Dominant energy condition and invertibility of the Dirac-Witten operator
24.10.2024, 16:15
– 2.09.0.14
Forschungsseminar Differentialgeometrie
Bernd Ammann (Regensburg)
The classical Atiyah-Singer theorem tells us: if is a closed spin manifold, carrying a metric of non-negative scalar curvature, then either the Dirac operator is invertible or carries a parallel spinor. The existence of a parallel spinor has many consequences: Ricci-flatness, special holonomy, stability properties, with compact, polynomial bounds on volume growth of balls etc.
From a point of view of general relativity, a major motivation to study non-negative scalar curvature is that it is a special case of the dominant energy condition. This is a condition on an "intial data pair" , where is a Riemannian metric and where is a symmetric 2-tensor that should be viewed as the second fundamental form for sitting as a spacelike hypersurface in some ambient Lorentzian manifold.
The classical Riemannian case corresponds to pairs . The Dirac operator generalizes to the Dirac-Witten operator, non-negative scalar curvature to the dominant energy condition. In case that the dominant energy condition holds, the Dirac-Witten operator is invertible, unless we have an imaginary generalized Killing spinor. We will discuss the impact of the existence of such a spinor on , e.g. to the fundamental group of .