[a]Roberta Arnaldi for the NA60+ Collaboration

#  Prospects for open heavy-flavour and quarkonium measurements with NA60+

###### Abstract

The high-intensity beams provided by the CERN SPS in a large range of energies
offer a unique opportunity to investigate the region of the QCD phase diagram
at high baryochemical potential. The NA60+ experiment, proposed for taking
data with heavy-ion collisions at the SPS in the next years, is in an ideal
position to provide new insights into the QCD phase diagram, measuring rare
probes via a Pb–Pb and p–A beam-energy scan, in the collisions energy interval
$\sqrt{s_{NN}}$= 6-17 GeV.

NA60+ plans to measure the production of hidden and open charm hadrons and
prospects on these measurements will be discussed. Open charm hadrons will be
measured from their decays into charged hadrons, reconstructed from the tracks
in the silicon detectors of the vertex telescope. This will enable high-
precision measurements of the yield of D0, D+, and D${}^{+}_{s}$ mesons, and
of $\Lambda_{c}^{+}$ baryons, thus allowing us to constrain the transport
properties of the QGP and the charm-quark hadronisation. Charmonium states,
J/$\psi$ and $\psi$(2S), will be measured through dimuon decays reconstructed
with the muon spectrometer. Hence, by measuring the charmonium yield in p–A
and Pb–Pb collisions at different collision energies, NA60+ will have a unique
opportunity to study the threshold energy for the onset of deconfinement.

##  1 Introduction

The study of the production of heavy-flavour (charm and beauty) hadrons both
at RHIC and LHC are providing very interesting insights on the properties of
the quark gluon plasma (QGP) at small baryochemical potential $\mu_{B}$.
However, these studies would certainly benefit if extended such as to include
new measurements in lower energy heavy-ion collisions. These new data would
allow one to probe the medium at lower temperatures with respect to those
obtained at LHC and RHIC colliders and to explore the region of finite
$\mu_{B}$, in which the baryons of the colliding nuclei are stopped in the
collision region. Very few results are available so far in this kinematic
region. Concerning the heavy-flavour hadrons, there is only a first indirect
open charm measurement by the NA60 Collaboration [1], with an uncertainty of
the order of 20$\%$, and an upper limit measurement on the $D^{0}$ production,
by NA49 [2]. Concerning quarkonium, many high precision results were reported
by the NA50 [3] and NA60 [4] Collaborations, but only at top SPS energies,
with the low energy range still to be investigated.

##  2 The NA60+ experiment

New high-precision measurements on heavy flavour hadrons will be within the
reach of NA60+ [5], a new experiment proposed at the CERN SPS. NA60+ will
investigate proton-nucleus and Pb-Pb collisions, with high luminosity beams
(i.e. $\sim 10^{6}$ Pb/s) in the energy range Elab = 20 - 158 AGeV,
corresponding to the centre-of-mass energies $\sqrt{s_{NN}}$ = 6 - 17.3 GeV.
The setup, shown in Fig. 1, will consist of a vertex spectrometer, made of
silicon sensors, for the precise measurement and production angle of the
produced charged particles, coupled with a muon spectrometer, to measure the
muon tracks filtered by a thick hadron absorber, placed downstream of the
vertex spectrometer. Details on the experimental apparatus can be found in
[5].

Figure 1: The NA60+ experimental setup. More details on the various detectors
composing the apparatus can be found in [5]

The NA60+ experiment will, then, be able to investigate the dimuon spectrum
from threshold to the charmonium mass region and to reconstruct the hadronic
decays of charm and strange hadrons. It should be pointed out that the scope
of the NA60+ experiment goes even beyond the study of open and closed heavy
flavours, including also the study of electromagnetic probes of the QGP, via
the measurement of the muon pair spectrum. More details can be found in [6].

##  3 Open heavy flavour measurements

By investigating the production of open heavy flavours in nucleus-nucleus
collisions at low collision energies, one can access the charm diffusion
coefficient, extract information on charm thermalization and on the
hadronization mechanism and eventually measure the total charm cross-section.
The first parameter depends on the temperature of the medium and it is
expected to be larger in the hadronic phase than in the QGP one, with a
minimum around the pseudocritical temperature ($T_{\rm{pc}}\sim 155$ MeV).
Hence, at the energies reached at the SPS accelerator, we should be able to
explore the QGP properties closer to $T_{\rm{pc}}$. Furthermore, since at SPS
energies the hadronic phase represents a larger fraction of the collision
evolution with respect to the colliders case, the sensitivity to the hadronic
interactions can be investigated. This measurement will also provide an input
for precision estimates of the heavy-quark diffusion coefficient at collider
energies. Studying the transverse momentum ($p_{T}$) distributions and the
azimuthal anisotropy of D mesons, insights on the charm thermalization in a
short-lived medium can also be addressed. Furthermore, the measurement of the
relative abundances of the different charm-hadron species (i.e. comparing the
production of strange and non-strange mesons or of baryons and mesons) can
provide information on the hadronisation mechanisms. Finally, an accurate
reconstruction of mesons and baryons ground states, such as the
$D^{0},D^{+},D^{+}_{s},\Lambda_{c}^{+}$ and possibly the $\Xi_{c}^{0,+}$,
should also provide a precise measurement of the $c\bar{c}$ cross section,
representing, therefore, a crucial reference for charmonium studies. It should
be noted that, so far, the total charm production cross section has never been
measured with high precision at top SPS energies or lower, because the yields
at these energies are very small.

Open-charm studies will be extremely relevant not only in Pb–Pb collisions,
but also in proton-nucleus interactions, using nuclear targets with different
atomic mass numbers, ranging from Be to Pb. These studies will open up the
possibility to investigate the nuclear PDFs via D meson production. NA60+ will
provide measurements in the the Bjorken-x range $0.1<x_{Bj}<0.3$ at $Q^{2}\sim
10-40$ GeV2, i.e. in a kinematic range so far poorly constrained by data,
dominated by anti-shadowing and EMC effects.

NA60+ will measure the open-charm hadrons reconstructing, in the vertex
telescope, their decays into two or three charged hadrons. The huge
combinatorial background, underlying these decays, can be reduced via
geometrical selections on the displaced decay-vertex topology, exploiting the
fact that the mean proper decay lengths $c\tau$ of open-charm hadrons are of
the order of 60-310 $\mu$m. It should be noted that open-charm hadron studies
are those which impose the strongest constraints on the design of the vertex
telescope, since a very good resolution on the track parameters is required.
Performance studies show that in one month of data taking, corresponding to
$\sim 10^{11}$ minimum bias Pb-Pb collisions, it will be possible to measure
the $D^{0}$ production through the decay channel $D^{0}\rightarrow
K^{-}\pi^{+}$ with a statistical precision of the order of $1\%$ even at the
lowest SPS energies. Similarly, the more challenging three-prongs decays will
be investigated, such as the $D^{+}_{s}\rightarrow\phi\pi^{+}\rightarrow
K^{-}K^{+}\pi^{+}$, with a statistical precision of the order of few percent,
also in this case. Example of invariant mass spectra are shown in Fig. 2. The
$\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+}$ measurement will also be
accessible. The measure is quite challenging, given the short lifetime of the
$\Lambda_{c}^{+}$ ($c\tau$ = 60 $\mu$m), and the inclusion, under study, of
timing layers in the vertex telescope might help in the improvement of the
NA60+ particle identification capabilities.

Figure 2: $D^{0}$ and $D^{+}_{s}$ invariant mass spectra, in central Pb-Pb
collisions, at $E_{lab}$ = 60 GeV (left) and 158 GeV (right)

##  4 Quarkonium measurements

The suppression of quarkonium states has been considered, since almost forty
years, one of the key signatures of the QGP formation [7]. Experimental
results obtained at top SPS energies confirm that the $J/\psi$ production is
indeed suppressed and the observed $\sim$30$\%$ suppression is interpreted as
due to the feed-down of the $\psi$(2S) and $\chi_{c}$ excited states. At the
higher RHIC and LHC energies, a new mechanism, called regeneration, has been
observed, leading to an enhancement of the quarkonium production,
counterbalancing, at least partially, the suppression process. Quarkonium
production is also affected by the cold nuclear matter effects, such as the
nuclear modification of the PDFs (e.g. (anti-)shadowing) and the absorption of
the (pre-)resonant state while crossing the nuclear matter. This latter
mechanism is strongly energy-dependent and it is particularly sizable at SPS
energies, because of the much larger crossing time, of the order of 0.5 fm/c
at mid-rapidity, to be compared with $10^{-5}-10^{-2}$ fm/c at LHC energies,
depending on the rapidity range. The precise determination of cold nuclear
matter effects is crucial to understand the size of the hot matter effects at
play and it is usually extensively studied via proton-nucleus collisions.

