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Towards exact field theory results for theStandard Model on fractional D6-branes

Gabriele Honecker (JG|U Mainz)

based on arXiv:1109.XXXX (MZ-TH/11-26) by G.H.to appear next week

XVII European Workshop on String Theory 2011, Padua7 September 2011

Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes

Motivation

our 4D world

U(1)

QL

R R

U(1)

u ,d eR N,

SU(3)

SU(2)L

R

I Explicit globally consistent models on fractional D6-braneson toroidal orbifolds T 6/Z2N G.H., Ott ‘04; Gmeiner, G.H. ‘07-09

rigid D6-branes expected on T 6/Z2 × Z2M Forste, G.H. et al. in progress; . . .

I Field theory @ 1-loop: gauge & Yukawa couplings, Kahlermetrics . . . known for untilted six-torus only (& partial results

for untilted T 6/Z2 × Z2 without/with torsion) Cremades, Ibanez, Marchesano ‘03;

Cvetic, Papadimitriou ‘03; Abel, Schofield/Goodsell ‘04/05; Lust, (Mayr, Richter), Stieberger ‘03(‘04);

Akerblom, Blumenhagen, Lust, Schmidt-Sommerfeld ‘07; . . .

I need generalisation to study low-energy regime of SM vacuaGabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes

Motivation

I need exact results for some vanishing angle in Z2 presence& tilted tori starting point: gauge thresholds in Gmeiner, G.H. ‘09

I (anti)symmetric matter contributes differently toholomorphic gauge kinetic function on tilted tori

I one vanishing angle: only N = 1 SUSY if Z2 acts

I T 6/Z2N : the SM on T 6/Z′6 and T 6/Z6

I T 6/Z2 × Z2M with discrete torsion: expected models on

T 6/Z2 × Z6 and T 6/Z2 × Z′6I different number of complex structure moduli (e.g. less bulk

from orbifolded six-torus, several exceptional ones) changeKahler metrics

I compact expressions, which are valid for any known torusorbifold, admit direct comparisons

I need gauge kinetic function for physical U(1)s likeU(1)Y = 1

2 (U(1)B−L + U(1)c) with U(1)B−L = 13 U(1)a + U(1)d

. . . and SO(2M) or Sp(2M) groups


Gauge coupling at 1-loop in string theoryStarting point:

I N = 1 SUSY result of 1-loop stringcalculation:

8π2

g 2a (µ)

=8π2

g 2a,string

+ba

2ln

(M2

string

µ2

)+

∆a

2

I tree-level 1g2

a,string∼ Vol(D6)a

I ba: from massless strings ∼ 1ε in

∫dl lε

I with gauge threshold ∆a from massivestrings

D6

SU(N)

SU(N)

D6

a

a

a

D6a

SU(N)

SU(N)

a

a

O6

b

via magnetic background field method developped by Abouelsaood, Callan, Nappi, Yost ‘87 &

Bachas, Porrati ‘92 . . . ; applied to D6-branes on untilted tori for T 6: Lust, Stieberger ‘03; Akerblom et al. ‘07; D6

on untilted tori for T 6/Z2 × Z2 with torsion: partial results by Blumenhagen, Schmidt-Sommerfeld ‘07;

T 6/Z2N and tilted tori: Gmeiner, G.H. ‘09; applied to magnetised D9/D5-systems e.g. by Billo, Frau, Pesando,

Di Vecchia, Lerda, Marotta‘07 Angelantonj, Condeescu, Dudas, Lennek ‘09 . . .


Gauge couplings in string & field theory

I N = 1 SUSY result of 1-loop string calculation:

8π2

g 2a (µ)

=8π2

g 2a,string

+ba

2ln

(M2

string

µ2

)+

∆a

2

needs to be matched with . . .

