Open String Wavefunctions in Flux Compactifications. Fernando Marchesano

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1 Open String Wavefunctions in Flux Compactifications Fernando Marchesano

2 Open String Wavefunctions in Flux Compactifications Fernando Marchesano In collaboration with Pablo G. Cámara

3 Motivation Two popular lines of research in type II vacua are Closed strings: Flux vacua CY 3 Open strings: D-brane models G 3 Moduli stabilization de Sitter vacua Inflation Warping... Chirality MSSM/GUT spectra Yukawa couplings Instanton effects...

4 Motivation Both subjects have greatly evolved in the past few years, but mostly independently Some overlapping research has shown that fluxes can have interesting effects on D-branes Soft-terms/moduli stabilization D-terms and superpotentials Instanton zero mode lifting Warping effects Cámara, Ibáñez, Uranga 03 Lüst, Reffert, Stieberger 04 Gomis, F.M., Mateos Martucci 06 Tripathy, Trivedi 05 Saulina 05 Kallosh, Kashani-Poor, Tomasiello Shiu s Talk

5 Motivation Both subjects have greatly evolved in the past few years, but mostly independently Some overlapping research has shown that fluxes can have interesting effects on D-branes The most interesting sector is however still missing G3 Understood Not understood

6 The problem The chiral sector of a D-brane model arises from open strings with twisted boundary conditions We do not know the precise effect of fluxes and warping microscopically CFT tricky because of RR flux Full D-brane action not available beyond U(1) gauge theories

7 The strategy Idea: Consider Type I/Heterotic strings in the field theory limit Twisted open strings can be understood as wavefunctions Their coupling to fluxes can be read from the 10D action type IIB type I

8 Type I flux vacua The particle content of type I theory is bosons fermions gravity g MN,C MN, φ ψ M, λ closed st. vector A α M χ α open st.

9 Type I flux vacua The particle content of type I theory is bosons fermions gravity g MN,C MN, φ ψ M, λ closed st. Fluxes vector A α M χ α open st. Torsion F 3 = dc 2 + ω 3 F 2 = da

10 Type I flux vacua The particle content of type I theory is bosons fermions gravity g MN,C MN, φ ψ M, λ vector A α M χ α open st. Torsion closed st. Fluxes Open string e.o.m. (/D + 14 ) eφ/2 /F 3 χ = 0 F 3 = dc 2 + ω 3 F 2 = da D K F KP eφ/2 F 2 MNF MNP = 0

11 Type I flux vacua The gravity background is of the form ds 2 = Z 1/2 ds 2 R 1,3 + ds 2 M 6 with M6 an SU(3)-structure manifold ( forms Jmn, Ωmnp) such that Ze φ g s = const. g 1/2 s e φ/2 F 3 = M6 e 3φ/2 d(e 3φ/2 J) d ( e φ J J ) = 0 Hull 86 Strominger 86 If M6 is complex N=1 SUSY vacuum If M6 is not complex N=0 no-scale vacuum Schulz 04 Cámara & Graña 07 Lüst, F.M., Martucci, Tsimpis 08

12 Twisted tori Ansatz for M6: elliptic fibration ds 2 M 6 = Z 1/2 (e a ) 2 + Z 3/2 ds 2 B 4 a Π 2 simplest examples (warped) twisted tori (B4 = T 4 ) B4 : base Π2 : fiber They can be described as: i) S 1 bundles ii) Coset manifolds Parallelizable Explicit metric Γ \ G G : nilpotent Lie group Γ : discrete subgroup

13 Twisted tori Ansatz for M6: elliptic fibration ds 2 M 6 = Z 1/2 (e a ) 2 + Z 3/2 ds 2 B 4 a Π 2 simplest examples (warped) twisted tori (B4 = T 4 ) B4 : base Π2 : fiber For instance: ds 2 B 4 = m=1,2,4,5 (R m dx m ) 2 ds 2 Π 2 = [ (R 3 dx 3 ) 2 +(R 6 ẽ 6 ) 2] F 3 = N(dx 1 dx 2 + dx 4 dx 5 ) ẽ 6 gs 1 T 4 dz 2 ẽ 6 = dx 6 + M 2 (x1 dx 2 x 2 dx 1 + x 4 dx 5 x 5 dx 4 )

