Chiral matter wavefunctions in warped compactifications

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1 Chiral matter wavefunctions in warped compactifications Paul McGuirk University of Wisconsin-Madison In collaboration with: Fernando Marchesano and Gary Shiu arxiv:1011.xxxx and arxiv: Presented at: The Ohio State University, SVP Fall Meeting, Nov. 7,

2 Outline 1.Motivation and setup 2.Unmagnetized branes 3.Magnetized branes 4.Warped kinetic terms 5.Conclusions 2

3 Outline 1.Motivation and setup 2.Unmagnetized branes 3.Magnetized branes 4.Warped kinetic terms 5.Conclusions 3

4 Motivation Warping plays an important role in string models: Generation of the electroweak hiearchy [RS; GKP] Stabilization of moduli [KKLT;...] Late-time acceleration [KKLT;...] and inflation [KKLMMT;...] Gauge/gravity duality In type II theories, realistic models require open strings A 4D effective action is valuable for detailed phenomenology 4

5 Motivation (cont.) Alternatively, consider an F-theory compactification In such constructions must satisfy the D3-tadpole condition χ (X) 24 = N D3 + B 3 H (3) F (3) Additional ingredients will cause warping which will modify 4D EFT D3s 5

6 Warped effective field theory In a flat background, CFT techniques can be used to determine an EFT (Kähler metrics,yukawa couplings [Lüst et. al.; Cvetic ˇ et. al.; Ibáñez et. al;...]) However, warping in type II usually involves Ramond- Ramond fluxes and it is difficult to make use of CFT techniques 6

7 Warped effective field theory Alternative: Dimensional reduction of a higher dimensional EFT 1 2g 2 8 W d 8 x gf 2 Requires knowledge of wavefunctions of light degrees of freedom 1 4g 2 8 A µ (x, y) =A µ (x) s (y) R 1,3 d 4 xf µν F µν S 4 d 4 y gs 2 Here, my focus is on open string modes (see B. Underwood s talk for closed strings) 1 g 2 4 = V S 4 g 2 8 7

8 Setup For simplicity, compactify IIB on T 6 = T 2 1 T 2 2 T 2 3 ds 2 10 = e 2α dx 2 4 +ee 2α dz m d z m F (5) =(1+ 10 ) F (5) ext F (5) ext =ee 4α dvol R 1,3 G (3) =0 Add two probe D7 branes D7 1 : z 3 = M 3 z 2 D7 2 : z 3 = M 3 z 2 Gives a U (1) U (1) gauge symmetry 8

9 Outline 1.Motivation and setup 2.Unmagnetized branes 3.Magnetized branes 4.Warped kinetic terms 5.Conclusions 9

10 Single D-brane For a single D7-brane, the 4D bosonic d.o.f. are Gauge boson Wilson lines A a A µ Worldvolume deformations Φ Dynamics described by the DBI+CS action S D7 =S DBI D7 S DBI D7 = τ D7 S CS D7 =τ D7 + S CS D7 W W P d 8 x Im τ 1 Mαβ C e B(2) (2) e λf λ =2πα M αβ =P g αβ + B αβ + λfαβ 10

11 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) correspond to background D7 positions φ a = λ 1 M 3 z 2 φ b = λ 1 M 3 z 2 11

12 Myers action Bosonic fluctuations governed by the Myers action S D7 = SD7 DBI + SD7 CS symmetrization Im 1 τ S DBI D7 = τ D7 S CS D7 = τ D7 where: W W Str d 8 x Str P interior product non-abelian pullback: P v α = vα + λv i D α Φ i det M αβ det Q i j e iλι Φι Φ C e B(2) e λf (2) M αβ =P E αβ + Im τ 1/2 Eαi Q 1 δ ij Ejβ + λ Im τ 1/2Fαβ E MN = g MN + Im τ 1/2 BMN Q i j = δ i j iλ Φ i, Φ k Im τ 1/2 Ekj 12

13 Myers 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) need small angle Leading order in α, action is Higgsed warped Yang-Mills Equations of motion are second order and hard to solve in general 13

14 Fermionic action To get first order equations, can use fermionic action d.o.f. encoded in two 10D M-W spinors Abelian case: [Martucci, Rosseel, Van denbleeken, Van Proeyen;...] S F D7 = 1 g 2 8 Θ = θ1 d 8 x Im τ 1 D7 det M αβ ΘP M 1 αβ Γβ D α O Θ θ 2 Involves Γ D7 δ Ψ M =D M δ λ =O 14

