CFT Duals for Black Rings and Black Strings

Alexandra Chanson111E-mail<EMAIL_ADDRESS>Victoria Martin222E-mail:
<EMAIL_ADDRESS>
Maria J. Rodriguez333E-mail: majo.rodriguez.b@gmail.coma,c and Luis Fernando
Temoche444E-mail<EMAIL_ADDRESS>
a Department of Physics, Utah State University, 4415 Old Main Hill Road, UT
84322, USA
b University of Iceland, Science Institute, Dunhaga 3, 107 Reykjavik, Iceland
c Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid,13-15
Calle Nicolas Cabrera, 28049 Madrid, Spain

###### Abstract

Holographic dualities between certain gravitational theories in four and five
spacetime dimensions and 2D conformal field theories (CFTs) have been proposed
based on hidden conformal symmetry exhibited by the radial Klein-Gordon (KG)
operator in a so-called near-region limit. In this paper, we examine hidden
conformal symmetry of black rings and black strings solutions, thus
demonstrating that the presence of hidden conformal symmetry is not linked to
the separability of the KG-equation (or the existence of a Killing-Yano
tensor). Further, we will argue that these classes of non-extremal black holes
have a dual 2D CFT. New revised monodromy techniques are developed to
encompass all the cases we consider.

###### Contents

  1. 1 Introduction
  2. 2 Methods
    1. 2.1 Revised monodromy technique
    2. 2.2 Building conformal coordinates
  3. 3 Warm-up: Kerr Black Hole
  4. 4 Dipole Black Ring and Black String
    1. 4.1 Monodromy analysis
    2. 4.2 Boosted charged black string
    3. 4.3 Conformal coordinates
  5. 5 Doubly-spinning Black Ring and Kerr Black String
    1. 5.1 Monodromy analysis
    2. 5.2 Boosted Kerr black string
    3. 5.3 Conformal coordinates
  6. 6 Discussion
  7. A Black Ring Metrics and Identities
    1. A.1 Dipole Black Ring
    2. A.2 Doubly-spinning Black Ring

##  1 Introduction

The Kerr black hole metric is considered one of the most relevant solutions of
General Relativity due to its astrophysical applications, and also celebrated
for its mathematical structure and symmetry. The solution describes a rotating
asymptotically flat black hole in four spacetime dimensions, and is famously
conjectured to be dual to a two dimensional conformal field theory (2D CFT).
This conjectured duality holds both in the extremal case (via a near-horizon
limit in the extremal Kerr geometry, or NHEK limit) [1] and the non-extremal
case (via a near-horizon limit in the dynamics that reveals hidden conformal
symmetry of the radial Klein-Gordon (KG) equation) [2].

One remarkable feature of the Kerr metric is that it leads to a separable KG-
equation [3]. We will call black hole solutions with a separable KG-equation
simply separable black hole metrics. It was later shown that such classes of
separable black hole solutions are uniquely determined by requiring the
existence of a principal closed conformal Killing-Yano form [4] (and
references therein). The geometry possesses a high degree of symmetry encoded
in the existence of a tower of Killing vectors and tensors. Unlike Killing
vectors, these higher-rank Killing tensors do not generate explicit isometries
of the metric, and for this reason they are often called “hidden” symmetry
generators [5]. An interesting question is to determine to what extent these
hidden symmetries of separability are related to the hidden conformal
symmetries found in geometries like Kerr [2].

Many miraculous properties of the Kerr solution survive in higher dimensions.
The KG-equations for many higher-dimensional solutions of the Einstein
equations are separable, and the geometry possesses a high degree of symmetry
encoded in a tower of Killing vectors and tensors [4, 6]. However, this
exceptional symmetry does not extend to all higher dimensional black hole
solutions of General Relativity. In particular, the KG-equations for generic
black ring solutions [7] are believed to be non-separable.

In the context of the nonextremal Kerr/CFT correspondence [2], hidden
conformal symmetry leads to the proposal that D-dimensional black holes are
dual to 2D CFTs (in contrast to D$-1$ CFTs) in a certain limit, stemming from
the analysis of the separable KG-equations. As previously mentioned, one
pressing question that remains to be clarified is whether the hidden conformal
symmetries relevant for the CFT description of nonextremal black holes are
bound to the existence of higher degrees of symmetry encoded in Killing
tensors, which are ultimately responsible for KG-equation separability.

We will argue that the CFT interpretation for generic black holes is valid
even without full separability of the KG-equation. Moreover, we conjecture
that the existence of hidden symmetry structures in the KG-equation will be
guaranteed for black hole metrics with two horizons – an outer event horizon
$r=r_{+}$ and Cauchy inner horizon $r=r_{-}$. In other words, black hole
metrics containing a smooth extremal limit $r_{ext}=r_{+}=r_{-}$. This
property for black holes seems to be the key for the CFT interpretations
rather than separability and existence of Killing-Yano forms.

Other techniques involving the CFT interpretation of black hole horizons
include the so-called monodromy technique [8, 9, 10]. In this approach,
expressions involving the monodromies are interpreted by making explicit the
relationship between the monodromy of the solutions and greybody factors for
fully separable KG-equations. One then wonders: are the solution monodormies,
hidden symmetries of KG separability and hidden conformal symmetries from the
dynamics related? We find that separability is not required, and therefore not
linked to the hidden conformal symmetries in the KG equation. Indeed, the
hidden conformal symmetry structure can be obtained from the monodromies,
allowing a 2D CFT interpretation even for black holes exhibiting non-separable
KG equations. With appropriate modifications to the monodormy technique, we
will be able to compute explicit relations that can be interpreted as
effective temperatures of a 2D CFT for generic black hole solutions of vacuum
Einstein’s equations with at least two horizons.

In this article, we pay special attention to two black ring solutions, the
dipole black ring [11] and the doubly-spinning black ring [12, 13]. For both
solutions, we construct the non-separable KG equation, take a constant angular
slice $\theta_{o}$ (as us done for NHEK), and keep only the pole terms. This
approach should be contrasted with the so-called “near-region limit” of [2]
and others. In those works, they take a limit $\omega M<<1$, $\omega r<<1$,
which has been endowed with a physical interpretation involving soft hair
[14]. However, the same hidden symmetry arguments can be obtained without such
a specific limit and simply focusing on the poles of the radial equation.
Indeed, we will find that the black ring solutions would require a more
complicated limit than that proposed by [2].

From the radial pole terms of our KG equations, we can extract the monodromies
and build the temperatures $T_{L}$ and $T_{R}$ for the purported dual 2D CFT.
We then move on to construct conformal coordinates for these spacetimes, to
further exhibit the hidden conformal symmetry structure. We show a difficulty
arises in constructing an appropriate radial function for the conformal
coordinates, reminiscent of what was found in [15] for higher-dimensional
spacetimes (D$>$5). We make progress by considering two limits that we
describe: a black string limit and a near-horizon limit in the conformal
coordinates. In this way we succeed in constructing conformal coordinates for
these systems.

This article is organized as follows. In Section 2 we outline our formalism,
namely the revised monodromy technique and the construction of conformal
coordinates. In Section 3 we demonstrate these techniques on the Kerr black
hole before considering our spacetimes of interest: the dipole black ring
(Section 4) and the doubly-spinning black ring (Section 5). In our Discussion
Section 6 we review our results and conclusions. We relegate some metric
definitions and properties to Appendix A.

##  2 Methods

Here we outline some of the techniques that we use to probe hidden conformal
symmetry in black ring solutions. While much of this section is review, some
of the analysis is, to the best of our knowledge, new.

###  2.1 Revised monodromy technique

We consider the KG equation for a massless scalar evaluated on a D-dimensional
spacetime background

$\nabla^{2}\Psi(x)={\frac{1}{\sqrt{-g}}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu})\Psi(x)=0.$
(1)

Let us assume that the background has $n+1$ Killing vectors. We choose a basis
where the vectors are simply labelled by the coordinates
$K=(\partial_{t},\partial_{\phi_{k}})$ with $k=1,\cdots,n$. The presence of
Killing vectors allows us to decompose the wave function as

$\Psi(x)=\exp(-i\omega
t+i\sum^{n}_{k=1}m_{\phi_{k}}\phi_{k})\,\Phi(r,\theta)\,.$ (2)

The full separability in $(r,\theta)$ coordinates via $\Phi(r,\theta)$ does
not follow from symmetries realized explicitly in the black hole background
and will be the subject of our paper. We will be mostly interested in
solutions that have Cauchy horizons, such as rotating black holes and black
rings. In the appropriate coordinate system these horizons are characterized
by being simple poles of a “radial” coordinate. If we denote this coordinate
by $r$, the horizons are defined by the zeroes of $g^{rr}=0$.

The basic observation for the revised monodromy technique starts as follows.
Horizons are regular surfaces, and in particular we expect that the wave
equation is well behaved around each pole. Hence, instead of analyzing the
behavior of the KG equation and solutions around each pole, we can instead
simply keep the kinetic radial part and pole terms for each horizon
$\\{r_{-},r_{+}\\}$ to identify the monodromy and corresponding 2D CFT
temperatures. This can be regarded as a near zone expansion555Again, this near
zone expansion need not coincide with the near-region limit defined in [2].,
with a local hidden $SL(2,R)\times SL(2,R)$ symmetry – this symmetry does not
correspond to an isometry of the background.

For example, the near zone limit $r\to{r_{j}=\\{r_{-},r_{+}\\}}$ of the black
holes we consider leads to

$\partial_{r}\left[\Delta(r)\partial_{r}\Phi(r)\right]+\left[{F_{+}(\omega,m_{\phi_{k}})^{2}\over(r-r_{+})}+{F_{-}(\omega,m_{\phi_{k}})^{2}\over(r-r_{-})}\right]\Phi(r)=0,$
(3)

where $\Delta(r)=(r-r_{+})(r-r_{-})f(r)$ for a regular function $f(r)$ at
$r_{+},r_{-}$. The solutions to this effective differential equation with a
non-trivial monodromy $\alpha_{j}$ around $(r-r_{j})$ take the form

$\Phi(r)\sim(r-r_{j})^{i\,\alpha_{j}}+(r-r_{j})^{-i\alpha_{j}}~{},\quad\alpha_{j}=\frac{F_{j}(\omega,m_{\phi_{k}})}{\lim_{r\rightarrow
r_{j}}(\Delta(r)/(r-r_{j}))^{1/2}}.$ (4)

The associated monodromy matrix is

$M_{j}=\left(\begin{array}[]{cc}e^{2\pi\alpha_{j}}&0\\\
0&e^{-2\pi\alpha_{j}}\end{array}\right).$ (5)

The curiosity here is that we write $m_{\phi_{k}}$ as a function of
$\alpha_{j}$, such that the coefficients in this relation can be interpreted
as effective temperatures of a 2D CFT. More explicitly, take
$\phi_{k}\to\phi_{k}+2\pi$, and then the wave function changes as [16]

$\Psi(\phi_{k}+2\pi)=e^{2\pi
im_{\phi_{k}}}\Psi(\phi_{k})=e^{-i4\pi^{2}(-T_{L,\phi_{k}}\omega_{L}+T_{R,\phi_{k}}\omega_{R})}\Psi(\phi_{k})\,,$
(6)

where $T_{j,\phi_{k}}$ is determined by rewriting (4) as
$m_{k}=m_{k}(\alpha_{j})$. As we explicitly compute below, it seems in general
that

$\displaystyle\omega_{R}=\alpha_{+}+\alpha_{-}\,,\qquad\omega_{L}=\alpha_{+}-\alpha_{-},$
(7)

involving the monodromies around the outer (+) and inner (-) horizons. These
relations seem to indicate that $T_{R/L,\phi_{k}}$’s are the relevant
temperatures to describe the black hole entropy as a Cardy formula for a 2D
CFT.

###  2.2 Building conformal coordinates

In this subsection we describe a streamlined technique for constructing
conformal coordinates, and also express the monodromy parameters
$\alpha_{\pm}$ in terms of these. We will use the technique outlined here to
attempt to construct coordinates that reproduce the conformal structure of the
near zone black ring solutions.

Following [2, 17], we seek a coordinate transformation of the following form

$\begin{split}w^{+}&=f(r)e^{\alpha\phi+\beta t}\\\
w^{-}&=f(r)e^{\gamma\phi+\delta t}\\\
y&=g(r)e^{1/2((\alpha+\gamma)\phi+(\beta+\delta)t)}.\end{split}$ (8)

Note that we have selected a single angular direction $\phi$. The primary
purpose of these conformal coordinates is to construct an $SL(2,R)$ Casimir

$\mathcal{H}^{2}=\frac{1}{4}\left(y^{2}\partial_{y}^{2}-y\partial_{y}\right)+y^{2}\partial_{+}\partial_{-},$
(9)

which is proportional to the near zone radial KG operator acting on our probe
scalar field $\Phi=R(x)e^{i(k\phi-\omega t)}$. A secondary feature that we
would like these coordinates to possess is that near the bifurcation surface
($w^{\pm}=0$) the metric becomes warped $AdS_{3}$ to leading order.

