An index theorem for non-standard Dirac operators

We consider supersymmetric quantum mechanical systems in arbitrary dimensions on curved spaces with nontrivial gauge fields. The square of the Dirac operator serves as Hamiltonian. We derive a relation between the number of supercharges that exist and restrictions on the geometry of the underlying spaces as well as the admissible gauge field configurations. From the superalgebra with two or more real supercharges we infer the existence of integrability conditions and obtain a corresponding superpotential. This potential can be used to deform the supercharges and to determine zero modes of the Dirac operator. The general results are applied to the Kähler spaces È n .

show abstract

“…Our ansatz is motivated by previous results in [16,17,18,19] and the simple observation that both the free Dirac operator / ∂ on flat space and…”

Section: Square Roots Ofmentioning

confidence: 99%

Extended supersymmetries and the Dirac operator

2005

View full text Add to dashboard Cite

show abstract

“…In the Taub-NUT case, the pseudo-classical approach sets the spin contributions to the angular momentum, "relative electric charge" (20) and Runge-Lenz vector (23) [24,25,26,27].…”

Section: Spinning Taub-nut Spacementioning

confidence: 99%

Fermion on Curved Spaces, Symmetries, and Quantum Anomalies

Visinescu¹

2006

SIGMA

View full text Add to dashboard Cite

Abstract. We review the geodesic motion of pseudo-classical spinning particles in curved spaces. Investigating the generalized Killing equations for spinning spaces, we express the constants of motion in terms of Killing-Yano tensors. Passing from the spinning spaces to the Dirac equation in curved backgrounds we point out the role of the Killing-Yano tensors in the construction of the Dirac-type operators. The general results are applied to the case of the four-dimensional Euclidean Taub-Newman-Unti-Tamburino space. The gravitational and axial anomalies are studied for generalized Euclidean Taub-NUT metrics which admit hidden symmetries analogous to the Runge-Lenz vector of the Kepler-type problem. Using the Atiyah-Patodi-Singer index theorem for manifolds with boundaries, it is shown that the these metrics make no contribution to the axial anomaly.

show abstract

“…In the last yers a huge effort was devoted for analyzing the importance of (KY) tensors [9,10,11,12,13,14,15] in several areas but there are relatively few manifolds of physical interest admitting these tensors. This drawback is mainly because (KY) tensors are antisymmetric and their equations impose restrictions on the manifold structure [8].…”

Section: Introductionmentioning

confidence: 99%