Uplifting dyonic AdS$_4$ black holes on seven-dimensional Sasaki-Einstein manifolds

We revisit non-rotating, dyonically charged, and supersymmetric AdS$_4$ black holes, which are solutions of $\mathcal{N}=2$ gauged supergravity with vector- and hyper-multplets. Uplifting the near horizon solutions to $D=11$ supergravity on seven-dimensional Sasaki-Einstein manifolds, we show that dyonic AdS$_4$ black holes correspond to AdS$_2$ solutions with electric and magnetic baryonic fluxes in $D=11$ supergravity. We identify the off-shell AdS$_4$ black hole solutions with parameters of $D=11$ AdS$_2$ solutions without imposing the equations of motion. We calculate the entropy of dyonic black holes, carefully analyzing the Page charge quantization conditions.


Introduction
The microscopic understanding of the Bekensteien-Hawking entropy of black holes has been a touchstone for any theory of quantum gravity. Recently, via AdS/CFT correspondence, the microscopic entropy was successfully computed for a class of supersymmetric and asymptotically AdS black holes. The breakthrough was triggered through two concrete examples. The topologically twisted index of ABJM theory at large N exactly matches the entropy of static, magnetically charged AdS 4 black holes in dual supergravity [1,2]. For the rotating, electrically charged AdS 5 black holes, on the other hand, the entropy was reproduced by computing the superconformal index of N = 4 super Yang-Mills theory [3][4][5]. See, for a review, [6] and the references therein for the related studies.
As it is very well known, the entropy of a black hole is computed by the Bekenstein-Hawking formula where A is the area of the event horizon and G (4) N is the four-dimensional Newton's constant. On the other hand, a new method to calculate the entropy was recently proposed in [7] and discussed further in [8][9][10][11][12][13]. In [7], the authors revisited AdS 2 × Y 9 solutions in eleven-dimensional supergravity [14,15]. By imposing the supersymmetry conditions and suspending the imposition of the equations of motion, they constructed the so-called supersymmetric action S SUSY . One can obtain the supersymmetric solutions by extremizing this nine-dimensional action S SUSY . Then, the authors considered magnetically charged AdS 4 black holes [16] in minimal gauged supergravity and uplifted the near horizon solutions to D = 11 supergravity on an arbitrary seven-dimensional Sasaki-Einstein manifold SE 7 [17]. Putting everything together, it was shown that S SUSY when extremized gives the entropy of this class of black holes. 1 This extremization principle provides a very powerful technique for calculating the black hole entropy when Y 9 is a fibration of seven-dimensional Sasaki-Einstein manifold over a two-dimensional Riemann surface Σ g with genus g. Because all the information required to calculate S SUSY is just the topological data, i.e., the toric vectors of cone C(SE 7 ), one can calculate the entropy without knowing the explicit form of the metric of black hole solutions. It was first investigated in AdS 3 solutions of IIB supergravity in [19] and applied to D = 11 supergravity for S 7 [8] and for other SE 7 manifolds [9][10][11].
In this paper, we are interested in the AdS black holes in four-dimensional gauged supergravity obtained as consistent truncations of D = 11 supergravity on SE 7 . Then, the four-dimensional massless vector fields are classified into two groups. Some of the vector fields are originated from the isometries of SE 7 while the others, called Betti vector fields, are from the reductions of three-form potential A 3 on non-trivial two-cycles in SE 7 . They are related to mesonic vs. baryonic flavor symmetries of dual boundary field theories, respectively.
First let us consider the simplest Sasaki-Einstein seven-manifold S 7 . Since there is no non-trivial two-cycle in it, all the massless vector fields come only from the isometries of S 7 . Then, the black holes have mesonic charges only. For example, it is well known that D = 11 supergravity is consistently truncated to D = 4, N = 2, U (1) 4 gauged supergravity where U (1) 4 ⊂ SO (8) are isometries of S 7 [20,21]. This theory is called STU supergravity, where the black hole solutions with the spherical horizon were studied in [22][23][24]. As we mentioned, the entropy of magnetically charged black hole is reproduced by computing the topologically twisted index [1] and also by solving the extremization problem [8]. Now let us consider topologically non-trivial SE 7 manifolds having non-trivial twocycles. In this case, the Betti vector fields come into play in addition to the vector fields from isometries and the black holes have both mesonic and baryonic charges, in general. However, to the best of our knowledge, neither black hole solutions nor the topologically twisted indices for general mesonic and baryonic charges has been computed.
On the one hand, explicit four-dimensional black hole solutions or AdS 2 solutions in M-theory with mesonic charges, except for that of graviphoton, are currently not available. However, if we assume that such a black hole solution exists, then we can compute the entropy by solving variational problems and show perfect agreement with the field theory computations [9][10][11]. 2 On the other hand, the explicit form of black holes with baryonic charges are well known. The consistent truncations, which keep the Betti vector fields as well as the graviphoton, are constructed in [29] and the four-dimensional AdS black hole solutions with baryonic charges were studied in [30]. Furthermore, AdS 2 solutions in Mtheory [31,32] also have baryonic charges. See also [33][34][35][36][37]. Given the explicit black hole solutions with baryonic charges, one can easily compute the Bekenstein-Hawking entropy, which can be successfully reproduced by solving variational problem [9,11]. However, it is not known how to calculate the entropy on the dual field theory side because the topologically twisted index at large N is independent of baryonic charges [38,39].
Hence, the correspondence between the black hole entropy and the topologically twisted index does not hold for the black holes with both baryonic and mesonic charges, because the entropy is known for black holes with baryonic charges only and the topologically twisted index depends on mesonic charges only. However, using the extremization principles, we can investigate the general cases with both baryonic and mesonic charges. Then, we expect that this method gives a guideline to resolve this puzzle and establish a precise correspondence.
Although the extremization principle provides a very powerful method to calculate the black hole entropy, it is applicable to a restricted class of black holes, i.e., four-dimensional magnetically charged black holes. One can think of various generalizations of this method to incorporate, for example, dyonically charged black holes, rotating black holes 3 , black holes in dimensions higher than four, etc. However, establishing the variational problems for these black holes from scratch is a daunting task and beyond the scope of this paper.
In this paper, instead of tackling the variational problem for dyonic black holes, we 2 For N = 2 Chern-Simons theories with chiral quiver, it is known that the matrix model to calculate the free energy on S 3 is not working [25]. An alternative prescription for calculating the free energy was proposed by [26,27] and further studied in [28,11]. 3 Recently, SSUSY for rotating AdS4 black holes is constructed in [40].
restrict ourselves to the explicit solutions of dyonic AdS 4 black holes. For the black holes in STU supergravity, whose massless vector fields come from the isometries of S 7 , the agreement between the entropy of the black hole and the topologically twisted index for ABJM theory also holds for the dyonic case [2]. However, the situation is more involved for the black holes which we revisit in this paper. The dyonically charged AdS 4 black holes in [30] were studied in four-dimensional N = 2 gauged supergravity, which is obtained by a consistent truncation of eleven-dimensional supergravity on a seven-dimensional homogenous manifold with SU (3)-structure [29]. One of such manifolds we will focus on is M 111 where the truncation yields two massless vector fields: a graviphoton and one Betti vector filed. Therefore, the black holes we are interested in have baryonic charges and their dual field theoretic computation of the entropy at large N is not known as we mentioned earlier.
Furthermore, while the STU black holes are known analytically, the black hole solutions we revisit in this paper can be studied numerically, which interpolate between AdS 4 and AdS 2 × Σ g . Only the near horizon solutions are known analytically [30]. Hence, we can uplift this near horizon solutions and obtain AdS 2 solutions in eleven-dimensions. On the other hand, there are previously well-known AdS 2 solutions in D = 11 supergravity discovered a decade ago [31,32]. These AdS 2 solutions, which we review in the appendix A, are classified into two groups according to the explicit form of four-form flux. 4 Though it was already mentioned in [30] that uplifting dyonic AdS 4 black holes 5 will be included in this class of AdS 2 solutions, we will explicitly study the uplifting and clarify the relations between them. In [11], we already showed that the metric uplift of magnetically charged black hole solution corresponds to that of AdS 2 solution with electric four-form flux [31,36]. In this paper, we complete the analysis by uplifting the flux as well. Then we move on to the dyonic black hole case. We identify the uplifted solution of dyonically charged black hole and AdS 2 solution with non-vanishing magnetic four-form flux [32,36]. It implies that turning on the additional electric charges on the magnetically charged AdS 4 black holes corresponds to turning on the magnetic, internal four-form flux of D = 11 supergravity.
We also establish the precise dictionary between the four-dimensional quantities of AdS black holes [30] and the parameters in AdS 2 solutions in M-theory [31,32]. These identifications lead to the following consequences. Firstly, it provides the expressions for the electric and magnetic charges of the four-dimensional black hole solutions in terms of the parameters, which describe the eleven-dimensional AdS 2 solutions. Secondly, the identifications can be extended to off-shell. For the parameters in AdS 2 solutions in Mtheory, we should impose a constraint between them, which solves the equation of motion [31,32]. Here off-shell means that the identifications hold without imposing this constraint. In this sense, the four-dimensional black hole solutions written in terms of the parameters in the eleven-dimensional solutions are dubbed as off-shell solutions. Thirdly, we calculate 4 These solutions are revisited in [36,37]. Recently, AdS2 solutions outside this classification have been studied in [40,41] as the near horizon solutions of the rotating AdS4 black holes uplifted to D = 11 supergravity. Furthermore, other AdS2 solutions were discussed in type IIB supergravity [42] and massive type IIA supergravity [43]. 5 Uplifting to D = 11 was also discussed in [44].
the off-shell expression for Bekenstein-Hawking entropy of the dyonically charged AdS 4 black holes. When we have the non-trivial internal four-form flux, we should consider the flux quantization with Page charges. We choose specific gauges where gauge potentials are well-defined on each coordinate patch and compute the Page charges. Using the integer N , introduced in the Page charge quantization with this gauge choice, we calculate the fourdimensional Newton's constant and the black hole entropy. Unlike the Maxwell charges of the magnetic black holes, the Page charges do not reproduce the constraint satisfied by the parameters. The paper is organized as follows. In section 2, we review the consistent truncations of the eleven-dimensional supergravity on a seven-dimensional Sasaki-Einstein manifold M 111 [29] and provide the uplifting formulae of four-dimensional AdS black holes [30] to elevendimensions. In section 3, we uplift the explicit near horizon solutions of black holes and compare them with previously known AdS 2 solutions in M-theory. In section 4, we present the off-shell identifications between the uplifting formulae and AdS 2 solutions. Then, we provide the off-shell expressions for black hole entropy. We discuss the flux quantization of Page charges in section 5. We conclude in section 6. In appendix A, we review AdS 2 solutions in D = 11 supergravity [31,32]. In appendix B, we summarize the near horizon solutions of black hole asymptotic to AdS 4 ×Q 111 studied in [30]. In appendix C, we provide the detailed calculation of Page charges. In appendix D, we discuss Newton's constant in four-and two-dimensions.

