Brane world effective actions from mixed amplitudes with general …personalpages.to.infn.it/~billo/seminars/billo_prin_pisa... · 2005. 11. 8. · In particular, intersecting brane

Brane world effective actions from mixedamplitudes with general fluxes

Marco Billò

D.F.T., Univ. Torino

Pisa, 4-5 Novembre 2005

Marco Billò ( D.F.T., Univ. Torino ) Brane worlds from mixed amplitudes Pisa, 4-5 Novembre 2005 1 / 35

Foreword

This talk is based on a work in progress:

M. Bertolini, M. B., A. Lerda, J.F. Morales and R. Russo, “Braneworld effective actions for D-branes with fluxes”, to appear (soon).


Plan of the talk

1 Brane-worlds scenarios

2 D9 branes with general fluxes

3 Effective supersymmetric actions

4 The Kähler metric from strings

5 Relation to the Yukawa couplings


Plan of the talk







Plan of the talk







Plan of the talk







Plan of the talk







Brane-worlds scenarios




Intersecting brane worlds

Four-dimensional field theories with many “realistic” features arisefrom type IIA or B superstring models on suitable configurations ofD-branes

[Bachas, 1995, Berkooz et al., 1996],...

In particular, intersecting brane worlds have received muchattention recently:

see, e.g., [Uranga, 2003]

I Type IIA on R1,3 × T6 (or, more generally, on a CY - Not discussedhere)

I D6 branes wrapping intersecting 3-cycles in T6 support, on theirnon-compact world-volume, gauge groups and chiral matter(the latter are localized at the intersection points in the internalspace)



Intersecting brane worlds

Four-dimensional field theories with many “realistic” features arisefrom type IIA or B superstring models on suitable configurations ofD-branes

[Bachas, 1995, Berkooz et al., 1996],...

In particular, intersecting brane worlds have received muchattention recently:

see, e.g., [Uranga, 2003]

I Type IIA on R1,3 × T6 (or, more generally, on a CY - Not discussedhere)

I D6 branes wrapping intersecting 3-cycles in T6 support, on theirnon-compact world-volume, gauge groups and chiral matter(the latter are localized at the intersection points in the internalspace)



Gauge groups and chiral matter from branes

Gauge groups from multiple branes, bifundamental chiral matterfrom “twisted” strings, replicas from multiple intersections

X6X4

X5 X9

X8

X7

U(2)L

3 families of l.m.quarks

U(3)

N.B. The torus T6 is assumed to be factorized as T2 × T2 × T2.



T-duality and magnetized branes

Upon T-duality (along one direction in each torus), IIA → IIB, andD6-branes intersecting on 3-cycles → D9 with magnetic fluxes

σ = 0σ = π

F(0)F(π)

twisted string

Strings connecting two D9 with differentfluxes feel different b.c.’s at their twoend-points. They are twisted.The twists θi are determined from thequantized values of the fluxes

F (σ)MN =

12π

pMN

qMN

pMN = Chern class, qMN = wrapping of the Dbrane around the cycle dX M ∧ dX N .



T-duality and magnetized branes

Upon T-duality (along one direction in each torus), IIA → IIB, andD6-branes intersecting on 3-cycles → D9 with magnetic fluxes

σ = 0σ = π

F(0)F(π)

twisted string

If the torus is factorized as T2 × T2 × T2,fluxes respecting this factorization arematrices in so(2)⊕ so(2)⊕ so(2). Abeliansituation: fluxes on different branescommute.General situation: fluxes on T6 representedby so(6) matrices. Oblique case: fluxes ondifferent branes do not commute.

I This generalization is important for in thecontext of the moduli stabilization problem

[Antoniadis-maillard, 2004, Bianchi-Trevigne, 2005]


D9 branes with general fluxes




Boundary conditions on magnetized branes

Bosonic part of the open string action: Back

(xM in the T6 directions, σ = 0, π denotes the end-point)