14 Twisted tori In our example dẽ 6 = M(dx 1 dx 2 + dx 4 dx 5 ) ( e de 6 = R 6 1 e 2 M + e4 e 5 ) R 1 R 2 R 4 R 5 In general dẽ a = 1 2 f a bcẽ b ẽ c de a = 1 2 f a bce b e c

15 Twisted tori In our example dẽ 6 = M(dx 1 dx 2 + dx 4 dx 5 ) ( e de 6 = R 6 1 e 2 M + e4 e 5 ) R 1 R 2 R 4 R 5 In general dẽ a = 1 2 f a bcẽ b ẽ c de a = 1 2 f a bce b e c f a bc : structure constants of a 6D Lie algebra g generators of g : ˆ a e a α (x) x α [ ˆ b, ˆ c ]= f a bc ˆ a

16 Twisted tori In our example dẽ 6 = M(dx 1 dx 2 + dx 4 dx 5 ) ( e de 6 = R 6 1 e 2 M + e4 e 5 ) R 1 R 2 R 4 R 5 In general dẽ a = 1 2 f a bcẽ b ẽ c de a = 1 2 f a bce b e c f a bc : structure constants of a 6D Lie algebra g generators of g : ˆ a e a α (x) x α exp(g) =H 5 R M 6 = Γ H5 \H 5 Z\R [ ˆ b, ˆ c ]= f a bc ˆ a G = exp(g) M 6 = Γ\G (For Z 1)

17 Dimensional reduction Following Cremades, Ibáñez, F.M. 04 Consider a U(N) gauge group (i.e., N D9-branes) The bosonic d.o.f. come from the 10D gauge boson AM A M = B α M U α + W αβ M e αβ U α : Cartan subalgebra As usual B α m 0 = U(N) α U(n α )=G unbr

18 Dimensional reduction Following Cremades, Ibáñez, F.M. 04 Consider a U(N) gauge group (i.e., N D9-branes) The bosonic d.o.f. come from the 10D gauge boson AM A M = B α M U α + W αβ M e αβ U α : Cartan subalgebra As usual B α m 0 = U(N) α U(n α )=G unbr We can expand the bosonic fields as B(x µ,x i ) = b µ (x µ ) B(x i ) dx µ + b m (x µ )[ B m + ξ m ](x i ) e m U(n α ) Adj. m W (x µ,x i ) = w µ (x µ ) W (x i ) dx µ + m... and similarly for fermions w m (x µ ) Φ m (x i ) e m Φ m e m ( n α,n β ) bif.

19 Laplace and Dirac eqs. The e.o.m for the adjoint fields read (Z 1) ˆ ˆ a a B = m 2 BB ( Γ a ˆ a + 1 ) 2 /fp B 4 + χ 6 = m χ B6χ 6 gauge bosons fermions... scalars P B 4 + = 1 2 (1 ± Γ B 4 ) B 6 = 6D Maj. matrix For bifundamental fields: ˆ a ˆ a i( B α m B β m ) see Cámara s Talk

20 Recap We want to understand the effect of fluxes on non-abelian gauge theories Nice framework: type I/heterotic flux vacua 10D field theory Simplest examples in terms of twisted tori The effect of fluxes appears in the modified Dirac and Laplace equations. For adjoint fields and Z 1: ˆ ˆ a a B = m 2 BB ( Γ a ˆ a + 1 ) 2 /fp B 4 + χ 6 = m χ B6χ 6

