# Growing Hair on the extremal $BTZ$ black hole

## TL;DR

This paper demonstrates a new local symmetry in a nonlinear sigma-model coupled to gravity in asymptotic AdS_3 space, enabling the generation of infinitely many hairy extremal BTZ black hole solutions.

## Contribution

It introduces a novel local symmetry that maps extremal BTZ black holes to a family of hairy black holes in an asymptotically AdS_3 setting.

## Key findings

- Existence of a local symmetry in the nonlinear sigma-model with gravity.
- Generation of infinitely many hairy black hole solutions from extremal BTZ black holes.
- The symmetry involves simultaneous twisting of fields and metric transformations.

## Abstract

We show that the nonlinear $\sigma-$model in an asymptotically $AdS_3$ space-time admits a novel local symmetry. The field action is assumed to be quartic in the nonlinear $\sigma-$model fields and minimally coupled to gravity. The local symmetry transformation simultaneously twists the nonlinear $\sigma-$model fields and changes the space-time metric, and it can be used to map an extremal $BTZ$ black hole to infinitely many hairy black hole solutions.