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

TL;DR

This work introduces Bipartite Field Theories (BFTs), a broad class of 4d N=1 quiver gauge theories defined by bipartite graphs on Riemann surfaces, linking D-brane setups, mirror symmetry, cluster integrable systems, and leading singularities in scattering amplitudes. It develops a comprehensive dictionary between graph data (faces, edges, nodes) and gauge-theory ingredients, and introduces perfect matchings and zig-zag paths as core tools. A generalized Kasteleyn framework is presented to efficiently determine master and mesonic Calabi–Yau moduli spaces, revealing that these moduli spaces are toric CY manifolds and are invariant under Seiberg duality (square moves). The paper also demonstrates loop-reduction mechanisms, Higgsing relations to boundary operations in Grassmannian cells, and extends the analysis beyond planar graphs, underscoring the central role of Calabi–Yau geometry in leading singularities and their potential broader impact on amplitude methods and geometry of the moduli space.

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.

Figures (44)

• Figure 1: An artistic representation of a bipartite graph on a Riemann surface defining a BFT.
• Figure 2: Examples of internal and external faces in a BFT.
• Figure 3: A bipartite graph and its quiver dual. Blue nodes in the quiver represent gauge groups and green squares represent global symmetry groups. Plaquettes in the quiver correspond to superpotential terms.
• Figure 4: Zig-zag paths in double line notation. This example has five zig-zag paths going through external legs and an internal one, shown in purple.
• Figure 5: The untwisting map.