I N = 1 SUSY field theory:

8π2

g 2a (µ)

=8π2 <(fa) +ba

2ln

(M2

Planck

µ2

)+

ba + 2 C2(Ga)

2K

+ C2(Ga) ln[g−2a (µ2)]−

∑a

C2(Ra) ln det KRa(µ2)

I derive holomorphic gauge kinetic function fa, Kahlermetrics KRa with representations Ra ∈ (Na,Nb), (Na,Nb),Antia, Syma, Kahlerpotential K by matching string & field theory expressions


Tree-Level Gauge Couplings & Kahler potential

I tree-level:

1g2

a= e−φ4

2π

√∏3i=1 V

(i)aa

SUSY=

8<:S X0

a −P3

i=1 Ui Xia T 6

S X0a − U X1

a T 6/Z′6S X0

a T 6/Z6

9=;= <(ftreea )

I with definition of dilaton & complex structure moduli

(S ,Ui )tree ≡ e−φ4

2π ×

8>>><>>>:“

1/qQ3

i=1 ri ,q

rj rk/ri

”T 6 hbulk

21 = 3„clattice√

r, 4

33/2clattice

√r

«T 6/Z′6 1

(clattice , ∅) T 6/Z6 0

I 1-loop massless strings:ba

2 ln(

Mstringµ

)2

↔ ba

2

[ln(

MPlanckµ

)2

+K]

provided by(MstringMPlanck

)2

= e2φ4 ≡ e2φ10

VolCY3and

K = − ln(S)α −∑

k ln(Uk)α −∑3

i=1 ln Ti + . . . with α =

8<:1 T 6

2 T 6/Z′64 T 6/Z6

I 1-loop massive strings provideI Kahler metrics & correction to hol. gauge kinetic functionsI field redefinitions of moduli


1-Loop Results from Massive Strings on e.g. T 6/Z′6

e.g. Z2 subgroup: N~v = ( 12 , 0,−

12 ) , distinguish relative angles

(~φab):

I (~0), N = 2 SUSY : parallel D6-branes contribute lattice sum:

holomorphic part δbf1−loopa ; non-hol. Kahler metrics KRa

I (φ, 0,−φ), N = 2 : hol. δbf1−loopa ; non-hol. KRa

I (0, φ,−φ), N = 1 : lattice sum on T 2(1) ( δbf

1−loopa ,KRa)

& field redefinitions

I (~φab), N = 1 : D6-branes at angles non-hol. ln(Γ(φab))terms & field redefinitions

I crucial for interpretation: global prefactor ba in gaugethreshold ∆ab identical KRa for each multiplet(Ra ∈ Antia, Syma, (Na,Nb), (Na,Nb))


Results for gauge thresholds I: BifundamentalsI lattice sums:

I Λ0,0(v ; V ) ≡ ln(2πvV ) + [2 ln η(iv) + c .c .]

with two-torus volume v and (length)2 of 1-cycle V

I Λ(σ, τ, v) ≡ −π2 vσ2 +[lnϑ1( τ−ivσ

2 ,iv)

η(iv) + c .c .]

with relative displacements & Wilson lines (σ, τ)I define ba ≡ ba δσi ,0

δτ i ,0

for D6-branes parallel on two-torus T 2(i)

bSU(Na) and gauge thresholds for bifundamentals and adjoints on T 6/Z2N with Z2 ≡ Z(2)2

SU

SY

(φ(1)ab , φ

(2)ab , φ

(3)ab )

bZ2

SU(Na) =∑

b bAab + . . .

=∑

bNb2 ϕab + . . .

∆Z2N

SU(Na) = Nb ∆ab + . . .

2 (0, 0, 0) − IZ2,(1·3)ab Nb

2 δσ2ab,0δτ2

ab,0

−bAab Λ0,0(v2; V(2)ab )

−bAab

(1− δσ2

ab,0δτ2

ab,0

)Λ(σ2

ab, τ2ab, v2)

2 (φ, 0,−φ) Nb2 δσ2

ab,0δτ2

ab,0

(|I (1·3)

ab | − IZ2,(1·3)ab

) −bAab Λ0,0(v2; V(2)ab )

−bAab

(1− δσ2

ab,0δτ2

ab,0

)Λ(σ2

ab, τ2ab, v2)

1 (0, φ,−φ) Nb2 δσ1

ab,0δτ1

ab,0|I (2·3)

ab |−bAab Λ0,0(v1; V

(1)ab )

−bAab

(1− δσ1

ab,0δτ1

ab,0

)Λ(σ1

ab, τ1ab, v1)

