1 Introduction↩︎

When the classical general relativity theory was found decades ago, Hawking and Bekenstein separately conjectured that any BH in thermal equilibrium has an entropy and a temperature. The entropy is known to be solely proportional to the quarter area of the BH. The exact expression of the entropy of the BH on the event horizon (at \(r=r_+\)) is explicitly given by [1]–[4] \[S_+ = \frac{k _BA_+}{4l^2_{P}},\] where the Planck length \(l^2_P = G\hbar/c^3\) and \(A_+\) is the area of the event horizon. The BH temperature is proportional to the surface gravity at \(r_+\) as given by \[T_+ = \frac{\hbar\kappa_+}{2\pi k_B}.\] where \(\kappa_+\) denotes the surface gravity of the BH on \(r_+\).

The presence of \(l^2_{P}\) is the result coming from the non-perturbative quantum gravity. Moreover, the presence of Boltzmann constant \(k_B\) shows the point of view of the entropy as the logarithm of density of states of thermodynamical system. The computation done by Hawking and Bekenstein provides us the relation between quantum and gravitational theories represented by BHs.

Since the above formulation of entropy and temperature occurs on the event (outer) horizon \(r_+\), it is natural to question whether such relations are also valid on the Cauchy (inner) horizon (at \(r_-\)). Now it is acclaimed that such relations also hold on \(r_-\), resulting the following similar formula, \[S_- = \frac{k _BA_-}{4l^2_{P}}, ~~~T_- = \frac{\hbar\kappa_-}{2\pi k_B},\] where \(A_-\) and \(\kappa_-\) are the area and surface gravity on the inner horizon.

Some speculations about the the area of the BHs is being quantized appears in Refs. [5]–[8] while it is also supported by the calculation for some specific BH solutions, for example, in string theory and M-theory [9]–[11]. For asymptotically flat BPS BHs in four and higher dimensions, the quantization rule becomes \[S_\pm = 2\pi \ell_{\rm pl}^2\left(\sqrt{N_1}\pm \sqrt{N_2}\right),\] or one can find the entropy product as [12]–[21] \[S_+S_- = (2\pi \ell_{\rm pl}^2)^2 N, ~~~N, N_1, N_2\in \mathbb{N}.\label{eq:entropyproduct}\tag{1}\] As we can observe that the above relations are mass independent or universal. Such universal thermodynamic products have attracted considerable attention because they provide direct probes of the microscopic structure of BH entropy via BH/CFT correspondence.

The Kerr-Sen BH [22] is an exact rotating charged solution in four-dimensional low-energy heterotic string theory which carries mass \(M\), angular momentum \(J\), and electric charge \(Q\). It reduces to the Kerr BH in the uncharged limit. In the Einstein frame, the entropy (area) product is universal while the entropy (area) sum depends on the ADM mass [23].

The dyonic generalization of the Kerr-Sen BH arises naturally in Einstein-Maxwell-Dilaton-Axion (EMDA) theory, where the BH simultaneously carries both electric charge \(Q\) and magnetic charge \(P\) [24]. This dyonic Kerr-Sen BH has attracted growing interest in recent years [25]–[30]. Its richer charge structure introduces qualitatively new features at the level of multi-horizon thermodynamics: the interplay between \(Q\) and \(P\) modifies the thermodynamic properties of the solution such as Smarr and Christodoulou-Ruffini irreducible mass [24].

Recently, the investigation of holographic pictures of BHs has grown rapidly in order to find the alternative and non-perturbative way of grasping its quantum properties. It is conjectured in Ref. [31] that the BH partition function is dual to the 2D CFT partition function which leads to the matching between Bekenstein-Hawking entropy of Kerr BH and Cardy entropy formula from 2D CFT. Then there are subsequent development to study more BH solutions in Einstein [32], [33] or beyond Einstein theory [34]. Furthermore, the radial wave equation on BH’s background also captures hidden conformal symmetry [35] that generalizes the result from [31] for both sectors in 2D CFT. For some development of the hidden conformal symmetry, see Refs. [36]–[38].

Another interesting perspective of the dual CFT of BH is to consider the 2D CFT thermodynamics from a BH thermodynamic in multi-horizon perspective [17]–[19], [39], [40]. It has opened a new window and shed considerable light on the black hole/CFT correspondence. It has long been recognized that the inner horizon may play a pivotal role in the microscopic counting of entropy associated with the outer horizon. In particular, it was pointed out in Refs. [17]–[19] that the mass-independence of the entropy product \(\mathcal{S}_{+}\mathcal{S}_{-}\) carries non-trivial implications for the holographic description of black holes. The mass-independence of the entropy product \(\mathcal{S}_{+}\mathcal{S}_{-}\) is precisely equivalent to the condition \[T_{+}\mathcal{S}_{+} + T_{-}\mathcal{S}_{-}=0.\label{eq:TSuniversality}\footnote{ Note that in some papers such as \cite{ChenLiuJHEP2012,ChenXueJHEP2013,ChenZhangJHEP2013}, they use T_{+}\mathcal{S}_{+} =T_{-}\mathcal{S}_{-} instead because they cast out the negative sign on T_-. }\tag{2}\] Interestingly, building upon these foundational results, it was proposed also that this thermodynamic method can be applied successfully to certain regular BHs [41] and acoustic BHs [42].

