Chiral matter wavefunctions in warped compactifications Paul McGuirk University of Wisconsin-Madison Based on: arxiv:0812.2247,1012.2759 (F. Marchesano, PM, G. Shiu) Presented at: Cornell University, April 27, 2011 1
Outline 1.Motivation and setup 2.The adjoint sector: 7-7 strings 3.The chiral sector: 7-7 strings 4.Application: 4d effective field theories 5.Caveats and conclusions 2
Outline 1.Motivation and setup 2.The adjoint sector: 7-7 strings 3.The chiral sector: 7-7 strings 4.Application: 4d effective field theories 5.Caveats and conclusions 3
Warping and its benefits In this talk, I ll focus on warped geometries in string theory ds 2 10 = e 2α dx 2 4 + e 2α dy6 2 warp factor α = α y Such geometries have interesting phenomenology - Address Hierarchy problem [RS; GKP] - Late-time acceleration [KKLT;...] - Inflation [KKLMMT;...] - Sequestering [Luty, Sundrum; Kachru, McAllister, Sundrum; Berg, Marsh, McAllister, Pajer;...] -... 4
Moduli stabilization Warping also a generic feature of string models String compactifications usually come with moduli: zeroenergy deformations of the internal space Such moduli determine 4d effective field theory and result in fifth forces so must be lifted 5
Moduli stabilization (cont.) Lifting can be achieved by the addition of fluxes W IIB = Ω G (3) W IIA = Ω H (3) + e J F Backreaction of fluxes complicates geometry (e.g. spoils Calabi-Yau) Least amount of complication: type IIB with ISD flux and constant dilaton allows for warped Calabi-Yau NB: non-perturbative effects needed to stabilize Kähler structure [KKLT] will break this condition [Koerber, Martucci; Heidenreich, McAllister, Torroba] 6
Open strings In type II theories, gauge groups and charged matter come from the open string excitations of D-branes Example: In flat space, the excitations of N coincident Dp-branes described by maximally supersymmetric (16 supercharges) U(N) Yang-Mills in (p+1)- dimensions 7
Open strings (cont.) Bifundamental matter comes from the intersection of D- branes (or from branes at singularities) Without fluxes, the number of families is the number of intersections of the branes 8
Chiral matter We will focus on D7-branes in warped IIB geometries Two D7s intersect on a 2-cycle in the 6d internal space and so to get a chiral spectrum in 4d, the branes need to be magnetized F (2) =0 9
4d effective field theories Intersecting, magnetized D7 branes in flux compactifications are thus promising for building realistic string models (though other ingredients needed) A 4d effective field theory is useful to discuss long wavelength phenomenology Two routes to an eft: Conformal field theory techniques Dimensional reduction 10
4d effective field theories (cont.) N 4 =1warped compactifications in type IIB necessarily involve Ramond-Ramond fluxes: ds 2 F (5) 10 = e 2α dx 2 4 + e 2α dy6 2 = 1+ F (5) F (5) =de 4α dvol R 1,3 6 G (3) =ig (3) This makes quantization of the string difficult and so the 4d eft cannot be easily extracted from stringy amplitudes 11
4d effective field theories (cont.) Alternate method: Higher dimensional 1 4g 2 8 W gf 2 actions are highly constrained so use dimensional reduction 1 4g 2 8 R 1,3 d 4 xf 2 Σ 4 d 4 y gs 2 Requires solutions of higher-dimensional equations of motion A µ x, y = Aµ x s y 1 g 2 4 = V S 4 g 2 8 wavefunction 12
Outline 1.Motivation and setup 2.The adjoint sector: 7-7 strings 3.The chiral sector: 7-7 strings 4.Application: 4d effective field theories 5.Caveats and conclusions 13
