Gravity amplitudes from n-space

Abstract

We identify a hidden GL(n, $\mathbb{C}$) 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, $\mathbb{C}$), but linearly under its notable subgroups, the little group and the permutation group S _n . Using GL(n, $\mathbb{C}$) 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−2MHV amplitudes and speculate on the deeper origins of GL(n, $\mathbb{C}$).