Eigenvalue estimates for Dirac operators in geometries with torsion

ISSN： 0232-704X

Source： Annals of Global Analysis and Geometry, Vol.37, Iss.1, 2010-01, pp. : 33-71

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Abstract

On a Riemannian spin manifold (M ⁿ, g), equipped with a non-integrable geometric structure and characteristic connection ▽^c with parallel torsion ▽^c T ^c = 0, we can introduce the Dirac operator D ^1/3, which is constructed by lifting the affine metric connection with torsion 1/3 T ^c to the spin structure. D ^1/3 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 (M ⁿ, g), we investigate the first eigenvalue of the operator $${left(D^{1/3} right)^{2}}$$ . As a main tool, we use Weitzenböck formulas, which express the square of the perturbed operator D ^1/3 + 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 Kähler and nearly parallel G ₂-spaces.