AdS 4 flux vacua and black holes flux vacua and domain walls

60 Outline Heterotic string compactifications Type II orientifolds models Intersecting brane models and their statistics D-instantons: non-perturbative couplings The string landscape: fluxes and branes AdS 2 AdS 4 flux vacua and black holes flux vacua and domain walls

61 Non-perturbative D-instantons: (M. Dine, N. Seiberg, X. Wen, E. Witten; K. Becker, M. Becker, A. Strominger; M. Green, M. Gutperle; J. Harvey, G. Moore; M. Billo, M. Frau, F. Fucito, A. Lerda, I. Pesandro; N. Dorey, T. Hollowwod, V. Khoze;... Recent review: M. Bianchi, S. Kovacs, G. Rossi)

62 Non-perturbative D-instantons: (M. Dine, N. Seiberg, X. Wen, E. Witten; K. Becker, M. Becker, A. Strominger; M. Green, M. Gutperle; J. Harvey, G. Moore; M. Billo, M. Frau, F. Fucito, A. Lerda, I. Pesandro; N. Dorey, T. Hollowwod, V. Khoze;... Recent review: M. Bianchi, S. Kovacs, G. Rossi) Perturbative effective action:

63 Non-perturbative D-instantons: (M. Dine, N. Seiberg, X. Wen, E. Witten; K. Becker, M. Becker, A. Strominger; M. Green, M. Gutperle; J. Harvey, G. Moore; M. Billo, M. Frau, F. Fucito, A. Lerda, I. Pesandro; N. Dorey, T. Hollowwod, V. Khoze;... Recent review: M. Bianchi, S. Kovacs, G. Rossi) Perturbative effective action: SM-sector: Contains global U(1) symmetries that forbid certain couplings (neutrino masses, Yukawa couplings)

64 Non-perturbative D-instantons: (M. Dine, N. Seiberg, X. Wen, E. Witten; K. Becker, M. Becker, A. Strominger; M. Green, M. Gutperle; J. Harvey, G. Moore; M. Billo, M. Frau, F. Fucito, A. Lerda, I. Pesandro; N. Dorey, T. Hollowwod, V. Khoze;... Recent review: M. Bianchi, S. Kovacs, G. Rossi) Perturbative effective action: SM-sector: Contains global U(1) symmetries that forbid certain couplings (neutrino masses, Yukawa couplings) Hidden sector: Moduli stabilization by fluxes. (Gukov, Vafa, Witten (2000); Taylor, Vafa; Mayr; (2000) Curio, Lüst, Klemm, Theisen (2000); Giddings, Kachru, Polchinski (2002) Lüst, Reffert, Stieberger (2004); Lüst, Tsimpis (2004); Derendinger, Petropoulos, Kounnas, Zwirner (2004); DeWolfe, Giryavets, Kachru, Taylor (2005); Camara, Font, Ibanez (2005); Villadoro, Zwirner (2007);...) Very often, not all moduli are stabilized.

65 Non-perturbative D-instantons: (M. Dine, N. Seiberg, X. Wen, E. Witten; K. Becker, M. Becker, A. Strominger; M. Green, M. Gutperle; J. Harvey, G. Moore; M. Billo, M. Frau, F. Fucito, A. Lerda, I. Pesandro; N. Dorey, T. Hollowwod, V. Khoze;... Recent review: M. Bianchi, S. Kovacs, G. Rossi) Perturbative effective action: SM-sector: Contains global U(1) symmetries that forbid certain couplings (neutrino masses, Yukawa couplings) Hidden sector: Moduli stabilization by fluxes. (Gukov, Vafa, Witten (2000); Taylor, Vafa; Mayr; (2000) Curio, Lüst, Klemm, Theisen (2000); Giddings, Kachru, Polchinski (2002) Lüst, Reffert, Stieberger (2004); Lüst, Tsimpis (2004); Derendinger, Petropoulos, Kounnas, Zwirner (2004); DeWolfe, Giryavets, Kachru, Taylor (2005); Camara, Font, Ibanez (2005); Villadoro, Zwirner (2007);...) Very often, not all moduli are stabilized. Unbroken space-time supersymmetry.

66 Non-perturbative D-instantons: (M. Dine, N. Seiberg, X. Wen, E. Witten; K. Becker, M. Becker, A. Strominger; M. Green, M. Gutperle; J. Harvey, G. Moore; M. Billo, M. Frau, F. Fucito, A. Lerda, I. Pesandro; N. Dorey, T. Hollowwod, V. Khoze;... Recent review: M. Bianchi, S. Kovacs, G. Rossi) Perturbative effective action: SM-sector: Contains global U(1) symmetries that forbid certain couplings (neutrino masses, Yukawa couplings) Hidden sector: Moduli stabilization by fluxes. (Gukov, Vafa, Witten (2000); Taylor, Vafa; Mayr; (2000) Curio, Lüst, Klemm, Theisen (2000); Giddings, Kachru, Polchinski (2002) Lüst, Reffert, Stieberger (2004); Lüst, Tsimpis (2004); Derendinger, Petropoulos, Kounnas, Zwirner (2004); DeWolfe, Giryavets, Kachru, Taylor (2005); Camara, Font, Ibanez (2005); Villadoro, Zwirner (2007);...) Very often, not all moduli are stabilized. Unbroken space-time supersymmetry. Take into account non-perturbative instanton corrections to the effective action!

67 String instanton corrections:

68 String instanton corrections: String instantons arise in a geometric way from branes wrapped around internal cycles.

69 String instanton corrections: String instantons arise in a geometric way from branes wrapped around internal cycles. It is possible to compute string instanton corrections from open string CFT (not only guess them from field theory). (R. Blumenhagen, M. Cvetic, T. Weigand, hep-th/ )

70 String instanton corrections: String instantons arise in a geometric way from branes wrapped around internal cycles. It is possible to compute string instanton corrections from open string CFT (not only guess them from field theory). (R. Blumenhagen, M. Cvetic, T. Weigand, hep-th/ ) String instantons can break axionic shift symmetries and hence global U(1) symmetries.

71 String instanton corrections: String instantons arise in a geometric way from branes wrapped around internal cycles. It is possible to compute string instanton corrections from open string CFT (not only guess them from field theory). (R. Blumenhagen, M. Cvetic, T. Weigand, hep-th/ ) String instantons can break axionic shift symmetries and hence global U(1) symmetries. Can generate new moduli dependent terms in the superpotential and hence are important for the moduli stabilization. (S. Kachru, R. Kallosh, A. Linde, S. Trivedi (2003);F. Denef, M. Douglas, B. Florea, A. Grassi, S. Kachru (2005); D.L., S. Reffert, E. Scheidegger, W. Schulgin, S. Stieberger(2006))

72 String instanton corrections: String instantons arise in a geometric way from branes wrapped around internal cycles. It is possible to compute string instanton corrections from open string CFT (not only guess them from field theory). (R. Blumenhagen, M. Cvetic, T. Weigand, hep-th/ ) String instantons can break axionic shift symmetries and hence global U(1) symmetries. Can generate new moduli dependent terms in the superpotential and hence are important for the moduli stabilization. (S. Kachru, R. Kallosh, A. Linde, S. Trivedi (2003);F. Denef, M. Douglas, B. Florea, A. Grassi, S. Kachru (2005); D.L., S. Reffert, E. Scheidegger, W. Schulgin, S. Stieberger(2006)) Can generate new matter couplings (Majorana masses, Yukawa couplings) see in a moment.

73 Geometry: Non space-time filling Euclidean D- branes:

74 Geometry: Non space-time filling Euclidean D- branes: Two kinds of string instantons:

75 Geometry: Non space-time filling Euclidean D- branes: Two kinds of string instantons: World-sheet instantons: exp( J/α ). Tree-level in CFT

76 Geometry: Non space-time filling Euclidean D- branes: Two kinds of string instantons: World-sheet instantons: Space-time instantons: exp( 1/g s ) exp( J/α ). Tree-level in CFT

77 Geometry: Non space-time filling Euclidean D- branes: Two kinds of string instantons: World-sheet instantons: Space-time instantons: exp( 1/g s ) exp( J/α ). Tree-level in CFT These are non-space time filling D(p)=E(p) branes wrapped around internal (p+1)-cycles:

78 Geometry: Non space-time filling Euclidean D- branes: Two kinds of string instantons: World-sheet instantons: Space-time instantons: exp( 1/g s ) exp( J/α ). Tree-level in CFT These are non-space time filling D(p)=E(p) branes wrapped around internal (p+1)-cycles: IIA: special lagrangian submanifolds: E2 on 3-cycles Mirror symmetry (SYZ) IIB: points, (complex lines), divisors, (CY) E(-1) (E1) E3 (E5)

79 Open string instanton CFT: (R. Blumenhagen, M. Cvetic, T. Weigand)

80 Open string instanton CFT: (R. Blumenhagen, M. Cvetic, T. Weigand) Zero mode fields: massless open strings on E(p)-branes or at intersection of E(p) and D(p )-branes.