Uplifting AdS black holes to eleven-dimensions
In this paper, we revisit AdS black holes in four-dimensional gauged supergravity theory [30], which is obtained by consistent truncations of eleven-dimensional supergravity on seven-dimensional homogeneous manifolds with SU (3)-structures such as Q 111 , M 111 , V 5,2 , etc. [29]. Here, let us focus on Q 111 , M 111 . Then, the resulting four-dimensional theory is N = 2 gauged supergravity coupled to two (Q 111 )/one (M 111 ) massless vector multiplets, a massive vector multiplet and one hypermultiplet, respectively. The massless vector field which is originated from the reduction of three-form flux on the non-trivial two-cycle of the internal manifold is called Betti vector field and corresponds to baryonic symmetry in dual field theory.
The AdS black hole solutions in these N = 2 gauged supergravity theories were studied in [30]. They are static, supersymmetric and asymptotically AdS 4 black holes with electric and magnetic charges. Their near horizon geometry has a form of AdS 2 × Σ g where Σ g is two-dimensional Riemann surface with genus g = 1. The full black hole solutions interpolating AdS 4 and AdS 2 × Σ g were studied only numerically. However, the near horizon solutions are known analytically and can be written in terms of the values of vector multiplet scalars at the horizon. In this note, we restrict ourselves to the near horizon solutions of AdS black holes.
In [30], it was already noted that the embedding of these AdS black holes in M-theory correspond to known AdS 2 solutions constructed by using wrapped branes [31,32]. For the case of magnetically charged black holes which asymptote to AdS 4 × M 111 , the metric uplift was explicitly studied in [11] and shown to be the same as that of AdS 2 solutions in M-theory with the electric four-form flux [31,36]. We extend this analysis to the fluxes. Then, we uplift the dyonic black hole solutions.

