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

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.

71+23 pages. v2: minor corrections, references added, matches published JHEP version

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• On Chiral Splitting and the Ambitwistor String
• Unified Formalism for 6D Superamplitudes Based on a Symplectic Grassmannian
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• Ambitwistor Strings in Six and Five Dimensions
• D3-Brane Loop Amplitudes from M5-Brane Tree Amplitudes
• Bootstrapping solutions of scattering equations
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• Scattering Amplitudes and BCFW in $\mathcal{N}=2^{\ast }$ Theory
• Loop Amplitudes in the Coulomb Branch of $\mathcal{N}=4$ Super-Yang-Mills Theory
• Yangian Symmetry in Five Dimensions
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• Symplectic Grassmannians, dual conformal symmetry and 4-point amplitudes in 6D