Eigenvalue estimates for Dirac operators in geometries with torsion

Abstract

On a Riemannian spin manifold (Mn, g), equipped with a non-integrable geometric structure and characteristic connection ▽c with parallel torsion ▽cT c = 0, we can introduce the Dirac operator D1/3, which is constructed by lifting the affine metric connection with torsion 1/3 T c to the spin structure. D1/3 is a symmetric elliptic differential operator, acting on sections of the spinor bundle and can be identified in special cases with Kostant’s cubic Dirac operator or the Dolbeault operator. For compact (Mn, g), we investigate the first eigenvalue of the operator \({\left(D^{1/3} \right)^{2}}\) . As a main tool, we use Weitzenbock formulas, which express the square of the perturbed operator D1/3 + S by the Laplacian of a suitable spinor connection. Here, S runs through a certain class of perturbations. We apply our method to spaces of dimension 6 and 7, in particular, to nearly Kahler and nearly parallel G2-spaces.