Fluxes, branes and black holes - how to find the right string vacuum?

Transcription

1 Fluxes, branes and black holes - how to find the right string vacuum? Dieter Lüst, LMU (ASC) and MPI München in collaboration with Ralph Blumenhagen, Gabriel L. Cardoso, Florian Gmeiner, Daniel Krefl, Gabriele Honecker, Jan Perz, Susanne Reffert, Emmanuel Scheidegger, Waldemar Schulgin, Stephan Stieberger, Prasanta Tripathy and Timo Weigand

2 I) Introduction Superstring theory in 10 dimensions is (almost) unique! Continuous moduli φ Fluxes,... " Moduli stabilization: < φ >= φ 0 Compactification There exist a huge number of lower (4-dimensional) groundstates: String landscape problem! Count the number of consistent string solutions! Vast landscape with N sol = discrete vacua! (Lerche, Lüst, Schellekens (1986), Douglas (2003))

3 I) Introduction Is there a vacuum selection mechanism?! (Unknown) dynamics! Unique vacuum?! 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 probability function in moduli space, ψ land (φ) 2 = e S land(φ) ψ land (φ) 2 and see if is peaked, i.e. has maxima with good phenomenological properties.

4 I) Introduction String theory and black holes: Compute the statistical entropy of (CY)-black holes: S bh (p, q) = A horizon 4 Attractor φ horizon = φ(p, q) S bh (φ) There exist many black hole solutions with different electric and magnetic charges! Where are the maxima of S bh? Conjecture: (Ooguri, Vafa Verlinde (2005)) S bh = S land ψ top (φ) (OVV)

5 Outline! Type II orientifolds models and moduli stabilization (Blumenhagen, Gmeiner, Honecker, Lüst, Weigand, hep-th/ , ; Lüst, Reffert, Schulgin, Stieberger, hep-th/ ; Lüst, Reffert, Scheidegger, Schulgin, Stieberger, to appear)! Vacuum selection by black hole entropy maximization (Cardoso, Lüst, Perz, hep-th/ )

6 (Intersecting) D-brane model building (Bachas (1995); Blumenhagen, Görlich, Körs, Lüst (2000); Angelantonj, Antoniadis, Dudas Sagnotti (2000); Ibanez, Marchesano, Rabadan (2001); Cvetic, Shiu, Uranga (2001);...) Local D-brane module of the Standard Model by 4 stacks of intersecting D-branes: D-brane compactifications: Wrapping the D-brane modules around cycles of the compact background space: Standard Model module Hidden sector module G=SU(3)xSU(2) xu(1) SM Soft SUSY breaking HS Moduli stabilization

7 (Intersecting) D-brane statistics How many orientifold models exist which come close to the (spectrum Example: of the) MSSM? (Blumenhagen, Gmeiner, Honecker, Lüst, Weigand, hep-th/ , hep-th/ ; related work: Marchesano, Shiu, hepth/ ; Dijkstra, Huiszoon, Schellekens, hep-th/ ; Anastasopoulos, Dijkstra, Kiritsis, Schellekens, hep-th/ ; Douglas, Taylor, hep-th/ ) M 6 = T 6 /(Z 2 Z 2 ) Systematic (complete) computer search: IIA orientifold: There exist about susy D-brane models on this orbifold (with restricted complex structure)!

8 (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!

9 Moduli stabilization Hidden sector module SM HS Moduli stabilization

10 Moduli Most string theory compactifications come with a large number of moduli. Since they are in conflict with experiment, a lot of effort is dedicated to the subject of moduli stabilization. Example: Compactification on a 1-dimensional circle: The radius R of the circle is a free undetermined parameter = modulus Realistic compactifications: M 6 is a 6-dim CY-space: 2 kinds of moduli:! Sizes of 4-cycles : Kähler moduli Σ 4 h (2,1) h (1,1)! Sizes of 3-cyles : complex structure moduli U i Σ 3 T i

