[2306.09933] A classification of supersymmetric Kaluza-Klein black holes with a single axial symmetry
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Abstract
We extend the recent classification of five-dimensional, supersymmetric asymptotically flat black holes with only a single axial symmetry to black holes with Kaluza-Klein asymptotics. This includes a similar class of solutions for which the supersymmetric Killing field is generically timelike, and the corresponding base (orbit space of the supersymmetric Killing field) is of multi-centred Gibbons-Hawking type. These solutions are determined by four harmonic functions on $\mathbb{R}^3$ with simple poles at the centres corresponding to connected components of the horizon, and fixed points of the axial symmetry. The allowed horizon topologies are $S^3$, $S^2\times S^1$, and lens space $L(p, 1)$, and the domain of outer communication may have non-trivial topology with non-contractible 2-cycles. The classification also reveals a novel class of supersymmetric (multi-)black rings for which the supersymmetric Killing field is globally null. These solutions are determined by two harmonic functions on $\mathbb{R}^3$ with simple poles at centres corresponding to horizon components. We determine the subclass of Kaluza-Klein black holes that can be dimensionally reduced to obtain smooth, supersymmetric, four-dimensional multi-black holes. This gives a classification of four-dimensional asymptotically flat supersymmetric multi-black holes first described by Denef et al.
Summary
This paper provides a mathematical classification of supersymmetric black holes in Kaluza-Klein theory (five-dimensional supergravity) possessing a single axial symmetry. It establishes rigorous conditions and structures for such solutions, contributing to the theoretical understanding of higher-dimensional black hole uniqueness theorems.
Key Points
- •Classifies supersymmetric black hole solutions in 5D Kaluza-Klein/supergravity settings with minimal symmetry assumptions
- •Extends black hole uniqueness theorems to higher-dimensional supersymmetric contexts with a single axial Killing vector
- •Uses techniques from differential geometry and integrable systems to characterize solution spaces
- •Relevant to string theory and M-theory compactifications where Kaluza-Klein black holes arise naturally
- •Provides foundational mathematical structure for understanding exotic black hole topologies in extra dimensions
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# A classification of supersymmetric Kaluza-Klein black holes with a single axial symmetry
David Katona111d.katona@sms.ed.ac.uk
School of Mathematics and Maxwell Institute for Mathematical Sciences,
University of Edinburgh, King’s Buildings, Edinburgh, EH9 3JZ, UK
###### Abstract
We extend the recent classification of five-dimensional, supersymmetric asymptotically flat black holes with only a single axial symmetry
to black holes with Kaluza-Klein asymptotics. This includes a similar class of solutions for which the supersymmetric Killing field
is generically timelike, and the corresponding base (orbit space of the supersymmetric Killing field) is of multi-centred
Gibbons-Hawking type. These solutions are determined by four harmonic functions on ℝ3superscriptℝ3\\mathbb{R}^{3} with simple poles at the centres
corresponding to connected components of the horizon, and fixed points of the axial symmetry. The allowed horizon topologies are S3superscript𝑆3S^{3},
S2×S1superscript𝑆2superscript𝑆1S^{2}\\times S^{1}, and lens space L(p,1)𝐿𝑝1L(p,1), and the domain of outer communication may have non-trivial topology with non-contractible 2-cycles.
The classification also reveals a novel class of supersymmetric (multi-)black rings for which the supersymmetric Killing field is globally
null. These solutions are determined by two harmonic functions on ℝ3superscriptℝ3\\mathbb{R}^{3} with simple poles at centres corresponding to
horizon components. We determine the subclass of Kaluza-Klein black holes that
can be dimensionally reduced to obtain smooth, supersymmetric, four-dimensional multi-black holes. This gives a classification of
four-dimensional asymptotically flat supersymmetric multi-black holes first described by Denef et al.
## 1 Introduction
Black holes are in the focus of gravitational research. In four-dimensional vacuum gravity, or Einstein-Maxwell theory,
asymptotically flat black holes have a surprisingly simple moduli space due to the well-known uniqueness theorems
(see e.g. \[ [1](https://ar5iv.labs.arxiv.org/html/2306.09933#bib.bib1 ""), [2](https://ar5iv.labs.arxiv.org/html/2306.09933#bib.bib2 "")\]). In contrast, higher dimensional general relativity has a much richer
structure, and black hole uniqueness does not hold even in the asymptotically flat vacuum case (for review see e.g. \[ [3](https://ar5iv.labs.arxiv.org/html/2306.09933#bib.bib3 "")\]), which
became clear with the discovery of rotating vacuum black holes with S2×S1superscript𝑆2superscript𝑆1S^{2}\\times S^{1} horizon topology, known as black rings \[ [4](https://ar5iv.labs.arxiv.org/html/2306.09933#bib.bib4 "")\].
Rather surprisingly, for a range of asymptotic charges, black rings coexist with the spherical Myers-Perry black holes \[ [5](https://ar5iv.labs.arxiv.org/html/2306.09933#bib.bib5 "")\],
providing an explicit example of non-uniqueness.
Much is known in general about higher dimensional stationary black holes. The topol
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