D3-D7 on warped throat

Transcription

1 D3-D7 on warped throat Anatoly Dymarsky Stanford University From Strings to Things, INT, May 2008

2 Applications of warped throat geometries Gauge/Gravity duality of N = 1 theories I. Klebanov and M. Strassler Confiment and chiral symmetry breaking Physics of glueballs SUSY breaking Compact manifolds with conic singularities Models of early Universe and stringy inflation An important step would be to understand gauge/gravity duality in a more general settings A. Dymarsky D3-D7 on warped throat 2/29

3 Geometry of Klebanov-Strassler solution ds 2 = h 1/2 dx 2 + h 1/2 ds 2 6 Geometry is a warped product of a flat Minkowski space and non-compact CY z1 2 + z2 2 + z2 3 + z2 4 = ɛ2 geometrical realization of confiment h(0) = const chiral symmetry / Ψ is broken explicitly non-trivial flux through S 3 cycle counts D3 branes background is ISD and dilaton is constant metric is Ricci-flat unbroken Z 2 symmetry A. Dymarsky D3-D7 on warped throat 3/29

4 Baryonic branch of KS solution KS solution is a part of 1-dimensional family of solutions parametrized by baryonic condensate U Geometrical Z 2 symmetry exchanges U U Non-KS solutions U 0 are non-cy SU(3)-structure geometries All solutions share the same complex structure. U 0 corresponds to pseudo-kahler deformation. N = 1 SUSY is unbroken for all U Different geometries from the branch correspond to different IR vacua of field theory A. Butti, M. Grana, R. Minasian, M. Petrini, A. Zaffaroni Connection between geometry and field theory, relation between U and baryonic condensate A.D., I. Klebanov, N. Seiberg M.Benna, A.D., I. Klebanov A. Dymarsky D3-D7 on warped throat 4/29

5 Baryonic branch of KS solution Field theory interpretation SU(N) SU(N + M) theory fundamental supermultiplets A α, B β with SU(2) SU(2) symmetry M αβ = A α B β meson 2 2 N N matrix theory experiences duality cascade behavior (Seiberg duality) which turns SU(N) SU(N + M) gauge group into SU(N M) SU(N) superpotential at the last step of the cascade W = X(detM + AB Λ 4M 2M ) baryonic branch AB = const A. Dymarsky D3-D7 on warped throat 5/29

6 VEV of Baryon operator Expectation of baryon operator is given by action of Euclidean D5 wrapping internal manifold E. Witten A and B baryons have different gauge field VEV is function of geometry U < AB >= const M.Benna, A.D., I.Klebanov U is a flat modulus U and Goldstone boson of spontaneously broken U(1) Baryon form scalar supermultiplet QFT: m 2 = 0 due to Goldstone theorem U is a pseudokahler deformation. NO U(1) symmetry on gravity side! Nontrivial S 3 cycle guarantees massless vector in the conformal case A. Dymarsky D3-D7 on warped throat 6/29

7 Wave function of pseudo-kahler deformation In the non-compact limit wave-function of U is not normalazible EOM describe KK tower of states with m 2 n 2 δb 2 = χ(x, τ)dg 5 + µ σ(x, τ)dx µ g 5 δg 13 = δg 24 = 2 1/3 [sinh(2τ) 2τ] 1/3 h 1/2 z(x, τ) After change of variables z = zk sinh τ z 2 sinh 2 τ z + m2 I(τ) 9 z = K 2 m2 K(τ) w (τ) 4 2 2/3 w = ɛ4/3 g smα K 5 sinh 2 τσ w cosh2 τ+1 sinh 2 τ w + m 2 I(τ) K 2 (τ) w = 16 9 K(τ) z m 2 = 0 solution z = τ tanh τ 1 is not normalizable A. Dymarsky D3-D7 on warped throat 7/29

8 Uplifting massless modes U(1) baryonic symmetry is gauged in compact case Goldstone boson should become massive Compactification is modeled through: Coupling field theory to gravity h(τ) h(τ) + const τ UV is the radius where new warp-factor approaches constant Imposing boundary conditions on wave-function at τ Bulk > τ UV z(τ Bulk ) = w(τ Bulk ) = 0 We impose Dirichlet boundary condition on z, w Mass of goldstone mode U is model-dependent and typically not very small compared to field theory scale M. Benna, A. D., I. Klebanov, A. Solovyov M 2 U e(4τ UV 2τ Bulk )/3 A. Dymarsky D3-D7 on warped throat 8/29

