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Sven KrippendorfDAMTP, University of Cambridge
based on arXiv:1002.1790, with: M. Dolan, A. Maharana, F. Quevedo
Cambridge, 22nd April 2010
Phenomenology of D-branesat Toric Singularities
Disclaimer: This talk is somewhat independent of the LHC.
... string theorists could have talked about this 20 years ago
for more than 25 years...
for more than 25 years...
Heterotic String Theory
for more than 25 years...
Heterotic String Theory Type II brane models
for more than 25 years...
Heterotic String Theory
e.g. intersecting branes in IIA,magnetised branes in IIB
Type II brane models
F-theory
for more than 25 years...
Heterotic String Theory
e.g. intersecting branes in IIA,magnetised branes in IIB
Type II brane models
0502005, 0702094
F-theory
Heterotic String Theory
e.g. intersecting branes in IIA,magnetised branes in IIB
Type II brane models
0502005, 0702094
Heterotic String Theory
e.g. intersecting branes in IIA,magnetised branes in IIB
Type II brane models
0502005, 0702094
moduli stabilisation
Heterotic String Theory
e.g. intersecting branes in IIA,magnetised branes in IIB
Type II brane models
0502005, 0702094
moduli stabilisation
under control in Type IIB(KKLT, LVS)
Perspective: Local Brane Models (within IIB)
... bottom-up model building
e.g.: aldazabal, ibanez, quevedo, uranga (10 years ago),verlinde, wijnholt (5 years ago)
Modular String Model Building (in LARGE volume)
Standard (like) Modelswith fractional (D3/D7) branes
at singularities
and what can we get?
e.g.: aldazabal, ibanez, quevedo, uranga (10 years ago),verlinde, wijnholt (5 years ago)
E
Brane models: how good are they to date?
chiral matter, adjoint and (bi-)fundamental matter
U(n), SO(n), Sp(n) gauge groups (exceptional gauge groups in F-theory)
models with the correct matter content
BUT structure of yukawa couplings, e.g.: hierarchy of masses, ckm matrix
e.g. madrid model, 0105155
... this applies also to F-theory models
ContentWhy branes at singularities, what gauge theories do we get? ...background and history
Studying gauge theories with dimers
What general properties do gauge theories of the infinite class of toric singularities obey?
How good are the models regarding flavour physics, i.e. can we get the CKM-matrix?
ContentWhy branes at singularities, what gauge theories do we get? ...background and history
Studying gauge theories with dimers
What general properties do gauge theories of the infinite class of toric singularities obey?
How good are the models regarding flavour physics, i.e. can we get the CKM-matrix?
Upper bound of#families (≤3)
ContentWhy branes at singularities, what gauge theories do we get? ...background and history
Studying gauge theories with dimers
What general properties do gauge theories of the infinite class of toric singularities obey?
How good are the models regarding flavour physics, i.e. can we get the CKM-matrix?
Mass structure(0, m, M)
Upper bound of#families (≤3)
ContentWhy branes at singularities, what gauge theories do we get? ...background and history
Studying gauge theories with dimers
What general properties do gauge theories of the infinite class of toric singularities obey?
How good are the models regarding flavour physics, i.e. can we get the CKM-matrix?
Mass structure(0, m, M)
Upper bound of#families (≤3)
yes, we can...there is enough structure aroundto build conrete models with the right ckm.
Motivation for branes at singularities
Local models -> a lot of information without addressing moduli stabilisation
Effective field theory (although distances below string scale) well under control
Gauge coupling unification (in principle)
Powerful (dimer) techniques for toric singularities
Gauge theories highly restricted (unlike intersecting branes in IIA)
E
Motivation for branes at singularities
Local models -> a lot of information without addressing moduli stabilisation
Effective field theory (although distances below string scale) well under control
Gauge coupling unification (in principle)
Powerful (dimer) techniques for toric singularities
Gauge theories highly restricted (unlike intersecting branes in IIA)
E
Hanany et al.
Classic Example:
ni D3-branes: U(n1)xU(n2)xU(n3)
mi D7-branes: U(m1)xU(m2)xU(m3)
Arrows: bi-fundamental matter
Anomaly cancellation
Hypercharge:
. C3/Z3
3[(n1, n̄2, 1), (1, n2, n̄3), (n̄1, 1, n3)]
D3 matter content:m2 = 3(n3 ! n1) + m1,
m3 = 3(n3 ! n2) + m1
Qanomaly!free =!
i
Qi
ni
aiqu: 0005067
let’s build models!
