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Brane Tilings and New Horizons Beyond Them

Brane Tilings and New Horizons Beyond Them. Lecture 2. Calabi-Yau Manifolds, Quivers and Graphs. Sebastián Franco. Durham University. Outline: Lecture 2. Brane Tilings as Physical Brane Configurations. Graphical QFT Dynamics. Orbifolds.

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Brane Tilings and New Horizons Beyond Them

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

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

  3. BraneTilings as Physical Brane Configurations Sebastian Franco

  4. Brane Intervals • An alternative approach for engineering gauge theories using branes (dual to branes at singularities) • Hanany, Witten D4-branes • 4,5 • 6 • 7,8 NS5-branes NS NS NS NS N=2 SUSY N=1 SUSY • TD-brane ~ 1/gs NS’ NS’ • TNS5 ~ 1/gs2 NS NS Sebastian Franco • The field theory lives in the common dimensions. In this case: 4d • The relative orientation of the branes controls the amount of SUSY

  5. BraneTilings • Branetilings are a higher dimensional generalizations of this type of brane setups • Franco, Hanany, Kennaway, Vegh, Wecht x6 • D5-branes • NS5-brane x4 • Field theory dimensions Sebastian Franco • The NS5-brane wraps a holomorphic curve S given by: • Where x and y are complex variables that combine the x4, x5, x6 and x7 directions • P(x,y) = 0 • P(x,y) is the characteristic polynomial coming from the toric diagram

  6. QFT Dynamics, Tilings and Geometry Sebastian Franco

  7. Graphical Gauge Theory Dynamics • Gauge Theory Dynamics • Graph Transformations • We are mainly interested in the low energy (IR) limit of these theories Massive Fields • 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 X1 X2 P1(Xi) × P2(Xi) P1(Xi) P2(Xi) Sebastian Franco • The equations of motion of the massive fields become:

  8. Geometry and Seiberg Duality • Feng, Franco, Hanany, He Franco, Hanany, Kennaway, Vegh, Wecht • Brane Tiling • (Gauge Theory) • Calabi-Yau 3-fold • What happens if this map is not unique? • Quiver 1 • Quiver 2 D3s • F0 Sebastian Franco • This is a purely geometric manifestation of Seiberg duality of the quivers! Full equivalence of the gauge theories in the low energy limit

  9. Geometry and Seiberg Duality • 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 • For the F0 example, the two previous quivers theories correspond to the following branetilings • Theory 2 • Theory 1 • Seiberg duality corresponds to a local transformation of the graph: Urban Renewal • Franco, Hanany, Kennaway, Vegh, Wecht Sebastian Franco

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

  11. Dual Phases of del Pezzo 3 1 4 1 4 1 4 1 1 5 2 3 6 6 4 4 1 1 1 5 2 3 2 1 4 1 4 4 4 6 6 3 1 5 2 1 4 3 5 2 6 3 4 2 5 2 5 1 6 3 5 6 1 3 3 4 4 2 2 5 5 Sebastian Franco 2 3 6 2 4 5 2 3 6 2 4 1 1 6 6 3 4 1 6 5 3 5 2 1 6 5 2 6 3 2 4 5

  12. Orbifolds • Quotienting by a discrete group such as Nor N × M • Generate new geometries and gauge theories from known ones • Orbifolds • Example:N orbifolds of correspond to identifications under rotations by multiples of 2p/N on each plane Geometry • 3 • 2p/3 Gauge Theory Y • At the level of the quiver, it basically amounts to adding images for gauge groups and fields and projecting the superpotential onto invariant terms 1 X Z • : N=4 SYM • 5 Sebastian Franco 5 2 4 3

  13. Orbifolds and BraneTilings • From a branetiling perspective, the M× N orbifold corresponds to enlarging the unit cell to include M × N copies of the original one • 3× 3) 7 8 9 7 1 2 3 1 4 5 6 4 3 • N=4 super Yang-Mills 3 Y 1 7 8 9 7 21 3 3 9 2 X Z 8 3 3 3 • W = [X,Y]Z Sebastian Franco 4 7 1 1 5 6 1 1 • 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 branetilings as starting points

