1. Brane Tilings and New Horizons Beyond Them Calabi-Yau Manifolds, Quivers and Graphs Sebastián Franco Durham University Lecture 2

2. Sebastian Franco 2 Outline: Lecture 2  Brane Tilings as Physical Brane Configurations  Graphical QFT Dynamics  Orbifolds  Scale Dependence in QFT  Partial Resolution of Singularities and Higgsing  From Geometry to Brane Tilings  Orientifolds

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4. Brane Intervals  An alternative approach for engineering gauge theories using branes (dual to branes at singularities) 4 T D-brane ~ 1/g s T NS5 ~ 1/g s 2  The field theory lives in the common dimensions. In this case: 4d  The relative orientation of the branes controls the amount of SUSY NS NS’ NS NS’ 0123456789 D4 ××××× NS5 ×××××× NS5’ ×××××× 4,5 7,8 6 D4-branes NS5-branes N=2 SUSY N=1 SUSY Hanany, Witten

5. Sebastian Franco Brane Tilings  Brane tilings are a higher dimensional generalizations of this type of brane setups 5  The NS5-brane wraps a holomorphic curve  given by: Where x and y are complex variables that combine the x 4, x 5, x 6 and x 7 directions 0123456789 D5 ×××××× NS5 ××××  x4x4 x6x6 D5-branes NS5-brane Field theory dimensions P(x,y) = 0  P(x,y) is the characteristic polynomial coming from the toric diagram Franco, Hanany, Kennaway, Vegh, Wecht

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7. 7 Graphical Gauge Theory Dynamics P 1 (X i )P 2 (X i ) X1X1 X2X2 P 1 (X i ) ×P 2 (X i )  2-valent nodes map to mass terms in the gauge theory. Integrating out the corresponding massive fields results in the condensation of the two nearest nodes The equations of motion of the massive fields become: Massive Fields  We are mainly interested in the low energy (IR) limit of these theories Gauge Theory Dynamics Graph Transformations

8. Sebastian Franco Geometry and Seiberg Duality D3s  This is a purely geometric manifestation of Seiberg duality of the quivers! Full equivalence of the gauge theories in the low energy limit 8 Brane Tiling (Gauge Theory) Calabi-Yau 3-fold  What happens if this map is not unique? Quiver 1Quiver 2 F0F0 Feng, Franco, Hanany, He Franco, Hanany, Kennaway, Vegh, Wecht

9. Sebastian Franco Geometry and Seiberg Duality  For the F 0 example, the two previous quivers theories correspond to the following brane tilings  Seiberg duality corresponds to a local transformation of the graph: Urban Renewal Theory 2Theory 1  Seiberg duality is a fascinating property of SUSY quantum field theories. Sometimes, it allows us to trade a strongly coupled one for a weakly coupled, and hence computable, dual 9 Franco, Hanany, Kennaway, Vegh, Wecht

10. Sebastian Franco Geometry and Seiberg Duality  Seiberg dualizing twice, takes us back to the original theory  The Calabi-Yau geometry is automatically invariant under this transformation CY Invariance Cluster Transformation Seiberg Duality  From the perspective of the dual quiver, this corresponds to a quiver mutation SD 1 SD 2 We have generated massive fields and can integrate them out 10

11. Sebastian Franco Dual Phases of del Pezzo 3 1 2 2 2 2 3 3 3 4 4 5 5 5 6 6 1 1 4 6 1 4 6 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 3 31 1 4 4 4 4 6 66 6 2 22 2 55 5 55 5 56 6 2 2 2 2 3 33 3 41 4141 4141

12. Sebastian Franco Orbifolds  Generate new geometries and gauge theories from known ones GeometryGauge Theory  At the level of the quiver, it basically amounts to adding images for gauge groups and fields and projecting the superpotential onto invariant terms X Y Z 2  /3 Orbifolds 4 1 2 3 5

13. Sebastian Franco Orbifolds and Brane Tilings 13 7897 1231 4564 7897 11 11 3 3 33 3 3 21  The explicit action of the orbifold group maps to the choice of periodicity on the torus  We can orbifold arbitrary geometries, by taking the corresponding brane tilings as starting points X Y Z W = [X,Y] Z N=4 super Yang-Mills 1 2 9 5 8 6 7 3 4

14. Sebastian Franco Partial Resolution and Higgsing  Replacing points by 2-spheres and sending their size to infinity 14  Eliminating points in the toric diagram Partial Resolution Cone over dP 2 Cone over dP 1 U(N) × U(N) U(N) d  In the brane tiling, it corresponds to removing edges and merging faces 1 2 3 45 1 2 3 45 1 2 3 45 1 2 3 45 1 2 3 4/5 1 2 3 1 2 3 1 2 3  Removing and edge corresponds to giving a non-zero vacuum expectation value to a bifundamental field Higgs Mechanism  Example: Franco, Hanany, Kennaway, Vegh, Wecht