Standard-Model
let’s build models!Left-Right
Standard-Model
let’s build models!
Pati-Salam
Left-Right
Standard-Model
let’s build models!
Pati-Salam Trinification
Left-Right
Standard-Model
e.g. left-right model
3 families SU(3)xSU(2)LxSU(2)RxU(1)B-L
B-L normalisation 32/3:
+ 3 Higgses -> unification at 1012 GeV
Proton stability from global U(1) B-L
embedded in compact CY
sin2 ! =314
= 0.214
e.g. left-right model
3 families SU(3)xSU(2)LxSU(2)RxU(1)B-L
B-L normalisation 32/3:
+ 3 Higgses -> unification at 1012 GeV
Proton stability from global U(1) B-L
embedded in compact CY
sin2 ! =314
= 0.214
details in Aldazabal, Ibanez, Quevedo, Uranga hep-th/0005067
What’s bad about the model?
Problem: Yukawa couplings
W = !ijk QiLHj
uukR =
!
"Q1
LQ2
LQ3
L
#
$
!
"0 Z12 !Y12
!Z12 0 X12
Y12 !X12 0
#
$
!
"u1
Ru2
Ru3
R
#
$ .
Masses: (0, M, M)M2 = |X12|2 + |Y12|2 + |Z12|2
Resolution: Embed into singularities with more structure (in this case del Pezzo 1)
Problem: Yukawa couplings
W = !ijk QiLHj
uukR =
!
"Q1
LQ2
LQ3
L
#
$
!
"0 Z12 !Y12
!Z12 0 X12
Y12 !X12 0
#
$
!
"u1
Ru2
Ru3
R
#
$ .
Masses: (0, M, M)M2 = |X12|2 + |Y12|2 + |Z12|2
Resolution: Embed into singularities with more structure (in this case del Pezzo 1)
cmq: 0810.5660.C3/Z3 = dP0
What types of singularities are there?
Orbifold Singularities
del Pezzo singularities (P2 blown-up), Conifold -> TORIC SINGULARITIES
non-toric singularities
What types of singularities are there?
Orbifold Singularities
del Pezzo singularities (P2 blown-up), Conifold -> TORIC SINGULARITIES
non-toric singularities
few suitable for model building
What types of singularities are there?
Orbifold Singularities
del Pezzo singularities (P2 blown-up), Conifold -> TORIC SINGULARITIES
non-toric singularities
infinite class, techniques
few suitable for model building
What types of singularities are there?
Orbifold Singularities
del Pezzo singularities (P2 blown-up), Conifold -> TORIC SINGULARITIES
non-toric singularities
infinite class, techniques
limited techniques
few suitable for model building
Gauge theories probing toric singularities
Toric CY-Cone:represented as T3 fibrationover rational polyhedral cone
D3 branes (at the tip of the cone)
D7 branes (wrapping 4-cyclespassing through singularity)
Gauge theories probing toric singularities
Toric CY-Cone:represented as T3 fibrationover rational polyhedral cone
D3 branes (at the tip of the cone)
D7 branes (wrapping 4-cyclespassing through singularity)
D3 branes
Gauge theories probing toric singularities
Toric CY-Cone:represented as T3 fibrationover rational polyhedral cone
D3 branes (at the tip of the cone)
D7 branes (wrapping 4-cyclespassing through singularity)
toric diagram
Gauge theories probing toric singularities
Toric CY-Cone:represented as T3 fibrationover rational polyhedral cone
D3 branes (at the tip of the cone)
D7 branes (wrapping 4-cyclespassing through singularity)
Gauge theories of toric singularitiesare always quiver gauge theories!
Gauge theories probing toric singularities
Toric CY-Cone:represented as T3 fibrationover rational polyhedral cone
D3 branes (at the tip of the cone)
D7 branes (wrapping 4-cyclespassing through singularity)
Gauge theories of toric singularitiesare always quiver gauge theories!
uR
QL
Hu
Hd dR
1
32
m
6!m
3!m
Gauge theories probing toric singularities
Toric CY-Cone:represented as T3 fibrationover rational polyhedral cone
D3 branes (at the tip of the cone)
D7 branes (wrapping 4-cyclespassing through singularity)
Gauge theories of toric singularitiesare always quiver gauge theories!
uR
QL
Hu
Hd dR
1
32
m
6!m
3!mbi-fundamental matter
Gauge theories probing toric singularities
Toric CY-Cone:represented as T3 fibrationover rational polyhedral cone
D3 branes (at the tip of the cone)
D7 branes (wrapping 4-cyclespassing through singularity)
Gauge theories of toric singularitiesare always quiver gauge theories!