It will sometimes be cleaner to reparametrize the radial coordinate $r$ in the
following way

$\begin{split}w^{+}&=\sqrt{\frac{x-\frac{1}{2}}{x+\frac{1}{2}}}e^{\alpha\phi+\beta
t}\\\ w^{-}&=\sqrt{\frac{x-\frac{1}{2}}{x+\frac{1}{2}}}e^{\gamma\phi+\delta
t}\\\
y&=\sqrt{\frac{1}{x+\frac{1}{2}}}e^{1/2((\alpha+\gamma)\phi+(\beta+\delta)t)}.\end{split}$
(10)

The nice thing about the $x$-coordinate is that this form (10) is the same in
both four and five dimensions. For Kerr,
$x=\frac{2r-r_{-}-r_{+}}{2(r_{+}-r_{-})}$ and for the 5D Myers-Perry black
hole $x=\frac{r^{2}-1/2(r_{+}^{2}+r_{-}^{2})}{r_{+}^{2}-r_{-}^{2}}$. Indeed,
we will find that the conformal coordinates for the black ring and black
string solutions fit this form as well.

In terms of these general conformal coordinates (10), we can write down the
Casimir (9) as

$\begin{split}\mathcal{H}^{2}R(x)=&\left(\partial_{x}\Delta\partial_{x}+\frac{(\omega(\alpha+\gamma)+k(\beta+\delta))^{2}}{4\left(x-\frac{1}{2}\right)(\beta\gamma-\alpha\delta)^{2}}-\frac{(\omega(\alpha-\gamma)+k(\beta-\delta))^{2}}{4\left(x+\frac{1}{2}\right)(\beta\gamma-\alpha\delta)^{2}}\right)R(x),\end{split}$
(11)

where $\Delta=\left(x-\frac{1}{2}\right)\left(x+\frac{1}{2}\right)$. This is
what should be compared to the near zone radial KG operator to determine the
parameters $(\alpha,\beta,\gamma,\delta)$. The solutions to the equation
$\mathcal{H}^{2}R(x)=0$ are hypergeometric functions (by construction). This
analysis also gives us the monodromy data in terms the the conformal
coordinates. The monodromy around the outer horizon is

$R(x)\sim\left(x-\frac{1}{2}\right)^{i\alpha_{+}},\qquad\alpha_{+}=\frac{\omega(\alpha+\gamma)+k(\beta+\delta)}{2(\beta\gamma-\alpha\delta)}.$
(12)

The monodromy around the inner horizon is

$R(x)\sim\left(x+\frac{1}{2}\right)^{i\alpha_{-}},\qquad\alpha_{-}=\frac{\omega(\alpha-\gamma)+k(\beta-\delta)}{2(\beta\gamma-\alpha\delta)}.$
(13)

These expressions for the monodromies make direct contact with the parameters
in the Casimir. We can now identify the CFT temperatures as proposed in [17,
18]. The analysis of the periodicities yields

$\displaystyle\alpha=2\pi T_{R},\qquad\gamma=2\pi T_{L}\,,$ (14)

or equivalently

$\displaystyle T_{R}=\alpha/(2\pi)\,,\qquad T_{L}=\gamma/(2\pi)\,.$ (15)

Via a Cardy formula for a 2D CFT, these will reproduce the expressions for the
inner and outer black hole entropies.

##  3 Warm-up: Kerr Black Hole

The simplest case where we can exhibit the formalism of Section 2 is the Kerr
black hole. For Kerr, with generic mass $M$ and angular momentum $J=Ma$, the
full radial KG equation is666For the Kerr metric and more details regarding
the resulting KG equation, please see for example [2].

$\displaystyle\left[\partial_{r}\left((r-r_{+})(r-r_{-})\partial_{r}\right)+{({2Mr_{+}\omega}-{a}m)^{2}\over(r-r_{+})(r_{+}-r_{-})}\right.$
(16)
$\displaystyle\left.-{({2Mr_{-}\omega}-{a}m)^{2}\over(r-r_{-})(r_{+}-r_{-})}+(r^{2}+2M(r+2M))\omega^{2}\right]\Phi(r)=K_{\ell}\,\Phi(r)~{}.$
(17)

In the near zone expansion we have

$\left[\partial_{r}\left((r-r_{+})(r-r_{-})\partial_{r}\right)+{({2Mr_{+}\omega}-{a}m)^{2}\over(r-r_{+})(r_{+}-r_{-})}-{({2Mr_{-}\omega}-{a}m)^{2}\over(r-r_{-})(r_{+}-r_{-})}\right]\Phi=0.$
(18)

In this case, the near zone expansion (18) is identical to the near-region
limit reported in [2]. Employing the definition (4), we can now compute the
monodromies for the outer horizon $r=r_{+}$

$\Phi\sim(r-r_{+})^{\pm\,i\alpha_{+}}~{},\quad\alpha_{+}={({2Mr_{+}\omega}-{a}m)\over(r_{+}-r_{-})}.$
(19)

Similarly, for the inner horizon $r=r_{-}$ it gives

$\Phi\sim(r-r_{-})^{\pm\,i\alpha_{-}}~{},\quad\alpha_{-}={({2Mr_{-}\omega}-{a}m)\over(r_{+}-r_{-})}.$
(20)

Solving for $m$ in terms of $\alpha_{\pm}$ we get

$-m={r_{+}\over a}\alpha_{-}-{r_{-}\over a}\alpha_{+}.$ (21)

Taking $\phi\to\phi+2\pi$, the wave function (97) changes as

$\displaystyle\Psi(\phi+2\pi)=\exp\left(-2\pi i{r_{+}\over a}\alpha_{-}+2\pi
i{r_{-}\over a}\alpha_{+}\right)\Psi(\phi).$ (22)

Changing the basis to $\omega_{R,L}=\alpha_{+}\pm\alpha_{-}$ we get777This is
related to the fact that the identification $\phi\rightarrow\phi+2\pi$ is
generated by the group element
$e^{-i4\pi^{2}(T_{R}H_{0}+T_{L}\bar{H}_{0})}=e^{2\pi\partial_{\phi}}$.

$\displaystyle\Psi(\phi+2\pi)=e^{i4\pi^{2}\,(T_{L}\,\omega_{L}-\,T_{R}\,\omega_{R})}\Psi(\phi),$
(23)

where

$T_{L}={r_{+}+r_{-}\over 4\pi a}~{},\quad T_{R}={r_{+}-r_{-}\over 4\pi a}.$
(24)

These are the same temperatures computed in [2] using the hypergeometric
equation. The CFT microstate degeneracy inferred from the Cardy formula with
central charge $c_{L}=c_{R}=12J$ agrees exactly with the Bekenstein-Hawking
area law. Further, the product of the inner and outer entropy, representing a
minimal phase-volume, is quantized as a full spin surface area:

$S_{+}S_{-}=4\pi^{2}J^{2}.$ (25)

##  4 Dipole Black Ring and Black String

The black ring solutions in this and the upcoming section distinguish
themselves from other black objects due to their non-trivial $S^{1}\times
S^{2}$ horizon topology. Known as $D=\\{10,11\\}$ supergravity solutions [19],
they can also be found as systems in minimal $\mathcal{N}=1,D=5$ supergravity.
A classification of those solutions can be found in [20]. The IR limit of the
latter leads to the identification of black rings as exact solutions of
General Relativity [21] in $D=5$, erecting themselves as clear violations of
the uniqueness theorem for black holes [22]. The dipole black ring in
particular is a generalization of [21], being an exact solution of Einstein-
Maxwell-dilaton theory. We will be considering the case where the dilaton
$\phi$ is decoupled, thus giving us pure Einstein-Maxwell theory with winding
number $N=3$. More precisely, we will focus on the neutral (non-
supersymmetric) dipole spinning black ring specified by the three physical
parameters $(M,J_{\psi},q)$: the mass, angular momentum along $S^{1}$ and
dipole magnetic charge, respectively. An in-depth description of the dipole
black ring solution is given in Appendix A. The black ring solution [21] is
parametrized by a scale $R$, the dipole parameter $\mu$ and $\lambda,\nu$
within $0<\nu\leq\lambda<1$ and $0\leq\mu<1$.

Although the $(x,y,\psi)$-coordinates discussed in Appendix A are useful for a
compact expression for the metric, it is instructive to consider
$(r,\theta,z)$-coordinates defined by

$\displaystyle x=\cos\theta,\qquad y=-R/r\,,\qquad\text{
and}\qquad\psi=z/R\,,$ (26)

as well as the constants $\nu=r_{0}/R$, $\lambda=(r_{0}/R)\cosh^{2}\sigma$ and
$\mu=(r_{0}/R)\sinh^{2}\gamma$, where $\gamma$ gives a convenient
parametrization of the charge. We label the functions $s^{2}_{X}=\sinh^{2}X$
and $c^{2}_{X}=\cosh^{2}X$ from here onwards. This is a convenient set of
coordinates to take the straight black string limit $R\rightarrow\infty$. In
these coordinates the outer horizon is at $r=r_{0}$ and the inner horizon is
at $r=0$. The two horizons coincide when $\nu=0$, which defines the extremal
limit, and thus $\nu$ can be regarded as a non-extremality parameter.

The expressions for the entropy were computed in [23] yielding

$\displaystyle S_{+}$ $\displaystyle=$
$\displaystyle{2\pi^{2}r_{0}^{2}R\,c_{\gamma}^{3}\,c_{\sigma}\over
G_{5}}{(1+r_{0}s^{2}_{\gamma}/R)^{3}(1-(r_{0}c^{2}_{\sigma}/R)^{2})^{1/2}\over(1-r_{0}/R)^{2}(1+r_{0}/R)}$
(27) $\displaystyle S_{-}$ $\displaystyle=$
$\displaystyle{2\pi^{2}r_{0}^{2}R\,s_{\gamma}^{3}\,s_{\sigma}\over
G_{5}}{(1+r_{0}s^{2}_{\gamma}/R)^{3}(1-(r_{0}c^{2}_{\sigma}/R)^{2})^{1/2}\over(1-r_{0}/R)^{2}}\,.$
(28)

In addition, we have the angular momentum

$\displaystyle J_{\psi}=\frac{\pi
R^{2}r_{0}c_{\sigma}s_{\sigma}}{2G_{5}}\frac{\left(1+\frac{r_{0}s^{2}_{\gamma}}{R}\right)^{9/2}\left(1+\frac{r_{0}c^{2}_{\sigma}}{R}\right)^{1/2}}{\left(1-\frac{r_{0}}{R}\right)^{2}}\,,$
(29)

and dipole charge

$q^{3}={2\pi^{2}r_{0}^{3}s_{\gamma}c_{\gamma}^{3}\over
G_{5}}{(1+r_{0}s^{2}_{\gamma}/R)^{3}\over(1-r_{0}/R)^{3}}\left({1-(r_{0}c^{2}_{\sigma}/R)\over
1-r_{0}s^{2}_{\gamma}/R}\right)^{3/2}\,.$ (30)

Interestingly, as observed in [23], it is the case that a the product of the
inner and outer entropies is independent of the mass of the black hole and
therefore depends solely on the quantized charges

$S_{+}S_{-}=4\pi^{2}\,J_{\psi}\,q^{3}.$ (31)

We now use the revised monodromy technique to analyze the KG-equation for a
massless scalar field on the dipole black ring geometry in the
$(r,\theta,z)$-coordinates. This allows us to compute temperatures of the
proposed dual 2D CFT.

###  4.1 Monodromy analysis

In eigenmodes the scalar field is of the form

$\displaystyle\Psi(t,r,\theta,\phi,z)=e^{-it\omega+im\phi+inz}\,\left(1+\frac{r}{R}\cos\theta\right)\,\Phi(r,\theta).$
(32)

Notice that both $m$ and $n$ are not integers because $(\phi,z)$ do not have
periodicity $2\pi$. Below we will account for this. The classical wave
equation for $\Phi(r,\theta)$ becomes

$\displaystyle\partial_{r}\left(r\,(r-r_{0})\left(1-\frac{r^{2}}{R^{2}}\right)\partial_{r}\,\Phi\right)+\frac{1}{\sin\theta}\,\partial_{\theta}\left(\left(1+\frac{r_{0}}{R}\cos\theta\right)\sin\theta\,\partial_{\theta}\,\Phi\right)$
(33)
$\displaystyle+\frac{(r+r_{0}\,s_{\gamma}^{2})^{3}}{r\,(r_{0}-r)\left(1-\frac{r^{2}}{R^{2}}\right)(r-r_{0}c_{\sigma}^{2})}\left(\omega\,r_{0}c_{\sigma}s_{\sigma}\left(1-\frac{r}{R}\right)\sqrt{\frac{1+\frac{r_{0}}{R}c_{\sigma}^{2}}{1-\frac{r_{0}}{R}c_{\sigma}^{2}}}-n(r-r_{0}c_{\sigma}^{2})\right)^{2}\Phi$
$\displaystyle+\omega^{2}\frac{(1+\frac{r_{0}}{R}c_{\sigma}^{2}\cos\theta)^{2}(r+r_{0}\,s_{\gamma}^{2})^{3}}{(1+\frac{r}{R}\cos\theta)^{2}(r-r_{0}c_{\sigma}^{2})}\,\Phi-m^{2}\frac{(1+\frac{r_{0}}{R}c_{\sigma}^{2}\cos\theta)(1-\frac{r_{0}}{R}s_{\gamma}^{2}\cos\theta)^{3}}{(1+\frac{r_{0}}{R}\cos\theta)\sin^{2}\theta}\,\Phi$
$\displaystyle+\left(\frac{r_{0}}{R}\,\cos\theta-\frac{r}{R}(2r-r_{0})\right)\,\Phi=0$