The NA60+ experiment aims to study the J/$\psi$ production in Pb-Pb
collisions, by performing an energy scan. Correlating the onset of the
J/$\psi$ suppression with the temperature of the system, as measured via the
thermal dimuons (this is one of the main topics that will be investigated by
the NA60+ experiment [6]), the threshold temperature for the melting of the
resonance can be accessed. Proton-nucleus collisions, collected at the same
collision energy as the A-A data, will allow a detailed study of the cold
nuclear matter effects, whose contribution is expected to increase, as already
mentioned, when the collision energy is decreased. It will be important not
only to disentangle the various cold nuclear matter effects at play, but also
to define a precise reference to understand the role of hot matter effects on
charmonium. Finally, the study of the J/$\psi$ production at SPS energies will
also be a testing ground for the observation of the intrinsic charm component
in the nucleon wavefunction. This effect should lead to an enhanced charm
production at large $x_{F}$. While at collider energy the region where the
intrinsic charm should be observed is at very forward rapidity, in fixed
target experiments, at SPS energies, it should be closer to mid-rapidity, and
hence more easily detectable. Calculations [8] indicate that already with a
probability of intrinsic charm contribution in the proton of 0.1%, the effect
of the intrinsic charm should be dominant.

J/$\psi$ and $\psi$(2S) will be measured in NA60+ through their decay in a
pair of muons, reconstructed by matching the tracks in the muon spectrometer
with those reconstructed in the vertex telescope. The expected mass resolution
will be of the order of 30 MeV/c. The $\chi_{c}$ will be measured in the decay
channel $\chi_{c}\rightarrow J/\psi\gamma$, with the $\gamma$ measured via its
conversion into a lepton pair in the vertex telescope. To cope with the low
production cross sections at low $\sqrt{s}$, a high luminosity is required. As
shown in Fig. 3 (left), performance studies based on one month of data taking,
with $I_{\rm{beam}}=10^{7}$ Pb/spill, and a total luminosity $L_{\rm{int}}$ =
24 $nb^{-1}$, assuming a factor 3 overall suppression for quarkonium, as
educated guess, we will collect $10^{4}-10^{5}$ J/$\psi$, depending on the
collision energy. This statistics will allow us to perform an accurate study
of the J/$\psi$ suppression from top SPS energies, down to the lowest
available ones. As shown in Fig. 3 (right), we should be able to determine the
onset for the suppression even at low $\sqrt{s_{NN}}$.

Figure 3: Number of expected J/$\psi$ collected under the assumptions
described in the text, as a function of the luminosity (left); performance
study of the J/$\psi$ nuclear modification factor $R_{AA}$ as a function of
centrality, at $E_{\rm{lab}}$ = 50 GeV. The light blue band corresponds to the
cold nuclear matter effects, as obtained from a performance study based on
realistic expectations for the p-A data taking.

A similar study on the $\psi$(2S) will also be within reach, at least down to
$E_{\rm{lab}}$ = 100 GeV. Reaching precise results also at lower beam energies
would require larger beam intensities or longer data taking periods. Studies
for the detection of the $\chi_{c}\rightarrow\mu\mu\gamma$ decays are ongoing
and they’ll enrich the NA60+ charmonium program.

##  5 Conclusions

Open charm and quarkonium physics has proven to be of extreme interest in the
understanding the properties of the QGP. However, while plenty of results are
available at the RHIC and LHC collider energies, we are so far lacking results
below top SPS energies. Hence, measurements carried on between
$6<\sqrt{s_{NN}}<17$ GeV/c will be mandatory to expand our understanding
towards the region of high-$\mu_{B}$. The NA60+ experiment, proposed at CERN
SPS starting from 2029, after the LHC Long Shutdown 3, is certainly well
positioned to perform this kind of studies and will certainly help to shed
further light on the heavy-flavour physics, complementing the understanding so
far achieved after a few decades of RHIC and LHC results.

## References

  * [1] R. Arnaldi et al. [NA60 Collaboration], Eur. Phys. J. C 59 (2009), 607-623 [arXiv:0810.3204 [nucl-ex]].
  * [2] C. Alt et al. [NA49 Collaboration], Phys. Rev. C 73 (2006), 034910 [arXiv:nucl-ex/0507031 [nucl-ex]].
  * [3] B. Alessandro et al. [NA50 Collaboration], Eur. Phys. J. C 39 (2005), 335-345 [arXiv:hep-ex/0412036 [hep-ex]].
  * [4] R. Arnaldi et al. [NA60 Collaboration], Conf. Proc. C 060726 (2006), 430-434 [arXiv:0706.4361 [nucl-ex]].
  * [5] C. Ahdida et al. [NA60+ Collaboration], [arXiv:2212.14452 [nucl-ex]].
  * [6] E. Scomparin, these proceedings, [arXiv:2307.11473 [nucl-ex]].
  * [7] T. Matsui and H. Satz, Phys. Lett. B 178 (1986), 416-422
  * [8] R. Vogt, Phys. Rev. C 103 (2021) no.3, 035204 [arXiv:2101.02858 [hep-ph]].


# Entropy Product Function and Central charges in NUT Geometry

Parthapratim<EMAIL_ADDRESS>
Department of Physics
Hiralal Mazumdar Memorial College For Women
Dakshineswar, Kolkata-700035, India

###### Abstract

We define an _entropy product function_ (EPF) for Taub-Newman-Unti-Tamburino
(TNUT) black hole (BH) following the prescription suggested by Wu et al. [1]
[PRD 100, 101501(R) (2019)]. The prescription argues that a generic four-
dimensional TNUT spacetime might be expressed in terms of three or four
different types of thermodynamic hairs. They can be defined as the Komar mass
($M=m$), the angular momentum ($J_{n}=mn$), the gravitomagnetic charge
($N=n$), the dual (magnetic) mass $(\tilde{M}=n)$. Taking this prescription
and using the _EPF_ , we derive the _central charges_ of dual CFT (conformal
field theory) via Cardy’s formula. Remarkably, we _find_ that for TNUT BH
there exists a relation between the _central charges and EPF_ as
$c=6\left(\frac{\partial{\cal F}}{\partial{\cal N}_{i}}\right)$, where ${\cal
F}$ is EPF and ${\cal N}_{i}$ is one of the integer-valued charges i.e. the
NUT charges ($N$) or any new conserved charges ($J_{N}$). We reverify these
results by calculating the exact values of different thermodynamic parameters.
We define the EPF ${\cal F}$ from the first law of thermodynamics of both
horizons. Moreover, we write the first laws of both the horizons for left-
moving and right-moving sectors. Introducing the Bézout’s identity, we show
that for TNUT BH one can generate more holographic descriptions described by a
pair of integers $(a,b)$. More holographic pictures have a great significance
towards understanding the holographic nature of quantum gravity. Furthermore,
using the _EPF_ we derive the central charges for Reissner-Nordström-NUT
(RNNUT) BH, Kerr-Taub-NUT (KNUT) BH and Kerr-Newman-NUT (KNNUT) BH. Finally we
prove that they are equal in both sectors provided that the EPF is mass-
independent (or universal).

##  1 Introduction

The Kerr/CFT correspondence [2] is a great discovery in BH physics to
understand the entropy of a BH. It is the macroscopic entropies of BHs which
has been reproduced exactly from the microscopic states counting in the dual
2D CFT via Cardy formula. The fact started for many years that the microscopic
origin of the BH entropy could be holographically encoded in a 2D CFT,
particularly after successfully counting of the Bekenstein-Hawking entropies
of a extremal BH in string theory [3].

One aspect is that the greybody factors for rotating black holes in four
dimensions were derived by Cvetic and Larsen [4] to study the Hawking
radiation. The string excitations in both sectors i.e. left and right moving
sectors generate two distinct contributions of entropy. Similarly, the
emission spectrum of the string is identified by left moving and right moving
which is independent of temperatures. Consequently the appearance of two
independent sets of thermodynamic variables is related to the presence of
event horizon and Cauchy horizons. For instance, the two contributions to the
entropy are proportional to the sum and difference of the Cauchy and event
horizon area, respectively. This geometic interpretation of the thermodynamic
variables gives access to features of the underlying microscopic theory.

