Network and Seiberg Duality

TL;DR

This work constructs a new class of 4d $\mathcal{N}=1$ SCFTs from planar bipartite networks (plabic graphs) and shows that IR fixed points, modulo Seiberg dualities realized as square moves, are classified by decorated permutations corresponding to cells of the totally non-negative Grassmannian $(\mathrm{Gr}_{k,n})_{\ge 0}$. It extends the framework to networks on bordered Riemann surfaces via open pants decompositions and ideal triangulations, where duality is encoded as flips and triangulation moves, yielding a geometry-driven, duality-frame-insensitive description of the IR physics. The paper provides a concrete network-quiver dictionary, derives $R$-charges from zig-zag path angles, and discusses the superconformal index and its relation to integrable 2d spin systems, highlighting deep links to higher Teichmüller theory and Grassmannian stratifications. Together, these results offer a geometric, combinatorial approach to a broad class of $\mathcal{N}=1$ SCFTs and suggest avenues for brane realizations, gravity duals, and connections to scattering amplitudes and wall-crossing phenomena.

Abstract

We define and study a new class of 4d N=1 superconformal quiver gauge theories associated with a planar bipartite network. While UV description is not unique due to Seiberg duality, we can classify the IR fixed points of the theory by a permutation, or equivalently a cell of the totally non-negative Grassmannian. The story is similar to a bipartite network on the torus classified by a Newton polygon. We then generalize the network to a general bordered Riemann surface and define IR SCFT from the geometric data of a Riemann surface. We also comment on IR R-charges and superconformal indices of our theories.

Figures (28)

• Figure 1: An example of a planar bicolored network on a disc with four boundary marked points.
• Figure 2: Move 1: a square move flips the color of the vertices around a square.
• Figure 3: Move 2: merging/unmerging of the same-colored vertices does not change the definition of the gauge theory. Similar operation exists for black-colored vertices.
• Figure 4: Move 3: a degree two vertices can be removed.
• Figure 5: Reduction: bubble reduction.