A#'$8,#',$?,5#,-$B-3'C 0'1234,5#-67$89:$;*55,< (...

Global Embedding ofD-branes at singularities

Trieste, SUSY, August 2013

Sven Krippendorf

!"#$%$&#'$()#*+,-$./!.0'1"234,5#-67$89:$;*55,<!"#$%&%'()*+))*,-./)0"1)!23345

(-=952425=$)*772##,,6"'")7314'&23892:"&);'<::=>43)?'<$$4312'50#:&42@)?A"4) ) ) ) )!"3&)B4%4')C<DD4&) ) ) ) ) )E<F9"4D)G"%H) ) ) ) ) ) ))))))>,)-,#9-1="@#'")=F9I54'JC":4@<)))))))))))))))))))))))))))))))C()C(B"%'<F@)K"#1'4>"3L4))))))))))))))))))))))))))))))))))))MF%$JN2'@&92$O#3<JM233(14

?*5<,-,5),$@,+"2#,P)MF%$(#3<JM233(14QN2'@&92$R.SRQ

A#'$8,#',$?,5#,-$B*-3"'*C

D52<2)9#2*5$95:$>#-25=$E',*-1

based on collaboration with

Michele Cicoli, Christoph Mayrhofer,Fernando Quevedo, Roberto Valandro

1304.0022, 1206.5237

Motivation

Old Challenge:Construct an explicit viable string vacuum

satisfying all constraints from particle physicsand cosmological observations.

Status/Approach in type IIB?

various mechanisms havebeen constructed

... the bottom-up approach to string model buildingAIQU: hep-th/0005067



Moduli stabilisationKähler moduli: perturbative + non-perturbative corrections (e.g. KKLT, LVS)Complex structure + dilaton: fluxes (e.g. GKP)





Inflationopen stringclosed string






D-brane SM modelmagnetised branes

D-branes at singularities


1002.1790, 1102.1973, 1106.6039








chirality (BPM), SUSY

re-heating, dark radiation

control of eft,uv sensitivity

1002.1790, 1102.1973, 1106.6039

Can these mechanismsbe combined leading to

a realistic string vacuum?




Moduli stabilisationKähler moduli: perturbative + non-perturbative correctionsComplex structure: fluxes








Moduli stabilisationKähler moduli: perturbative + non-perturbative correctionsComplex structure: fluxes



Focus on combining moduli stabilisationwith D-brane model building


Content

• Geometric requirements for combining both mechanisms

• Models with(out) flavour branes

• Models with explicit flux stabilisation (scan)

Geometric Requirements

• Visible sector with D-branes at del Pezzo singularities(2 dPn’s mapped on top of each other with O-involution)

• Mechanism for explicit Kähler moduli stabilisation: here LVS(1 additional divisor allowing for non-perturbative effect)

• No intersection between the two sectors(BPM, chirality + moduli stabilisation)

• CYs with h11≥4 (h11-≥1), search in available list of CYs (Kreuzer-Skarke list)