In the present work, we explicitly extend this multi-horizon thermodynamic framework to the dyonic Kerr-Sen black hole, which carries both electric charge \(Q\) and magnetic charge \(P\) in addition to mass \(M\) and angular momentum \(J\). We demonstrate that the entropy product \(\mathcal{S}_{+}\mathcal{S}_{-}\) for the DKSBH is mass-independent, thereby satisfying the universality criterion. We use this property to derive the central charges \(c_{L} = c_{R}\) of the dual two-dimensional CFT in the Kerr/CFT correspondence via the Cardy entropy formula. These results provide further microscopic evidence for the holographic description of the entropy of the DKSBH. We briefly compare the results from thermodynamic method of the dyonic Kerr-Sen BH to those of Kerr-Newman BH. Moreover, we also derive the CFT temperatures and central charge of dyonic dilaton BH which is the static version of dyonic Kerr-Sen BH.

By studying the radial scalar wave on dyonic Kerr-Sen background with low-frequency limit, we find exactly the same results of the CFT temperature derived from thermodynamic method. Furthermore, the existence of dual CFT for dyonic Kerr-Sen BH allows us to compute the QNM spectra in the dyonic Kerr-Sen background due to the fact that the radial scalar wave equation captures \(SL(2,R)\times SL(2,R)\) isometry.

The paper is organized as follows. Section 2 presents the dyonic Kerr-Sen BH solution and its thermodynamics. Section 3 derives the thermodynamic products and exhibits the universal properties of dyonic Kerr-Sen BH thermodynamics. In Section 4, we compute the CFT thermodynamic quantities and subsequently compare it with those of Kerr-Newman BH. We also compute the CFT thermodynamics of dyonic dilaton BH in Section 5 and the QNM specturm from the dual CFT in Section 6. Finally, Section 7 summarizes our results. Throughout this work, the natural units \(G = \hbar = c = k_B = 1\) are used.

2 Dyonic Kerr-Sen Black Hole↩︎

2.1 Spacetime metric↩︎

An exact rotating dyonically and electrically charged BH solution in four dimensional Einstein-Maxwell-Dilaton-Axion theory is given by the following metric in Boyer-Lindquist coordinates [24], \[ds^2 = - \frac{\Delta}{\varrho^2 } X^2+ \frac{\varrho ^2}{\Delta}dr^2 + \varrho ^2 d\theta^2 + \frac{\sin^2\theta}{\varrho ^2}Y^2,\label{eq:dKSmetric} \;\tag{3}\] where \[X = dt - a \sin^2\theta d\phi, ~~ Y = adt- (r^2-2dr-k^2+a^2 )d\phi,\] \[\Delta = r^2-2dr-2m(r-d) -k^2+a^2+p^2+q^2,\] \[\varrho^2 = r^2-2dr -k^2 + a^2\cos^2\theta.\label{eq:metricfunctiondKS}\;\tag{4}\] The parameters \(m, a, q, p, d, k\) are mass, spin, electric charge, magnetic (dyonic) charge, dilaton charge, and axion charge of the black hole, respectively. The dilaton and axion charges explicitly depend on the electromagnetic charges with the following relations \[d= \frac{p^2 -q^2}{2m}, ~~~~~ k =\frac{pq}{m}.\;\] As we can observe, when one of the electromagnetic charges vanishes, the axion charge will also vanish. When both electromagnetic charges possess the same value, the dyonic charge will vanish.

The electromagnetic field, its dual, dilaton scalar, and axion pseudoscalar fields of the dyonic Kerr-Sen BH (3 ) are given in the following exquisite forms \[\boldsymbol{A} = \frac{q(r-p^2/m)}{\varrho^2}X-\frac{p\cos\theta}{\varrho^2}Y,\label{eq:AdKS}\;\tag{5}\] \[\boldsymbol{B} = \frac{p(r-p^2/m)}{\varrho^2}X+\frac{q\cos\theta}{\varrho^2}Y, \label{eq:BdKS}\;\tag{6}\] \[e^{\phi} = \frac{r^2+(k+a\cos\theta)^2}{\varrho^2}, \label{eq:dilatondKS}\tag{7}\] \[\chi = 2\frac{kr-d(k+a\cos\theta)}{r^2+(k+a\cos\theta)^2}. \label{eq:axiondKS}\tag{8}\] For non-dyonic solution, one can use results in [43] and turn off \(p\). The original derivation for Kerr-Sen BH is given in [22].

