SUSY Breaking in D-brane Models

Beyond Orbifold Singularities

Sebastián FrancoDurham University

José F. MoralesINFN - Tor Vergata

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Why D-branes at Singularities?

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Local approach to String Phenomenology

QFTs dynamics gets geometrized:

Basic setup giving rise to the AdS/CFT, or more generally gauge/gravity, correspondence. New tools for dealing with strongly coupled QFTs in terms of weakly coupled gravity

Gauge symmetry, matter content, superpotential

UV completion, gravitational physics

Duality Confinement Dynamical supersymmetry breaking

New perspectives for studying quantum field theories (QFTs) and geometry

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Quivers from Geometry …

On the worldvolume of D-branes probing Calabi-Yau singularities we obtain quiver gauge theories

Example: Cone over dP3

1 2

3

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6W = X12X23X34X56X61 - X12X24X45X51 - X23X35X56X62

- X34X46X62X23 + X13X35X51 + X24X46X62

… and Geometry from Quivers

D3s

CY

Quiver

Calabi-Yau

Starting from the gauge theory, we can infer the ambient geometry by computing its moduli space

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Toric Calabi-Yau Cones

Toric Varieties Admit a U(1)d action, i.e. Td fibrations Described by specifying shrinking cycles and relations

We will focus on non-compact Calabi-Yau 3-folds which are complex cones over 2-complex dimensional toric varieties, given by T2 fibrations over the complex plane

(p,q) Web Toric Diagram

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Complex plane 2-sphere

(1,0)

(1,-1)(-1,-1)

(-1,2)Cone over del Pezzo 1

2-cycle

4-cycle

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Quivers from Toric Calabi-Yau’s

We will focus on the case in which the Calabi-Yau 3-fold is toric

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D3s

ToricCY

Recall F-term equations are given by:

The resulting quivers have a more constrained structure:

Toric Quivers The F-term equations are of the form monomial = monomial

The superpotential is a polynomial and every arrow in the quiver appears in exactly two terms, with opposite signs

Feng, Franco, He, Hanany

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Brane Tilings

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Periodic Quivers

It is possible to introduce a new object that combines quiver and superpotential data

W = X1113 X2

32 X221 - X12

13 X232 X1

21 - X2113 X1

32 X221 + X22

13 X132 X1

21

- X1113 X2

34 X241 + X12

13 X234 X1

41 + X2113 X1

34 X241 - X22

13 X134 X1

41

1 2

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Periodic QuiverPlanar quiver drawn on the surface of a 2-torus such that every

plaquette corresponds to a term in the superpotential

Franco, Hanany, Kennaway, Vegh, Wecht

Example: Conifold/2 (cone over F0)

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=

F-term eq.:X2

34 X241 = X2

32 X221

Unit cell

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Brane Tilings

In String Theory, the dimer model is a physical configuration of branes

Franco, Hanany, Kennaway, Vegh, Wecht

Periodic Quiver Take the dual graph It is bipartite (chirality)

Dimer Model

Field Theory Periodic Quiver DimerU(N) gauge group node face

bifundamental (or adjoint) arrow edge

superpotential term plaquette node

3

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1 1

2 2

4

4

8

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Perfect Matchings

Perfect matching: configuration of edges such that every vertex in the graph is an endpoint of precisely one edge

p1 p2 p3 p4 p5

p6 p7 p9p8

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Perfect matchings are natural variables parameterizing the moduli space. They automatically satisfy vanishing of F-terms

Franco, Hanany, Kennaway, Vegh, Wecht Franco, Vegh

(n1,n2)(0,0)

(1,0) (0,1)

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Solving F-Term Equations via Perfect Matchings The moduli space of any toric quiver is a toric CY and perfect matchings simplify its

computation

P1(Xi) P2(Xi)

X0 =

W=X 0P1 ( X i )− X0P2 ( X i )+⋯ 𝜕𝑋 0W=0

⇔P1 ( X i )=P2 (X i )

For any arrow in the quiver associated to an edge in the brane tiling X0:

Graphically:

This parameterization automatically implements the vanishing F-terms for all edges!

Consider the following map between edges Xi and perfect matchings pm:

1 if

0 if =X i=∏

μpμPi μ

∏𝑖∈ 𝑃1

∏μpμ=∏

𝑖∈ 𝑃2

∏μpμPi μ Pi μ

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Perfect Matchings and Geometry

This correspondence trivialized formerly complicated problems such as the computation of the moduli space of the SCFT, which reduces to calculating the determinant of an adjacency matrix of the dimer model (Kasteleyn matrix)

There is a one to one correspondence between perfect matchings and GLSM fields describing the toric singularity (points in the toric diagram)

p1, p2, p3, p4, p5

p8

p6

p9

p7

K =

white nodes

black nodes

Kasteleyn Matrix Toric Diagram

det K = P(z1,z2) = nij z1i z2

j Example: F0

Franco, Hanany, Kennaway, Vegh, WechtFranco, Vegh

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Other Interesting Developments

Flavors D7-branes

The state of the art in local model building: exquisite realizations of the Standard Model, including CKM and leptonic mixing matrix

