We give a microscopic derivation of the semiclassical entropy of static black holes in 3d Lovelock gravities which are certain 3d Horndeski theories that were recently discovered from higherdimensional Lovelock gravities via various methods and admit novel black hole solutions analogous to higherdimensional ones. Assuming the ground state is described by the soliton obtained from the black hole solution by performing a double Wick rotation we reproduce the semiclassical entropy from Cardy formula without central charges for the asymptotic growth of the number of states. We explain in detail how the mass of the soliton which is needed in the Cardy formula can be computed in the minisuperspace formalism.
['High Energy Physics - Theory', 'General Relativity and Quantum Cosmology']
Microscopic entropy of static black holes in 3d Lovelock gravities
{'Gökhan Alkaç, a Luis Guajardo b and Hikmet Özşahin c': '- a Department of Aerospace Engineering, Faculty of Engineering,  \nAtılım University, 06836 Ankara, Türkiye  \nInstituto de Investigación Interdisciplinario, Vicerrectoría Académica, Universidad de Talca, 3465548 Talca, Chile  \n- c Department of Physics, Faculty of Arts and Sciences,  \nSinop University, 57000 Sinop, Türkiye  \nE-mail: alkac@mail.com , luis.guajardo.r@gmail.com , h.ozsahin@sinop.edu.tr', 'Abstract': 'We give a microscopic derivation of the semi-classical entropy of static black holes in 3d Lovelock gravities, which are certain 3d Horndeski theories that were recently discovered from higher-dimensional Lovelock gravities via various methods and admit novel black hole solutions analogous to higherdimensional ones. Assuming the ground state is described by the soliton obtained from the black hole solution by performing a double Wick rotation, we reproduce the semi-classical entropy from Cardy formula without central charges for the asymptotic growth of the number of states. We explain in detail how the mass of the soliton, which is needed in the Cardy formula, can be computed in the mini-superspace formalism.', 'Contents': '- b', '1 Introduction': "One of the most important challanges for a quantum theory of gravity is to provide a microscopic derivation of black hole entropy. Among succesful examples, Strominger's derivation in [1] is arguably the simplest since it does not require string theory or supersymmetry. It is based on the observation of Brown and Hanneaux [2] that the global charges corresponding to asymptotic symmetries of 3d general relativity (GR) with a negative cosmological constant yield a Poisson bracket algebra with a central extension, which turn out to be two copies of the Virasoro algebra with the central charges  \nc ± = 3 /lscript 2 G , (1.1)  \nwhere /lscript is the AdS radius and G is Newton's constant. Since this is the algebra satisfied by the generators of conformal transformations on the Hilbert space of 2d conformal field theories (CFTs), a consistent quantum theory of gravity on AdS 3 can be assumed to be a 2d CFT. The entropy of a 2d CFT can be obtained by means of Cardy formula for the asymptotic growth of the number of states, whose standard form reads [3]  \nS Cardy = 2 π √ c + 6 ˜ ∆ + +2 π √ c -6 ˜ ∆ -, (1.2)  \n˜ L ± 0 = L ± 0 -c ± 24 . (1.3)  \nwhere ˜ ∆ ± are the eigenvalues of the shifted Virasoro operators  \nBy assuming the additive constants in ˜ ∆ ± vanish for the zero mass and angular momentum black hole ( M = J = 0) such that  \nStrominger showed that one can reproduce the semi-classical entropy of the Bañados-TeitelboimZanelli (BTZ) black hole [4,5] from Cardy formula.  \n˜ ∆ ± = 1 2 ( M/lscript ± J ) , (1.4)  \nLet us see how this works for the static BTZ black hole whose line element is given by  \nd s 2 b = -r 2 -r 2 + /lscript 2 d t 2 b + /lscript 2 r 2 -r 2 + d r 2 + r 2 d θ 2 b , (1.5)  \nwhere r + is the event horizon radius. This is a useful exercise since we will be interested in static black hole solutions in this paper. By Euclidean methods of [6], it is easy to get the inverse temperature β , the mass M and the entropy S of the black hole as follows  \nβ = 2 π/lscript 2 r + , M = r 2 + 8 G/lscript 2 , S = A 4 G = πr + 2 G , (1.6)  \nsatisfying the first law δS = βδM . Using the value of the central charge given in (1.1) and ˜ ∆ ± = 1 2 M/lscript (since the angular momentum vanishes, i.e, J = 0), one sees that Cardy formula (1.2) precisely reproduces the semi-classical entropy given in (1.6), i.e., S Cardy = S .  \nLater, an intriguing puzzle emerged in [7] from the study of a static hairy black hole solution that is obtained from minimal coupling of a scalar field to AdS 3 gravity with a particular potential. Although the generators of the asymptotic symmetries acquire a nontrivial contribution from the  \nscalar coupling, the Poisson bracket algebra of the global charges remains the same as that of GR with the central charges in (1.1) [8]. However, when the entropy of the hairy black hole solution is expressed as a function of its mass, it differs from that of the BTZ black hole. Therefore, it seems impossible to reproduce the semi-classical entropy from the Cardy formula (1.2).  \nAs shown in [7], this issue can be handled by a careful consideration of the Cardy formula from scratch (see [9] for a pedagogical derivation). When the lowest eigenvalues ∆ ± 0 of the Virasoro operators L ± 0 are non-vanishing (∆ ± 0 = 0), it is given by  \nS Cardy = 2 π √ c + eff 6 ˜ ∆ + +2 π √ c -eff 6 ˜ ∆ -, (1.7)  \n/negationslash  \nwhere c ± eff are the effective central charges given by  \nc ± eff = c ± -24 ˜ ∆ ± 0 . (1.8)  \nThis allows us to rewrite the Cardy formula without any reference to central charges c ± as  \nS Cardy = 4 π √ -˜ ∆ + 0 ˜ ∆ + +4 π √ -˜ ∆ -0 ˜ ∆ -, (1.9)  \nwhere ˜ ∆ ± 0 are the lowest eigenvalues of the shifted Virasoro operators ˜ L ± 0 that we defined in (1.3). Once we assume that the BTZ black hole and the hairy black hole belong to disconnected sectors with distinct ground states, the Cardy formula in (1.9) might reproduce the semi-classical entropy. In [7], this was achieved by assuming the ground state of each sector is described by the soliton solution obtained from the corresponding static black hole solution by performing a double Wick rotation. The validity of this approach was succesfully tested for static black hole solutions arising from different self-interactions of the minimally-coupled scalar field in [10] and for a rotating hairy black hole in [11].  \nThere exist various generalizations of the Cardy formula for warped AdS 3 black holes, Lifshitz black holes and hyper-scaling violation black holes (both in 3d and higher dimensions) where it is also possible to incorporate rotation and electric charge. The reader is referred to [12-24] for details. Later, it was realized that the semi-classical entropy of the Schwarzschild black hole (with a spherical horizon) in d -dimensions might be reproduced by a proper modification of it [25].  \nFor 3d static black holes, which we will be interested in here, one has ˜ ∆ ± 0 = 1 2 M 0 /lscript , where M 0 is the mass of the soliton, and the Cardy formula (1.9) simplifies into  \nS Cardy = 4 π/lscript √ -M 0 M. (1.10)  \nIf the mass M and the semi-classical entropy S of the black hole are known, it suffices to determine the mass of the corresponding soliton solution M 0 to check whether the Cardy formula (1.10) yields a microscopic derivation of the entropy.  \nThe soliton configuration corresponding to the static BTZ black hole with the line element (1.5) obtained by double Wick rotation  \nt b → i/lscriptθ s , θ b → i t s /lscript , (1.11)  \nis just the AdS 3 soliton [26] whose line element reads  \nd s 2 s = -r 2 /lscript 2 d t 2 s + /lscript 2 r 2 -/lscript 2 d r 2 +( r 2 -/lscript 2 ) d θ 2 s . (1.12)  \nBy a redefinition of the radial coordinate r 2 -/lscript 2 → r 2 , it becomes the line element of the AdS 3 spacetime in global coordinates  \nd s 2 = -[ 1 + r 2 /lscript 2 ] d t 2 + [ 1 + r 2 /lscript 2 ] -1 d r 2 + r 2 d θ 2 , (1.13)  \nwhich shows that the ground state for the BTZ black hole is the global AdS 3 spacetime. The mass of the AdS 3 soliton in (1.12) can be easily computed to be 1  \nM 0 = -1 8 G , (1.14)  \nwhich reproduces the semi-classical entropy given in (1.6) when used in Cardy formula (1.10).  \nIn order to understand why Strominger's derivation works, one can check the Cardy formula with the central charges (1.2) for static black holes  \nS Cardy = 2 π √ c ± 3 M/lscript. (1.15)  \nSince the mass of the AdS 3 soliton (1.14) can be expressed in terms of the central charges of GR (1.1) as  \nM 0 = -c ± 12 /lscript , (1.16)  \ntwo versions of Cardy formula (1.15, 1.10) become equivalent 2 . However, the soliton mass M 0 and the central charges c ± of a generic theory are generally not related in the same manner. This shows why Strominger's derivation works for the BTZ black hole in GR, and also justifies the need to use the Cardy formula without central charges (1.10) in more complicated cases.  \nIn this paper, we aim to provide further non-trivial evidence for the role played by the soliton solution in the microscopic derivation of the entropy of hairy black holes. For this purpose, we will focus on certain examples of 3d Horndeski theories, which are scalar-tensor theories with 2nd-order field equations [27,28]. They were recently obtained from 2nd- and 3rd-order Lovelock Lagrangians, which normally give non-trivial contributions to field equations only in d > 4 and d > 6 respectively, via various methods (see [29-33]). Lovelock gravities are the most natural generalizations of GR that are pure gravity theories with 2nd-order field equations [34,35] (see [36] for a review) admitting exact static black hole solutions [37, 38]. After a regularized Kaluza-Klein reduction or a technique called the 'Weyl trick', one obtains Horndeski theories admitting novel hairy black hole solutions in d = 3. We will call them 3d Lovelock gravities because of their higher-dimensional origin.  \nIn previous works, either the Regge-Teitelboim method [39] (by using the full form of the Hamiltonian of the theory under consideration) or the quasi-local generalization of the Abbot-Deser-Tekin method [40-42] given in [43,44] have been used to find the soliton mass. However, both become quite cumbersome for 3d Lovelock gravities 3 . By considering a mini-superspace of static field configurations  \n/negationslash  \nin the Regge-Teitelboim method, which can also be used to study the thermodynamics of the black hole solutions, we will be able to obtain the soliton mass and verify that it has zero entropy by an appropriate choice of mini-superspace variables. We would like to emphasize that this is the first time the whole analysis is performed via the mini-superspace formalism, and therefore, our work shows that even more complicated scenarios such as higher-dimensional planar black holes with rotation and charge can be studied similary.", '2 3d Einstein-Gauss-Bonnet theory': "We start our analysis with 3d Einstein-Gauss-Bonnet (EGB) theory whose action is given by  \nI = 1 16 πG ∫ d 3 x √ -g [ R + 2 L 2 + α 2 L 2 L 2 ] , (2.1)  \nL 2 = 4 G µν φ µ φ ν -4 X ✷ φ +2 X 2 , (2.2)  \nwhere α 2 is a dimensionless coupling constant, L is a constant with dimension of a length and L 2 is the 3d version of the well-known 2nd-order Lovelock Lagrangian, also known as the Gauss-Bonnet invariant, which reads  \nG = R 2 -4 R µν R µν + R µνρσ R µνρσ . (2.3)  \nFor the terms with the scalar field φ , we use the following definitions  \nφ µ ≡ ∂ µ φ, φ µν ≡ ∇ µ ∇ ν φ, /square φ ≡ g µν φ µν , X ≡ g µν φ µ φ ν . (2.4)  \nIn order to study static field configurations, which cover both the black hole and the soliton solutions, we assume a line element of the form  \nd s 2 = -N 2 r 2 L 2 F d t 2 + L 2 r 2 F d r 2 + R 2 d θ 2 , (2.5)  \nwhere the functions ( N , F , R ) characterizing the spacetime metric, together with the scalar field φ , depend only the radial coordinate r . By inserting them directly into the action (2.1), one can obtain a reduced action for the functions ( N , F , R , φ ) and obtain the equations satisfied by them from the Euler-Lagrange equations of the resulting reduced action.  \nIn order to study the thermodynamics of the solutions, one can identify the partition function for a thermodynamic ensemble with the Euclidean path integral evaluated by the saddle-point approximation around the Euclidean continuation of the classical solution [6]. Assuming β and η are the periodicities of the Euclidean time τ = it and the angular coordinate θ respectively, the Euclidean reduced action can be put into the following form  \nI E = -ηβ 16 πGL 2 ∫ d r N [ 2 Rr [ r F ' R ' +2 F ( R ' + r R '' ) ] +2 α 2 r 3 ξ F [ rξ 3 FR +2 ξ 2 R (2 F + r F ' ) -4 r F ξ ' R ' + ξ ( -3 r F ' R ' + F (4 r R ξ ' -6 R ' -2 r R '' ) ) ] ] + B, (2.6)  \nwhere ξ = φ ' and B is a boundary term. This form of the Euclidean reduced action is obtained by  \nperforming integration by parts successively until there remains