Abstract
Abstract We introduce and initiate the investigation of a general class of 4d, $\mathcal{N}=1$ 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 $\mathcal{N}=4$ 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.
Highlights
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
In this paper 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
The correspondence we have introduced implies that every Bipartite Field Theories (BFTs) has a quiver diagram living on Σ, which is dual to the bipartite graph G as illustrated with an example in figure 3
Summary
In this paper 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. The equivalence between different multi-loop diagrams has a striking manifestation in terms of a single underlying Calabi-Yau manifold, arising as the moduli space of the BFTs. Section 3.3 explains how the boundary operator on a cell in the positive Grassmannian maps to the Higgs mechanism in the corresponding BFT. Note added: while this paper was being finalized, we became aware of [57], which has some overlap with this work
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