Solution to the Ward Identities for Superamplitudes

TL;DR

This work systematically solves the SUSY and R-symmetry Ward identities for on-shell amplitudes in the maximally supersymmetric theories ${ m N}=4$ SYM and ${ m N}=8$ supergravity. It rewrites NMHV and general N$^K$MHV superamplitudes as sums of manifestly SUSY- and ${ m SU}({ m N})_R$-invariant Grassmann polynomials multiplying a linearly independent algebraic basis of amplitudes, whose size is given by the dimension of the corresponding ${ m SU}(n-4)$ irrep associated with a rectangular Young diagram. Cyclic and reflection symmetries in gauge theory reduce the functional basis of color-ordered amplitudes, while gravity, being unordered, yields even larger reductions in practice; the framework extends to tree and loop levels. The paper provides explicit NMHV and ${ m N^2MHV}$ constructions in both theories, develops a rigorous basis-counting via Young tableaux and Kostka numbers, and develops a recursive, invariant construction of the corresponding ${ m Z}$-polynomials that encode all amplitudes in a compact form. These results streamline higher-loop computations, clarify the structure of amplitude relations, and offer a combinatorial bridge between on-shell SUSY Ward identities and representation theory.

Abstract

Supersymmetry and R-symmetry Ward identities relate on-shell amplitudes in a supersymmetric field theory. We solve these Ward identities for (Next-to)^K MHV amplitudes of the maximally supersymmetric N=4 and N=8 theories. The resulting superamplitude is written in a new, manifestly supersymmetric and R-invariant form: it is expressed as a sum of very simple SUSY and SU(N)_R-invariant Grassmann polynomials, each multiplied by a "basis amplitude". For (Next-to)^K MHV n-point superamplitudes the number of basis amplitudes is equal to the dimension of the irreducible representation of SU(n-4) corresponding to the rectangular Young diagram with N columns and K rows. The linearly independent amplitudes in this algebraic basis may still be functionally related by permutation of momenta. We show how cyclic and reflection symmetries can be used to obtain a smaller functional basis of color-ordered single-trace amplitudes in N=4 gauge theory. We also analyze the more significant reduction that occurs in N=8 supergravity because gravity amplitudes are not ordered. All results are valid at both tree and loop level.