SUSY Breaking in D-brane Models Beyond Orbifold Singularities Sebastián Franco Durham University...
-
Upload
marilynn-banks -
Category
Documents
-
view
223 -
download
1
Transcript of SUSY Breaking in D-brane Models Beyond Orbifold Singularities Sebastián Franco Durham University...
SUSY Breaking in D-brane Models
Beyond Orbifold Singularities
Sebastián Franco
Durham University
José F. Morales
INFN - Tor Vergata
Seb
asti
an F
ran
coWhy D-branes at Singularities?
2
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
Seb
asti
an F
ran
coQuivers from Geometry …
On the worldvolume of D-branes probing Calabi-Yau singularities we obtain quiver gauge theories
Example: Cone over dP3
1 2
3
45
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
Seb
asti
an F
ran
coToric 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
4
Complex plane 2-sphere
(1,0)
(1,-1)(-1,-1)
(-1,2)Cone over del Pezzo 1
2-cycle
4-cycle
Seb
asti
an F
ran
coQuivers from Toric Calabi-Yau’s
We will focus on the case in which the Calabi-Yau 3-fold is toric
5
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
Seb
asti
an F
ran
coPeriodic 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
34
Periodic Quiver
Planar 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)
7
=
F-term eq.:X2
34 X241 = X2
32 X221
Unit cell
Seb
asti
an F
ran
coBrane 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 Dimer
U(N) gauge group node face
bifundamental (or adjoint)
arrow edge
superpotential term plaquette node
3
11
1 1
2 2
4
4
8
Seb
asti
an F
ran
coPerfect 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
9
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)
Seb
asti
an F
ran
co
10
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=∏
μ
pμPi μ
∏𝑖∈ 𝑃1
∏μ
pμ=∏𝑖∈ 𝑃2
∏μ
pμPi μ Pi μ
Seb
asti
an F
ran
co
11
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
Seb
asti
an F
ran
co
12
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, Sparks
Franco, Hanany, Martelli, Sparks, Vegh, Wecht
Franco, Uranga - Franco, Uranga
Franco, Hanany, Krefl, Park, Vegh
Seb
asti
an F
ran
co1. Retrofitting the Simplest SUSY Breaking Models
14
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
Seb
asti
an F
ran
co
15
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
Seb
asti
an F
ran
co2. Dynamical SUSY Breaking Models
16
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
11
Example: PdP4
Seb
asti
an F
ran
co3. Geometrization of SUSY Breaking
17
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
Seb
asti
an F
ran
co
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 , Pinansky
Bertolini, Bigazzi, Cotrone Intriligator, Seiberg
(-1,2)
(1,-1)(-1,-1)
(1,0)
IR bottom of cascade
Example: dP1
Seb
asti
an F
ran
co4. Metastable SUSY from Obstructed Deformations
19
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
Seb
asti
an F
ran
co5. Dynamically Generated ISS
20
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 conifold
Argurio, 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 PM
1 2 3
Seb
asti
an F
ran
coConclusions
21
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