# Principal Tensor Strikes Again: Separability of Vector Equations with Torsion

## TL;DR

This paper demonstrates how a torsion-generalized principal Killing--Yano tensor enables the separability of various field equations in black hole spacetimes with 3-form fields, extending known symmetries to more complex backgrounds.

## Contribution

It introduces the role of torsion-modified principal tensors in separability of field equations in black hole spacetimes with 3-form fields, including explicit examples in higher dimensions.

## Key findings

- Separable vector field equations in specific black hole backgrounds.
- Explicit formulas for Proca field separability in higher-dimensional spacetimes.
- Identification of torsion's role in hidden symmetries and integrability.

## Abstract

Many black hole spacetimes with a 3-form field exhibit a hidden symmetry encoded in a torsion generalization of the principal Killing--Yano tensor. This tensor determines basic properties of such black holes while also underlying the separability of the Hamilton--Jacobi, Klein--Gordon, and (torsion-modified) Dirac field equations in their background. As a specific example, we consider the Chong--Cveti\v{c}--L\"u--Pope black hole of $D=5$ minimal gauged supergravity and show that the torsion-modified vector field equations can also be separated, with the principal tensor playing a key role in the separability ansatz. For comparison, separability of the Proca field in higher-dimensional Kerr--NUT--AdS spacetimes (including new explicit formulae in odd dimensions) is also presented.