Dimer models and orientifolds Sebastián Franco Princeton University September 2007 Based on: arXiv:0707.0298: Franco, Hanany, Krefl, Park, Uranga and Vegh

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

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

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

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

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

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

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.

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 Sebastian Franco To preserve a common SUSY with the D3-branes: W - W

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.

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: (+,+,+,-) Sebastian Franco Gauge group Matter SO(N) + +

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

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 (-+++):

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

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.

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 2 + The geometric action is determined using mesonic operators. Consistent with all known examples and partial resolution.

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

Examples L131 L121 Orientifolds of Laba theories a+b even Sebastian Franco 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

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.

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 10 + 5 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.

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.

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?