+NbI

Z2ab

2 (sgn(φ)− 2φ) ln(2)

1(φ(1), φ(2), φ(3))

with∑3k=1 φ

(k) = 0

Nb4

(|Iab|+ sgn(Iab) I Z2

ab

) bAab sgn(Iab)∑3

i=1 ln

(Γ(|φ(i)

ab |)Γ(1−|φ(i)

ab |)

)sgn(φ(i)ab )

+Nb2 I Z2

ab

(sgn(Iab) + sgn(φ

(2)ab )− 2φ

(2)ab

)ln(2)

Table:Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes

From gauge thresholds to Kahler metrics I: Bifundamentals

~φab = (~0), (0i , (φ)j , (−φ)k) : has N = 2 or 1 SUSY on T 6/Z2N

I 1-loop correction to gauge kinetic function for arbitrary (σ, τ)

δbf1−loopa (vi ) = −

bAab4π2×

ln η(iv) (σiab, τ

iab) = (0, 0)

12 ln

ϑ1(τ iab−ivi σ

iab

2,iv)

η(ivi )6= (0, 0)

I Kahler metrics for (~0), f (S,U) =`SQ

l Ul´−α

4 , r.h.s. also (0i , φ,−φ),

K(2)Adja

=

√2πf (S ,U)

2 v2

√√√√V(1)aa V

(3)aa

V(2)aa

and K(i)

(Na,Nb)= f (S ,U)

√2πV

(i)aa

vjvk

I Comparison with other orbifolds:I T 6 and T 6/Z2 × Z2M without torsion: N = 4 for ~φab = (~0):

no δbf1−loopa , K

(i)Adja

derived from scattering amplitudes√

I T 6/Z2 × Z2 with discrete torsion: N = 1 for ~φab = (~0):present result differs from Blumenhagen, Schmidt-Sommerfeld ‘07


Bifundamentals cont’d

other (φ(1)ab , φ

(2)ab , φ

(3)ab ) is always N = 1 SUSY

I Kahler metrics:

K(Na,Nb) = f (S ,U)

√√√√∏3i=1

1vi

(Γ(|φ(i)

ab |)Γ(1−|φ(i)

ab |)

)− sgn(φ(i)ab

)

sgn(Iab)

I formula for bifundamentals on various orbifolds differs only in

f (S ,U) =(

S∏hbulk

21l=1 Ul

)−α4

due to hbulk21 = 3, 1, 0 and α = 1, 2, 4

I adjoints at some angles have identical formula

I same for (anti)symmetrics of SU(N)

I adjoints on identical branes differ (see previous slide)

I same (anti)symmetrics of SO(2M) and Sp(2M)(up to normalisation on identical branes)

I hierarchies of couplings can be due to unisotropic tori &particle generations at different intersections


Intermezzo I: Kahler metrics

I K(i)Adja

=

√2πf (S ,U)

ca vi

√√√√V(j)aa V

(k)aa

V(i)aa

I with ca = 1, 2, 4 for bulk, fractional or rigid D6-branesI identical formulas on all orbifolds:

Kahler metrics for bifundamental matter (Na,Nb) on various orbifolds

(φ(1)ab , φ

(2)ab , φ

(3)ab )

T 6 andT 6/Z2 × Z2M

without torsionT 6/Z2N

T 6/Z2 × Z2M

with discrete torsion

(0, 0, 0) − f (S ,Ul)

√2πV

(2)ab

v1v3

f (S ,Ul)

√2πV

(i)ab

vjvk

(ijk) ' (1, 2, 3) cyclic

(0(i), φ(j), φ(k))

φ(j) = −φ(k) f (S ,Ul)

√2πV

(i)ab

vjvk

(φ(1), φ(2), φ(3))∑3i=1 φ

(i) = 0f (S ,Ul)

√√√√∏3i=1

1vi

(Γ(|φ(i)

ab |)Γ(1−|φ(i)

ab |)

)− sgn(φ(i)ab

)

sgn(Iab)


Intermezzo II: holomorphic gauge kinetic functions

I depend on number of angles and number of Z2 symmetriesI details of orbifold & model encoded in beta function

coefficients bAabI additional angle dependent term in presence of Z2 or Z2×Z2

<(δb f1-loop

SU(Na)(φ(i)ab ))

=Nb

16π2 ca

3∑i=1

IZ(i)

2

ab

(2φ

(i)ab − sgn(φ

(i)ab )− sgn(Iab)

)might possibly be absorbed in 1-loop field redefinitions (???)