81 Open string instanton CFT: (R. Blumenhagen, M. Cvetic, T. Weigand) Zero mode fields: massless open strings on E(p)-branes or at intersection of E(p) and D(p )-branes. Compute open string amplitudes between matter fields and zero mode fields.

82 Open string instanton CFT: (R. Blumenhagen, M. Cvetic, T. Weigand) Zero mode fields: massless open strings on E(p)-branes or at intersection of E(p) and D(p )-branes. Compute open string amplitudes between matter fields and zero mode fields. Finally, one has to integrate over all bosonic and fermionic zero modes localised on E(p)-branes

83 Open string instanton CFT: (R. Blumenhagen, M. Cvetic, T. Weigand) Zero mode fields: massless open strings on E(p)-branes or at intersection of E(p) and D(p )-branes. Compute open string amplitudes between matter fields and zero mode fields. Finally, one has to integrate over all bosonic and fermionic zero modes localised on E(p)-branes Not all instantons can contribute to the effective action: - F-terms: E(p)-brane is half-bps: need two fermionic zero modes θ i - D-terms: E(p)-brane breaks all 4 supersymmetries: need four fermionic zero modes θ i, θ i

84 E(2)-instantons in IIA: In the following we will consider the contribution of a E(2)- instanton to the superpotential in type IIA orientfolds.

85 E(2)-instantons in IIA: In the following we will consider the contribution of a E(2)- instanton to the superpotential in type IIA orientfolds. The E(2)-brane is wrapping a 3-cycle Instanton action: W np e S E = exp ( 2π l 3 s ( 1 g s Ξ Ξ Re(Ω) i of the CY-space. Ξ C 3 )) ) e U Ξ

86 E(2)-instantons in IIA: In the following we will consider the contribution of a E(2)- instanton to the superpotential in type IIA orientfolds. The E(2)-brane is wrapping a 3-cycle Instanton action: W np e S E = exp ( 2π l 3 s ( 1 g s Ξ Ξ Re(Ω) i of the CY-space. Ξ C 3 )) ) e U Ξ In general the instanton action W np is not U(1) a gauge invariant (shifts in C 3 ): C 3 C 3 + N a Q aξ, Q aξ = I[Π a Ξ] Ξ Ξ

87 E(2)-instantons in IIA: In the following we will consider the contribution of a E(2)- instanton to the superpotential in type IIA orientfolds. The E(2)-brane is wrapping a 3-cycle Instanton action: W np e S E = exp ( 2π l 3 s ( 1 g s Ξ Ξ Re(Ω) i of the CY-space. Ξ C 3 )) ) e U Ξ In general the instanton action W np is not U(1) a gauge invariant (shifts in C 3 ): C 3 C 3 + N a Q aξ, Q aξ = I[Π a Ξ] Ξ W np Ξ must contain charged matter fields: W np Φ a e S E (U(1) selection rules will follow from fermionic zero modes.)

88 There are two possible cases:

89 There are two possible cases: E2 is wrapping a 3-cycle different from the gauge group 3-cycle, i.e. E2-brane describes a genuine string instanton.

90 There are two possible cases: E2 is wrapping a 3-cycle different from the gauge group 3-cycle, i.e. E2-brane describes a genuine string instanton. E2 is wrapping the same 3-cycle as the gauge group 3-cycle, i.e. Ξ = Π a E2-brane corresponds to a gauge instanton. additional bosonic zero modes 4N c b, b, b, b.

91 There are two possible cases: E2 is wrapping a 3-cycle different from New the matter gauge group 3-cycle, i.e. couplings E2-brane describes a genuine string instanton. E2 is wrapping the same 3-cycle as the gauge group 3-cycle, i.e. Ξ = Π a E2-brane corresponds to a gauge instanton. additional bosonic zero modes 4N c b, b, b, b.

92 There are two possible cases: E2 is wrapping a 3-cycle different from New the matter gauge group 3-cycle, i.e. couplings E2-brane describes a genuine string instanton. E2 is wrapping the same 3-cycle as the gauge group 3-cycle, i.e. Ξ = Π a E2-brane corresponds to a gauge instanton. additional bosonic zero modes 4N c b, b, b, b. Compute the 1-instanton Veneziano-Yankielowicz/ Affleck-Dine-Seiberg (VY/ADS) superpotential from string theory. (N. Akerblom, R. Blumenhagen, D.L., E. Plauschinn, M. Schmidt-Sommerfeld)

93

94 W n.p. = VY/ADS non-perturbative superpotential: = ( ) 1 det[ Φ ab Φ ab ] exp f a,tree + f a,1 loop (M) ( ) 1 C(T ) exp f a,tree (S, U) det[ Φ ab Φ ab ] SU(N c ), N f = N c

95 W n.p. = VY/ADS non-perturbative superpotential: = ( ) 1 det[ Φ ab Φ ab ] exp f a,tree + f a,1 loop (M) ( ) 1 C(T ) exp f a,tree (S, U) det[ Φ ab Φ ab ] SU(N c ), N f = N c

96 W n.p. = VY/ADS non-perturbative superpotential: = ( ) 1 det[ Φ ab Φ ab ] exp f a,tree + f a,1 loop (M) ( ) 1 C(T ) exp f a,tree (S, U) det[ Φ ab Φ ab ] SU(N c ), N f = N c From integration over bosonic zero modes, ADHM constraints.

97 W n.p. = VY/ADS non-perturbative superpotential: = ( ) 1 det[ Φ ab Φ ab ] exp f a,tree + f a,1 loop (M) ( ) 1 C(T ) exp f a,tree (S, U) det[ Φ ab Φ ab ] We also computed the ADS superpotential for SO(N c ), N f = N c 3 and USp(2N c ), N c = N f SU(N c ), N f = N c

98 W n.p. = VY/ADS non-perturbative superpotential: = ( ) 1 det[ Φ ab Φ ab ] exp f a,tree + f a,1 loop (M) ( ) 1 C(T ) exp f a,tree (S, U) det[ Φ ab Φ ab ] We also computed the ADS superpotential for SO(N c ), N f = N c 3 and USp(2N c ), N c = N f SU(N c ), N f = N c Possible generalizations: SU(N c ), N f N c 1, due to gaugino condensation? Non-rigid cycles (b 1 (Ξ) 0)? This corresponds to adjoint chiral fields in gauge theory, which must get a mass by fluxes.

99 Non-perturbative Yukawa couplings: (R. Blumenhagen, M. Cvetic, D. Lüst, R. Richter, T. Weigand, arxiv: )

100 Non-perturbative Yukawa couplings: (R. Blumenhagen, M. Cvetic, D. Lüst, R. Richter, T. Weigand, arxiv: ) SU(5) GUT intersecting D6-brane models:

101 Non-perturbative Yukawa couplings: (R. Blumenhagen, M. Cvetic, D. Lüst, R. Richter, T. Weigand, arxiv: ) SU(5) GUT intersecting D6-brane models: Consider two stack a and b of D6-branes: G = U(5) a U(1) b = SU(5) a U(1) a U(1) b Open strings: sector number U(5) a U(1) b reps. U(1) X (a 1, a) 3 + (1, 1) 10 (2,0) 2 (a, b) 3 5 ( 1,1) 3 2 (b 5, b) 3 1 (0, 2) 2 (a, b) 1 5 H (1,1) + 5H ( 1, 1) ( 1) + (1)

102 Non-perturbative Yukawa couplings: (R. Blumenhagen, M. Cvetic, D. Lüst, R. Richter, T. Weigand, arxiv: ) SU(5) GUT intersecting D6-brane models: Consider two stack a and b of D6-branes: G = U(5) a U(1) b = SU(5) a U(1) a U(1) b Open strings: sector number U(5) a U(1) b reps. U(1) X (a 1, a) 3 + (1, 1) 10 (2,0) 2 (a, b) 3 5 ( 1,1) 3 2 (b 5, b) 3 1 (0, 2) 2 (a, b) 1 5 H (1,1) + 5H ( 1, 1) ( 1) + (1) Abelian symmetries: U(1) a U(1) b One anomalous linear combination: global symmetry One anomaly free linear combination U(1) X