uR
QL
Hu
Hd dR
1
32
m
6!m
3!mbi-fundamental matter
change rank byaddition of fractional branes
Gauge theories probing toric singularities
Toric CY-Cone:represented as T3 fibrationover rational polyhedral cone
D3 branes (at the tip of the cone)
D7 branes (wrapping 4-cyclespassing through singularity)
Gauge theories of toric singularitiesare always quiver gauge theories!
uR
QL
Hu
Hd dR
1
32
m
6!m
3!m
WARNINGonly matter content,
dimer for superpotential
Question:
How to determine the gauge theory associated to a given singularity (inverse problem)? -> most efficient solution via dimers
Aside: Dimer techniques useful in understanding gauge theories of M2 branes at singularities (no inverse algorithm)
0503149, 0504110, 0505211, 0511063, 0511287, 0604136, 0706.1660, 0803.4474Hanany, Kennaway, Franco, Vegh, Wecht, Martelli, Sparks, Feng, He, Vafa, Yamazaki
The Dimer LanguageDimers visualise the gauge theory of toric singularities.
Geometry: Toric Diagram Gauge Theory: Dimerinverse slopes in toric diagram
winding numbers of zigzag paths
The Dimer LanguageDimers visualise the gauge theory of toric singularities.
Geometry: Toric Diagram Gauge Theory: Dimerinverse slopes in toric diagram
winding numbers of zigzag paths
The Dimer LanguageDimers visualise the gauge theory of toric singularities.
Geometry: Toric Diagram Gauge Theory: Dimerinverse slopes in toric diagram
winding numbers of zigzag paths
(1,1)
The Dimer LanguageDimers visualise the gauge theory of toric singularities.
Geometry: Toric Diagram Gauge Theory: Dimerinverse slopes in toric diagram
winding numbers of zigzag paths
(1,-1)
The Dimer LanguageDimers visualise the gauge theory of toric singularities.
Geometry: Toric Diagram Gauge Theory: Dimerinverse slopes in toric diagram
winding numbers of zigzag paths
(0,-1)
The Dimer LanguageDimers visualise the gauge theory of toric singularities.
Geometry: Toric Diagram Gauge Theory: Dimerinverse slopes in toric diagram
winding numbers of zigzag paths
(-2, 1)
The Dimer LanguageDimers visualise the gauge theory of toric singularities.
Geometry: Toric Diagram Gauge Theory: Dimerinverse slopes in toric diagram
winding numbers of zigzag paths
Dimer Language IIReading off the gauge theory
Faces = gauge groups
Intersection of zigzag paths = bi-fundamental matter
Vertices (faces orbited by zigzag paths) = superpotential terms
Dimer Language IIReading off the gauge theory
Faces = gauge groups
Intersection of zigzag paths = bi-fundamental matter
Vertices (faces orbited by zigzag paths) = superpotential terms
Dimer Language IIReading off the gauge theory
Faces = gauge groups
Intersection of zigzag paths = bi-fundamental matter
Vertices (faces orbited by zigzag paths) = superpotential terms
W = X13X32X21 !X14X43X32X21
merge zigzag paths according to cutting of toric diagram
caveat: additional crossings, concrete prescription to be avoided by precise operations
Dimer Language III: How do I get a dimer?
embed toric singularity in orbifold of conifold whose dimer is known (chess-board).
collaps cycles in singularity(= cutting toric diagram)
Gulotta 0807.3012
merge zigzag paths according to cutting of toric diagram
caveat: additional crossings, concrete prescription to be avoided by precise operations
Dimer Language III: How do I get a dimer?
embed toric singularity in orbifold of conifold whose dimer is known (chess-board).
collaps cycles in singularity(= cutting toric diagram)
Gulotta 0807.3012
merge zigzag paths according to cutting of toric diagram
caveat: additional crossings, concrete prescription to be avoided by precise operations
Dimer Language III: How do I get a dimer?
embed toric singularity in orbifold of conifold whose dimer is known (chess-board).
collaps cycles in singularity(= cutting toric diagram)
Gulotta 0807.3012
1 2 3 1
4 5 6
7 8 9 7
1 2 3 1
10 11 12
! ! !
" " " "
! ! !
" " " "
! ! !
merge zigzag paths according to cutting of toric diagram
caveat: additional crossings, concrete prescription to be avoided by precise operations
Dimer Language III: How do I get a dimer?
embed toric singularity in orbifold of conifold whose dimer is known (chess-board).
collaps cycles in singularity(= cutting toric diagram)
Gulotta 0807.3012
1 2 3 1
4 5 6
7 8 9 7
1 2 3 1
10 11 12
! ! !