11 KKLT We will consider type IIB orientifolds with D3/D7-branes. KKLT-Proposal: Step 1: Fix all moduli (preserving SUSY) Dilaton and complex structure moduli U are stabilized with 3-form fluxes, Kähler moduli T are fixed by nonperturbative effects " SUSY AdS vacuum. Step 2: Lift the minimum of the potential to a positive value by introducing D3 branes " metastable ds vacuum. (Kachru, Kallosh, Linde, Trivedi, hep-th/ ) V Re(T) Can the KKLT scenario be realized in concrete orientifold models? -2

12 Calabi-Yau orientifolds! Consider IIB compactification on Calabi-Yau manifold Y 6! N=2 supersymmetry in D=4! Include orientifold action O = ( 1) F L Ω σ! IIB CY orientifold X 6 with O3/O7-planes and with N=1 Susy in D=4 Here: σω = Ω, σj = J! Add set of D3/D7-branes to cancel the orientifold tadpoles

13 Orientifold action σ acts holomorphically split of (co)homology of : X 6 H (p,q) (X 6 ) = H (p,q) + (X 6 ) H (p,q) (X 6 ) = J = h (+) (1,1) (X 6) k=1 t k ω k, Ω = h ( ) (2,1) (X 6) λ=0 X λ α λ F λ β λ = B 2 = V ol(x 6 ) = 1 6 h ( ) (1,1) (X 6) a=1 b a ω a, C 2 = X 6 J J J = 1 6 h ( ) (1,1) (X 6) a=1 h (+) (1,1) (X 6) i,j,k=1 c a ω a K ijk t i t j t k

14 Orientfold Moduli Moduli of Calabi-Yau orientfold X 6 : 1 dilaton S chiral multiplet h ( ) (2,1) (X 6) CS moduli u λ chiral multiplets X 6 Ω G 3 X 6 Ω G 3 TV/GVW TV/GVW h (+) (1,1) (X 6) Kähler moduli t k chiral/linear multiplets e T KKLT h ( ) (1,1) (X 6) add. moduli b a, c a chiral multiplets C 4 J B 2 Calibration h (+) (2,1) (X 6) add. vectors V j µ vector multiplets - - Moduli the N=1 supergravity basis (chiral N=1 superfields): T j = 3 4 K jkl t k t l eφ 10 K jbc G b (G+G) c i Cj, j = 1,..., h (+) (1,1) (X 6) G a = i c a + S b a, a = 1,..., h ( ) (1,1) (X 6)

15 Effective action Lowest order Kähler potential: K = ln(s + S) ln α -expansion at tree level (supergravity approximation): X 6 Ω Ω 2 ln 1 6 X 6 J J J Gauge kinetic function for D7-brane wrapped on divisor C j 4 : Superpotential: f j = T j + (S, U) tree level W = W 0 (S, U) + h (+) (1,1) (X 6) j=1 loop corrections γ j (S, U) e a j T j Flux superpotential (GVW,TV) non-perturbative superpotential

16 Superpotentials Non-perturbative superpotential W γ j (S, U) e a j T j :! Euclidean D3-brane wrapped on 4-cycle C 4 with volume T : W e 2π T a j = 2π Condition for non-vanishing superpotential: N F 2 = χ(c 4) = h 0,0 (C 4 ) h 1,0 (C 4 ) + h 2,0 (C 4 ) = 1 (Arithmetic genus χ also depends on O7-plane and 3-form flux!)! Gaugino condensation in gauge theory on D7-branes wrapped on 4-cycle : C 4 W e T/b a, b SU(M) = M 2π a j = 2π M Condition for non-vanishing superpotential: no massless adjoint chiral multiplets (open string moduli of D7!) (Witten; Tripathy, Trivedi; Kallosh, Kashani-Poor, Tomasiello; Saulina; E. Bergshoeff, R. Kallosh, A. K. Kashani-Poor, D. Sorokin, A. Tomasiello; J.Park) (Gomis, Mateos, Marchesano)