103 Two different cases: U(1) X massive: Georgi-Glashow model U(1) X massless: flipped SU(5) model

104 Two different cases: U(1) X massive: Georgi-Glashow model U(1) X massless: flipped SU(5) model From U(1) X charges it follows that: Perturb. allowed couplings(e.g. u,c,t-quarks): 10 (2,0) 5 ( 1,1) 5 H ( 1, 1), Perturb. forbidden couplings (e.g. d,s,b-quarks): 10 (2,0) 10 (2,0) 5 H (1,1)

105 Two different cases: U(1) X massive: Georgi-Glashow model U(1) X massless: flipped SU(5) model From U(1) X charges it follows that: Perturb. allowed couplings(e.g. u,c,t-quarks): 10 (2,0) 5 ( 1,1) 5 H ( 1, 1), Perturb. forbidden couplings (e.g. d,s,b-quarks): 10 (2,0) 10 (2,0) 5 H (1,1) These can be generated by E2-instantons: The instanton has to wrap a rigid 3-cycle Ξ invariant under the orientfold projection Ω σ and carrying gauge group O(1)

106 U(1) X charge of Yukawa coupling: one needs 6 fermionic zero modes: Instanton intersection numbers: [Ξ Π a ] + = [Ξ Π b ] + = 0, [Ξ Π a ] = [Ξ Π b ] = 1 E2 D6 a E2 D6 a E2 D6 b 10! 1 3 [5]! [5] # $! 2 [5] 10 4! [5] 5 H " [!1] 5! [5] D6 a D6 a D6 a

107 U(1) X charge of Yukawa coupling: one needs 6 fermionic zero modes: Instanton intersection numbers: [Ξ Π a ] + = [Ξ Π b ] + = 0, [Ξ Π a ] = [Ξ Π b ] = 1 E2 D6 a E2 D6 a E2 D6 b 10! 1 3 [5]! [5] # $! 2 [5] 10 4! [5] 5 H " [!1] 5! [5] Instanton action: Yukawa coupling: D6 a D6 a ( exp( S inst ) = exp Cα 10 D6 a ) 10 α λ [ij] i λ j + C 5 5 m λ m ν W Y = Y αβ H ɛ ijklm 10 α ij 10 β kl 5H m e S E2 e Z

108 Two features of non-perturbative Yukawa coupling:

109 Two features of non-perturbative Yukawa coupling: (i) depend on complex structure moduli: Suppression factor Vol E2 /Vol D6 = (R E2 /R D6 ) 3 with R D6 = 7 2 R E2 one gets exp ( S E2 )

110 Two features of non-perturbative Yukawa coupling: (i) depend on complex structure moduli: Suppression factor Vol E2 /Vol D6 = (R E2 /R D6 ) 3 with R D6 = 7 2 R E2 one gets exp ( S E2 ) (ii) Yukawa coupling factorizes into Y αβ H = Y α Y β Unit rank mass matrix!

111 Two features of non-perturbative Yukawa coupling: (i) depend on complex structure moduli: Suppression factor Vol E2 /Vol D6 = (R E2 /R D6 ) 3 with R D6 = 7 2 R E2 one gets exp ( S E2 ) (ii) Yukawa coupling factorizes into Y αβ H = Y α Y β Unit rank mass matrix! E2-instantons also relevant for doublet-triplet splitting, Majorana masses, masses for exotic states,... (Ibanez, Uranga, hep-th/ ; Cvetic, Richer, Weigand, hep-th/ ; Ibanez, Schellekens, Uranga, arxiv: ; Antusch, Ibanez, Macri, arxiv: ; Bianchi, Kiritsis, arxiv: ; Bianchi, Fucito, Morales, arxiv: )

112 String model buiding: What do we like to learn?

113 String model buiding: What do we like to learn? Where is the fundamental string scale?

114 String model buiding: What do we like to learn? Where is the fundamental string scale? Where is the scale of supersymmetry breaking?

115 String model buiding: What do we like to learn? Where is the fundamental string scale? Where is the scale of supersymmetry breaking? Where is the scale of the extra dimensions?

116 String model buiding: What do we like to learn? Where is the fundamental string scale? Where is the scale of supersymmetry breaking? Where is the scale of the extra dimensions? Where are we in the landscape?

117 String model buiding: What do we like to learn? Where is the fundamental string scale? Where is the scale of supersymmetry breaking? Where is the scale of the extra dimensions? Where are we in the landscape? Plan for the next years:

118 String model buiding: What do we like to learn? Where is the fundamental string scale? Where is the scale of supersymmetry breaking? Where is the scale of the extra dimensions? Where are we in the landscape? Plan for the next years: Bottom-up approach: Work upstairs from experimatal data.

119 String model buiding: What do we like to learn? Where is the fundamental string scale? Where is the scale of supersymmetry breaking? Where is the scale of the extra dimensions? Where are we in the landscape? Plan for the next years: Bottom-up approach: Work upstairs from experimatal data. Top-down approach: Develop high scale theory from mathematical consistency.

120 String model buiding: What do we like to learn? Where is the fundamental string scale? Where is the scale of supersymmetry breaking? Where is the scale of the extra dimensions? Where are we in the landscape? Plan for the next years: Bottom-up approach: Work upstairs from experimatal data. Top-down approach: Success? Develop high scale theory from mathematical consistency.

121 String model buiding: What do we like to learn? Where is the fundamental string scale? Where is the scale of supersymmetry breaking? Where is the scale of the extra dimensions? Where are we in the landscape? Plan for the next years: Bottom-up approach: Work upstairs from experimatal data. Top-down approach: Success? Develop high scale theory from mathematical consistency. (How good is the chain between fundamental theory and the data?)

122 Outline Heterotic string compactifications Type II orientifolds models Intersecting brane models and their statistics D-instantons: non-perturbative couplings The string landscape: fluxes and branes AdS 2 AdS 4 flux vacua and black holes flux vacua and domain walls

123 Outline Heterotic string compactifications Type II orientifolds models Intersecting brane models and their statistics D-instantons: non-perturbative couplings The string landscape: fluxes and branes AdS 2 AdS 4 flux vacua and black holes flux vacua and domain walls

124 String landscape problem (Weinberg, Bousso, Polchinski, Susskind, Linde, Schellekens,...) (Review: D. Lüst, arxiv:0707:2305)

125 String landscape problem (Weinberg, Bousso, Polchinski, Susskind, Linde, Schellekens,...) (Review: D. Lüst, arxiv:0707:2305) Superstring theory in 10 dimensions is (almost) unique!

126 String landscape problem (Weinberg, Bousso, Polchinski, Susskind, Linde, Schellekens,...) (Review: D. Lüst, arxiv:0707:2305) Superstring theory in 10 dimensions is (almost) unique! Compactification

127 String landscape problem (Weinberg, Bousso, Polchinski, Susskind, Linde, Schellekens,...) (Review: D. Lüst, arxiv:0707:2305) Superstring theory in 10 dimensions is (almost) unique! Compactification There exist a huge number of lower dim. groundstates:

128 String landscape problem (Weinberg, Bousso, Polchinski, Susskind, Linde, Schellekens,...) (Review: D. Lüst, arxiv:0707:2305) Superstring theory in 10 dimensions is (almost) unique! Compactification There exist a huge number of lower dim. groundstates: Many possibilities for different particle physics models!

129 String landscape problem (Weinberg, Bousso, Polchinski, Susskind, Linde, Schellekens,...) (Review: D. Lüst, arxiv:0707:2305) Superstring theory in 10 dimensions is (almost) unique! Compactification There exist a huge number of lower dim. groundstates: Many possibilities for different particle physics models! Many possibilities for different cosmological models!

130 String landscape problem (Weinberg, Bousso, Polchinski, Susskind, Linde, Schellekens,...) (Review: D. Lüst, arxiv:0707:2305) Superstring theory in 10 dimensions is (almost) unique! Compactification There exist a huge number of lower dim. groundstates: Many possibilities for different particle physics models! Many possibilities for different cosmological models! How to make any prediction in string theory, i.e. how to determine the correct string vacuum state?

131 Moduli: The landscape problem is closely related to the moduli problem:

132 Moduli: The landscape problem is closely related to the moduli problem: String compactifications contain many massless moduli fields with flat potential: (Geometrical) background parameters of the compactification.

133 Moduli: The landscape problem is closely related to the moduli problem: String compactifications contain many massless moduli fields with flat potential: (Geometrical) background parameters of the compactification. However undetermined moduli result in uncalculable couplings (and new forces)!