" " " "
! ! !
" " " "
! ! !
merge zigzag paths according to cutting of toric diagram
caveat: additional crossings, concrete prescription to be avoided by precise operations
Dimer Language III: How do I get a dimer?
embed toric singularity in orbifold of conifold whose dimer is known (chess-board).
collaps cycles in singularity(= cutting toric diagram)
Gulotta 0807.3012
1 2 3 1
4 5 6
7 8 9 7
1 2 3 1
10 11 12
! ! !
" " " "
! ! !
" " " "
! ! !
merge zigzag paths according to cutting of toric diagram
caveat: additional crossings, concrete prescription to be avoided by precise operations
Dimer Language III: How do I get a dimer?
embed toric singularity in orbifold of conifold whose dimer is known (chess-board).
collaps cycles in singularity(= cutting toric diagram)
Gulotta 0807.3012
1 2 3 1
4 5 6
7 8 9 7
1 2 3 1
10 11 12
! ! !
" " " "
! ! !
" " " "
! ! !
merge zigzag paths according to cutting of toric diagram
caveat: additional crossings, concrete prescription to be avoided by precise operations
Dimer Language III: How do I get a dimer?
embed toric singularity in orbifold of conifold whose dimer is known (chess-board).
collaps cycles in singularity(= cutting toric diagram)
Gulotta 0807.3012
1 2 3 1
4 5 6
7 8 9 7
1 2 3 1
10 11 12
! ! !
" " " "
! ! !
" " " "
! ! !
Operations on the dimer
(1,0) + (0,1) -> (1,1)Operation 1:
Operations on the dimer
(1,0) + (0,1) -> (1,1)Operation 1:
Operations on the dimer
(1,0) + (0,1) -> (1,1)Operation 1:
Operations on the dimer
(1,0) + (0,1) -> (1,1)
(1,0) + (0,1) & (-1,0) + (0,-1)
I(1,1),(!1,!1) = 0Iab = namb !manb
Operation 1:
Operation 2:
-> (1,1) + (-1,-1)
Operations on the dimer
(1,0) + (0,1) -> (1,1)
(1,0) + (0,1) & (-1,0) + (0,-1)
I(1,1),(!1,!1) = 0Iab = namb !manb
Operation 1:
Operation 2:
-> (1,1) + (-1,-1)
Operations on the dimer
(1,0) + (0,1) -> (1,1)
(1,0) + (0,1) & (-1,0) + (0,-1)
I(1,1),(!1,!1) = 0Iab = namb !manb
Operation 1:
Operation 2:
-> (1,1) + (-1,-1)
e.g. del Pezzo 31 2 1
3 4
5 6 5
1 2 1
2 6
! !
" " "
! !
" "
! !
WdP3 = !X12Y31Z23 !X45Y64Z56 + X45Y31Z14!53 + X12Y25Z56!61
+X36Y64Z23"42 !X36Y25Z14!53!61"42
=
!
"X45
Y23
Z25
#
$
!
"0 Z14!53 !Y64
!Z14!53!61"42 0 X12!61
Y64"42 !X12 0
#
$
!
"X36
Y31
Z56
#
$ .
1 2 1
3 4
5 6 5
1 2 1
7 8
! !
" " "
! !
" " "
! !
e.g. del Pezzo 31 2 1
3 4
5 6 5
1 2 1
2 6
! !
" " "
! !
" "
! !
WdP3 = !X12Y31Z23 !X45Y64Z56 + X45Y31Z14!53 + X12Y25Z56!61
+X36Y64Z23"42 !X36Y25Z14!53!61"42
=
!
"X45
Y23
Z25
#
$
!
"0 Z14!53 !Y64
!Z14!53!61"42 0 X12!61
Y64"42 !X12 0
#
$
!
"X36
Y31
Z56
#
$ .
1 2 1
3 4
5 6 5
1 2 1
7 8
! !
" " "
! !
" " "
! !
1 2 1
3 4
5 6 5
1 2 1
2 6
! !
" " "
! !
" "
! !
1 2 1
3 2
3 6 3
1 2 1
2 6
! !
" " "
! !
" "
! !
Application of Gulotta’s algorithmto toric del-Pezzo surfaces
1 2 1
3 2
3 1 3
1 2 1
2 1
! !
" " "
! !
" "
! !
1 2 1
3 4
3 6 3
1 2 1
2 6
! !