Dimers provide the largest classification of 4d N=1 SCFTs and connect them to their gravity duals

Other Directions: mirror symmetry, crystal melting, cluster algebras, integrable systems

Orientifolds of non-orbifold singularities

Krippendorf, Dolan, Maharana, Quevedo

Krippendorf, Dolan, Quevedo

Dimer models techniques have been extended to include:

Benvenuti, Franco, Hanany, Martelli, SparksFranco, Hanany, Martelli, Sparks, Vegh, Wecht

Franco, Uranga - Franco, Uranga

Franco, Hanany, Krefl, Park, Vegh

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The SUSY Breaking ZOO

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1. Retrofitting the Simplest SUSY Breaking Models

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Remarkably, branes at singularities allow us to engineer the “simplest” textbook SUSY breaking models

Non-chiral orbifolds of the conifold provide a flexible platform for engineering interesting theories

Fractional branes Anomaly-free rank assignments

i i+1i-1

W = Xi-1,i Xi,i+1 Xi+1,i Xi,i-1 + …

i i+1i-1

W = Xi,i-1 Xi-1,i fi,i - fi,i Xi,i+1 Xi+1,i + …

Using Seiberg duality, two possible types of nodes:

Aharony, Kachru, Silverstein

NSNS

NSNS’

NS’

NS’

D4

Conifold/3

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Polonyi

Fayet

W = L1 X23 X32

1 101 2 3

X23

X32

a

b

X23 and X32 are neutral under U(1)(2) + U(1) (3)

SUSY is broken once we turn on an FI term for U(1)(2) – U(1) (3)

W = L12 X

101 2

a

b

General Strategy: consider wrapped D-instanton over orientifolded empty node

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2. Dynamical SUSY Breaking Models

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It is possible to engineer standard gauge theories with DSB. A detailed understanding of orientifolds of non-orbifold singularities provides additional tools.

G5 × U(n1) × U(n2) × U(n4)

For n1 = n4 = 0, n5 = 1, n1 = 5, we can obtain and SO(1) × U(5) gauge theory with matter:

Controlled by signs of fixed points

This theory breaks SUSY dynamically.

Franco, Hanany, Krefl, Park, Vegh

5 5

5 5

33

76

4

2

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Example: PdP4

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3. Geometrization of SUSY Breaking

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We are familiar with the behavior of N = k M regular and M fractional branes at the conifold

Logarithmic cascading RG flow Gravity dual based on a complex deformation of the conifod In the IR: confinement and chiral

symmetry breaking Klebanov, Strassler

The deformation can be understood in terms of gauge theory dynamics at the bottom of the cascade Nf = Nc gauge group with quantum moduli space

Complex Deformations and Webs

Complex deformation decomposition of (p,q) web into subwebs in equilibrium

(0,1)(-1,1)

(-1,0)

(0,-1) (1,-1)

(1,0)

S3

conifold

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This theory dynamically breaks SUSY with a runaway

Fractional Branes Deformation N=2

3M

M2M

Admits fractional branes and a duality cascade but no complex deformationFranco, Hanany, UrangaEjaz, Klebanov, Herzog

Dynamical SUSY breaking (due to ADS superpotential)Franco, Hanany, Saad, Uranga

Franco, Hanany, Saad ,UrangaBerenstein, Herzog, Ouyang , PinanskyBertolini, Bigazzi, Cotrone Intriligator, Seiberg

(-1,2)

(1,-1)(-1,-1)

(1,0)

IR bottom of cascade

Example: dP1

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4. Metastable SUSY from Obstructed Deformations

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Low Energies: interesting generalization of ISS including massless flavors

Crucial superpotential couplings are indeed generated by the geometry

Obstructed runaway models

Metastable SUSY breaking

Franco, Uranga

Adding massive flavors from D7-branes

We add massive flavors to the Nf < Nc gauge group to bring it to the free-magnetic range

3M

M2M

SU(3M) with 2M massless flavors

D7-branes

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5. Dynamically Generated ISS

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There are various similarities between anti-branes in a Klebanov-Strassler throat and ISS

Consider P M 3/2 P . In the L1» L3» L2 regime:

Node 1 has Nc=Nf Quantum Moduli Space

On the mesonic branch:

Masses from Quantum Moduli Space

Is there some (holographic) relation between the two classes of meta-stable states?

Intriligator, Seiberg, ShihKachru, Pearson, Verlinde

Let us engineer the following gauge theory with branes at an orbifold of the conifoldArgurio, Bertolini, Franco, Kachru

W = X21 X12 X23 X32

The gauge theory on node 3 becomes an ISS model with dynamically generated masses. Metastability of the vacuum requires P=M.

det M – BB = L12M

W = M X23 X32W = M X23 X32

M PM1 2 3

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Conclusions

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We reviewed gauge theories on D-branes probing orbifold and non-orbifold toric singularities and their orientifolds

We discussed non-perturbative D-brane instanton contributions to such gauge theories and the conditions under which they arise

Local D-brane models lead to a wide range of SUSY breaking theories, from retrofitted simple models to geometrized dynamical SUSY breaking

Dimer models provide powerful control of the connection between geometry and gauge theory

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THANK YOU!