alternative setups: magnetised branesmodels with D-branes @ singularities

1002.1790, 1102.1973, 1106.6039

1206.5237

h1,1= 4 : 1197 polytopes ⌃ dP0 dP1 dP2 dP3 dP4 dP5 dP6 dP7 dP8

There are 2 dPn + O-involution 82 9 5 - - - 2 10 31 25

The 2 dPn do not intersect 68 9 2 - - - 2 10 27 18

Further rigid divisor 21 3 - - - - - 4 9 5

Search in Kreuzer-Skarke database

h1,1= 5 : 4990 polytopes ⌃ dP0 dP1 dP2 dP3 dP4 dP5 dP6 dP7 dP8

There are 2 dPn & O-involution 386 27 60 21 7 3 13 40 121 94

The 2 dPn do not intersect 327 27 55 7 3 1 11 39 112 72

Further rigid divisor 168 14 16 - - - 5 28 68 37

1206.5237

more Kähler moduli = more computing time

let’s take one example,add some D-branes,

check consistency conditions,and stabilise Kähler moduli

dP0 example: h1,1=4, h1,2=112• charge matrix, SR-ideal

• basis of divisors

• triple intersection form, volume

• 3 dP0’s at z4=0, z7=0, z8=0

z1 z2 z3 z4 z5 z6 z7 z8 DeqX

1 1 1 0 3 3 0 0 90 0 0 1 0 1 0 0 20 0 0 0 1 1 0 1 30 0 0 0 1 0 1 0 2

SR = {z4 z6, z4 z7, z5 z7, z5 z8, z6 z8, z1 z2 z3}

�b = D4 +D5 = D6 +D7, �q1 = D4, �q2 = D7, �s = D8

I3 = 27�3b + 9�3

q1 + 9�3q2 + 9�3

s V =1

9

r2

3

h⌧3/2b �

p3⇣⌧3/2q1 + ⌧3/2q2 + ⌧3/2s

⌘i

Orientifold projection

• We take an O-involution exchanging two (shrinking) dP0s:

z4 ↔ z7 and z5 ↔ z6 (h1,1-=1 and h1,1+=3)

• This exchanges the two dP0s: Dq1=D4 and Dq2=D7

• There are no O3-planes and 2 O7-planes:O71: z4z5 - z6z7 = 0 → [O71]=Db. O72: z8 = 0 → [O72]=Ds.

• O7-planes do not intersect the shrinking dP0s and each other

Model without flavour branes• dP0 trinification model (N=3 D3-branes)

• to cancel D7 tadpole: 4 D7s (+images) on top of each O7-plane

• hidden sector: SO(8)xSO(8)

• FW flux: Fs=-Ds/2 cancelled by choosing B=-Ds/2 ➛ Fs=Fs-B=0 → pure SO(8) SYM on Ds (gaugino condensate)

• FW flux: Fb=-Db/2 also cancelled by choosing B=-Db/2 -Ds/2 ➛ adjoint scalars; can be lifted by flux but SO(8)→SU(4)xU(1)(special! no intersection of Γs & Γb, so cancellation on both possible)

• Non-perturbative superpotential

• D5 tadpole cancelled as F = -F ’.

• QD3 = -60 + 2ND3 = -54 (Whitney brane: QD3=-432, no g.c. on Γb)freedom to turn on three-form fluxes H3 & F3.

3C

3R3L

W = W0 +Ase�asTs(+Abe

�abTb) as =⇡

3ab =

⇡

2

1206.5237

Moduli Stabilisation

• complex structure assumed to be stabilised with 3-form fluxes (D3 tadpole allows to turn on fluxes.)

• EFT:

• singularity stabilisation: D-term minimum at ξi=0 and Ci=0 (soft-masses), F-terms sub-leading

K = �2 ln

V +

⇣

g3/2s

!+

(T+ + T+ + q1V1)2

V +(G+ G+ q2V2)2

V +CiCi

V2/3,

V =1

9

r2

3

⇣⌧3/2b �

p3⌧3/2s

⌘

W = Wlocal

+Wbulk

= W0

+ yijk CiCjCk +As e

�⇡3 Ts +Ab e

�⇡2 Tb

VD =1

Re(f1)

X

i

q1iKiCi � ⇠1

!2

+1

Re(f2)

X

i

q2iKiCi � ⇠2

!2

,

⇠1 = �4q1⌧+V

⇠2 = �4q2b

V

Moduli Stabilisation

• F-term potential

• FW flux on large four-cycle (matter fields), D-term potential

• can account for dS/Minkowski minima...

• KKLT: τb>τs requires ab<as but not realised here...

4!106

6!106

8!106

1!107

Vol

2.!10"23

4.!10"23

6.!10"23

8.!10"23

1.!10"22

V

VF ' 8

3(asAs)

2p⌧se�2 as⌧s

V � 4 asAsW0⌧se�as⌧s

V2+

3

4

⇣W 20

g3/2s V3

hVi '3W0

p⌧s

4asAseash⌧si h⌧si '

✓3⇣

2

◆2/3 1

gs

Vtot

= VD + VF ' p1

V2/3

0

@X

j

qbj |�c,j |2 �p2

V2/3

1

A2

+X

j

W 2

0

2V2

|�c,j |2 + VF (T )

W0 ' 0.2gs ' 0.03As ' 1

⇣ ' 0.522

Gravity/moduli mediated SUSY breaking• no flavour branes ➔ no redefinitions of moduli

