Quantum mechanics of Yano tensors: Dirac equation in curved spacetime

Abstract

In spacetimes admitting Yano tensors the classical theory of the spinning particle possesses enhanced worldline supersymmetry. Quantum mechanically generators of extra supersymmetries correspond to operators that in the classical limit commute with the Dirac operator and generate conserved quantities. We show that the result is preserved in the full quantum theory, that is, Yano symmetries are not anomalous. This was known for Yano tensors of rank two, but our main result is to show that it extends to Yano tensors of arbitrary rank. We also describe the conformal Yano equation and show that is invariant under Hodge duality. There is a natural relationship between Yano tensors and supergravity theories. As the simplest possible example, we show that when the spacetime admits a Killing spinor then this generates Yano and conformal Yano tensors. As an application, we construct Yano tensors on maximally symmetric spaces: they are spanned by tensor products of Killing vectors.

Hidden Symmetries of Higher-Dimensional Rotating Black Holes

“…We expect that, similar to the 4-dimensional case [32], [126], in higher dimensions also the separability of the Dirac equation can be characterized by the corresponding symmetry operators. These operators are well known [16], [25].…”

“…An alternative (equivalent) definition of a rank-p CKY tensor naturally generalizes the definition (2.2) [219], [114], [25]. It reads 12) wherek (obtained again by tracing procedure) is given by (2.9).…”

“…Using the relation 25) it is easy to show that under the Hodge duality the exterior part transforms into the divergence part and vice versa. Indeed, we find…”

Dirac-Type Operators on Curved Spaces and the Role of Killing-Yano Tensors

“…which means that the quantum operator K does not necessarily define a genuine quantum mechanical symmetry [12]. On a generic curved space-time, we therefore have a gravitational quantum anomaly proportional to a contraction of the SK tensor k µν with the Ricci tensor R µν .…”

Nonlinear symmetries on spaces admitting Killing tensors

“…It was observed [4] that a K-Y tensor generates additional supercharges in the dynamics of pseudo-classical spinning particles [5] being the natural geometrical objects to be coupled with the fermionic degrees of freedom [4,6]. In this way it was realized the significant connection between K-Y tensors and non-standard supersymmetries.…”