There are two horizons for dyonic Kerr-Sen BH, namely event horizon (\(r_+\)) and Cauchy horizon (\(r_-\)). Their radii can be determined by solving \(\Delta=0\) as the horizons are given by \[r_\pm = \left(m+d\right)\pm \sqrt{m^2+d^2+k^2-p^2-q^2-a^2}.\label{eq:rpm}\tag{9}\] From Eq. (9 ), the requirement for the BH to have horizons is when \[a^2+p^2+q^2 \leq m^2+d^2+k^2.\label{eq:nonakedsing}\tag{10}\] Hence, the extremal limit of the dyonic Kerr-Sen BH corresponds to \[a^2+p^2+q^2 = m^2+d^2+k^2, ~~~ r_\pm=m+d.\label{eq:extremalcon}\;\tag{11}\]

2.2 Thermodynamics↩︎

The thermodynamic quantities and properties on \(r_+\) of this BH are derived in [24]. For the thermodynamic quantities and properties on \(r_-\), one can replace \(r_+ \rightarrow r_-\). Thus, we can obtain the thermodynamic quantities on both horizons. The physical mass, electric charge, magnetic charge, and angular momentum are given by \[M=m,~ Q= q, ~ P = p, ~J=ma.\label{eq:MQPJ}\tag{12}\] The area of this BH is explicitly \[A_\pm = 4\pi(r_\pm^2 - 2dr_\pm -k^2 +a^2).\] Thus the entropy is given by \[S_\pm = \pi(r_\pm^2 - 2dr_\pm -k^2 +a^2).\label{eq:entropy}\tag{13}\]

The angular velocity on the horizon is \[\Omega_\pm = \frac{a}{r_\pm^2 - 2dr_\pm -k^2 +a^2},\label{eq:angvelocity}\tag{14}\] and the surface gravity is given by \[\kappa_\pm = \frac{r_\pm-(m+d)}{r_\pm^2 - 2dr_\pm -k^2 +a^2}.\label{eq:surface}\tag{15}\] The temperature can be expressed in terms of surface gravity which reads as \[T_\pm = \frac{\kappa_\pm}{2\pi}.\label{eq:temperature}\tag{16}\] It is worth noting that \(T_+>T_-\) and \(T_-<0\).

The first law of thermodynamics of dyonic Kerr-Sen BH is \[dM = T_\pm S_\pm + \Omega_\pm dJ + \Phi_\pm dQ +\Psi_\pm dP.\label{eq:firstlaw}\tag{17}\] The first law can also be rewritten in terms of black hole’s area, the surface gravity and the potentials, \[dM = \frac{\kappa_\pm}{8\pi} dA_\pm + \Omega_\pm dJ + \Phi_\pm dQ +\Psi_\pm dP. \label{eq:firstlaw1}\tag{18}\] It is easy to derive the Smarr formula that is given by \[M =2 T_\pm S_\pm +2\Omega_\pm J + \Phi_\pm Q +\Psi_\pm P,\label{eq:Smarr}\tag{19}\] and the Christodoulou-Ruffini square mass formula explicitly reads as \[M^2 = \frac{A_\pm}{16\pi}+\frac{4\pi J^2}{A_\pm}+\frac{P^2+Q^2}{2}.\label{eq:Christodouloumass}\tag{20}\] One can derive the following expressions for the thermodynamic quantities, \[\frac{\kappa_\pm}{8\pi} = \frac{\partial M}{\partial A_\pm} =\frac{1}{M}\left(\frac{1}{32\pi} -\frac{2\pi J^2}{A_\pm^2}\right),\] \[\Omega_\pm = \frac{\partial M}{\partial J} = \frac{4\pi J}{M A_\pm},\] \[\Phi_\pm = \frac{\partial M}{\partial Q} = \frac{Q}{2M},\label{eq:ElPot}\tag{21}\] \[\Psi_\pm = \frac{\partial M}{\partial P} = \frac{P}{2M}.\label{eq:MagPot}\tag{22}\] Notably, the electric and magnetic potentials are the same for both inner and outer horizons.

3 Entropy Quantization↩︎

It has been observed that the product of horizon entropies for many BH solutions is mass independent as given in Eq. (1 ) where \(N\) depends only on the quantized charges like angular momentum and electric charge. This universality encodes the quantum nature of the entropy. Motivated by this universality, we will derive the similar relation for dyonic Kerr-Sen BH which will benefit us to study the dual CFT in the next section.

Firstly, we start by computing the product and sum of the entropies which are given by \[S_-S_+ = (2\pi J)^2. \label{eq:entropyproductdKS}\tag{23}\] and \[S_-+S_+ = 2\pi(2M^2-Q^2-P^2). \label{eq:entropysumdKS}\tag{24}\] As we can see that the entropy product is quantized which depends on the angular momentum only. It is somehow different from the dyonic Kerr-Newman BH where the entropy product also depends on the charges. The universal entropy product implies that we can derive the CFT thermodynamics from this relation.