• FTSM=0 ➔ sequestered soft-masses

• gravitino mass:

• remaining soft-masses receive contributions from Ftb, Fs, gauge kinetic function f=Re(S), after many no-scale cancellations

• Assumption: no D-term contribution [soft scalar masses after gauge breaking will need further study]

• Note: lightest modulus (mTb~Mp/V3/2) heavier than TeV soft-masses [➔cosmological moduli problem]

• Pheno: particular slice of CMSSM, resp. Mini Split-SUSY

0906.3297

Mgaugino

m3/2

Vm

scalar

m3/2pV or

m3/2

VM

string

MPpV

V 10

6�7

m3/2 = eK/2|W | ⇠ MP |W0|V

let’s add flavour D7-branes

1304.2771

Geometric summary

• Same geometric background and orientifold as before. Can have different D7 brane setup, leading to flavour branes...

• dP0: left-right symmetric model (n0=n2=2, m0=m2=3, m1=m=0) as in AIQU hep-th/0005067

• visible sector gauge group: SU(3)cxSU(2)LxSU(2)RxU(1)B-L

• hidden sector D7-D7 matter

3

22

11

��

�

�

+-�

�

-

+

+�-dP'dP 00

D72flav

D72flav'

D70flavD70

flav'

D7SU(2) D7SU(2)'

Moduli stabilisation (with flavour branes)

• D-term potential: quiver D-terms same as before (soft-scalar masses); bulk D-terms some fields receive vev, flat directions at tree-level (need to be fixed by higher order F-term contributions)

• F-term potential for matter fields (leading contribution from soft-masses)

• Plug in D-term constraints, reasonable assumptions for Kähler matter metric lead to effective matter uplifting contribution

• Overall F-term potential

• After minimising at Large Volume:

⇤ ⌘ hV i = W 20

hVi3

s

ln

✓hViW0

◆(� 3

4 a3/2s

+p

9

hVi1/3

[ln (V/W0)]5/2

✓1� 6

ln (V/W0)

◆)

VF ' 8

3(asAs)

2p⌧se�2 as⌧s


V2+

3

4

⇣W 20

g3/2s V3+ p

W 20

V8/3 [ln (V/W0)]2

V matterF ' p

W 20

V8/3 [ln (V/W0)]2 , with p := (2c' + c�)

✓9

2

◆2/3

,

V matterF ' W 2

0

V2 [ln (V/W0)]2

X

i

c⇢i |⇢i|2 ,with: Kmatter ' ⌧ c⇢s |⇢|2/V2/3

Scales and gauge coupling unification• Soft masses (re-defs due to flavour branes):

• Required: Soft masses @ TeV, correct gs, and CC=0 by 2 parameters

MZ Mbreaking MString

!3"1

!2"1

!y"1

10 1000 105 107 109 1011 1013x !GeV"

20

40

60

80

100

!"1

11 12 13 14 15x

1

2

3

4

!

p=5.45W0=1

W0=10-4

W0=10-7

W0=10-14

c� = 1, c� =1

2as = ⇡/3

Msoft

'm

3/2

ln�MP /m

3/2

� ' W0

MP

V ln (V/W0

)

V/W0 ' 5 · 1013,⇤ = 0 ) W0 ' 0.01,V ' 5 · 1011

⇣ ' 0.522 ) gs ' 0.015 ' 1/65,Ms ' 1012GeV

↵�1unif = Re(S)|Zfrac| = g�1

s /3

Conclusion/Outlook

• Successful and explicit combination of moduli stabilisation with D-branes at singularities (with and without flavour branes), satisfying all known consistency conditions

• dS moduli stabilisation with TeV soft-masses (sequestered soft-masses no flavour branes)

• Can the dP0 left-right model pass all low-energy tests?

• Include inflationary sector

Thank you!