1-loop contribution to the gauge kinetic function δb f1-loopSU(Na)(vi ) from bifundamentals on various orbifolds

(φ(1)ab , φ

(2)ab , φ

(3)ab )

T 6 andT 6/Z2 × Z2M


T 6/Z2 × Z2M


(0, 0, 0) −

− bAab4π2 ln η(iv2)

− bAab8π2 (1− δσ2

ab,0δτ2

ab,0)×

× ln

(e−π(σ2

ab)2v2/4 ϑ1(τ2ab−iσ2

abv22

,iv2)

η(iv2)

)−∑3

i=1bA,(i)ab4π2 ln η(ivi )

−∑3

i=1bA,(i)ab8π2 (1− δσi

ab,0δτ i

ab,0)×

× ln

(e−π(σi

ab)2vi/4 ϑ1(τ iab−iσi

abvi2

,ivi )

η(ivi )

)

(0(i), φ,−φ) − bAab4π2 ln η(ivi )−

bAab8π2 (1− δσi

ab,0δτ i

ab,0) ln

(e−π(σi

ab)2vi/4 ϑ1(τ iab−iσi

abvi2

,ivi )

η(ivi )

)(φ(1), φ(2), φ(3)) −


Intermezzo III: Field redefinitions

I 1-loop field redefinitions (here: Volume(T 2(i))≡ vi ≡ Ti )

cf. Akerblom, Blumenhagen, Lust, Schmidt-Sommerfeld ‘07

I dilaton and bulk compl. structures (i = 3, 1, ∅ for T 6, T 6/Z′6, T 6/Z6)

SL = S− 1

4π2ca

Xb

NbY0b

3Xj=1

φ(j)b ln Tj UL

i = Ui +1

4π2ca

Xb

NbYib

3Xj=1

φ(j)b ln Tj

with Πb =Ph21

i=0

`X i

b Πeveni + Y i

bΠoddi

´I only verified for bulk D6-branes or at three anglesI problem: chiral matter exists also for (0, φ,−φ) in presence of

Z2 symmetriesI exceptional complex structures cf. Angelantonj, Condeescu, Dudas, Lennek ‘09

W Lα = Wα + 1

4π2ca

∑b NbY α

b

∑3j=1 φ

(j)b ln Tj

I incorporated in Kahler metrics by f (S ,U)→ f (SL,UL, φ(i)) and

V (i)(Ui )→ V (i)(ULi ) - trivial for hbulk

21 = 1, e.g. T 6/Z′6I might absorb angle-dependent term in hol. gauge kinetic function (?)


Examples for Bifundamentals & Adjoints

U(1)

QL

R R

U(1)

u ,d eR N,

SU(3)

SU(2)L

R

On the ABa lattice on T 6/Z′6 (Gmeiner, G.H.

‘07-09)

I aa sector ~φ = (~0) : one adjoint 8 + 1

of U(3) KAdja =√

π r2

f (S,Ul )v2

I ac sector π(0, 12 ,−

12 ) : 2 uR

K(3,1) = f (S ,U)√

4π√3 v2v3

I a(θ2c) sector π(−13 ,−

16 ,

12 ) : 1 uR

K(3,1) = f (S ,U)√

10v1v2v3

I ab′ sector π( 16 ,−

12 ,

13 ) : 2 QL K(3,2) = f (S ,U)

√10

v1v2v3

I a(θb′) sector π(−12 ,

16 ,

13 ) : 2 QL same

I a(θ2b′) sector π(−16 ,−

16 ,

13 ) : 1 QL f (S ,U)

√25

2 v1v2v3

I . . . all possible configurations of angles appear!I r : ratio of radii on a - vi volumes of two-toriGabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes

Kahler metrics and Yukawa hierarchies

I size of Yukawa interactions does not only depend onworldsheet polygons

I Kahler metrics from unisotropic tori and some vanishingintersection angle contribute to hierarchy