15. Sebastian Franco 1 1 2 2 3 3 2 2 3 3 p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 X 11 11 X 12 11 X 21 11 X 23 1 X 32 1 X 31 11 X 13 11 P = 1 1 2/3 1 1 2 2 3 3 2 2 3 3 p1p1 p 2, p 3 p4p4 p5p5 p6p6 Suspended Pinch Point p1p1 p 2, p 3 p4p4 p5p5 Remove p 6 Possible partial resolutions = possible sub-toric diagrams Remove X 23 12/3 2) Conifold Remove p 1 Also remove e.g. p 2 Remove e.g. X 12 1/23 1 1 2 2 3 3 2 2 3 3 3 3 3 3 p2p2 p4p4 p5p5 p6p6

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17. Sebastian Franco Running Couplings in QFT in QFT, coupling constants generically depend on the energy scale  (they run) 17 Standard Model log 10  (GeV)  i -1 5101520 0 60 80 40 20 0 U(1) SU(2) SU(3)  Remarkably, in SUSY field theories we know exact expressions for the  -functions: R i : superconformal R- charge of chiral multiplets  The models we will study are strongly coupled Superconformal Field Theories (SCFTs). This implies they are independent of the energy scale Renormalization Group  The running of any coupling is controlled by its  -function:

18. Sebastian Franco Geometry of the Tiling and Conformal Invariance  In a SCFT, the beta functions for all superpotential and gauge couplings must vanish. When all ranks are equal:  Conformal invariance constraints the geometry of the tiling embedding 18 Superpotential couplings For every node: Gauge couplings For every face: N faces + N nodes - N edges =  (  ) = 0  We will focus on the torus. It would be interesting to investigate whether bipartite graphs on the Klein bottle have any significance in String Theory  We conclude that conformal invariance requires the tiling to live on either a torus or a Klein bottle Summing over the entire tiling Franco, Hanany, Kennaway, Vegh, Wecht

19. Sebastian Franco Isoradial Embedding and R-charges  The vanishing of the beta functions now becomes: 19 Superpotential couplings For every node: Gauge couplings For every face:  R-charges can be traded for angles in the isoradial embedding Isoradial Embedding: every face of the brane tiling is inscribed in a circle of equal radius

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21. Sebastian Franco Zig-Zag Paths  They can be efficiently implemented using a double line notation (alternating strands) 21 oriented paths on the tiling that turn maximally left at white nodes and maximally right at black nodes Zig-Zag Paths Feng, He, Kennaway, Vafa  Example: F 0 clockwise/counterclockwise around white/black nodes  They provide an alternative way for connecting brane tilings to geometry every intersection gives rise to an edge

22. Sebastian Franco Brane Tilings from Geometry  Question: given a toric diagram, how do we determine the corresponding brane tiling(s)? 22  Answer: the vectors normal to the external faces of the toric diagram determine the homology of zig-zag paths in the brane tiling Hanany, Vegh (1,1) (1,-1) (-2,1) (0,-1) 12 34 Del Pezzo 1  Seiberg duality corresponds to relative motion of the zig-zag paths 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 Feng, He, Kennaway, Vafa

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24. Sebastian Franco Orientifolds 24  Dimer models solve the problem of finding the gauge theory on D-branes probing an arbitrary toric Calabi-Yau 3-fold singularity Quotient by the action of:   : worldsheet orientation reversal (in the quiver, it conjugates the head or tail of arrows)   : involution of the Calabi-Yau  F L : left-moving fermion number  At the level of the gauge theory, it adds new possibilities:  New representations for fields: e.g. symmetric and antisymmetric  New gauge groups: symplectic and orthogonal Orientifold Projection  (-1) F L  The correspondence can be extended to more general geometries  Orientifold Planes: fixed point loci of . Closed cousins of D-branes

25. Sebastian Franco Orientifolding Dimers 25  There are two classes of orientifolds: Fixed pointsFixed lines  Fixed points: preserve U(1) 2 mesonic flavor symmetry  Fixed lines: projects U(1) 2 to a U(1) subgroup  Fixed points and lines correspond to orientifold planes and come with signs that determine their type  There is a global constraint on signs for orientifolds with fixed points Orientifolding Franco, Hanany, Krefl, Park, Vegh

26. Sebastian Franco Orientifold Rules: Fixed Points 26 Gauge groupMatter Signs: (+,+,+,-) SO(N) ++  Superpotential: project parent superpotential  Supersymmetry constrains sign parity to be (-1) k for dimers with 2k nodes  O + /O - on face projects gauge group to SO(N)/Sp(N/2)  O + /O - on edge project bifundamental to /  Assign a sign to every orientifold pointO + /O -

27. Sebastian Franco Examples 27  All these theories contain gauge anomalies unless the ranks of the gauge groups are restricted or (anti)fundamental matter is added. For (-,+,+,+):

28. Sebastian Franco Examples 28  Orientifolds of L 1,5,2  Orientifolds of SPP