Jan Gutowski

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∞-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.

We classify supersymmetric solutions of D = 5 ( 3 , 2 ) signature supergravity with either vanishing or imaginary gauge coupling constant preserving the minimal N = 2 supersymmetry. We prove that the geometry of such solutions is characterized by a nilpotent integrable endomorphism, and obtain the necessary and sufficient conditions on the fluxes imposed by supersymmetry. We also construct examples of supersymmetric domain wall solutions for which the nilpotent integrable endomorphism is associated with a Killing–Yano two-form, as well as a new descendant ‘preon’ solution which preserves N = 6 supersymmetry. This is notable as such N = 6 descendant solutions do not exist in the standard signature D = 5 supergravity.

We classify all warped dS ₅ backgrounds in D = 11 supergravity with enhanced supersymmetry. We show that backgrounds preserving N = 16 supersymmetries consist of either a stack of M5 branes with transverse space R 5 , or a generalized M5-brane configuration with transverse space R × N 4 , where N ₄ is a hyper-Kähler manifold and the M5-brane harmonic function is determined by a hyper-Kähler potential on N ₄. Moreover, we find that there are no backgrounds preserving exactly N = 24 supersymmetries. Backgrounds preserving N = 32 supersymmetries correspond to either R 1 , 10 or A d S 7 × S 4 .

We determine the necessary and sufficient conditions for warped product dS _n solutions, 5 ⩽ n ⩽ 10, to preserve supersymmetry in D = 11 supergravity, without assuming factorization of the Killing spinors. We prove that for 7 ⩽ n ⩽ 10, all such solutions are flat, with vanishing four-form. We also show that the only warped product dS ₆ solutions are either the maximally supersymmetric AdS ₇ × S ⁴ solution, or R 1 , 6 × N 4 where N ₄ is hyperKähler, with vanishing four-form. Supersymmetric warped product dS ₅ solutions are then classified; it is shown that all such solutions are generalized M5-brane configurations, for which the transverse space is R × N 4 , and N ₄ is a hyperKähler manifold. If the four-form is covariantly constant, then N ₄ admits a hyperKähler potential.

The supersymmetry of near-horizon geometries in heterotic supergravity is considered. A necessary and sufficient condition for a solution to preserve more than the minimal N = 2 supersymmetry is obtained. A supersymmetric near-horizon solution is constructed which is a U(1) fibration of AdS ₃ over a particular Aloff–Wallach space. It is proven that this solution preserves the conditions required for N = 2 supersymmetry, but does not satisfy the necessary condition required for further supersymmetry enhancement. Hence, there exist supersymmetric near-horizon heterotic solutions preserving exactly N = 2 supersymmetry.

The supersymmetry of near-horizon geometries in heterotic supergravity is considered. A necessary and sufficient condition for a solution to preserve more than the minimal N=2 supersymmetry is obtained. A supersymmetric near-horizon solution is constructed which is a U(1) fibration of AdS3 over a particular Aloff-Wallach space. It is proven that this solution preserves the conditions required for N=2 supersymmetry, but does not satisfy the necessary condition required for further supersymmetry enhancement. Hence, there exist supersymmetric near-horizon heterotic solutions preserving exactly N=2 supersymmetry.