Covariant representation theory of the Poincaré algebra and some of its extensions

TL;DR

This work develops a covariant, higher-dimensional generalization of the spinor-helicity formalism by constructing covariant representations of the Poincaré (and SUSY) algebras in $D>4$, enabling a complete on-shell description of massless and massive states through polarization vectors and a vector–spinor dictionary built from pure spinors. It introduces a covariant spin-algebra framework, derives on-shell SUSY Ward identities in any dimension, and demonstrates concrete amplitude consequences, including higher-dimensional vanishing amplitudes and their implications for YM, gravity, and string theories via KLT. The approach unifies four-dimensional spinor-helicity results with higher-dimensional physics, reveals a natural role for pure spinors as a geometric bridge to twistor-like spaces, and provides practical tools for on-shell computations and potential extensions to coherent-state amplitudes and self-dual sectors. Together, these results offer a covariant, dimension-agnostic toolkit for on-shell amplitude analysis with broad implications for field theory and string theory in flat backgrounds.

Abstract

There has been substantial calculational progress in the last few years for gauge theory amplitudes which involve massless four dimensional particles. One of the central ingredients in this has been the ability to keep precise track of the Poincare algebra quantum numbers of the particles involved. Technically, this is most easily done using the well-known four dimensional spinor helicity method. In this article a natural generalization to all dimensions higher than four is obtained based on a covariant version of the representation theory of the Poincare algebra. Covariant expressions for all possible polarization states, both bosonic and fermionic, are constructed. For the fermionic states the analysis leads directly to pure spinors. The natural extension to the representation theory of the on-shell supersymmetry algebra results in an elementary derivation of the supersymmetry Ward identities for scattering amplitudes with massless or massive legs in any integer dimension from four onwards. As a proof-of-concept application a higher dimensional analog of the vanishing helicity-equal amplitudes in four dimensions is presented in (super) Yang-Mills theory, Einstein (super-)gravity and superstring theory in a flat background.

Figures (3)

• Figure 1: the $\mathcal{N}=1$, $D=6$ vector multiplet in weight space with a degeneracy lifted for illustration purposes
• Figure 2: the $\mathcal{N}=1$, $D=10$ vector multiplet in (projected) weight space
• Figure 3: the $\mathcal{N}=1$, $D=11$ multiplet in projected weight space, ignoring particle multiplicities