134 Moduli stabilization: Moduli can be stabilized, i.e. fixed by creating a (static) potential for them:

135 Moduli stabilization: Moduli can be stabilized, i.e. fixed by creating a (static) potential for them: Tree level: background fluxes

136 Moduli stabilization: Moduli can be stabilized, i.e. fixed by creating a (static) potential for them: Tree level: background fluxes Non-perturbatively: string instantons

137 Moduli stabilization: Moduli can be stabilized, i.e. fixed by creating a (static) potential for them: Tree level: background fluxes Non-perturbatively: string instantons In this way one can obtain a discrete set of vacua with either negative cosological constant, Λ < 0 (AdS vacua) zero cosmological constant, Λ = 0 (Minkowski vacua) positive cosmological constant, Λ > 0 (ds vacua) and various possibilities for gauge and matter fields

138 Effective moduli potential:

139 Effective moduli potential: Flat moduli potential before turning on fluxes and nonperturbative effects: V (φ) V (φ) 0 φ

140 Effective moduli potential:

141 Effective moduli potential: Non-flat moduli potential after turning on fluxes and nonperturbative effects: V (φ) φ

142 Effective moduli potential: Non-flat moduli potential after turning on fluxes and nonperturbative effects: V (φ) ds-vacuum φ

143 Effective moduli potential: Non-flat moduli potential after turning on fluxes and nonperturbative effects: V (φ) ds-vacuum AdSvacuum φ

144 Effective moduli potential: Non-flat moduli potential after turning on fluxes and nonperturbative effects: V (φ) ds-vacuum Minkowski vacuum AdSvacuum φ

145 Two possible solutions of the landscape problem:

146 Two possible solutions of the landscape problem: String statistics (Anthropic answer): determine the fraction of vacua with good phenomenological properties: (Λ/M P lanck ) , G = SU(3) SU(2) U(1)

147 Two possible solutions of the landscape problem: String statistics (Anthropic answer): determine the fraction of vacua with good phenomenological properties: (Λ/M P lanck ) , G = SU(3) SU(2) U(1) Entropy of string vacua (Entropic answer): determine a wave function (probability function) in moduli space, and see if is peaked, i.e. has maxima with good phenomenological properties. ψ 2 ψ(φ)

148 Wave function: Two possible ways to compute the wave function:

149 Wave function: Two possible ways to compute the wave function: (i) Computation of transition amplitudes: V (φ) (Hartle, Hawking, Coleman, De Luccia, Linde, Vilenkin,...) φ

150 Wave function: Two possible ways to compute the wave function: (i) Computation of transition amplitudes: V (φ) Tunneling: (Hartle, Hawking, Coleman, De Luccia, Linde, Vilenkin,...) ( Γ M P exp 24π2 MP 4 ) Λ(φ) ψ(φ) 2 e Λ(φ) φ

151 Wave function: Two possible ways to compute the wave function: (i) Computation of transition amplitudes: V (φ) (Hartle, Hawking, Coleman, De Luccia, Linde, Vilenkin,...) Thermal fluctuation: Γ M P e E T H φ T H = Λ(φ) M P

152 Wave function: Two possible ways to compute the wave function: (i) Computation of transition amplitudes: V (φ) (Hartle, Hawking, Coleman, De Luccia, Linde, Vilenkin,...) Thermal fluctuation: Γ M P e E T H φ T H = Λ(φ) M P (ii) Stringy computation of the entropy of string vacua: ψ(φ) 2 = e S string(φ)

153 How to define an entropy for string vacua?

154 How to define an entropy for string vacua? Similar problem in quantum gravity:

155 How to define an entropy for string vacua? Similar problem in quantum gravity: Compute the statistical entropy of black holes and compare it with the Bekenstein/Hawking entropy:

156 How to define an entropy for string vacua? Similar problem in quantum gravity: Compute the statistical entropy of black holes and compare it with the Bekenstein/Hawking entropy: S bh (p, q) = A horizon 4

157 How to define an entropy for string vacua? Similar problem in quantum gravity: Compute the statistical entropy of black holes and compare it with the Bekenstein/Hawking entropy: S bh (p, q) = A horizon 4 There exist many black hole solutions with different electric and magnetic charges!

158 How to define an entropy for string vacua? Similar problem in quantum gravity: Compute the statistical entropy of black holes and compare it with the Bekenstein/Hawking entropy: S bh (p, q) = A horizon 4 There exist many black hole solutions with different electric and magnetic charges! Which one has the largest possible entropy?

159 Conjecture: (Ooguri, Vafa, Verline, hep-th/ ) Black hole (flux) landscape correspondence:

160 Conjecture: (Ooguri, Vafa, Verline, hep-th/ ) Black hole (flux) landscape correspondence: Probability distribution for AdS flux vacua in 2 dimensions is the same as the entropy of certain charged (4D) supersymmetric black holes:

161 Conjecture: (Ooguri, Vafa, Verline, hep-th/ ) Black hole (flux) landscape correspondence: Probability distribution for AdS flux vacua in 2 dimensions is the same as the entropy of certain charged (4D) supersymmetric black holes: 3 kinds of entropies:

162 Conjecture: (Ooguri, Vafa, Verline, hep-th/ ) Black hole (flux) landscape correspondence: Probability distribution for AdS flux vacua in 2 dimensions is the same as the entropy of certain charged (4D) supersymmetric black holes: 3 kinds of entropies: (i) Thermodynamic black hole entropy: S thermo (p, q)

163 Conjecture: (Ooguri, Vafa, Verline, hep-th/ ) Black hole (flux) landscape correspondence: Probability distribution for AdS flux vacua in 2 dimensions is the same as the entropy of certain charged (4D) supersymmetric black holes: 3 kinds of entropies: (i) Thermodynamic black hole entropy: S thermo (p, q) (ii) Microscopic stringy entropy of wrapped D-branes: S brane (p, q)

164 Conjecture: (Ooguri, Vafa, Verline, hep-th/ ) Black hole (flux) landscape correspondence: Probability distribution for AdS flux vacua in 2 dimensions is the same as the entropy of certain charged (4D) supersymmetric black holes: 3 kinds of entropies: (i) Thermodynamic black hole entropy: S thermo (p, q) (ii) Microscopic stringy entropy of wrapped D-branes: S brane (p, q) (iii) Flux vacuum: S flux (φ)?

165 Conjecture: (Ooguri, Vafa, Verline, hep-th/ ) Black hole (flux) landscape correspondence: Probability distribution for AdS flux vacua in 2 dimensions is the same as the entropy of certain charged (4D) supersymmetric black holes: 3 kinds of entropies: (i) Thermodynamic black hole entropy: S thermo (p, q) (ii) Microscopic stringy entropy of wrapped D-branes: S brane (p, q) (iii) Flux vacuum: S flux (φ)? Conjecture: S thermo (p, q) = S brane (p, q) = S flux (φ)

166 Conjecture: (Ooguri, Vafa, Verline, hep-th/ ) Black hole (flux) landscape correspondence: Probability distribution for AdS flux vacua in 2 dimensions is the same as the entropy of certain charged (4D) supersymmetric black holes: 3 kinds of entropies: (i) Thermodynamic black hole entropy: S thermo (p, q) (ii) Microscopic stringy entropy of wrapped D-branes: S brane (p, q) (iii) Flux vacuum: S flux (φ)? Conjecture: S thermo (p, q) = S brane (p, q) = S flux (φ) (Strominger, Vafa)

167 Conjecture: (Ooguri, Vafa, Verline, hep-th/ ) Black hole (flux) landscape correspondence: Probability distribution for AdS flux vacua in 2 dimensions is the same as the entropy of certain charged (4D) supersymmetric black holes: 3 kinds of entropies: (i) Thermodynamic black hole entropy: S thermo (p, q) (ii) Microscopic stringy entropy of wrapped D-branes: S brane (p, q) (iii) Flux vacuum: S flux (φ)?? Conjecture: S thermo (p, q) = S brane (p, q) = S flux (φ) (Strominger, Vafa)

168 Entropy of string (flux) vacua: use the correspondence between flux potentials and attractors of supersymmetric charged black holes coupled to scalars:

169 Entropy of string (flux) vacua: use the correspondence between flux potentials and attractors of supersymmetric charged black holes coupled to scalars: Replace the fluxes by their correspond D-branes.

