On Integrability of spinning particle motion in higher-dimensional black hole spacetimes

TL;DR

This paper demonstrates the integrability of spinning particle motion in higher-dimensional rotating black hole spacetimes, extending previous results on geodesic motion to include spin degrees of freedom and hidden symmetries.

Contribution

It introduces explicit integrals of motion for spinning particles in higher dimensions, showing their involution in certain cases, and conjectures universal integrability across all dimensions.

Findings

Identifies n independent integrals of motion in all dimensions.

Proves integrability in 4-7 dimensions for the bosonic part.

Suggests integrability likely holds in all higher dimensions.

Abstract

We study the motion of a classical spinning particle (with spin degrees of freedom described by a vector of Grassmann variables) in higher-dimensional general rotating black hole spacetimes with a cosmological constant. In all dimensions n we exhibit n bosonic functionally independent integrals of spinning particle motion, corresponding to explicit and hidden symmetries generated from the principal conformal Killing--Yano tensor. Moreover, we demonstrate that in 4-, 5-, 6-, and 7-dimensional black hole spacetimes such integrals are in involution, proving the bosonic part of the motion integrable. We conjecture that the same conclusion remains valid in all higher dimensions. Our result generalizes the result of Page et. al. [hep-th/0611083] on complete integrability of geodesic motion in these spacetimes.