If we only keep the kinetic radial part and the single poles for each horizon,
the wave function becomes

$\displaystyle\partial_{r}\left({r\,(r-r_{0})(1-\frac{r^{2}}{R^{2}})}\,\partial_{r}\,\Phi\right)+{r_{0}^{3}\over
r-r_{0}}c_{\gamma}^{6}{1-{r_{0}\over R}\over 1+{r_{0}\over
R}}\left({s_{\sigma}\over 1-{r_{0}\over
R}}n+{c_{\sigma}}K\omega\right)^{2}\Phi$ (34) $\displaystyle-{r_{0}^{3}\over
r}s_{\gamma}^{6}\left(c_{\sigma}n+s_{\sigma}K\omega\right)^{2}\Phi=0.$

The calculation of the monodromies employing the prescription (4) is
straightforward. Around the outer event horizon $r\to r_{0}$:

$\Phi\sim(r-r_{0})^{\pm
i\alpha_{+}}~{},\quad\alpha_{+}={r_{0}}{c_{\gamma}^{3}\over 1+{r_{0}\over
R}}\left({s_{\sigma}\over 1-{r_{0}\over R}}n+{c_{\sigma}}K\omega\right).$ (35)

We can also compute the monodromy around the inner horizon $r\to 0$:

$\Phi\sim r^{\pm
i\alpha_{-}}~{},\quad\alpha_{-}={r_{0}}s_{\gamma}^{3}\left(c_{\sigma}n+s_{\sigma}K\omega\right).$
(36)

Here we have an expression for $\alpha_{\pm}$ as a function of $\omega$ and
$n$. Solving for $n$ gives

$n={1-{r_{0}\over R}\over 1-c_{\sigma}^{2}{r_{0}\over R}}{1\over
2r_{0}s_{\gamma}^{3}c_{\gamma}^{3}}\left[\left(c_{\sigma}c_{\gamma}^{3}-s_{\sigma}s_{\gamma}^{3}(1+{r_{0}\over
R})\right)\,\omega_{L}+\left(c_{\sigma}c_{\gamma}^{3}+s_{\sigma}s_{\gamma}^{3}(1+{r_{0}\over
R})\right)\,\omega_{R}\right]$ (37)

with $\omega_{R,L}=\alpha_{+}\pm\alpha_{-}$. The wave function transforms as

$\displaystyle\Psi(z+2\pi\Delta z)=e^{2\pi\Delta
z\,in}\Psi(z)=e^{-i4\pi^{2}\left(-T_{L}\omega_{L}+T_{R}\omega_{R}\right)}\Psi(z)$
(38)

with conformal temperatures

$\displaystyle T_{R}$ $\displaystyle=$ $\displaystyle{\Delta z\over
2\pi}{1-{r_{0}\over R}\over 1-c_{\sigma}^{2}{r_{0}\over R}}{1\over
2r_{0}s_{\gamma}^{3}c_{\gamma}^{3}}\left(c_{\sigma}c_{\gamma}^{3}-s_{\sigma}s_{\gamma}^{3}(1+{r_{0}\over
R})\right)$ (39) $\displaystyle T_{L}$ $\displaystyle=$ $\displaystyle{\Delta
z\over 2\pi}{1-{r_{0}\over R}\over 1-c_{\sigma}^{2}{r_{0}\over R}}{1\over
2r_{0}s_{\gamma}^{3}c_{\gamma}^{3}}\left(c_{\sigma}c_{\gamma}^{3}+s_{\sigma}s_{\gamma}^{3}(1+{r_{0}\over
R})\right)$ (40)

and

$\displaystyle\Delta
z=2\pi\frac{(1+r_{0}s_{\gamma}^{2}/R)^{3/2}(1-r_{0}c_{\sigma}^{2}/R)^{1/2}}{(1-r_{0}/R)}\,.$
(41)

Here $\Delta z$ is the periodicity of the azimuthal direction in the black
ring solution. Fixing this condition represents a balance between forces in
the ring that can be achieved when there are no conical singularities. See
e.g. [11] for more details. Our results here for the dipole black ring apply
for both situations, with or without balance. For our computations, in order
to link the results between the black ring and strings, it will be useful to
not fix the bound. Note that to be able to take the black string limit
$R\rightarrow\infty$ one needs to to keep the quantity $\Delta z$ unfixed.

A straightforward calculation using these results shows that the black ring
satisfies a Cardy entropy formula [24]:

$\displaystyle S_{\pm}$ $\displaystyle=$
$\displaystyle\frac{c\pi^{2}}{3}(\,T_{L}\pm\,T_{R})\,$ (42)

with central charge

$\displaystyle c=6\,q^{3}\,$ (43)

and the entropies reported in (27). In analogy to the supersymmetric cases
[7], the entropy is independent of the mass and a power of the dipole charge.
The central charge agrees with that of [11]. In these cases the Bekenstein-
Hawking entropy of a large extremal ring can be reproduced through a
microscopic calculation.

It is worth noticing that the extremal black ring configuration
$\nu\rightarrow 0$ has $T_{R}=0$. The extremal system $\nu=0$ is regarded as
the ground state of the ring with finite radius. The matching to the results
previously obtained from the thermodynamical characterization of the dual CFT
is remarkable. In the supersymmetric case, the black ring exhibits a set of
three dipole charges $q_{i}$, which when quantized become $n_{i}$:

$n_{i}=\Big{(}\frac{\pi}{4G_{5}}\Big{)}^{1/3}q_{i}.$ (44)

For both dipole black ring and its supergravity cousin, extremality is reached
when $\nu\rightarrow 0$. For the latter case, the extremal configuration can
be regarded as the ground state of a ring with large but finite radius [11].
Then in the extremal, large string limit, the central charge becomes

$c=6{n}_{1}{n}_{2}{n}_{3}{n}_{p}.$ (45)

Further, for these configurations $n_{p}$ corresponds to the chiral momentum
excitations. Determined through a Komar integral, its value is:

$n_{p}=J$ (46)

where $J$ is the ADM value of the angular momentum of the ring and $n_{i}$ can
be identified with the number of each type of M5 branes forming the ring,

$n_{i}=\left(\frac{2\pi}{G}\right)^{1/3}Q_{i}\,.$ (47)

###  4.2 Boosted charged black string

The black string solution appears as a higher dimensional solution of
Einstein-Maxwell theory. These black objects attracted great interest
regarding their event horizon stability. The well-known Gregory-Laflamme
instability [25] was described first in the context of black strings. Since
then, further studies of the instability and the fate of black strings
evolving into regular higher dimensional black hole configurations have
populated the literature [26, 27].

We introduce the metric and gauge field for the charged boosted black string

$\displaystyle
ds^{2}=-\frac{\hat{f}}{h}\Big{[}dt-\frac{{r_{0}}{c_{\sigma}}{s_{\sigma}}}{r\hat{f}}dz\Big{]}^{2}+\frac{f}{\hat{f}h}dz^{2}+h^{2}\Big{[}\frac{dr^{2}}{f}+r^{2}d{\Omega}^{2}_{2}\Big{]}$
(48) $\displaystyle
A_{\phi}=\sqrt{3}{r_{0}}s_{\gamma}c_{\gamma}(1+\cos({\theta})),$ (49)

where ${d\Omega_{2}}^{2}={d\theta}^{2}+{\sin{\theta}}^{2}{d\phi}^{2}$ and

$\displaystyle f=1-\frac{r_{0}}{r}$ (50)
$\displaystyle{\hat{f}}=1-\frac{{r_{0}}{c_{\sigma}}^{2}}{r}$ (51)
$\displaystyle h=1+\frac{{r_{0}}{s_{\gamma}}^{2}}{r}.$ (52)

In the metric (48) the coordinates that we will use to construct our conformal
coordinates will be $(t,r,z)$. The entropy of the outer and inner horizons is
given by [28]

$S_{+}=\frac{2{\pi}^{2}{r}_{0}^{2}R{c}_{\gamma}^{3}c_{\sigma}}{G_{5}},\qquad
S_{-}=\frac{2{\pi}^{2}{r}_{0}^{2}R{s}_{\gamma}^{3}s_{\sigma}}{G_{5}}.$ (53)

By introducing the coordinate transformation $x=\frac{r}{r_{0}}-\frac{1}{2}$
on the metric, it is possible to identify the outer/inner horizon at
$x=\pm\frac{1}{2}$. Then, we calculate the KG equation of the system using the
ansatz

$\Psi(t,x,\theta,\phi,z)=e^{-it\omega+im\phi+inz}R(x)\Omega(\theta)$ (54)

and we verify that the equation is separable. The radial part is given by:

$\displaystyle\Big{[}{\partial}_{x}[\Delta{\partial}_{x}]$ (55)
$\displaystyle+\frac{r_{0}^{2}(c_{\gamma}^{2}+s_{\gamma}^{2}+2x)^{3}[{\omega}^{2}(c_{\sigma}^{2}+s_{\sigma}^{2}+2x)+n^{2}(c_{\sigma}^{2}+s_{\sigma}^{2}-2x)+4{\omega}nc_{\sigma}s_{\sigma}]}{16(x-1/2)(x+1/2)}\Big{]}R(x)$
$\displaystyle=K_{l}R(x)$

where $\Delta=(x-\frac{1}{2})(x+\frac{1}{2})$. The angular component is

$\displaystyle\frac{1}{\sin{\theta}}{\partial}_{\theta}\Big{[}\sin\theta{\partial}_{\theta}\Omega\Big{]}-\frac{m^{2}}{{\sin}^{2}\theta}\Omega=-K_{l}\Omega.$
(56)

We will focus on the radial equation and isolate the pole terms in (55). The
wave equation becomes:

$\Big{[}{\partial}_{x}[\Delta{\partial}_{x}]+{r_{0}}^{2}\Big{(}\frac{{c_{\gamma}}^{6}}{x-\frac{1}{2}}(ns_{\sigma}+\omega
c_{\sigma})^{2}-\frac{{s_{\gamma}}^{6}}{x+\frac{1}{2}}(nc_{\sigma}+\omega
s_{\sigma})^{2}\Big{)}\Big{]}R(x)=0.$ (57)

This agrees with the $R\rightarrow\infty$ limit of (34).

Around the inner horizon $r\rightarrow 0$, the monodromy is given by:

$\Phi\sim r^{\pm
i\alpha_{-}}~{},\quad\alpha_{-}=r_{0}{s}_{\gamma}^{3}(n{c}_{\sigma}+\omega{s}_{\sigma})$
(58)

and with respect to the outer horizon $r\rightarrow r_{0}$:

$\Phi\sim r^{\pm
i\alpha_{+}}~{},\quad\alpha_{+}=r_{0}{c}_{\gamma}^{3}(n{s}_{\sigma}+\omega{c}_{\sigma}).$
(59)

Solving for $n$ gives

$n=\frac{1}{2{r}_{0}{c}_{\gamma}^{3}{s}_{\gamma}^{3}}\Big{[}-({c}_{\sigma}{c}_{\gamma}^{3}+{s}_{\sigma}{s}_{\gamma}^{3}){\omega}_{L}+({c}_{\sigma}{c}_{\gamma}^{3}-{s}_{\sigma}{s}_{\gamma}^{3}){\omega}_{R}\Big{]}$
(60)

with $\omega_{R,L}=\alpha_{+}\pm\alpha_{-}$. The wave function transforms as

$\displaystyle\Psi(z+2\pi\Delta z)=e^{2\pi\Delta
z\,in}\Psi(z)=e^{\left(i4\pi^{2}T_{L}\omega_{L}-i4\pi^{2}T_{R}\omega_{R}\right)}\Psi(z).$
(61)

We can read off the left and right temperatures

$T_{L}=\frac{\Delta
z}{2\pi}\frac{({c}_{\sigma}{c}_{\gamma}^{3}+{s}_{\sigma}{s}_{\gamma}^{3})}{2{r}_{0}{c}_{\gamma}^{3}{s}_{\gamma}^{3}},\qquad
T_{R}=\frac{\Delta
z}{2\pi}\frac{({c}_{\sigma}{c}_{\gamma}^{3}-{s}_{\sigma}{s}_{\gamma}^{3})}{2{r}_{0}{c}_{\gamma}^{3}{s}_{\gamma}^{3}}.$
(62)

Again, assuming a Cardy formula holds, we recover the entropies (53) via
completely different means. The central charge is in this case $c=6q^{3}$,
with

$q^{3}={2\pi^{2}r_{0}^{3}s_{\gamma}c_{\gamma}^{3}\over G_{5}}.$ (63)

###  4.3 Conformal coordinates

We attempt to find conformal coordinates of the form (8) that reproduce the
near zone, constant theta slice Laplacian for the dipole black ring. Our first
objective is to determine suitable radial dependence of the functions $f(r)$
and $g(r)$. The procedure that we follow is outlined in Section 3.2 of [15].
Defining the function $h(r)\equiv f(r)/g(r)$, the authors of [15] found that
the Casimir operator in Boyer-Lindquist coordinates takes the form

