[Paper Review] All superalgebras for warped AdS$_2$ and black hole near horizon geometries

Ulf Gran, J. Gutowski+1 more|View|Dec 21, 2017

Black Holes and Theoretical Physics24 references3 citations

TL;DR

This paper classifies all symmetry superalgebras of near-horizon geometries and warped AdS₂ backgrounds in 10- and 11-dimensional supergravities under smoothness and compactness assumptions. It shows that either the even subalgebra decomposes as 𝔰𝔩(2,ℝ)⊕𝔱₀ with 𝔟₀/𝔠 acting transitively on spheres, or the superalgebra is nilpotent; crucially, no such backgrounds preserve more than 16 supersymmetries.

ABSTRACT

We identify all symmetry superalgebras $\mathfrak{g}$ of near horizon geometries of black holes with a Killing horizon, assuming the solution is smooth and that the spatial cross section of the event horizon is compact without boundary. This includes all warped AdS$_2$ backgrounds with the most general allowed fluxes in 10- and 11-dimensional supergravities. If the index of a particular Dirac operator vanishes, we find that the even symmetry subalgebra decomposes as $\mathfrak{g}_0=\mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{t}_0$, where $\mathfrak{t}_0/\mathfrak{c}$ is the Lie algebra of a group that acts transitively and effectively on spheres, and $\mathfrak{c}$ is the center of $\mathfrak{g}$. If the Dirac operator index does not vanish, then the symmetry superalgebra is nilpotent with one even generator. We also demonstrate that there are no near horizon geometries, and also therefore no warped AdS$_2$ backgrounds, in 10- and 11-dimensions that preserve more than 16 supersymmetries.

Motivation & Objective

• To identify all possible symmetry superalgebras of near-horizon geometries of black holes with a Killing horizon, assuming smooth fields and compact, boundaryless spatial horizon sections.
• To classify the symmetry superalgebras of warped AdS₂ backgrounds with general fluxes in 10- and 11-dimensional supergravities as special cases of near-horizon geometries.
• To determine the maximum number of preserved supersymmetries in such backgrounds under the horizon conjecture and geometric constraints.
• To establish that no smooth, compact, flux-bearing near-horizon geometries or warped AdS₂ solutions in 10D and 11D supergravity preserve more than 16 supersymmetries.
• To demonstrate that the structure of the even subalgebra 𝔤₀ depends on the index of a Dirac operator: either 𝔰𝔩(2,ℝ)⊕𝔱₀ or a 1D nilpotent algebra.

Proposed method

• Utilizes the horizon conjecture to relate the number of preserved supersymmetries to the index of a twisted Dirac operator 𝔇̸, assuming smooth fields and compact horizon sections.
• Applies the geometric method of defining superalgebra brackets via Killing spinor bilinears and spinorial Lie derivatives, as developed in prior work on warped AdSₙ backgrounds.
• Employs the homogeneity theorem of [9] to show that spatial horizon sections and internal manifolds of warped AdS₂ backgrounds admit transitive group actions with Lie algebras 𝔟₀⊕𝔰𝔬(2) and 𝔟₀, respectively.
• Analyzes the structure of the even subalgebra 𝔤₀, showing it decomposes as 𝔰𝔩(2,ℝ)⊕𝔱₀ when Index 𝔇̸ = 0, with 𝔟₀/𝔠 acting transitively and effectively on a sphere.
• Uses classification results of homogeneous spaces in 8 and 9 dimensions to rule out higher supersymmetry for compact internal manifolds.
• Performs explicit field equations and KSE (Killing spinor equation) analysis in IIB, massive IIA, and heterotic supergravities to rule out N > 16 supersymmetries on S⁸ or other symmetric spaces.

Experimental results

Research questions

• RQ1 What are the complete sets of symmetry superalgebras for near-horizon geometries of black holes with compact, boundaryless spatial horizon sections in 10- and 11-dimensional supergravities?
• RQ2 How does the index of the Dirac operator 𝔇̸ determine the structure of the even subalgebra 𝔤₀ in these superalgebras?
• RQ3 Can warped AdS₂ backgrounds in 10- and 11-dimensional supergravity preserve more than 16 supersymmetries under smoothness and compactness conditions?
• RQ4 What constraints do the geometry of the spatial horizon section and the internal space impose on the possible isometry algebras and supersymmetry algebras?
• RQ5 Under what conditions does the symmetry superalgebra become nilpotent, and how does this relate to the index of the Dirac operator?

Key findings

• When the index of the Dirac operator vanishes, the even subalgebra of the symmetry superalgebra decomposes as 𝔰𝔩(2,ℝ)⊕𝔱₀, where 𝔟₀/𝔠 is the Lie algebra of a group acting transitively and effectively on a sphere.
• When the index of the Dirac operator is non-zero, the symmetry superalgebra is nilpotent with a one-dimensional even subalgebra, and the non-vanishing anti-commutators are explicitly given in equation (3.94).
• There are no smooth, compact, flux-bearing near-horizon geometries or warped AdS₂ backgrounds in 10- and 11-dimensional supergravity that preserve more than 16 supersymmetries.
• The spatial horizon sections of generic near-horizon geometries admit a transitive isometry action by a group with Lie algebra 𝔟₀⊕𝔰𝔬(2), while internal spaces of warped AdS₂ backgrounds admit such an action with Lie algebra 𝔟₀.
• For N > 16 supersymmetries, the requirement of compactness and smoothness leads to contradictions in Ricci curvature on spheres like S⁸, ruling out such solutions in IIB and massive IIA supergravity.
• The results complete the classification of symmetry superalgebras for warped AdSₙ backgrounds in 10- and 11-dimensional supergravities for n = 2, 4, 5, under standard assumptions, with no solutions exceeding 16 supersymmetries.