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

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Sebastian Franco 2 Outline: Lectures 1, 2 and 3 Introduction and Motivation Supersymmetric Gauge Theories Quantum Field Theory and Geometry Cluster Integrable Systems Bipartite Field Theories Recent Developments Brane Tilings

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Sebastian Franco QFT from D-branes and Geometry 3 D3s Lives in 10 dimensions Quantum field theories arise on the worldvolume of D-branes 6 3+1 # of dimensions 5d Sasaki-Einstein manifold Calabi-Yau 3-fold Quantum Field Theory Contains membrane-like objects (submanifolds) called Dp-branes (e.g. D3-branes: 3 space and 1 time dimensions) String Theory The geometry of the extra dimensions around D-branes determines the structure of the quantum field theories living on them 10 + D3 a D3 b D3 c Fractional brane

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Sebastian Franco Motivation 4 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

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Sebastian Franco 5

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N=1 Supersymmetric Gauge Theories Supersymmetry (SUSY): symmetry relating bosons (integer spin) and fermions (semi- integer spin) 6 Complex scalar (spin 0) + Weyl spinor (spin 1/2) + auxiliary field F Chiral Superfields matter : electrons, quarks, Higgs, etc and their superpartners Gauge bosons (spin 1) + gauginos (spin 3/2) + auxiliary field D Vector Superfields forces : electromagnetism, strong interactions, etc SUSY QFTs It is possible to construct supersymmetric quantum field theories There are several reasons motivating its study, e.g.: phenomenology, fundamental ingredient of String Theory and exact results in quantum field theory (QFT) Superfields Fields Superfields efficiently package ordinary fields with different spin

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Sebastian Franco N=1 Supersymmetric Gauge Theories 7 A SUSY QFT is fully specified, i.e. we can write down its Lagrangian, by providing: In particular, the scalar potential is given by: The list of chiral superfields and how they are charged under the gauge symmetries (vector superfields) The superpotential W(X i ) Holomorphic, gauge invariant function of chiral superfields X i Superpotential Charges, assuming all gauge groups are U(1) F-terms D-terms F-terms: a contribution from each chiral superfield X i D-terms: a contribution from each gauge group U(1) a

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Sebastian Franco Supersymmetric Vacua Two qualitative possibilities for the scalar potential: V( ) V( ) This is what we want. In SUSY theories, typically not an isolated point, but a moduli space of vacua Moduli Space (V = 0)

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Sebastian Franco 9

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Quivers from Geometry The quantum field theories on D-branes probing Calabi-Yau singularities have the following structure: 10 In this talk, we will mainly focus on the case in which nodes are U(1) groups A quiverWith a superpotential Oriented graph Holomorphic function of closed, oriented loops (gauge invariance) 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 = U(N) gauge group = bifundamental (matrix) chiral multiplet complex variable X ij electromagnetism, etc electron, quark, etc

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Sebastian Franco … and Geometry from Quivers (for Physicists) 11 D3s CY Quiver Calabi-Yau If we are confined to the D-branes, we can infer the ambient geometry by computing the moduli space of the quiver gauge theory The moduli space corresponds to vanishing of the scalar potential. For a quiver with U(1) gauge groups, this means: F-terms: For every arrow X i in the quiver D-terms: For every node U(1) a in the quiver

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Sebastian Franco … and Geometry from Quivers (for Mathematicians) 12 Path algebra of the quiver: multiplication given by path concatenation Subject to the ideal of relations coming from the superpotential We can also think about the superpotential as an efficient way of encoding relations Closed loops (i.e. gauge invariant operators) subject to relations (i.e. F-terms = 0) × = 123 X 12 123 X 23 123 X 12 X 23 The relations identify paths in the quiver with common endpoints Moduli Space Center of the Algebra Superpotential Algebra of the Quiver

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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 13 Complex plane 2-sphere (1,0) (1,-1)(-1,-1) (-1,2) Cone over del Pezzo 1 2-cycle 4-cycle

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Sebastian Franco Quivers from Toric Calabi-Yaus We will focus on the case in which the Calabi-Yau 3-fold is toric 14 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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Sebastian Franco Quivers from Toric Calabi-Yaus The Calabi-Yau singularity fractionizes the D3-branes 15 In addition to D3-branes, Type IIB string theory contains D-branes of other dimensionalities, such as D5 and D7-branes. By wrapping them over vanishing 2 or 4- cycles we also obtain 3+1 dimensional objects D3s CY D5s CY D7s CY 4-cycle 2-cycle Gauge groups (nodes in quiver) Fractional branes Fractional branes: bound states of D3, D5 and D7-branes The number of gauge groups in the quiver is equal to the number of fractional branes, which is given by the Euler characteristic of the Calabi-Yau: = b 0 + b 2 + b 4

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Sebastian Franco 16

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Sebastian Franco 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 17 = Relation: X 2 34 X 2 41 = X 2 32 X 2 21 Unit cell

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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 18