170 Entropy of string (flux) vacua: use the correspondence between flux potentials and attractors of supersymmetric charged black holes coupled to scalars: Replace the fluxes by their correspond D-branes. Black hole BPS (SUSY) conditions: (attractor mechanism) A z (r) A A z =z (p I Hor, q I) z A i1 DZ p,q (φ) = 0 φ = φ(p, q), S(p, q) = S(φ(p, q)) z A i2 r =0 Hor r

171 Entropy of string (flux) vacua: use the correspondence between flux potentials and attractors of supersymmetric charged black holes coupled to scalars: Replace the fluxes by their correspond D-branes. Black hole BPS (SUSY) conditions: (attractor mechanism) A z (r) A A z =z (p I Hor, q I) z A i1 DZ p,q (φ) = 0 φ = φ(p, q), S(p, q) = S(φ(p, q)) z A i2 Conjecture: S(φ(p, q)) is entropy function in the flux landscape. r =0 Hor r

172 Entropy of string (flux) vacua: use the correspondence between flux potentials and attractors of supersymmetric charged black holes coupled to scalars: Replace the fluxes by their correspond D-branes. Black hole BPS (SUSY) conditions: (attractor mechanism) A z (r) A A z =z (p I Hor, q I) z A i1 DZ p,q (φ) = 0 φ = φ(p, q), S(p, q) = S(φ(p, q)) z A i2 Conjecture: S(φ(p, q)) is entropy function in the flux landscape. r =0 Hor r (The black hole BPS conditions are very similar to the flux SUSY conditions, replacing the central charge Z by the superpotential W!)

173 Consider IIB on and through q I Example: S 2 CY with Ramond 5-form fluxes p I S 2 α I (β I ) (α I, β I CY )

174 Consider IIB on and through q I Example: S 2 CY with Ramond 5-form fluxes p I S 2 α I (β I ) (α I, β I CY ) These flux vacua correspond to supersymmetric charged black holes with electric/magnetic charges and p I : q I

175 Consider IIB on and through q I Example: S 2 CY with Ramond 5-form fluxes p I S 2 α I (β I ) (α I, β I CY ) These flux vacua correspond to supersymmetric charged black holes with electric/magnetic charges and p I : 3-branes around 3-cycles 5-form fluxes through 5-cycles q I

176 Consider IIB on and through q I Example: S 2 CY with Ramond 5-form fluxes p I S 2 α I (β I ) (α I, β I CY ) These flux vacua correspond to supersymmetric charged black holes with electric/magnetic charges and p I : 3-branes around 3-cycles 5-form fluxes through 5-cycles B.h. charges p and q Fluxes p and q q I

177 Consider IIB on and through q I Example: S 2 CY with Ramond 5-form fluxes p I S 2 α I (β I ) (α I, β I CY ) These flux vacua correspond to supersymmetric charged black holes with electric/magnetic charges and p I : 3-branes around 3-cycles 5-form fluxes through 5-cycles B.h. charges p and q Fluxes p and q q I Entropy Cosmol. constant S(φ) Λ(φ)

178 Consider IIB on and through q I Example: S 2 CY with Ramond 5-form fluxes p I S 2 α I (β I ) (α I, β I CY ) These flux vacua correspond to supersymmetric charged black holes with electric/magnetic charges and p I : 3-branes around 3-cycles 5-form fluxes through 5-cycles B.h. charges p and q Fluxes p and q q I Entropy Cosmol. constant S(φ) Near horizon geometry Vacuum geometry Λ(φ) AdS 2 S 2 CY

179 Application: entropy maximization (Cardoso, Lüst, Perz, hep-th/ ; related work: Gukov, Saraikin, Vafa, hep-th/ ; Fiol, hep-th/ ; Belluci, Ferrara, Marrani, hep-th/

180 Application: entropy maximization (Cardoso, Lüst, Perz, hep-th/ ; related work: Gukov, Saraikin, Vafa, hep-th/ ; Fiol, hep-th/ ; Belluci, Ferrara, Marrani, hep-th/ Consider N = 2 flux vacua at the Calabi-Yau conifold point in the moduli space, where additional states become massless:

181 Application: entropy maximization (Cardoso, Lüst, Perz, hep-th/ ; related work: Gukov, Saraikin, Vafa, hep-th/ ; Fiol, hep-th/ ; Belluci, Ferrara, Marrani, hep-th/ Consider N = 2 flux vacua at the Calabi-Yau conifold point in the moduli space, where additional states become massless: ( ) S(φ) = π const. β 2π φ2 2β π φ2 log φ

182 Application: entropy maximization (Cardoso, Lüst, Perz, hep-th/ ; related work: Gukov, Saraikin, Vafa, hep-th/ ; Fiol, hep-th/ ; Belluci, Ferrara, Marrani, hep-th/ Consider N = 2 flux vacua at the Calabi-Yau conifold point in the moduli space, where additional states become massless: ( ) For S(φ) = π β < 0 (β > 0) const. β 2π φ2 2β π φ2 log φ a maximum (minimum) at : φ = 0 we get that the entropy exhibits

183 Application: entropy maximization (Cardoso, Lüst, Perz, hep-th/ ; related work: Gukov, Saraikin, Vafa, hep-th/ ; Fiol, hep-th/ ; Belluci, Ferrara, Marrani, hep-th/ Consider N = 2 flux vacua at the Calabi-Yau conifold point in the moduli space, where additional states become massless: ( ) For S(φ) = π β < 0 (β > 0) const. β 2π φ2 2β π φ2 log φ a maximum (minimum) at : φ = 0 we get that the entropy exhibits Following the entropic principle infrared free theories seem to be preferred!

184 Outline Heterotic string compactifications Type II orientifolds models Intersecting brane models and their statistics D-instantons: non-perturbative couplings The string landscape: fluxes and branes AdS 2 AdS 4 flux vacua and black holes flux vacua and domain walls

185 Outline Heterotic string compactifications Type II orientifolds models Intersecting brane models and their statistics D-instantons: non-perturbative couplings The string landscape: fluxes and branes AdS 2 flux vacua and black holes AdS 4 flux vacua and domain walls by C. Kounnas, D. Lüst, M. Petropoulos, D. Tsimpis, arxiv:

186 Outline Heterotic string compactifications Type II orientifolds models Intersecting brane models and their statistics D-instantons: non-perturbative couplings The string landscape: fluxes and branes AdS 2 AdS 4 flux vacua and black holes flux vacua and domain walls by C. Kounnas, D. Lüst, M. Petropoulos, D. Tsimpis, arxiv: (Related work: Cowdall, Lu, Pope, Stelle, Townsend, hep-th/ ; Boonstra, Peeters, Skenderis, hep-th/ ; Behrndt, Cardoso, Lüst, hep-th/ ; Ceresole, Dall Agata, Kallosh, Van Proeyen, hep-th/ ; Behrndt, Cvetic, hep-th/ ; Lüst, Tsimpis, hep-th/ ; Ceresole, Dall Agata, Giryavets, Kallosh, Linde, hep-th/ ;...)

187 Q: Can one associate an entropy for 4D AdS 4 flux vacua?

188 Q: Can one associate an entropy for 4D AdS 4 flux vacua? 1st. step: Replace fluxes by their corresponding branes (sources).

189 Q: Can one associate an entropy for 4D AdS 4 flux vacua? 1st. step: Replace fluxes by their corresponding branes (sources). 2nd step: Compute stringy entropy of branes.

190 Q: Can one associate an entropy for 4D AdS 4 flux vacua? 1st. step: Replace fluxes by their corresponding branes (sources). 2nd step: Compute stringy entropy of branes. AdS 4 : Like a domain wall in 4D (co-dimension one object): Consider branes that fill 2 spatial dimensions in 4D.

191 Q: Can one associate an entropy for 4D AdS 4 flux vacua? 1st. step: Replace fluxes by their corresponding branes (sources). 2nd step: Compute stringy entropy of branes. AdS 4 : Like a domain wall in 4D (co-dimension one object): Consider branes that fill 2 spatial dimensions in 4D. Two requirements:

192 Q: Can one associate an entropy for 4D AdS 4 flux vacua? 1st. step: Replace fluxes by their corresponding branes (sources). 2nd step: Compute stringy entropy of branes. AdS 4 : Like a domain wall in 4D (co-dimension one object): Consider branes that fill 2 spatial dimensions in 4D. Two requirements: Domain wall interpolates between and, AdS 4 R 4

193 Q: Can one associate an entropy for 4D AdS 4 flux vacua? 1st. step: Replace fluxes by their corresponding branes (sources). 2nd step: Compute stringy entropy of branes. AdS 4 : Like a domain wall in 4D (co-dimension one object): Consider branes that fill 2 spatial dimensions in 4D. Two requirements: Domain wall interpolates between and, All moduli are fixed to finite values on AdS 4 R 4 AdS 4 horizon.

194 Q: Can one associate an entropy for 4D AdS 4 flux vacua? 1st. step: Replace fluxes by their corresponding branes (sources). 2nd step: Compute stringy entropy of branes. AdS 4 : Like a domain wall in 4D (co-dimension one object): Consider branes that fill 2 spatial dimensions in 4D. Two requirements: Domain wall interpolates between and, All moduli are fixed to finite values on AdS 4 R 4 AdS 4 Is there an attractor mechanism for domain walls? horizon.