The other important relation is Eq. (2 ). The relation can be straightforwardly derived from entropy (13 ) and temperature (16 ). Moreover, one can also show that \[\frac{\Omega_+}{T_+}+\frac{\Omega_-}{T_-}=0,\label{eq:OmegaTuniversality}\tag{25}\] which is also universal.

Now we want to derive the entropy bound of for both horizons which is exactly Penrose-like inequality for event horizon. From Eq. (10 ), we obtain \[M^2 \geq J+\frac{P^2+Q^2}{2}.\] Since \(r_+\geq r_-\), therefore \(S_+\geq S_-\geq 0\) and from the entropy product (1 ), we obtain \[S_+ \geq \sqrt{S_+S_-}=2\pi J \geq S_-.\] We know that \[S_+ + S_- \geq S_+ \geq \frac{S_++S_-}{2}\geq S_-,\] thus we have \[2\pi(2M^2-P^2-Q^2)\geq S_+ \geq \pi(2M^2-P^2-Q^2)\geq S_-.\] Then we can find the entropy bound as follows \[\pi(2M^2-P^2-Q^2) \leq S_+ \leq 2\pi(2M^2-P^2-Q^2).\label{eq:entropybound43}\tag{26}\] and \[0 \leq S_- \leq 2\pi J.\label{eq:entropybound-}\tag{27}\] Note that in the limit \(Q =P= 0\), we obtain the Kerr entropy bound.

The other fruitful relations beside the entropy product and sum are the temperature product and sum. It is straightforward that we can obtain \[T_-T_+ = -\frac{(2M^2-P^2 -Q^2)^2-4J^2}{(8\pi JM)^2},\] and \[T_-+T_+ = -\frac{(2M^2-P^2-Q^2)^2-4J^2}{8\pi MJ^2}.\] Both temperature product and sum are not universal.

4 CFT Thermodynamics↩︎

In this section, we will derive the CFT thermodynamics from BH thermodynamics on \(r_\pm\) from the fact that dyonic Kerr-Sen BH satisfies the universality of the entropy product. Thus we want to prove that through thermodynamic method, we can construct the dual CFT where we can derive exactly the same central charges of the left- and right-moving sectors.

In 2D CFT, the first law and energy formula for both left- and right-moving sectors are given by [17], [18] \[dE_{L,R} = T_{L,R} dS_{L,R} + \Omega_{L,R} dJ + \Phi_{L,R} dQ+\Psi_{L,R} dP,\] \[E_{L,R}= 2T_{L,R} S_{L,R} + 2\Omega_{L,R} J + \Phi_{L,R} Q+\Psi_{L,R} P,\] respectively. The relations between the BH thermodynamics and 2D CFT thermodynamics read as [17], [18] \[E_{L,R}= \frac{M}{2}, ~~~\frac{2\Omega_{L,R}}{T_{L,R}}=\frac{\Omega_+}{T_+}\pm\frac{\Omega_-}{T_-},\label{eq:massOmrel}\tag{28}\] \[\frac{1}{T_{L,R}}= \frac{1}{T_+}\pm \frac{1}{T_-}, ~~~ S_{L,R}=\frac{S_+ \pm S_-}{2},\label{eq:TSrel}\tag{29}\] \[\frac{2\Phi_{L,R}}{T_{L,R}}=\frac{\Phi_+}{T_+}\pm\frac{\Phi_-}{T_-},~~\frac{2\Psi_{L,R}}{T_{L,R}}=\frac{\Psi_+}{T_+}\pm\frac{\Psi_-}{T_-}.\label{eq:PhiPsirel}\tag{30}\]

Using the above relations, we can find \[T_L = \frac{1}{8\pi M},~~ T_R = \frac{\sqrt{\left(M^2-\frac{(P^2+Q^2)}{2}\right)^2-J^2}}{4\pi M(2M^2-P^2-Q^2)}\label{eq:TLR}\tag{31}\] \[S_L = \pi(2M^2-P^2-Q^2),\label{eq:SL}\tag{32}\] \[S_R = 2\pi \sqrt{\left(M^2-\frac{(P^2+Q^2)}{2}\right)^2-J^2},\label{eq:SR}\tag{33}\] \[\Omega_L = 0, ~~~ \Omega_R = \frac{J}{2M(2M^2-P^2-Q^2)}, \label{eq:OmegaLR}\tag{34}\] \[\Phi_{L,R}= \frac{Q}{4M}, ~~ \Psi_{L,R}= \frac{P}{4M}.\label{eq:PhiPsiLR}\tag{35}\]