$\begin{split}\mathcal{H}^{2}=&\frac{h^{2}+1}{4(h^{\prime})^{2}}\partial_{r}^{2}+\left(\frac{1+h^{2}}{4hh^{\prime}}\left[\frac{(h^{\prime})^{2}-h^{\prime\prime}h}{(h^{\prime})^{2}}\right]+\frac{h}{2h^{\prime}}\right)\partial_{r}+F_{1}(r)\partial_{t}^{2}+F_{2}(r)\partial_{t}\partial_{\phi}+F_{3}(r)\partial_{\phi}^{2}.\end{split}$
(64)

In comparing this to the near zone KG operator (34), we can determine the
radial dependence of the conformal coordinates $h(r)$ by solving the following
differential equation

$\frac{\frac{h^{2}+1}{4h^{\prime}h}\frac{d}{dr}\left(\frac{h}{h^{\prime}}\right)+\frac{h}{2h^{\prime}}}{\frac{h^{2}+1}{4h^{\prime
2}}}=\frac{\frac{d\Delta}{dr}}{\Delta},$ (65)

where for the dipole black
$\Delta=r(r-r_{0})\left(1-\frac{r^{2}}{R^{2}}\right)$. This has the following
solution

$h(r)=\frac{\exp\left(c_{2}-c_{1}\left(\frac{\log(r-r_{0})}{r_{0}^{3}-R^{2}r_{0}}+\frac{\log(r)}{R^{2}r_{0}}+\frac{\log(r-R)}{2R^{2}(R-r_{0})}-\frac{\log(r+R)}{2R^{2}(R+r_{0})}\right)\right)}{\sqrt{1-\exp\left(2\left(c_{2}-c_{1}\left(\frac{\log(r-r_{0})}{r_{0}^{3}-R^{2}r_{0}}+\frac{\log(r)}{R^{2}r_{0}}+\frac{\log(r-R)}{2R^{2}(R-r_{0})}-\frac{\log(r+R)}{2R^{2}(R+r_{0})}\right)\right)\right)}},$
(66)

where $c_{1}$ and $c_{2}$ are constants. We see here that we are not able to
choose a good $c_{1}$ that eliminates all of the branch cut behavior for us.
This is the same problem that occurs in higher dimensions, as pointed out in
[15]: a solution $h(r)$ that is free of branch cuts (besides the overall
square root) can only exist if the highest power of $\Delta\sim r^{2}$ (or in
the special case where one can write
$\Delta\propto(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})$).

There are two ways to make progress here. The first is to focus only on the
outer horizon and take a near horizon limit in the conformal coordinates
themselves. This is a reasonable thing to do, as it essentially is taking a
near horizon limit in the dynamics, which is close in spirit to the standard
hidden conformal symmetry analyses [2, 14]. To do this, we choose the
constants in equation (66) to be $c_{2}=0$ and
$c_{1}=r_{0}(R^{2}-r_{0}^{2})/2$, in analogy to what was suggested in [15].
Then we expand the result around $r=r_{0}$ to obtain

$h(r)=f(r_{0},R)\sqrt{\frac{r-r_{0}}{r_{0}}},$ (67)

where $f(r_{0},R)$ is an uninteresting function that is independent of $r$.
You will notice that this is reminiscent of the Kerr answer
$h(r)=\sqrt{\frac{r-r_{+}}{r_{+}-r_{-}}}$, with $r_{+}=r_{0}$ and $r_{-}=0$.

The second way to make progress is to consider the black string limit
$R\rightarrow\infty$. In that case, we would like to solve the differential
equation

$\frac{\frac{h^{2}+1}{4h^{\prime}h}\frac{d}{dr}\left(\frac{h}{h^{\prime}}\right)+\frac{h}{2h^{\prime}}}{\frac{h^{2}+1}{4h^{\prime
2}}}=\frac{2r-r_{0}}{r(r-r_{0})}.$ (68)

This has the solution

$h(r)=\frac{e^{c_{2}}r^{\frac{c_{1}}{r_{0}}}}{\sqrt{(r-r_{0})^{\frac{2c_{1}}{r_{0}}}-e^{2c_{2}}r^{\frac{2c_{1}}{r_{0}}}}}.$
(69)

We have freedom to choose $c_{1}$ and $c_{2}$. One clean choice is to take
$c_{1}=r_{0}/2$ and $c_{2}=0$. Then our radial functions take the particularly
simple form

$h(r)=\sqrt{-\frac{r}{r_{0}}},\qquad g(r)=\sqrt{-\frac{r_{0}}{r-r_{0}}},\qquad
f(r)=gh=\sqrt{\frac{r}{r-r_{0}}}.$ (70)

The fact that two of these radial functions are imaginary does not cause any
problems. The important thing is that the Casimir

$\mathcal{H}^{2}=\frac{1}{4}\left(y^{2}\partial_{y}^{2}-y\partial_{y}\right)+y^{2}\partial_{+}\partial_{-}$
(71)

is still real. The radial functions (70) admit the same $x$-coordinate
structure as presented in (10). For the dipole black ring, we see that
$x=\frac{r}{r_{0}}-\frac{1}{2}$.

At last we are ready to attempt to find the angular conformal coordinate
variables $(\alpha,\beta,\gamma,\delta)$. To do this, we would like to compare
the Casimir

$\begin{split}\mathcal{H}^{2}R(x)=&\left(\partial_{x}\Delta\partial_{x}+\frac{(\omega(\alpha+\gamma)+n(\beta+\delta))^{2}}{4\left(x-\frac{1}{2}\right)(\beta\gamma-\alpha\delta)^{2}}-\frac{(\omega(\alpha-\gamma)+n(\beta-\delta))^{2}}{4\left(x+\frac{1}{2}\right)(\beta\gamma-\alpha\delta)^{2}}\right)R(x)\end{split}$
(72)

and the $R\rightarrow\infty$ limit of the dipole black ring solution (34),
written in the $x$-coordinate:

$\displaystyle\partial_{x}\left({\Delta}\,\partial_{x}\,\Phi\right)+r_{0}^{2}\left({c_{\gamma}^{6}\over
x-\frac{1}{2}}\left({s_{\sigma}}n+{c_{\sigma}}\omega\right)^{2}-{s_{\gamma}^{6}\over
x+\frac{1}{2}}\left(c_{\sigma}n+s_{\sigma}\omega\right)^{2}\right)\Phi=0,$
(73)

where $\Delta=\left(x-\frac{1}{2}\right)\left(x+\frac{1}{2}\right)$. We can
compare $\omega$ and $n$ terms in equations (72) and (73) to find the
conformal coordinate parameters $(\alpha,\beta,\gamma,\delta)$, as in (10):

$\begin{split}w^{+}&=\sqrt{\frac{x-\frac{1}{2}}{x+\frac{1}{2}}}e^{\alpha
z+\beta t}\\\ w^{-}&=\sqrt{\frac{x-\frac{1}{2}}{x+\frac{1}{2}}}e^{\gamma
z+\delta t}\\\
y&=\sqrt{\frac{1}{x+\frac{1}{2}}}e^{1/2((\alpha+\gamma)z+(\beta+\delta)t)}.\end{split}$
(74)

Due to the squared terms in (72) and (73), there are 16 possible combinations
of $(\alpha,\beta,\gamma,\delta)$ that work. We choose a branch based on two
criteria. First, we would like $\alpha=2\pi T_{R}$ and $\gamma=2\pi T_{L}$,
where $T_{R}$ and $T_{L}$ are given by the $R\rightarrow\infty$ limit of (39).
Second, when we plug the conformal coordinates (74) into the black string
metric (48), we would like to see a warped AdS3 factor near the bifurcation
surface $w^{\pm}=0$. The result is

$\begin{split}\alpha=\frac{(c_{\sigma}\,c_{\gamma}^{3}-s_{\sigma}\,s_{\gamma}^{3})}{2r_{0}\,s_{\gamma}^{3}c_{\gamma}^{3}}\,\qquad\beta=\frac{(-c_{\sigma}\,s_{\gamma}^{3}+s_{\sigma}\,c_{\gamma}^{3})}{2r_{0}\,s_{\gamma}^{3}c_{\gamma}^{3}}\\\
\gamma=\frac{(s_{\sigma}\,s_{\gamma}^{3}+c_{\sigma}\,c_{\gamma}^{3})}{2r_{0}\,s_{\gamma}^{3}c_{\gamma}^{3}}\qquad\delta=\frac{(c_{\sigma}\,s_{\gamma}^{3}+s_{\sigma}\,c_{\gamma}^{3})}{2r_{0}\,s_{\gamma}^{3}c_{\gamma}^{3}}\
.\end{split}$ (75)

As desired, the metric near the bifurcation surface has a warped AdS3 factor:

$ds^{2}=4r_{0}^{2}c_{\gamma}^{4}\left(\frac{dw^{+}dw^{-}+\frac{s_{\gamma}^{6}}{c_{\gamma}^{6}}dy^{2}}{y^{2}}+...\right).$
(76)

The correction terms in (76) are at least second order in $w^{\pm}$.

##  5 Doubly-spinning Black Ring and Kerr Black String

In this section we study the hidden conformal symmetries of the doubly-
spinning black ring. The metric is a vacuum solution to the five-dimensional
Einstein equations with event horizon topology $S^{1}\times S^{2}$. The
regular solution was constructed by Pomeransky and Senkov [12] and is
parametrized by mass and two angular momenta $(M,J_{\phi},J_{\psi})$
respectively. It is a single black ring configuration balanced by angular
momentum $J_{\psi}$ in the plane of the ring, but with angular momentum
$J_{\phi}$ also in the orthogonal plane, corresponding to rotation of the
$S^{2}$ sphere (see [29] for a detailed analysis of the physical properties).
A more general version of the doubly-spinning black ring solution,
corresponding to an unbalanced ring with conical singularities was later found
in [13]. It is in fact this more general unbalanced black ring solution that
contains the Kerr black string and 5-dimensional Myers-Perry black hole as a
“collapse” limit of the balanced ring solution. To keep our analysis as
general as possible we obtain our results for the solutions [13].

In order to set up the notation, we give some details of the doubly spinning
black ring in Appendix A.2. The black ring solution has been given in
different related $(x,y)$-forms. We can rewrite this same foliation of space
in a manner that is particularly appropriate in the region near the ring. The
$(r,\theta)$-coordinates employed are a transformation from the
$(x,y)$-coordinates where

$\displaystyle x=\cos\theta,\qquad y=-R/r\,,\qquad\text{
and}\qquad\psi=-z/R\,,$ (77)

and has four independent physical parameters $\mu,\nu,\lambda,R$, of which the
first three are dimensionless and the last sets the scale of the solution
which are required to satisfy

$\displaystyle 0\leq\nu\leq\mu\leq\lambda<1\,.$ (78)

The single spinning black ring [7] with rotation in only one plane is found in
the limit $\nu\rightarrow 0$. The radial and angular coordinates take the
ranges $0\leq r<R$ and $0\leq\theta<\pi$.888One can also redefine the
constants as $\nu=(M-\sqrt{M^{2}-a^{2}})/R=r_{-}/R$,
$\mu=(M+\sqrt{M^{2}-a^{2}})/R=r_{+}/R$ and $\lambda=r_{+}c_{\sigma}/R$. Taking
the ring radius $R$ much larger than the ring thickness $\mu R$ gives the
boosted Kerr-black string metric.Note that $\sqrt{2}R=\varkappa$ with respect
to the definitions of [13].. The KG equation of the doubly-spinning black ring
solution has a hidden phase space symmetry that we argue can be linked to a 2D
CFT.