Warped spherical cow Our goal is to learn something about the 4d eft from the 8d eft via dimensional reduction Since our focus is to learn about the effects of warping, I ll focus here on a simple background ds 2 10 = e 2α dx 2 4 + e 2α dy6 2 F (5) T 6 = 1+ F (5) F (5) =de 4α dvol R 1,3 G (3) =0 As a warmup, consider the adjoint matter on a single D7 wrapping T 4 T 6 14
Bosonic modes The 8d bosonic degrees of freedom are A α 8d gauge boson A µ 4d gauge boson Φ i transverse fluctuations A a, Φ i 4d scalars The bosonic action is the DBI+CS action SD7 DBI = τ D7 d 8 x det (P [g αβ ]+λf αβ ) W S CS λ =2πα D7 = τ D7 P C (4) (2) e λf W P g αβ = gαβ + λ 2 g ij α Φ i β Φ j 15
Bosonic modes (cont.) In this case, warping does not effect the zero modes R 1,3Φ i + e 4α T 4Φ i =0 e 4α R 1,3A µ + T 4A µ =0 R 1,3A a + b F ba + abcd b e 4α F cd =0 e 2α e 2α The zero modes satisfy and so are constant R 1,3X =0 16
Fermionic modes The 8d fermionc degrees of freedom are encoded in a pair of 10d Majorana-Weyl fermions subject to a gauge redundancy called κ-symmetry (c.f. GS superstring) Θ = θ1 θ 2 Θ Θ + P D7 κ P± D7 = 1 2 1 ±iγ(8) iγ (8) 1 The 4d degrees of freedom are fermionic superpartners of the gauge boson and complexified scalars ψ 0 A µ ψ 1,2 A 1,2 ψ 3 Φ gaugino Wilsonini modulino 17
Fermionic modes (cont.) The fermionic action is [Martucci, Rosseel, Van denbleeken, Van Proeyen] SD7 f = τ D7 det (g αβ ) ΘP D7 Γ α α + 1 F 16 / (5) Γ α iσ 2 Θ Then W For the zero modes: 4-cycle chirality operator / R + / 1,3 T + 1 4 2 / T α 4 1+2ΓT 4 θ =0 ψ 0 = η 0 ψ 1,2 = η 1,2 e 3α/2 e α/2 e 3α/2 ψ 3 = η 3 constant spinors 18
Fermionic modes (cont.) Since the Γ matrices are warped, these are consistent with supersymmetry δa µ Γ µ ψ 0 δa = e α/2 η a Γ a ψ a 0 δφ Γ i ψ 3 Killing spinor This analysis can be generalized (see 0812.2247) - Abelian magnetic flux - Calabi-Yau - Bulk fluxes (see also [Cámara, Marchesano]) We ll return to the above wavefunctions later 19
Outline 1.Motivation and setup 2.The adjoint sector: 7-7 strings 3.The chiral sector: 7-7 strings 4.Application: 4d effective field theories 5.Caveats and conclusions 20
Bifundamental matter Chiral matter is much more phenomenologically interesting, but more involved. Consider first vector-like bifundamental matter: T 6 = T 2 1 T 2 2 T 2 3 matter curve D7 1 : z 3 = M 3 z 2 D7 2 : z 3 = M 3 z 2
Intersections as Higgsing When the branes are coincident, the symmetry is enhanced to U (2). The transverse fluctuations are promoted to an adjoint-valued scalar Φ vevs for φ a,b Position of D7 1 Position of D7 2 φ a Φ = φ + φ φ b - strings (bifundamental) 7 2 7 1 correspond to background D7 positions φ a = λ 1 M 3 z 2 φ b = λ 1 M 3 z 2 small angle needed to neglect α corrections 22
Non-Abelian bosonic action Must use the non-abelian action [Myers] S DBI D7 = τ D7 S CS D7 = τ D7 W W Str det d 8 x Str (Mαβ )det Q i e j iλι Φι Φ C (4) e λf (2) P symmetrization P v α = vα + λd α Φ i v i M αβ =P g αβ + g αi Q 1 δ ij gjβ + λfαβ Q i j = δ i j iλ Φ i, Φ k g kj interior product 23
D7 bosonic action (cont.) Bulk fields given as a non-abelian Taylor expansion adjoint valued neutral Ψ Φ = n=0 λ n n! Φi1 Φ i n i1 in Ψ 0 = Ψ 0 + O (λ) Leading order in α, action is warped Yang-Mills Equations of motion are second order and hard to solve in general 24
Non-Abelian fermionic action Abelian analogue of Martucci action not known S F D7 = 1 g 2 8 d 8 x θ e α / R 1,3 +e α / T 4 +e α 1 2 / T 4α 1+2Γ T 4 θ To leading order in α [Wynants] / /D δl = i θe α Γ i Φ i, θ 25