Sbos = − 14πα′

∫d2ξ

[∂αxM∂αxNGMN + iεαβ∂αxM∂βxNBMN

]− i

∑σ

qσ

∫Cσ

dxMAσM

In presence of constant G,B and field-strengths Fσ, the boundaryconditions read

∂xM∣∣∣σ=0,π

= (Rσ)MN ∂xN

∣∣∣σ=0,π

where the reflection matrix Rσ is givenb by

Rσ =(G −Fσ

)−1 (G + Fσ

), Fσ = B + 2πα′ Fσ



Boundary conditions on magnetized branes

Bosonic part of the open string action: Back

(xM in the T6 directions, σ = 0, π denotes the end-point)

Sbos = − 14πα′

∫d2ξ

[∂αxM∂αxNGMN + iεαβ∂αxM∂βxNBMN

]− i

∑σ

qσ

∫Cσ

dxMAσM

In presence of constant G,B and field-strengths Fσ, the boundaryconditions read

∂xM∣∣∣σ=0,π

= (Rσ)MN ∂xN

∣∣∣σ=0,π

where the reflection matrix Rσ is givenb by

Rσ =(G −Fσ

)−1 (G + Fσ

), Fσ = B + 2πα′ Fσ



Twisted world-sheet fields

The above b.c.’s can be solved interms of a holomorphic, multivaluedfield X M(z) defined all over thecomplex z plane (doubling trick):

X M(e2πiz) = RMN X N(z) , R = R−1

π R0

CzImz

∂X(z) = ∂x(z)

Rez

Both R0 and Rπ, and hence R, preserve the metric: tR G R = GWe can go to a complex basis Z =

(Z i , Z i) = EX , where

R ≡ E R E−1 = diag(

e2iπθ1 , · · · , e2iπθd , e−2iπθ1 , · · · , e−2iπθd)

for 0 ≤ θi < 1 (d = 3 in our case). Back






π R0

CzImz

∂X(z) = ∂x(z)

Rez

∂X(z) = R0∂x(z)











π R0

CzImz

∂X(z) = ∂x(z)

Rez

∂X(z) = R0∂x(z)

∂X(z) = R−1π R0∂x(z)











π R0

CzImz

∂X(z) = ∂x(z)

Rez

∂X(z) = R0∂x(z)

∂X(z) = R−1π R0∂x(z)











π R0

CzImz

∂X(z) = ∂x(z)

Rez

∂X(z) = R0∂x(z)

∂X(z) = R−1π R0∂x(z)








The open string basis

The open string complex, multivalued, fields Z i(z) (and thecorresponding w.s fermions Ψi(z) have mode expansions shiftedby θi .The θi play exactly the same role as the angles betweenintersecting D6. They represent the 3 “open string moduli” whichdetermine the open string CFT properties.The vacuum |θ〉 is created by bosonic and fermionic twist fields

|θ〉 = limz→0

d∏i=1

σθi (z) sθi (z) |0〉

The physical vertices contain (excited) twist fields



Dependence of the twists on the closed moduli

The d open string twists θi depend on the 4d2 closed stringmoduli GMN and BMN and on the quantized fluxes F MN

0,π (or on thewrapping numbers for the intersecting branes)






πθ

U2

U

1

For intersecting D-branes, the θi depend onthe moduli describing the shape of the torus:

tan(πθ) =U2 n

m + U1n






For general magnetized branes, from their definition aseigenvalues of the monodromy R we obtain Back

2πi∂θi

∂m=

12

(E G−1 ∂(G − B)

∂m[Rπ − R0] E−1

)ii

− 12

(E

[R−1

π − R−10

]G−1 ∂(G + B)

∂mE−1

)ii

where m is a generic closed string modulus, built out of G and B.







2πi∂θi

∂m=

12

(E G−1 ∂(G − B)

∂m[Rπ − R0] E−1

)ii

− 12

(E

[R−1

π − R−10

]G−1 ∂(G + B)

∂mE−1

)ii

where m is a generic closed string modulus, built out of G and B.Applies to general toroidal configurations with any G and B, and togeneric (i.e. non-abelian) fluxes Fσ







2πi∂θi

∂m=

12

(E G−1 ∂(G − B)

∂m[Rπ − R0] E−1

)ii

− 12

(E

[R−1

π − R−10

]G−1 ∂(G + B)

∂mE−1

)ii

where m is a generic closed string modulus, built out of G and B.Crucial formula to reconstruct the Kähler metric for the twistedscalars from mixed open/closed amplitudes







2πi∂θi

∂m=

12

(E G−1 ∂(G − B)

∂m[Rπ − R0] E−1

)ii

− 12

(E

[R−1

π − R−10

]G−1 ∂(G + B)

∂mE−1

)ii

where m is a generic closed string modulus, built out of G and B.In the factorized case, and upon T -duality, reproduces thedependence of the angles just described


Effective supersymmetric actions




Supersymmetric brane-worlds?