Review of uplifting formulae
We begin by reviewing the uplifting formulae for the metric and the four-form flux from the consistent truncation studied in [29]. The eleven-dimensional metric ansatz is The seven-dimensional internal space is written as a U (1)-fibration over a six-dimensional base B 6 . This space is characterized by SU (3)-structure i.e., a real one-form η = e V θ, a real two-form J and a complex three-form Ω. Here K is related to the Kähler potential K as K ≡ e −K /8. From now on, we focus on M 111 case where B 6 is a product space of CP 2 and S 2 . Then the metric is written as Here, we normalized the scalar curvature of S 2 and CP 2 to be R(S 2 ) = 2 and R(CP 2 ) = 4. The one-form θ has a form of θ = dψ + η where ψ has periodicity π/2 and dη = ( We also have K = v 2 1 v 2 . Then, the eleven-dimensional metric is written as From the viewpoint of four-dimensional N = 2 gauged supergravity, v 1 , v 2 are the imaginary part of the scalars in two vector multiplets and φ is the one of scalars in the hypermultiplet. A 0 is a graviphoton. Now let us turn to the four-form flux in eleven-dimensional supergravity. The ansatz for three-form potential has a form of Here C 3 is a three-form, B is a two-form, A i are n V one-forms. w i are n V two-forms and α A , β A are 2n H three-forms in B 6 . For M 111 case, we have i = 1, 2 and A = 1. The real scalars b i are the real part of the scalars in the vector multiplets. The real scalars ξ A ,ξ A and scalar a dual to three-form B are the hypermultiplet scalars. For the black hole solutions [30] we are interested in, there is only one nontrivial hypermultiplet scalar φ, whereas the other scalars ξ =ξ = 0 and a can be set to zero. Then the four-form flux reduces to where with I = 0, 1, 2. Here the two-, four-forms are defined as and they satisfy the following relations We also used the electric and magnetic gauging parameters as e I = (e 0 , 0, 0) and m I = (0, 2, 2). (2.10) The coefficient K ijk is defined as Since the values of the scalar fields are constant on the horizon, we are interested in a class of solutions satisfying db i = 0. (2.12) Then, the final expression of the four-form flux is given by 6 (2.14)

Uplifting near horizon solutions of AdS black holes
Now let us consider the uplifting of the near horizon solutions [30] of AdS black holes. The four-dimensional space-time metric has a form of (2.15) 6 Throughout this paper, we have used b0 = 1. The Freund-Rubin parameter e0 is related to the AdS4 radius and the non-trivial scalar fields at the AdS4 vacuum [30] as (2.13) Here we set e0 = 6 for simplicity.
Then, the eleven-dimensional metric (2.4) becomes The solutions in [30] i.e., R 2 1 , R 2 2 , φ are written in terms of v 1 and v 2 . We will substitute the explicit solutions into this uplifting formula (2.16) and write down the AdS 2 solutions in eleven-dimensional supergravity in the next section.
For the uplifting of the four-flux, let us begin with the ansatz as where κ = 1 for S 2 and κ = −1 for H 2 . Then, the dual field strength becomes Here R IJ , I IJ are the real, imaginary part of the period matrix N IJ where F IJ = ∂F ∂X I ∂X J . F is the prepotential and X I are the homogeneous coordinates. The electric and magnetic charges can be obtained by In the consistent truncations investigated by Cassani, Koerber and Varela (which we abbreviate to CKV) in [29], they worked out in the frame where the electric and magnetic gauging parameters are e I = (e 0 , 0, 0) and m I = (0, 2, 2). (2.21) In this frame, the prepotential and the homogeneous coordinates are following However, the four-dimensional black hole solutions studied by Halmagyi, Petrini and Zaffaroni (which we abbreviate to HPZ) in [30], the authors worked out in the purely electric gauging as follows e I = (e 0 , 2, 2) 7 and m I = (0, 0, 0). (2.23) The prepotential and the homogeneous coordinates are The purely electrically gauged frame (HPZ) can be obtained by a symplectic rotation of the dyonically gauged frame (CKV) as follows Under a symplectic rotation, X I , F I and F I , G I also transform. More specifically, the field strengths in the two different frames are related as As a result, the four-form flux (2.14) becomes (2.28)

