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
Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes
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 . . .
Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes
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
Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes
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
Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes
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))
Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes
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
Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes
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
Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes
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)
Table:Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes
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
without torsionT 6/Z2N
T 6/Z2 × Z2M
with discrete torsion
(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)) −
Table:Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes
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 (?)
Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes
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−
Table:Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes
Results for gauge thresholds II: (Anti)Symmetrics
bSU(Na) and gauge thresholds for symmetrics and antisymmetrics: T 6/Z2N with Z2 ≡ Z(2)2
SU
SY
(φ(1)aa′ , φ
(2)aa′ , φ
(3)aa′)
bZ2N
SU(Na) = bAaa′ + bMaa′ + . . .
= Na2
(ϕSyma + ϕAntia
)+(ϕSyma − ϕAntia
)+ . . .
∆Z2N
SU(Na) = Na ∆aa′ + ∆a,O6 + . . .
2(0, 0, 0)↑↑ ΩR
−δσ2aa′ ,0
δτ2aa′ ,0×(NaI
Z2,(1·3)
aa′2 + |I ΩRZ2,(1·3)
a |) −
(bAaa′ + bMaa′
)Λ0,0(v2; V
(2)aa′ )− (4 b2) bMaa′ ln
(η(2iv2)ϑ4(0,2iv2)
)− 2 bMaa′ ln(2)
−(
1− δσ2aa′ ,0
δτ2aa′ ,0
) [bAaa′ Λ(σ2
aa′ , τ2aa′ , v2) + bMaa′ Λ(σ2
aa′ , τ2aa′ , v2)
]2
(0, 0, 0)
↑↑ ΩRZ(2)2
−δσ2aa′ ,0
δτ2aa′ ,0×(Na I
Z2,(1·3)
aa′2 + |I ΩR,(1·3)
a |) −
(bAaa′ + bMaa′
)Λ0,0(v2; V
(2)aa′ )− (4 b2) bMaa′ ln
(η(2iv2)ϑ4(0,2iv2)
)− 2 bMaa′ ln(2)
−(
1− δσ2aa′ ,0
δτ2aa′ ,0
) [bAaa′ Λ(σ2
aa′ , τ2aa′ , v2) + bMaa′ Λ(σ2
aa′ , τ2aa′ , v2)
]1
(0, 0, 0)⊥ ΩR on T1 × T2
−(Na I
Z2,(1·3)
aa′ δσ2
aa′,0δτ2aa′,0
2
+|I ΩR,(1·2)a |+ |I ΩRZ2,(2·3)
a |) −bAaa′ Λ0,0(v2; V
(2)aa′ ) + |I ΩR,(1·2)
a |Λ0,0(v3; 2V(3)aa′ ) + |I ΩRZ2,(2·3)
a |Λ0,0(v1; 2V(1)aa′ )
−bAaa′(
1− δσ2aa′ ,0
δτ2aa′ ,0
)Λ(σ2
aa′ , τ2aa′ , v2)
1(0, 0, 0)
⊥ ΩR on T2 × T3
−Na I
Z2,(1·3)
aa′ δσ2
aa′,0δτ2aa′,0
2
+IΩR,(2·3)a + I
ΩRZ2,(1·2)a
−bAaa′ Λ0,0(v2; V(2)aa′ ) + |I ΩR,(2·3)
a |Λ0,0(v1; 2 V(1)aa′ ) + |I ΩRZ2,(1·2)
a |Λ0,0(v3; 2 V(3)aa′ )
−bAaa′(
1− δσ2aa′ ,0
δτ2aa′ ,0
)Λ(σ2
aa′ , τ2aa′ , v2)
2
(φ, 0,−φ)φ 6=± 12
↑↑(
ΩR+ ΩRZ(2)2
)on T2
δσ2aa′ ,0
δτ2aa′ ,0
[Na
“|I (1·3)
aa′ |−IZ2,(1·3)
aa′
”2
−|I ΩR,(1·3)a | − |I ΩRZ2,(1·3)
a |] −
(bAaa′ + bMaa′
)Λ0,0(v2; V
(2)aa′ )− (4 b2) bMaa′ ln
(η(2iv2)ϑ4(0,2iv2)
)− 2 bMaa′ ln(2)
−(
1− δσ2aa′ ,0
δτ2aa′ ,0
)×[bAaa′ Λ(σ2
aa′ , τ2aa′ , v2) + bMaa′ Λ(σ2
aa′ , τ2aa′ , v2)
]
1
(φ, 0,−φ)φ 6=± 12
⊥(
ΩR+ ΩRZ(2)2
)on T2
Na
“|I (1·3)
aa′ |−IZ2,(1·3)
aa′
”2 δσ2
aa′ ,0δτ2
aa′ ,0−bAaa′ Λ0,0(v2; V
(2)aa′ )− bAaa′
(1− δσ2
aa′ ,0δτ2
aa′ ,0
)Λ(σ2
aa′ , τ2aa′ , v2)
+(|I ΩR
a |+ |I ΩRZ2a |
)ln(2)
1(0, φ,−φ)↑↑ ΩR on T1
Na2 |I
(2·3)aa′ | − |I
ΩR,(2·3)a |
−(bAaa′ + bMaa′
)Λ0,0(v1; V
(1)aa′ )− (4 b1) bMaa′ ln
(η(2iv1)ϑ4(0,2iv1)
)+(Na I
Z2aa′
2 (sgn(φ)− 2φ) + |I ΩRZ2a |+ 2 |I ΩR,(2·3)
a |)
ln(2)
1(0, φ,−φ)⊥ ΩR on T1
Na2 |I
(2·3)aa′ | − |I
ΩRZ2,(2·3)a |
−(bAaa′ + bMaa′
)Λ0,0(v1; V
(1)aa′ )− (4 b1) bMaa′ ln
(η(2iv1)ϑ4(0,2iv1)
)+(Na I
Z2aa′
2 (sgn(φ)− 2φ) + |I ΩRa |+ 2 |I ΩRZ2,(2·3)
a |)
ln(2)
1
(φ(1), φ(2), φ(3))
0 < |φ(i ,j)| ≤ |φ(k)| < 1
sgn(φ(i)) = sgn(φ(j))
6= sgn(φ(k))
Na
“|Iaa′ |+sgn(Iaa′ )I
Z2aa′
”4
+ cΩRa |IΩR
a |+ cΩRZ2a |IΩRZ2
a |2
(bAaa′ + bMaa′
)sgn(Iaa′)
∑3i=1 ln
(Γ(|φ(i)
aa′ |)
Γ(1−|φ(i)
aa′ |)
)sgn(φ(i)
aa′ )
+(Na I
Z2aa′
2
(sgn(Iaa′) + sgn(φ
(2)aa′)− 2φ
(2)aa′
)+ |I ΩR
a |+ |I ΩRZ2a |
)ln(2)
Table:Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes
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
Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes
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
Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes
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
without torsionT 6/Z2N
T 6/Z2 × Z2M
with discrete torsion
(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)) −
Table:Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes
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
Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes
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 )))
−
Table:Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes
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
Gabriele Honecker (JG|U Mainz) Towards exact field theory results for the Standard Model on fractional D6-branes
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