Transcript Document
SUSY Breaking in D-brane Models
Beyond Orbifold Singularities
Sebastián Franco
José F. Morales
Durham University
INFN - Tor Vergata
Why D-branes at Singularities?
Local approach to String Phenomenology
UV completion,
gravitational physics
Gauge symmetry, matter
content, superpotential
New perspectives for studying quantum field theories (QFTs) and geometry
Basic setup giving rise to the AdS/CFT, or more generally gauge/gravity, correspondence.
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New tools for dealing with strongly coupled QFTs in terms of weakly coupled gravity
QFTs dynamics gets geometrized:
Duality
Confinement
Dynamical supersymmetry breaking
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Quivers from Geometry …
On the worldvolume of D-branes probing Calabi-Yau singularities we obtain
quiver gauge theories
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2
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3
W = X12X23X34X56X61 - X12X24X45X51 - X23X35X56X62
- X34X46X62X23 + X13X35X51 + X24X46X62
Example: Cone over dP3
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… and Geometry from Quivers
Calabi-Yau
Starting from the gauge theory, we can infer the
CY
ambient geometry by computing its moduli space
D3s
Quiver
Toric Calabi-Yau Cones
Admit a U(1)d action, i.e. Td fibrations
Toric Varieties
Described by specifying shrinking cycles and relations
Complex plane
2-sphere
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
Cone over del Pezzo 1
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(p,q) Web
(-1,2)
Toric Diagram
4-cycle
(1,0)
2-cycle
(-1,-1)
(1,-1)
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Quivers from Toric Calabi-Yau’s
Feng, Franco, He, Hanany
We will focus on the case in which the Calabi-Yau 3-fold is toric
Toric
CY
D3s
The resulting quivers have a more constrained structure:
Toric Quivers
The F-term equations are of the form
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Recall F-term equations are given by:
𝜕𝑊
𝜕𝑋𝑖𝑗
monomial = monomial
=0
The superpotential is a polynomial and every arrow in the quiver
appears in exactly two terms, with opposite signs
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Periodic Quivers
Franco, Hanany, Kennaway, Vegh, Wecht
It is possible to introduce a new object that combines quiver and superpotential data
Periodic Quiver
Planar quiver drawn on the surface of a 2-torus such that every
plaquette corresponds to a term in the superpotential
Example: Conifold/ℤ2 (cone over F0)
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Unit cell
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F-term eq.:
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X234 X241 = X232 X221
W = X1113 X232 X221 - X1213 X232 X121 - X2113 X132 X221 + X2213 X132 X121
- X1113 X234 X241 + X1213 X234 X141 + X2113 X134 X241 - X2213 X134 X141
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Brane Tilings
Franco, Hanany, Kennaway, Vegh, Wecht
Periodic Quiver
Take the dual graph
Dimer Model
It is bipartite (chirality)
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1
2
3
2
1
4
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In String Theory, the dimer model is a physical configuration of branes
Field Theory
Periodic Quiver
Dimer
U(N) gauge group
node
face
bifundamental
(or adjoint)
arrow
edge
superpotential term
plaquette
node
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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
(n1,n2)
(0,0)
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p1
(1,0)
p6
p2
p3
p7
p4
p8
p5
(0,1)
p9
Perfect matchings are natural variables parameterizing the moduli space. They
automatically satisfy vanishing of F-terms
Franco, Hanany, Kennaway, Vegh, Wecht
Franco, Vegh
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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
For any arrow in the quiver associated to an edge in the brane tiling X0:
𝜕𝑋0 W = 0
W = X0 P1 Xi − X0 P2 Xi + ⋯
P1 Xi = P2 Xi
Graphically:
X0
P1(Xi)
=
P2(Xi)
Consider the following map between edges Xi and perfect matchings pm:
pμPiμ
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Xi =
1 if Xi ∈ pμ
Piμ =
μ
0 if Xi ∉ pμ
This parameterization automatically implements the vanishing F-terms for all edges!