On the other hand the solution of wave equation for most general spacetime
background is derived to compute greybody factors in four and five dimensions
[5]. Interestingly the wave equation has an exact symmetry that interchanges
the Cauchy (or inner) horizon and event (or outer) horizons. Importantly, this
symmetry has a discrete in nature. The modes in the vicinity of the event
horizon give rise to the Hawking radiation having Hawking temperature
$T_{H}=\frac{T_{L}+T_{R}}{2T_{R}T_{L}}$. Similarly, from the modes in the
vicinity of the Cauchy horizon give rise to a characteristic temperature
$T_{C}=\frac{T_{L}-T_{R}}{2T_{R}T_{L}}$. The temperatures $T_{R}$ and $T_{L}$
that appear in these formulas agree precisely with those that follow from
thermodynamics.

One of the interesting feature in the holographic description of KN class of
BHs is that the central charges of the dual CFT are independent of the mass of
BHs. The attribute could be related to the fact that the entropy product of
${\cal H}^{\pm}$ of these BHs are also mass-independent.

In the context of general relativity, it has been examined that for a four-
dimensional Kerr-Newman BH [6], the entropy product of ${\cal H}^{\pm}$ i.e.

$\displaystyle{\cal S}_{+}{\cal S}_{-}$ $\displaystyle=$ $\displaystyle
4\pi^{2}\left(J^{2}+\frac{Q^{4}}{4}\right)$ (1)

is mass-independent. On the other hand in the context of string theory, it has
been argued that for a BPS class of BHs, the entropy product of ${\cal
H}^{\pm}$ should be quantized as [7]

$\displaystyle{\cal S}_{+}{\cal S}_{-}$ $\displaystyle=$
$\displaystyle(2\pi)^{2}\left[\sqrt{N_{1}}+\sqrt{N_{2}}\right]\left[\sqrt{N_{1}}-\sqrt{N_{2}}\right]=(2\pi)^{2}N,\,\,N\in{\mathbb{N}},N_{1}\in{\mathbb{N}},N_{2}\in{\mathbb{N}}~{}.$
(2)

It is an well-established fact that the thermodynamics method 222The
conventional way to determine the cental charges of dual CFT by using
asymptotic symmetry group (ASG) of near-horizon geometry of extremal BH in
either BBC (Branich-Brandt-Compere) formalism‘[2] or equivalently the
stretched horizon formalism [28] is a powerful mechanism to determine the dual
CFT in Kerr/CFT correspondence [9]. In case of Kerr/CFT correspondence it has
been proved that more universal information of the dual CFT including the dual
temperatures, and the central charges of both left-moving sectors and right-
moving sectors are fully encoded in the thermodynamics of the event (or outer)
horizon (EH) (${\cal H}^{+}$) and Cauchy (or inner) horizon (CH) (${\cal
H}^{-}$) of the Kerr-Newman (KN) BH. Also it has been suggested in [9] that
the thermodynamics method is universal to determine the dual holographic
picture of BH.

For KN BH [9] it has been explicitly proved that the first laws of BH
thermodynamics satisfied for EH and CH horizons as

$\displaystyle dM$ $\displaystyle=$ $\displaystyle\pm{T}_{\pm}\,d{\cal
S}_{\pm}+\Omega_{\pm}\,dJ+\Phi_{\pm}\,dQ~{}.$ (3)

where $T_{\pm}$, $\Omega_{\pm}$ and $\Phi_{\pm}$ represents the Hawking
temperature, the angular velocity and the electric potential of ${\cal
H}^{\pm}$ calculated on the horizons for KN BH. Under the exchange of
$r_{+}\leftrightarrow r_{-}$ and using the symmetry of outer (inner) horizons
$r_{\pm}$, one would get the Hawking temperature of the inner horizon

$\displaystyle T_{-}$ $\displaystyle=$ $\displaystyle-
T_{+}\Big{|}_{r_{+}\leftrightarrow r_{-}}$ (4)

while the other quantites are changed under the symmetry of $r_{\pm}$ as

$\displaystyle\mathcal{S}_{-}$ $\displaystyle=$
$\displaystyle\mathcal{S}_{+}\Big{|}_{r_{+}\leftrightarrow
r_{-}},\,\,\Omega_{-}=\Omega_{+}\Big{|}_{r_{+}\leftrightarrow
r_{-}},\,\,\Phi_{-}=\Phi_{+}\Big{|}_{r_{+}\leftrightarrow r_{-}}$ (5)

An another universal relation exists for KN BH

$\displaystyle T_{+}{\cal S}_{+}=T_{-}{\cal S}_{-}~{}~{}$ (6)

which must says that the entropy product is _mass independent_.

In recent times, it has been proposed that [1] a generic four dimensional TNUT
spacetime has four thermodynamical hairs. They could be defined as the Komar
mass ($M=m$), the angular momentum ($J_{n}=mn$), the gravitomagnetic charge
($N=n$), and or the dual (magnetic) mass $(\tilde{M}=n)$. Using this
formalism, _in this work_ we prove that there exists a relation between _EPF
and the central charges of dual CFT for TNUT BH_. First, we define the EPF
from the first law of BH thermodynamics for both the horizons and then using
this function we determine the central charges of dual CFT. Moreover, we
calculate the thermodynamic parameters in left-moving sectors and right-moving
sectors. Furthermore, we compute the first laws of thermodynamics in left-
moving sectors and right-moving sectors. Finally, we examine the holographic
picture of TNUT BH. Since the BH carries NUT parameter ($N$) 333NUT charge
should be considered as a thermodynamic variable since it is independently
varied in the full cohomogeneity of first law in BH thermodynamics. and new
conserved charges ($J_{N}$) hence there exists two elementary holographic
pictures, which we call the $N$ picture and the $J_{N}$ picture.

Like four dimensional dyonic RN BH here in the context of _NUT class of BHs_
we prove that there exists a relation between the EPF (${\cal F}$) and the
central charges ($c$) as

$\displaystyle c^{i}_{L,R}=6\left(\frac{\partial{\cal F}}{\partial{\cal
N}_{i}}\right)~{}$ (7)

where ${\cal N}_{i}$ is one of the integer-valued charges appearing in the
first laws, and should be angular momentum, or another conserved charges. For
instance i.e. for RN BH, the EPF [9] is defined as

$\displaystyle\mathcal{F}=\frac{{\cal S}_{+}{\cal
S}_{-}}{4\pi^{2}}=\frac{Q^{4}}{4}.$ (8)

Since the EPF is mass-independent hence $T_{+}{\cal S}_{+}=T_{-}{\cal S}_{-}$.
Therfore one can derive the central charges for $Q$ picture by using Eq. (7)
as

$\displaystyle c_{L}^{Q}$ $\displaystyle=$ $\displaystyle 6Q^{3}$ (9)
$\displaystyle c_{R}^{Q}$ $\displaystyle=$ $\displaystyle 6Q^{3}~{}$ (10)

They are equal for left moving and right-moving sectors that means
$c_{L}^{Q}=c_{R}^{Q}$. The central charges are equal means that the EPF is
universal (mass-independent). This result is completely agreement with the
central charges derived by asymptotic symmetry group (ASG) analysis [27].
Where it was proved by uplifting the four-dimensional compactification
solution to a five-dimensional solution.

In the literature [15, 16, 17], it has been shown that the area (or entropy)
products in NUT geometry i.e. TNUT BH, KTNUT BH and KNTNUT have “mass-
dependent” characteristics. Secondly, the first laws of thermodynamics in the
left moving sectors and the right moving sectors do not satisfied like Eq.
(30). In that situation, the EPF is mass-dependent that means

$\displaystyle T_{+}{\cal S}_{+}\neq T_{-}{\cal S}_{-}~{}~{}$ (11)

Moreover, we cannot read the information of dual CFT that means we cannot
derived the exact left-moving and right-moving temperatures of dual CFT.
Furthermore, the central charges of left-moving sectors and right-moving
sectors are not equal i.e.

