Gravity Amplitudes from n-Space

TL;DR

This paper reveals a hidden GL(n,C) symmetry in tree-level n-point MHV gravity amplitudes, providing new geometric formulas that exhibit various symmetric forms and redundancies, and discusses potential extensions and deeper symmetries.

Contribution

It introduces a novel GL(n,C) covariant formulation of MHV gravity amplitudes using auxiliary n-space variables, leading to new symmetric expressions and insights into amplitude redundancies.

Findings

Derived a simple, geometric formula for MHV amplitudes using GL(n,C) symmetry.

Reproduced known symmetric forms of amplitudes by fixing redundancies.

Presented new S_n symmetric amplitude expressions that are not manifestly homogeneous.

Abstract

We identify a hidden GL(n,C) symmetry of the tree level n-point MHV gravity amplitude. Representations of this symmetry reside in an auxiliary n-space whose indices are external particle labels. Spinor helicity variables transform non-linearly under GL(n,C), but linearly under its notable subgroups, the little group and the permutation group S_n. Using GL(n,C) covariant variables, we present a new and simple formula for the MHV amplitude which can be derived solely from geometric constraints. This expression carries a huge intrinsic redundancy which can be parameterized by a pair of reference 3-planes in n-space. Fixing this redundancy in a particular way, we reproduce the S_{n-3} symmetric form of the MHV amplitude of [1], which is in turn equivalent to the S_{n-2} symmetric form of [2] as a consequence of the matrix tree theorem. The redundancy of the amplitude can also be fixed in a way that fully preserves S_n, yielding new and manifestly S_n symmetric forms of the MHV amplitude. Remarkably, these expressions need not be manifestly homogenous in spinorial weight or mass dimension. We comment on possible extensions to N^{k-2}MHV amplitudes and speculate on the deeper origins of GL(n,C).