no term with any derivative of the function N . The boundary term B is fixed by requiring that the Euclidean action I E has an extremum on-shell 4 , i.e., δI E | on-shell = 0. In our case, this condition gives the variation of the boundary term B as  \nδB = ηβ 16 πGL 2 [ r ( 1 + 2 α 2 r 2 ξ 2 F ) ( r NF ' +2 F ( N + r N ' ) ) δ R2 r 2 FN ( 1 + 2 α 2 r 2 ξ 2 F ) δ R ' -r 2 N ( R ' -4 α 2 r 2 ξ 3 FR +6 α 2 r 2 ξ 2 FR ' ) δ F +8 α 2 r 4 ξ F 2 N ( ξ R-R ' ) δξ ] . (2.7)  \nNote that the Euler-Lagrange equation following from the variation of the reduced action I E with respect to the function N , which is δI E δ N = 0, guarantees that there will be no bulk contribution to the on-shell Euclidean action. Therefore, the on-shell action can be found from the boundary term as  \nI E ∣ ∣ ∣ ∣ on-shell = B ( ∞ ) -B ( r min ) , (2.8)  \nwhere r min is the minimum value of the radial coordinate r . Since it is related to the free energy as I E | on-shell = βF = βM -S , the mass M and the semi-classical entropy S of the solution can be calculated from the following relations  \nM = ∂ β I E ∣ ∣ ∣ ∣ on-shell , S = ( β∂ β -1) I E ∣ ∣ ∣ ∣ on-shell . (2.9)", '2.1 Novel black hole solution': "An ansatz for the line element of a static black hole solution can be written as  \nd s 2 b = -N 2 r 2 L 2 f b d t 2 b + L 2 r 2 f b d r 2 + r 2 d θ 2 b , (2.10)  \nwhich corresponds to the following choice of the mini-superspace variables  \nN = N, F = f b , R = r. (2.11)  \nWhen N = 1, which will be the case for the solutions that we study here, it describes an asymptotically AdS 3 black hole solution provided that the function f b has at least one zero and its value at infinity is a positive constant [ f ∞ ≡ f b ( r →∞ ) > 0]. The AdS radius in such a case is given by  \n/lscript = L √ f ∞ . (2.12)  \nThe periodicity β of the Euclidean time τ b = it b , which is the inverse temperature of the black hole, is found by avoiding a conical singularity at the horizon. We can take the periodicity of the angular coordinate η to be 2 π . Defining  \nh b ≡ r 2 L 2 f b , (2.13)  \nβ = 4 π h ' b ( r + ) = 4 πL 2 r 2 + f ' b ( r + ) , η = 2 π, (2.14)  \nwe have  \nwhere r + is the largest positive root of the equation f b ( r ) = 0 (or equivalently h b ( r ) = 0).  \nBy using the choices (2.11) in the reduced action of 3d EGB gravity (2.6), the following field configuration can be deduced [46]  \nN = 1 , φ = log ( r /lscript ) , 1 -f b -α 2 f 2 b = ( r + r ) 2 . (2.15)  \nThe value of the function f b at infinity f ∞ can be found by solving the equation  \n1 -f ∞ -α 2 f 2 ∞ = 0 , (2.16)  \nfrom which the AdS radius /lscript can be determined by using (2.12). This solution is the analogue of the higher-dimensional black hole solution of the EGB theory that exist in d > 4 [47].  \nThe thermodynamic analysis of the black hole solutions that we consider here via the Euclidean methods was given in [48] but we will review the points that are relevant for the analysis of the corresponding soliton solutions. As discussed in [48], the quadratic equation for the function f b can be solved. However, its explicit form is not needed for the thermodynamic analysis. This will be particularly useful when we study the novel black hole solution in 3d cubic Lovelock gravity where the function f b satisfies a cubic equation. By using f b ( r + ) = 0 in the derivative of the equation satisfied by the function f b (2.15), one gets f ' b ( r + ) = 2 r + . From this, the inverse temperature β of the black hole follows as  \nβ = 2 πL 2 r + . (2.17)  \nWhen the solution for the scalar field in (2.15) is used as ξ = 1 r , the variation of the boundary term B b simplifies into  \nδB b = -βr 2 8 GL 2 (1 + 2 α 2 f b ) δf b , (2.18)  \nwhere we have used η = 2 π .  \nThe next step is to evalute it at infinity and the event horizon to find the value of the boundary term B b at these points, which will yield the on-shell value of the action through (2.8) ( r min = r + ). The variation of the function f b at these values of the radial coordinate r are as follows,  \nδf b ∣ ∣ ∣ ∣ ∞ = -2 r + (1 + 2 α 2 f b ) r 2 δr + ∣ ∣ ∣ ∣ ∞ , δf b ∣ ∣ ∣ ∣ r + = -f ' b ( r + ) δr + = -4 πL 2 βr 2 + δr + . (2.19)  \nWhile the first one directly follows from the equation satisfied by the function f b in (2.15), the second one requires a special care. The variation of the function h b at the horizon should respect the conditions h b ( r + ) = 0 and ( h b + δh b )( r + + δr + ) = 0, from which one gets  \nδh b ∣ ∣ ∣ ∣ r + = -h ' b ( r + ) δr + = -4 π β δr + . (2.20)  \nwhere we have used (2.14). Using this with the relation between the functions h b and f b (2.13) yields the result in (2.19).  \nWith these at hand, the on-shell action can be obtained easily as  \nI b E ∣ ∣ ∣ ∣ on-shell = B b ( ∞ ) -B b ( r + ) = βr 2 + 8 GL 2 -πr + 2 G , (2.21)  \nfrom which the mass M and the entropy S are found by using 5 (2.9) as  \nM = r 2 + 8 f ∞ G/lscript 2 , S = πr + 2 G . (2.22)  \nHere, we have used (2.12) to write the mass M in terms of the AdS radius /lscript . This completes our review of the thermodynamics of the black hole solution.  \nThe holographic studies of 3d Lovelock gravities was initiated in [49], where it was shown that, unlike a generic Horndeski theory, it is possible to construct a monotonic c-function along a renormalization group flow induced by matter fields and to establish a holographic c-theorem along the lines of [50, 51]. The value of the c-function in the UV ( r →∞ ) is proportional to the Weyl anomaly coefficient, from which the central charges for the 3d EGB theory can be read as 6  \nc ± = 3 /lscript 2 G (1 -2 α 2 f ∞ ) . (2.23)  \nThis result was also confirmed by a direct calculation of the Weyl anomaly coefficient of the 2d boundary CFT by considering the theory on S 2 . It is obvious that the Cardy formula with central charges for static black holes (1.15) fails to reproduce the semi-classical entropy in (2.22). Therefore, we need to construct the soliton solution describing the ground state.", '2.2 Soliton corresponding to the novel black hole solution': "As mentioned in the introduction, the soliton solution is obtained by applying a double Wick rotation, which we perform as  \nt b → iLθ s , θ b → i t s L . (2.24)  \nWhen applied to the ansatz (2.10) that we used to obtain the black hole solution, one gets  \nd s 2 s = -r 2 L 2 d t 2 s + L 2 r 2 f s d r 2 + N 2 r 2 f s d θ 2 s , (2.25)  \nThe bulk piece of the reduced action remains the same after the Wick rotation and one obtains the following field configuration describing the soliton  \nN = 1 , φ = log ( r /lscript ) , 1 -f s -α 2 f 2 s = ( r + r ) 2 . (2.26)  \nThe event horizon radius r + is technically an integration constant and it is known that the soliton solution describing the ground state should be devoid of integration constants [7,10]. We will see that it will be fixed by matching the boundary geometries of both the black hole and the soliton solutions.  \nFirst of all, we should still demand the regularity of the spacetime at r = r + . Denoting the periodicities of the Euclidean time τ s = it s and the angular coordinate θ s as β s and η s respectively, and defining  \nh s ≡ r 2 L 2 f s , (2.27)  \nwe see that it does not constrain β s but we get  \nη s = 4 π h ' s ( r + ) L = 4 πL r 2 + f ' s ( r + ) . (2.28)  \nFor our solution, f ' s ( r + ) = 2 r + and we obtain  \nη s = 2 πL r + . (2.29)  \nThe boundary geometries of both the black hole and the soliton solutions are the same provided that  \n( √ -g 00 β ) b = ( √ -g 00 β ) s , as r →∞ , (2.30)  \n( √ g 33 β ) b = ( √ g 33 β ) s , as r →∞ , (2.31)  \nβ s = √ f ∞ β, η s = 2 π √ f ∞ , η s β s = 2 πβ. (2.32)  \nwhich imply  \nUsing this value of the angular periodicity η s in (2.29) fixes the integration constant r + as  \nr + = √ f ∞ L for the soliton. (2.33)  \nSince the field variations are taken with respect to the integration constants, one has to be careful in using this result while calculating the thermodynamical properties of the soliton. We can use (2.33) only after integrating the boundary term and obtaining the expressions for the mass M 0 and the entropy S 0 in terms of the integration constant r + from the on-shell action.  \nFor the thermodynamical analysis, we should first note that the soliton ansatz (2.25) corresponds to the following choice of the minisuperspace variables  \nN = 1 √ f s , F = f s , R = Nr √ f s . (2.34)  \nUsing them together with N = 1 and ξ = φ ' = 1 r in (2.7) gives the variation of the boundary term for the soliton as  \nδB s = -βr 2 8 GL 2 d d r [ r ( 1 + 2 α 2 f s ) δf s ] . (2.35)  \nThe variation of the function f s at infinity is obtained from the equation satisfied by it in (2.26) as follows  \nAt r = r + , we make use of that the variation of the function h s should respect the conditions  \nδf s ∣ ∣ ∣ ∣ ∞ = -2 r + (1 + 2 α 2 f s ) r 2 δr + ∣ ∣ ∣ ∣ ∞ . (2.36)  \n( h s + δh s )( r + + δr + ) = 0 , ( h ' s + δh ' s )( r + + δr + ) = 4 π η s L , (2.37)  \nδh s ∣ ∣ ∣ ∣ r + = -h ' s ( r + ) δr + , δh ' s ∣ ∣ ∣ ∣ r + = -h '' s ( r + ) δr + . (2.38)  \nwhich imply  \nTogether with the relation (2.27), they yield the following relations for the function f s  \nδf s ∣ ∣ ∣ ∣ r + = -f ' s ( r + ) δr + , δf ' s ∣ ∣ ∣ ∣ r + = -[ f '' s ( r + ) + 2 r + f ' ( r + ) ] δr + . (2.39)  \nWe should keep in mind that the integration constant r + will be fixed according to (2.33) at the end. After the integration of the boundary term, one gets  \nB s ( ∞ ) = -r 2 + 8 GL 2 , B s ( r + ) = 0 , (2.40)  \nf ' s ( r + ) = 2 r + , f '' s ( r + ) = -2(3 + 4 α 2 ) r 2 + , (2.41)  \nwhere we have used  \nfor our solution.  \nUsing the value of the integration constant from (2.33), we obtain the on-shell action for the soliton as  \nThe mass M 0 and the entropy S 0 for the soliton follows from the relations (2.9) easily as  \nI s E ∣ ∣ ∣ ∣ on-shell = -βf ∞ 8 G . (2.42)  \nM 0 = -f ∞ 8 G , S 0 = 0 . (2.43)  \nThe vanishing of the entropy verifies the non-degeneracy of the ground state. More importantly, using the mass of the soliton M 0 in the Cardy formula without central charges for static black holes (1.10) reproduces the semi-classical entropy that we have derived in (2.22). The formula with the central charges (1.15) is not succesful since the soliton mass is not related to the central charges as given in (1.16).", '3 3d Cubic Lovelock gravity': "Now, we would like to study a more complicated scenario where two distinct static black hole solutions exist. With this aim, we consider 3d cubic Lovelock gravity with the action [33]  \nI = 1 16 πG ∫ d 3 x √ -g [ R + 2 L 2 + α 2 L 2 L 2 + α 3 L 4 L 3 ] , (3.1)  \nL 2 = 4 G µν φ µ φ ν -4 X ✷ φ +2 X 2 , (3.2)  \nL  \n-  \n-  \n48  \nR  \nµν  \nφ  \nµν  \nX  \n48  \nR  \nµν  \nφ  \nµ  \nφ  \nν  \nX  \n+24  \nRX  \n✷  \nφ  \n+6  \nRX  \n2  \n+96  \nφ  \nµν  \nφ  \nµ  \nφ  \nν  \n✷  \nφ  \n+48 φ µν φ µν ✷ φ -24 φ µν φ µν X -144 φ µν φ µ φ ν X -96 φ µ φ ν φ µρ φ ρ ν -32 φ µν φ µρ φ ρ ν -16( ✷ φ ) 3 +24 X ( ✷ φ ) 2 -24 X 3 , (3.3)  \nwhere we have again employed the definitions (2.4). L 2 is the 3d version of the Gauss-Bonnet invariant as discussed before and L 3 is the 3d version of the cubic Lovelock Lagrangian given by  \nC = 1 8 δ µ 1 ν 1 µ 2 ν 2 µ 3 ν 3 ρ 1 σ 1 ρ 2 σ 2 ρ 3 σ 3 R ρ 1 σ 1 µ 1 ν 1 R ρ 2 σ 2 µ 2 ν 2 R ρ 3 σ 3 µ 3 ν 3 . (3.4)  \nα 3 is another dimensionless coupling constant just as α 2 .  \nThe static black hole solutions of this theory were found in [33] and their thermodynamics were  \n3  \n=  \npresented in [48] by finding the reduced action of the theory for the black hole ansatz (2.10) and φ = φ ( r ). However, as we have seen in the previous section, one can start with the reduced action for our more general mini-superspace ansatz (2.5). Then, it is enough to make proper choices for the mini-superspace variables to derive both the black hole and the soliton solutions, and also their thermodynamics. After quite a lengthy calculation, one finds the Euclidean reduced action of 3d cubic Lovelock gravity as  \nI E = -ηβ 16 πGL 2 ∫ d r N { 2 Rr [ r F ' R ' +2 F ( R ' + r R '' ) ] +2 α 2 r 3 ξ F [ rξ 3 FR +2 ξ 2 R (2 F + r F ' ) -4 r F ξ ' R ' -ξ ( 3 r F ' R ' -F (4 r R ξ ' -6 R ' -2 r R '' ) ) ] -6 α 3 r 5 ξ 3 F 2 [ 4 rξ 3 FR24 r F ξ ' R ' +12 ξ 2 R (2 F + r F ' ) -3 ξ ( 5 r F ' R ' -2 F (4 r R ξ ' -5 R ' -r R '' ) ) ] } + B, (3.5)  \nwhere the variation of the boundary term, which follows from the condition δI E | on-shell = 0, is given by  \nδB = ηβr 16 πGL 2 [ ( 1 + 2 α 2 r 2 ξ 2 F 18 α 3 r 4 ξ 4 F 2 )( r NF ' +2 F ( N + r N ' ) ) δ R +2 r FN ( -1 -2 α 2 r 2 ξ 2 F +18 α 3 r 4 ξ 4 F 2 ) δ R ' -r N ( R ' -2 α 2 r 2 ξ 2 F (2 ξ R3 R ' ) + 18 α 3 r 4 ξ 4 F 2 (4 ξ R5 R ' ) ) δ F +8 r 3 N ξ F 2 ( α 2 -18 r 2 α 3 ξ 2 F ) ( ξ R-R ' ) δξ ] . (3.6)", '3.1 Novel black hole solution': '3d cubic Lovelock gravity admits a novel black hole solution analogous to the higher-dimensional black hole solution of cubic Lovelock gravity that exist in d > 6 [52], where the functions in our black hole ansatz (2.10) and