###  5.1 Monodromy analysis

We now turn to a general analysis of the KG-equation (1) for a massless scalar
with the ansatz

$\tilde{\Psi}(t,r,\theta,\phi,z)=e^{-it\omega+im\phi+inz}\,\left(1+\frac{r}{R}\cos\theta\right)\,\tilde{\Phi}(r,\theta)$
(79)

in the background of the doubly spinning black ring [12, 13]

$\displaystyle\partial_{r}\left(\left(1-\frac{r^{2}}{R^{2}}\right)(r-\mu\,R)(r-\nu\,R)\,\partial_{r}\,\tilde{\Phi}\right)$
(80)
$\displaystyle+\frac{1}{\sin\theta}\,\partial_{\theta}\left((1+\mu\cos\theta)(1+\nu\cos\theta)\sin\theta\,\partial_{\theta}\,\tilde{\Phi}\right)$
$\displaystyle+\left(\frac{C_{\mu\nu\lambda}(1-\mu\nu)^{2}\left(1+\frac{r}{R}\cos\theta\right)^{2}(r/R)^{2}\,\tilde{K}(r,\theta)}{\left(1-\frac{r^{2}}{R^{2}}\right)(r-\mu\,R)(r-\nu\,R)(1+\mu\cos\theta)(1+\nu\cos\theta)\sin^{2}\theta}+\tilde{f}_{r}+\tilde{f}_{\theta}\right)\tilde{\Phi}=0$

where

$\displaystyle\tilde{K}(r,\theta)$ $\displaystyle=$ $\displaystyle
n^{2}R^{2}F_{r\theta}-m^{2}F_{\theta r}+2\,m\,\omega(F_{\theta
r}\,\omega_{\phi}-J_{r\theta}\,\omega_{\psi})+2\,n\,R\,\omega(F_{r\theta}\,\omega_{\psi}+J_{r\theta}\,\omega_{\phi})$
(81) $\displaystyle-2\,n\,m\,R\,J_{r\theta}+\omega^{2}\left(\frac{H_{\theta
r}^{2}}{H_{r\theta}}\,\tilde{F}^{2}+F_{r\theta}\,\omega_{\psi}^{2}-F_{\theta
r}\,\omega_{\phi}^{2}+2\,\omega_{\phi}\,\omega_{\psi}\,J_{r\theta}\right)$
$\displaystyle\tilde{F}\equiv\left(\frac{F_{\theta
r}F_{r\theta}+J_{r\theta}^{2}}{H_{\theta
r}H_{r\theta}}\right)^{1/2}={\left(r^{2}-R^{2}\right)^{1/2}(r-\mu\,R)^{1/2}(r-\nu\,R)^{1/2}G(\cos\theta)^{1/2}\over(1-\nu\mu)(1+{r\over
R}\cos\theta)^{2}}$ (82)
$C_{\mu\nu\lambda}=\frac{(1-\mu)^{2}(1-\nu)}{(1-\mu\nu)(1-\lambda)\Phi\Psi}$
(83) $F_{r\theta}\equiv F(-R/r,\cos\theta)\,,\qquad F_{\theta r}\equiv
F(\cos\theta,-R/r)$ (84) $H_{r\theta}\equiv H(-R/r,\cos\theta)\,,\qquad
H_{\theta r}\equiv H(\cos\theta,-R/r)$ (85) $J_{r\theta}\equiv
J(-R/r,\cos\theta)$ (86)
$\displaystyle\tilde{f}_{r}=\frac{(\mu+\nu)\,r}{R}-\frac{2\,r^{2}}{R^{2}}\qquad\tilde{f}_{\theta}=-\cos\theta\,(\mu+\nu+2\,\mu\,\nu\cos\theta)\,,$
(87)

and the definitions of $F,G,H,J,\Phi,\Psi,\Sigma,\omega_{\psi},\omega_{\phi}$
can be found in Appendix A.2. Recall that the doubly spinning black ring
satisfies the constraints (78) ensuring that the quantities $\Phi$, $\Psi$ and
$\Sigma$ are positive. Under these conditions, the solution has a regular
event horizon located at $r=\mu\,R$. In addition, there is an inner Cauchy
horizon at $r=\nu\,R$. The two horizons coincide when $\nu=0$, which defines
the extremal limit, hence $\nu$ can be regarded as a non-extremality
parameter. In general, a ring-shaped ergosurface is present at $r=\lambda R$.

Focusing on a constant $\theta$-slice of the region near the ring
$r\rightarrow\\{\mu\,R,\nu\,R\\}$ while maintaining all the terms in the KG-
equation that have a singular behavior at the location of the horizon, we find
that the equation (80) takes the form

$\displaystyle\partial_{r}\left(\left(1-\frac{r^{2}}{R^{2}}\right)(r-\mu\,R)(r-\nu\,R)\,\partial_{r}\,\tilde{\Phi}\right)$
(88)
$\displaystyle-\frac{C_{\mu\nu\lambda}\,(1-\mu\nu)}{(r-\mu\,R)(\mu-\nu)^{2}}\left(\sqrt{\frac{\nu\Sigma\Psi}{\Phi}}\,m-\frac{(\mu+\nu)\Sigma\,R}{(1+\mu)}\sqrt{\frac{\lambda(1+\lambda)}{(1-\mu\nu)(1-\lambda)}}\,\omega\right)^{2}\tilde{\Phi}$
$\displaystyle-\frac{C_{\mu\nu\lambda}\,(\mu+\nu)^{2}(1-\mu\nu)(1-\lambda^{2})\nu^{2}}{(r-\nu\,R)(1-\nu^{2})(\mu-\nu)^{2}}\times$
$\displaystyle\left(\frac{\mu\,(1-\lambda\mu)+\nu^{2}(\lambda-\mu)}{(\mu+\nu)}\sqrt{\frac{1}{\nu\Phi}}\,m-(1-\mu)R\sqrt{\frac{\lambda(1+\lambda)\Sigma}{(1-\lambda)(1-\mu\nu)\Psi}}\,\omega\right)^{2}\tilde{\Phi}=0$

for the $n=0$ sector in with analogy to the Myers-Perry black hole solution in
[9, 8]. We find that taking $m=0$ is not possible. This is in contrast with
the Myers-Perry black hole solution, that is $\phi,\psi$ symmetric.

The regular singular points of the equation (88) correspond to the outer
horizon $r=\mu\,R$, inner horizon $r=\nu\,R$ and $r=\pm R$. Now we can proceed
with the monodromy technique in the usual way. The solutions to this effective
differential equation have a non-trivial monodromy $\alpha_{\pm}$ around each
horizon. Near the outer horizon $r\rightarrow\mu\,R$ we have

$\tilde{\Phi}\sim(r-\mu\,R)^{\alpha_{+}}$ (89) $\displaystyle\alpha_{+}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{C_{\mu\nu\lambda}\,(1-\mu\nu)}{(\mu-\nu)^{2}}}\left(\frac{(\mu+\nu)\Sigma\,R}{(1+\mu)}\sqrt{\frac{\lambda(1+\lambda)}{(1-\mu\nu)(1-\lambda)}}\,\omega-\sqrt{\frac{\nu\Sigma\Psi}{\Phi}}\,m\right).$

Around the inner Cauchy horizon $r\rightarrow\nu\,R$ we find

$\tilde{\Phi}\sim(r-\nu\,R)^{\alpha_{-}}\,,$ (90)

where

$\displaystyle\alpha_{-}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{C_{\mu\nu\lambda}\,(\mu+\nu)^{2}(1-\mu\nu)(1-\lambda^{2})\nu^{2}}{(1-\nu^{2})(\mu-\nu)^{2}}}\times$
$\displaystyle\left((1-\mu)R\sqrt{\frac{\lambda(1+\lambda)\Sigma}{(1-\lambda)(1-\mu\nu)\Psi}}\,\omega-\frac{\mu\,(1-\lambda\mu)+\nu^{2}(\lambda-\mu)}{(\mu+\nu)}\sqrt{\frac{1}{\nu\Phi}}\,m\right).$

To analyze the temperatures of the proposed dual CFT we are obliged to
consider each spin sector separately in analogy to [30]. We begin writing the
above monodromy parameters as

$\alpha_{+}=A\omega+Bm,\qquad\alpha_{-}=C\omega+Dm.$ (92)

Solving for $\omega$ and $m$, we find

$m=\frac{C\alpha_{+}-A\alpha_{-}}{BC-AD}.$ (93)

The wave function changes under $\phi\rightarrow\phi+2\pi$ as:

$\tilde{\Psi}(\phi+2\pi)=\tilde{\Psi}(\phi)e^{2\pi im}.$ (94)

Using (93) and

$\omega_{R}=\alpha_{+}+\alpha_{-},\qquad\omega_{L}=\alpha_{+}-\alpha_{-},$
(95)

we find that

$\tilde{\Psi}(\phi+2\pi)=\tilde{\Psi}(\phi)e^{\frac{\pi i}{BC-
AD}\left((A+C)\omega_{L}-(A-C)\omega_{R}\right)}.$ (96)

We can compare this to the relationship

$\tilde{\Psi}(\phi+2\pi)=e^{2\pi
im}\tilde{\Psi}(\phi)=e^{i4\pi^{2}(T_{L}\omega_{L}-T_{R}\omega_{R})}\tilde{\Psi}(\phi)\,,$
(97)

and from this we have the temperatures:

$T_{L}=\frac{A+C}{4\pi(BC-AD)},\qquad T_{R}=\frac{A-C}{4\pi(BC-AD)}.$ (98)

Thus, we find that the temperatures for the double spinning black ring are:

$\begin{split}T_{L}&=\frac{\sqrt{\Phi}(\mu-\nu)\left(-\left(1-\mu^{2}\right)^{3/2}\nu+\Psi\sqrt{C_{\mu\nu\lambda}(\nu+1)\Sigma\Phi(1-\mu\nu)}\right)}{4\pi\left(\Sigma-\left(1-\mu^{2}\right)\nu\right)\sqrt{C_{\mu\nu\lambda}(\lambda+1)\left(1-\mu^{2}\right)\nu\Sigma\Psi(1-\mu\nu)}}\\\
T_{R}&=\frac{\sqrt{\Phi}(\mu-\nu)\left(\left(1-\mu^{2}\right)^{3/2}\nu+\Psi\sqrt{C_{\mu\nu\lambda}(\nu+1)\Sigma\Phi(1-\mu\nu)}\right)}{4\pi\left(\Sigma-\left(1-\mu^{2}\right)\nu\right)\sqrt{C_{\mu\nu\lambda}(\lambda+1)\left(1-\mu^{2}\right)\nu\Sigma\Psi(1-\mu\nu)}}.\\\
\end{split}$ (99)

In the $R\rightarrow\infty$ these reduce to

$T_{L}=\frac{r_{+}+r_{-}}{4\pi a},\qquad T_{R}=\frac{r_{+}-r_{-}}{4\pi a},$
(100)

as expected.

Now, we can check that we can also map these quantities via the Cardy formula
to the Bekenstein-Hawking entropy for the balanced doubly spinning black
rings:

$\displaystyle
S_{+}\equiv\frac{\pi^{2}}{3}(c_{\phi}T_{L}+c_{\phi}T_{R})=\frac{16\sqrt{2}\pi^{2}R^{3}\mu(1+\nu)(\mu+\nu)}{(1-\mu)(1-\mu\nu)^{2}}\,,$
(101)

and the corresponding value of the inner horizon area is
$S_{-}\equiv\frac{\pi^{2}}{3}(c_{\phi}T_{L}-c_{\phi}T_{R})$

$\displaystyle
S_{-}\equiv\frac{\pi^{2}}{3}(c_{\phi}T_{L}-c_{\phi}T_{R})=\frac{16\sqrt{2}\pi^{2}R^{3}\nu(1+\nu)(\mu+\nu)}{(1-\nu)(1-\mu\nu)^{2}}.$
(102)

Then the natural choice for the central charge of the black ring (in analogy
to Kerr) is

$\displaystyle c_{\phi}=12J_{\phi}\,.$ (103)

with angular momentum

$\displaystyle J_{\phi}=\frac{8\pi
R^{3}(\mu+\nu)(1-\mu)}{(1-\mu\nu)^{3/2}}\sqrt{\frac{\nu\lambda(1+\lambda)\Sigma}{(1-\lambda)\Phi\Psi}}\,.$
(104)

This is an appealing, simple picture. In this case, the angular momentum
$J_{\phi}$ is present as in the Kerr black hole. Our results are consistent
with the CFT identifications for the extremal doubly rotating black ring [31].

Recall from [13] that the balance condition is

$\lambda=\frac{2\,\mu}{1+\mu^{2}}\,,$ (105)

and to obtain exactly the form for the entropy as in [12] new parameters have
to be defined: $\tilde{\lambda}=\mu+\nu$, $\tilde{\nu}=\mu\nu$ and
$\tilde{k}=R/\sqrt{2}$.

Finally, we observe that the area product is quantized,

$S_{+}S_{-}=4\pi^{2}J_{\phi}^{2}\,,$ (106)

One feature of this system is a lack of symmetry between the angular
directions; one simply cannot take eigenvalue in the wave function $m=0$ to
analyze other sector. Indeed, when the rotation of the $S^{2}$ is not present
it is unclear how to justify the central charge value of $c_{\phi}$ that is
needed to reproduce the entropy of black rings.

###  5.2 Boosted Kerr black string

In this subsection we will focus on a family of exact solutions to vacuum
5-dimensional General Relativity that is translationally symmetric. This type
of solution is called a Kerr black string. It generalizes a black hole
solution but it also extends along a linear $z$-direction in which it can be
boosted. The physical parameters include mass, spin and linear momentum
$(M,J,P_{z})$. The line element for the boosted Kerr black string is given by
[13, 12, 32]

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2Mr\cosh^{2}\sigma}{\Sigma}\right)dt^{2}+\left(1+\frac{2Mr\sinh^{2}\sigma}{\Sigma}\right)dz^{2}$
$\displaystyle+\frac{(\Delta+2Mr)^{2}-\Delta
a^{2}\sin^{2}\theta}{\Sigma}\sin^{2}\theta
d\phi^{2}+\Sigma\left(\frac{dr^{2}}{\Delta}+d\theta^{2}\right)$
$\displaystyle+\frac{4Mr}{\Sigma}\left[\frac{1}{2}\sinh 2\sigma
dtdz-a\sin^{2}\theta\left(\cosh\sigma dt-\sinh\sigma dz\right)d\phi\right],$

where $\Delta=r^{2}+a^{2}-2Mr$, $\Sigma=r^{2}+a^{2}\cos^{2}\theta$. Here $M$
and $a$ can be regarded as the mass and spin and $\sigma$ is the boost
parameter. As in Kerr also $M=(r_{+}+r_{-})/2$, $a=\sqrt{r_{+}r_{-}}$ or
$J=Ma$. This geometry can be obtained from the unbalanced doubly-spinning
black ring solution taking the large $R\rightarrow\infty$ limit.