Equations of motion For the bifundamental zero modes, take the ansatz ψ0,3 =e 3α/2 e χ ψ 0,3 1,2 =eα/2 e χ 1,2 gaugino, modulino wilsonini Equations of motion from Fermionic action (no flux yet) e 4α 0= 1 χ 1 + 2χ 2 +e 4α D 3 χ 3 0= 1 χ 0 + 2χ 3 D± 3 χ 2 0= 1χ 3 2χ 0 D± 3 χ 1 e 4α 0= 1χ 2 2χ 1 +e 4α ˆD 3 χ 0 D 3 = im 3 z 2 26
BPS conditions For a single D7 brane, the equations of motion follow from F- and D-flatness conditions: [Jockers, Louis; Martucci] fundamental 3-form W = S 4 P γ e λf dγ = Ω e B(2) (2) D = S 4 P Im η e λf η =e 2α Im τ e ij e B(2) Comparing to the CS-action, the non-abelian version W = T 4 T 4 should be (see also [Butti et. al.]) Str P e iλι Φι Φ γ e λf (2) S 4 e 2α D = S T 4 T S 4 (2) warped Kähler form P e iλι Φι Φ Im η e λf (2) These yield the previous equations of motion with ψ 0 =0 J 27
Unwarped zero mode In the absence of warping, the zero modes are exponentially localized on the intersection ψ0,1 =0 ψ2 =σ x µ e M 3 z 2 2 ψ3 = ± iσ x µ e M 3 z 2 2 4D field Mixture of deformation modulus and Wilson lines of the un-higgsed theory 28
Warped zero mode For arbitrary warping, no simple analytic solution In the weak warping case, can treat the warping as a perturbation e 4α =1+β 1 Can then expand the warped zero mode in terms of the unwarped massive modes 29
Unwarped spectrum The equation of motion for the massive modes is D X λ = m λx ± λ D = 0 1 2 D3 1 0 D 3 ± 2 2 D ± 3 0 1 D 3 2 1 0 X λ = χ 0 χ 1 χ 2 χ 3 Easiest to work in a rotated basis J = 1 1 1/ 2 i/ 2 i/ 2 1/ 2 1 1 2 3 X = J 1 X ˆD 2 = 1 2 2 ± M 3 z 2 ˆD 3 = i 2 2 M 3 z 2 30
Unwarped spectrum (cont.) Boundary conditions: Periodicity along matter curve Localized on intersection T 2 1 - sector modes built from ladder operators (giving two simple harmonic oscillator algebras) and Fourier modes lowering ˆD 2 i i ˆD+ 3 i ˆD+ 2 i ˆD 3 ϕ mnpl = h mn z 1, z 1 i ˆD+ l pe 2 i ˆD κ 3 raising M 3 z 2 2 Fourier mode 31
Unwarped spectrum (cont.) Unwarped spectrum: m 2 λ = m 2 + n 2 + M 3 l + p +1 Φ λ = ϕ mnlp, 0, 0, 0 T m 2 λ = m 2 + n 2 + M 3 l + p +1 Φ λ = 0, ϕ mnlp, 0, 0 T m 2 λ = m 2 + n 2 + M 3 l + p m 2 λ = m 2 + n 2 + M 3 l + p +2 Φ λ Φ λ = 0, 0, ϕ mnlp, 0 = 0, 0, 0, ϕ mnlp T T 32
Expanding the warped zero mode Write the warped zero mode as X = Φ 0 + λ c λ Φ λ m 2 λ unwarped modes To leading order c βk λ = 1 d 4 y Φ λ D + 0 Φ 0 S 4 where D = D 0 + βk + O 2 Examples given in [1012.2759] 33
Chirality Without magnetic flux, the spectrum is vector-like In order to have a chiral theory, the intersection must be magnetized 1 2π T 2 F (2) = M 1 σ 3 T 2 SUSY requires [Marino, Minasian, Moore, Strominger;...] F (2) = 4 F (2) Hodge- on S 4 34
Unwarped zero modes For example, if M 1 > 0, only the -sector has zero modes Due to magnetic flux, wavefunction are quasi-periodic [Cremades, Ibáñez, Marchesano;...] =e κ z 2 2 e 0 2πiM 1z 1 Im z 1 j/2m1 2M1 ϑ z 1, i2m 0 1 ϕ j, κ = M1 2 2 + M 2 3 j =0,...,2M 1 1 d 4 y ϕ j, ϕ k, families orthogonal: = δ 0 0 kj TS 4 Each family is a Gaussian peak at a different location on the matter curve 35
Warped zero modes As in unmagnetized case, warped zero mode has no general simple analytic solution Again, expand in unwarped massive modes Spectrum built from three QSHO algebras ϕ j, nlp = id1 n i D + l 2 i ˆD pϕ j, 3 0 D1 = 1 M 1 z 1 D2 2 ± κ z 2 D3 i 2 κ z 2 36
Warped zero modes (cont.) Expand warped zero mode in terms of unwarped massive modes X j, = Φ j, 0 + k,λ c k λφ k, λ then c k λ = 1 m 2 λ d 4 y Φ k, λ S 4 unwarped modes D + βk 0 Φ j, 0 family mixing is generic Examples given in [1012.2759] 37
Outline 1.Motivation and setup 2.The adjoint sector: 7-7 strings 3.The chiral sector: 7-7 strings 4.Application: 4d effective field theories 5.Caveats and conclusions 38
