1. Dimer models and orientifolds
Sebastián Franco
Princeton University
September 2007
Based on: arXiv: : Franco, Hanany, Krefl, Park, Uranga and Vegh

2. Outline Motivation Background Orientifolding dimers
D-branes over toric singularities
Brane dimers
Orientifolds
Orientifolding dimers
Fixed points and fixed lines
Rules
Geometric action
Examples
The mirror perspective
Applications
Sebastian Franco
SUSY breaking
D-brane instantons
Conclusions

3. Motivation Branes at singularities:
Bottom-up approach to string phenomenology
Local constructions of SM-like theories
Many features are independent of details of the compactification
Interesting strong gauge dynamics gets geometrized
Confinement
Klebanov and Strassler
SUSY breaking with runaway
Berenstein et. al., Franco et. al.,
Bertolini et. al.
Meta-stable SUSY breaking
Franco and Uranga
Extensions of the AdS/CFT to theories with reduced (super)symmetry
Sebastian Franco
Infinite families of Sasaki Einstein metrics (Ypq, Labc) and their field theory duals
Franco et. al., Benvenuti et. al.,
Butti et. al
Precision matching of geometry and gauge theory

4. solved the problem of finding the gauge theory on
Dimer models:
solved the problem of finding the gauge theory on
D-branes probing an arbitrary toric singularity.
Hanany and Kennaway - Franco, Hanany, Kennaway, Vegh and Wecht
Orientifolds add new possibilities
Interesting spectrum (matter representations and gauge groups)
Non-conformal theories
SUSY breaking without runaway
New interactions due to stringy instantons
Available techniques are very limited
Orbifolds
Non-orbifold singularities : partial resolution in a few cases and singularities with T-dual Hanany-Witten setups
Sebastian Franco
Need for tools to compute and classify orientifolds for D-branes over general singularities

5. D-branes over toric singularities
Toric CY3
N D3-branes
On the worldvolume of the D3-branes:
N=1 superconformal field theory
Gauge group: P U(N)
every field appears exactly twice
in W with opposite signs
(toric condition)
F-terms: monomial = monomial
Sebastian Franco
Toric singularities provide a simple, yet extremely rich set of geometries
and gauge theories. E.g.: orbifolds, dPn (n ≤ 3), Yp,q, La,b,c, etc

6. Combining quiver and superpotential data:
the periodic quiver
Periodic
quiver
planar quiver drawn on the surface of a 2-torus such that every plaquette corresponds to a term in the superpotential
Toric condition: every field belongs to two plaquettes
Sign of superpotential term: orientation of the plaquette
Example: complex cone over F0
Sebastian Franco

7. bifundamental (or adjoint)
Brane dimers
Periodic quiver
we consider the dual graph
Dimer graph
it is bipartite (orientation)
Gauge theory
Periodic quiver
Dimer graph
U(N) gauge group
node
face
bifundamental (or adjoint)
arrow
edge
superpotential term
plaquette
Sebastian Franco

8. Perfect matchings Perfect matching:
subset of edges such that every vertex in the graph
is an endpoint of precisely one edge in the set
Sebastian Franco
Perfect matchings are in one to one correspondence with GLSM fields. Their position in the toric diagram is given by the slope of the height function.
Franco and Vegh
Straightforward to count them using Kasteleyn matrix.

9. Orientifolds w s (-1)FL W - W
We study D3-branes probing a toric CY singularity in the presence of an orientifold quotient:
w s (-1)FL
w: worldsheet orientation reversal
s: involution of M
FL: left-moving fermion number
Orientifold planes
fixed point loci of s
O+-plane
RR charge
O- -plane
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To preserve a common SUSY with the D3-branes:
W W

10. Orientifolding dimers
Z2 identification in the dimer
Two classes of orientifolds:
Fixed points
Fixed lines
Fixed points: preserve U(1)2 mesonic flavor symmetry
Fixed lines: projects U(1)2 to a U(1) subgroup
Sebastian Franco
Fixed points and lines correspond to O-planes and come with signs that give rise to different orientifolds.
There is a global constraint on signs for orientifolds with fixed points.