The left and right temperatures from (31 ) are of dimension \(\mathcal{L}^{-1}\) and proportional to the microscopic temperatures obtained from the hidden conformal symmetry in the low-frequency scattering, up to a scale factor. The scale factor can be thought of as the size of the box in which the microscopic CFT lives. It can be determined from the following relation [17]–[19] \[R=\frac{T_-^2 -T_+^2}{T_- T_+ (\Omega_+ -\Omega_-)}.\label{eq:scalefactor}\tag{36}\] Thus for dyonic Kerr-Sen BH, we have \[R= \frac{2M(2M^2-P^2-Q^2)}{J}.\label{eq:scalefactor1}\tag{37}\] Applying the scale factor to temperatures (31 ) as \(\bar{T}_{L,R}=R T_{L,R}\), the dimensionless left- and right-moving CFT temperatures are then given by \[\bar{T}_L = \frac{2M^2-P^2-Q^2}{4\pi J}, ~~~\bar{T}_R= \frac{\sqrt{\left(M^2-\frac{(P^2+Q^2)}{2}\right)^2-J^2}}{2\pi J}. \label{eq:dimTLR}\;\tag{38}\]

The central charges in left- and right-moving sectors of the 2D CFT can be computed via the Cardy formula as \[S_{L,R} = \frac{\pi^2}{3}c_{L,R}\bar{T}_{L,R}.\] Therefore the central charges of dual CFT are given as follows \[c_L = c_R = 12J.\] which is exactly the same as Kerr-Sen, Kerr-Newman, and Kerr BHs. This results show that dyonic Kerr-Sen BH is dual to a \(c_L=c_R=12J\) of 2D CFT at temperature \((T_L,T_R)\).

In the extremal limit where both horizons coincide \(r_+=r_-\) which is equivalent to \(M^2 -\frac{P^2+Q^2}{2}= J\), the 2D CFT thermodynamic quantities reduce to \[\bar{T}_L =\frac{1}{2\pi},\qquad \bar{T}_R=0,\] \[S_L = 2\pi J,\qquad S_R=0,\] \[\Omega_L=0,\qquad \Omega_R = \frac{1}{4M},\] while \(\Phi_{L,R}, \Psi_{L,R}, T_L\) have similar forms as the non-extremal condition. Therefore, we can obtain the microscopic entropy via Cardy formula in the dual CFT \[S_{CFT} = \frac{\pi^2}{3}c_L T_L = 2\pi J.\] which is in perfect agreement with macroscopic Bekenstein-Hawking entropy of the extremal dyonic Kerr-Sen BH [25].

4.1 Comparison to Kerr-Newman BH↩︎

Let us compare our dyonic Kerr-Sen BH results with Kerr-Newman BH thermodynamics given in Ref. [17]. The entropy products of both dyonic Kerr-Sen BH and Kerr-Newman BH are independent of mass. However, the Kerr-Newman entropy product \[S_{+}S_{-} = \pi^{2}(4J^{2}+Q^{4}),\] also depends on the charge \(Q\) while the entropy product of dyonic Kerr-Sen BH (23 ) is independent of both charges \(P, Q\) and depends only on the angular momentum \(J\). There is a fundamental difference between the dyonic Kerr-Sen BH and Kerr-Newman BH, turning off \(P\) charge will not reduce dyonic Kerr-Sen BH to the Kerr-Newman spacetime.

The sums of entropy, on the other hand, have very similar formula. For Kerr-Newman BH, \(S_{+}+S_{-} = 2\pi (2M^{2}-Q^{2})\), while the dyonic Kerr-Sen BH entropy sum (24 ) only has extra contribution from the magnetic charge \(P\) with the same dependence as the electric charge \(Q\).

For CFT thermodynamics of Kerr-Newman BH, by using (29 ), we obtain \[\begin{align} S_{L}=&&\pi(2M^{2}-Q^{2}),\quad S_{R}=2\pi M\sqrt{M^{2}-\left(\frac{J^{2}}{M^{2}}+Q^{2}\right)},\notag\\ T_{L}=&&\frac{1}{8\pi M},\quad T_{R}=\frac{\sqrt{M^2 -\left(\frac{J^{2}}{M^{2}}+Q^{2}\right)}}{4\pi(2 M^2 - Q^2)}. \end{align}\] Using (36 ), the scale factor is \[R=\frac{2 M (2 M^2 -Q^{2})}{J}.\] This leads to the dimensionless left- and right-moving CFT temperatures \[\bar{T}_{L}=\frac{2 M^2-Q^{2}}{4 \pi J},\quad \bar{T}_{R}=\frac{M \sqrt{M^2 -\left(\frac{J^{2}}{M^{2}}+Q^{2}\right)}}{2 \pi J}.\] And the central charges are determined to be \(c_{L,R}=12J\) [17]. Notably, even though \(\bar{T}_{L}, S_{L}\) of dyonic Kerr-Sen BH reduce to Kerr-Newman BH when \(P=0\), the right-moving sector \(\bar{T}_{R}, S_{R}\) of dyonic Kerr-Sen BH do NOT reduce to Kerr-Newman results. The fundamental difference of CFT thermodynamics between dyonic Kerr-Sen BH and Kerr-Newman BH is essentially in the right-moving sector.