$\displaystyle c_{L}\neq c_{R}$ (12)

However incorporating the fromalism stated in [1], we should be able to
defined the EPF for NUT class of BHs because the entropy product of ${\cal
H}^{\pm}$ is universal i.e. mass-independent [20]. In this work we will
consider the case $J_{N}=MN$ and the case $J_{N}\neq MN$ is already been
discussed in the following work [15, 17, 16]. Moreover in this work we derive
the central charges from entopy product function and compared it with the
result obtained from the ASG analysis for NUT class of BHs.

There are two ways one can compute the central charges. The first one is that
ASG of near horizon geometry of extremal BH and the second one is that via EPF
method i.e. thermodynamics method. In this work we will particularly emphasize
on EPF method. Also we will see that _EPF method is more covenient to derive
the central charges than the ASG method_.

The outline of the Letter is as follows. In the next section (2), we derive
the relation between EPF and central charges for NUT class of BH i.e. TNUT BH.
In Sec. (3), we derive the exact calculations of central charges by direct
calculation. In Sec. (4), we derive the central charges for RNNUT BH by using
EPF. In Sec. (5), we derive the central charges for KNUT BH by using EPF. In
Sec. (6), we derive the central charges for KNNUT BH by using EPF. In Sec.
(7), we have given our conclusions. In Appendix-A (Sec. 8), we evaluate the
central charges from EPF for RN BH, Kerr BH and KN BH. In Appendix-B (Sec. 9),
we provide the calculation of central charges by using near horizon geometry
of extremal KN BH.

##  2  TNUT BH and EPF

The Lorentzian TNUT BH [18, 19] is a solution of the Einstein equation. The
metric is defined as

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-{\cal
A}\,\left(dt+2n\cos\theta d\phi\right)^{2}+\frac{dr^{2}}{{\cal
A}}+\left(r^{2}+n^{2}\right)\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)~{},$ (13)

where the function ${\cal A}$ is

$\displaystyle{\cal A}$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}+n^{2}}\left[r^{2}-n^{2}-2mr\right]$ (14)

Under the new prescription [1] for NUT class of BHs, the global conserved
charges computed via conformal completion method [8] could be defined as

$\displaystyle\mbox{Komar mass}:\,\,m$ $\displaystyle=$ $\displaystyle M~{}$
$\displaystyle\mbox{Angular momentum}:m\,n$ $\displaystyle=$ $\displaystyle
J_{n}~{}$ $\displaystyle\mbox{Gravitomagnetic charge}:\,\,n$ $\displaystyle=$
$\displaystyle N$ (15) $\displaystyle\mbox{Dual~{}(magnetic)
mass}:\,\,\tilde{M}$ $\displaystyle=$ $\displaystyle n~{}.$ (16)

It means that NUT charge is a thermodynamic multihair. It further implies that
simultaneously it has both rotation-like and electromagnetic charge-like
properties.

Taking the effect of Eq. (16), the metric can be written as

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-{\cal
A}\,\left(dt+2N\,\cos\theta d\phi\right)^{2}+\frac{dr^{2}}{{\cal
A}}+\left(r^{2}+N^{2}\right)\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)~{},$ (17)

where the function ${\cal A}$ is given by

$\displaystyle{\cal A}$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}+N^{2}}\left(r^{2}-N^{2}-2Mr\right)$ (18)

The BH horizons are at

$\displaystyle r_{\pm}=M\pm\sqrt{M^{2}+N^{2}}\,\,\mbox{and}\,\,r_{+}>r_{-}$
(19)

$r_{+}$ is called EH and $r_{-}$ is called CH. Thus the entropy [20, 21] of
the BH is computed under the formalism stated in Eq. (16)

$\displaystyle{\cal S}_{\pm}$ $\displaystyle=$ $\displaystyle
2\pi\left[M^{2}+N^{2}\pm\sqrt{M^{4}+J_{N}^{2}}\right]~{}.$ (20)

One can compute the BH temperature by defining partition function of a well-
defined microcanonical ensemble. The fact that a BH solution having a
topological charge should be described by the BH thermodynamics. It was
started by Gibbons and Hawking in 1977 [22]. They proposed that the partition
function of a canonical ensemble for BHs should be calculated with its
Euclidean action in the form of the gravitional path integral. It is described
by

$\displaystyle\cal{Z}$ $\displaystyle=$ $\displaystyle e^{-\beta F}=\int
D[g]e^{-\frac{I}{h}}\sim e^{-\frac{I}{h}},$ (21)

where $I$ and $F$ is the Euclidean action and the free energy of the BH. The
period $\beta$ of the Euclidean time is the inverse of the BH temperature i.e.
$\beta=\frac{1}{T}$. For Taub-NUT BH and for both the horizons ${\cal
H}^{\pm}$, $\frac{I_{\pm}}{\beta_{\pm}}=F_{\pm}=\frac{m}{2}$,
$\beta_{\pm}=\frac{1}{T_{\pm}}$ is the interval of the time coordinate [23].

In our earlier work [21], we proved that mass parameter can be expressed as a
function of area (or entropy), new conserved charges and NUT parameter for
both the horizons i.e. $M=M({\cal S}_{\pm},J_{N},N)$. Then the first law of
thermodynamics is satisfied for both the horizons as

$\displaystyle dM$ $\displaystyle=$ $\displaystyle T_{+}d{\cal
S}_{+}+\omega_{+}dJ_{N}+\psi_{+}dN,~{}$ (22) $\displaystyle=$ $\displaystyle-
T_{-}d{\cal S}_{-}+\omega_{-}dJ_{N}+\psi_{-}dN~{}$ (23)

where

$\displaystyle T_{\pm}$ $\displaystyle=$
$\displaystyle\frac{r_{\pm}-M}{2\pi\left(r_{\pm}^{2}+N^{2}\right)},\,\,\omega_{\pm}=\frac{N}{r_{\pm}^{2}+N^{2}},\,\,\psi_{\pm}=-\frac{2N\,r_{\pm}}{r_{\pm}^{2}+N^{2}}~{}$
(24)

Like KN BH, there exists a symmetry for NUT class of BHs between the following
physical quantites under exchange of two physical horizons:

$\displaystyle T_{-}$ $\displaystyle=$ $\displaystyle-
T_{+}|_{r_{+}\leftrightarrow r_{-}}$ (25) $\displaystyle{\cal S}_{-}$
$\displaystyle=$ $\displaystyle{\cal S}_{+}|_{r_{+}\leftrightarrow r_{-}}$
(26) $\displaystyle\omega_{-}$ $\displaystyle=$
$\displaystyle\omega_{+}|_{r_{+}\leftrightarrow r_{-}}$ (27)
$\displaystyle\psi_{-}$ $\displaystyle=$
$\displaystyle\psi_{+}|_{r_{+}\leftrightarrow r_{-}}~{}$ (28)

This indicates that if the first law of thermodynamics satisfied on the event
horizon then it must be satisfied on the Cauchy horizon. Using Eq. (22) and
Eq. (23) we find

$\displaystyle d\left({\cal S}_{+}{\cal S}_{-}\right)=\left(\frac{{\cal
S}_{-}T_{-}-{\cal S}_{+}T_{+}}{T_{+}T_{-}}\right)\,dM+\left(\frac{{\cal
S}_{+}T_{+}\omega_{-}-{\cal
S}_{-}T_{-}\omega_{+}}{T_{+}T_{-}}\right)\,dJ+\left(\frac{{\cal
S}_{+}T_{+}\psi_{-}-{\cal S}_{-}T_{-}\psi_{+}}{T_{+}T_{-}}\right)\,dQ$ (29)

It can be rewritten as

$\displaystyle d\mathcal{F}=\left(\frac{{\cal S}_{-}T_{-}-{\cal
S}_{+}T_{+}}{4\pi^{2}\,T_{+}T_{-}}\right)\,dM+\left(\frac{{\cal
S}_{+}T_{+}\omega_{-}-{\cal
S}_{-}T_{-}\omega_{+}}{4\pi^{2}\,T_{+}T_{-}}\right)\,dJ+\left(\frac{{\cal
S}_{+}T_{+}\psi_{-}-{\cal
S}_{-}T_{-}\psi_{+}}{4\pi^{2},T_{+}T_{-}}\right)\,dQ~{}.$ (30)

where

$\displaystyle\mathcal{F}$ $\displaystyle=$ $\displaystyle\frac{{\cal
S}_{+}{\cal S}_{-}}{4\pi^{2}}~{}$ (31)

