An extremization principle for the entropy of rotating BPS black holes in AdS5
JHE
BPS black holes in AdS5
Seyed Morteza Hosseini 0 1 3
Kiril Hristov 0 1 2
Alberto Zaffaroni 0 1 3
0 Tsarigradsko Chaussee 72 , 1784 Sofia , Bulgaria
1 I-20126 Milano , Italy
2 Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences
3 Dipartimento di Fisica, Universita` di Milano-Bicocca
We show that the Bekenstein-Hawking entropy of a class of BPS electrically charged rotating black holes in AdS5 × S5 can be obtained by a simple extremization principle. We expect that this extremization corresponds to the attractor mechanism for BPS rotating black holes in five-dimensional gauged supergravity, which is still unknown. The expression to be extremized has a suggestive resemblance to anomaly polynomials and the supersymmetric Casimir energy recently studied for N = 4 super Yang-Mills.
AdS-CFT Correspondence; Black Holes in String Theory; Supergravity Mod-
-
HJEP07(21)6
1 Introduction
2 Supersymmetric AdS5 black holes in U(
1
)3 gauged supergravity
2.1
2.2
The asymptotic AdS5 vacuum
Properties of the solution
3 An extremization principle for the entropy
4 Dimensional reduction in the limiting case: Jφ = J
ψ
4.1
4.2
4.3
The near-horizon geometry
Dimensional reduction on the Hopf fibres of squashed S3
Attractor mechanism in four dimensions
4.4 Comparison with five-dimensional extremization
5 Discussion and future directions
A Five-dimensional N = 2 gauged supergravity
B Four-dimensional N = 2 gauged supergravity
C Generalities about the supersymmetric Casimir energy
the supersymmetric partition function on S3 ×S1 is equal to the superconformal index only
up to a multiplicative factor e−βESUSY , where the supersymmetric Casimir energy ESUSY is
of order N 2 [10–16]. However, it is not clear what the average energy of the vacuum should
have to do with the entropy, which is the degeneracy of ground states. The analogous
problem for static, asymptotically AdS4 dyonic black holes was recently solved in [17, 18].
It was shown in [17, 18] that the topologically twisted index for three-dimensional gauge
– 1 –
theories [19]1 has no large cancellation between bosons and fermions at large N , it scales
like N 3/2 [17, 24, 25] (see also [26]) and correctly reproduces the entropy of a class of
BPS black holes in AdS4 × S7. More precisely, the topologically twisted index Ztwisted is a
function of magnetic charges pi and fugacities Δi for the global symmetries of the theory.
The entropy of the black holes with electric charges qi is then obtained as a Legendre
transform of log Ztwisted:
S(qi, pi) = log Ztwisted(pi, Δi) − i X qiΔi ¯
Δi
,
i
(1.1)
where Δ¯ i is the extremum of I(Δi) = log Ztwisted(pi, Δi) − iqiΔi. This procedure has been
called I−extremization in [17, 18] and shown to correspond to the attractor mechanism in
gauged supergravity [27–33].
It is natural to ask what would be the analogous of this construction in five dimensions.
In this paper, we humbly look attentively at the gravity side of the story and try to
understand what kind of extremization can reproduce the entropy of the supersymmetric
rotating black holes. Unfortunately, the details of the attractor mechanism for rotating
black holes in five-dimensional gauged supergravity are not known but we can nevertheless
find an extremization principle for the entropy. The final result is quite surprising and
intriguing.
We consider the class of supersymmetric rotating black holes found and studied in [1–5].
They are asymptotic to AdS5 × S5 and depend on three electric charges QI (I = 1, 2, 3),
associated with rotations in S5, and two angular momenta Jφ, Jψ in AdS5. Supersymmetry
actually requires a constraint among the charges and only four of them are independent.
We show that the Bekenstein-Hawking entropy of the black holes can be obtained as the
Legendre transform of the quantity2
(1.2)
(1.3)
E = −iπN 2 Δ1Δ2Δ3
ω1ω2
where ΔI are chemical potentials conjugated to the electric charges QI and ω1,2 chemical
potentials conjugated to the angular momenta Jφ, Jψ. The constraint among charges is
reflected in the following constraint among chemical potentials,
Δ1 + Δ2 + Δ3 + ω1 + ω2 = 1 .
To further motivate the result (1.2) we shall consider the case of equal angular momenta
Jψ = Jφ. In this limit, the black hole has an enhanced SU(2) × U(
1
) isometry and it can
be reduced along the U(
1
) to a static dyonic black hole in four dimensions. We show that,
upon dimensional reduction, the extremization problem based on (1.2) coincides with the
attractor mechanism in four dimensions, which is well understood for static BPS black
holes [27–33].
charges.
1For further developments see [20–23].
2Notice that one can write the very same entropy as the result of a different extremization in the context
× S1 and the superconformal index I. Both the partition function and the
superconformal index are functions of a set of chemical potentials ΔI (I = 1, 2, 3) and ωi
(i = 1, 2) associated with the R-symmetry generators U(
1
)3 (...truncated)