1. SUSY Breaking in D-brane Models Beyond Orbifold Singularities Sebastián Franco Durham University José F. Morales INFN - Tor Vergata

2. Sebastian Franco Why 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

3. Sebastian Franco Quivers from Geometry …  On the worldvolume of D-branes probing Calabi-Yau singularities we obtain quiver gauge theories Example: Cone over dP 3 1 2 3 45 6 W = X 12 X 23 X 34 X 56 X 61 - X 12 X 24 X 45 X 51 - X 23 X 35 X 56 X 62 - X 34 X 46 X 62 X 23 + X 13 X 35 X 51 + X 24 X 46 X 62 … 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

4. Sebastian Franco Toric Calabi-Yau Cones Toric Varieties  Admit a U(1) d action, i.e. T d 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 T 2 fibrations over the complex plane (p,q) WebToric Diagram 4 Complex plane 2-sphere (1,0) (1,-1)(-1,-1) (-1,2) Cone over del Pezzo 1 2-cycle 4-cycle

5. Sebastian Franco Quivers from Toric Calabi-Yau’s  We will focus on the case in which the Calabi-Yau 3-fold is toric 5 D3s Toric CY  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

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7. Periodic Quivers  It is possible to introduce a new object that combines quiver and superpotential data W = X 11 13 X 2 32 X 2 21 - X 12 13 X 2 32 X 1 21 - X 21 13 X 1 32 X 2 21 + X 22 13 X 1 32 X 1 21 - X 11 13 X 2 34 X 2 41 + X 12 13 X 2 34 X 1 41 + X 21 13 X 1 34 X 2 41 - X 22 13 X 1 34 X 1 41 12 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 7 = F-term eq.: X 2 34 X 2 41 = X 2 32 X 2 21 Unit cell

8. Sebastian Franco Brane 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 TheoryPeriodic QuiverDimer U(N) gauge groupnodeface bifundamental (or adjoint) arrowedge superpotential termplaquettenode 3 1 1 1 1 2 2 4 4 8

9. Sebastian Franco Perfect Matchings  Perfect matching: configuration of edges such that every vertex in the graph is an endpoint of precisely one edge p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 p9p9 p8p8 9  Perfect matchings are natural variables parameterizing the moduli space. They automatically satisfy vanishing of F-terms Franco, Hanany, Kennaway, Vegh, Wecht Franco, Vegh (n 1,n 2 ) (0,0) (1,0)(0,1)

10. Sebastian Franco 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 P 1 (X i )P 2 (X i ) X0X0 =  For any arrow in the quiver associated to an edge in the brane tiling X 0 : Graphically:  This parameterization automatically implements the vanishing F-terms for all edges!  Consider the following map between edges X i and perfect matchings p  :

11. Sebastian Franco 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) p 1, p 2, p 3, p 4, p 5 p8p8 p6p6 p9p9 p7p7 K = white nodes black nodes Kasteleyn MatrixToric Diagram det K = P(z 1,z 2 ) =  n ij z 1 i z 2 j Example: F 0 Franco, Hanany, Kennaway, Vegh, Wecht Franco, Vegh

12. Sebastian Franco 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

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14. Sebastian Franco 1. 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 branesAnomaly-free rank assignments ii+1i-1 W = X i-1,i X i,i+1 X i+1,i X i,i-1 + … ii+1i-1 W = X i,i-1 X i-1,i  i,i -  i,i X i,i+1 X i+1,i + …  Using Seiberg duality, two possible types of nodes: Aharony, Kachru, Silverstein NS NS’ D4

15. Sebastian Franco 15 PolonyiFayet W =  1 X 23 X 32 110 1 2 3 X23X23 X32X32    X 23 and X 32 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 =  1 2 X 10 1 2    General Strategy: consider wrapped D-instanton over orientifolded empty node

16. Sebastian Franco 2. 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. G 5 × U(n 1 ) × U(n 2 ) × U(n 4 )  For n 1 = n 4 = 0, n 5 = 1, n 1 = 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 55 55 3 3 7 6 4 2 1 1 Example: PdP 4

17. Sebastian Franco 3. 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 N f = N c 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) S3S3 conifold

18. Sebastian Franco  This theory dynamically breaks SUSY with a runaway Fractional Branes  Deformation  N=2 3M M 2M  Admits fractional branes and a duality cascade but no complex deformation Franco, Hanany, UrangaEjaz, Klebanov, Herzog  Dynamical SUSY breaking (due to ADS superpotential) Franco, Hanany, Saad, Uranga Berenstein, Herzog, Ouyang, Pinansky Bertolini, Bigazzi, CotroneIntriligator, Seiberg (-1,2) (1,-1)(-1,-1) (1,0) IR bottom of cascade Example: dP 1

19. Sebastian Franco 4. 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 N f < N c gauge group to bring it to the free-magnetic range 3M M 2M  SU(3M) with 2M massless flavors  D7-branes

20. Sebastian Franco 5. 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  1 »  3 »  2 regime: Node 1 has N c =N f 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 = X 21 X 12 X 23 X 32  The gauge theory on node 3 becomes an ISS model with dynamically generated masses. Metastability of the vacuum requires P=M. det M – BB =  1 2M W = M X 23 X 32 W =  M  X 23 X 32 MPM 1 2 3

21. Sebastian Franco Conclusions 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

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