The S matrix of 6D super Yang-Mills and maximal supergravity from rational maps

Abstract

Abstract We present new formulas for n-particle tree-level scattering amplitudes of six-dimensional $$ \mathcal{N}=\left(1,1\right) $$ N = 1 1 super Yang-Mills (SYM) and $$ \mathcal{N}=\left(2,2\right) $$ N = 2 2 supergravity (SUGRA). They are written as integrals over the moduli space of certain rational maps localized on the (n − 3)! solutions of the scattering equations. Due to the properties of spinor-helicity variables in six dimensions, the even-n and odd-n formulas are quite different and have to be treated separately. We first propose a manifestly supersymmetric expression for the even-n amplitudes of $$ \mathcal{N}=\left(1,1\right) $$ N = 1 1 SYM theory and perform various consistency checks. By considering soft-gluon limits of the even-n amplitudes, we deduce the form of the rational maps and the integrand for n odd. The odd-n formulas obtained in this way have a new redundancy that is intertwined with the usual SL(2, ℂ) invariance on the Riemann sphere. We also propose an alternative form of the formulas, analogous to the Witten-RSV formulation, and explore its relationship with the symplectic (or Lagrangian) Grassmannian. Since the amplitudes are formulated in a way that manifests double-copy properties, formulas for the six-dimensional $$ \mathcal{N}=\left(2,2\right) $$ N = 2 2 SUGRA amplitudes follow. These six-dimensional results allow us to deduce new formulas for five-dimensional SYM and SUGRA amplitudes, as well as massive amplitudes of four-dimensional $$ \mathcal{N}=4 $$ N = 4 SYM on the Coulomb branch.