Uplifted AdS 2 solutions in D = 11 supergravity
In this section, we consider the explicit near horizon solutions of black holes studied in [30] and uplift them to the eleven-dimensions. Let us briefly summarize the near horizon solutions of dyonically charged black hole asymptotic to AdS 4 × M 111 [30]. There are two unknown function: the AdS radius R 1 and the size R 2 of two-dimensional Riemann surface Σ g . We also have two complex scalars b i + iv i in vector multiplets and one hypermultiplet scalar φ. The black holes have electric charges q 0 , q i and magnetic charges p 0 , p i . The solutions are described in terms of two parameters v 1 , v 2 . We write down the solutions for readers' convenience.
We substitute the above solutions e 2φ , R 1 , R 2 into the eleven-dimensional metric (2.16) and obtain .
For the four-form flux (2.28), the solution b 1 , b 2 are also needed. Since the explicit expression of the flux is quite messy, we will not write down here. The Bekenstein-Hawking entropy of the black hole can be easily calculated from the area law as The black hole entropy is expressed in terms of the vector multiplet scalars v 1 and v 2 . Since the magnetic and electric charges of the black holes also can be written by using these scalar fields, one can write down the entropy in terms of black hole charges, in principle. However, the expressions are quite non-linear and not easily invertible apparently. One can also reproduce the entropy by calculating the two-dimensional Newton's constant following the method described in [7]. In [36], the authors argued that the entropy can be written by using the three-sphere free energy as 9 where u corresponds to R 2 2 /R 2 1 in our convention. While the Bekenstein-Hawking entropy is given by the on-shell action of Euclidean AdS 4 whose boundary is S 3 is where the radius of vacuum AdS 4 is 1/2. Then, we obtain See the equation (4.24) in [36].
For the black holes with a universal twist, 10 the value of R 1 becomes 1/4 and the equation (3.5) holds. But, for the general solutions, R 1 is a non-trivial function of v 1 and v 2 (3.1). Hence, our expression (3.8) does not match with (3.5).
In the subsequent sections, we consider the uplifting of the magnetic and dyonic black holes separately and compare them to the known AdS 2 solutions in the literatures. The AdS 2 solutions in M-theory was classified by considering M2-branes wrapped on two-cycles in Calabi-Yau five-folds in [14]. The nine-dimensional internal space is a U (1)-fibration over the eight-dimensional Kähler manifold. We focus on a particular class of solutions where the eight-dimensional Kähler manifolds are products of two-dimensional Kähler-Einstein manifolds [31,32]. Recently, these solutions were revisited in [36,37]. We will show that the uplifts of magnetic, dyonic black holes correspond to AdS 2 solutions in M-theory with electric and dyonic baryonic fluxes, respectively, studied in [31,32,36,37].

Uplifting magnetic black holes
First, let us consider a simple case, i.e. the magnetically charged black holes. The electric charges of the black holes can be turned off by setting v 2 = (3 − v 2 1 )/(2v 1 ). In this case, the solutions can be written in terms of one parameter v 1 only. The metric uplift was discussed in the appendix of [11]. From (3.2), we have The uplift of flux (2.28) is very simple for the magnetically charged black hole since the real part of the vector multiplet scalars b i vanish.
We compute the dual field strengths as Here we used and for the magnetic black holes. Then the explicit form of the four-form flux becomes (3.14) 10 We can turn off the baryonic charges by setting v1 = v2 = 1 and consider a universal twist.
We note that the magnetic black hole solutions in four-dimensions correspond to the AdS 2 solutions with electric flux in eleven-dimensions.

Flux quantization
Before we proceed further, let us consider the various cycles in eleven-dimensional background we are interested in. For the detailed discussions, see, for example, [9,33,45].
We begin with two-and five-cycles in M 111 . A seven-dimensional Sasaki-Einstein manifold M 111 is a U (1)-fibration over S 2 × CP 2 . The first two-cycle S 1 is the two-sphere S 2 and the second two-cycle S 2 is obtained by fixing a point in S 2 ⊂ CP 2 . Since the Kähler form J = J S 2 + J CP 2 is trivial we have the exact two-form J. Then two 2-cycles S 1 and S 2 are not independent. Using the relation we can find an independent two-cycle S, which can be used as a basis of the second homology Here we follow the convention of [9]. The volumes of S 2 and CP 2 are given by We also have Here CP 2 can be replaced by other Kähler-Einstein four-manifold, for example, CP 1 × CP 1 .
The integers M, m, n a is determined by the topology of KE + 4 . The data for Σ a are two-cycles in Kähler-Einstein four-manifold. For KE + 4 = CP 2 case, we have Σ 1 = S 2 . We refer to [9] for details.
We also have the two five-cycles H 1 , H 2 obtained by fixing a point in S 2 and S 2 ⊂ CP 2 in M 111 , respectively. These five-cycles are not independent and satisfy the following relation 11 2H 1 + 3H 2 = 0, (3.20) which corresponds to a Poincaré dual to the exact two-form J. Now we consider the four-and seven-cycles in eleven-dimensional background AdS 2 ×Y 9 where Y 9 is obtained by fibering M 111 space over Σ g . The independent four-cycleH can be obtained by multiplying Σ g to the two-cycle S as (3.21) The seven-cycles Y 7 , C 1 , C 2 can be obtained by fixing a point on AdS 2 and Σ g , S 2 , S 2 ⊂ CP 2 , respectively. Then, we integrate the Hodge dual of four-form flux (3.14) over these seven-cycles and consider the flux quantization as 12 Here N, N 1 , N 2 are all integers. p 1 and p 2 are the magnetic charges of the black hole. We note that these seven-cycles are not independent and satisfy the following relation 13 We integrate the flux * 11 F 4 on this cycle using (3.22) and obtain which is a consequence of the Dirac quantization condition studied in [30]. Now let us more elaborate on the independent seven-cycle associated to the Betti vector field. In addition to the exact two-form J = J S 2 + J CP 2 in M 111 , there is a harmonic two-form ω = J CP 2 − 2J S 2 . We rewrite the relevant part of the three-form potential (2.5) in terms of two-form J and w as These particular linear combinations of vector fields in the consistent truncation of [29] can be explained as follows. The vector 2A 1 + A 2 mixes with the graviphoton A 0 to give a massless and a massive vector field [46]. The vector A 1 − A 2 is nothing but the Betti vector field we are interested in. 14 We find that its field strength is proportional to the difference 12 Here we used Vol(Σg) = 4π(g − 1). 13 This relation in Y9 can be obtained by d where Σa are the sevencycles. The twisting parameter n is given by 2(1 − g)(1, 0, 0, 1). See [9] for more detailed explanation.
14 For Q 111 case, the detailed discussions are given in [35].
of the magnetic charges p 1 − p 2 by using the equation (3.11). 15 Then, the charge can be calculated by using (3.22) as Hence, we have a non-trivial seven-cycle H associated to the Betti vector field as Now we calculate the electric and magnetic fluxes as where the seven-and four-cycle for M 111 are 30) and the period of ψ is ∆ ψ = π/2.