Simplest models with standard-model-like features break all susy.Preserving some susy requires some tuning, in the closed and inthe open string sector.In the closed, bulk sector:

I T6 compact −→ cancel RR tadpolesI cancel NS-NS tadpoles for susy −→ orientifolds;

In the open sector, i.e. on the branes:I Susy generically broken for the open strings connecting two

different D-branes: angles θi −→ twists in the CFT −→ mass splitbetween R and NS spectrum

I Susy (partially) preserved for particular values of the twists



Supersymmetric configurations

The SUSY preserved on the twisted strings can be described inthe space of the θi ’s, which we take in [0,1).

3

1

2

B

C

O

A

θ3

θ1

θ2

For θ1 = θ2 = θ3 = 0, N = 4 susyspectrum (like for strings betweenparallel branes in flat space) Back





3

1

2

B

C

O

A

θ3

θ1

θ2

When one θ vanishes,we get an N = 2 hyper-multiplet:

I two massless scalars from NSI two massless fermions from R sector





3

1

2

B

C

O

A

θ3

θ1

θ2

On the faces, e.g., for∑

j 6=i θj − θi = 0(which we will write as

∑j εj(i)θj = 0)

we have N = 1 chiral multiplets Φi

I one massless scalar φi from NSI a chiral fermion χi from R sector)





3

1

2

B

C

O

A

θ3

θ1

θ2

Nearly N = 1 point

distance of order (α′)

We will consider softly broken N = 1 bytaking θ’s close to a face:

θi = θ(0)i + 2α′εi ,

∑j

εj(i)θ(0)j = 0

with θ(0)i and εi fixed in the limit α′ → 0.





3

1

2

B

C

O

A

θ3

θ1

θ2

Nearly N = 1 point



θi = θ(0)i + 2α′εi ,

∑j

εj(i)θ(0)j = 0


The scalar φi gets a mass M2 = 12α′ =

∑j εj(i)θj =

∑j εj(i)εj





3

1

2

B

C

O

A

θ3

θ1

θ2

Nearly N = 1 point



θi = θ(0)i + 2α′εi ,

∑j

εj(i)θ(0)j = 0


Amounts to soft susy breaking á la FI from v.e.v.’s of the auxiliaryfields D. We’re working on a direct description of this via stringdiagrams.





3

1

2

B

C

O

A

θ3

θ1

θ2

In the interior of the tetrahedron, we stillhave a chiral massless fermion from Rsector, but only massive scalars.





3

1

2

B

C

O

A

θ3

θ1

θ2

Outside the tetrahedron, the scalarswould become tachyonic.



Effective action in the N=1 case

The l.e.e.a is an N = 1 SUGRA coupled with gauged mattercoming from diferent sectors:from the closed string sector, upon usual T6 compactification.

I For instance, 62 moduli m from NS-NS bkg fields GMN ,BMNdescribing the stringy shape of the T6.

from the open string sector, gauge + matter fields living on theD-branes.

I In particular, chiral multiplets Φi (“twisted” matter) from stringsstretching between different D-branes (localized at theirintersections)

















Effective N=1 action for twisted matter

Regarding the moduli as fixed, the Kähler potential for the twistedchiral matter will be of the form

K = Kφiφi (m)φiφi + O(φ4)

(easy to check that there’s no mixing between φi and φj with i 6= jin our cases).This corresponds to a lagrangian kinetic term Back

L = −Kφiφi (m)(∂µφi∂µφi + M2φiφi)

The dependence of the“metric” Kφiφi on the closed string moduli mcan be determined from mixed open/closed amplitudes.


The Kähler metric from strings




Mixed amplitudes and the Kähler metric

VmVφ

Vφ

Let Vm be the closed string NS-NS vertex forthe modulus m. The amplitude Back

Aφiφi m ∼ 〈Vφi Vm Vφi 〉

is related to the derivative w.r.t. m of thescalar kinetic term.