  14. Partial Resolution and Higgsing • Replacing points by 2-spheres and sending their size to infinity • Eliminating points in the toric diagram • Partial Resolution • Example: • Cone over dP2 • Cone over dP1 • In the brane tiling, it corresponds to removing edges and merging faces 2 2 2 2 2 2 2 2 • Franco, Hanany, Kennaway, Vegh, Wecht 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 5 5 5 5 4/5 4/5 4/5 4/5 4 4 4 4 Sebastian Franco • U(N)d • U(N) × U(N) • Removing and edge corresponds to giving a non-zero vacuum expectation value to a bifundamental field Higgs Mechanism

  15. Any toric geometry can be obtained by partial resolution of a M × N) orbifold for sufficiently large M and N Suspended Pinch Point 2 2 2 2/3 3 3 3 2 2 2 2/3 • P = 1 1 1 1 3 3 3 2 2 2 2/3 1 1 1 1 3 3 3 2 2 2 2/3 3 3 3 • Possible partial resolutions = possible sub-toric diagrams • Remove X23 • Also remove e.g. p2 • Remove e.g. X12 • 1)2 × • 2)Conifold • Remove p6 • Remove p1 • 1 • 1/2 • 2/3 • 3 p5 p5 p5 p6 p6 Sebastian Franco 1/2 3 1/2 p1 p1 p2, p3 p2, p3 p2 p4 p4 p4 3 1/2 3 1/2 3 1/2

  16. Scale Dependence in QFT Sebastian Franco

  17. Running Couplings in QFT • in QFT, coupling constants generically depend on the energy scale L (they run) • Renormalization Group • Standard Model • ai-1 • 80 • 20 • 40 • 60 • U(1) • SU(2) • SU(3) • 0 • 15 • 10 • 5 • 20 • 0 • log10 L (GeV) • The running of any coupling l is controlled by its b-function: • Remarkably, in SUSY field theories we know exact expressions for the b-functions: • Gauge couplings (NSVZ): Sebastian Franco • Ri: superconformal R-charge of chiral multiplets • Superpotential couplings: • The models we will study are strongly coupled Superconformal Field Theories (SCFTs). This implies they are independent of the energy scale

  18. Geometry of the Tiling and Conformal Invariance • Franco, Hanany, Kennaway, Vegh, Wecht • Conformal invariance constraints the geometry of the tiling embedding • In a SCFT, the beta functions for all superpotential and gauge couplings must vanish. When all ranks are equal: • Superpotential couplings • For every node: • Gauge couplings • For every face: • Summing over the entire tiling Sebastian Franco • We conclude that conformal invariance requires the tiling to live on either a torus or a Klein bottle • Nfaces+ Nnodes- Nedges= () = 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

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

  20. From Geometry to BraneTilings Sebastian Franco

  21. Zig-Zag Paths • oriented paths on the tiling that turn maximally left at white nodes and maximally right at black nodes • Zig-Zag Paths • They can be efficiently implemented using a double line notation (alternating strands) • Feng, He, Kennaway, Vafa • Example: F0 • clockwise/counterclockwise around white/black nodes • every intersection gives rise to an edge Sebastian Franco • They provide an alternative way for connecting branetilings to geometry

  22. BraneTilings from Geometry • Question: given a toric diagram, how do we determine the corresponding branetiling(s)? • 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 • Feng, He, Kennaway, Vafa Del Pezzo 1 • (1,1) • (-2,1) 4 4 1 1 • (1,-1) 2 2 3 3 • (0,-1) • 1 • 2 4 4 1 1 Sebastian Franco 2 2 3 3 • 4 • 3 • Seiberg duality corresponds to relative motion of the zig-zag paths

  23. Applications and Extensions: Orientifolds Sebastian Franco

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

  25. Orientifolding Dimers • Franco, Hanany, Krefl, Park, Vegh 2 identification in the dimer • Orientifolding • There are two classes of orientifolds: Fixed points Fixed lines • Fixed points: preserve U(1)2mesonic flavor symmetry • Fixed lines: projects U(1)2 to a U(1) subgroup Sebastian Franco • 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

  26. Orientifold Rules: Fixed Points • 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 • Supersymmetryconstrains sign parity to be (-1)k for dimers with 2k nodes Orientifold of Signs:(+,+,+,-) Sebastian Franco SO(N) + +

  27. Examples • Orientifolds of • Orientifolds of 3 Sebastian Franco • All these theories contain gauge anomalies unless the ranks of the gauge groups are restricted or (anti)fundamental matter is added. For (-,+,+,+):

  28. Examples • Orientifolds of SPP • Orientifolds of L1,5,2 Sebastian Franco

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