21 Gauge Bosons Laplace equation ˆ a ˆ a B = m 2 BB In our example: R 1 ˆ 1 = x 1 + M 2 x2 x 6 R 4 ˆ 4 = x 4 + M 2 x5 x 6 R 2 ˆ 2 = x 2 M 2 x1 x 6 R 5 ˆ 5 = x 5 M 2 x4 x 6 R 3 ˆ 3 = x 3 R 6 ˆ 6 = x 6 If B does not depend on x 6 ˆ a = a B = e 2πi k x k =(k1,k 2,k 3,k 4,k 5 ) If B depends on x 6 like e 2πik 6x 6 eq. of a W-boson in a magnetized T 4, with magnetic flux k6m F cl 2 = k 6 M(dx 1 dx 2 + dx 4 dx 5 )

22 Gauge Bosons Laplace equation ˆ a ˆ a B = m 2 BB KK modes on the S 1 fiber are analogous to magnetized open strings B = θ-functions & sums of Hermite functions Fluxes freeze moduli extra degeneracies M =0 M 6= 0 m 2 B = k ( ) 2 ( 6M k6 k3 (n + 1) + + πr 1 R 2 R 6 R 3 ) k 2 6 =2 } 2j"j=R 6 k 6 =1 } j"j=r 6 k 6 =0

23 Gauge Bosons Laplace equation ˆ a ˆ a B = m 2 BB KK modes on the S 1 fiber are analogous to magnetized open strings B = θ-functions & sums of Hermite functions Fluxes freeze moduli extra degeneracies Wavefunctions are localized

24 Group Manifolds While the previous example was quite simple, one can solve the Laplace eq. for more general manifolds of the form Γ \ G A natural object to consider is the non-abelian Fourier transform ˆf ω ϕ( s) = G B(g)π ω (g)ϕ( s)dg unirrep of G auxiliary Hilbert space H

25 Group Manifolds While the previous example was quite simple, one can solve the Laplace eq. for more general manifolds of the form Γ \ G Let us consider the function B ϕ,ψ ω (g) = (π ω (g)ϕ, ψ) scalar product in H Note that (π ω (g)ϕ, ψ) = (π ω (g)π ω ( )ϕ, ψ) So we can take Ψ = δ-function and φ eigenfunction Finally we can impose Γ-invariance via B ω (g) = γ Γ π ω (γg)ϕ( s 0 )

26 Group Manifolds While the previous example was quite simple, one can solve the Laplace eq. for more general manifolds of the form Γ \ G By construction, we have a correspondence of unirreps of G and families of wavefunctions in Γ \ G Previous example H 2p+1 Heisenberg group π k z u( s) = e 2πik z [z+ x y/2+ y s] u( s + x) π k x, k y = e 2πi( k x x + k y y) ( x, y, z) fiber KK modes base KK modes

27 Fermions Dirac equation Squared Dirac eq. (D + F) (D + F)Ψ = m χ 2 Ψ i(d + F)Ψ = m χ Ψ D Γ a ˆ a F 1 2 /fp B 4 + Moduli lifting info.

28 Fermions Dirac equation Squared Dirac eq. (D + F) (D + F)Ψ = m χ 2 Ψ i(d + F)Ψ = m χ Ψ D Γ a ˆ a F 1 2 /fp B 4 + Moduli lifting info. Previous example: F = 0 D D = ˆ m ˆ m ˆ m ˆ m ε ˆ ε ˆ 6 ˆ m ˆ m ˆ m ˆ m ε = flux density All entries of the matrix commute standard diagonalization

29 Fermions Dirac equation Squared Dirac eq. (D + F) (D + F)Ψ = m χ 2 Ψ i(d + F)Ψ = m χ Ψ D Γ a ˆ a F 1 2 /fp B 4 + Moduli lifting info. Previous example: F = 0 0 ξ 3 = 0 B 1 1 ξ ± = ±i B 0 2 k 3 ;k 6 +3 j"k 6j R 6 2 k 3 ;k 6 +2 j"k 6j R 6 2 k 3 ;k 6 + j"k 6j R 6 2 k 3 ;k 6 0»» 3» + n =1 + ; 3 n =2 + ; 2 n =0 + ; 2 n =1 + ; 1 n =0 ; 2 + n =1 n =0 n =0 + ; 0 n =1 ; 3 + n =1 + n = x 8 x 4 x 2 x

30 Fermions Squared Dirac eq. (D + F) (D + F)Ψ = m χ 2 Ψ More involved example: F 0 ˆ m ˆ m (D + F) (D + F) = 0 ˆ m ˆ m ε ˆ z 3 ε ˆ z 2 0 ε ˆ z 3 ˆ m ˆ m ε ˆ z 1 0 ε ˆ z 2 ε ˆ z 1 ˆ m ˆ m ε 2 Entries no longer commute!!

