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SUSY Breaking in D-brane Models

This paper discusses the use of D-branes at singularities to study string phenomenology, UV completion, gauge symmetry, matter content, superpotential, and the AdS/CFT correspondence. It also explores new tools for dealing with strongly coupled quantum field theories in terms of weakly coupled gravity. The paper includes examples and applications of quivers, toric Calabi-Yau cones, periodic quivers, and dimer models.

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SUSY Breaking in D-brane Models

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

  2. 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. New tools for dealing with strongly coupled QFTsin terms of weakly coupled gravity Sebastian Franco • QFTs dynamics gets geometrized: • Duality • Confinement • Dynamical supersymmetry breaking

  3. Quivers from Geometry … • On the worldvolume of D-branes probing Calabi-Yau singularities we obtain quiver gauge theories • 1 • 2 • W = X12X23X34X56X61 - X12X24X45X51 - X23X35X56X62 • - X34X46X62X23 + X13X35X51 + X24X46X62 Calabi-Yau • 6 • 3 • Example: Cone over dP3 Quiver • 5 • 4 … and Geometry from Quivers CY Sebastian Franco • Starting from the gauge theory, we can infer the ambient geometry by computing its moduli space D3s

  4. Toric Calabi-YauCones • Admit a U(1)daction, i.e. Tdfibrations • 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 • (-1,2) • (p,q) Web • Toric Diagram Sebastian Franco • 4-cycle • (1,0) • 2-cycle • (-1,-1) • (1,-1)

  5. 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 • The resulting quivers have a more constrained structure: • Toric Quivers The F-term equations are of the form monomial = monomial • Recall F-term equations are given by: Toric CY Sebastian Franco D3s • The superpotential is a polynomial and every arrow in the quiver appears in exactly two terms, with opposite signs

  6. BraneTilings Sebastian Franco

  7. 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 • 1 • 2 • Example: Conifold/2 (cone over F0) • Unit cell • 4 • 3 = • F-term eq.: Sebastian Franco • 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

  8. BraneTilings • Franco, Hanany, Kennaway, Vegh, Wecht • Take the dual graph • It is bipartite (chirality) • Periodic Quiver • Dimer Model • 4 • 1 • 1 • 2 • 2 • 3 • 1 • 1 • 4 • In String Theory, the dimer model is a physical configuration of branes Sebastian Franco

  9. 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) • p1 • p2 • p3 • p4 • p5 • (1,0) • (0,1) • p6 • p7 • p8 • p9 Sebastian Franco • Perfect matchings are natural variables parameterizing the moduli space. They automatically satisfy vanishing of F-terms • Franco, Vegh • Franco, Hanany, Kennaway, Vegh, Wecht

  10. Solving F-Term Equations via Perfect Matchings • Themoduli space of any toric quiver is a toricCY and perfect matchings simplify its computation • For any arrow in the quiver associated to an edge in the brane tiling X0: • Graphically: X0 = • Consider the following mapbetween edges Xi and perfect matchings pm: P1(Xi) P2(Xi) Sebastian Franco 1 if • This parameterization automatically implements the vanishing F-termsfor all edges! = 0 if

  11. Perfect Matchings and Geometry • There is a one to one correspondence between perfect matchingsand 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 • p8 white nodes • p1, p2, p3, p4, p5 black nodes • K = • p6 • p7 Sebastian Franco • Example: F0 • det K = P(z1,z2) =  nij z1i z2j • p9

  12. 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: • Flavors D7-branes • Franco, Uranga - Franco, Uranga • 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 • Krippendorf, Dolan, Maharana, Quevedo • Krippendorf, Dolan, Quevedo Sebastian Franco • Other Directions: mirror symmetry, crystal melting, cluster algebras, integrable systems

  13. The SUSY Breaking ZOO Sebastian Franco

  14. 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’ D4 NS’ NS • Conifold/3 NS • Fractional branes • Anomaly-free rank assignments • Using Seiberg duality, two possible types of nodes: Sebastian Franco • i-1 • i-1 • i • i • i+1 • i+1 • W = Xi-1,i Xi,i+1Xi+1,i Xi,i-1 + … • W = Xi,i-1 Xi-1,ifi,i- fi,iXi,i+1 Xi+1,i + …

  15. General Strategy: consider wrapped D-instanton over orientifolded empty node Polonyi Fayet • X23 • a • a Sebastian Franco • 2 • 2 • 3 • 1 • 1 • X32 • b • b • 0 • 0 • 1 • 1 • 1 • X23and X32 are neutral under U(1)(2) + U(1)(3) • W = L1X23 X32 • W = L12X • SUSY is broken once we turn on an FI term for U(1)(2) – U(1)(3)

  16. 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 2 5 5 • Controlled by signs of fixed points • Example:PdP4 Sebastian Franco • 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.

  17. 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 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 subwebsin equilibrium • (-1,1) • (0,1) S3 Sebastian Franco • (-1,0) • (1,0) conifold • (0,-1) • (1,-1)

  18. Deformation • Fractional Branes • N=2 • Dynamical SUSY breaking (due to ADS superpotential) • Franco, Hanany, Saad, Uranga 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 • 2M Sebastian Franco • (-1,-1) • (1,-1) • IR bottom of cascade • This theory dynamically breaks SUSY with a runaway Berenstein, Herzog, Ouyang , Pinansky Franco, Hanany, Saad ,Uranga Bertolini, Bigazzi, Cotrone Intriligator, Seiberg

  19. 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 • M • 2M Sebastian Franco • 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

  20. 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 • W = X21X12X23 X32 • 1 • 2 • 3 • Consider P  M  3/2 P . In the L1» L3» L2regime: Node 1 has Nc=Nf Quantum Moduli Space det M – BB = L12M Sebastian Franco On the mesonic branch: • W = M X23 X32 • W = M X23 X32 • M • M • P • The gauge theory on node 3 becomes an ISS model with dynamically generated masses. Metastability of the vacuum requires P=M.

  21. 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 • Local D-brane models lead to a wide range of SUSY breaking theories, from retrofitted simple models to geometrized dynamical SUSY breaking Sebastian Franco

  22. Thank you! Sebastian Franco

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