This $\mathcal{F}$ is defined as _EPF_ for TNUT BH. We first defined the EPF
from the first law of thermodynamics of both the horizons. Again we know for
TNUT BH the entropy product of ${\cal H}^{\pm}$ [20] is

$\displaystyle\frac{{\cal S}_{+}{\cal S}_{-}}{4\pi^{2}}$ $\displaystyle=$
$\displaystyle J_{N}^{2}+N^{4}~{}$ (32)

Thus

$\displaystyle\mathcal{F}$ $\displaystyle=$ $\displaystyle J_{N}^{2}+N^{4}~{}$
(33)

where ${\cal F}$ is function of $J_{N}$ and $N$. The term _EPF_ was first
introduced in [11, 10] for investigation of the holographic pictures of four
dimensional dyonic RN BH. The Eq. (30) implies that $\mathcal{F}$ is
independent of the mass parameter indicates that

$\displaystyle T_{+}{\cal S}_{+}$ $\displaystyle=$ $\displaystyle T_{-}{\cal
S}_{-}.~{}~{}$ (34)

To derive the Smarr formula for TNUT BH first we have to consider the
dimensions of the following thermodynamic parameters:

$\displaystyle[M]=[N]=L,\,\,[{\cal S}_{+}]=[{\cal S}_{-}]=[J_{N}]=L^{2}$ (35)

Thus taking $({\cal S}_{+},J_{N},N)$ as independent parameters and $M$ is the
homogenous function of $({\cal S}_{+}^{\frac{1}{2}},J_{N}^{\frac{1}{2}},N)$
with degree 1. Then Euler’s homogenous function theorem indicates that

$\displaystyle M$ $\displaystyle=$ $\displaystyle 2{\cal S}_{+}\frac{\partial
M}{\partial{\cal S}_{+}}+2J_{N}\frac{\partial M}{\partial
J_{N}}+N\frac{\partial M}{\partial N}.$ (36)

Hence the Smarr formula is

$\displaystyle M$ $\displaystyle=$ $\displaystyle 2\left(T_{+}{\cal
S}_{+}+\omega_{+}J_{N}\right)+\Psi_{+}N.$ (37)

Similarly, the Smarr formula for the Cauchy horizon is given by

$\displaystyle M$ $\displaystyle=$ $\displaystyle 2\left(-T_{-}{\cal
S}_{-}+\omega_{-}J_{N}\right)+\Psi_{-}N.$ (38)

Using the Smarr formulae for the event horizon and Cauchy horizons (37) and
(38), we can see that (29) implies that

$\displaystyle\mathrm{d}\ln(\mathcal{F})$ $\displaystyle=$
$\displaystyle\frac{(\omega_{-}-\omega_{+})\mathrm{d}J_{N}+(\Psi_{-}-\Psi_{+})\mathrm{d}N}{2\pi^{2}(\omega_{-}-\omega_{+})J_{N}+\pi^{2}N(\Psi_{-}-\Psi_{+})}.$
(39)

When $J_{N}\neq 0$ and $N\neq 0$, in general we can write

$\displaystyle\frac{\omega_{-}-\omega_{+}}{\Psi_{-}-\Psi_{+}}$
$\displaystyle=$ $\displaystyle g(J_{N},N),$ (40)

where $g(J_{N},N)$ is some unknown function of $(J_{N},N)$. Thus we have

$\displaystyle\mathrm{d}\ln(\mathcal{F})=\frac{g\,\mathrm{d}J_{N}+\mathrm{d}N}{2\pi^{2}g\,J_{N}+N\pi^{2}}.$
(41)

Consistency requires that

$\displaystyle\partial_{N}\frac{g}{2J_{N}g+N}=\partial_{J_{N}}\frac{1}{2J_{N}\,g+N},$
(42)

which leads to

$\displaystyle 2J_{N}\partial_{J_{N}}\,g+N\partial_{N}g=-g.$ (43)

This means that $g$ is the homogeneous function of $(J_{N}^{\frac{1}{2}},N)$
with degree $-1$, thus we may write $g(J_{N},N)=N^{-1}f(N^{2}/J_{N})$.
Integrating (41) we get

$\displaystyle\mathcal{F}\propto J_{N}^{2}\,f(N^{2}/J_{N}),$ (44)

where

$\displaystyle\ln f(x)=\int^{x}_{0}\frac{\mathrm{d}y}{2f(y)+y}.$ (45)

Hence the EPF $\mathcal{F}$ is the homogeneous function of $(J_{N},N^{2})$
with degree 2, thus quasi-homogeneous function of $(J_{N},N)$.

Following Refs. [4, 5], we can define the thermodynamic parameters in the
left-moving and right-moving sectors for TNUT BH as

$\displaystyle T_{L,R}$ $\displaystyle=$
$\displaystyle\frac{T_{+}T_{-}}{T_{-}\pm T_{+}},$ $\displaystyle{\cal
S}_{L,R}$ $\displaystyle=$ $\displaystyle\frac{{\cal S}_{+}\pm{\cal
S}_{-}}{2},$ $\displaystyle\omega_{L,R}$ $\displaystyle=$
$\displaystyle\frac{T_{-}\,\omega_{+}\pm T_{+}\,\omega_{-}}{2(T_{-}\pm
T_{+})}$ $\displaystyle\psi_{L,R}$ $\displaystyle=$
$\displaystyle\frac{T_{-}\,\psi_{+}\pm T_{+}\,\psi_{-}}{2(T_{-}\pm T_{+})}~{}$
(46)

It was demonstrated in Ref. [4, 5] the first law of thermodynamics for left-
moving and right-moving sectors are satisfied separately in the context of
rotating BH in four dimensions, and also there exists separate Smarr formulae
for two sectors. Here, we _show_ that such type of relation exists for _NUT
class of BH_ also. Thus in terms of left and right moving modes of dual CFT,
the first law of BH thermodynamics can be rewritten as

$\displaystyle\frac{dM}{2}$ $\displaystyle=$ $\displaystyle T_{L}\,d{\cal
S}_{L}+\omega_{L}\,dJ_{N}+\psi_{L}\,dN$ (47) $\displaystyle=$ $\displaystyle
T_{R}\,d{\cal S}_{R}+\omega_{R}\,dJ_{N}+\psi_{R}\,dN~{}.$

Now we shall keep $J_{N}$ as a invariant quantity and taking the perturbation
of type $(dN,dJ_{N})=dN(1,0)$ then we should get from the first law

$\displaystyle dN=\frac{T_{L}}{\psi_{R}-\psi_{L}}\,d{\cal
S}_{L}-\frac{T_{R}}{\psi_{R}-\psi_{L}}\,d{\cal S}_{R}~{}.$ (48)

From the above computations, we can read the information of the dual CFT.
Also, we can predict two important facts:
(a) ${\cal S}_{L}$ and ${\cal S}_{R}$ in Eq. (46) are the exact entropies of
the left and right moving sectors of dual CFT [24, 4, 5, 25].
(b) When we keep $J_{N}$ is fixed, we get the $N$ picture i.e the NUT picture.
There exists the first law of thermodynamics

$\displaystyle dN$ $\displaystyle=$ $\displaystyle T_{L}^{N}d{\cal
S}_{L}-T_{R}^{N}d{\cal S}_{R},~{}.$ (49)

where $T_{L}^{N}$ and $T_{R}^{N}$ are the exact left-moving and right-moving
temperatures of the dual CFT. It was first reported in [9, 11] for four
dimensional KN BH.

Let us assume that the CFT entropies should be reduced by Cardy’s formula

$\displaystyle\mathcal{S}_{L}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}}{3}c_{L}^{N}T_{L}^{N}~{}.$ (50)

and

$\displaystyle\mathcal{S}_{R}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}}{3}c_{R}^{N}T_{R}^{N}~{}.$ (51)

from which we can derive the central charges. It should be easily proved that
if the entropy product function is mass-independent then the left-moving and
right-moving sector central charges must be equal [9] i.e.