195 Q: Can one associate an entropy for 4D AdS 4 flux vacua? 1st. step: Replace fluxes by their corresponding branes (sources). 2nd step: Compute stringy entropy of branes. AdS 4 : Like a domain wall in 4D (co-dimension one object): Consider branes that fill 2 spatial dimensions in 4D. Two requirements: Domain wall interpolates between and, All moduli are fixed to finite values on AdS 4 R 4 AdS 4 Is there an attractor mechanism for domain walls? horizon.

196 AdS 4 flux vacua with all moduli fixed:

197 AdS 4 flux vacua with all moduli fixed: Type IIB with tree level 3-form fluxes:

198 AdS 4 flux vacua with all moduli fixed: Type IIB with tree level 3-form fluxes: Normally, only complex structure moduli are fixed!

199 AdS 4 flux vacua with all moduli fixed: Type IIB with tree level 3-form fluxes: Normally, only complex structure moduli are fixed! To fix all Kähler moduli one needs:

200 AdS 4 flux vacua with all moduli fixed: Type IIB with tree level 3-form fluxes: Normally, only complex structure moduli are fixed! To fix all Kähler moduli one needs: D3-instantons wrapped around 4-cycles,

201 AdS 4 flux vacua with all moduli fixed: Type IIB with tree level 3-form fluxes: Normally, only complex structure moduli are fixed! To fix all Kähler moduli one needs: D3-instantons wrapped around 4-cycles, geometrical (metric) fluxes,

202 AdS 4 flux vacua with all moduli fixed: Type IIB with tree level 3-form fluxes: Normally, only complex structure moduli are fixed! To fix all Kähler moduli one needs: D3-instantons wrapped around 4-cycles, geometrical (metric) fluxes, consider compactifications without Kähler moduli (see later)

203 Tree level type IIA flux vacua:

204 Tree level type IIA flux vacua: (Curio, Klemm, Lüst, Theisen, hep-th/ ; Derendinger, Kounnas, Petropoulos, Zwirner, hep-th/ ; Villadoro, Zwirner, hep-th/ ; De Wolfe, Giryavets, Kachru, Taylor, hep-th/ ; Camara, Font, Ibanez, hep-th/ ; Villadoro, Zwirner, arxiv: )

205 Tree level type IIA flux vacua: (Curio, Klemm, Lüst, Theisen, hep-th/ ; Derendinger, Kounnas, Petropoulos, Zwirner, hep-th/ ; Villadoro, Zwirner, hep-th/ ; De Wolfe, Giryavets, Kachru, Taylor, hep-th/ ; Camara, Font, Ibanez, hep-th/ ; Villadoro, Zwirner, arxiv: ) Consider compactifcation on CY space Y with Hodge numbers h 2,1 h1,1

206 Tree level type IIA flux vacua: (Curio, Klemm, Lüst, Theisen, hep-th/ ; Derendinger, Kounnas, Petropoulos, Zwirner, hep-th/ ; Villadoro, Zwirner, hep-th/ ; De Wolfe, Giryavets, Kachru, Taylor, hep-th/ ; Camara, Font, Ibanez, hep-th/ ; Villadoro, Zwirner, arxiv: ) Consider compactifcation on CY space Y with Hodge numbers h 2,1 h1,1 Superpotential: W IIA = W H + W F + W geom

207 Tree level type IIA flux vacua: (Curio, Klemm, Lüst, Theisen, hep-th/ ; Derendinger, Kounnas, Petropoulos, Zwirner, hep-th/ ; Villadoro, Zwirner, hep-th/ ; De Wolfe, Giryavets, Kachru, Taylor, hep-th/ ; Camara, Font, Ibanez, hep-th/ ; Villadoro, Zwirner, arxiv: ) Consider compactifcation on CY space Y with Hodge numbers h 2,1 h1,1 Superpotential: W IIA = W H + W F + W geom W F (T ) = e J c F R = Y 1 m 0 (J c J c J c ) Y Y ( F R 2 J c J c ) + Y F R 4 J c + = i m 0 F 0 (T ) m i F i (T ) + iẽ i T i + ẽ 0. (i = 1,..., h 1,1 ) Y F R 6 0-form R-flux (massive IIA SUGRA) 2-form R-flux 4-form R-flux 6-form R-flux

208 Tree level type IIA flux vacua: (Curio, Klemm, Lüst, Theisen, hep-th/ ; Derendinger, Kounnas, Petropoulos, Zwirner, hep-th/ ; Villadoro, Zwirner, hep-th/ ; De Wolfe, Giryavets, Kachru, Taylor, hep-th/ ; Camara, Font, Ibanez, hep-th/ ; Villadoro, Zwirner, arxiv: ) Consider compactifcation on CY space Y with Hodge numbers h 2,1 h1,1 Superpotential: W IIA = W H + W F + W geom W F (T ) = e J c F R = Y 1 m 0 (J c J c J c ) Y Y ( F R 2 J c J c ) + Y F R 4 J c + = i m 0 F 0 (T ) m i F i (T ) + iẽ i T i + ẽ 0. (i = 1,..., h 1,1 ) Y F R 6 0-form R-flux (massive IIA SUGRA) W H (S, U) = Y 2-form R-flux 4-form R-flux 6-form R-flux 3-form NS-flux Ω c H 3 = iã 0 S + i c m U m, (m = 1,..., h 2,1 )

209 Tree level type IIA flux vacua: (Curio, Klemm, Lüst, Theisen, hep-th/ ; Derendinger, Kounnas, Petropoulos, Zwirner, hep-th/ ; Villadoro, Zwirner, hep-th/ ; De Wolfe, Giryavets, Kachru, Taylor, hep-th/ ; Camara, Font, Ibanez, hep-th/ ; Villadoro, Zwirner, arxiv: ) Consider compactifcation on CY space Y with Hodge numbers h 2,1 h1,1 Superpotential: W IIA = W H + W F + W geom W F (T ) = e J c F R = Y 1 m 0 (J c J c J c ) Y Y ( F R 2 J c J c ) + Y F R 4 J c + = i m 0 F 0 (T ) m i F i (T ) + iẽ i T i + ẽ 0. (i = 1,..., h 1,1 ) Y F R 6 0-form R-flux (massive IIA SUGRA) 2-form R-flux 4-form R-flux 6-form R-flux 3-form NS-flux W H (S, U) = Ω c H 3 = iã 0 S + i c m U m, (m = 1,..., h 2,1 ) Y Metric fluxes (twisted W geom (S, T, U) = i Ω c dj = ã i ST i d tori) im T i U m, Y

210 The fluxes induce a C7 tadpole: Ñ flux = (C7 df 2 + C 7 (ã 0 H 3 + d F 2 ) ) = h 1,1 I=0 ã I m I. D6-brane charge, need D6-branes, O6-planes.

211 IIA flux solution with all moduli fixed:

212 IIA flux solution with all moduli fixed: Compactification on 6D torus: Y = T 2 1 T 2 2 T 2 3 ( h 1,1 = h 2,1 = 3)

213 IIA flux solution with all moduli fixed: Compactification on 6D torus: Y = T 2 1 T 2 2 T 2 3 Prepotential: F = T 1 T 2 T 3 ( h 1,1 = h 2,1 = 3)

214 IIA flux solution with all moduli fixed: Compactification on 6D torus: Y = T 2 1 T 2 2 T 2 3 Prepotential: F = T 1 T 2 T 3 Kähler potential: K = log(s + S) ( h 1,1 = h 2,1 = 3) 3 (T i + T 3 i ) (U m + Ūm) i=1 m=1

215 IIA flux solution with all moduli fixed: Compactification on 6D torus: Y = T 2 1 T 2 2 T 2 3 Prepotential: F = T 1 T 2 T 3 Kähler potential: K = log(s + S) Flux superpotential: ( h 1,1 = h 2,1 = 3) 3 (T i + T 3 i ) (U m + Ūm) i=1 m=1 W IIA = iẽ i T i + i m 0 T 1 T 2 T 3 + iã 0 S + i c m U m. (no geometrical fluxes)

216 IIA flux solution with all moduli fixed: Compactification on 6D torus: Y = T 2 1 T 2 2 T 2 3 Prepotential: F = T 1 T 2 T 3 Kähler potential: K = log(s + S) Flux superpotential: ( h 1,1 = h 2,1 = 3) 3 (T i + T 3 i ) (U m + Ūm) i=1 m=1 W IIA = iẽ i T i + i m 0 T 1 T 2 T 3 + iã 0 S + i c m U m. SUSY solution: (no geometrical fluxes) DW IIA = 0 AdS 4 vacuum.