5 Static Case: dyonic dilaton BH↩︎

In this section, we consider the static case for dyonic Kerr-Sen BH that we call as dyonic dilaton BH. One can easily obtain this solution by setting \(a=0\). The spacetime metric of this BH is given by \[ds^2 = - \frac{\Delta}{\varrho^2 } dt^2+ \frac{\varrho ^2}{\Delta}dr^2 + \varrho ^2d\theta^2 + \varrho ^2 \sin^2\theta d\phi^2,\label{eq:ddmetric} \;\tag{39}\] where \[\Delta = r^2-2dr-2m(r-d) -k^2+p^2+q^2,\] \[\varrho^2 = r^2-2dr -k^2.\label{eq:metricfunctiondd}\;\tag{40}\] When the electromagnetic charges vanish, we obtain Schwarzschild BH solution. Remarkably, even for a generic charged dyonic BH, we cannot have an extremal solution since \(\varrho^2\) has a zero at \(r_c = M+d = P^2/M\) for \(M^2 =\frac{P^2+Q^2}{2}\), signifying a curvature singularity. This is a spherical singularity (see [44]) locating at exactly the same position as the extremal horizon, i.e., the root of \(\Delta=0\), and thus there is no BH.

The entropy is then given by \[S_\pm = \pi(r_\pm^2 - 2dr_\pm -k^2),\label{eq:entropydd}\tag{41}\] and the temperature is \[T_\pm = \frac{r_\pm-(m+d)}{2\pi(r_\pm^2 - 2dr_\pm -k^2)}.\label{eq:temperaturedd}\tag{42}\] The electric and magnetic potentials are still in the similar form as (21 )-(22 ). The first law of thermodynamics of dyonic dilaton BH is given by \[dM = T_\pm S_\pm + \Phi_\pm dQ +\Psi_\pm dP,\label{eq:firstlawdd}\tag{43}\] and the Smarr formula now is \[M =2 T_\pm S_\pm + \Phi_\pm Q +\Psi_\pm P,\label{eq:Smarrdd}\tag{44}\] Moreover, the Christodoulou-Ruffini square mass formula in this case explicitly reads \[M^2 = \frac{A_\pm}{16\pi}+\frac{P^2+Q^2}{2}.\label{eq:Christodouloumassdd}\tag{45}\]

For this BH, it is clear that the entropy product is zero while the temperature product is singular. This is different from the Reissner-Nordström BH where the entropy product still depends on the electric charge and the temperature product is not singular [19]. The singularity of the temperature product corresponds to the singular dimensionless left- and right-moving CFT temperatures since the scale factor \(R\) is singular. However, we can observe that the left- and right-moving CFT temperatures and entropies coincide to each other and non-vanishing as, \[T_{L,R}= \frac{1}{8\pi M},~~~ S_{L,R}=\pi(2M^2-P^2-Q^2),\label{eq:TLRSLRdd}\tag{46}\] Assuming Cardy entropy formula must be valid using above \(T_{L,R}\) for this BH, one can derive the central charge \[c_{L}=c_R=24M(2M^2-P^2-Q^2).\label{eq:cLRdd}\tag{47}\] Interestingly, the results for dyonic dilaton BH match to that of Schwarzschild BH when the electromagnetic charges vanish. It has been computed in [45], [46] that, even though Schwarzschild BH has only a single horizon, it can be represented by two-sectors in CFT by using soft-hair method. Yet, another computation using hidden conformal symmetry [47] shows that the Schwarzschild BH can be represented by a single CFT sector with the central charge \(c=96M^3\). For this case, the central charge can be interpreted as the total central charge from left and right sectors.

