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

A bstract We present a formulation of the three- and four-point amplitudes on the Coulomb branch of N N = 4 SYM as integrals over the symplectic Grassmannian. We demonstrate that their kinematic spaces are equivalent to symplectic Grassmannians SpGr( n , 2 n ). 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 N = (1, 1) SYM amplitude in terms of four-dimensional variables in a form that makes its symplectic Grassmannian structure manifest.