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 \(M\) is a closed spin manifold, carrying a metric \(g\) of non-negative scalar curvature, then either the Dirac operator is invertible or \((M,g)\) carries a parallel spinor. The existence of a parallel spinor has many consequences: Ricci-flatness, special holonomy, stability properties, \(\widetilde M=\mathbb{R}^k\times N\) with \(N\) 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" \((g,k)\), where \(g\) is a Riemannian metric and where \(k\) is a symmetric 2-tensor that should be viewed as the second fundamental form for \(M\) sitting as a spacelike hypersurface in some ambient Lorentzian manifold.

The classical Riemannian case corresponds to pairs \((g,0)\). 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 \((M,g,k)\), e.g. to the fundamental group of \(M\).