17 Superpotentials Non-perturbative superpotential W γ j (S, U) e a j T j :! Euclidean D3-brane wrapped on 4-cycle C 4 γ j (S, U) may comprise one-loop effects and further with volume T : W e 2π T a j = 2π instanton effects (from (-1)-branes): Condition for non-vanishing superpotential:! one-loop N F corrections to the gauge coupling: 2 = χ(c 4) = h 0,0 (C 4 ) h 1,0 (C 4 ) + h 2,0 (C 4 ) = 1 (Arithmetic genus χ also depends on O7-plane and 3-form flux!) γ j η(u)! Gaugino condensation in gauge theory on D7-branes! additional wrapped instantons 4-cycle in the : D7-gauge theory: C 4 W e T/b a, b SU(M) = M 2π a j = 2π F F C M 4 Condition for non-vanishing superpotential: no massless adjoint chiral multiplets (open string moduli of D7!) γ j e S

18 All together: V = e κ2 4 K ( D S W 2 + i AdS vacua Total effective N=1 superpotential: W = W flux (S, U) + W np (T ) = κ 2 10 Kähler potential: K = ln(s S) ln Ω Ω + 2 ln V F-term scalar potential: V = 1 6 J J J D T iw 2 + j D U j W 2 3 W 2 ) Supersymmetric vacua! impose F-term SUSY-conditions: G 3 Ω + h (+) (1,1) (X 6) j=1 γ j (S, U) e a j T j D i W = i W + κ 2 4 W i K = 0, i = S, U i, T i All known SUSY vacua with all moduli fixed are AdS.

19 Stability of vacua: Stability Breitenlohner and Freedman say:... an AdS background is stable... not only when the critical point is a relative minimum, but also when it is a maximum or a saddle point provided the eigenvalues of are not too negative. But: We want stability after the uplift to ds space, so we must require the eigenvalues of the scalar mass matrix to be > 0. 2 V M N > 0, (M, N = S, U j, T i ) determinants of all upper-left sub-matrices must be positive Theorem for general Kähler potentials of the Kähler moduli: V ij For models without complex structure moduli, no stable vacuum is possible!

20 D-term Recall: we have h ( ) moduli G a = ic a + S b a (1,1) : G aw 0 a = 1,..., h ( ) (1,1) In the covering space Y 6 some divisors are noninvariant under the orientifold action: There is a calibration condition (for wrapped D3/D7-brane): C a 4 σ C a 4 = C a 4 B 2 J = 0 b a 2 = 0 This requirement may be derived from an effective D-term: V i D 1 Re(T i ) (K abb b ) 2

21 Resolved orbifolds Can the moduli stabilization be realized in specific models? (Lüst, Reffert, Schulgin, Stieberger, hep-th/ ; Lüst, Reffert, Schulgin, Scheidegger, Stieberger, to appear; related work by Choi, Falkowski, Nilles, Olechowski, Pokorski, hep-th/ ; Denef, Douglas, Florea, Grassi, Kachru, hep-th/ ) We consider IIB orientifolds with D3/D7-branes, compactified on M 6 = T 6 /Z N, T 6 /(Z N Z M ) orbifolds or their blown-up versions (smooth Calabi-Yau!) Orbifold: Abelian point groups, no discrete torsion, no vector structure. N=2 SUSY # SU(3), crystallographic action: Z N Z 3 Z 4 Z 6 I Z 6 II Z 7 Z 12 I Z 12 II :,,,,, Z 8 I, Z 8 II,,. : Z 2 Z 2,, Z 2 Z 4, Z 4 Z 4,,,,. ZN Z M Z 3 Z 3 Z 2 Z 6 Z 2 Z 6 Z 3 Z 6 Z 6 Z 6