217 γ i T i = IIA flux solution with all moduli fixed: Compactification on 6D torus: Y = T 2 1 T 2 2 T 2 3 Prepotential: F = T 1 T 2 T 3 Kähler potential: K = log(s + S) Flux superpotential: SUSY solution: DW IIA = 0 5 γ 1 γ 2 γ 3 3 m 2 0 AdS 4 3 (T i + T i ) vacuum., S = 2 γ i T i, c m U m = 2 γ i T i. 3 m 0 ã 0 3 m 0 (γ i = m 0 ẽ i ) ( h 1,1 = h 2,1 = 3) W IIA = iẽ i T i + i m 0 T 1 T 2 T 3 + iã 0 S + i c m U m. i=1 3 (U m + Ūm) m=1 (no geometrical fluxes)

218 T-dual type IIB description:

219 T-dual type IIB description: c m 0 : (NS 3-form fluxes in IIA) Need geometric (metric) fluxes on the IIB side.

220 T-dual type IIB description: c m 0 : (NS 3-form fluxes in IIA) Need geometric (metric) fluxes on the IIB side. Simple situation: IIA: No complex structure moduli: h2,1 = 0 : W IIA = W F + W H = iẽ i T i + i m 0 T 1 T 2 T 3 + iã 0 S

221 T-dual type IIB description: c m 0 : (NS 3-form fluxes in IIA) Need geometric (metric) fluxes on the IIB side. Simple situation: IIA: No complex structure moduli: h2,1 = 0 : W IIA = W F + W H = iẽ i T i + i m 0 T 1 T 2 T 3 + iã 0 S IIB: No Kähler moduli: (LG description) Then all IIB moduli can be fixed by 3-form flux superpotential: W IIB = X Ω ( F R 3 + SH NS 3 (Becker, Becker, Vafa, Walcher, hep-th/ ) ) = iei U i + im 0 U 1 U 2 U 3 + ia 0 S

222 Replace fluxes by (2+p)-dim. branes: 2 common directions in uncompatified space AdS 4 domain wall.

223 Replace fluxes by (2+p)-dim. branes: 2 common directions in uncompatified space AdS 4 domain wall. Internal space: Brane flux dictionary:

224 Replace fluxes by (2+p)-dim. branes: 2 common directions in uncompatified space AdS 4 domain wall. Internal space: Brane flux dictionary: (i) Ramond flux: Fn R flux through Σ n D(2+6-n)-brane wrapped around Σ 6 n

225 Replace fluxes by (2+p)-dim. branes: 2 common directions in uncompatified space AdS 4 domain wall. Internal space: Brane flux dictionary: (i) Ramond flux: Fn R flux through Σ n D(2+6-n)-brane wrapped around Σ 6 n (ii) NS 3-form flux H 3 through Σ 3 Σ 3 NS5-brane wrapped around

226 Replace fluxes by (2+p)-dim. branes: 2 common directions in uncompatified space AdS 4 domain wall. Internal space: Brane flux dictionary: (i) Ramond flux: Fn R flux through Σ n D(2+6-n)-brane wrapped around Σ 6 n (ii) NS 3-form flux H 3 through Σ 3 Σ 3 NS5-brane wrapped around (iii) Geometrical flux Kaluza-Klein (KK) monopole

227 Replace fluxes by (2+p)-dim. branes: 2 common directions in uncompatified space AdS 4 domain wall. Internal space: Brane flux dictionary: (i) Ramond flux: Fn R flux through Σ n D(2+6-n)-brane wrapped around Σ 6 n (ii) NS 3-form flux H 3 through Σ 3 Σ 3 NS5-brane wrapped around (iii) Geometrical flux Kaluza-Klein (KK) monopole T-duality

228 SUSY conditions (calibrated sources): ( ) 0 = µ H µp + eφ 16 n F nγ µ P n ɛ, ( ) 0 = Φ HP + eφ 8 n ( 1)n (5 n)f n P n ɛ.

229 SUSY conditions (calibrated sources): ( ) 0 = µ H µp + eφ 16 n F nγ µ P n ɛ, ( ) 0 = Φ HP + eφ 8 n ( 1)n (5 n)f n P n Bianchi identities: df + H F = Q j, d F H F = Q α(j). ɛ.

230 SUSY conditions (calibrated sources): ( ) 0 = µ H µp + eφ 16 n F nγ µ P n ɛ, ( ) 0 = Φ HP + eφ 8 n ( 1)n (5 n)f n P n Bianchi identities: df + H F = Q j, d F H F = Q α(j). Theorem: Supersymmetric brane configurations that satisfy Bianchi id. will automatically satisfy the dilaton and Einstein eq. of motion. (Koerber, Tsimpis, arxiv: )) ɛ.

231 SUSY conditions (calibrated sources): ( ) 0 = µ H µp + eφ 16 n F nγ µ P n ɛ, ( ) 0 = Φ HP + eφ 8 n ( 1)n (5 n)f n P n Bianchi identities: df + H F = Q j, d F H F = Q α(j). Theorem: Supersymmetric brane configurations that satisfy Bianchi id. will automatically satisfy the dilaton and Einstein eq. of motion. (Koerber, Tsimpis, arxiv: )) Properties of our solutions: ɛ.

232 SUSY conditions (calibrated sources): ( ) 0 = µ H µp + eφ 16 n F nγ µ P n ɛ, ( ) 0 = Φ HP + eφ 8 n ( 1)n (5 n)f n P n Bianchi identities: df + H F = Q j, d F H F = Q α(j). Theorem: Supersymmetric brane configurations that satisfy Bianchi id. will automatically satisfy the dilaton and Einstein eq. of motion. (Koerber, Tsimpis, arxiv: )) Properties of our solutions: Branes are smeared in transverse direction. ɛ.

233 SUSY conditions (calibrated sources): ( ) 0 = µ H µp + eφ 16 n F nγ µ P n ɛ, ( ) 0 = Φ HP + eφ 8 n ( 1)n (5 n)f n P n Bianchi identities: df + H F = Q j, d F H F = Q α(j). Theorem: Supersymmetric brane configurations that satisfy Bianchi id. will automatically satisfy the dilaton and Einstein eq. of motion. (Koerber, Tsimpis, arxiv: )) Properties of our solutions: Branes are smeared in transverse direction. Near horizon geometry is AdS 4 ɛ.

234 SUSY conditions (calibrated sources): ( ) 0 = µ H µp + eφ 16 n F nγ µ P n ɛ, ( ) 0 = Φ HP + eφ 8 n ( 1)n (5 n)f n P n Bianchi identities: df + H F = Q j, d F H F = Q α(j). Theorem: Supersymmetric brane configurations that satisfy Bianchi id. will automatically satisfy the dilaton and Einstein eq. of motion. (Koerber, Tsimpis, arxiv: )) Properties of our solutions: Branes are smeared in transverse direction. Near horizon geometry is AdS 4 Moduli are stabilized at horizon (attractor behavior) ɛ.

235 SUSY conditions (calibrated sources): ( ) 0 = µ H µp + eφ 16 n F nγ µ P n ɛ, ( ) 0 = Φ HP + eφ 8 n ( 1)n (5 n)f n P n Bianchi identities: df + H F = Q j, d F H F = Q α(j). Theorem: Supersymmetric brane configurations that satisfy Bianchi id. will automatically satisfy the dilaton and Einstein eq. of motion. (Koerber, Tsimpis, arxiv: )) Properties of our solutions: Branes are smeared in transverse direction. Near horizon geometry is AdS 4 Moduli are stabilized at horizon (attractor behavior) Non-vanishing C7 tadpole, same as for fluxes ɛ.

236 IIA Examples: (i) Not all moduli are stabilized: Ramond 4-form flux superpotential: W IIA = J = iẽ i T i DW IIA = 0 Y F R 4 S and U-fields are not fixed!