Comparison to known AdS 2 solutions
Now let us come back to the uplifting of magnetic black holes. In this section, we will show that these uplifted solutions correspond to the AdS 2 solutions in eleven-dimensional supergravity without magnetic four-form flux [31]. Once we identify v 1 in the metric (3.9) and the flux (3.14) as we reproduce the metric and the flux in [31] 16 where e −3A = 3 + 2 1 + 2 1 2 + 1 .
Here we have compared to the explicit solutions in [9], instead of expressions in [31]. Let us compare the fluxes through the various seven-cycles (3.22) and those in [9]. Then, one can easily read off the magnetic charges of black holes in terms of 1 as Using the identification (3.31), these magnetic charges reduce to the solution of Halmagyi, Petrini and Zaffaroni [30] as . (3.35) These AdS 2 solutions in eleven-dimensional supergravity were revisited rather recently [36]. As discussed in [37], the solutions with the electric baryonic charges studied in the section 4 of [36] correspond to those of [31]. With the following identifications one can check the explicit form of the metric and the four-form flux 17 reduce to (3.32). The electric and magnetic fluxes are 18 Let us identify the electric fluxes of the uplifted solution (3.28) and the solution (3.38). Then we can easily read off the magnetic charges of the Betti vector field as .

(3.40)
It is consistent with the previous result (3.35). See also the equation (5.21) in [11]. 17 Here, we follow the notation used in [36]. See the equation (4.25) in that paper 18 When CP 2 is replaced by CP 1 × CP 1 , this electric flux reduce to result of [36] with H here = 8HABCMZ, ψ here = ψABCMZ/4. See the equation (4.28) in that paper.

Uplifting dyonic black holes
In this section, we study the uplift of the dyonically charged black hole solutions and identify them to known AdS 2 solutions in M-theory with the internal four-form flux studied in [32]. As in the case of magnetic black holes, these AdS 2 solutions were also revisited in [36,37]. Because the form of the uplifted solutions, particularly the form of the flux, is quite messy, let us consider simple cases to make a comparison more explicitly. Then, we reproduce the solutions studied in the section 4.3.2 in [36]. We will also reproduce these solutions along the line of [32] at the end of the appendix A.2. Now, we focus on the cases where the electric baryonic charge vanishes in the eleven-dimensional solutions. We note that we can turn off the electric baryonic charge by adjusting the ratio between the size of S 2 , Σ g and CP 2 . 19 First, let us consider a case where the ratio of the size of S 2 and CP 2 in (3.2) becomes one. Then we have v 1 = v 2 , κ = 1. The metric (3.2) reduce to Here we usedS 2 instead of Σ g for κ = 1. If we change a variable as v 1 ≡ 1/ √ 1 + w 2 , then the metric and flux (2.28) have a form of where 2 = ±. They successfully reproduce the known results. See the equation (4.32) of [36] and the equation (3.25b) in [37], respectively. The electric and magnetic fluxes are computed as The entropy of this black hole is (3.44) Now, let us consider the second case where the ratio of the size of Σ g and CP 2 in (3.2) becomes one. Then we have v 2 = (9 − 4v 2 1 + 3 9 − 8v 2 1 )/(8v 1 ), κ = −1. Now let us identify v 1 ≡ √ 1 + 3w 2 /(1 + w 2 ) and we reporduce the metric and the flux as See the equation (4.29) of [36] and the equation (3.25b) in [37], respectively. 20 The fluxes become The entropy of this black holes is where 0 < w < 1/ √ 3. So far, we have considered simple cases where the electric baryonic charge vanishes. In general, we note that the uplift of dyonic four-dimensional black holes corresponds to the AdS 2 solution with dyonic baryonic charge in eleven-dimensions.

Reinterpretation of AdS black holes : Off-shell solutions
In this section, we compare the uplift formulae of dyonic AdS 4 black holes and the AdS 2 solutions in M-theory studied in [31,32]. We identify all the quantities needed for describing four-dimensional black hole solutions, i.e., the metric, scalar fields and the electric and magnetic charges, with the parameters appearing in eleven-dimensional solutions.
As we have seen in the previous section, the near horizon solutions of dyonic AdS black holes associated with M 111 are described by using two scalar fields v 1 and v 2 . On the other hand, the AdS 2 solutions in M-theory are written in terms of i , m ij as we will review in detail in the appendix A. Furthermore, there is a rescaling symmetry. Then, the number of the independent parameters are reduced from five to two, which is the same as the dimension of the solution space (v 1 , v 2 ). We note that the AdS 2 solutions in [31,32] can be treated off-shell in a sense that one can write down the metric and flux without imposing the constraint (A.24), which is required by the equation of the motion. These off-shell description have two advantages. As we mentioned, the black hole solutions in [30] are described by v 1 , v 2 for M 111 case. In this off-shell description, we have more parameters, i.e., 1 , 2 , 4 , m 12 , m 14 before we impose the constraints and consider rescaling symmetry. It enables us to reinterpret the AdS black hole solutions in a more symmetric way. In addition to this, we obtain the expression for the entropy before imposing the equation of the motion. It certainly follows the spirit of the extremization principles proposed in [7], where one imposes the conditions for supersymmetry and relaxes the equation of the motion for the fluxes.

Black holes asymptotic to AdS 4 × M 111
In this section, we study the off-shell near horizon geometries of the black holes which are asymptotically AdS 4 ×M 111 . Based on this, we will generalize to the black holes asymptotic to AdS 4 × Q 111 in the next section.

Magnetic black holes
Now let us compare the uplift formulae of the four-dimensional magnetically charged black holes, i.e., the metric (2.16) and the flux (3.10) to AdS 2 solutions with electric flux in eleven-dimensional supergravity (A.11) and (A.12). That allows us to identify the fourdimensional black hole solutions in terms of i s as The magnetic charges of the black holes are Here we have 21 This off-shell solution is eventually described by a single independent parameter among 1 , 2 , 4 . When we set 2 = 1 by rescaling and solve the constraint (A.9), we have Then the magnetic black hole solutions studied in the section 3.1 can be reproduced with the identification as The entropy of the magnetic black holes is given by (4.8) When we consider rescaling and impose the constraint as above, we can reproduce the previously known results [9,11].
Here, we provide a quite simple observation. In the study of the AdS black holes in N = 2 gauged supergravity [30], it is known that the Dirac quantization conditions give the constraints on the magnetic charges of the black hole as 22 Once we substitute the relation between the magnetic charges and the parameters i (4.4) into the second constraint above, we obtain 2 1 2 + 1 4 + 2 2 + 2 2 4 = 0, (4.10) which is nothing but the constraint (A.9) required by the equation of motion.