9 Physics of glueballs EOM of superpartners are related via SQM Q 1 Q 2 ψ 1 = m 2 ψ 1 Q 2 Q 1 ψ 2 = m 2 ψ 2 factorization of coupled fluctuations m 2 n 2 for each U non-susy degeneration of spectra m U for large U Gravity Multiplet along: transverse traceless graviton and U(1) chiral current A.D., D. Melnikov Accidental degeneration of spectra for some U? Parametrical mass gap A. Dymarsky D3-D7 on warped throat 9/29

10 Explicit model: D3 D7 D3 on the warped deformed conifold together with D. Baumann, I. Klebanov, L. McAllister, P. Steinhardt A. Dymarsky D3-D7 on warped throat Explicit model: D3 D3 D7 10/29

11 Motivation Conical singularity is typical for CY Warping suppresses forces between branes (flat potential) and helps resolve hierarchy problem Kahler moduli are fixed via non-perturbative effects Complex moduli are fixed by flux Potential is uplifted by brane-antibrane interaction Geometry of throat is explicit hence low-energy physics is controllable A. Dymarsky D3-D7 on warped throat Explicit model: D3 D3 D7 11/29

12 Explicit model: D3 D7 D3 on the warped deformed conifold 0 r0 rμ r bulk CY D3 D7 D3 warped throat Branes on the KS throat mobile SUSY D3 SUSY D7 brane fixed by fluxes anti-d3 brane at the bottom of conifold A. Dymarsky D3-D7 on warped throat Explicit model: D3 D3 D7 12/29

13 KKLT/KKLMMT proposal: Fixing Moduli Complex moduli are fixed by fluxes Kahler moduli are fixed by non-perturbative effects on warped D3-D7 branes Potential is uplifted by D3 brane antibrane interaction W = W 0 + Ae ρ S. Kachru, R. Kallosh, A. Linde, J. Maldacena, L. McAllister, S. Trivedi V = e K (g a bd a W D b W 3 W 2 ) O. DeWolf, S. Giddings A. Dymarsky D3-D7 on warped throat Explicit model: D3 D3 D7 13/29

14 D3 backreaction on four-cycle volume D3 affects D7 through the change of warped volume of four-cycle W = W 0 + A(ϕ)e aρ A exp( T 3δV n ) O. Ganor; M. Berg, M. Haack, B. Kors; S. Giddings, A. Maharana D3 is treated as a source D7 is treated as a probe For D7 embedded along Σ 4 f(w i ) = 0 the superpotential is A(w i ) [f(w i )] 1/n D. Baumann, A. D., I. Klebanov, J. Maldacena, L. McAllister, A. Murugan A. Dymarsky D3-D7 on warped throat Explicit model: D3 D3 D7 14/29

15 Landscape of effective potential 0.6 n 7 =8 W 0 = a= Γ = 0.01 α=2 s= V eff /V AdS φ φ 1 A. Dymarsky D3-D7 on warped throat Explicit model: D3 D3 D7 15/29

16 Effective potential and stable trajectories Search for flat trajectories in 6 dimensions Radial motion stable with respect to angular directions C. Burgess, J. Cline, K. Dasgupta, H. Firouzjahi; also A. Krause, E. Pajer Potential along angular directions O. DeWolfe, L. McAllister, G. Shiu, B. Underwood Several inflaton fields S. Panda, M. Sami, S. Tsujikawa Search through different embeddings with different number of D7 outlined particular scenario most suitable for inflation D. Baumann, A. D., I. Klebanov, L. McAllister A. Dymarsky D3-D7 on warped throat Explicit model: D3 D3 D7 16/29

17 Effective potential and inflation Inflation is possible near fine-tuned inflection point V(φ) x V(φ) x s= s= V(φ) x φ/φ μ V(φ) x φ/φ μ s= s= φ/φ μ φ/φ μ A. Dymarsky D3-D7 on warped throat Explicit model: D3 D3 D7 17/29

18 Inflation around inflection point Effective cubic potential V = V 0 + λ 1 φ + 1 3! λ 3φ V Total number of e-foldings N tot = π 2V 2 0 λ 1 λ 3 Spectral index n S 1 can be red or blue n S 1 = 4π N tot cot( πn CMB N tot ) n s φ/φ μ N tot With N tot > 120, n S is compatible with experimental data A. Dymarsky D3-D7 on warped throat Explicit model: D3 D3 D7 18/29

19 Explicit D3 D3 D7 model: an outline Despite generality of the setup inflation is difficult to achieve Only certain embeddings can lead to inflationary potential Landscape Flat potential for a wide range of inflaton field is not possible with any amount of fine tuning Fine tuning Is this result typical? Other models of stringy inflation A. Dymarsky D3-D7 on warped throat Explicit model: D3 D3 D7 19/29