String amplitudes would give canonical kinetic terms, so Back

Vφi →√

Kφiφi Vφi , Vφi →√

Kφiφi Vφi

We have then Back

Aφiφi m = i K−1φiφi

∂

∂m∂

∂φi∂

∂φiL = i K−1

φiφi

∂

∂m

[Kφiφi

(k1 k2 − M2

)]Marco Billò ( D.F.T., Univ. Torino ) Brane worlds from mixed amplitudes Pisa, 4-5 Novembre 2005 19 / 35



VmVφ

Vφ





Vφi →√


Kφiφi Vφi

We have then Back


∂

∂m∂

∂φi∂

∂φiL = i K−1

φiφi

∂

∂m

[Kφiφi

(k1 k2 − M2




VmVφ

Vφ





Vφi →√


Kφiφi Vφi

We have then Back


∂

∂m∂

∂φi∂

∂φiL = i K−1

φiφi

∂

∂m

[Kφiφi

(k1 k2 − M2



Closed string moduli vertices

The vertex for the insertion of a genericmodulus m reads Recall

Vm(z, z) =∂

∂m(G − B)MN V M

L (z) V NR (z)

where

V ML (z) =

[∂X M

L (z) + i(kL ·ΨL)ΨM(z)

]ei kL·XL(z) ,

V NR (z) =

[∂X N

R (z) + i(kR ·ΨR)ΨN(z)]

ei kR ·XR(z)

VmVφ

Vφ



The form of the amplitude

The amplitude Aφiφi m reads Back

Aφiφi m =

[∂

∂m(G − B) · R0

]MN

EMa EN

b Aab





Aφiφi m =

[∂

∂m(G − B) · R0

]MN

EMa EN

b Aab

I Impose the boundary identification V MR (z; kR) = RM

0 N V NL (z; kR)





Aφiφi m =

[∂

∂m(G − B) · R0

]MN

EMa EN

b Aab

I Switch to the open string complex basis Za = EaMX M





Aφiφi m =

[∂

∂m(G − B) · R0

]MN

EMa EN

b Aab

The matrix Aab is the CFT correlator

Aab =e−πiα′s/2

8πα′2〈Vφi V a

L V bL Vφi 〉





Aφiφi m =

[∂

∂m(G − B) · R0

]MN

EMa EN

b Aab




L V bL Vφi 〉

I Overall normalization





Aφiφi m =

[∂

∂m(G − B) · R0

]MN

EMa EN

b Aab




L V bL Vφi 〉

k1 k2

kL, kRI Cocycle to put off-shell in a controlled way

the closed string vertex

s = (k1 + k2)2 = (kL + kR)2

= 2(k1 · k2 −M2) = 2kL · kR





Aφiφi m =

[∂

∂m(G − B) · R0

]MN

EMa EN

b Aab




L V bL Vφi 〉

I Vertices in the open string complex basis Za



The CFT correlatorIt is easy to see that the correlator Aab has the matrix form

A ≡(

0 Aj δij

Aj δij 0

), with Aj =

e−πiα′s/2

8πα′2〈Vφi V j

L VjL Vφi 〉

Now we must:I insert the explicit form of the vertices Vφi (x1) and Vφi (x2)I integrate their positions x1,2 over the real axis and the position z of

the closed vertex V jL(z) over the upper half plane, up to SL(2,R)

We get

Aj =i εj(i)

4πα′eiπθj sin

[π(θj + α′s/2

)] Γ(α′s + 1)Γ(1− θj − α′s/2)

Γ(1− θj + α′s/2)

=i εj(i)

4πα′eiπθj sin(πθj)(1−

12α′ s ρj) + O

(α′s2

)I We have defined ρj = ψ(1− θj) + ψ(θj) + 2γE




A ≡(

0 Aj δij

Aj δij 0

), with Aj =

e−πiα′s/2


L VjL Vφi 〉



We get

Aj =i εj(i)

4πα′eiπθj sin

[π(θj + α′s/2

)] Γ(α′s + 1)Γ(1− θj − α′s/2)

Γ(1− θj + α′s/2)

=i εj(i)