11. Orientifold rules + + SO(N) Assign a sign to every orientifold point
O+/O-
O+/O- on edge project bifundamental to /
O+/O- on face projects gauge group to SO(N)/Sp(N/2)
Superpotential: project parent superpotential
These rules are consistent with all known examples and with partial resolution
proof by induction
SUSY: constrains sign parity to be (-1)k for dimers with 2k nodes
Example: orientifold of C3
Signs: (+,+,+,-)
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Gauge group
Matter
SO(N) + +

12. Tadpoles/anomalies and extra flavors
The resulting field theories are in general chiral
gauge anomalies
The system can be rendered consistent by choosing the ranks of gauge groups or by adding fundamental flavors (non-compact D7-branes)
Flavor D7-branes and dimers:
Franco and Uranga
Conifold:
D7-branes are in one-to-one correspondence with edges
Sebastian Franco
They introduce Qa, Qb and a cubic coupling:
2-index representation
Qa Xab Qb
X Q Q
(anti)fundamental
orientifold

13. Examples Orientifolds of C3 Orientifolds of C3/Z3
Sebastian Franco
All these spectra are anomalous unless the ranks of the gauge groups are restricted or (anti)fundamental matter is added.
For (-+++):

14. Examples
Orientifolds of SPP
Orientifolds of L1,5,2
Sebastian Franco

15. Geometric action Orientifold action: w s (-1)FL geometric action
Chan-Paton action
Look at action on mesonic operators (coordinates in moduli space). They correspond to closed paths on the dimer.
1) Every time a path crosses an O+/-, it gets a +/- sign.
2) Homologically trivial paths pick a (-) per enclosed node.
3) Path mapped to image (i.e. not on top of O-plane) picks a sign for each node and O- in the strip.
C2/Z2×C:
(1)
(2)
(3)
Sebastian Franco
Consistent with all known examples.

16. Orientifolds with fixed lines
Fixed lines consistent with two classes of unit cell:
two fixed lines
one fixed line
rectangle
rhombus
Assign a +/- sign to each fixed line (no constraint on signs)
O+/O- on edge project bifundamental to /
O+/O- on face projects gauge group to SO(N)/Sp(N/2)
Orientifold ofC3:
+ sign
SO(N) O7
Sebastian Franco
The geometric action is determined using mesonic operators.
Consistent with all known examples and partial resolution.

17. Examples Orientifolds of C2/ZN × C: even N
Orientifolds of C2/ZN × C: odd N
Sebastian Franco

18. Examples L131 L121 Orientifolds of Laba theories a+b even
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a+b even
rectangular unit cell
T-dual to HW with O4 or O8
a+b odd
rhombus unit cell
T-dual to HW with O8

19. The mirror perspective
The mirror geometry is a double fibration over the complex plane: C*
z = u v
z = P(x1,x2)
Riemann surface S
D3-branes on the CY singularity
D6-branes on special Lagrangian
3-cycles (1-cycles on S)
Intersections
matter
Feng, He, Kennaway
and Vafa
Disk instantons
superpotential terms
SPP:
Sebastian Franco
Different orientifolds follow from signs on O-plane branches.

20. Applications: SUSY breaking
Fractional branes on singularities lead to interesting IR dynamics:
Confinement
Runaway/meta-stable SUSY breaking
Orientifolds add new possibilities.
Ex.: SU(5) with
SUSY breaking vacuum
Affleck, Dine and Seiberg
Z6’
PdP4
Sebastian Franco
n0=1 n2=5 n1=n3=0
n5=1 n1=5 n2=n4=0
The U(1) in U(5) is anomalous and becomes massive.
Simple to generalize and classify.

21. Applications: D-brane instantons
Euclidean D-branes can induce superpotential terms:
Field theory instanton
depending on structure in
internal space
Stringy instanton
Requires only two uncharged zero-modes (there are generically four)
Project out additional zero-modes using O--plane
C3/(Z2×Z2)
conifold/Z5
Instanton
Argurio, Bertolini, Franco and Kachru
Sebastian Franco
Spacefilling
Argurio, Bertolini, Ferretti, Lerda and Petersson
Straightforward to use techniques to study stringy instantons generating superpotentials in general orientifold singularities.

22. Conclusions Further directions
Introduced dimer model techniques to study orientifolds of general toric singularities.
Most comprehensive clasification of such models
Previously known examples become trivial
Gave a flavor of some interesting applications:
SUSY breaking
D-brane instantons
Further directions
Systematic study of resulting field theories (for example SUSY breaking ones)
Sebastian Franco
Can we get further insight on SUSY breaking from geometric realization of theories?