  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.
