Symplectic Grassmannian description of the Coulomb branch three and four point amplitudes

TL;DR

This paper reformulates three- and four-point amplitudes on the Coulomb branch of N=4 SYM as integrals over symplectic Grassmannians, revealing their geometric structure and extending the approach to six-dimensional SYM amplitudes.

Contribution

It introduces a novel symplectic Grassmannian integral formulation for Coulomb branch amplitudes in N=4 SYM and extends this framework to six-dimensional SYM.

Findings

Amplitudes are expressed as integrals over SpGr(n,2n).

The four-point amplitude matches known results up to a kinematic factor.

Six-dimensional amplitudes are reformulated in four-dimensional variables highlighting symplectic Grassmannian structure.

Abstract

We present a formulation of the three- and four-point amplitudes on the Coulomb branch of N=4 SYM as integrals over the symplectic Grassmannian. We demonstrate that their kinematic spaces are equivalent to symplectic Grassmannians SpGr(n,2n). For the three-point case, we express the amplitude as an integral over the symplectic Grassmannian in a specific little group frame. In the four-point case, we show that the integral yields the amplitude up to a known kinematic factor. Building on the four-dimensional analysis, we also express the six-dimensional N = (1,1) SYM amplitude in terms of four-dimensional variables in a form that makes its symplectic Grassmannian structure manifest.