Angles, beta function coefficients and Kahler metrics for branes a and d

Brane a

y (~φay ) bAay K(3a,Ny ) (~φa(θy)) bAa(θy) K(3a,Ny ) (~φa(θ2y)) bAa(θ2y) K(3a,Ny )

a (0, 0, 0) −6KAdja =√π r2

f (S ,Ul )v2

±( 13 ,−

13 , 0) 3 KAdja = f (S ,Ul)

√2π rv1v2

a′ (−13 ,

13 , 0)

bAaa′ + bMaa′= 3

2 − 1

KAntia =

f (S ,Ul)√

2π rv1v2

(0, 0, 0)bAa(θa′) + bMa(θa′)

= 6− 4 = 2

KAntia =

f (S ,Ul)√

4π√3 v1v3

( 13 ,−

13 , 0)

bAa(θ2a′) + bMa(θ2a′)

= 32 − 1

KAntia =

f (S ,Ul)√

2π rv1v2

b ( 12 ,−

16 ,−

13 ) 0 − (−1

6 ,12 ,−

13 ) 0 − ( 1

6 ,16 ,−

13 ) 0 −

b′ ( 16 ,−

12 ,

13 ) 2 f (S ,U)

√10

v1v2v3(−1

2 ,16 ,

13 ) 2 f (S ,U)

√10

v1v2v3(−1

6 ,−16 ,

13 ) 1 f (S ,U)

√25

2 v1v2v3

c (0, 12 ,−

12 ) 2 f (S ,U)

√4π√

3 v2v3( 1

3 ,16 ,−

12 ) 0 − (−1

3 ,−16 ,

12 ) 1 f (S ,U)

√10

v1v2v3

d ( 12 ,−

12 , 0) 2 f (S ,U)

√2π rv1v2

(−16 ,

16 , 0) 1

2 f (S ,U)√

2π rv1v2

( 16 ,−

16 , 0) 1

2 f (S ,U)√

2π rv1v2

d ′ ( 16 ,−

16 , 0) 1

2 f (S ,U)√

2π rv1v2

( 12 ,−

12 , 0) 2 f (S ,U)

√2π rv1v2

(−16 ,

16 , 0) 1

2 f (S ,U)√

2π rv1v2

h3 (0, 0, 0) 0 − ( 13 ,−

13 , 0) 3 · 03 − (−1

3 ,13 , 0) 3 · 03 −

Brane d

x (~φxd) bAdx K(Nx ,1d ) (~φx(θd)) bA(θd)x K(Nx ,1d ) (~φx(θ2d)) bA(θ2d)x K(Nx ,1d )

b (0,−13 ,

13 ) 3 · 01 − ( 1

3 ,−23 ,

13 ) 6 f (S ,U)

√8

v1v2v3(−1

3 , 0,13 ) 4 f (S ,U)

√4π√

3v1v3

c ( 12 , 0,−

12 ) 2 f (S ,U)

√4π√

3v1v3

(−16 ,−

13 ,

12 ) 3 f (S ,U)

√10

v1v2v3( 1

6 ,13 ,−

12 ) 0 −

h3 ( 12 ,−

12 , 0) 24 · 03 − (−1

6 ,16 , 0) 6 · 03 − ( 1

6 ,−16 , 0) 6 · 03 −

x (~φxd ′) bAd ′x K(Nx ,1d ) (~φx(θd ′)) bA(θd ′)x K(Nx ,1d ) (~φx(θ2d ′)) bA(θ2d ′)x K(Nx ,1d )

b (−13 , 0,

13 ) 2 f (S ,U)

√4π√

3v1v3

(0,−13 ,

13 ) 3 · 01 − ( 1

3 ,−23 ,

13 ) 3 f (S ,U)

√8

v1v2v3

d ′ (−13 ,

13 , 0)

2bAdd ′ + bMdd ′

= 9− 9 = 0− (0, 0, 0)

2bAd(θd ′) + bMd(θd ′)

= 4− 4 = 0− ( 1

3 ,−13 , 0)

2bAd(θ2d ′) + bMd(θ2d ′)

= 9− 9 = 0−


Gauge thresholds → Kahler metrics II: Anti/Sym of SU(N)