22 Fixing all moduli in resolved orbifolds! Take some toroidal orbifold. Describe the local patches near the singularities with toric methods. Resolve the singularities via blow-ups.! Glue the patches to get a smooth Calabi-Yau! Perform an orientifold projection on the smooth CY O-planes and D-branes! Determine divisor topologies to decide whether a n.p.-superpotential is generated! Determine the Kähler potential: K = ln(s S) ln Ω Ω + 2 ln V! Fix the dilaton and the complex structure moduli with 3-form flux! Look for critical points of the scalar potential

23 Hodge numbers after orientifold action: Z N Lattice T 6 h (1,1) h (1,1) Z 3 SU(3) Z 4 SU(4) Z 4 SU(2) SU(4) SO(5) 27 0 Z 4 SU(2) 2 SO(5) Z 6 I (G 2 SU(3) 2 ) 25 6 Z 6 I SU(3) G Z 6 II SU(2) SU(6) 25 6 Z 6 II SU(3) SO(8) 29 6 Z 6 II (SU(2) 2 SU(3) SU(3)) Z 6 II SU(2) 2 SU(3) G Z 7 SU(7) 24 9 Z 8 I (SU(4) SU(4)) 24 0 Z 8 I SO(5) SO(9) 27 0 Z 8 II SU(2) SO(10) 27 0 Z 8 II SO(4) SO(9) 31 0 Z 12 I E Z 12 I SU(3) F Z 12 II SO(4) F

24 Hodge numbers after orientifold action: Z N Lattice T 6 h (1,1) h (1,1) Z 2 Z 2 SU(2) Z 2 Z 4 SU(2) 2 SO(5) Z 2 Z 6 SU(2) 2 SU(3) G Z 2 Z 6 SU(3) G Z 3 Z 3 SU(3) Z 3 Z 6 SU(3) G Z 4 Z 4 SO(5) Z 6 Z 6 G

25 Results on Orientifold Models: Twist Γ Lattice h untw. (1,1) h untw. (2,1) h twist. (1,1) h twist. (2,1) Z 3 SU(3) Z 4 SU(2) 2 SO(5) Z 4 SU(2) SU(4) SU(5) Z 4 SU(4) Z 6 I SU(3) G Z 6 I G 2 SU(3) Z 6 II SU(2) 2 SU(3) G Z 6 II SU(3) SO(8) Z 6 II SU(2) 3 SU(3) Z 6 II SU(2) SU(6) Z 7 SU(7) Z 8 I SO(5) SO(9) Z 8 I SU(4) 2 (gen. Cox) Z 8 II SO(4) SO(9) Z 8 II SU(2) SO(10) Z 12 I SU(3) F Z 12 I E Z 12 II SO(4) F

26 Results on Orientifold Models: Twist Γ Lattice h untw. (1,1) h untw. (2,1) h twist. (1,1) h twist. (2,1) Z 3 SU(3) Z 4 SU(2) 2 SO(5) Z 4 SU(2) SU(4) SU(5) Z 4 SU(4) Z 6 I SU(3) G Z 6 I G 2 SU(3) Z 6 II SU(2) 2 SU(3) G Z 6 II SU(3) SO(8) Z 6 II SU(2) 3 SU(3) Z 6 II SU(2) SU(6) Z 7 SU(7) Z 8 I SO(5) SO(9) Z 8 I SU(4) 2 (gen. Cox) Z 8 II SO(4) SO(9) Z 8 II SU(2) SO(10) Z 12 I SU(3) F Z 12 I E Z 12 II SO(4) F ??

27 Results on Orientifold Models: Twist Γ h untw. (1,1) h untw. (2,1) h twist. (1,1) h twist. (2,1) Z 2 Z Z 2 Z Z 2 Z Z 2 Z Z 3 Z Z 3 Z Z 4 Z Z 6 Z

28 Results on Orientifold Models: Twist Γ h untw. (1,1) h untw. (2,1) h twist. (1,1) h twist. (2,1) Z 2 Z Z 2 Z Z 2 Z Z 2 Z Z 3 Z Z 3 Z Z 4 Z Z 6 Z Denef et al.

