1 / 28

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.

christoffer-carus
Download Presentation

Brane Tilings and New Horizons Beyond Them

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related