# Black holes, hidden symmetries, and complete integrability

## TL;DR

This paper reviews how hidden symmetries and the principal tensor underpin the integrability and separability properties of higher-dimensional black holes, revealing deep geometric structures similar to four-dimensional Kerr black holes.

## Contribution

It introduces the principal tensor as a unifying object generating explicit and hidden symmetries, explaining the integrability and separability of equations in higher-dimensional black hole spacetimes.

## Key findings

- Principal tensor generates all symmetries of higher-dimensional black holes.
- Complete integrability of geodesic motion is guaranteed.
- Separability of key equations like Hamilton-Jacobi, Klein-Gordon, and Dirac is demonstrated.

## Abstract

The study of higher-dimensional black holes is a subject which has recently attracted a vast interest. Perhaps one of the most surprising discoveries is a realization that the properties of higher-dimensional black holes with the spherical horizon topology and described by the Kerr-NUT-(A)dS metrics are very similar to the properties of the well known four-dimensional Kerr metric. This remarkable result stems from the existence of a single object called the principal tensor. In our review we discuss explicit and hidden symmetries of higher-dimensional black holes. We start with the overview of the Liouville theory of completely integrable systems and introduce Killing and Killing-Yano objects representing explicit and hidden symmetries. We demonstrate that the principal tensor can be used as a `seed object' which generates all these symmetries. It determines the form of the black hole geometry, as well as guarantees its remarkable properties, such as special algebraic type of the spacetime, complete integrability of geodesic motion, and separability of the Hamilton-Jacobi, Klein-Gordon, and Dirac equations. The review also contains a discussion of different applications of the developed formalism and its possible generalizations.