On timelike supersymmetric solutions of Abelian gauged 5-dimensional supergravity
HJE
On timelike supersymmetric solutions of Abelian
Samuele Chimento 0
Instituto de F´ısica Teo´rica UAM/CSIC 0
0 C/ Nicol ́as Cabrera , 13-15, C.U. Cantoblanco, E-28049 Madrid , Spain
We consider 5-dimensional gauged N = 1 supergravity coupled to Abelian vector multiplets, and we look for supersymmetric solutions for which the 4-dimensional K¨ahler base space admits a holomorphic isometry. Taking advantage of this isometry, we are able to find several supersymmetric solutions for the ST[2, nv + 1] special geometric model with arbitrarily many vector multiplets. Among these there are three families of solutions with nv + 2 independent parameters, which for one of the families can be seen to correspond to nv + 1 electric charges and one angular momentum. These solutions generalize the ones recently found for minimal gauged supergravity in JHEP 1704 (2017) 017 and include in particular the general supersymmetric asymptotically-AdS5 black holes of Gutowski and Reall, analogous black hole solutions with non-compact horizon, the three near horizon geometries themselves, and the singular static solutions of Behrndt, Chamseddine and Sabra.
Black Holes; Black Holes in String Theory; Supergravity Models
-
2
3
4
5
1
1 Introduction
3.1
Summary
Solutions
4.1
4.2
4.3
Ansatz
Introduction
Abelian gauged N = 1, d = 5 supergravity
2.1
Timelike supersymmetric solutions
Timelike supersymmetric solutions of Abelian gauged N
supergravity with one additional isometry
Solutions for the ST[2, nv + 1] model
Supersymmetric black holes
Conserved charges
Static solutions
still an open problem for rotating supersymmetric black holes in AdS5, see e.g. [2–4]).
However, while assuming unbroken supersymmetry makes the problem more tractable,
it is usually not enough to find explicit solutions, and one has to make some additional
assumptions or to impose a specific ansatz in order to solve the equations.1
An approach that has proven to be very successful in ungauged 5-dimensional
supergravity, with or without vector multiplets, is to assume that the 4-dimensional base space,
which for that theory has to be hyperKa¨hler, admits one triholomorphic isometry. In this
case the base space has a Gibbons-Hawking metric [6, 7], and it turns out that the
solutions can be completely characterized in terms of a small number of building blocks, namely
harmonic functions on 3-dimensional flat space [8, 9]. The same ansatz has also been
effective for N = 1, d = 5 supergravity with vector multiplets and non-Abelian gaugings [10],
but without Fayet-Iliopoulos terms, in which case the base space is again a 4-dimensional
hyperKa¨hler space.
Recently [11] a similar ansatz was applied to the case of minimal d = 5 gauged
supergravity, where a U(1) subgroup of the SU(2) R-symmetry group is gauged by adding
a Fayet-Iliopoulos term to the bosonic action. In this case the base space is just K¨ahler,
instead of hyperKa¨hler, and the ansatz consists in assuming that it admits a holomorphic
isometry. The metric of the base space can then be written in terms of two functions [12]
in a form that generalizes the Gibbons-Hawking metrics, and the problem of finding
supersymmetric solutions is reduced to that of solving a system of fourth order differential
equations for these two functions plus a third one.
The aim of this paper is to apply the same ansatz in the case of N
gauged supergravity [11]. They are studied in some detail in subsection 4.3, where the
conserved charges are computed for one of the families, and it is shown that they include as
particular cases black holes with compact or non-compact horizon, as well as static singular
solutions. In subsection 4.4 we give the explicit expression of the fields for supersymmetric
black holes not included in the solutions of subsection 4.3, despite being very similar to a
subcase of them. We conclude in section 5 with some final remarks.
1For a comprehensive review of supersymmetric solutions of supergravity theories with many references
see, e.g. ref. [5].
2By superficially asymptotically-AdS we mean that the metric components approach those of AdS in an
appropriate limit, which however does not guarantee that the solutions are globally asymptotically-AdS.
– 2 –
HJEP07(21)59
Abelian gauged N
= 1, d = 5 supergravity
In this section we give a brief description of the bosonic sector of a general theory of
N = 1, d = 5 supergravity coupled to nv vector multiplets in which a U(1) subgroup of the
SU(2) R-symmetry group has been gauged by the addition of Fayet-Iliopoulos (FI) terms.
The U(1) subgroup to be gauged and the gauge vector used in the gauging are determined
by the tensor PI r, as we are going to explain.3 Our conventions are those in refs. [
13, 14
]
which are those of ref. [
15
] with minor modifications.
The supergravity multiplet is constituted by the graviton eµa , the gravitino ψµi and the
graviphoton Aµ . All the spinors are symplectic Majorana spinors and carry a fundamental
SU(2) (...truncated)