Brane Tilings as On-shell Diagrams

TL;DR

This paper explores the connection between brane tilings in conformal field theories and the positive Grassmannian structures used in scattering amplitude computations in N=4 SYM, aiming to uncover deeper theoretical links.

Contribution

It introduces a novel approach to analyze brane tilings using the mathematics of the positive Grassmannian, bridging two areas of theoretical physics.

Findings

Identified structural similarities between brane tilings and on-shell diagrams.

Generated data suggesting potential mathematical connections.

Proposed a framework for further exploration of the link between BFTs and amplitude calculations.

Abstract

A new way of computing scattering amplitudes in a certain very important QFT (N=4 SYM) has recently been developed, in which an algebraic structure called the positive Grassmannian plays a very important role. The mathematics of the positive Grassmannian involve, among other things, bipartite graphs, which also appear in the formulation of a certain class of conformal field theories that are currently being generalized into Bipartite Field Theories (BFT). The fact that the same structures appear in two such different realms of physics suggests a deeper connection between the two that is yet to be fully unveiled. Here we explore that potential connection by looking at the graphs of a certain class of BFTs, the brane tilings, in terms of the new mathematics developed for the computation of the amplitudes. This way we produce a set of data that will hopefully be useful in the development of that connection.