Extremal Black Holes from Homotopy Algebras

TL;DR

This work develops a perturbative, homotopy-algebraic framework to construct extremal black hole solutions compatible with a fixed near-horizon geometry. By encoding the deformation problem in a cyclic $L_\infty$-algebra and employing the homotopy Maurer–Cartan formalism together with the homological perturbation lemma, the authors reduce the problem to solving a minimal-model Maurer–Cartan equation whose solutions are governed by the lowest-order data, yielding a finite-dimensional moduli space. The extremal Kerr horizon is treated explicitly, with concrete first- and next-to-lowest-order deformations computed via Green’s functions and gauge fixing, and the Kerr solution recovered as a specific point in the deformation moduli. Overall, the paper provides a rigorous algebraic-perturbative pathway to classify and understand extremal horizon geometries, linking horizon rigidity and uniqueness to an explicit, perturbative deformation theory.

Abstract

The uniqueness and rigidity of black holes remain central themes in gravitational research. In this work, we investigate the construction of all extremal black hole solutions to the Einstein equation for a given near-horizon geometry, employing the homotopy algebraic perspective, a powerful and increasingly influential framework in both classical and quantum field theory. Utilising Gaußian null coordinates, we recast the deformation problem as an analysis of the homotopy Maurer-Cartan equation associated with an $L_\infty$-algebra. Through homological perturbation theory, we systematically solve this equation order by order in directions transverse to the near-horizon geometry. As a concrete application of this formalism, we examine the deformations of the extremal Kerr horizon. Notably, this homotopy-theoretic approach enables us to characterise the moduli space of deformations by studying only the lowest-order solutions, offering a systematic way to understand the landscape of extremal black hole geometries.