~φaa′ = (~0), (0i , (φ)j , (−φ)k) : has N = 2 or 1 SUSY on T 6/Z2N

I gauge kinetic function on tilted tori: v ≡ v1−b (b ∈ 0, 1

2)

δa′ f1−loopa (vi ) =

−1

4π2

bAaa′ ln η(ivi ) + bMaa′ ln η(i vi ) (σ, τ)iaa′ = (0, 0)

bAaa′2 ln

ϑ1( τ−ivσ2 ,iv)

η(iv) +bM

aa′2 ln

ϑ1(τ iaa′−i viσ

1aa′

2 ,i vi )

η(i vi )6= (0, 0)

with e.g. ln η(i v) = ln η(iv)+4b ln η(2iv)ϑ4(0,2iv)

extra contribution from Mobius strip besides (bAaa′ + bMaa′) ln η(iv)

I Kahler metrics: V ≡ 2(1−b)V ln(2v V ) = ln(vV ) + 2 ln 2gives same shape as for bifundamentals:

K(i)Antia/Syma

= f (S ,U)

√2πV

(i)aa

vjvk


Anti/Sym cont’d

other ~φaa′ is always N = 1

I ln(Γ(|φ|) terms identical to bifundamental case sameKahler metrics for (anti)symmetrics:

KAnti/Syma= f (S ,U)

√√√√∏3i=1

1vi

(Γ(|φ(i)

aa′ |)

Γ(1−|φ(i)

aa′ |)

)− sgn(φ(i)

aa′)

sgn(Iaa′ )

I same shape for Z2 and angle dependent partI one-loop redefinition of dilaton and complex structure moduliI or extra terms in hol. gauge kinetic function

I new constant term in gauge threshold from O6-planes

<(δa′ f

1-loopSU(Na)

)⊃ 1

8π2 ca

∑3m=0 ηΩRZ(m)

2

|I ΩRZ(m)2

z | ln 2

I constant shift in dilaton and complex structuresI or constant term in ho. gauge kinetic functionI never noticed before


1-loop gauge kinetic functions from (anti)symmetrics

I need to distinguish position relative to O6-planesI annulus terms as for bifundamentalsI new terms from Mobius strip

Comparison of one-loop contribution to the gauge kinetic function δa′ f1-loop(vi )SU(Na) from (anti)symmetric sectors on various orbifolds

(φ(1)aa′ , φ

(2)aa′ , φ

(3)aa′) T 6 T 6/Z2 × Z2M


T 6/Z2 × Z2M


(0, 0, 0)↑↑ ΩR −

−∑3

i=1

bM,(i)

aa′4π2 ln η(i vi )

−∑3

i=1

[bM,(i)

aa′8π2 (1− δσi

aa′ ,0δτ i

aa′ ,0)×

× ln

(e−π(σi

aa′ )2vi/4 ϑ1(

τ iaa′−iσi

aa′ vi2

,i vi )

η(i vi )

)]−bA

aa′4π2 ln η(iv2)− bM

aa′4π2 ln η(i v2)

−(1− δσ2aa′ ,0

δτ2aa′ ,0

)×

×

[bA

aa′8π2 ln

(e−π(σ2

aa′ )2v2/4 ϑ1(

τ2aa′−iσ2

aa′ v22

,iv2)

η(iv2)

)

+bM

aa′8π2 ln

(e−π(σ2

aa′ )2v2/4 ϑ1(

τ2aa′−iσ2

aa′ v22

,i v2)

η(i v2)

)]−∑3

i=1

bA,(i)aa′4π2 ln η(ivi )

−∑3

i=1

bM,(i)

aa′4π2 ln η(i vi )

(0, 0, 0)

↑↑ ΩRZ(i)2

−bMaa′

4π2 ln η(i vi )

− bMaa′

8π2 (1− δσiaa′ ,0

δτ iaa′ ,0

)×

× ln

(e−π(σi

aa′ )2vi/4 ϑ1(

τ iaa′−iσi

aa′ vi2

,i vi )

η(i vi )

) same as ↑↑ ΩR

i = 2 : same as ↑↑ ΩRi = 1, 3 : − bA

aa′4π2 ln η(iv2)