" " "
! !
" "
! !
Gulotta’s dimers = Traditional dimers
Philosophy: Use this algorithmic view of gauge theories to find general features for gauge theories probing toric singularities!
Are there properties revealed that are not apparent from looking at the superpotential
and are they useful for model building?
Restricting the # of familiesThe dimers obtained with the operations of Gulotta are highly restricted (otherwise: inconsistent dimers)
example: additional crossing (-> mass term)
without add. crossings 3 (left),with add. crossings 4 but unique (right)
A B
Restricting the # of familiesThe dimers obtained with the operations of Gulotta are highly restricted (otherwise: inconsistent dimers)
example: additional crossing (-> mass term)
without add. crossings 3 (left),with add. crossings 4 but unique (right)
A B
e.g.1 2
3 4
4 6
7 8
! !
" "
! !
" "
Restricting the # of familiesThe dimers obtained with the operations of Gulotta are highly restricted (otherwise: inconsistent dimers)
example: additional crossing (-> mass term)
without add. crossings 3 (left),with add. crossings 4 but unique (right)
A B
e.g.1 2
3 4
4 6
7 8
! !
" "
! !
" "
Restricting the # of familiesThe dimers obtained with the operations of Gulotta are highly restricted (otherwise: inconsistent dimers)
example: additional crossing (-> mass term)
without add. crossings 3 (left),with add. crossings 4 but unique (right)
A B
e.g.1 2
3 4
4 6
7 8
! !
" "
! !
" "
Restricting the # of familiesThe dimers obtained with the operations of Gulotta are highly restricted (otherwise: inconsistent dimers)
example: additional crossing (-> mass term)
without add. crossings 3 (left),with add. crossings 4 but unique (right)
A B
e.g.1 2
3 4
4 6
7 8
! !
" "
! !
" "
Restricting the # of familiesThe dimers obtained with the operations of Gulotta are highly restricted (otherwise: inconsistent dimers)
example: additional crossing (-> mass term)
without add. crossings 3 (left),with add. crossings 4 but unique (right)
A B
e.g.1 2
3 4
4 6
7 8
! !
" "
! !
" "
maximum of3 fields in/out forany gauge group
-> 3 families
Restricting the # of familiesThe dimers obtained with the operations of Gulotta are highly restricted (otherwise: inconsistent dimers)
example: additional crossing (-> mass term)
without add. crossings 3 (left),with add. crossings 4 but unique (right)
A B
e.g.1 2
3 4
4 6
7 8
! !
" "
! !
" "
maximum of 4 fields(no add. branches)
between 2 gauge groups
-> 4 families
Restricting the # of familiesThe dimers obtained with the operations of Gulotta are highly restricted (otherwise: inconsistent dimers)
example: additional crossing (-> mass term)
without add. crossings 3 (left),with add. crossings 4 but unique (right)
A B
e.g.1 2
3 4
4 6
7 8
! !
" "
! !
" "
maximum of 4 fields(no add. branches)
between 2 gauge groups
-> 4 families
Restricting the # of familiesThe dimers obtained with the operations of Gulotta are highly restricted (otherwise: inconsistent dimers)
example: additional crossing (-> mass term)
without add. crossings 3 (left),with add. crossings 4 but unique (right)
A B
e.g.1 2
3 4
4 6
7 8
! !
" "
! !
" "
maximum of 4 fields(no add. branches)
between 2 gauge groups
-> 4 families(unique F0)
Upper bound on the number of families given by the physically
observed number with one exception (4 families in F0).
Upper bound on the number of families given by the physically
observed number with one exception (4 families in F0).
Can you break this bound?
Upper bound on the number of families given by the physically
observed number with one exception (4 families in F0).
Can you break this bound?
requires to go to non-toric phases/singularities. there are examples with more families.
Mass Hierarchies
Known result: dP0 (0, M, M); dP1 (0, m, M)
the zero eigenvalue is present in all toric dPs
appearing due to vanishing determinant of Yukawa matrix
Is the structure (0, m, M) generic in models at toric singularities?