Dyonic black holes
As we have explained in the previous sections, the eleven-dimensional uplifts of the dyonic black holes correspond to AdS 2 solutions with non-vanishing internal four-form flux reviewed in the appendix A.2. In addition to i , new parameters m ij are introduced due to the presence of the non-trivial internal four-form flux. Here we only impose the self-duality 21 The L used in [31] is denoted as LGKW . It is related to L used in (4.5) by L 3 GKW = L 3 ( 1 + 2 2 + 4). 22 See the equation (B.14) in the appendix for Q 111 case.
conditions (A.20) and identify 2 = 3 , m 12 = m 13 for KE 4 case. Then the black hole solutions can be expressed in terms of the five parameters 1 , 2 , 4 , m 12 and m 14 . . (4.12) The electric and magnetic charges of the black holes are written as 24 (4.14) 23 The L used in [32] is denoted as LDGK . It is related to L used in (4.12) by L 3 DGK = L 3 ( 1 + 2 2 + 4). 24 Here we usedq 0 0 = 0.
In [30], the authors presented the solutions associated to M 111 in terms of two scalars (v 1 , v 2 ). Alternatively, one can also consider an off-shell solutions in terms of four scalars (v i , b i ), which satisfy two constraints (B.7), (B.8) as the authors did in Q 111 model. Using the dictionary (4.11), we note that the first constraint (B.7) corresponds to the primitive condition 2m 12 + m 14 = 0, (4.16) while the second one (B.8) is satisfied automatically.

Black holes asymptotic to AdS 4 × Q 111
So far, we have mainly focused on the asymptotically AdS 4 × M 111 black holes. However, based on the results of the previous section, we can generalize the off-shell black hole solutions to the Q 111 case by manifesting the symmetries between 1 , 2 and 3 .

Magnetic black holes
For the magnetically charged black holes, we identify as follows (4.17) where When we set 2 = 3 , we have v 2 = v 3 , we can reproduce the M 111 solution. 25 25 Note that the conventions used in the literatures [30,9]

Dyonic black holes
For the dyonically charged black holes, we have only imposed the self-dual condition on m ij (A.20) as M 111 case. Then we have The electric and magnetic charges are Here we use (i, j, k) = (1, 2, 3) in cyclic order. Note that there is no summation in the repeated indices. Upon rescaling of i and imposing the constraint (A.19), the remaining primitivity condition (A.21) reduces the number of the independent parameters from 7 to 4. It is consistent to the black hole solution analysis in [30] where it was shown that (4.23)

Page charge quantization and black hole entropy
In this section we consider the flux quantization in AdS 2 solutions of eleven-dimensional supergravity studied in [31,32], which we review in the appendix A. There are two classes of AdS 2 solutions depending on whether the internal four-form fluxes vanish or not. First let us consider the case where the internal four-form fluxes vanish (A.12). Then we integrate * 11 F 4 over the various seven-cycles Y 7 , C 1 , C 2 , which we studied in section 3.1.1, and impose the flux quantization conditions as Here, N, N 1 and N 2 are all integers. As we mentioned, the seven-cycles Y 7 , C 1 , C 2 are not independent and satisfy the relation (3.23) Now we integrate * 11 F 4 along this trivial-cycle and obtain which reduce to 2 1 2 + 1 4 + 2 2 + 2 2 4 = 0, (5.4) by using the flux quantization conditions (5.1). It is just the constraint equations on i imposed by the equation of motion (A.9). We recall that this constraint also appears as the consequence of Dirac quantization conditions of four-dimensional black hole we explained at the end of the section 4.1.1. Furthermore, when we identify n 4 , n 5 as then, (5.3) reduces to 3n 4 + 2n 5 = κ2 (g − 1), which is a twisting condition. See, for example, [8]. These magnetic fluxes, together with the components of the Reeb vector field, appear in the expression of the entropy functional studied in [8][9][10][11]. After extremizing with respect to the Reeb vector, the entropy functional for M 111 depends on the magnetic fluxes only and has a form of 26 3 (3n 4 + 2n 5 ) 18n 4 + 81n 2 4 − 72n 4 n 5 − 48n 2 5 , (5.6) with 3n 4 + 2n 5 = 2(1 − g). We substitute the fluxes (5.5) into the entropy functional (5.6) and obtain it in terms of i s. For the magnetic black holes, this expression agrees to the black hole entropy (4.8) when we set 2 = 1 by rescaling and impose the constraint 4 = −(1 + 2 1 )/(2 + 1 ) on both sides. 27 When we integrate * 11 F 4 over the seven-cycle H (3.27) associated to the Betti vector field, we have the electric baryonic charge as Then one can easily see that the electric baryonic charge vanishes when 1 = 2 or 2 = − 4 . These two cases were studied in the section 3.2.
Now, let us move on to the second class of AdS 2 solutions, which are associated with non-vanishing internal four-form fluxes (A. 16). In this case, the equation of the motion for the flux becomes The second term vanishes when there is the electric flux only. However, it becomes nontrivial when the magnetic flux is turned on. Hence, we integrate * 11 F 4 + 1 2 A 3 ∧ F 4 over various seven-cycles and calculate the so-called Page charge, which is the correct quantity to be quantized [47]. 28 Let us begin with the ansatz for the three-form potential (2.5) where Here P is the potential, which satisfies In contrast to the magnetic black hole solutions, the dyonic black holes have non-zero b i . Then, the last term in the three-form potential ansatz (5.9) gives non-trivial contributions to M 7 A 3 ∧ F 4 where M 7 = Y 7 , C 1 , C 2 . Furthermore, as explained in [49], we do not have gauge potentials which are globally well-defined. Instead, we only have gauge potentials which are well-defined on each coordinate patch and related to each other via gauge transformations. Since the gauge potential (5.9) is well-defined on Y 7 , we should find the expressions for two gauge potentials which are well-defined on C 1 , C 2 , respectively. Then we integrate * 11 F 4 + 1 2 A 3 ∧ F 4 over these seven-cycles and compute the Page charges as 29 L DGK l p where we define G 7 as Here, note that G 7 is not a differential form and does not define a cohomology class though it is closed, because it transforms under gauge transformations. Then, the charges obtained by integrating G 7 over homologous cycles are not equal. It implies that integrating G 7 over the trivial seven-cycle ( 3 π 2 κ. (5.14) Then the primitivity condition 2m 12 + m 14 = 0 enables us to define a trivial four-cycle as As we mentioned at the end of the section 4.1.2, the primitivity condition of AdS 2 solutions in M-theory corresponds to the constraint (B.7), which should be satisfied by the scalar fields of the black holes. Certainly, one can check this correspondence also holds for the uplifted four-form flux. Given the four-form flux (2.28), we integrate F 4 on this trivial cycle (5.15) and obtain v 1 (q 2 − b 2 p 0 κ) + 2v 2 (q 1 − b 1 p 0 κ) = 0. (5.16) 29 In the appendix C, we calculate the Page charges for Q 111 case in detail. Here, we obtain the result for M 111 case by identifying 2 = 3, m12 = m13 and replacing the volumes of Kähler-Einstein manifolds appropriately. One can refer to the section 7.1.4 in [45] for the connections on the various coordinate patches in M 111 .
In addition, imposing the self-dual condition on the internal part of the four-form flux (2.28), we have Using these two equations, we easily reproduce the constraint (B.7).