The KG equation for the boosted Kerr black string solution is separable under
the ansatz

$\tilde{\Psi}(t,r,\theta,\phi,z)=e^{-it\omega+im\phi+inz}\,\,\tilde{\Phi}(r,\theta)\equiv
e^{-it\omega+im\phi+inz}\,\,\Phi(\theta)\Psi(r).$ (108)

The radial wave equation was found in [33]

$\displaystyle\Delta\partial_{r}(\Delta\partial_{r}\Psi)-\Delta\left(n^{2}r^{2}+a^{2}\omega^{2}-2\omega
mc_{\sigma}+\lambda_{lm}\right)\Psi\qquad\qquad\qquad\qquad\qquad$ (109)
$\displaystyle+(2Mrc_{\sigma})^{2}\Bigg{[}\left(\frac{\omega(\Delta+2Mr)}{2Mrc_{\sigma}}-\frac{ma}{2Mr}\right)^{2}-\frac{m^{2}a^{2}\tanh^{2}_{\sigma}}{(2Mr)^{2}}\qquad\qquad\qquad\qquad$
$\displaystyle\qquad\qquad+\frac{\Delta+2Mr}{2Mr}\left((\omega-n\,\tanh\sigma)^{2}-\frac{\omega^{2}}{c_{\sigma}^{2}}+\frac{2\,n\,m\,a\,\tanh\sigma}{c_{\sigma}(\Delta+2Mr)}\right)\Bigg{]}\Psi=0.$

We are again interested in the near zone limit of (109) (that is, we focus
only on the pole terms). The result is

$\displaystyle\Delta\partial_{r}(\Delta\partial_{r}\Psi)+(2Mr\cosh\sigma)^{2}\Bigg{[}\omega-n\tanh\sigma-\frac{ma}{2Mr\cosh\sigma}\Bigg{]}^{2}\Psi=0.\qquad\qquad\qquad$
(110)

We can extract the monodromy data analyzing the poles of the differential
equation

$\displaystyle\Psi_{\pm}\sim(r-r_{\pm})^{\pm
i\alpha_{\pm}}\qquad\Rightarrow\qquad\alpha_{\pm}=\frac{2Mr_{\pm}\cosh\sigma}{(r_{+}-r_{-})}\Bigg{[}\omega-n\tanh\sigma\pm\frac{ma}{2Mr_{\pm}\cosh\sigma}\Bigg{]}$

and we can rewrite the KG equation as

$\displaystyle\partial_{r}(\Delta\partial_{r}\Psi)+(r_{+}-r_{-})\left[\frac{\alpha^{2}_{+}}{r-r_{+}}-\frac{\alpha^{2}_{-}}{r-r_{-}}\right]\Psi=0.$
(111)

Having computed the monodromies, considering $n=0$,999It is worth noting that
when $\sigma=0$ the eigenvalue $n$ does not affect the thermodynamics. This
can be compared to the case of the full doubly-spinning black ring, in which
the condition $\lambda\to\mu$ results in the Figueras black ring, in which the
momentum $J_{\psi}=0$ [13]. Note this reasoning may also be applied to Section
4, except there it is the $m$ eigenvalue that decouples in the near zone
analysis. See the end of Section 4 in [13], as well as Section 4 of [34], for
further discussion.  we are able to the define the frequencies

$\displaystyle\omega_{L}=2Mc_{\sigma}\,\omega\,,\qquad\omega_{R}=\frac{2M}{r_{+}-r_{-}}\left(2Mc_{\sigma}\,\omega-\frac{ma}{M}\right)\,.$
(112)

Likewise, fixing the periodicities along the compact $\phi$-direction

$\displaystyle\Psi(\phi+2\pi)=e^{2\pi
im}\Psi(\phi)=e^{4\pi^{2}i(T_{R}w_{R}-T_{L}w_{L})}\Psi(\phi)$ (113)

yields a prescription for the left and right conformal field theory
temperatures

$\displaystyle T_{R}=\frac{r_{+}-r_{-}}{4\pi a}\qquad
T_{L}=\frac{r_{+}+r_{-}}{4\pi a}.$ (114)

The temperatures obey the Cardy relation, with a particular central charge
given as:

$\displaystyle S_{\pm}=\frac{c_{\phi}\pi^{2}}{3}\left(T_{L}\pm
T_{R}\right)\qquad\text{where}\qquad
c_{\phi}=12aMc_{\sigma}R=12J_{\text{BS}}\,.$ (115)

For a black string (BS) of length $R$, entropy $S_{\pm}=2\pi
MR\,r_{\pm}c_{\sigma}$ and and spin $J_{\text{BS}}=aMc_{\sigma}R$ (see e.g.
[33, 32]). While the temperatures are independent of the boost parameter, our
results show that the central charge picks up this dependence. And, while the
results are directly comparable to the $D=4$ Kerr case (24) when the boost
parameter vanishes $(\sigma=0)$, our results indicate that the solution has a
more general CFT interpretation than for Kerr. Finally, it is also worth
emphasizing that these identifications in the large $R$ limit are also in full
agreement with the results derived in the previous section which further
supports our proposals.

Note the system is quantized:

$\displaystyle S_{+}S_{-}={4\pi^{2}J_{\text{BS}}^{2}}.$ (116)

###  5.3 Conformal coordinates

From our discussion of the dipole black ring, we know right away that we will
not be able to find conformal coordinates that reproduce (80) unless one of
two limits is taken: (i) zooming in on one of the black ring horizons or (ii)
the black string limit $R\rightarrow\infty$. Limit (i) proceeds exactly as in
the dipole black ring case (specifically the discussion surrounding (67)), and
so we will not repeat that discussion here. In this section we will focus on
the second limit $R\rightarrow\infty$.

Let us consider the KG-equation for the doubly-spinning black ring (88).
Before taking the $R\rightarrow\infty$ limit, it is convenient to employ a
reparametrization

$\displaystyle\nu=r_{-}/R,\qquad\mu=r_{+}/R,\qquad\lambda=r_{+}c_{\sigma}/R,$
(117)

where we have introduced the Kerr horizons $r_{\pm}$ and $c_{\sigma}$ that
will represent the boost parameter in the black string. Using these
definitions and taking the large $R$ limit, the wave equation (88) can be
written as

$\displaystyle\left[\partial_{r}\left(\Delta\,\partial_{r}\,\right)+\frac{\left(c_{\sigma}r_{+}(r_{+}+r_{-})\,\omega-\sqrt{r_{+}r_{-}}\,m\right)^{2}}{(r-r_{+})(r_{+}-r_{-})}-\frac{\left(c_{\sigma}r_{-}(r_{+}+r_{-})\,\omega-\sqrt{r_{+}r_{-}}\,m\right)^{2}}{(r-r_{-})(r_{+}-r_{-})}\right]\tilde{\Phi}=0\,$

with $\Delta=(r-r_{+})(r-r_{-})$. This is none other than the near-region KG-
equation for the Kerr black hole that was reported in [2], but modified by the
boost parameter $c_{\sigma}$. Changing coordinates to
$x=\frac{2r-(r_{+}+r_{-})}{2(r_{+}-r_{-})}$, we have

$\displaystyle\left[\partial_{x}\bar{\Delta}\,\partial_{x}+\frac{\left(c_{\sigma}r_{+}(r_{+}+r_{-})\,\omega-\sqrt{r_{+}r_{-}}\,m\right)^{2}}{(x-1/2)(r_{+}-r_{-})^{2}}-\frac{\left(c_{\sigma}r_{-}(r_{+}+r_{-})\,\omega-\sqrt{r_{+}r_{-}}\,m\right)^{2}}{(x+1/2)(r_{+}-r_{-})^{2}}\right]\tilde{\Phi}=0\,.$

where $\bar{\Delta}=x^{2}-\frac{1}{4}$. Comparing with the Casimir (72), we
obtain conformal coordinates (10) that reproduce the near zone radial
equation, with, for example,

$\alpha=\frac{r_{+}-r_{-}}{2\sqrt{r_{+}r_{-}}},\qquad\beta=0,\qquad\gamma=\frac{r_{+}+r_{-}}{2\sqrt{r_{+}r_{-}}},\qquad\delta=-\frac{1}{c_{\sigma}(r_{+}+r_{-})}.$
(118)

Note that these identifications for the conformal coordinates agree with those
of Kerr [17] by setting the boost parameter $\sigma=0$ where the (unboosted)
Kerr black string geometry becomes $ds^{2}=ds^{2}_{Kerr}+dz^{2}$. At this
stage, there is an ambiguity in fixing the conformal coordinates. However, we
can make an argument that in general $\alpha=2\pi T_{R}$ and $\gamma=2\pi
T_{L}$. Assuming a Cardy formula and comparing with the entropy for the Kerr
black string (see above) we are able to determine the central charge of the
CFT dual

$\displaystyle c=12aMc_{\sigma}R=12J_{\text{BS}}\,.$ (119)

##  6 Discussion

We have studied the presence of hidden conformal symmetry in five-dimensional
systems in which the KG-equation seems non-separable (in the dipole black ring
and the doubly spinning black ring cases) as well as their separable large $R$
black string counterparts (in particular the boosted charged black string and
the boosted Kerr black string). In analogy to the conjectured non-extremal
Kerr/CFT correspondence of [2], we propose that a dual CFT exists in a near
zone limit of the black ring and black string solutions, and we use a revised
monodromy technique to work out the associated CFT left/right temperatures
$(T_{L},T_{R})$ respectively. The present approach provides a derivation of
these temperatures based on monodromy data, without the need for a low energy
limit. Furthermore, we develop a set of conformal coordinates for each
solution we consider, and show that they are related to $T_{L}$ and $T_{R}$ in
a natural way. Assuming a Cardy entropy formula and comparing with the
Bekenstein-Hawking entropy formula we are able to determine the corresponding
central charge. This includes the dipole black ring or string configurations
with $c=6q^{3}$ and and doubly-spinning black ring or boosted Kerr black
string with $c=12J$.

The identification of left- and right-moving sectors in terms of the monodromy
coefficients in higher dimensions proceeded almost exactly as in the Kerr
black hole example. The only subtlety is for black ring and strings that the
identifications gives a unique way of realizing the hidden conformal symmetry.
This is not linked to 5-dimensional spacetimes but rather to the isometries
involved. The presence of two commuting $U(1)$ isometries in the 5D Myers-
Perry black hole solution gives two inequivalent ways of realizing the hidden
conformal symmetry [30, 8]. Likewise, the central charge associated to the
associated to the $\psi$ circle is $c_{\psi}=6J_{\phi}$ and
$c_{\phi}=6J_{\psi}$ for the $\phi$ circle.

We also have the phenomenological observation that the entropy product is
independent of the mass, and we can define for black rings and black strings
the central charge $c$ as

$\displaystyle\frac{S_{+}S_{-}}{4\pi^{2}}=\mathcal{F}(J)\rightarrow c\equiv
c_{R}=c_{L}=6\frac{\partial\mathcal{F}}{\partial J}\,.$ (120)

The analysis of present work was inspired in part by the following question:
Are all black holes dual to a 2D CFT? Much evidence exists for a Kerr/CFT
correspondence both at [1] and away from [2] extremality. Furthermore, through
studying scattering amplitudes on black hole backgrounds, there is evidence to
suggest that 2D CFT duals might exist for more general black hole spacetimes
[35, 36]. Studying the presence of hidden conformal symmetry in more exotic
solutions such as black rings and black strings is an ideal arena to push the
boundaries of how general a phenomenon it is to have a proposed Near-
Horizon/CFT duality.

The black ring and string solutions are also great systems to study the
interplay between hidden conformal symmetry and separability. That is, does
the presence of a tower of Killing tensors (responsible for the separability
of the KG-equation) play a direct role in the presence of hidden conformal
symmetry in the near zone radial wave equation? We are able to show that a
consistent hidden conformal symmetry analysis is possible in non-separable
systems, provided that we focus on the outer horizon. Focusing only on the
outer horizon is essentially what is done to find globally defined hidden
symmetry generators, as in [37].

We would like to stress that the soft hair interpretation of hidden conformal
symmetry, as presented for example in [14], is not viable in more general
contexts, such as black ring solutions. Rather than taking a frequency
dependent limit in the wave equation, such as $\omega M<<1$ and $\omega r<<1$,
one instead needs to discard all terms of the wave equation except for the
poles. For Kerr these two perspectives coincide, but in more general scenarios
they do not.

## Acknowledgements

This work was supported by the NSF grant PHY-2012036 at Utah State University.
The work of ABC is partially supported through a Utah NASA Space Consortium
Grant fellowship, as well a USU Howard L. Blood Fellowship. VLM is supported
by the Icelandic Research Fund under grants 195970-053 and 228952-051 and by
the University of Iceland Research Fund. LFT acknowledges support from USU
PDRF fellowship and USU Howard L. Blood Fellowship. MJR is partially supported
through the NSF grant PHY-2012036, RYC-2016-21159, CEX2020-001007-S and
PGC2018-095976-B-C21, funded by MCIN/AEI/10.13039/501100011033.