$\displaystyle c_{L}^{N}=c_{R}^{N}~{}$ (52)

Putting ${\cal S}_{+}={\cal S}_{L}+{\cal S}_{R}$ and ${\cal S}_{-}={\cal
S}_{L}-{\cal S}_{R}$ in Eq. (30) and take variations on both sides of the
equations while keeping $J_{N}$ is constant, and we obtain

$\displaystyle\left(\frac{\partial{\cal F}}{\partial N}\right)dN$
$\displaystyle=$ $\displaystyle\frac{{\cal S}_{L}\,d{\cal S}_{L}-{\cal
S}_{R}\,d{\cal S}_{R}}{2\pi^{2}}~{}$ (53)

Taking the advantage from Eq. (50), Eq. (51), Eq. (52) and Eq. (49), we get
the central charges

$\displaystyle c_{L}^{N}$ $\displaystyle=$ $\displaystyle
6\left(\frac{\partial{\cal F}}{\partial N}\right)=24N^{3}$ (54) $\displaystyle
c_{R}^{N}$ $\displaystyle=$ $\displaystyle 6\left(\frac{\partial{\cal
F}}{\partial N}\right)=24N^{3}~{}$ (55)

These are the central charges of dual CFT in the NUT picture and we proved
that they are equal. Similarly, we should keep $N$ is constant and find the
$J_{N}$ picture in which the dual CFT is that of the central charges

$\displaystyle c_{L}^{J_{N}}$ $\displaystyle=$ $\displaystyle
6\left(\frac{\partial{\cal F}}{\partial J_{N}}\right)=12J_{N}$ (56)
$\displaystyle c_{R}^{J_{N}}$ $\displaystyle=$ $\displaystyle
6\left(\frac{\partial{\cal F}}{\partial J_{N}}\right)=12J_{N}~{}$ (57)

From the above analysis, remarkably we obtain a relation between the EPF and
the central charges as

$\displaystyle c_{L}^{i}$ $\displaystyle=$ $\displaystyle
6\left(\frac{\partial F}{\partial{\cal N}_{i}}\right)$ (58) $\displaystyle
c_{R}^{i}$ $\displaystyle=$ $\displaystyle 6\left(\frac{\partial
F}{\partial{\cal N}_{i}}\right)~{}$ (59)

This is the master formula of this work. Where ${\cal N}_{i}$ is one of the
integer-valued charges appering in the first laws. It should be a NUT charges
or another new conserved charges i.e. $J_{N}$. Thus

$\displaystyle c_{L}^{i}$ $\displaystyle=$ $\displaystyle c_{R}^{i}~{}$ (60)

It must be mentioned that the Eq. (47) indicates how the BH responds to
various types of perturbations. If we could consider the perturbations
carrying only NUT charges, or more precisely $(dN,dJ_{N})=dN(1,0)$, the
corresponding first law [Eq. 49] gives the exact left-moving and right-moving
temperatures, $T_{L}^{N}$ and $T_{R}^{N}$ of dual CFT and then the central
chrages $c_{L,R}^{N}$ in the NUT picture.

On the other side, if we consider the perturbations carrying only new
conserved charges, $J_{N}$ or more precisely $(dN,dJ_{N})=dJ_{N}(0,1)$, the
first law gives the exact left-moving and right-moving temperatures,
$T_{L}^{J_{N}}$ and $T_{R}^{J_{N}}$ of dual CFT in the $J_{N}$ picture and
then the central chrages $c_{L,R}^{J_{N}}$.

Now if we consider the perturbations simultaneously i.e. $(dN,dJ_{N})=d{\cal
N}(a,b)$ with $a$, $b$ being two coprime integers. Then from the first laws we
could write

$\displaystyle\frac{dM}{2}$ $\displaystyle=$ $\displaystyle T_{L}\,d{\cal
S}_{L}+\left(a\,\omega_{L}\,dJ_{N}+b\,\psi_{L}\,dN\right)$ (61)
$\displaystyle=$ $\displaystyle T_{R}\,d{\cal
S}_{R}+\left(a\,\omega_{R}\,dJ_{N}+b\,\psi_{R}\,dN\right)~{}.$

Using a similar procedure we get the dual picture i.e. $(a,b)$ picture, in
which the CFT is that of the central charges

$\displaystyle c_{L}^{(a,b)}$ $\displaystyle=$ $\displaystyle
a\,c_{L}^{N}+b\,c_{L}^{J_{N}}$ (62) $\displaystyle c_{R}^{(a,b)}$
$\displaystyle=$ $\displaystyle a\,c_{R}^{N}+b\,c_{R}^{J_{N}}$ (63)

Now we will introduce the Bézout’s identity which states that every pair of
coprime integers $a$, $b$ there exist other pairs of coprime integers $c$, $d$
such that $ad-bc=1$. Thus the $(a,b)$ picture should be viewed as being
generated from two elementary (1,0) and (0,1) pictures by a $SL(2,Z)$
transformation:

$\displaystyle\left(\begin{array}[]{c}c_{L,R}^{(a,b)}\\\
c_{L,R}^{(c,d)}\end{array}\right)=\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\left(\begin{array}[]{c}c_{L,R}^{(1,0)}\\\
c_{L,R}^{(0,1)}\end{array}\right),~{}~{}~{}\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in SL(2,Z).$ (72)

So for NUT class of BH, we can generate more holographic descriptions labeled
by a pair of coprime integers $(a,b)$. These pictures are constructed on two
elementary pictures and related to each other by $SL(2,Z)$ duality. More
holographic pictures indicates towards multiple holographic duals which play a
key role to understanding the quantum nature of gravity.

##  3 Exact calculation of central charges for TNUT BH

In the previous section we derived the central charges by using EPF. In the
present section we will reverify the above result by direct calculation in
both sectors. To derive the microscopic entropy via the Cardy formula we have
to calculate the following important thermodynamic parameters in left-moving
sectors and right-moving sectors:

$\displaystyle T_{L}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi\left(r_{+}+r_{-}\right)},\,\,\,T_{R}=\frac{1}{4\pi\left(r_{+}-r_{-}\right)}$
$\displaystyle\mathcal{S}_{L}$ $\displaystyle=$
$\displaystyle\frac{\pi(r_{+}-r_{-})^{2}}{2},\,\,\,\mathcal{S}_{R}=\frac{\pi(r_{+}^{2}-r_{-}^{2})}{2}$
$\displaystyle\omega_{L}$ $\displaystyle=$ $\displaystyle
0,\,\,\,\omega_{R}=\frac{N}{(r_{+}-r_{-})^{2}}$ $\displaystyle\Psi_{L}$
$\displaystyle=$
$\displaystyle-\frac{N}{r_{+}-r_{-}},\,\,\,\Psi_{R}=-\frac{N(r_{+}+r_{-})}{(r_{+}-r_{-})^{2}}~{}.$
(73)

There exists two holographic picture for TNUT BH. They are defined as $N$
picture and $J_{N}$ picture. Now we have to calculate the dimensionless
temperature of the left and right moving sectors of the dual CFT in $J_{N}$
picture. It could be found as

$\displaystyle T_{L}^{J_{N}}$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
N}\frac{\left(r_{+}-r_{-}\right)^{2}}{\left(r_{+}+r_{-}\right)}~{}.$ (74)

&

$\displaystyle T_{R}^{J_{N}}$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
N}\left(r_{+}-r_{-}\right)~{}.$ (75)

These are exactly the microscopic temperature of dual CFT in TNUT spacetime.

Now we are ready to determine the central charges in left and right moving
sectors of the TNUT/CFT correspondence via the Cardy formula

$\displaystyle\mathcal{S}_{L}^{J_{N}}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}}{3}c_{L}^{J_{N}}T_{L}^{J_{N}},\,\,\,\mathcal{S}_{R}^{J_{N}}=\frac{\pi^{2}}{3}c_{R}^{J_{N}}T_{R}^{J_{N}}~{}.$
(76)

Hence the central charges of dual CFT becomes

$\displaystyle c_{L}^{J_{N}}$ $\displaystyle=$ $\displaystyle
12J_{N},\,\,\,c_{R}^{J_{N}}=12J_{N}~{}.$ (77)

It implies that the central charges of left moving sectors and right moving
sectors of dual CFT are same for TNUT BH. This is a remarkable result for
_TNUT BH_. This is possible only due to introduction of new conserved charges
i.e. $J_{N}$. The result is exactly same as we have seen in case of Kerr BH
[13] and KN BH [9]. This kind of observation tells us that TNUT BH is dual to
$c_{L}^{J_{N}}=c_{R}^{J_{N}}=12J_{N}$ of 2D CFT at temperature
$(T_{L}^{J_{N}},T_{R}^{J_{N}})$ for each value of $M$ and $J_{N}$.