CMQ: 0810.5660
m = |Y12|2 +|!16|2
!2(|X62|2 + |Z62|2)
M = |Y12|2 + |X62|2 + |Z62|2
Mass Hierarchies
Known result: dP0 (0, M, M); dP1 (0, m, M)
the zero eigenvalue is present in all toric dPs
appearing due to vanishing determinant of Yukawa matrix
Is the structure (0, m, M) generic in models at toric singularities?
uR
QL
Hu
Hd dR
1
32
m
6!m
3!mCMQ: 0810.5660
m = |Y12|2 +|!16|2
!2(|X62|2 + |Z62|2)
M = |Y12|2 + |X62|2 + |Z62|2
Mass hierarchies II
How do we choose “quarks”?one left & right handed quark in every coupling with quarks.-> every superpotential term has two quarks-> quarks aligned in closed lines
Connected or disconnected lines?connected to be able to higgs to common gauge group
Maximal or non-maximal number of quarks?after Higgsing the same result of vanishing determinant
1 2 1
3 4
5 6 5
1 2 1
7 8
! !
" " "
! !
" " "
! !
Mass hierarchies II
How do we choose “quarks”?one left & right handed quark in every coupling with quarks.-> every superpotential term has two quarks-> quarks aligned in closed lines
Connected or disconnected lines?connected to be able to higgs to common gauge group
Maximal or non-maximal number of quarks?after Higgsing the same result of vanishing determinant
1 2 1
3 4
5 6 5
1 2 1
7 8
! !
" " "
! !
" " "
! !
1 2 1
3 4
5 6 5
1 2 1
7 8
! !
" " "
! !
" " "
! !
Mass hierarchies II
How do we choose “quarks”?one left & right handed quark in every coupling with quarks.-> every superpotential term has two quarks-> quarks aligned in closed lines
Connected or disconnected lines?connected to be able to higgs to common gauge group
Maximal or non-maximal number of quarks?after Higgsing the same result of vanishing determinant
1 2 1
3 4
5 6 5
1 2 1
7 8
! !
" " "
! !
" " "
! !
1 2 1
3 4
5 6 5
1 2 1
7 8
! !
" " "
! !
" " "
! !
general structure (0, m, M)
general structure (0, m, M)
non-vanishing mass?
general structure (0, m, M)
non-vanishing mass?
corrections to Kähler potential, deformation of singularity (non-toric)
... back to model building
Flavour mixing: CKMAim: construct models with the correct flavour mixing among quarks
2 types of models: a) up & down from D3D3 states
... we have a non-trivial superpotential (Yukawa
structure) in these singularity models.
What does this imply for model building?
VCKM =
!
"1 ! !3
! 1 !2
!3 !2 1
#
$ .
b) up from D3D3 states & down from D3D7 states
2R2 2R
32L
E
Flavour mixing: CKMAim: construct models with the correct flavour mixing among quarks
2 types of models: a) up & down from D3D3 states
... we have a non-trivial superpotential (Yukawa
structure) in these singularity models.
What does this imply for model building?
VCKM =
!
"1 ! !3
! 1 !2
!3 !2 1
#
$ .
b) up from D3D3 states & down from D3D7 states
uR
QL
Hu
Hd dR
1
32
m
6!m
3!m2R2 2R
32L
E
A left-right modelwith the right CKM
2R2 2R
32L
W =
!
"XL
23
Y L23
ZL23
#
$
!
"0 Z12 !Y62
!Z12!61" 0 X12
!61"
Y62 !X12 0
#
$
!
"X36
Y31
Z36
#
$ .
E
A left-right modelwith the right CKM
!61
Y31R
B
X36R ,Z36
R
A
X23L , Y23
L , Z23L
X12, Z12
Y62
B"
A"
2R 2R2
32L
6
2
1
4
1 2
W =
!
"XL
23
Y L23
ZL23
#
$
!
"0 Z12 !Y62
!Z12!61" 0 X12
!61"
Y62 !X12 0
#
$
!
"X36
Y31
Z36
#
$ .
E
A left-right modelwith the right CKM
!61
Y31R
B
X36R ,Z36
R
A
X23L , Y23
L , Z23L
X12, Z12
Y62
B"
A"
2R 2R2
32L
6
2
1
4
1 2
W =
!
"XL
23
Y L23
ZL23
#
$
!
"0 Z12 !Y62
!Z12!61" 0 X12
!61"
Y62 !X12 0
#
$
!
"X36
Y31
Z36
#
$ .
After breaking of U(2)R
W =
!
"XL
23
Y L23
ZL23
#
$
!
"0 Zu
12 !Y u62
!Zu12
!! 0 Xu
12!!
Y u62 !Xu
12 0
#
$
!
"Xu
36
Y u31
Zu36
#
$ +
!
"Xd
23
Y d23
Zd23
#
$
!
"0 Zd
12 !Y d62
!Zd12
vd! 0 0
Y d62 0 0
#
$
!
"Xd
36
Y d31
Zd36
#
$
E
A left-right modelwith the right CKM
The CKM is given in terms of ratios of Higgs vevs.
