Cluster Transformations from Bipartite Field Theories

TL;DR

This paper derives cluster transformations of face weights in bipartite field theories using gauge theory properties, linking them to Grassmannian coordinates and expanding combinatorial understanding of BFT dynamics.

Contribution

It introduces a gauge-theoretic derivation of cluster transformations in BFTs, connecting them to Grassmannian coordinate properties and expanding combinatorial objects associated with BFTs.

Findings

Cluster transformations follow from gauge theory properties.

Transformations relate to Grassmannian coordinates.

Adds new combinatorial objects to Grassmannian theory.

Abstract

Bipartite field theories (BFTs) are a new class of 4d N=1 quantum field theories defined by bipartite graphs on bordered Riemann surfaces. In this paper we derive, purely in terms of the gauge theory, the cluster transformations of face weights under square moves in the graph. In this context, we obtain them by connecting regular parametrizations of the master space of the associated BFTs. For BFTs on a disk, these transformations follow from the properties of coordinates in the Grassmannian. This represents a new addition to the list of combinatorial objects for the Grassmannian, such as matching and matroid polytopes, that have been shown to emerge from BFT dynamics.