Piμ
pμ
𝑖∈𝑃1 μ
Piμ
=
pμ
𝑖∈𝑃2 μ
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Perfect Matchings and Geometry
There is a one to one correspondence between perfect matchings and GLSM fields
describing the toric singularity (points in the toric diagram)
Franco, Hanany, Kennaway, Vegh, Wecht
Franco, Vegh
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)
Kasteleyn Matrix
Toric Diagram
white nodes
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black nodes
K =
p8
det K = P(z1,z2) = nij z1i z2j
p1, p2, p3, p4, p 5
p6
p7
p9
Example: F0
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Other Interesting Developments
Dimers provide the largest classification of 4d N=1 SCFTs and connect them
to their gravity duals
Benvenuti, Franco, Hanany, Martelli, Sparks
Franco, Hanany, Martelli, Sparks, Vegh, Wecht
Dimer models techniques have been extended to include:
Franco, Uranga - Franco, Uranga
Flavors D7-branes
Orientifolds of non-orbifold singularities
Franco, Hanany, Krefl, Park, Vegh
The state of the art in local model building: exquisite realizations of the
Standard Model, including CKM and leptonic mixing matrix
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Krippendorf, Dolan, Maharana, Quevedo
Krippendorf, Dolan, Quevedo
Other Directions: mirror symmetry, crystal melting, cluster algebras,
integrable systems
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1. Retrofitting the Simplest SUSY Breaking Models
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
Aharony, Kachru, Silverstein
NS
NS’
NS’
Conifold/ℤ3
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Fractional branes
NS
D4
NS’
NS
Anomaly-free rank assignments
Using Seiberg duality, two possible types of nodes:
i-1
i
i+1
W = Xi-1,i Xi,i+1 Xi+1,i Xi,i-1 + …
i-1
i
i+1
W = Xi,i-1 Xi-1,i fi,i - fi,i Xi,i+1 Xi+1,i + …
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General Strategy: consider wrapped D-instanton over orientifolded empty node
Polonyi
a
1
0
b
2
1
W = L12 X
Fayet
a
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1
0
b
X23
2
1
X32
3
1
W = L1 X23 X32
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)
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2. Dynamical SUSY Breaking Models
It is possible to engineer standard gauge theories with DSB. A detailed understanding
of orientifolds of non-orbifold singularities provides additional tools.
Franco, Hanany, Krefl, Park, Vegh
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G5 × U(n1) × U(n2) × U(n4)
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1
4
5
2
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Controlled by signs of fixed points
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Example: PdP4
For n1 = n4 = 0, n5 = 1, n1 = 5, we can obtain and SO(1) × U(5) gauge theory with
matter:
This theory breaks SUSY dynamically.
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3. Geometrization of SUSY Breaking
We are familiar with the behavior of N = k M regular and M fractional branes at the
conifold
Logarithmic cascading RG flow
In the IR: confinement and chiral
symmetry breaking
Gravity dual based on a complex
deformation of the conifod
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
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(-1,1)
decomposition of (p,q) web into subwebs in equilibrium
(0,1)
(-1,0)
S3
(1,0)
conifold
(0,-1)
(1,-1)
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Deformation
Fractional Branes
Franco, Hanany, Saad, Uranga
N=2
Dynamical SUSY breaking (due to ADS superpotential)
Example: dP1
Admits fractional branes and a duality cascade but no complex deformation
Ejaz, Klebanov, Herzog
Franco, Hanany, Uranga
(-1,2)
3M
(1,0)
M
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2M
(-1,-1)
(1,-1)
IR bottom of cascade
This theory dynamically breaks SUSY with a runaway
Berenstein, Herzog, Ouyang , Pinansky
Bertolini, Bigazzi, Cotrone
Franco, Hanany, Saad ,Uranga
Intriligator, Seiberg
4. Metastable SUSY from Obstructed Deformations
Franco, Uranga
Obstructed
runaway models
Metastable SUSY
breaking
Adding massive flavors
from D7-branes
3M
SU(3M) with 2M massless flavors
D7-branes
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2M
M
We add massive flavors to the Nf < Nc gauge group to bring it to the free-magnetic range
Low Energies: interesting generalization of ISS including massless flavors
Crucial superpotential couplings are indeed generated by the geometry
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5. Dynamically Generated ISS
There are various similarities between anti-branes in a Klebanov-Strassler throat and ISS
Kachru, Pearson, Verlinde
Intriligator, Seiberg, Shih
Is there some (holographic) relation between the two classes of meta-stable states?
Masses from Quantum Moduli Space
Argurio, Bertolini, Franco, Kachru
Let us engineer the following gauge theory with branes at an orbifold of the conifold
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M
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Consider P
2
M
M
3
P
W = X21 X12 X23 X32
3/2 P . In the L1» L3» L2 regime:
Node 1 has Nc=Nf
On the mesonic branch:
Quantum Moduli Space
det M – BB = L12M
WW= = M X23
X23
X32
X32
The gauge theory on node 3 becomes an ISS model with dynamically generated masses.
Metastability of the vacuum requires P=M.
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Conclusions
We reviewed gauge theories on D-branes probing orbifold and non-orbifold
toric singularities and their orientifolds
Dimer models provide powerful control of the connection between
geometry and gauge theory
We discussed non-perturbative D-brane instanton contributions to such
gauge theories and the conditions under which they arise
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Local D-brane models lead to a wide range of SUSY breaking theories, from
retrofitted simple models to geometrized dynamical SUSY breaking
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