Analogously, for $N$ picture, the dimensionless temperature of the left and
right moving sectors of the dual CFT are

$\displaystyle T_{L}^{N}$ $\displaystyle=$
$\displaystyle\frac{\left(r_{+}-r_{-}\right)^{2}}{16\pi N^{3}}~{}.$ (78)

and

$\displaystyle T_{R}^{N}$ $\displaystyle=$
$\displaystyle\frac{\left(r_{+}^{2}-r_{-}^{2}\right)}{16\pi N^{3}}~{}.$ (79)

These are exactly the microscopic temperature of dual CFT in $N$ picture for
TNUT spacetime. Therefore the central charges of dual CFT are computed to be

$\displaystyle c_{L}^{N}$ $\displaystyle=$ $\displaystyle
24N^{3},\,\,\,c_{R}^{N}=24N^{3}~{}.$ (80)

These results are completely agreement with the result obtained in previous
section. With the above thermodynamic parameters one can easily check that for
TNUT BH

$\displaystyle\frac{\mathcal{S}_{L}}{\mathcal{S}_{R}}=\frac{T_{L}}{T_{R}}$
(81)

which is definitely satisfied. Where $\mathcal{S}_{L,R}$ are the left moving
entropies and right moving entropies of the dual CFT, and $T_{L,R}$ are the
CFT temperatures. Now we will give more examples for derivations of central
charges by using _EPF_. To do that we have to consider first RN-NUT BH.

##  4  Central charges for RN-NUT BH from EPF

For RN-NUT BH, the metric function ${\cal A}$ should be

$\displaystyle{\cal A}$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}+N^{2}}\left(r^{2}-N^{2}-2Mr+Q^{2}\right)$ (82)

where $Q$ is the purely electric charge. Now the BH horizons are located at

$\displaystyle r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}+N^{2}}$ (83)

Now the entropy [20, 21] of ${\cal H}^{\pm}$ is computed under the criterion
stated in Eq. (16)

$\displaystyle{\cal S}_{\pm}$ $\displaystyle=$
$\displaystyle\pi\left[2\left(M^{2}+N^{2}\right)-Q^{2}\pm
2\sqrt{M^{4}-M^{2}Q^{2}+J_{N}^{2}}\right]~{}.$ (84)

The EPF for this BH is

$\displaystyle{\cal F}$ $\displaystyle=$ $\displaystyle
J_{N}^{2}+\left(\frac{Q^{2}}{2}-N^{2}\right)^{2}~{}$ (85)

For $J_{N}$ picture, we get the central charges for left moving and right-
moving sectors

$\displaystyle c_{L}^{J_{N}}$ $\displaystyle=$ $\displaystyle
6\left(\frac{\partial{\cal F}}{\partial J_{N}}\right)=12J_{N}$ (86)
$\displaystyle c_{R}^{J_{N}}$ $\displaystyle=$ $\displaystyle
6\left(\frac{\partial{\cal F}}{\partial J_{N}}\right)=12J_{N}~{}$ (87)

It proves that central charges are equal in both sectors i.e.

$\displaystyle c_{L}^{J_{N}}$ $\displaystyle=$ $\displaystyle
c_{R}^{J_{N}}~{}$ (88)

Similarly for $Q$ picture, we get the equal central charges for left moving
and right-moving sectors

$\displaystyle c_{L}^{Q}$ $\displaystyle=$ $\displaystyle
12Q\left(\frac{Q^{2}}{2}-N^{2}\right)$ (89) $\displaystyle c_{R}^{Q}$
$\displaystyle=$ $\displaystyle 12Q\left(\frac{Q^{2}}{2}-N^{2}\right)~{}$ (90)

Analogously for $N$ picture, the equal central charges for left moving and
right-moving sectors are

$\displaystyle c_{L}^{N}$ $\displaystyle=$ $\displaystyle
24N\left(N^{2}-\frac{Q^{2}}{2}\right)$ (91) $\displaystyle c_{R}^{N}$
$\displaystyle=$ $\displaystyle 24N\left(N^{2}-\frac{Q^{2}}{2}\right)~{}$ (92)

Alternatively, we could say that the central charges are equal means the EPF
is universal (mass-independent).

##  5  Central charges for KNUT BH from EPF

The metric of KNUT BH is

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{\Delta_{r}}{\rho^{2}}\,\left[dt-
Xd\phi\right]^{2}+\frac{\sin^{2}\theta}{\rho^{2}}\,\left[(r^{2}+a^{2}+N^{2})\,d\phi-a\,dt\right]^{2}+\rho^{2}\,\left[\frac{dr^{2}}{\Delta_{r}}+d\theta^{2}\right]~{}.$
(93)

where

$\displaystyle a$ $\displaystyle\equiv$
$\displaystyle\frac{J}{M},\,\rho^{2}\equiv r^{2}+(N+a\,\cos\theta)^{2}$ (94)
$\displaystyle\Delta_{r}$ $\displaystyle\equiv$ $\displaystyle
r^{2}-2Mr+a^{2}-N^{2}$ (95) $\displaystyle X$ $\displaystyle\equiv$
$\displaystyle a\,\sin^{2}\theta-2N\,\cos\theta~{}.$ (96)

and the global conserved charges are the Komar mass $M$, angular momentum
$J=aM$ and gravitomagnetic charge or dual mass or NUT parameter $N$.

The horizons are located at

$\displaystyle r_{\pm}$ $\displaystyle=$ $\displaystyle
M\pm\sqrt{M^{2}-a^{2}+N^{2}}$ (97)

Introducing $J_{N}=MN$, the entropy of ${\cal H}^{\pm}$ should be derived to
be [20, 21]

$\displaystyle{\cal S}_{\pm}$ $\displaystyle=$ $\displaystyle
2\pi\left[(M^{2}+N^{2})\pm\sqrt{M^{4}+J_{N}^{2}-J^{2}}\right]~{}.$ (98)

The EPF for KNUT BH is defined as

$\displaystyle{\cal F}$ $\displaystyle=$ $\displaystyle
J^{2}+J_{N}^{2}+N^{4}~{}$ (99)

Hence for $J$ picture, we find the equal central charges for left moving and
right-moving sectors

$\displaystyle c_{L}^{J}$ $\displaystyle=$ $\displaystyle 12J$ (100)
$\displaystyle c_{R}^{J}$ $\displaystyle=$ $\displaystyle 12J~{}$ (101)

Similarly for $J_{N}$ picture, we find the equal central charges for left
moving and right-moving sectors

$\displaystyle c_{L}^{J_{N}}=c_{R}^{J_{N}}=12J_{N}~{}$ (102)

Analogously for $N$ picture, the equal central charges for left moving and
right-moving sectors are

$\displaystyle c_{L}^{N}=c_{R}^{N}=24N^{3}~{}$ (103)

##  6  Central charges for KNNUT BH from EPF

The metric for KNNUT BH can be written as

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{\Delta_{r}}{\rho^{2}}\,\left[dt-
Xd\phi\right]^{2}+\frac{\sin^{2}\theta}{\rho^{2}}\,\left[(r^{2}+a^{2}+N^{2})\,d\phi-a\,dt\right]^{2}+\rho^{2}\,\left[\frac{dr^{2}}{\Delta_{r}}+d\theta^{2}\right]~{}.$
(104)

where

$\displaystyle a$ $\displaystyle\equiv$
$\displaystyle\frac{J}{M},\,\rho^{2}\equiv r^{2}+(N+a\,\cos\theta)^{2}$ (105)
$\displaystyle\Delta_{r}$ $\displaystyle\equiv$ $\displaystyle
r^{2}-2Mr+a^{2}+Q^{2}-N^{2}$ (106) $\displaystyle X$ $\displaystyle\equiv$
$\displaystyle a\,\sin^{2}\theta-2N\,\cos\theta~{}.$ (107)

and here the conserved charges are Komar mass $M$, angular momentum $J=aM$,
gravito-electric charge $Q$ and the gravitomagnetic charge or dual (magnetic)
mass or NUT charge $N$. The BH horizons are

$\displaystyle r_{\pm}$ $\displaystyle\equiv$ $\displaystyle
M\pm\sqrt{M^{2}-a^{2}-Q^{2}+N^{2}}$ (108)

Incorporating $J_{N}=MN$, the entropy of ${\cal H}^{\pm}$ should be derived as
[20, 21]