6 Hidden conformal symmetry and quasi-normal modes↩︎

6.1 Hidden conformal symmetry↩︎

Before studying the QNM spectrum on the dyonic Kerr-Sen BH background, we revisit the hidden conformal symmetry calculation on dyonic Kerr-Sen BH. It is worth noting that we use the spacetime metric (3 ) instead of the metric in shifted radial coordinate as it has been computed in [26], [28]. Now we consider massless neutral quantum scalar field where the solution has generic form \[\chi (t,r,\theta,\phi)=e^{-i\omega t+in\phi}P_l^n(\theta)R(r),\] where \(\omega\) is the asymptotic energy and \(n\) is the angular momentum of the scalar field. Plugging in above ansatz to the following Klein-Gordon equation, \[\nabla_{\alpha} \nabla^{\alpha}\chi = 0\label{eq:KG},\tag{48}\] one can find radial equation in low-frequency limit (\(\omega M\ll 1\)) that reads as \[\left[ \partial_r \left(\Delta \partial_r\right) + \frac{r_+ - r_-}{r - r_+} A + \frac{r_+ - r_-}{r - r_-} B -l(l+1) \right] R(r) = 0 , \label{eq:radeq} \;\tag{49}\] where \[\begin{align} &&A=\frac{\left[(2m r_+-2p^2)\omega - an \right]^2}{(r_+ -r_-)^2},\nonumber\\ &&B=-\frac{\left[(2m r_- -2p^2)\omega - an \right]^2}{(r_+ -r_-)^2}. \label{eq:AB}\; \end{align}\tag{50}\] where the separation constant \(l(l+1)\) is the eigenvalues on a sphere. The angular equation in low-frequency limit is just a Laplacian on the two-sphere [35]. The main reason we do the computation of the hidden conformal symmetry is to show that the left- and right-moving temperatures are explicitly the same with the results from the thermodynamic method as given in Eq. (38 ). Thus to reveal the hidden conformal symmetry, we perform the following coordinate transformations \[\begin{align} &&\omega^+ = \sqrt{\frac{r-r_+}{r-r_-}}\mathrm{e}^{2\pi \bar{T}_R \phi + 2 \bar{n}_R t}, \tag{51}\\ &&\omega^- = \sqrt{\frac{r-r_+}{r-r_-}}\mathrm{e}^{2\pi \bar{T}_L \phi + 2 \bar{n}_L t}, \tag{52}\\ &&y = \sqrt{\frac{r_+ -r_-}{r-r_-}}\mathrm{e}^{\pi (\bar{T}_L +\bar{T}_R) \phi + (\bar{n}_L + \bar{n}_R) t}, \tag{53}\; \end{align}\] where \(\bar{n}_{L,R}\) is the conserved charge in CFT. From those conformal coordinates, we can define three locally conformal operators in terms of the new conformal coordinates \(\omega^+,\,\omega^-\) and \(y\) as \[\begin{align} && H_1 = i \partial_+, \tag{54}\\ && H_0 = i \left(\omega^{+}\partial_+ + \frac{1}{2}y\partial_y \right), \tag{55}\\ && H_{-1} = i \left(\omega^{+2}\partial_+ + \omega^{+}y\partial_y - y^2 \partial_- \right), \tag{56} \end{align}\] as well as \[\begin{align} && \bar{H}_1 = i \partial_-, \tag{57}\\ && \bar{H}_0 = i\left(\omega^{-}\partial_- + \frac{1}{2}y\partial_y \right), \tag{58}\\ && \bar{H}_{-1} = i \left(\omega^{-2}\partial_- + \omega^{-}y\partial_y - y^2 \partial_+ \right). \tag{59} \end{align}\] Note that we have defined \(\partial_\pm = \frac{\partial}{\partial \omega^\pm}\). Each set of operators given in Eqs. (54 )-(56 ) and (57 )-(59 ) satisfies the \(SL(2,R)\) Lie algebra, \[\begin{align} \left[H_0,H_{\pm 1} \right] = \mp iH_{\pm 1}, ~~~ \left[H_{-1},H_1 \right]=-2iH_0. \end{align}\] Hence the quadratic Casimir operator can be formed from any of two sets of operators as \[\begin{align} \mathcal{H}^2 &=& \bar{\mathcal{H}}^2 = - H_0^2 + \frac{1}{2}(H_1 H_{-1} + H_{-1} H_{1})\nonumber\\ &=& \frac{1}{4}(y^2 \partial_y^2 - y\partial_y)+y^2\partial_+ \partial_-. \label{eq:quadraticCasimir}\; \end{align}\tag{60}\] Now the radial wave equation (49 ) can be represented as the Casimir operator (60 ) that represents \(SL(2,R)\times SL(2,R)\) isometry. Simply, we can write \[\begin{align} \mathcal{H}^2 R(r)=\bar{\mathcal{H}}^2 R(r)= l(l+1) R(r), \label{quadraticCasimirEq} \end{align}\tag{61}\] that is satisfied when the dimensionless left- and right-moving CFT temperatures are given by Eq. (38 ).

6.2 QNM Spectra↩︎

The radial wave equation (49 ) possesses two independent solutions with ingoing and outgoing boundary conditions. However, the relevant solution that we need to use to find the QNM spectra is the one with ingoing boundary condition. The ingoing solution is given by \[R(r)=c_1 \left(\frac{r-r_+}{r-r-}\right)^{c_2}(r-r_-)^{-l-1} {_2F_1}\left(c_3,c_4;c_5;z\right),\label{eq:radialsol}\tag{62}\] where \(_2F_1\) is a hypergeometric function, \(c_1\) is just an integration constant, and \[c_2 = -i\frac{\omega -n\Omega_+}{2\pi T_+}, ~~c_3=1+l-\frac{i\omega}{4\pi T_R}+\frac{in\Omega_+}{2\pi T_+},\nonumber\;\] \[c_4 =1+l -\frac{i\omega}{4\pi T_L}, ~~c_5=1-\frac{i\omega}{2\pi T_+}+\frac{in\Omega_+}{2\pi T_+},\nonumber\;\] \[z= \frac{r-r_+}{r-r_-}.\]

