# A Riemann-Hilbert approach to rotating attractors

## TL;DR

This paper develops a Riemann-Hilbert method to construct rotating extremal black hole solutions in gravity, explicitly solving the associated linear system and revealing integrable structures and transformations between solutions.

## Contribution

It introduces a vectorial Riemann-Hilbert factorization approach to explicitly solve for rotating black hole attractors and identifies Geroch group elements implementing solution-generating transformations.

## Key findings

- Explicit factorization of monodromy matrices with second order poles.
- Construction of rotating extremal black hole solutions.
- Explicit expression for the master potential encoding conserved currents.

## Abstract

We construct rotating extremal black hole and attractor solutions in gravity theories by solving a Riemann-Hilbert problem associated with the Breitenlohner-Maison linear system. By employing a vectorial Riemann-Hilbert factorization method we explicitly factorize the corresponding monodromy matrices, which have second order poles in the spectral parameter. In the underrotating case we identify elements of the Geroch group which implement Harrison-type transformations which map the attractor geometries to interpolating rotating black hole solutions. The factorization method we use yields an explicit solution to the linear system, from which we do not only obtain the spacetime solution, but also an explicit expression for the master potential encoding the potentials of the infinitely many conserved currents which make this sector of gravity integrable.