29 Resume: Orientifolds and Moduli stabilization:! Moduli stabilization is model dependent! However in particular examples discussed here, all moduli can be fixed!! One has still to glue together hidden module with Standard Model module. Tension between chirality and fluxes! (Marchesano, hep-th/ )! Moduli stabilization in Minkowski vacua: (Blano-Pillado, Kallosh, Linde, hep-th/ ; Krefl, Lüst, hepth/ )! Open problems: Calculate 1-loop determinant prefactor, incorporate uplift to ds by D-terms,...

30 III) Vacuum selection by entropy maximization Ooguri, Vafa, Verline, hep-th/ : Consider IIB on S 2 CY with Ramond 5-form fluxes through S 2 Σ 3 (Σ 3 CY ). These flux vacua correspond to supersymmetric charged black holes : Attractor values for the scalar fields at horizon S bh (φ(p, q)) = A hor(φ(p, q)) 4, ψ(φ(p.q)) 2 = e S bh(φ(p,q)) N = 2 black hole N = 1 landscape D3-branes wrapped around Σ 3 F 5 through S 2 Σ 3 Black hole charges (q I, p I ) 5-form fluxes (q I, p I ) Attractor Central charge Z(z) Superpotential W (z) Stabilization cond. D A Z = 0 Supersymmetry cond. D A W = 0 Entropy S Cosmological constant V 0 Near horizon geometry AdS 2 S 2 Vacuum space AdS 2 S 2

31 Vacuum selection by entropy maximization (Cardoso, Lüst, Perz, hep-th/ ; related work: Gukov, Saraikin, Vafa, hep-th/ ; Fiol, hep-th/ ; Belluci, Ferrara, Marrani, hep-th/ Simple toy model: Consider flux vacua (without D-branes) at particular (conifold) points in the moduli spaces, where additional states become massless:! Additional vector multiplets (β > 0)! Asymptotically free gauge theory! Additional hypermultiplets (β < 0)! Infrared free gauge theory Which of the 2 possibilities is more likely?

32 Vacuum selection by entropy maximization Entropy at two derivative level: ) S = π Im (Ȳ I F (0) I (Y ) Y I (0) F I (Ȳ ) One modulus example with conifold singularity at V " 0 : F (0) (V ) = i(y 0 ) 2 ( β 2π V 2 log V + a), V = Imz 1 = ImY 1 /Y 0 For β < 0 (β > 0) a maximum (minimum) at V=0: we get that the entropy exhibits Re V Im V

33 Vacuum selection by entropy maximization Entropy with higher curvature interactions: (Cardoso, de Wit, Mohaupt (1998)) ( S = π Im ( ) Ȳ I F I (Y, Υ) Y I FI (Ȳ, Ῡ)) + 4Im (Υ F Υ ), (Υ = 64) The full non-perturbative partition function is known from the topological string: (Gopakumar, Vafa (1998)) For real F (Y, Υ) = g 2 top = g top iυ 128π F top = iυ 128π n=1 π2 Υ, 16(Y 0 ) q = e g top, 2 Q = e 2πV n log (1 q n Q) the entropy has again a maximum: S Q Π 0 Π arg Q

34 Vacuum selection by entropy maximization Resume:! Following the entropic principle we find that infrared free theories are preferred! - Maximum of the entropy e S = Z top 2 e L - Minimum of the OSV free energy F top = log Z top! Entropy of non-supersymmetric black holes: Conifold is not any longer an attractor point! (Cardoso, Grass, Lüst, Perz, to appear; Kallosh, hep-th/ ; Tripathy, Trivedi, hep-th/051117; Sahoo, Sen, hep-th/ , Ferrara, Kallosh, hep-th/060347)

35 Conclusions We know several (perturbative) vacua of string theory. How does the string landscape really look like? Thank You!