12α′ s ρj) + O

(α′s2





A ≡(

0 Aj δij

Aj δij 0

), with Aj =

e−πiα′s/2


L VjL Vφi 〉



We get

Aj =i εj(i)

4πα′eiπθj sin

[π(θj + α′s/2

)] Γ(α′s + 1)Γ(1− θj − α′s/2)

Γ(1− θj + α′s/2)

=i εj(i)


12α′ s ρj) + O

(α′s2





A ≡(

0 Aj δij

Aj δij 0

), with Aj =

e−πiα′s/2


L VjL Vφi 〉



We get

Aj =i εj(i)

4πα′eiπθj sin

[π(θj + α′s/2

)] Γ(α′s + 1)Γ(1− θj − α′s/2)

Γ(1− θj + α′s/2)

=i εj(i)


12α′ s ρj) + O

(α′s2




The result for the amplitude

Altogether, one can write (up to 2-derivative terms, i.e. up to s2)the correlator Aab in matrix form as

A =12G−1 (R−1 − 1)H , H = i

(hj 00 −hj

)with

hj =εj(i)

4πα′(1− 1

2α′ s ρj) =

12π

K−1φiφi

∂

∂θj Kφiφi (k1 · k2 −M2)

and

Kφiφi = e2γE α′M2

√Γ(1− θi)

Γ(θi)

∏k 6=i

√Γ(θk )

Γ(1− θk )





A =12G−1 (R−1 − 1)H , H = i

(hj 00 −hj

)with

hj =εj(i)

4πα′(1− 1

2α′ s ρj) =

12π

K−1φiφi

∂


and


√Γ(1− θi)

Γ(θi)

∏k 6=i

√Γ(θk )

Γ(1− θk )

We used the kinematics s = 2(k1 · k2 −M2), the dependenceof M2 on θj and the fact that ψ(x) = d ln Γ(x)

dx .





A =12G−1 (R−1 − 1)H , H = i

(hj 00 −hj

)with

hj =εj(i)

4πα′(1− 1

2α′ s ρj) =

12π

K−1φiφi

∂


and


√Γ(1− θi)

Γ(θi)

∏k 6=i

√Γ(θk )

Γ(1− θk )

The exponential term goes to 1 in the field theory limit



The magic of the result

Substituting into the expression of the correlator Aφiφi mRecall we

get after some algebra

Aφiφi m =12E G−1 ∂

∂m(G − B) (Rπ − R0) E−1∣∣j

j hj − h.c.

Comparing with the expression of the dependence of the twists θifrom the moduli m Recall we can write

Aφiφi m = 2π∂θj

∂mhj = K−1

φiφi

∂θj

∂m∂


This is the expression we expected Recall if Kφiφi really is theKähler metric



The field theory Kähler metric

Summarizing, in the field theory limit the expression of the Kählermetric Kφiφi for the scalar φi depends on the moduli only throughthe open string twists

θ(0)i = lim

α′→0θi

in an N = 1 configuration. Explicitly,

Kφiφi =

√√√√Γ(1− θ(0)i )

Γ(θ(0)i )

∏k 6=i

√√√√ Γ(θ(0)k )

Γ(1− θ(0)k )

This holds for a general toroidal compactification, and witharbitrary magnetic fluxes, also non-commuting

Generalizes [Lust et al., 2004]


Relation to the Yukawa couplings




Stringy expression of the Yukawa couplings

In the stringy description, Yukawa couplings havethe form Yijk = Aijk Wijk , where

Wijk = classical contribution from extendedworld-sheets bordered by the intersectingbranes.

I Multiple intersections → familiesI different minimal world-sheets →

exponential hierarchy of couplings

(have counterparts in magnetized braneworlds [Cremades et al., 2004])Aijk = quantum fluctuations given by thecorrelator of the twisted vertices located atthe intersections. Back

χk(1)

χk(2)

φi

χj(1)

χj(2)

Vχj

Vφi

Vχk



Yukawa couplings and N=1 superpotential

In N = 1 susy, the Yukawa couplings arise from the superpotential∫d2θW (Φi) + c.c → . . .+

∂W∂φi∂φj χ

iχj + h.c. .