Black hole entropy
So far, we have calculated the entropies of magnetic and dyonic black holes for M 111 , Q 111 with the presence of the four-dimensional Newton's constant. Now the quantization conditions of Page charges lead us to express these black hole entropies in terms of the quantized integer N . Let us focus on the entropy of the dyonic AdS 4 black holes for Q 111 (4.23) as We calculate the four-dimensional Newton's constant in (D.7) Then the final expression of black hole entropy becomes In [37], the authors already considered the Page charge, but they chose a gauge where the second term in (5.13) does not contribute to the Page charge computation. As a result, they obtained the entropy depending on i only. See the equation (3.22) in [37]. However, we choose a gauge where the gauge potential is well-defined on Y 7 and obtain the non-trivial contribution from the second term in (5.13). As we can see in the equation (4.21), M is non-trivial for the dyonic black hole. Hence, our expression for the entropy of dyonic black holes (5.20) does not agree with the result of [37].

Discussion
In this paper, we have revisited dyonic AdS 4 black holes studied in [30] and uplifted their near horizon solutions to eleven-dimensional supergravity. Then, we identified the relations between AdS 4 black holes [30] and previously known AdS 2 solutions in M-theory [31,32]: the uplifts of magnetic, dyonic AdS 4 black holes correspond to the AdS 2 solutions with electric, dyonic four-form fluxes, respectively. Furthermore, comparing the uplift formulae and the AdS 2 solutions directly, we have identified the off-shell four-dimensional AdS black holes. More specifically, the quantities needed for describing AdS 4 black holes, i.e., the metric, the magnetic and electric charges and the scalars of vector-and hyper-multiplets at the horizon are determined in terms of the parameters i , m ij of AdS 2 solutions without imposing the constraint required by the equation of the motion. Hence, we have called it off-shell AdS 4 black hole solution and calculated the off-shell entropy. We also have considered the flux quantizations. For the uplift solution of the dyonic black holes, AdS 2 solutions in M-theory has non-trivial internal four-form fluxes. Then, one should consider the quantization conditions for the Page charges, which are conserved but transform under large gauge transformations. We have expressed the black hole entropy using this quantized integer.
As we mentioned in the introduction, AdS 2 solutions, which have been constructed so far, have baryonic charges. Then, it would be interesting to construct explicit AdS 2 solutions with nontrivial mesonic charges and compare the black hole entropy to the topologically twisted index of dual field theory. The study of new solutions with mesonic charges were tackled numerically in [37]. We plan to investigate it further in the near future.
Recently, the Euclidean supergravity solutions which are called black saddles were constructed in STU supergravity [50] and their on-shell action was shown to agree exactly with the topologically twisted index of ABJM theory. It is of interest to find black saddle solutions in four-dimensional gauged supergravity constructed in [29] and compare them to AdS 4 black holes constructed in [30] and revisited in this paper.
Generalizing the extremization principles along the line of [7] in several directions leads to many interesting open problems. For the dyonically charged AdS 4 black holes we have discussed, the first step in this program is generalizing the Gauntlett-Kim geometries [15] to include the transgression term [32]. There are also concrete examples of correspondence between the AdS 4 black holes in massive type IIA supergravity and dual field theories on D2-branes [51][52][53][54][55]. More interestingly, there have been intensive studies on the entropy of the rotating AdS black holes and their field theory computations in various dimensions. The generalization for rotating AdS 4 black holes was initiated in [40]. Furthermore, very recently, accelerating, rotating and dyonically charged AdS 4 black holes and their uplifts to D = 11 supergravity have been studied in [41]. 30 It would be very nice to formulate the extremization principles for these classes of black holes. developed in [31,32]. The eleven-dimensional metric ansatz is 31 Supersymmetry requires that the nine-dimensional internal space is a U (1)-fibration of eight-dimensional Kähler manifold. Here, we focus on a particular class of solution where the Kähler base is constructed from a product of Kähler-Einstein manifold S 2 or H 2 . Then the metric is given by The two-dimensional Kähler-Einstein manifold is normalized such that where KE 2(i) is T 2 , S 2 or H 2 with the scalar curvature R( KE 2(i) ) = 2. Here dP = R is the Ricci form for the eight-dimensional Kähler manifold and is given by where J i are Kähler forms of the ds 2 ( KE 2(i) ) metrics. 32 Note that the Kähler forms of eight-dimensional Kähler manifold is The warp factor is given by where R is the Ricci scalar for eight-dimensional Kähler manifold.