−bM,(1)

aa′4π2 ln η(i v1)− b

M,(3)

aa′4π2 ln η(i v3)

−(1− δσ2aa′ ,0

δτ2aa′ ,0

)×

× bAaa′

8π2 ln

(e−π(σ2

aa′ )2v2/4 ϑ1(

τ2aa′−iσ2

aa′ v22

,iv2)

η(iv2)

) same as ↑↑ ΩR

(0(i), φ(j), φ(k))

↑↑(

ΩR+ ΩRZ(i)2

)−bA

aa′4π2 ln η(ivi )−

bMaa′

4π2 ln η(i vi )

− bAaa′

8π2 (1− δσiaa′ ,0

δτ iaa′ ,0

) ln

(e−π(σi

aa′ )2vi/4 ϑ1(

τ iaa′−iσi

aa′ vi2

,ivi )

η(ivi )

)

− bMaa′

8π2 (1− δσiaa′ ,0

δτ iaa′ ,0

) ln

(e−π(σi

aa′ )2vi/4 ϑ1(

τ iaa′−iσi

aa′ vi2

,i vi )

η(i vi )

) i = 2 : same as (0, 0, 0) ↑↑ ΩRi = 1, 3 : − bA

aa′4π2 ln η(ivi )−

bMaa′

4π2 ln η(i vi )

−bAaa′

4π2 ln η(ivi )

−bMaa′

4π2 ln η(i vi )

(0(i), φ(j), φ(k))

↑↑(

ΩRZ(j)2 + ΩRZ(k)

2

) −bAaa′

4π2 ln η(ivi )

− bAaa′

8π2 (1− δσiaa′ ,0

δτ iaa′ ,0

)×

× ln

(e−π(σi

aa′ )2vi/4 ϑ1(

τ iaa′−iσi

aa′ vi2

,ivi )

η(ivi )

) same as ↑↑(

ΩR+ ΩRZ(i)2

)i = 2 : − bA

aa′4π2 ln η(iv2)

−(1− δσ2aa′ ,0

δτ2aa′ ,0

)×

× bAaa′

8π2 ln

(e−π(σ2

aa′ )2v2/4 ϑ1(

τ2aa′−iσ2

aa′ v22

,iv2)

η(iv2)

)

i = 1, 3 : same as ↑↑(

ΩR+ ΩRZ(i)2

)same as

↑↑(

ΩR+ ΩRZ(i)2

)

(φ(1), φ(2), φ(3)) −


Examples for (anti)symmetrics

U(1)

QL

R R

U(1)

u ,d eR N,

SU(3)

SU(2)L

R

The T 6/Z′6 example (Gmeiner, G.H. ‘07-09)

I ~φcc ′ = (~0), i.e. ↑↑ ΩR for brane (θc)

I π(−16 , 0,

16 ) for brane (θ2b)

I π(0, 12 ,−

12 ) for brane (θa)

I π( 12 , 0,−

12 ), i.e. ↑↑ ΩRZ(2)

2 for brane (θd)

I . . . all possible angles occur

I Kahler metrics for aa′ and dd ′ sectors given in table on a previous slideI example for gauge kinetic function @1-loop

δtotalf1-loopSU(3)a

(vi ) =2X

k=0

Xy=a,a′,b,b′,c,d,d′,h3

δ(θk y)f1-loopSU(3)a

(vi )

=1

2π2

“ln η(i v1)− ln η(iv1)

”− 2

π2ln η(iv3)− 3

4π2ln

e−πv3/4 ϑ1( 1−iv3

2, iv3)

η(iv3)

!I U(1)B−L has many more different terms


Special cases of ΩR invariant D6-branes: Sp(2N) groupsI ΩR-invariance depends on exceptional 3-cycles

I hidden sector Sp(6) ↑↑ a or Sp(2) ↑↑ dI Sp(2)c → U(1)c broken by continuous (σ2

c , τ2c )!