$\displaystyle{\cal S}_{\pm}$ $\displaystyle=$ $\displaystyle
2\pi\left[2(M^{2}+N^{2})-Q^{2}\pm
2\sqrt{M^{4}+J_{N}^{2}-J^{2}-M^{2}Q^{2}}\right]~{}.$ (109)

The EPF for KNNUT BH is defined to be

$\displaystyle{\cal F}$ $\displaystyle=$
$\displaystyle\left[J^{2}+J_{N}^{2}+\left(\frac{Q^{2}}{2}-N^{2}\right)^{2}\right]~{}$
(110)

Hence we find the equal central charges for left moving and right-moving
sectors for $J$ picture

$\displaystyle c_{L}^{J}=c_{R}^{J}=12J=12aM=12a\sqrt{a^{2}+Q^{2}-N^{2}}~{}$
(111)

This central charges in $J$ picture are completely agreement with the central
charges derived by using Brown-Henneaux technique which makes the use of
asymptotic symmetry group (ASG) [26][Shortly derived in Appendix C]. Similarly
we find the equal central charges for left moving and right-moving sectors for
$J_{N}$ picture

$\displaystyle c_{L}^{J_{N}}=c_{R}^{J_{N}}=12J_{N}~{}$ (112)

Analogously the equal central charges for left moving and right-moving sectors
are in $N$ picture,

$\displaystyle c_{L}^{N}=c_{R}^{N}=24N\left(N^{2}-\frac{Q^{2}}{2}\right)~{}$
(113)

Finally for $Q$ picture, we get the equal central charges for left moving and
right-moving sectors

$\displaystyle c_{L}^{Q}=c_{R}^{Q}=12Q\left(\frac{Q^{2}}{2}-N^{2}\right)~{}$
(114)

The above analysis tells us that using EPF one could derive central charges
for various pictures of conserved charges. It also remarkable that the central
charges are equal means the _EPF is mass-independent_. On the other hand the
central charges are unequal means the _EPF is mass-dependent_. This is an
another way one could verify whether the entropy product of ${\cal H}^{\pm}$
is mass-independent or not.

##  7  Conclusions

It was proposed that a generic four dimensional TNUT BH could be explicitly
expressed in terms of three or four different types of thermodynamic hairs.
They must be defined as the Komar mass ($M=m$), the angular momentum
($J_{n}=mn$), the gravitomagnetic charge ($N=n$), and or the dual (magnetic)
mass $(\tilde{M}=n)$. In this context, we defined the EPF for a TNUT BH. Using
this feature and the above proposal, we evaluated the _central charges_ of
dual CFT by virtue of Cardy’s formula.

Remarkably for NUT class of BH, we proved that there exists an _universal
relation between the central charges and EPF_ as $c=6\left(\frac{\partial{\cal
F}}{\partial{\cal N}_{i}}\right)$. We first showed the entropy product
function could be derived by using first law of thermodynamics of both the
horizons. Also, we computed the different thermodynamic parameters in the
left-moving sectors and right-moving sectors and reverified the results
obtained by using EPF. Moreover, we examined the first laws that satisfied for
both the horizons in case of left-moving and right-moving sectors.

Furthermore, we introduced the Bézout’s identity and showed that for NUT class
of BH we can generate more holographic pictures that described by a pair of
intgers $(a,b)$. Again more holographic picture of NUT class of BH indicates
toward understanding the holographic nature of quantum gravity. At the end we
showed that using EPF one could derive central charges for various NUT class
of BHs. Remarkably, we found that the central charges are equal in both
sectors and this immediately indicates that the _EPF is mass-independent_.
This is an another way one could verify whether the entropy product is mass-
independent or not. Finally, we showed that the EPF method is more convenient
to derive the central charges than the ASG method.

##  8 Appendix-A

In this appendix section, we will calculate the central charges from EPF for
four dimensional RN BH, Kerr BH and KN BH and contrast with the central
charges derived from the near-horizon geometry of the corresponding extremal
BH by using the asymptotic symmetry group (ASG) introduced by Brown and
Henneaux [12].

1) _Kerr BH_
The EPF for Kerr BH is

$\displaystyle{\cal F}$ $\displaystyle=$ $\displaystyle J^{2}~{}$ (115)

The equal central charges for left moving and right-moving sectors obtained
for $J$ picture as

$\displaystyle c_{L}^{J}$ $\displaystyle=$ $\displaystyle 12J$ (116)
$\displaystyle c_{R}^{J}$ $\displaystyle=$ $\displaystyle 12J~{}$ (117)

This result is completely agreement with the central charges obtained by ASG
analysis [2].

2)_KN BH_
The EPF for KN BH is given by

$\displaystyle{\cal F}$ $\displaystyle=$ $\displaystyle
J^{2}+\frac{Q^{4}}{4}~{}$ (118)

Hence for $J$ picture we find the central charges for left moving and right-
moving sectors are

$\displaystyle c_{L}^{J}$ $\displaystyle=$ $\displaystyle 12J~{}$ (119)
$\displaystyle c_{R}^{J}$ $\displaystyle=$ $\displaystyle 12J$ (120)

These results are completely agreement with the central charges obtained by
ASG analysis in Eq. (125). Similarly for $Q$ picture, one finds the equal
central charges for left moving and right-moving sectors

$\displaystyle c_{L}^{Q}$ $\displaystyle=$ $\displaystyle 6Q^{3}$ (121)
$\displaystyle c_{R}^{Q}$ $\displaystyle=$ $\displaystyle 6Q^{3}~{}$ (122)

Similarly, these results are also completely agreement with the central
charges obtained by ASG analysis [13].

##  9 Appendix B: Calculation of Central charges by using Near-Horizon
Geometry of Extremal KN BH

In this appendix section, we will review how to derive central charges of
extremal KN BH in the near-horizon limit by using ASG analysis followed by the
work of Hartman [13]. To do that we have to consider near horizon metric of an
extremal, stationary, rotationally symmetric BH of the following form

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\Upsilon\left[-r^{2}\,dt^{2}+\frac{dr^{2}}{r^{2}}+\zeta\,d\theta^{2}\right]+\chi\left(d\phi+kr\,dt\right)^{2}~{}$
(123)

where

$\Upsilon=\rho_{+}^{2},\,\zeta=1,\,\chi=\frac{(r_{+}^{2}+a^{2})^{2}\sin^{2}\theta}{\rho_{+}^{2}}$

and we have defined

$\rho_{+}^{2}=r_{+}^{2}+a^{2}\cos^{2}\theta,\,k=\frac{2ar_{+}}{r_{+}^{2}+a^{2}},\,r_{+}=m$

Then the central charges is derived to be in the $J$ picture

$\displaystyle c_{L}$ $\displaystyle=$ $\displaystyle
3k\int_{0}^{\pi}\sqrt{\Upsilon\,\zeta\,\chi}\,d\theta$ (124) $\displaystyle=$
$\displaystyle 12ar_{+}=12am=12J=12a\sqrt{a^{2}+Q^{2}}~{}$ (125)

##  10 Appendix C: Calculation of Central charges by using Near-Horizon
Geometry of Extremal KN-NUT BH

In this appendix section, we will review how to derive central charges of
extremal KN-NUT BH in the near-horizon limit by using ASG analysis followed by
the work [26]. To proceed we have to consider near horizon metric of an
extremal, stationary, rotationally symmetric BH of the following form

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\Upsilon\left[-y^{2}\,d\tau^{2}+\frac{dy^{2}}{y^{2}}+\zeta\,d\theta^{2}\right]+\chi\left(d\phi+p\,y\,d\tau\right)^{2}~{}$
(126)

where

$\Upsilon=\rho_{+}^{2},\,\zeta=1,\,\chi=\frac{[r_{+}^{2}+(a+n)^{2}]^{2}\sin^{2}\theta}{\rho_{+}^{2}}$

and we have defined

$\rho_{+}^{2}=r_{+}^{2}+(n+a\cos\theta)^{2},\,p=\frac{2ar_{+}}{r_{+}^{2}+(a+n)^{2}},\,r_{+}=m$

Then the central charges is derived to be in the $J$ picture

$\displaystyle c_{L}$ $\displaystyle=$ $\displaystyle
3p\int_{0}^{\pi}\sqrt{\Upsilon\,\zeta\,\chi}\,d\theta$ (127) $\displaystyle=$
$\displaystyle 12ar_{+}=12am=12J~{}$ (128)

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