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Sebastian Franco Anomaly Cancellation Chiral theories can contain gauge anomalies. Anomalies must vanish in order for the quantum theories to be consistent When the ranks of all faces are equal, the theories are automatically anomaly-free More general rank assignments can be solutions to the anomaly cancellation conditions. They corresponds to different types of wrapped D-branes Faces are even-sided polygons Alternating orientation of arrows In order for a theory to be anomaly-free, every gauge group must have the same number of fields in the fundamental and antifundamental representations Equal number of incoming and outgoing arrows at every face

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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 20 Perfect matchings are natural variables parameterizing the moduli space. They automatically implement the relations in the quiver path algebra. Franco, Hanany, Kennaway, Vegh, Wecht Franco, Vegh (n 1,n 2 ) (0,0) (1,0)(0,1)

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Sebastian Franco 21 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 relations from W for all edges! Consider the following map between edges X i and perfect matchings p :

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Sebastian Franco From Edges to Perfect Matchings: the F 0 Example p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 p8p8 p9p9 X 13 111 Y 13 111 Z 13 111 W 13 111 X 32 111 Y 32 111 X 21 111 Y 21 111 X 34 111 Y 34 111 X 41 111 Y 41 111 1 3 22 4 43 33 X 13 Y 13 Z 13 W 13 X 34 Y 34 X 41 Y 41 X 21 Y 21 X 32 Y 32 Let us compute the perfect matching matrix for an explicit example P =

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Sebastian Franco Efficient Determination of Perfect Matchings Franco, Hanany, Kennaway, Vegh, Wecht Franco, Vegh Finding the perfect matchings of a given brane tiling reduces to calculating the determinant of its Kasteleyn matrix The determination of the moduli space of a quiver theory, i.e. its associated Calabi-Yau geometry) becomes a combinatorial problem of bipartite graphs K = black nodes white nodes det K = P(z 1,z 2 ) = a mn (X ij ) z 1 m z 2 n Weighted adjacency matrix of the brane tiling Kasteleyn Matrix P(z 1,z 2 ): characteristic polynomial 23

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Sebastian Franco The Kasteleyn Matrix in an Explicit Example: F 0 24 p 1, p 2, p 3 p 4, p 5 p8p8 p6p6 p9p9 p7p7 z1z1 z2z2 1 3 22 4 43 33 X 13 Y 13 Z 13 W 13 X 34 Y 34 X 41 Y 41 X 21 Y 21 X 32 Y 32 1 2 3 4 5 6 7 8 z2z2 z 2 -1 z1z1 z 1 -1 5678 1 X 41 X 34 z 2 X 13 2 Y 13 Y 21 X 32 z 1 3 Y 34 z 2 -1 Z 13 Y 41 4 Y 32 z 1 -1 W 13 X 21 K = Perfect matchings indeed correspond to the points in the toric diagram of the Calabi-Yau 3-fold Q: charge matrix encoding linear relations in the toric diagram P(z 1,z 2 ) = det K = (X 21 X 41 Y 21 Y 41 – X 13 Y 13 Z 13 W 13 – X 32 X 41 Y 32 Y 41 – X 21 X 34 Y 21 Y 34 + X 32 X 34 Y 32 Y 34 ) + X 32 X 41 W 13 Z 13 z 1 + X 13 Y 13 Y 32 Y 41 z 1 -1 + X 13 Y 21 Y 34 W 13 z 2 -1 + X 21 X 34 Y 13 Z 13 z 2

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Sebastian Franco Example 1: Cone over del Pezzo 2 25 1 2 2 3 3 3 4 4 5 5 1 5 1 35 11 Toric diagram N faces + N nodes - N edges = 0N gauge + N superpot - N fields = 0 1) The number of faces in the brane tiling (gauge groups in the quiver) is equal to the Euler characteristic of the Calabi-Yau 3-fold For toric Calabi-Yaus: = # triangles in toric diagram 2) Since brane tilings live on a 2-torus: In this example: 5 + 8 - 13 = 0

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Sebastian Franco Example 2: Cone over del Pezzo 3 26 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 Toric diagram = N gauge = 6 N gauge + N superpot - N fields = 6 +12 - 18 = 0 Some general features:

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Sebastian Franco 27 In recent years, metrics for infinite families of 5d Sasaki-Einstein manifolds were constructed. These manifolds are denoted Y p,q and L a,b,c and have S 2 × S 3 topology. Volumes of the SE manifold and lower dimensional cycles were reproduced by non- perturbative computations in the quantum field theories The cones over these manifolds are toric Calabi-Yau 3-folds (-1,p-q) (0,p) (0,0)(1,0) The gauge theories for these infinite classes of CYs were determined using brane tilings Benvenuti, Franco, Hanany, Martelli, Sparks Gauntlett, Martelli, Sparks, Waldram Franco, Hanany, Martelli, Sparks, Vegh, Wecht Cvetic, Lu, Page, Pope p-q2q Example 3: Infinite Families of Calabi-Yaus

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