the scalar field φ takes the following form [33]  \nN = 1 , 1 -f b -α 2 f 2 b +6 α 3 f 3 b = ( r + r ) 2 , φ = log ( r /lscript ) . (3.7)  \nThis time, the value of the function f b at infinity f ∞ satisfies the cubic equation  \n1 -f ∞ -α 2 f 2 ∞ +6 α 3 f 3 ∞ = 0 . (3.8)  \nThe mass and the entropy of the solution can be obtained by taking the mini-superspace variables as given in (2.11) in the variation of the boundary term (3.6). The procedure was reviewed in the previous section and details can be found in [48]. The mass M and the entropy S the black hole are found as  \nM = r 2 + 8 f ∞ G/lscript 2 , S = πr + 2 G . (3.9)  \nThe central charges for the theory was computed in [49] to be  \nc ± = 3 /lscript 2 G ( 1 -2 α 2 f ∞ +6 α 3 f 2 ∞ ) . (3.10)  \nUsing them in the formula (1.15) obviously does not yield the semi-classical entropy S in (3.9).', '3.2 Soliton corresponding to the novel black hole solution': 'The corresponding soliton solution following from the double Wick rotation (2.24) of the novel black hole solution is as follows  \nN = 1 , 1 -f s -α 2 f 2 s +6 α 3 f 3 s = ( r + r ) 2 , φ = log ( r /lscript ) , r + = √ f ∞ L, (3.11)  \nwhere we fix the integration constant r + by obeying the constraints imposed by the regularity of the spacetime (2.29) and the boundary matching conditions (2.32). Note that we again have f s ( r + ) = 2 r + , which can be verified from the equation satisfied by the function f s . Using the choice (2.34) of the minisuperspace variables in (3.6) together with N = 1 and φ = log ( r /lscript ) from (3.11), we obtain the variation of the boundary term for the soliton as  \nδB s = -βr 2 8 GL 2 d d r [ r ( 1 + 2 α 2 f s -18 α 3 f 2 s ) δf s ] . (3.12)  \nWhile the variation of the function f s at infinity is given by  \nδf s ∣ ∣ ∣ ∣ ∞ = -2 r + (1 + 2 α 2 f s -18 α 3 f 2 s ) r 2 δr + ∣ ∣ ∣ ∣ ∞ , (3.13)  \nthe variation at r = r + can be easily calculated through relations (2.39) where the equations (2.41) are still valid. As a result, we get  \nM 0 = -f ∞ 8 G , S 0 = 0 , (3.14)  \nfor the soliton. Note that we exactly have the same structure as we had for the novel black hole in the 3d EGB theory [see (2.22) and (2.43)] and the semi-classical entropy is reproduced through the formula (1.10).', '3.3 BTZ black hole': 'Contrary to the 3d EGB theory, the BTZ black hole is admitted as a solution in 3d Cubic Lovelock gravity where the functions in our black hole ansatz (2.10) and the scalar field φ takes the following form [33]  \nwhere the value of the function f b at infinity is related to the coupling constants α 2 and α 3 as  \nN = 1 , f b = f ∞ [ 1 -r 2 + r 2 ] , φ = 1 3 √ α 2 2 α 3 f ∞ log    r -√ r 2 -r 2 + /lscript    , (3.15)  \nf ∞ = α 3 2 +972 α 2 3 54 α 3 ( α 2 2 +18 α 3 ) . (3.16)  \nThe mass and the entropy of the solution can be obtained by taking the mini-superspace variables as given in (2.11) in the variation of the boundary term (3.6). Details can be found in [48]. The mass and the entropy of the BTZ black hole in 3d cubic Lovelock gravity are found as  \nM = [ 1 + α 2 2 18 α 3 ] r 2 + 8 G/lscript 2 , S = [ 1 + α 2 2 18 α 3 ] πr + 2 G . (3.17)  \nWhen we attempt to reproduce the semi-classical entropy S in by using Cardy formula with central charges for static black holes (1.15), it does not work as expected. Hence, it is needed to construct the soliton describing the ground state as before.', '3.4 Soliton corresponding to the BTZ black hole': "After a double Wick rotation given in (2.24), we obtain the following configuration describing a soliton  \nN = 1 , f s = f ∞ [ 1 -r 2 + r 2 ] , φ = 1 3 √ α 2 2 α 3 f ∞ log    r -√ r 2 -r 2 + /lscript    , (3.18)  \nThis time, we have f ' s ( r + ) = 2 f ∞ r + , and consequently the angular periodicity η s becomes  \nη s = 2 πL f ∞ r + . (3.19)  \nWith this, the boundary matching condition in (2.32) fixes the integration constant r + as  \nr + = L √ f ∞ = /lscript for the soliton, (3.20)  \nwhich should be imposed only after integrating the boundary term. Using the choice (2.34) in (3.6) together with ξ = φ ' = 1 3 √ α 2 2 α 3 1 r √ f s from (3.18) , we obtain the variation of the boundary term for the soliton as  \nδB s = -βr 2 144 GL 2 [ 18 + α 2 2 α 3 ] d d r ( r δf s ) . (3.21)  \nThe variation of the function f s at infinity is as follows  \nδf s ∣ ∣ ∣ ∣ ∞ = -2 f ∞ r + r 2 δr + ∣ ∣ ∣ ∣ ∞ . (3.22)  \nM 0 = -1 8 G ( 1 + α 2 2 18 α 3 ) , S 0 = 0 . (3.23)  \nAt the horizon, we use the relations (2.39) with f ' s ( r + ) = 2 f ∞ r + and f '' s ( r + ) = -6 f ∞ r 2 + . With these, the on-shell Euclidean action for the soliton I s E can be easily found by integrating the variation of the boundary term in (3.21). After imposing (3.20), the relations (2.9) give the mass and the entropy of the soliton as  \nUsing the soliton mass M 0 in the Cardy formula for static black holes with no reference to central charges (1.10) reproduces the semi-classical entropy in (3.17)  \nWe have checked the validity of the Cardy formula in the context of 3d cubic Lovelock gravity. This theory is interesting because there are two different black hole solutions, with non-trivial scalar profiles. By means of the double Wick rotation, we have found two different ground states, and the  \nCardy formula holds for each one of them. This is not surprising, because the black hole solutions belong to disconnected sectors of the theory and, in the same way as in [7], there is no continuous way to deform one sector to the other, so each branch must possess its own ground state.", '4 Summary and outlook': "We have revisited the proposal of [7] for the microscopic derivation of the entropy of 3d black holes where the semi-classical entropy is reproduced from the Cardy formula without central charges (1.9) describing the asymptotic growth of the number of states. In this setup, the ground state is described by a soliton obtained through a double Wick rotation of the static black hole solution. We have checked the validity of the formula (1.10) for static black holes in 3d Lovelock gravities, which are 3d Horndeski theories that were recently obtained by various procedures from the higher-dimensional Lovelock Lagrangians. In order to deal with the relatively complicated structure of these theories compared to previous studies, we employed the mini-superspace formalism to calculate the mass of the soliton required in the Cardy formula (1.10). For all solutions we have considered, we have shown the validity of the proposal of [7], providing evidence in 3d Horndeski theories for the first time.  \nThe natural way to extend our results further is to add other global charges to the static solutions studied here. It is known that rotating versions of static black hole solutions in 3d can be obtained by 'a boost in the t -θ plane' [53,54] as follows:  \nt → 1 √ 1 -ω 2 ( t + ω/lscriptθ ) , θ → 1 √ 1 -ω 2 ( θ + ω t /lscript ) . (4.1)  \nFor the rotating black holes, the Cardy formula without central charges (1.9) reads  \nS Cardy = 2 π √ -M 0 /lscript ( M/lscript + J ) + 2 π √ -M 0 /lscript ( M/lscript -J ) , (4.2)  \nwhich we expect to hold for the rotating black holes resulting from the boost, as it was the case in [11].  \nAdding electric charge is another possibility, for which the soliton describing the ground state is magnetically charged and the Cardy formula is modified accordingly. The Cardy formula for charged AdS 3 black holes proposed in [17] reads  \nS Cardy = 4 π/lscript √ | -M 0 + α Φ m Q m || M -α Φ e Q e | , (4.3)  \nwhere (Φ e,m , Q e,m ) are the electric/magnetic potentials and charges, and α is a constant that depends on the nature of electromagnetic coupling. Remarkably, the Smarr relation is intimately related to the Cardy formula (see the discussion in [17]). Since the Smarr relation does not exist in 3d EinteinMaxwell theory due to the logarithmic behaviour of the electric potential, the authors used a conformal coupling to electromagnetism to test the validity of this formula. The same strategy can be used for black holes in 3d Lovelock gravities. Later, it was shown that the mass and the electric charge of the black hole solution in Einstein-Maxwell theory get modified when the 'holographic boundary conditions' [55] are imposed such that the Smarr relation holds [56]. Therefore, one might expect the charged Cardy formula (4.3) to be valid even with the usual Maxwell coupling when these boundary conditions are imposed.  \nAcknowledgements G. A. is supported by TÜBİTAK Grant No 124F058. L.G. thank Instituto de Matemáticas, Universidad de Talca, for its hospitality during the preparation of this work.", 'References': "- [1] A. 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