15 Fermionic action (cont.) In a warped geometry ( ), After κ-fixing S F D7 = 1 g 2 8 W d 8 x θ e α / R 1,3 +e α / T 4 +e α 1 2 / T 4α 1+2Γ S 4 θ α ) [Wynants]: Non-Abelian modification (leading / /D δl = i θe α Γ i Φ i, θ and take trace G (3) =0 D M = M / F (5) Γ M O =0 4-cycle chirality 15

16 Adjoint fields Warping effect on adjoint zero-mode wavefunctions are mild Vector multiplet: A µ const Wilson line multiplet: A m const Deformation multiplet: φ const ψ 0 e 3α/2 ψ 1,2 e α/2 ψ 3 e 3α/2 Consistent with supersymmetry 16

17 Equations of motion For the bifundamental modes, take the ansatz ψ 0,3 =e 3α/2 χ 0,3 ψ 1,2 =eα/2 χ 1,2 gaugino, modulino wilsonini Equations of motion: 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 17

18 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 should be (see also [Butti et. al.]) W = Str P e iλι Φι Φ γ e λf (2) S 4 D = S 4 S (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 18

19 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 line of the un-higgsed theory 19

20 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 20

21 Unwarped spectrum The equation of motion for the massive modes is D X λ = m λx ± λ D = D3 1 0 D 3 ± 2 2 D ± D 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/ X = J 1 X ˆD 2 = ± M 3 z 2 ˆD 3 = i 2 2 M 3 z 2 21

22 Unwarped spectrum (cont.) Boundary conditions: Periodicity along T 2 1 Localized on intersection - 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 2 i ˆD pe κ 3 raising z 2 2 Fourier mode 22

23 Unwarped spectrum (cont.) Unwarped spectrum: m 2 λ = m 2 + n 2 + M 3 l + p +1 Φ λ = ϕ mnlp, 0, 0, 0 m 2 λ = m 2 + n 2 + M 3 l + p +1 Φ λ = 0, ϕ mnlp, 0, 0 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 23

24 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 [1011.xxxx] 24

25 Outline 1.Motivation and setup 2.Unmagnetized branes 3.Magnetized branes 4.Warped kinetic terms 5.Conclusions 25

26 Chirality Without magnetic flux, the spectrum is vector-like In order to have a chiral theory, the intersection must be magnetized 1 F (2) = M 1 σ 3 2π T 2 SUSY requires [Marino, Minasian, Moore, Strominger;...] F (2) = 4 F (2) Hodge- on S 4 26

27 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;...] ϕ j, =e κ z 2 2 e 0 2πiM 1z 1 Im z 1 j/2m1 2M1 ϑ z 1, i2m 0 1 κ = M M 2 3 j =0,...,2M 1 1 families orthogonal: d 4 y ϕ j, ϕ k, = δ 0 0 kj S 4 27

28 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 28

29 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 [1011.xxxx] 29

30 Outline 1.Motivation and setup 2.Unmagnetized branes 3.Magnetized branes 4.Warped kinetic terms 5.Conclusions 30

31 Adjoint fields Recall the adjoint field wavefunctions ψ 0 e 3α/2 ψ 1,2 e α/2 ψ 3 e 3α/2 The resulting 4D kinetic terms are L K iī µ ζ i µ ζī + 1 4g 2 F µν F µν 4 K 1 1 = K 2 2 = 1 d 4 y K V 3 3 = 1 w S 4 g 2 8 V w = g 2 8 V w X 6 d 6 y e 4α 1 g 2 4 = 1 g 2 8 S 4 d 4 y e 4α S 4 d 4 y Im τ 1 e 4α 31

32 Chiral matter fields Warping modifications for chiral matter more complex 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 = 1 j k g8 2 V w d 4 y Im τ 1 X k, S 4 e #α X j, e #α =diag e 4α, 1, 1, e 4α X j, is the warped zero mode, not simply related to unwarped zero mode 32

33 Chiral matter fields (cont.) In the weak warping limit, the first order correction is δk j k = 1 V d 4 1 y Im τ χ k, βχ j, δv 3 3 S V 4 unwarped K j k 0 Generic warp factors will introduce off-diagonal terms in the Kähler metric Second order corrections make use of warped wavefunctions (work in progress...) 33

34 Summary and future directions Studied the wavefunctions for bifundamental matter in warped compactifications in both the chiral and non-chiral cases Needed to develop warped effective field theory and detailed phenomenology (though still work to be done!) Warping effects are more intricate than for adjoint matter; require a series expansion in unwarped massive modes General warp factors induce off-diagonal terms in the Kähler metric Extension to Calabi-Yau case is likely tricky... With a non-susy source (such as D3-branes), wavefunctions can be used to study soft terms (work in progress) 34