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Non-extremal Black Holes from the Generalised R-map

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Black Objects in Supergravity

Part of the book series: Springer Proceedings in Physics ((SPPHY,volume 144))

Abstract

We review the timelike dimensional reduction of a class of five-dimensional theories that generalises \(5D,\;\mathcal{{N}} = 2\) supergravity coupled to vector multiplets. As an application we construct instanton solutions to the four-dimensional Euclidean theory, and investigate the criteria for solutions to lift to static non-extremal black holes in five dimensions. We focus specifically on two classes of models: STU-like models, and models with a block diagonal target space metric. For STU-like models the second order equations of motion of the four-dimensional theory can be solved explicitly, and we obtain the general solution. For block diagonal models we find a restricted class of solutions, where the number of independent scalar fields depends on the number of blocks. When lifting these solutions to five dimensions we show, by explicit calculation, that one obtains static non-extremal black holes with scalar fields that take finite values on the horizon only if the number of integration constants reduces by exactly half.

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Notes

  1. 1.

    For a derivation see [29], and see [41] for a general formula for \(d \ge 4\) dimensions.

  2. 2.

    These solutions necessarily lift to BPS black holes. If the metric of the target manifolds allows for a field rotation matrix \(R^I_{\;\;K}\) that satisfies \(a_{IJ}R^I_{\;\;K} R^J_{\;\;L} = a_{KL}\) then one can generalise this ansatz to produce solutions which lift to non-BPS black holes [30, 42, 43].

  3. 3.

    By finite values we mean \(\phi ^x \longrightarrow / 0, \pm \infty \).

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Acknowledgments

The work of T.M. is supported in part by STFC grant ST/G00062X/1. The work of O.V. is supported by an STFC studentship and DAAD.

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  1. Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool, L69 7ZL, UK

    Thomas Mohaupt & Owen Vaughan

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  1. Thomas Mohaupt
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Correspondence to Thomas Mohaupt .

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  1. Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare, via E. Fermi 40, Frascati, 00044, Roma, Italy

    Stefano Bellucci

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Mohaupt, T., Vaughan, O. (2013). Non-extremal Black Holes from the Generalised R-map. In: Bellucci, S. (eds) Black Objects in Supergravity. Springer Proceedings in Physics, vol 144. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00215-6_6

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