phenomenology of d-branes at toric · pdf filephenomenology of d-branes at toric...
TRANSCRIPT
![Page 1: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/1.jpg)
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
![Page 2: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/2.jpg)
Disclaimer: This talk is somewhat independent of the LHC.
... string theorists could have talked about this 20 years ago
![Page 3: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/3.jpg)
![Page 4: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/4.jpg)
for more than 25 years...
![Page 5: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/5.jpg)
for more than 25 years...
Heterotic String Theory
![Page 6: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/6.jpg)
for more than 25 years...
Heterotic String Theory Type II brane models
![Page 7: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/7.jpg)
for more than 25 years...
Heterotic String Theory
e.g. intersecting branes in IIA,magnetised branes in IIB
Type II brane models
F-theory
![Page 8: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/8.jpg)
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
![Page 9: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/9.jpg)
Heterotic String Theory
e.g. intersecting branes in IIA,magnetised branes in IIB
Type II brane models
0502005, 0702094
![Page 10: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/10.jpg)
Heterotic String Theory
e.g. intersecting branes in IIA,magnetised branes in IIB
Type II brane models
0502005, 0702094
moduli stabilisation
![Page 11: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/11.jpg)
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)
![Page 12: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/12.jpg)
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)
![Page 13: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/13.jpg)
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
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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
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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?
![Page 16: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/16.jpg)
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)
![Page 17: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/17.jpg)
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)
![Page 18: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/18.jpg)
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.
![Page 19: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/19.jpg)
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
![Page 20: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/20.jpg)
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.
![Page 21: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/21.jpg)
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
![Page 22: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/22.jpg)
let’s build models!
Standard-Model
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let’s build models!Left-Right
Standard-Model
![Page 24: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/24.jpg)
let’s build models!
Pati-Salam
Left-Right
Standard-Model
![Page 25: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/25.jpg)
let’s build models!
Pati-Salam Trinification
Left-Right
Standard-Model
![Page 26: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/26.jpg)
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
![Page 27: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/27.jpg)
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
![Page 28: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/28.jpg)
What’s bad about the model?
![Page 29: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/29.jpg)
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)
![Page 30: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/30.jpg)
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
![Page 31: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/31.jpg)
What types of singularities are there?
Orbifold Singularities
del Pezzo singularities (P2 blown-up), Conifold -> TORIC SINGULARITIES
non-toric singularities
![Page 32: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/32.jpg)
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
![Page 33: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/33.jpg)
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
![Page 34: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/34.jpg)
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
![Page 35: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/35.jpg)
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)
![Page 36: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/36.jpg)
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
![Page 37: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/37.jpg)
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
![Page 38: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/38.jpg)
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!
![Page 39: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/39.jpg)
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
![Page 40: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/40.jpg)
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
![Page 41: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/41.jpg)
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
![Page 42: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/42.jpg)
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
![Page 43: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/43.jpg)
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
![Page 44: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/44.jpg)
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
![Page 45: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/45.jpg)
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
![Page 46: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/46.jpg)
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)
![Page 47: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/47.jpg)
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)
![Page 48: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/48.jpg)
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)
![Page 49: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/49.jpg)
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)
![Page 50: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/50.jpg)
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
![Page 51: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/51.jpg)
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
![Page 52: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/52.jpg)
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
![Page 53: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/53.jpg)
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
![Page 54: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/54.jpg)
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
![Page 55: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/55.jpg)
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
![Page 56: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/56.jpg)
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
! ! !
" " " "
! ! !
" " " "
! ! !
![Page 57: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/57.jpg)
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
! ! !
" " " "
! ! !
" " " "
! ! !
![Page 58: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/58.jpg)
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
! ! !
" " " "
! ! !
" " " "
! ! !
![Page 59: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/59.jpg)
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
! ! !
" " " "
! ! !
" " " "
! ! !
![Page 60: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/60.jpg)
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
! ! !
" " " "
! ! !
" " " "
! ! !
![Page 61: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/61.jpg)
Operations on the dimer
(1,0) + (0,1) -> (1,1)Operation 1:
![Page 62: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/62.jpg)
Operations on the dimer
(1,0) + (0,1) -> (1,1)Operation 1:
![Page 63: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/63.jpg)
Operations on the dimer
(1,0) + (0,1) -> (1,1)Operation 1:
![Page 64: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/64.jpg)
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)
![Page 65: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/65.jpg)
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)
![Page 66: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/66.jpg)
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)
![Page 67: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/67.jpg)
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
! !
" " "
! !
" " "
! !
![Page 68: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/68.jpg)
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
! !
" " "
! !
" " "
! !
![Page 69: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/69.jpg)
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
! !
" " "
! !
" "
! !
![Page 70: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/70.jpg)
Gulotta’s dimers = Traditional dimers
![Page 71: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/71.jpg)
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?
![Page 72: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/72.jpg)
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
![Page 73: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/73.jpg)
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
! !
" "
! !
" "
![Page 74: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/74.jpg)
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
! !
" "
! !
" "
![Page 75: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/75.jpg)
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
! !
" "
! !
" "
![Page 76: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/76.jpg)
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
! !
" "
! !
" "
![Page 77: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/77.jpg)
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
![Page 78: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/78.jpg)
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
![Page 79: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/79.jpg)
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
![Page 80: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/80.jpg)
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)
![Page 81: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/81.jpg)
Upper bound on the number of families given by the physically
observed number with one exception (4 families in F0).
![Page 82: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/82.jpg)
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?
![Page 83: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/83.jpg)
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.
![Page 84: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/84.jpg)
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
![Page 85: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/85.jpg)
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
![Page 86: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/86.jpg)
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
! !
" " "
! !
" " "
! !
![Page 87: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/87.jpg)
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
! !
" " "
! !
" " "
! !
![Page 88: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/88.jpg)
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
! !
" " "
! !
" " "
! !
![Page 89: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/89.jpg)
general structure (0, m, M)
![Page 90: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/90.jpg)
general structure (0, m, M)
non-vanishing mass?
![Page 91: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/91.jpg)
general structure (0, m, M)
non-vanishing mass?
corrections to Kähler potential, deformation of singularity (non-toric)
![Page 92: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/92.jpg)
... back to model building
![Page 93: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/93.jpg)
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
![Page 94: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/94.jpg)
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
![Page 95: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/95.jpg)
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
![Page 96: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/96.jpg)
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
![Page 97: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/97.jpg)
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
![Page 98: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/98.jpg)
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
![Page 99: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/99.jpg)
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
![Page 100: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/100.jpg)
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
![Page 101: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/101.jpg)
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
![Page 102: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/102.jpg)
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
![Page 103: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/103.jpg)
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
![Page 104: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/104.jpg)
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
![Page 105: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/105.jpg)
Seiberg duality in quivers and dimers
![Page 106: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/106.jpg)
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
![Page 107: Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,](https://reader031.vdocuments.mx/reader031/viewer/2022011723/5a9d6afb7f8b9abd058ce8a0/html5/thumbnails/107.jpg)
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