The Casimir operator of a metric connection with skew-symmetric torsion

Abstract

For any triple $(M^n,g,\nabla )$ consisting of a Riemannian manifold and a metric connection with skew-symmetric torsion we introduce an elliptic, second order operator $\Omega$ acting on spinor fields. In case of a reductive space and its canonical connection our construction yields the Casimir operator of the isometry group. Several non-homogeneous geometries (Sasakian, nearly Kähler, cocalibrated ${\mathrm{G}}_2$-structures) admit unique connections with skew-symmetric torsion. We study the corresponding Casimir operator and compare its kernel with the space of $\nabla$-parallel spinors.

Latex2e, 15 pages

References in corpus (3)

• Killing spinor equations in dimension 7 and geometry of integrable G_2-manifolds
• On types of non-integrable geometries
• Connections on naturally reductive spaces, their Dirac operator and homogeneous models in string theory

Cited by in corpus (10)

• 3-Sasakian manifolds in dimension seven, their spinors and G_2 structures
• Almost Hermitian 6-Manifolds Revisited
• The Square of the Dirac and spin-Dirac Operators on a Riemann-Cartan Space(time)
• On the Ricci tensor in type II B string theory
• $G_2$-Manifolds With Parallel Characteristic Torsion
• Killing and twistor spinors with torsion
• Spin(7)-manifolds with parallel torsion form
• Cocalibrated G_2-manifolds with Ricci flat characteristic connection
• Twistorial eigenvalue estimates for generalized Dirac operators with torsion
• An integral formula for $G_2$-structures