• D-term potential: quiver D-terms same as before (soft-scalar masses); bulk D-terms some fields receive vev, flat directions at tree-level (need to be fixed by higher order F-term contributions)

• F-term potential for matter fields :

V bulkD =

1

Re(fflav,0)D2

flav,0 +1

Re(fflav,2)D2

flav,2 +1

Re(fD7O71)D2

D7O71

Dflav,0 =12X

i=1

|'i+|2 +

9X

i=1

|�i+|2 �

6X

i=1

|'i�|2 � ⇠flav,0

Dflav,2 =12X

i=1

|'i+|2 +

6X

i=1

|'i�|2 +

18X

i=1

|�i+|2 +

9X

i=1

|�i�|2 � ⇠flav,2

DD7O71=

9X

i=1

|�i+|2 +

18X

i=1

|�i+|2 + 2

9X

i=1

|⇡i|2 �9X

i=1

|�i�|2 � ⇠D7O71

⇠D7O71=

9

2

tbV =

✓9

2

◆2/3 1

V2/3, ⇠flav,0 = ⇠D7O71

+

p⌧q

2V , ⇠flav,2 = 3 ⇠flav,0 .

K � ⌧ c⇢s |⇢|2/V2/3

��

�

�

+-�

�

-

+

+�-dP'dP 00

D72flav

D72flav'

D70flavD70

flav'

D7SU(2) D7SU(2)'

V matterF ' W 2

0

V2 [ln (V/W0)]2

"c'

6X

i=1

|'i+|2 +

12X

i=1

|'i�|2!

+ c�

9X

i=1

|�i+|2

+c�

18X

i=1

|�i+|2 +

9X

i=1

|�i�|2!

+ c⇡

9X

i=1

|⇡i|2#


• F-term for matter fields after using D-flatness condition

• Minimum for given

• Overall F-term potential

• After minimising at Large Volume:

V matterF ' W 2

0

V2 [ln (V/W0)]2

"(c' + c� � c�)

9X

i=1

|�i+|2 + 2 (c� � c')

9X

i=1

|�i�|2

+(2c' + c⇡ � 2c�)9X

i=1

|⇡i|2 + (2c' + c�) ⇠

#

c� > c� � c' > 0 , c⇡ > 2 (c� � c') > 0

V matterF ' p

W 20

V8/3 [ln (V/W0)]2 , with p := (2c' + c�)

✓9

2

◆2/3

,

VF ' 8

3(asAs)

2p⌧se�2 as⌧s


V2+

3

4

⇣W 20

g3/2s V3+ p

W 20

V8/3 [ln (V/W0)]2

⇤ ⌘ hV i = W 20

hVi3

s

ln

✓hViW0

◆(� 3

4 a3/2s

+p

9

hVi1/3

[ln (V/W0)]5/2

✓1� 6

ln (V/W0)

◆)

K � ⌧ c⇢s |⇢|2/V2/3

�i+ = �i

� = ⇡i = 0

Restrictions on mi• Number of chiral intersections determine local flavour brane charges

• Dflav|dP0 =aH is 2-form Poincare dual to the intersection Dflav ∩DdP0

• If Dflav is connected and Dflav ∩DdP0 is an effective curve X (described as vanishing locus of hom. eqn. of degree a) ⇒ [X]=aH, a > 0.

• The locally induced D7-charge of a flavour brane must be a positive multiple of H [note m1=m]

• Not all local models realised globally in this class, e.g.: n0=n1=n2 ⇒ mi=0.

n2n0

n1

m1

m0m2�loc

D70= (m+ 3(n

1

� n0

))H

✓1 +

1

2H

◆

�loc

D71= �2mH (1 +H)

�loc

D72= (m+ 3(n

1

� n2

))H

✓1 +

3

2H

◆

0 �m 3(n1 �max{n0, n2})

h�1

,�2

i ⌘Z

X

⇣��(2�form)

1

^ �(4�form)

2

+ �(2�form)

2

^ �(4�form)

1

⌘

Strategy for model building

• gauge theory via favourite technique (e.g. dimers), quiver gauge theory [bi-fundamentals]

• phenomenology: choose your favourite gauge groups (non GUT, e.g. Pati-Salam) to embed SM matter content; break it to the SM, study flavour structure of couplings, proton decay etc.