##  Appendix A Black Ring Metrics and Identities

###  A.1 Dipole Black Ring

The following Dipole Black Ring identities are developed from [11]. For
simplicity, we have worked with the non-dilaton limit $(N=3)$, where $N$ is
the dipole charge winding number with magnetic source $A_{\phi}$. The line
element is given as:

$ds^{2}=-\frac{F(y)}{F(x)}\left(\frac{H(x)}{H(y)}\right)\left(dt+C(\nu,\lambda)\>R\>\frac{1+y}{F(y)}\>d\psi\right)^{2}\\\
+\frac{R^{2}}{(x-y)^{2}}\>F(x)\left(H(x)H(y)^{2}\right)\left[-\frac{G(y)}{F(y)H(y)^{3}}d\psi^{2}-\frac{dy^{2}}{G(y)}+\frac{dx^{2}}{G(x)}+\frac{G(x)}{F(x)H(x)^{3}}d\varphi^{2}\right]\
$ (121)

with functions defined as follows:

$\displaystyle F(\xi)$ $\displaystyle=1+\lambda\xi,\qquad
G(\xi)=(1-\xi^{2})(1+\nu\xi)$ (122) $\displaystyle H(\xi)$
$\displaystyle=1-\mu\xi.$ (123)

The curvature of this system involves the existence of energy-momentum sourced
by electromagnetic fields. The potential is given by

$A_{\phi}=\sqrt{3}C(\nu,-\mu)R\frac{1+\cos{\theta}}{H_{\theta}}+k_{1},$ (124)

where $H_{\theta}$ is given in (134). The constant $k_{1}$ is associated to
the motion of Dirac strings [7]. Given the Faraday tensor
$F_{\mu\nu}={\partial}_{\mu}A_{\nu}-{\partial}_{\nu}A_{\mu}$, we have that the
only non-zero components are

$F_{23}=-F_{32}=-{A^{\prime}}_{\phi}(\theta).$ (125)

With this result in hand, we can determine the stress-energy tensor

$T_{\alpha\beta}=F_{\mu\alpha}g^{\alpha\beta}F_{\beta\nu}-\frac{1}{4}g_{\mu\nu}F_{\sigma\alpha}g^{\alpha\beta}F_{\beta\phi}g^{\phi\sigma}$
(126)

with non zero components

$\displaystyle T_{00}$
$\displaystyle=\frac{F_{r}H^{3}_{\theta}{A^{\prime}}^{2}_{\phi}}{(r^{2}\sin{\theta})^{2}F^{3}_{\theta}H^{9}_{r}}$
(127) $\displaystyle T_{11}$
$\displaystyle=\frac{R^{2}(1+\frac{r\cos{\theta}}{R})^{6}H^{3}_{\theta}{A^{\prime}}^{2}_{\phi}}{(r^{2}\sin{\theta})^{4}F_{\theta}G_{r}H^{6}_{r}}$
(128) $\displaystyle T_{22}$
$\displaystyle=\frac{(1+\frac{r\cos{\theta}}{R})^{4}H^{3}_{\theta}\big{[}-G_{\theta}(1+\frac{r\cos{\theta}}{R})^{2}+2r^{2}F_{\theta}H_{\theta}H^{2}_{r}{\sin{\theta}}^{2}\big{]}{A^{\prime}}^{2}_{\phi}}{(r^{3}\sin{\theta})^{2}F_{\theta}G^{2}_{\theta}H^{6}_{r}{\sin{\theta}}^{2}}$
(129) $\displaystyle T_{33}$
$\displaystyle=\frac{(1+\frac{r\cos{\theta}}{R})^{4}G_{\theta}\big{[}2r^{2}G_{\theta}H^{2}_{r}-H^{2}_{\theta}(1+\frac{r\cos{\theta}}{R})^{2}\big{]}{A^{\prime}}^{2}_{\phi}}{(r^{3}\sin{\theta}^{2})^{2}F^{2}_{\theta}H^{2}_{\theta}H^{6}_{r}}$
(130) $\displaystyle T_{44}$
$\displaystyle=\frac{(1+\frac{r\cos{\theta}}{R})^{6}H^{3}_{\theta}\big{[}r^{2}F^{2}_{\theta}G_{r}H^{2}_{r}+R^{2}C(\nu,\lambda)(1+\frac{r\cos{\theta}}{R})^{2}(1-\frac{R}{r})^{2}\big{]}{A^{\prime}}^{2}_{\phi}}{(r^{2}\sin{\theta})^{4}F^{3}_{\theta}F_{r}H{9}_{r}}$
(131) $\displaystyle T_{04}$
$\displaystyle=T_{40}=\frac{RC(\nu,\lambda)(1-\frac{R}{r})(1+\frac{r\cos{\theta}}{R})^{8}H^{3}_{\theta}{A^{\prime}}^{2}_{\phi}}{(r^{2}\sin{\theta})^{4}F^{3}_{\theta}H^{9}_{r}}$
(132)

where

$F_{r}\equiv F(-R/r)\,,\qquad F_{\theta}\equiv F(\cos\theta)$ (133)
$H_{r}\equiv H(-R/r)\,,\qquad H_{\theta}\equiv H(\cos\theta)$ (134)
$G_{r}\equiv G(-R/r)\,,\qquad G_{\theta}\equiv G(\cos{\theta})$ (135)
$C({\sigma}_{1},{\sigma}_{2})=\sqrt{{\sigma}_{2}({\sigma}_{2}-{\sigma}_{1})\frac{1+{\sigma}_{2}}{1-{\sigma}_{2}}}.$
(136)

Now we address the stability condition for the ring, balancing the centripetal
force against the magnetic repulsion generated between the monopoles
distributed in the ring structure. From a geometric perspective, that balance
is reached by the avoidance of conical singularities in the $\phi,\psi$
directions, through the constraint

$\Delta\phi=2\pi\frac{(1+\mu)^{N/2}\sqrt{1-\lambda}}{1-\nu}$ (137)

and conical singularities at $x=-1$ and $y=-1$ , considering the condition

$\frac{1-\lambda}{1+\lambda}\Big{(}\frac{1+\mu}{1-\mu}\Big{)}^{N}=\Big{(}\frac{1-\nu}{1+\nu}\Big{)}^{2}.$
(138)

Now we define the extremality conditions. The event horizon and Cauchy horizon
for our ring solution are at $r=r_{0}$ and $r=0$, respectively. Extremality is
reached when $\nu=0$, that is $r_{0}=0$. As expected, this has consequences on
the thermodynamical description of the Dipole Black Ring. Having both the
temperature and horizon area given as

$\displaystyle T=\frac{1}{4\pi
R}\frac{{\nu}^{(N-1)/2}(1+\nu)}{(\mu+\nu)^{N/2}}\sqrt{\frac{1-\lambda}{\lambda(1+\lambda)}}$
(139)
$\displaystyle{\cal{A}}_{H}=8{\pi}^{2}R^{3}\frac{(1+\mu)^{N}{\nu}^{(3-N)/2}(\mu+\nu)^{N/2}\sqrt{\lambda(1-{\lambda}^{2})}}{(1-\nu)^{2}(1+\nu)},$
(140)

by simple inspection we determine that at extremality, $T=0$. In the non-
dilatonic limit, $N=3$, the area remains finite at extremality,
${\cal{A}}_{H}\neq 0$, leading to degeneracy of the horizon. Then, as a
consequence, even at extremality, the entropy $S$ is non-vanishing.

###  A.2 Doubly-spinning Black Ring

The doubly-spinning black ring identities are extracted from [13]. The full
metric may be represented as

$\displaystyle
ds^{2}=-\frac{H[y,x]}{H[x,y]}(dt-\omega_{\psi}d\psi-\omega_{\phi}d\phi)^{2}+\gamma[x,y]\left(\frac{dx^{2}}{G[x]}+\frac{dy^{2}}{G[y]}\right)$
$\displaystyle-\frac{1}{H[y,x]}\left(F[x,y]d\psi^{2}+2J[x,y]d\psi
d\phi+F[y,x]d\phi^{2}\right)\ \ $ (141)

with functions defined by101010Valid solutions to
$G_{\mu\nu}[g(\ast,\cdot)]=0$ are constrained under
$0\leq\nu\leq\mu\leq\lambda<1$ and $R>0$. The metric is independent of time
$-\infty<t<\infty$, angles $0\leq\psi,\phi<2\pi$; further, the C-metric-like
coordinates $(x,y)$ take ranges $-1<x<1$ and $-\infty<y<-1$. :

$\displaystyle\gamma[x,y]=\frac{R^{2}(1-\mu)^{2}(1-\nu)H[x,y]}{(1-\lambda)(1-\mu\nu)\Phi\Psi(x-y)^{2}}:=\frac{R^{2}C_{\mu\nu\lambda}H[x,y]}{(x-y)^{2}}$
(142) $\displaystyle G[x]=(1-x^{2})(1+\mu x)(1+\nu x)$ (143) $\displaystyle
F[x,y]=\frac{R^{2}}{\mu\nu(1-\mu\nu)\Phi(x-y)^{2}}\left(G[x]f_{1}[\lambda,\mu,\nu;y]+G[y]f_{2}[\lambda,\mu,\nu;x]\right)$
(144) $\displaystyle
H[x,y]=(1-\lambda)(1-\nu)\Psi\Phi+\lambda(\mu+\nu)\left((1-\lambda\mu)^{2}-(\lambda-\mu)^{2}\nu^{2}\right)(1+x)$
$\displaystyle-(1-x^{2})(1-y^{2})\nu\left(\frac{\lambda(\lambda-\mu)^{2}\nu(\mu+\nu)(1+x)-(1-\lambda)\Psi+(\mu+1-\Phi)}{1-x^{2}}\right.$
(145)
$\displaystyle\left.+\frac{\lambda(\lambda-\mu)(-1+\lambda\mu)(\mu+\nu)(1+y)+\lambda(\lambda-\mu)(1-\mu)(1-\lambda\mu)(\mu+\nu)+\Sigma\Psi}{1-y^{2}}\right.$
$\displaystyle\left.-\left(\Sigma\Psi+\lambda\mu(\lambda-\mu)(-1+\lambda\mu)(\mu+\nu)\right)\phantom{\frac{}{}}\right)$
$\displaystyle
J[x,y]=A[x,y]\left(1+\frac{\Sigma\Psi\mu}{\nu}+\frac{\mu(1+\Sigma\Psi)}{2}(x+y)\right.$
(146) $\displaystyle\left.+(1+\mu x)(1+\mu
y)\left(\Phi(\Phi-(1-\lambda\mu)(2+\lambda\mu))-(1-\lambda\mu)(\mu^{2}-1)\right)\phantom{\frac{}{}}\right)$
$\displaystyle
A[x,y]:=\nu\frac{R^{2}(\mu+\nu)\sqrt{\nu(\lambda-\mu)(1-\lambda\mu)}(1-x^{2})(1-y^{2})}{\mu^{2}(1-\mu\nu)\Phi(x-y)}$
(147)
$\displaystyle\omega_{\phi}[x,y]=\frac{R(\mu+\nu)}{H[y,x]}\sqrt{\frac{\nu\lambda(1-\lambda^{2})\Phi\Psi\Sigma}{1-\mu\nu}}y(1-x^{2})$
(148)
$\displaystyle\omega_{\psi}[x,y]=\frac{\omega_{\phi}[x,y]}{\Psi(1-\lambda)y(1-x^{2})}\sqrt{\frac{(\lambda-\mu)(1-\lambda\mu)}{\nu}}\left(\Phi(1+\nu
y)\right.$ (149)
$\displaystyle\left.+(1-\mu)\nu(1-y)(1+x\lambda)+y\nu(1-\Phi+\mu)(1-x^{2})\right).$

Expressions for the functions $f_{1}$ and $f_{2}$ can be found in [13]. We
also define

$\Phi\equiv
1+\mu\nu-\lambda(\mu+\nu),\qquad\Psi\equiv\mu(1+\nu)-\lambda(\nu+\mu^{2}),\qquad\Sigma\equiv\mu(1-\nu)+\lambda(\nu-\mu^{2}).$
(150)

Consider the massless, spinless wave equation111111The KG field selection
automatically projects onto the lowest spin-weight state(-tower) because the
(uncharged) KG field is spin self-dual. Having a fully extended, analytic
scalar field is useful in embedding-measured representations of boundary
(orbifolds/)manifolds, such as asympotic infinity or coordinate singularities;
in particular, the interplay between both (see [10, 14, 38] ).  in the
background of a doubly-spinning black ring. In particular, we will use the
coordinate singular points (event horizons) in order to put the wave equation
into a useful form for extracting the near-horizon quasi-normal modes. The KG
operator equation is:

$\displaystyle\delta\left[\mathcal{L}\left[[\cdot]_{\text{KG}}^{m=0}\right]\right]=0\
\Leftrightarrow\
\partial_{\mu}\left[\sqrt{|-g|}g^{\mu\nu}\partial_{\nu}[\cdot]\right]=0.$
(151)

Consider a 5D metric with global symmetries in
$x^{\\{\alpha,\beta\\}}\in\\{t,\phi,\psi\\}$; then
$\partial_{\alpha}[\sqrt{-|g_{T}|}g^{\mu\nu}]=0$ and, letting
$\\{i,j\\}\in\\{x,y\\}$ represent the additional coordinates, the (massless)
KG equation is:

$\displaystyle
0=\sqrt{|-g|}g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}[\cdot]+\partial_{i}\left[\sqrt{|-g|}g^{ij}\partial_{j}[\cdot]\right].$
(152)