Warped kinetic terms The warped wavefunctions are useful in deducing the warped effective field theory Example: kinetic terms for modulino θ 3 x µ,y a = e 3α/2 ψ 3 x µ η 3 y a W d 8 x g θ 3 Γ µ µ θ 3 = R 1,3 d 4 x ψ 3 / ψ 3 Γ µ e α d 4 g y e 4α η 3 η 3 S 4 g g = 39
Warped kinetic terms (cont.) Can infer that the Kähler metric behaves as K 3 3 V w w 4 V 6 = d 6 g y V w Y 6 6 V 4 = S 4 d 4 y g Agrees with bosonic analysis: Φ x µ,y a =const σ x µ d 8 x gg µν g ij µ Φ i ν Φ j W w R 1,3 d 4 x η µν µ σ µ σ e 4α e 4α d 4 ge y 4α S 4 40
Warped kinetic terms (cont.) For a more general Calabi-Yau, Φ x µ,y a = σ A x µ s A y +c.c. K A B 1 w e 4α m A m B m A = ι Ω SA V 6 S 4 Then borrowing from warped closed string results [Shiu, Torroba, Underwood, Douglas] and unwarped closed-open results [Jockers, Louis] K = log i S S A σ B 2iL A Bσ A σ B S = τ L A Bσ 41
Wilson lines Similarly, for Wilson lines In the Calabi-Yau case K a b 1 w V 6 S 4 g g a b A (1) I int = w x µ W I y +c.c. K I J 1 V 6 w S 4 P J W I harmonic (1,0)-forms W J unwarped Kähler form Kähler potential can be found in special cases 42
Bifundamentals Warping modifications for chiral matter more complex - much less is known about Kähler potential even in the unwarped toroidal case For non-chiral bifundamental matter S = 1 g 2 8 W d 8 x g 1 tr 2 ηµν g ab F µa F νb +e 4α η µν g ij µ Φ i ν Φ j K iī 1 V 6 w X e #α X S 4 d 4 y g warped zero mode e #α =diag e 4α, 1, 1, e 4α 43
Bifundamentals (cont.) Recall the bifundamental wavefunctions are exponentially localized in the weakly warped case Approximating the Gaussians as δ-functions, for weak warping Wilson line contribution w V 2 = w K iī V 1 + V 1 w V 6 M 3 Σ 2 d 2 y g Modulus contribution e 4α 44
Chiral matter For chiral matter, warping induces off-diagonal entries in Kähler metric K j k 1 d 4 g w y X k, e #α X j, V 6 S 4 Even to leading order in weak warping, the family orthogonality is spoiled d 4 y ϕ j, ϕ k, 0 0 e 4α = δ kj T 4 Requires care for model building of this sort 45
Soft terms All of the above analysis was performed in the supersymmetric case Non-supersymmetric perturbations will induce soft susybreaking terms that calculated using the above analysis (work in progress) Example (see also [Cámara, Ibáñez, Urangra;...]) δm 1/2 G (3) Ω + S 4 46
Probe approximation All of the above was discussed in the probe approximation where the gravitational backreaction of D7 is ignored There are situations where this is ok (e.g. Sen limit, quenched approximation) but generically questionable Progress has been made for the adjoint case in the unwarped limit [Grimm] Such effects likely lead to problematic soft terms 47
Moduli stabilization We were lead to consider warping from the consideration of moduli stabilization (i.e. fluxes warping) but effects were fluxes were largely neglected here Since fluxes generate a potential for D7 moduli as well, expect impact on wavefunctions and 4d physics Additionally, stabilization of Kähler moduli requires departure from conformally Calabi-Yau 48
Conclusions We analyzed the wavefunctions for open string fields coming from D7s and intersections of D7s in warped geometries Warping effect on adjoint matter simple, but not so for chiral matter Such wavefunctions are important for understanding the 4d effective field theory of such constructions Here, I talked about some of the effects on kinetic terms (more on kinetic terms and Yukawas detailed in 0812.2247,1012.2759) More realistic models will also include: - Fluxes - 7-brane backreaction - Non-perturbative effects (for stabilizing Kähler moduli) 49
Thank you!