Initial data rigidity via Dirac-Witten operators

In this paper we prove an initial data rigidity result à la Eichmair, Galloway and Mendes (arXiv:2009.09527) using Dirac operator techniques. It applies to initial data sets on spin bands that satisfy the dominant energy condition, a boundary condition for the future null expansion scalar and the $\widehat{A}$-obstruction for positive scalar curvature on one of the boundary pieces. Interestingly, these bands turn out to carry lightlike imaginary $W$-Killing spinors, which are connected to Lorentzian special holonomy and moduli spaces of Ricci-flat metrics. We also obtain slight generalizations of known rigidity results on Riemannian bands.