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.
Sebastian Franco
New tools for dealing with strongly coupled QFTs in terms of weakly coupled gravity
 QFTs dynamics gets geometrized:
 Duality
 Confinement
 Dynamical supersymmetry breaking
2
Quivers from Geometry …
 On the worldvolume of D-branes probing Calabi-Yau singularities we obtain
quiver gauge theories
1
2
6
3
W = X12X23X34X56X61 - X12X24X45X51 - X23X35X56X62
- X34X46X62X23 + X13X35X51 + X24X46X62
Example: Cone over dP3
5
4
Sebastian Franco
… 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
Sebastian Franco
(p,q) Web
(-1,2)
Toric Diagram
4-cycle
(1,0)
2-cycle
(-1,-1)
(1,-1)
4
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
Sebastian Franco
 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
5
Sebastian Franco
6
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)
Sebastian Franco
1
Unit cell
2
F-term eq.:
4
3
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
7
Brane Tilings
Franco, Hanany, Kennaway, Vegh, Wecht
Periodic Quiver
 Take the dual graph
Dimer Model
 It is bipartite (chirality)
1
4
1
2
3
2
1
4
1
Sebastian Franco
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
8
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)
Sebastian Franco
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
9
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
Sebastian Franco
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
10
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
Sebastian Franco
black nodes
K =
p8
det K = P(z1,z2) =  nij z1i z2j
p1, p2, p3, p4, p 5
p6
p7
p9
Example: F0
11
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
Sebastian Franco
Krippendorf, Dolan, Maharana, Quevedo
Krippendorf, Dolan, Quevedo
 Other Directions: mirror symmetry, crystal melting, cluster algebras,
integrable systems
12
Sebastian Franco
13
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
Sebastian Franco
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 + …
14
 General Strategy: consider wrapped D-instanton over orientifolded empty node
Polonyi
a
1
0
b
2
1
W = L12 X
Fayet
a
Sebastian Franco
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)
15
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
5
5
G5 × U(n1) × U(n2) × U(n4)
7
3
6
3
1
1
4
5
2
5
Controlled by signs of fixed points
Sebastian Franco
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.
16
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
Sebastian Franco
(-1,1)
decomposition of (p,q) web into subwebs in equilibrium
(0,1)
(-1,0)
S3
(1,0)
conifold
(0,-1)
(1,-1)
17
 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
Sebastian Franco
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
Sebastian Franco
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
19
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
1
M
Sebastian Franco
 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.
20
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
Sebastian Franco
 Local D-brane models lead to a wide range of SUSY breaking theories, from
retrofitted simple models to geometrized dynamical SUSY breaking
21
Sebastian Franco
22