Then, with a trial solution of $[\cdot]\to
e^{i\lambda_{\alpha}x^{\alpha}}[\cdot]$ $\Rightarrow\
\partial_{\alpha}\partial_{\beta}[\cdot]\to\lambda_{\alpha}\lambda_{\beta}[\cdot]$
the above may be written as:

$\displaystyle
0=-\sqrt{|-g|}g^{\alpha\beta}\lambda_{\alpha}\lambda_{\beta}[\cdot]+\partial_{i}\left[\sqrt{|-g|}g^{ij}\partial_{j}[\cdot]\right].$
(153)

Further, consider a solution which is poloidally factorized:
$[\cdot]\to\varphi_{T}[t,\phi,\psi]\varphi_{P}[x,y]$ where
$\varphi_{P}[x,y]=h[x,y]\tilde{\Phi}[x,y]$; then
$\partial_{a}\varphi_{P}=\varphi_{P}(\partial_{a}[\ln[h\tilde{\Phi}]])$, and
the poloidal piece of the KG equation becomes:

$\displaystyle\partial_{a}[\sqrt{-g}g^{ab}_{P}\partial_{b}\varphi]\ $
$\displaystyle=\
\frac{R^{2}\varphi_{T}}{1-\nu\mu}\partial_{a}\left[\frac{G[x^{a}]\varphi_{P}}{(x-y)^{2}}\partial_{a}[\ln[h\tilde{\Phi}]]\right]$
(154)
$\displaystyle=\frac{hR^{2}\varphi_{T}}{(1-\nu\mu)(x-y)^{2}}\left(\partial_{a}[G[x^{a}]\tilde{\Phi}_{,a}]+2G[x^{a}]\left(\partial_{a}\text{ln}{[h]}-\frac{\delta^{x}_{a}-\delta^{y}_{a}}{x-y}\right)\tilde{\Phi}_{,a}\right.$
$\displaystyle\left.+(\partial_{a}\ln
h)G[x^{a}]\left(\partial_{a}\ln{[h_{,a}G[x^{a}]]}-\frac{2(\delta^{x}_{a}-\delta^{y}_{a})}{x-y}\right)\tilde{\Phi}\right).$

In the above picture the free derivative terms have the same differential
envelope ($\partial_{a}[G[x^{a}][\cdot]]$), reminiscent of an isotropic fluid
polarization.

Note that $h\to x-y$ gives
$\left(\partial_{a}\ln{[\frac{h}{x-y}]}\right)\tilde{\Phi}_{,a}\to 0$ and
$h_{,aa}\to 0$. In this case, it can be shown that:

$\displaystyle\partial_{a}[\sqrt{-g}g^{ab}_{P}\partial_{b}\varphi]=\frac{R^{2}\varphi_{T}}{(1-\nu\mu)(x-y)}\left(\partial_{a}[G_{a}\tilde{\Phi}_{,a}]-\frac{G_{a}}{x-y}\left(\frac{2(\delta^{x}_{a}+\delta^{x}_{y}))}{x-y}-\frac{G_{a,a}}{G_{a}}(\delta^{x}_{a}-\delta^{y}_{a})\right)\tilde{\Phi}\right).\qquad\qquad$
(155)

Then, defining
$\Omega[x,y]=\frac{(x-y)^{2}(1-\nu\mu)}{R^{2}}\sqrt{|-g|}=\frac{R^{2}C_{\mu\nu\lambda}H[x,y]}{(x-y)^{2}}\equiv\sqrt{G[x]G[y]|g_{P}|}$
induces121212Note:
$\frac{\Omega_{xy}}{|-g_{T}|}=\frac{C_{\mu\nu\lambda}H_{xy}(1-\mu\nu)^{2}(x-y)^{2}}{R^{2}G_{x}G_{y}}$
the resultant KG-form:

$\displaystyle
0=\partial_{a}[G[x^{a}]\tilde{\Phi}_{,a}]-\Omega[x,y]g^{\alpha\beta}\lambda_{\alpha}\lambda_{\beta}\tilde{\Phi}$
(156)
$\displaystyle\qquad\qquad\qquad-\frac{G[x^{a}]}{x-y}\left(\frac{2(\delta^{x}_{a}+\delta^{x}_{y})}{x-y}-\left(\frac{-2x^{a}}{1-(x^{a})^{2}}+\frac{\mu}{1+\mu
x^{a}}+\frac{\nu}{1+\nu
x^{a}}\right)(\delta^{x}_{a}-\delta^{y}_{a})\right)\tilde{\Phi}.$

From this analysis, it can be shown that

$\displaystyle
0=\partial_{r}\left(\left(1-\frac{r^{2}}{R^{2}}\right)(r-\mu\,R)(r-\nu\,R)\,\partial_{r}\,\tilde{\Phi}\right)+\frac{1}{\sin\theta}\,\partial_{\theta}\left((1+\mu\cos\theta)(1+\nu\cos\theta)\sin\theta\,\partial_{\theta}\,\tilde{\Phi}\right)$
$\displaystyle\qquad+\left(\frac{C_{\mu\nu\lambda}(1-\mu\nu)^{2}\left(1+\frac{r}{R}\cos\theta\right)^{2}(r/R)^{2}\,\tilde{K}(r,\theta)}{\left(1-\frac{r^{2}}{R^{2}}\right)(r-\mu\,R)(r-\nu\,R)(1+\mu\cos\theta)(1+\nu\cos\theta)\sin^{2}\theta}+\tilde{f}_{r}+\tilde{f}_{\theta}\right)\tilde{\Phi}\
\ \ \ $ (157)

where

$\displaystyle\tilde{K}(r,\theta)$ $\displaystyle=$ $\displaystyle
n^{2}R^{2}F_{r\theta}-m^{2}F_{\theta r}+2\,m\,\omega(F_{\theta
r}\,\omega_{\phi}-J_{r\theta}\,\omega_{\psi})+2\,n\,R\,\omega(F_{r\theta}\,\omega_{\psi}+J_{r\theta}\,\omega_{\phi})$
(158) $\displaystyle-2\,n\,m\,R\,J_{r\theta}+\omega^{2}\left(\frac{H_{\theta
r}^{2}}{H_{r\theta}}\,\tilde{F}^{2}+F_{r\theta}\,\omega_{\psi}^{2}-F_{\theta
r}\,\omega_{\phi}^{2}+2\,\omega_{\phi}\,\omega_{\psi}\,J_{r\theta}\right)$ and
$\displaystyle\ \
\tilde{f}_{r}=\frac{(\mu+\nu)\,r}{R}-\frac{2\,r^{2}}{R^{2}}\qquad\tilde{f}_{\theta}=-\cos\theta\,(\mu+\nu+2\,\mu\,\nu\cos\theta)\,.$
(159)

## References

  * [1] M. Guica, T. Hartman, W. Song, and A. Strominger, The Kerr/CFT Correspondence, Phys. Rev. D 80 (2009) 124008, [arXiv:0809.4266].
  * [2] A. Castro, A. Maloney, and A. Strominger, Hidden Conformal Symmetry of the Kerr Black Hole, Phys. Rev. D 82 (2010) 024008, [arXiv:1004.0996].
  * [3] B. Carter, Global structure of the Kerr family of gravitational fields, Phys. Rev. 174 (1968) 1559–1571.
  * [4] V. P. Frolov and D. Kubiznak, Hidden Symmetries of Higher Dimensional Rotating Black Holes, Phys. Rev. Lett. 98 (2007) 011101, [gr-qc/0605058].
  * [5] V. Frolov, P. Krtous, and D. Kubiznak, Black holes, hidden symmetries, and complete integrability, Living Rev. Rel. 20 (2017), no. 1 6, [arXiv:1705.05482].
  * [6] D. Kubiznak and V. P. Frolov, Hidden Symmetry of Higher Dimensional Kerr-NUT-AdS Spacetimes, Class. Quant. Grav. 24 (2007), no. 3 F1–F6, [gr-qc/0610144].
  * [7] R. Emparan and H. S. Reall, Black Rings, Class. Quant. Grav. 23 (2006) R169, [hep-th/0608012].
  * [8] A. Castro, J. M. Lapan, A. Maloney, and M. J. Rodriguez, Black Hole Monodromy and Conformal Field Theory, Phys. Rev. D 88 (2013) 044003, [arXiv:1303.0759].
  * [9] A. Castro, J. M. Lapan, A. Maloney, and M. J. Rodriguez, Black Hole Scattering from Monodromy, Class. Quant. Grav. 30 (2013) 165005, [arXiv:1304.3781].
  * [10] A. B. Chanson, J. Ciafre, and M. J. Rodriguez, Emergent black hole thermodynamics from monodromy, Phys. Rev. D 104 (2021), no. 2 024055, [arXiv:2004.14405].
  * [11] R. Emparan, Rotating circular strings, and infinite nonuniqueness of black rings, JHEP 03 (2004) 064, [hep-th/0402149].
  * [12] A. Pomeransky and R. Sen’kov, Black ring with two angular momenta, arXiv preprint hep-th/0612005 (2006).
  * [13] Y. Chen, K. Hong, and E. Teo, Unbalanced Pomeransky-Sen’kov black ring, Phys. Rev. D 84 (2011) 084030, [arXiv:1108.1849].
  * [14] S. Haco, S. W. Hawking, M. J. Perry, and A. Strominger, Black Hole Entropy and Soft Hair, JHEP 12 (2018) 098, [arXiv:1810.01847].
  * [15] C. Keeler, V. Martin, and A. Priya, Hidden conformal symmetries from Killing towers with an application to large-D/CFT, SciPost Phys. 12 (2022), no. 5 170, [arXiv:2110.10723].
  * [16] A. Aggarwal, A. Castro, and S. Detournay, Warped Symmetries of the Kerr Black Hole, JHEP 01 (2020) 016, [arXiv:1909.03137].
  * [17] M. Perry and M. J. Rodriguez, Central charges for AdS black holes, Class. Quant. Grav. 39 (2022), no. 4 045009, [arXiv:2007.03709].
  * [18] M. J. Perry and M. J. Rodriguez, CFT duals of Kerr-Taub-NUT and beyond, arXiv:2205.09146.
  * [19] I. Bena and N. P. Warner, One ring to rule them all … and in the darkness bind them?, Adv. Theor. Math. Phys. 9 (2005), no. 5 667–701, [hep-th/0408106].
  * [20] J. P. Gauntlett, J. B. Gutowski, C. M. Hull, S. Pakis, and H. S. Reall, All supersymmetric solutions of minimal supergravity in five- dimensions, Class. Quant. Grav. 20 (2003) 4587–4634, [hep-th/0209114].
  * [21] R. Emparan and H. S. Reall, A Rotating black ring solution in five-dimensions, Phys. Rev. Lett. 88 (2002) 101101, [hep-th/0110260].
  * [22] S. W. Hawking, Black holes in general relativity, Commun. Math. Phys. 25 (1972) 152–166.
  * [23] A. Castro and M. J. Rodriguez, Universal properties and the first law of black hole inner mechanics, Phys. Rev. D 86 (2012) 024008, [arXiv:1204.1284].
  * [24] J. L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186–204.
  * [25] R. Gregory and R. Laflamme, Black strings and p-branes are unstable, Phys. Rev. Lett. 70 (1993) 2837–2840, [hep-th/9301052].
  * [26] T. Wiseman, From black strings to black holes, Class. Quant. Grav. 20 (2003) 1177–1186, [hep-th/0211028].
  * [27] G. T. Horowitz and K. Maeda, Fate of the black string instability, Phys. Rev. Lett. 87 (2001) 131301, [hep-th/0105111].
  * [28] G. Compere, S. de Buyl, S. Stotyn, and A. Virmani, A General Black String and its Microscopics, JHEP 11 (2010) 133, [arXiv:1006.5464].
  * [29] H. Elvang and M. J. Rodriguez, Bicycling Black Rings, JHEP 04 (2008) 045, [arXiv:0712.2425].
  * [30] C. Krishnan, Hidden Conformal Symmetries of Five-Dimensional Black Holes, JHEP 07 (2010) 039, [arXiv:1004.3537].
  * [31] B. Chen and J.-j. Zhang, Holographic Descriptions of Black Rings, JHEP 11 (2012) 022, [arXiv:1208.4413].
  * [32] J. G. Rosa, Boosted black string bombs, JHEP 02 (2013) 014, [arXiv:1209.4211].
  * [33] O. J. C. Dias, Superradiant instability of large radius doubly spinning black rings, Phys. Rev. D 73 (2006) 124035, [hep-th/0602064].
  * [34] R. C. Myers and M. J. Perry, Black Holes in Higher Dimensional Space-Times, Annals Phys. 172 (1986) 304.
  * [35] T. Hartman, W. Song, and A. Strominger, Holographic Derivation of Kerr-Newman Scattering Amplitudes for General Charge and Spin, JHEP 03 (2010) 118, [arXiv:0908.3909].
  * [36] M. Cvetic and F. Larsen, Greybody Factors and Charges in Kerr/CFT, JHEP 09 (2009) 088, [arXiv:0908.1136].
  * [37] P. Charalambous, S. Dubovsky, and M. M. Ivanov, On the Vanishing of Love Numbers for Kerr Black Holes, JHEP 05 (2021) 038, [arXiv:2102.08917].
  * [38] L. Pipolode Gioia and A.-M. Raclariu, Eikonal Approximation in Celestial CFT, arXiv:2206.10547.