A.1 Electric four-form flux
First, let us consider AdS 2 solutions with the electric four-form flux as 33 In this section, we use LGKW to distinguish them from L used in (3.2). They are related as L 3 = L 3 GKW /( 1 + 2 + 3 + 4). When we are dealing with the non-trivial magnetic four-form flux, LGKW is replaced by LDGK . 32 Our normalization is different to [31] as J GKW The sign of F4 is different to that of [31]. See the equation 6.13 of that paper.
The details can be found in the section 6.3 of [31]. The eight-dimensional Ricci-scalar R satisfies the equations of the motion as Computing this equation, we obtain the constraint satisfied by i s as i.e., The solutions we are interested in this paper has Y 9 as the M 111 fibration over Σ g . Hence, we consider the eight-dimensional Kähler manifold as a product of S 2 , CP 2 and Σ g . Here, we set 2 = 3 with R(CP 2 ) = 4. Then the metic reduces to The constraint reduces to Then the eleven-dimensional metric becomes ds 2 11 = L 2 GKW e 2A ds 2 (AdS 2 ) + (dz + P ) 2 It is the equation (A.5) of [11] and (C.3) of [9].

A.2 Magnetic four-form flux
Now let us turn to the case where the non-trivial internal four-form fluxes are turned on. See the section 3.2 of [32] for details. The four-form flux is where 34 Here matrix m ij is symmetric in i, j indices and their diagonal entries are zero. When the magnetic flux is included, the equations of the motion (A.8) is generalized to The equation of motion (A.18) reduces to the constraint satisfied by the parameters i and m ij as We set 2 = 1 by rescaling and impose the remaining primitive condition. Then solving the equation of the motion (A.24) gives where m 2 ≡ 2m 2 12 + m 2 14 = 6m 2 12 . The eleven-dimensional metric becomes ds 2 11 =L 2 DGK e 2A ds 2 (AdS 2 ) + (dz + P ) 2 34 In the equation 3.21 of [32], there is a typo in F2. It should be divided by a factor 2. This typo was pointed out in [35]. See the footnote 4 in that paper.
We can easily reproduce the solutions discussed in the section 3.2. When we have 1 = 1 and introduce a new variable w such that m 2 ≡ 3 2 w 2 , the metric reduces to (3.42) for κ = 1 case. When we have 1 = 1 + 2m 2 and m 2 ≡ 3 2 w 2 , the metric reduces to (3.45) for κ = −1 case. 35 B Q 111 solutions of Halmagyi, Petrini and Zaffaroni In [30], the authors studied the AdS black hole solutions in four-dimensional N = 2 gauged supergravity coupled to vector and hypermutiplets. As we mentioned in the main text, the prepotential F and the homogeneous coordinates X I are needed to describe the vector multiplet scalar manifold. In addition to that, one also need the Killing prepotential P x Λ and the Killing vectors k u Λ for the hypermultiplet scalar manifold. The authors considered only abelian gaugings of the hypemultiplet scalar manifold and focused on a case where For the Q 111 solution we will summarize in this section, the Killing prepotential and the Killing vector are Now let us explicitly record the near horizon solutions of black holes asymptotic to AdS 4 × Q 111 studied in [30] for readers' convenience. The solutions are parameterized by the scalars of three vector multiplets (v i , b i ), whose values are constants at the horizon, with two nontrivial constraints between them. In that sense, this solution is also off-shell. Here i = 1, 2, 3. The radius of AdS 2 and the Riemann surface Σ g , and a non-trivial hypermultiplet scalar φ are . (B.3) 35 The relations between these solutions and the solutions in [36] have been studied in [37].
The electric and magnetic charges are given by 36 The polynomial σ is defined as 37 The charges p 2 , p 3 , q 2 , q 3 are similarly given by. There are two constraints, which should be satisfied by scalar fields (v i , b i ) as These constraints follow from Here L Λ is the symplectic section 36 We have corrected the typographical errors in [30] using blue color. 37 Here σ(1) implies that the index 1 follows the permutation rules given by the elements of the symmetric group S3. For example, σ(v 2 where K is the Kähler potential defined as The phase ψ is found to be fixed to π/2. The Dirac quantization conditions of this class of black holes are Then, they give some constraints on the magnetic charges as (B.14) Throughout this paper, we have chosen the upper sign. The solutions for M 111 can be obtained by identifying b 3 = b 1 , v 3 = v 1 etc.

C Page charges
In this section, we provide the computational details for the quantization of the Page charges. We follow the prescription for calculating the Page charges described in [49]. The gauge connections cannot be globally defined. They are well-defined on each coordinate patches and related to each other via gauge transformations. For simplicity, we consider AdS 2 solution in M-theory with four two-spheres, i.e. uplifting AdS 2 × S 2 near horizon solution of dyonic black holes to seven-dimensional Sasaki-Einstein manifold Q 111 . First let us identify the four seven-cycles C (i) 7 obtained by fixing a point on AdS 2 and S 2 i where i = 1, · · · , 4. We integrate * 11 F 4 + 1 2 A 3 ∧ F 4 on these cycles and compute the Page charges.
We have a globally well-defined one-form as Dψ ≡ dψ + 1 4 (P 1 + P 2 + P 3 + P 4 ) , (C.1) where dP i = J i . Let us focus on the internal part, which do not include AdS 2 part, of the four-form flux and the three-form potential obtained from the uplifting formula as Here, we observe that the gauge potential A is well-defined on C 7 , which is obtained by a U (1)-fibration over S 2 1 × S 2 2 × S 2 3 . Hence we can integrate A The first thing to do is finding three-form potential A (124) 3 , which is well-defined on the C where Then, we can easily read off the potential well-defined on C Similarly, we find the three-form potential A where In the case of the dyonic black hole, the quantity should be quantized is a Page charge as we have calculated in (C.17) For the eleven-dimensional metric in a form of (2.16), we calculate the black hole entropy from the two-dimensnional Newton's constant following the method described in the appendix A of [7]. The two-dimensional Newton's constant is related to the fourdimensional Newton's constant as where z = 4ψ, and reproduce the Bekenstein-Hawking entropy (5.20).