237 IIA Examples: (i) Not all moduli are stabilized: Ramond 4-form flux superpotential: W IIA = J = iẽ i T i DW IIA = 0 Y F R 4 S and U-fields are not fixed! The corresponding sources are intersecting D4-branes: ξ 0 ξ 1 ξ 2 y 1 y 2 y 3 x 1 x 2 x 3 x 4 D4 D4

238 ds 2 10 = 1 η µν dξ µ dξ ν + H 1 H 2 δ ij dy i dy j H2 + H1 H 2 H 1 2 (dx a ) 2 + a=1 H1 e 2φ = H 1 H 2, F x3 x 4 y i y j = δkl ε ijk y lh 1, F x1 x 2 y i y j = δkl ε ijk y lh 2, where 10 D metric and dilaton: H 2 4 (dx a ) 2 a=3 4πH α = 1 + c α y + d α 2y 2, α = 1, 2

239 ds 2 10 = 10 D metric and dilaton: 1 η µν dξ µ dξ ν + H 1 H 2 δ ij dy i dy j H2 + H1 H 2 H 1 2 (dx a ) 2 + a=1 H1 e 2φ = H 1 H 2, F x3 x 4 y i y j = δkl ε ijk y lh 1, F x1 x 2 y i y j = δkl ε ijk y lh 2, H 2 4 (dx a ) 2 where 4πH α = 1 + c α y + d α 2y 2, α = 1, 2 H are not harmonic in overall transverse directions. Smeared sources: j α := 2 H α = c α δ 3 (y) d α 4πy 4 a=3

240 ds 2 10 = 10 D metric and dilaton: 1 η µν dξ µ dξ ν + H 1 H 2 δ ij dy i dy j H2 + H1 H 2 H 1 2 (dx a ) 2 + a=1 H1 e 2φ = H 1 H 2, F x3 x 4 y i y j = δkl ε ijk y lh 1, F x1 x 2 y i y j = δkl ε ijk y lh 2, H 2 4 (dx a ) 2 where 4πH α = 1 + c α y + d α 2y 2, α = 1, 2 H are not harmonic in overall transverse directions. Smeared sources: j α := 2 H α = c α δ 3 (y) d α Near horizon limit y 0 : a=3 4πy 4 Metric of ds 2 NH = y 2 D1 D 2 η µν dξ µ dξ ν + D 1 D 2 dy 2 AdS 4 S 2 T 4 + D2 D 1 y 2 + D 1 D 2 dω (dx a ) 2 D1 4 + (dx a ) 2 a=1 D 2 a=3

241 ds 2 10 = 1 η µν dξ µ dξ ν + H 1 H 2 δ ij dy i dy j H2 + H1 H 2 H 1 2 (dx a ) 2 + a=1 H1 e 2φ = H 1 H 2, F x3 x 4 y i y j = δkl ε ijk y lh 1, F x1 x 2 y i y j = δkl ε ijk y lh 2, H 2 4 (dx a ) 2 where 4πH α = 1 + c α y + d α 2y 2, α = 1, 2 H are not harmonic in overall transverse directions. Smeared sources: j α := 2 H α = c α δ 3 (y) d α Metric of String coupling 10 D metric and dilaton: Near horizon limit y 0 : ds 2 NH = y 2 D1 D 2 η µν dξ µ dξ ν + D 1 D 2 dy 2 AdS 4 S 2 T 4 g s = e φ 0 + D2 D 1 a=3 4πy 4 y 2 + D 1 D 2 dω (dx a ) 2 D1 4 + (dx a ) 2 a=1 D 2 Runaway behavior a=3

242 (ii) All moduli are stabilized:

243 (ii) All moduli are stabilized: W IIA = iẽ i T i + i m 0 T 1 T 2 T 3 + iã 0 S + i c m U m DW IIA = 0 All moduli are stabilized!

244 (ii) All moduli are stabilized: W IIA = iẽ i T i + i m 0 T 1 T 2 T 3 + iã 0 S + i c m U m DW IIA = 0 All moduli are stabilized! The corresponding sources are intersecting D4, NS5 and D8-branes: ξ 0 ξ 1 ξ 2 y x 1 x 2 x 3 x 4 x 5 x 6 D4 D4 D4 NS5 NS5 NS5 NS5 D8

245 ds 2 10 = Explicit form of the solution: e 2φ = { ( 3 H D8 α=1 HD4 α ) ( 4 H D4 2 HD4 3 H D4 α=1 HNS5 α {H NS5 1 HD8 3 H NS5 )} 1 2 η µν dξ µ dξ ν {H D8 ( 3 α=1 HD4 α 4 (dx 1 ) 2 + H NS5 ) } 1 2 dy 2 1 H NS5 2 (dx 2 ) 2 } H1 D4HD4 3 {H NS5 H2 D4HD8 2 H3 NS5 (dx 3 ) 2 + H1 NS5 H4 NS5 (dx 4 ) 2 } H1 D4HD4 2 {H NS5 H3 D4HD8 2 H4 NS5 (dx 5 ) 2 + H1 NS5 H3 NS5 (dx 6 ) 2 }; ( 4 ) ( 3 ) 1 2 ( ) H D8 5 2 ; α=1 HNS5 α α=1 HD4 α H x2 x 4 x 6 H x1 x 3 x 6 F x3 x 4 x 5 x 6 = yh NS5 1 = yh NS5 3 ( H D8 ) 1 ; Hx2 x 3 x 5 = yh NS5 ( H D8 ) 1 ; Hx1 x 4 x 5 = yh NS5 4 = yh D4 1 ; F x1 x 2 x 5 x 6 = yh D4 2 ; 2 ( H D8 ) 1 ; ( H D8 ) 1 ; F x1 x 2 x 3 x 4 ( = 4 yh3 D4 ; F = y H D8 α=1 HNS5 α ) 1.

246 Properties of this solution:

247 Properties of this solution: Smeared (thick) branes: H D4 α = { ( c D4 α y{1 1 y 2 y 0 )}, y < y cd4 α y 0, y y 0

248 Properties of this solution: Smeared (thick) branes: H D4 α = { ( c D4 α y{1 1 y 2 y 0 )}, y < y cd4 α y 0, y y 0 Near horizon limit y 0 : AdS 4 T 6

249 Properties of this solution: Smeared (thick) branes: H D4 α = { ( c D4 α y{1 1 y 2 y 0 )}, y < y cd4 α y 0, y y 0 Near horizon limit y 0 : AdS 4 T 6 For y : R 3,1 T 6

250 Properties of this solution: Smeared (thick) branes: H D4 α = { ( c D4 α y{1 1 y 2 y 0 )}, y < y cd4 α y 0, y y 0 Near horizon limit y 0 : AdS 4 T 6 For y : R 3,1 T 6 All moduli take finite, fixed values at horizon, agree with those from minimizing the flux superpotential.

251 Properties of this solution: Smeared (thick) branes: H D4 α = { ( c D4 α y{1 1 y 2 y 0 )}, y < y cd4 α y 0, y y 0 Near horizon limit y 0 : AdS 4 T 6 For y : R 3,1 T 6 All moduli take finite, fixed values at horizon, agree with those from minimizing the flux superpotential. Non-vanising tadpole: need (smeared) D6-branes and O6-plane.

252 T-dual type II B description:

253 T-duality in T-dual type II B description: x 1 direction: W IIB = i(ẽ 1 U 1 + c 2 U 2 + c 3 U 3 + c 1 T 1 + ẽ 2 T 2 + ẽ 3 T 3 + m 0 U 1 T 2 T 3 + ã 0 S) This includes geometrical fluxes.

254 T-duality in T-dual type II B description: x 1 direction: W IIB = i(ẽ 1 U 1 + c 2 U 2 + c 3 U 3 + c 1 T 1 + ẽ 2 T 2 + ẽ 3 T 3 + m 0 U 1 T 2 T 3 + ã 0 S) This includes geometrical fluxes. Corresponding branes: ξ 0 ξ 1 ξ 2 y x 1 x 2 x 3 x 4 x 5 x 6 D3 D5 D5 D7 NS5 NS5 KK KK

255 T-duality in T-dual type II B description: x 1 direction: W IIB = i(ẽ 1 U 1 + c 2 U 2 + c 3 U 3 + c 1 T 1 + ẽ 2 T 2 + ẽ 3 T 3 + m 0 U 1 T 2 T 3 + ã 0 S) This includes geometrical fluxes. Corresponding branes: ξ 0 ξ 1 ξ 2 y x 1 x 2 x 3 x 4 x 5 x 6 D3 D5 D5 D7 NS5 NS5 KK KK Near horizon: AdS 4 N, (Nilmanifold)

256 Resume: We found new supersymmetric brane solutions that correspond to AdS 4 flux vacua.

257 Resume: We found new supersymmetric brane solutions that correspond to AdS 4 flux vacua. Open problems:

258 Resume: We found new supersymmetric brane solutions that correspond to AdS 4 flux vacua. Open problems: Entropy count of brane configurations.

259 Resume: We found new supersymmetric brane solutions that correspond to AdS 4 flux vacua. Open problems: Entropy count of brane configurations. Uplift to ds and its brane description.

260 Resume: We found new supersymmetric brane solutions that correspond to AdS 4 flux vacua. Open problems: Entropy count of brane configurations. Uplift to ds and its brane description. Characterization of brane configurations via generalized geometry.

261 Conclusions 3rd RTN-workshop,Valencia

262 Conclusions How does the string landscape really look like? 3rd RTN-workshop,Valencia

263 Conclusions How does the string landscape really look like? 3rd RTN-workshop,Valencia

264 Conclusions How does the string landscape really look like? Thank you for organizing this conference! 3rd RTN-workshop,Valencia