For W = WijkΦiΦjΦk , the Wijk are the Yukawa couplings in thebasis where the kinetic terms are determined by the Kählerpotential K Recall

When realized in string compactifications, non-renormalizationproperty: W gets no perturbative α′ correctionsIn the brane-world context, we identify therefore the Wijk as theclassical world-sheet instanton contributions





∂W∂φi∂φj χ

iχj + h.c. .







∂W∂φi∂φj χ

iχj + h.c. .





Kähler metric and quantum Yukawas

The N = 1 holomorphic couplings Wijk are related to the physicalones, Yijk (the ones provided by the string computation) byrescaling the fields φi , χj , χk to give them canonical kinetic terms.

Recall

One has thus

Yijk = (Kφiφi Kφjφj Kφk φk )−1/2 Wijk

We had already found Recall

Yijk = Aijk Wijk

Hence, the amplitude Aijk for the three twisted vertices should befactorizable into

Aijk = (Kφiφi Kφjφj Kφk φk )−1/2



The abelian case

Vχj

Vφi

Vχk

In the case of a factorized torus withcommuting angles (or for D-branes atangles) the direct computation of the stringamplitude Aijk is possibleIt involves in particular the correlator of threebosonic twist fields on the torus which aresimultaneously expressible in terms of twistangles {θi}, {νi}, {λi}

This correlator is computable by factorization of the 4-twistamplitude, and its dependence on the three sets of anglesfactorizesIn the end, one indeed finds


in agreement with the non-renormalization theorem[Cvetic-Papadimitriou, 2003, Lust et al., 2004]



The abelian case

σλ1σλ2

σλ3

σθ1σθ2

σθ3

σν1 σν2 σν3







The abelian case

σλ1σλ2

σλ3

σθ1σθ2

σθ3

σν1 σν2 σν3







The abelian case

σλ1σλ2

σλ3

σθ1σθ2

σθ3

σν1 σν2 σν3







The non-abelian case?

We have considered the general case in which the reflectionmatrices at the various boundaries do not commute, and shownthat the Kähler metric remains the sameHence the monodromy matrices Rθ,ν,λ induced by the the threetwist operators cannot, in general, be simultaneously diagonalizedWe have thus to deal with (“non-abelian twist fields”), whose3-point CFT correlators are not known. Their computationrepresents a challenge.The non-renormalization theorem, however, suggests that thecorrelator still factorizes and depends on the three sets {θi}, {νi},{λi} of monodromy eigenvalues















Some references



A few ref.s on magnetized and intersecting branes

C. Bachas, arXiv:hep-th/9503030.

M. Berkooz, M. R. Douglas and R. G. Leigh, Nucl. Phys. B 480(1996) 265 [arXiv:hep-th/9606139].

R. Rabadan, Nucl. Phys. B 620 (2002) 152 [arXiv:hep-th/0107036].

A. M. Uranga, Class. Quant. Grav. 20 (2003) S373[arXiv:hep-th/0301032].



A few ref.s on Yukawas in brane-worlds

D. Cremades, L. E. Ibanez and F. Marchesano, JHEP 0307 (2003)038 [arXiv:hep-th/0302105].

M. Cvetic and I. Papadimitriou, Phys. Rev. D 68 (2003) 046001[Erratum-ibid. D 70 (2004) 029903] [arXiv:hep-th/0303083].

S. A. Abel and A. W. Owen, Nucl. Phys. B 663 (2003) 197[arXiv:hep-th/0303124].

D. Cremades, L. E. Ibanez and F. Marchesano, JHEP 0405 (2004)079 [arXiv:hep-th/0404229].



A few other ref.s (mixed amplitudes, oblique fluxes, ...)

D. Lust, P. Mayr, R. Richter and S. Stieberger, Nucl. Phys. B 696(2004) 205 [arXiv:hep-th/0404134].

I. Antoniadis and T. Maillard, Nucl. Phys. B 716 (2005) 3[arXiv:hep-th/0412008].

M. Bianchi and E. Trevigne, arXiv:hep-th/0506080; M. Bianchi andE. Trevigne, JHEP 0508 (2005) 034 [arXiv:hep-th/0502147].


Documents

Brane world effective actions from mixed amplitudes with general …personalpages.to.infn.it/~billo/seminars/billo_prin_pisa... · 2005. 11. 8. · In particular, intersecting brane