!61
Y31R
B
X36R ,Z36
R
A
X23L , Y23
L , Z23L
X12, Z12
Y62
B"
A"
2R 2R2
32L
6
2
1
4
1 2
W =
!
"XL
23
Y L23
ZL23
#
$
!
"0 Z12 !Y62
!Z12!61" 0 X12
!61"
Y62 !X12 0
#
$
!
"X36
Y31
Z36
#
$ .
After breaking of U(2)R
W =
!
"XL
23
Y L23
ZL23
#
$
!
"0 Zu
12 !Y u62
!Zu12
!! 0 Xu
12!!
Y u62 !Xu
12 0
#
$
!
"Xu
36
Y u31
Zu36
#
$ +
!
"Xd
23
Y d23
Zd23
#
$
!
"0 Zd
12 !Y d62
!Zd12
vd! 0 0
Y d62 0 0
#
$
!
"Xd
36
Y d31
Zd36
#
$
dP1:
E
A left-right modelwith the right CKM
The CKM is given in terms of ratios of Higgs vevs.
!61
Y31R
B
X36R ,Z36
R
A
X23L , Y23
L , Z23L
X12, Z12
Y62
B"
A"
2R 2R2
32L
6
2
1
4
1 2
W =
!
"XL
23
Y L23
ZL23
#
$
!
"0 Z12 !Y62
!Z12!61" 0 X12
!61"
Y62 !X12 0
#
$
!
"X36
Y31
Z36
#
$ .
After breaking of U(2)R
W =
!
"XL
23
Y L23
ZL23
#
$
!
"0 Zu
12 !Y u62
!Zu12
!! 0 Xu
12!!
Y u62 !Xu
12 0
#
$
!
"Xu
36
Y u31
Zu36
#
$ +
!
"Xd
23
Y d23
Zd23
#
$
!
"0 Zd
12 !Y d62
!Zd12
vd! 0 0
Y d62 0 0
#
$
!
"Xd
36
Y d31
Zd36
#
$
VCKM = V †u Vd =
!
""#
Xu12
Zu12
!Y u12
Zu12"61
1!Y u
12Xu12
(Zu12)
2"61
1 ! !Y u12
Zu12"61
1 0 !Xu12
Zu12
$
%%& .
!
""#
0 0 1a !Y d
12Zd
12"61a 0
a !a !Y d12
Zd12"61
0
$
%%& .
normalisation
=
!
"1 ! !3
! 1 !2
!3 !4 1
#
$ .
dP1:
E
A left-right modelwith the right CKM
The CKM is given in terms of ratios of Higgs vevs.
!61
Y31R
B
X36R ,Z36
R
A
X23L , Y23
L , Z23L
X12, Z12
Y62
B"
A"
2R 2R2
32L
6
2
1
4
1 2
W =
!
"XL
23
Y L23
ZL23
#
$
!
"0 Z12 !Y62
!Z12!61" 0 X12
!61"
Y62 !X12 0
#
$
!
"X36
Y31
Z36
#
$ .
After breaking of U(2)R
W =
!
"XL
23
Y L23
ZL23
#
$
!
"0 Zu
12 !Y u62
!Zu12
!! 0 Xu
12!!
Y u62 !Xu
12 0
#
$
!
"Xu
36
Y u31
Zu36
#
$ +
!
"Xd
23
Y d23
Zd23
#
$
!
"0 Zd
12 !Y d62
!Zd12
vd! 0 0
Y d62 0 0
#
$
!
"Xd
36
Y d31
Zd36
#
$
VCKM = V †u Vd =
!
""#
Xu12
Zu12
!Y u12
Zu12"61
1!Y u
12Xu12
(Zu12)
2"61
1 ! !Y u12
Zu12"61
1 0 !Xu12
Zu12
$
%%& .
!
""#
0 0 1a !Y d
12Zd
12"61a 0
a !a !Y d12
Zd12"61
0
$
%%& .=
!
"1 ! !3
! 1 !2
!3 !4 1
#
$ .
In dP1 we get almost the right CKM.
dP1:
E
A left-right modelwith the right CKM
The CKM is given in terms of ratios of Higgs vevs.
After breaking of U(2)R
W =
!
"XL
23
Y L23
ZL23
#
$
!
"0 Zu
12 !Y u62
!Zu12
!! 0 Xu
12!!
Y u62 !Xu
12 0
#
$
!
"Xu
36
Y u31
Zu36
#
$ +
!
