The S Matrix of 6D Super Yang-Mills and Maximal Supergravity from Rational Maps

TL;DR

This work develops a comprehensive connected-formulation program for tree-level amplitudes in six-dimensional maximal supersymmetric theories, deriving explicit rational-map based formulas for both even and odd particle multiplicities in ${ m N}=(1,1)$ SYM and, via double copy, ${ m N}=(2,2)$ SUGRA. A key advance is the separation of even and odd cases in 6D spinor-helicity language, with odd points requiring a novel T-shift redundancy and a new ${ m oldhat{A}}_n$ matrix whose reduced Pfaffian acts as a quarter-integrand. The authors construct both nonlinear (map-based) and linear (Witten–RSV-like) formulations, establish soft-limit relations, and demonstrate CHY-compatibility and dimensional reductions to 5D and 4D massive sectors, including Coulomb-branch amplitudes. The results yield new five-dimensional and four-dimensional amplitudes and illuminate the structure of double-copy relations in higher dimensions, offering avenues for loop-level generalizations and potential links to Grassmannian/Veronese formulations. Overall, the paper provides a unified, highly symmetric framework for computing a broad class of tree amplitudes across dimensions.

Abstract

We present new formulas for $n$-particle tree-level scattering amplitudes of six-dimensional $\mathcal{N}=(1,1)$ super Yang-Mills (SYM) and $\mathcal{N}=(2,2)$ 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 $\mathcal{N}=(1,1)$ 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 $\text{SL}(2, \mathbb{C})$ 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 $\mathcal{N}=(2,2)$ 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 $\mathcal{N}=4$ SYM on the Coulomb branch.