Single-Minus Graviton Amplitudes from Matrix Theory

The $n$-point single-minus graviton amplitudes are nonzero in a special kinematic region that preserves some supersymmetry when embedded in supergravity. We compute these amplitudes within the BFSS matrix theory by relating them to an index problem considered by A. Sen. More precisely, the single-minus amplitudes are obtained from an index that counts BPS states in the Coulomb branch of four-dimensional ${\cal N}=4$ super Yang-Mills. We explicitly check this relation in a specific kinematic region by computing the index using BPS wall-crossing formulae and matching it to the gravity result. We further employ BPS wall-crossing to show that the index satisfies a tower of $w_{1+\infty}$ soft theorems in any kinematic regime, again matching gravity. Finally, we show that for certain kinematics, the single-minus amplitudes can be realized by scattering a graviton off a plane wave.