The ingoing solution at asymptotic infinity (\(r\gg M\)) but still satisfying \(r\omega\ll 1\) behaves as \(R\sim D r^l\) where \[D= \frac{\Gamma(1+2l)\Gamma\left(1-i\frac{\omega -n\Omega_+}{2\pi T_+}\right)}{\Gamma\left(1+l -\frac{i\omega}{4\pi T_L}\right)\Gamma\left(1+l-\frac{i\omega}{4\pi T_R}+\frac{in\Omega_+}{2\pi T_+}\right)}.\label{eq:D}\tag{63}\] The above constant encodes the information of the absorption cross-section which is given by [26], [28] \[\begin{align} P_{abs} &\sim& |D|^{-2} =\sinh\left(\frac{\omega -n\Omega_+}{2T_+}\right)\bigg|\Gamma\left(1+l -\frac{i\omega}{4\pi T_L}\right)\bigg|^2\nonumber\\ &\times& \bigg|\Gamma\left(1+l-\frac{i\omega}{4\pi T_R}+\frac{in\Omega_+}{2\pi T_+}\right)\bigg|^2.\label{eq:Pabs}\; \end{align}\tag{64}\] The QNM spectra can be determined from the poles of the absorption cross-section. The poles of gamma functions are given when the value in the bracket is zero or negative integers. Thus from the absorption cross-section (64 ), we can find two different QNM spectra as follows \[\begin{align} \omega_L &=& -4\pi i T_L (1+l +n_L), \tag{65}\\ \omega_R &=& -4\pi i T_R (1+l+n_R) + \frac{2n\Omega_+ T_R}{T_+}, \tag{66}\; \end{align}\] where \(n_{L,R}=0,1,2,...\) In fact, the family of modes given by the second family of modes (66 ) has identical frequencies to those of [48] where the QNM spectra take the form \[\omega = -2\pi i T_R (1+l+n) + 2n\Omega_+.\label{eq:QNMresonance}\tag{67}\] which is valid in the regime of Im\((\omega)\ll\) Re\((\omega) \ll 1/M\). It has been noted in [49], [50] that the purely imaginary QNM spectrum (65 ) is related to the natural frequency of oscillation and represents the modes of excitation of the black hole and dual to the spectrum of excitations of CFT.

7 Summary↩︎

In this work, we have investigated the inner and outer horizon thermodynamics of the dyonic Kerr-Sen black hole in four-dimensional Einstein-Maxwell-Dilaton-Axion theory, which carries mass \(M\), angular momentum \(J\), electric charge \(Q\), and magnetic charge \(P\). We have carried out the thermodynamic quantities such as entropy, temperature, angular velocity, electric potential, and magnetic potential on both the event horizon and the Cauchy horizon. We have explicitly verified the first law and Smarr formula on each horizon.

We have computed the entropy product and sum for the dyonic Kerr-Sen black hole and demonstrated that the entropy product \(S_-S_+ = (2\pi J)^2\) is mass-independent and depends solely on the angular momentum, establishing the universality of the entropy (or area) product. This universality is equivalent to the condition \(T_+ S_+ + T_- S_-=0\), which we also verified. From the entropy sum and product, we derived Penrose-like entropy bounds for both horizons.

Exploiting the universal entropy product, we constructed the dual two-dimensional CFT thermodynamics through the Kerr/CFT correspondence via the thermodynamic method. The central charges of the left- and right-moving sectors were found to be \(c_L=c_R=12J\), identical to those of the Kerr, Kerr-Newman, and Kerr-Sen black holes. In the extremal limit, the microscopic Cardy entropy \(S_{CFT}=2\pi J\) is in perfect agreement with the macroscopic Bekenstein-Hawking entropy. Comparing to the results from Kerr-Newman BH, when we turned off the magnetic charge of dyonic Kerr-Sen BH, the left-moving sector has similar results while the right-moving sector is fundamentally different.

We also carried out the static solution from the dyonic Kerr-Sen BH and its thermodynamics. It has been shown that the entropy product vanishes thus one could not reproduce the regular dimensionless CFT temperatures and non-vanishing central charges from that of the rotating case. However, from the thermodynamic method, we have found that the left- and right-moving CFT temperatures are regular and by using Cardy entropy formula, it has been found that \(c_{L,R}=24M(2M^2-P^2-Q^2)\) which reflects the non-vanishing left- and right-sectors in 2D CFT.

Finally, we revisited the hidden conformal \(SL(2,R)\times SL(2,R)\) symmetry of the radial scalar wave equation in the low-frequency limit, confirming that the CFT temperatures derived thermodynamically are consistent with those from the hidden conformal symmetry analysis. From the poles of the greybody absorption cross-section, we obtained two families of QNM spectra providing a complete analytic expression for the QNM spectra of the dyonic Kerr-Sen black hole in terms of its physical parameters.

References↩︎