1-loop gauge kinetic functions and Kahler metrics for SO(2Mx) and Sp(2Mx)

(φ(1)xx ′ , φ

(2)xx ′ , φ

(3)xx ′) S

US

Yδx f1-loop

SO/Sp(2Mx ) K(i)Antix

,K(i)Symx

T 6 and T 6/Z3

(0, 0, 0)↑↑ ΩR N

=4

− K(i),i=1,2,3Antix

= f (S ,Ul)(2π)1/2

21/3 vi

√V

(j)xx V

(k)xx

V(i)xx

(0, 0, 0)⊥ ΩR onT 2

(j) × T 2(k)

N=

2

1π2 ln

(22/3 η(i vi )

) K(i)Symx

= f (S ,Ul)√

2π21/3 vi

√V

(j)x V

(k)x

V(i)x

K(j)Antix

= f (S ,Ul)√

2π21/3 vj

√V

(i)x V

(k)x

V(j)x

K(k)Antix

= f (S ,Ul)√

2π21/3 vk

√V

(i)x V

(j)x

V(k)x

T 6/Z2N

(0, 0, 0)

↑↑ ΩR or ΩRZ(2)2 N

=2

12π2

(Mx ln

(21/6 η(iv2)

)+ ln

(22/3 η(i v2)

))K

(2)Symx

= f (S ,Ul)√

2π24/3 v2

√V

(1)x V

(3)x

V(2)x

(0, 0, 0)

⊥ ΩR and ΩRZ(2)2

on T 2(2)

N=

1

12π2

(Mx ln

(21/6 η(iv2)

)+∑

j=1,3 ln(211/12 η(i vj)

))K

(2)Antix

= f (S ,Ul)√

2π24/3 v2

√V

(1)xx V

(3)xx

V(2)xx

T 6/Z2 × Z2M without discrete torsion

(0, 0, 0)

N=

1

12π2

(Mx ln 2−1/2 +

∑3i=1 ln (2 η(i vi ))

)K

(i),i=1,2,3Antix

= f (S ,Ul)√

2π24/3 vi

√V

(j)xx V

(k)xx

V(i)xx

T 6/Z2 × Z2M with discrete torsion

(0, 0, 0)

↑↑ ΩRZ(k)2

k = 0 . . . 3 N=

1

14π2

(Mx∑3

i=1 ln(√

2η(ivi ))

+∑3

i=1 ln (2η(i vi )))

−


Physical linear combinations of U(1)s

I 1-loop contribution to a single U(1)a ⊂ U(Na) factor

δtotal f1-loopU(1)a

≡ δa′ f1-loopU(1)a

+∑

b 6=a,a′

δb f1-loopU(1)a

= 2Na

2 δa′ f1-loop,ASU(Na) + δa′ f

1-loop,MSU(Na) +

∑b 6=a,a′

δb f1-loopSU(Na)

remember also ftree

U(1)a= 2Na ftree

SU(Na)

I massless linear combination U(1)X =∑

i xiU(1)i

δtotal f1-loopU(1)X

=∑

i

x2i δtotalf

1-loopU(1)i

+4∑i<j

xixjNi

(−δj f1-loop

SU(Ni )+ δj′ f

1-loopSU(Ni )

)remember ftree

U(1)X=∑

i x2i ftree

U(1)i

I cross-check of Kahler metrics√

I hyper charge and B-L are of this type

I relevant for kinetic mixing between observable and hidden U(1)s


Conclusions & Outlook

I completed list of gauge kinetic functions & Kahler metrics@ 1-string loop G.H. to appear in the next days (MZ-TH/11-26)

I for arbitrary (untilted & tilted) background latticesI with arbitrary (discrete & continuous) Wilson lines &

displacement moduliI for all possible (non)-vanishing SUSY anglesI for physical U(1)sI including all const. (Kahler metrics/gauge kinetic fctn)

OutlookI Selection rules & Yijk ∼ e−Areaijk dependence of Yukawa

couplings for T 6/Z′6 model G.H., Vanhoof to appear soon ‘11

I complete worldsheet instanton sum for holomorphic partof Yukawas with Z2 symmetry still missing

I study (numerical) parameter dependence of couplings inview of phenomenology at Mweak

I new models on T 6/Z2 × Z(′)6 Forste, G.H. et al. in progress

I more on kinetic mixing of U(1)sGabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes

towards exact field theory results for the standard model ... · towards exact eld theory results...

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