"Xd
23
Y d23
Zd23
#
$
!
"0 Zd
12 !Y d62
!Zd12
vd! 0 0
Y d62 0 0
#
$
!
"Xd
36
Y d31
Zd36
#
$
VCKM = V †u Vd
In dP1 we get almost the right CKM.
3
2L
2L2
2R
2R2
W =
!
"XL
43
Y L23
ZL23
#
$
!
"0 Z14 !Y64
!Z14!61"42
#2 0 X12!61#
Y64"42# !X12 0
#
$
!
"X36
Y31
Z36
#
$ .
E
A left-right modelwith the right CKM
The CKM is given in terms of ratios of Higgs vevs.
After breaking of U(2)R
W =
!
"XL
23
Y L23
ZL23
#
$
!
"0 Zu
12 !Y u62
!Zu12
!! 0 Xu
12!!
Y u62 !Xu
12 0
#
$
!
"Xu
36
Y u31
Zu36
#
$ +
!
"Xd
23
Y d23
Zd23
#
$
!
"0 Zd
12 !Y d62
!Zd12
vd! 0 0
Y d62 0 0
#
$
!
"Xd
36
Y d31
Zd36
#
$
VCKM = V †u Vd
In dP1 we get almost the right CKM.
3
2L
2L2
2R
2R2
=
!
"1 ! !3
! 1 !2
!3 !2 1
#
$ .=
!
"""#
Xu12!61Y u64"
Xu12Zu
14
(Y u64)
2 1
1 !Zu14!61"Y u
64!Xu
12!61"Y u
64Zu
14!61Y u64"
1 !Xu12Zu
14
(Y u64)
2
$
%%%&.
!
"#0 0 1
a "Y d64
Zd14!61
!bZd14!61
"Y d64
0a b 0
$
%& .
W =
!
"XL
43
Y L23
ZL23
#
$
!
"0 Z14 !Y64
!Z14!61"42
#2 0 X12!61#
Y64"42# !X12 0
#
$
!
"X36
Y31
Z36
#
$ .
dP2:
E
A left-right modelwith the right CKM
The CKM is given in terms of ratios of Higgs vevs.
After breaking of U(2)R
W =
!
"XL
23
Y L23
ZL23
#
$
!
"0 Zu
12 !Y u62
!Zu12
!! 0 Xu
12!!
Y u62 !Xu
12 0
#
$
!
"Xu
36
Y u31
Zu36
#
$ +
!
"Xd
23
Y d23
Zd23
#
$
!
"0 Zd
12 !Y d62
!Zd12
vd! 0 0
Y d62 0 0
#
$
!
"Xd
36
Y d31
Zd36
#
$
VCKM = V †u Vd
3
2L
2L2
2R
2R2
=
!
"1 ! !3
! 1 !2
!3 !2 1
#
$ .=
!
"""#
Xu12!61Y u64"
Xu12Zu
14
(Y u64)
2 1
1 !Zu14!61"Y u
64!Xu
12!61"Y u
64Zu
14!61Y u64"
1 !Xu12Zu
14
(Y u64)
2
$
%%%&.
!
"#0 0 1
a "Y d64
Zd14!61
!bZd14!61
"Y d64
0a b 0
$
%& .
W =
!
"XL
43
Y L23
ZL23
#
$
!
"0 Z14 !Y64
!Z14!61"42
#2 0 X12!61#
Y64"42# !X12 0
#
$
!
"X36
Y31
Z36
#
$ .
In dP1 we get almost the right CKM. In dP2 we get the right CKM.CP violation: with correct CKM, Jarlskog invariant J≈ε6
dP2:
E
Summary
D-branes at toric singularities interesting class of models:
Upper bound of 3 families in toric singularities
Mass Hierarchies are possible, generic structure (0, m, M).
Sufficient structure for realistic CKM-matrix & CP-violation (concrete models with this structure)
Open questions: compact models, a completely realistic local model...
kdmq: 1002.1790
Seiberg duality in quivers and dimers
The zeroth Hirzebruch surface
1 2
1 4 1
4 3
3 2 3
1 2
! ! !
"
! ! !
"
!! ! 3
1
4
2
1
3 4
2
! !
" " "
! !
" " "
! !
3 4 3
3 2
1 2 1
2 3
33 4
The zeroth Hirzebruch surface
1 2
1 4 1
4 3
3 2 3
1 2
! ! !
"
! ! !
"
!! ! 3
1
4
2
1
3 4
2
! !
" " "
! !
" " "
! !
3 4 3
3 2
1 2 1
2 3
33 4
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