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

We present new formulas for n-particle tree-level scattering amplitudes of sixdimensional N = (1,1) super Yang-Mills (SYM) and 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 Al = (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 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 N = 4 SYM on the Coulomb branch.