31 Fermions Squared Dirac eq. (D + F) (D + F)Ψ = m χ 2 Ψ More involved example: F 0 ˆ m ˆ m (D + F) (D + F) = 0 ˆ m ˆ m ε ˆ z 3 ε ˆ z 2 0 ε ˆ z 3 ˆ m ˆ m ε ˆ z 1 0 ε ˆ z 2 ε ˆ z 1 ˆ m ˆ m ε 2 Entries no longer commute!! Eigenvectors: ξ 3 ˆ z 1 ˆ z 2 ˆ z 3 B ξ ± ˆ z 3 ˆ z 1 + m ξ± ˆ z 2 ˆ z 3 ˆ z 2 m ξ± ˆ z 1 B ˆ z 3 ˆ z 3 + m 2 ξ ± m 2 ξ 3 = m 2 B m 2 ξ ± = 1 4 ( ) 2 ε µ ± ε 2 µ +4m 2 B

32 Recap II We have computed the spectrum of KK modes in several type I vacua based on twisted tori If one assumes the hierarchy then one has Vol 1/2 B 4 Vol Π2 ε = m flux m KK base m KK fib Massless modes and lifted moduli ψ = const like in T 6 Base KK modes ψ like in T 4 Fiber KK modes Exotic, localized ψ

33 About warping In the above we have assumed a constant warping One can check that 2 T 4 Z 2 = ε So for Vol 1/2 Vol we have ε m KK Π2 and B 4 Z = const. is a good approximation base However, for Vol 1/2 B 4 Vol Π2 we have Warping effects Fiber modes more localized should dominate

34 Type IIB T-dual We can take our models to type IIB by T-duality on the fiber coordinates: N D9-branes N D7-branes KK mode on B 4 (T 2 ) 1 (T 2 ) 2 KK mode on (T 2 ) 1 (T 2 ) 2 KK mode on Π 2 (T 2 ) 3 Winding mode on (T 2 ) 3 D7 " # = # + $ 0 # = # 0 " 3 2 (T ) 1 2 (T ) 2 2 (T ) 3 2! R % 2

35 Conclusions We have considered type I flux vacua in order to see the effect of fluxes on open strings via field theory calculations Assuming constant Z, one can compute exactly the massless and massive spectrum of wavefunctions for models based on twisted tori and group quotients Γ \ G The techniques used here for adjoint fields also work for bifundamental chiral multiplets see Cámara s Talk Computing 4D couplings via wavefunctions, we can compare with the ones from 4D sugra. They indeed agree for ε small For ε not small, however, we expect new phenomena, in part due to warping and in part due to exotic KK modes

36 Outlook As a byproduct, we have developed a method for computing wavefunctions on group manifolds and quotients Γ \ G This is not only useful for type I compactifications, but also for the KK spectrum of type IIA flux vacua de Sitter vacua AdS vacua Silverstein 07 Haque, Underwood,Shiu, van Riet 08 Lüst & Tsimpis 04 see Villadoro s & Zagermann s Talks

37 Outlook As a byproduct, we have developed a method for computing wavefunctions on group manifolds and quotients Γ \ G This is not only useful for type I compactifications, but also for the KK spectrum of type IIA flux vacua de Sitter vacua AdS vacua We have also seen that the effect of RR fluxes is very simple once that the background eom have been applied ( Γ a ˆ a [ /f + e φ/2 /F 3 ] ) χ 6 Silverstein 07 Haque, Underwood,Shiu, van Riet 08 ( Γ a ˆ a /fp B 4 + Lüst & Tsimpis 04 see Villadoro s & Zagermann s Talks ) χ 6...hint for a CFT computation?