Bipartite Field Theories: from D-Brane Probes to Scattering Amplitudes

Abstract

We introduce and initiate the investigation of a general class of 4d, N=1 quiver gauge theories whose Lagrangian is defined by a bipartite graph on a Riemann surface, with or without boundaries. We refer to such class of theories as Bipartite Field Theories (BFTs). BFTs underlie a wide spectrum of interesting physical systems, including: D3-branes probing toric Calabi-Yau 3-folds, their mirror configurations of D6-branes, cluster integrable systems in (0+1) dimensions and leading singularities in scattering amplitudes for N=4 SYM. While our discussion is fully general, we focus on models that are relevant for scattering amplitudes. We investigate the BFT perspective on graph modifications, the emergence of Calabi-Yau manifolds (which arise as the master and moduli spaces of BFTs), the translation between square moves in the graph and Seiberg duality and the identification of dual theories by means of the underlying Calabi-Yaus, the phenomenon of loop reduction and the interpretation of the boundary operator for cells in the positive Grassmannian as higgsing in the BFT. We develop a technique based on generalized Kasteleyn matrices that permits an efficient determination of the Calabi-Yau geometries associated to arbitrary graphs. Our techniques allow us to go beyond the planar limit by both increasing the number of boundaries of the graphs and the genus of the underlying Riemann surface. Our investigation suggests a central role for Calabi-Yau manifolds in the context of leading singularities, whose full scope is yet to be uncovered.

58 pages. Typos fixed

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• The Standard Model Quiver in de Sitter String Compactifications
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• The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning
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• N=1 Geometries via M-theory
• Orbifold Reduction and 2d (0,2) Gauge Theories
• New Directions in Bipartite Field Theories
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• Unoriented Quivers with Flavour
• Higher laminations, webs and N=2 line operators
• Bipartite Field Theories from D-Branes
• A Dimer ABC
• A Comprehensive Survey of Brane Tilings
• Integrability from 2d N=(2,2) Dualities
• The Geometry of On-Shell Diagrams
• Twistor actions for gauge theory and gravity
• Grassmannians and form factors with $q^2=0$ in N=4 SYM theory
• $3d$ Printing of $2d$ $\mathcal{N}=(0,2)$ Gauge Theories
• Wilson lines, Grassmannians and Gauge Invariant Off-shell Amplitudes in N=4 SYM
• Surface defects as transfer matrices
• Gauge Theories and Dessins d'Enfants: Beyond the Torus
• Internally Calabi-Yau algebras and cluster-tilting objects
• Scattering Amplitudes and Toric Geometry
• On the refined counting of graphs on surfaces
• Network, cluster coordinates and N = 2 theory II: Irregular singularity
• Double Handled Brane Tilings
• Infrared Duality in Unoriented Pseudo del Pezzo
• Cluster Transformations from Bipartite Field Theories
• Twin Theories, Polytope Mutations and Quivers for GTPs
• Mutation of frozen Jacobian algebras
• Mass Deformations of Unoriented Quiver Theories
• Graded Quivers, Generalized Dimer Models and Toric Geometry
• Unsupervised Machine Learning Techniques for Exploring Tropical Coamoeba, Brane Tilings and Seiberg Duality
• D-brane Instantons as Gauge Instantons in Orientifolds of Chiral Quiver Theories
• Web of Seiberg-like dualities for 3d $\mathcal{N}=2$ quivers
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• Bipartite field theories and D-brane instantons
• Non-Planar On-Shell Diagrams
• Gauge Theories, Tessellations & Riemann Surfaces
• BFT${}_2$: a General Class of $2d$ $\mathcal{N}=(0,2)$ Theories, 3-Manifolds and Toric Geometry
• Anatomy of the Amplituhedron
• Three dimensional Seiberg-like duality and tropical cluster algebra
• Quivers, Lattice Gauge Theories and Fractons
• Some Open Questions in Quiver Gauge Theory
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• Perfect matching modules, dimer partition functions and cluster characters
• Consistent Dimer Models on Surfaces with Boundary
• From frieze patterns to cluster categories
• Bipartite Graphs, On-Shell Diagrams, and Bipartite Field Theories: a Computational Package in Mathematica
• Rooted Clusters for Graph LP Algebras
• Hilbert Series of Bipartite Field Theories
• Brane Tilings as On-shell Diagrams