20 Explicit D3 D3 D7 model: an outline Yet certain fine-tuned inflationary scenario is possible Inflation is possible near fine-tuned inflection point with spectral index within observable bounds Existence proof of stingy inflation A. Dymarsky D3-D7 on warped throat Explicit model: D3 D3 D7 20/29

21 D7 brane on KS and baryonic branch A. Dymarsky D3-D7 on warped throat D7-brane 21/29

22 Baryonic branch: another way to uplift potential Compactification fixes some non-zero value of U... D3 breaks SUSY in a controllable way and uplifts the potential A.D., I. Klebanov, N. Seiberg Does D7 brane preserve SUSY for non-zero U? A. Dymarsky D3-D7 on warped throat D7-brane 22/29

23 SUSY D-branes and Generalized Calibration Action of D7 brane is calibrated by closed form W C L.Martucci, P.Smith S DBI + S CS Vol 4 W C [F = da] Calibration condition is saturated for SUSY embedding D5 wrapping minimal S 3 of conifold Constant tension along the branch W C = Ω (3,0) φ(t) = φ(a, v, t) A.D., Y. Tachikawa, I. Klebanov Σ A. Dymarsky D3-D7 on warped throat D7-brane 23/29

24 Calibration condition det(g + M) (PfJ PfM) 2 + (J M) 2 Inequality is saturated when M is of (1, 1) type. (PfJ PfM) 2 + (J M) 2 R ( e iθ (PfJ PfM + ij M) ) Inequality is saturated when I ( e iθ (PfJ PfM + ij M) ) = 0 Kappa-symmetry equation for D7 mutually supersymmetric with background: cos θ = e φ M is of (1, 1) type 1 2 (J J M M) Σ = e2a U J M Σ A. Dymarsky D3-D7 on warped throat D7-brane 24/29

25 Calibration form e 4A e φ det(g + M) + C e M Σ W C Σ Using 6 F 3 = e 2φ H 3 we find RR fields C 4 = e 4A Vol 4 C 6 = e 4A Vol 4 U 2 B C 8 = e 4A Vol 4 U 2 2 B B W C = 1 2 (e2a J) (e 2A J) U(e 2A J) M + U 2 B M U 2 2 B B Using that e 2A J = UB d [U(π + χ)g 5 ] and M = B + F we get dw C = 0 On-shell action S = U 2 [ Vol 4 Σ A(π + χ) dg (π + ] χ)2 g 5 dg 5 Σ g 5 dg 5 = const for large t A. Dymarsky D3-D7 on warped throat D7-brane 25/29

26 When embedding Σ is SUSY? For which holomorphic Σ there is such A satisfying kappa-symmetry condition? Field theory interpretation of arbitrary Σ is not clear Geometrical interpretation of calibration does not provide an answer Extra identity d(e 2A φ J e 2A φ J) = 0 guarantees that kappa-symmetry equation for D7 and for Euclidean D5 (baryion) is integrable. A. Dymarsky D3-D7 on warped throat D7-brane 26/29

27 Ouyang embedding An interesting example is Ouyang embedding Σ : z 1 + iz 2 = const in the KS case F = da if of (1, 1) type M = B + F is primitive on Σ: J M Σ = 0 Different approaches to find A Use unbroken U(1) U(1) to express general F = da (1, 1) form through one function f(θ 1, θ 2 ) The problem is to satisfy primitivity condition: second order differential equation for f(θ 1, θ 2 ) this a non-trivial RHS Integrate kappa-symmetry condition A e 2A J = w 3 g(t) + d(..) The problem is to make F = da of (1, 1) type A. Dymarsky D3-D7 on warped throat D7-brane 27/29

28 Kuperstein embedding Kuperstein embedding z 4 = µ A = 0 satisfies kappa-symmetry in the KS case S.Kuperstein Generalization to the baryonic branch (U 0) µ = 0: D7 touches the tip. Usual conifld variables θ i, φ i, ψ, t applicable µ 0: D7 stretches to ɛ 2 + µ 2 + µ 2 A = ζ(t)g 5 solves kappa-symmetry equation w ζ = [ (ζ + χ) 2 + 2π(ζ + χ) + (h 2 2 sinh2 (t)e 2φ π 2 (e 2φ 1)) ] g 5 dg 5 dw ζ Σ = 0 ζ + χ Ut2 e 2/3t and S r 4 + U 2 log 2 r A. Dymarsky D3-D7 on warped throat D7-brane 28/29

29 Outline Warped throat geometries provide explicit description for low energy physics of dual gauge theories... massless modes and moduli space glueball physics soft breaking SUSY and play important role in understanding stringy inflation explicit model of stringy inflation around inflection point variety of scenarios A. Dymarsky D3-D7 on warped throat Outline 29/29