Eigenvalue Estimates of the spin^cDirac Operator and Harmonic Forms on Kähler-Einstein Manifolds

Abstract

Abstract. We establish a lower bound for the eigenvalues of the Dirac operator defined on a compact Kähler-Einstein manifold of positive scalar curvature and endowed with particular spin c structures. The limiting case is characterized by the existence of Kählerian Killing spin c spinors in a certain subbundle of the spinor bundle. Moreover, we show that the Clifford multiplication between an effective harmonic form and a Kählerian Killing spin c spinor field vanishes. This extends to the spin c case the result of A. Moroianu stating that, on a compact Kähler-Einstein manifold of complex dimension 4 +3 carrying a complex contact structure, the Clifford multiplication between an effective harmonic form and a Kählerian Killing spinor is zero.

A projective Dirac operator on $\mathbb{C}P^n$ and extended SUSY

“…It is well known that CP n is a symmetric riemannian space G/H, and hence there is a standard way to construct, at least in principle, the Dirac operator using the right actions of the Lie algebra of G [12,13]. This approach was also followed in [11] where a bound was found for the spectrum of the twisted Dirac operator in Kählerian submanifolds of CP n for any spin c structure, more recent estimates were found for the eigenvalues of the Dirac operator on Kähler-Einstein manifolds in [14].…”