• bulk effects: couplings, massless hypercharge (simple solution: origin of hypercharge in non-abelian gauge theory)

• choose your favourite singularity and try (dP0 no mass hierarchy, dP1/dP2 no suff. flavour structure, dP3 flavour structure+diagonal kin. terms)

[Buican, Malyshev, Morrison, Verlinde, Wijnholt]

1106.6039,1102.1973, 1002.1790

dP0 = C3/Z3

• the gauge theory

• anomaly cancellation

• examples: trinification/left-right symmetric models

n2n0

n1

m1

m0m2

m0 = m+ 3(n1 � n0) m1 = m m2 = m+ 3(n1 � n2)

3C

3R3L 3

22

11

W = ✏ijkXiY jZk + ybulkijk XiY jZk +WD3D7

D-brane charges• D-brane charges encoded in the Mukai charge vector:

• Fractional branes described by well-chosen collection of sheaves, supported on shrinking cycle (7-branes). For C3/Z3:

(H is hyperplane class for dP0)

• Flavour branes are D7-branes wrapping holomorphic 4-cyclesthat pass through the singularity (i.e. ):

Local models only describe the local charges of D7flav

�E = D ^ ch(E) ^

sTd(TD)

Td(ND)

ch(F0) = �1 +H � 12 H ^H , ch(F1) = 2� H � 1

2 H ^H , ch(F2) = �1

2-form ↔ D7-charge4-form ↔ D5-charge6-form ↔ D3-charge

D = DdP0

Dflav · DdP0 6= 0Dflav

�loc

D7i⌘ �Di

flav

��dP0

= aiH + biH ^H

�D7(Dflav,F) = Dflav

✓1 + F +

1

2F ^ F +

c2(Dflav)

24

◆

Diaconescu, Gomis; Douglas, Fiol, Romelsberger;Hanany, Herzog, Vegh

Model with flavour branes

• dP0: left-right symmetric model (n0=n2=2, m0=m2=3, m1=m=0)

• gauge group: SU(3)cxSU(2)LxSU(2)RxU(1)B-L

• Local D7 charges

• Which divisor? Dflav|Dq1=3H

3

22

11

�loc

D70= 3H(1 +

1

2H) , �loc

D72= 3H(1 +

3

2H)

(3D1 +Ds +Dq2 + ↵bDb)��Ds,Dq2

= 0

Dflav = 3D1 + ↵bDb + ↵sDs + ↵q2Dq2

↵sand ↵b

fixed by

D(0,2)flav = 3D1 +Ds +Dq2 + ↵b

(0,2)Db = (1 + ↵b(0,2))Db �Dq1


• dP0: left-right symmetric model (n0=n2=2, m0=m2=3, m1=m=0)

• gauge group: SU(3)cxSU(2)LxSU(2)RxU(1)B-L


• D7 charge:

• D5 charge: F0|Dq1=H/2 and F2|Dq1=3H/2. Pullback of Ds and Dq2 on divisor Dflav are trivial. Hence

• D3 charge:

3

22

11

�loc

D70= 3H(1 +

1

2H) , �loc

D72= 3H(1 +

3

2H)

F (0)flav =

1

2D1 + �b

0Db F (2)flav =

3

2D1 + �b

2Db

Dflav ·1

2F2

flav +c2(Dflav)

24

�

D(0,2)flav = 3D1 +Ds +Dq2 + ↵b

(0,2)Db = (1 + ↵b(0,2))Db �Dq1



• Overall charges from the quiver system

• Total charge of quiver system:

• To saturate D7 tadpole, still need two D7s on Db (+images)

• D7- and D5-charges cancel globally. Net D3-charge

3

22

11

�loc

D70= 3H(1 +

1

2H) , �loc

D72= 3H(1 +

3

2H)

�fracD3 = 2�F0 + 3�F1 + 2�F2 = 2Dq1 + 2Dq1 ^D1 � 32dVol

0X

�D7flav0= (Db �Dq1) + (Db �Dq1) ^ 1

2D1 + 5 dVol0X�D7flav2

= (Db �Dq1) + (Db �Dq1) ^ 32D1 + 7 dVol0X

�

z4=0quiver = 2Db + 2Db ^D1 +

�272 � 3

�dVol0X

Z

XdVol0X = 1

↵b(0,2) = 0 �b

(0,2) = 0

Qtot

D3

= Qz4=0

D3, quiver +Qz7=0

D3, quiver +QSU(2)

D3

+QSO(8)

D3

= �63

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