Spindle black holes and mass-deformed ABJM

By extending the method developed by Arav, Gauntlett, Roberts and Rosen, we construct supersymmetric $AdS_2\times{\Sigma}$ solutions of gauged $\mathcal{N}=8$ supergravity which are asymptotic to the $SU(3)\times{U}(1)$-invariant Warner fixed point, where ${\Sigma}$ is a spindle. The Warner fixed point is dual to the mass-deformed ABJM theory. The solutions are in the anti-twist class. We numerically calculate the Bekenstein-Hawking entropy of the presumed black holes with the $AdS_2\times{\Sigma}$ horizon.

The orbifold solutions with a single conical deficit angle, namely, a half spindle, is topologically a disk.The AdS 5 solutions from M5-branes wrapped on a topological disk were first constructed in [17,18], and were proposed to be the gravity dual to a class of 4d N = 2 Argyres-Douglas theories, [19].See also [20,21] for further studies.Solutions from other branes wrapped on a topological disk were constructed: D3-branes, [22,23], M2-branes, [24,9], and D4-branes, [25].The disk solutions are locally identical to spindle solutions, but originate from a different global completion.See also [26] for more disk solutions and [27] for defect solutions from the other different completion of global solutions.
A common feature of the examples we have listed is that they could asymptote to the maximally supersymmetric vacuum in their respective dimensions.Via the AdS/CFT correspondence, [35], they are dual to 3d ABJM theory, 4d N = 4 super Yang-Mills theory, 5d U Sp(2N ) gauge theory and 6d N = (2, 0) theory, respectively.However, in [13] by Arav, Gauntlett, Roberts and Rosen, it was shown that we can construct orbifold solutions asymptote to the AdS vacua with less supersymmetries: AdS 3 × Σ solution asymptotic to the N = 2 AdS 5 vacuum, [36,37,38], dual to the Leigh-Strassler SCFTs, [39], was constructed.Furthermore, holographically, central charge was computed and matched with the field theory result from the Leigh-Strassler theory on R 1,1 × Σ.
In this work, we construct supersymmetric AdS 2 × Σ solutions asymptotic to the N = 2 AdS 4 vacuum, [40,41], which is dual to the mass-deformed ABJM theory, [42,43].The holographic RG flow from the N = 8 AdS 4 vacuum dual to the ABJM theory to the N = 2 AdS 4 vacuum, [40,41], dual to the mass-deformed ABJM was constructed in [44,45] and uplifted to eleven-dimensional supergravity in [46].In light of the discovery of ABJM theory, [47], the mass-deformed ABJM theory was further understood in [42,43,48].See [49] also for the gravity calculation of threesphere free energy of mass-deformed ABJM theory.The supersymmetric black hole solutions interpolating the N = 2 AdS 4 vacuum and the horizon of AdS 2 × Σ g was constructed in [50].
We employed the U (1) 2 -invariant truncation, [50], of SO(8)-gauged N = 8 supergravity in four dimensions, [51], which is a consistent truncation of eleven-dimensional supergravity, [52], on a seven-dimensional sphere, [53].We derive the BPS equations for the AdS 2 × Σ solutions asymptotic to the N = 2 AdS 4 vacuum dual to the mass-deformed ABJM theory.We study the BPS equations and find the flux quantizations through the spindle, Σ.We find all the necessary algebraic constraints to determine the boundary conditions to construct solutions explicitly.Due to the complexity of the expressions of constraints, we cannot solve for the boundary conditions analytically.We resort to the numerical determination of boundary conditions for each choice of the spindle numbers, n N,S , t N,S , and the flavor charges, p F 1 , p F 2 .Using these boundary conditions, we explicitly construct the solutions numerically.We also numerically calculate the Bekenstein-Hawking entropy of the presumed black hole with the spindle horizon.As a check, for the case of no flavor symmetries, p F 1 = p F 2 = 0, the numerical values of Bekenstein-Hawking entropy precisely match with the one of solutions from minimal gauged supergravity, [6].
For the structure of the work, we will closely follow [13] as it is well organized and also to facilitate the comparison.
In section 2 we review the U (1) 2 -invariant truncation.In section 3 we study the BPS equations and calculate the Bekenstein-Hawking entropy.In section 4, we numerically construct the solutions.In section 5 we conclude.In appendix A we review the construction of the U (1) 2invariant truncation of gauged N = 8 supergravity.In appendix B we present the derivation of the BPS equations.
From the AdS/CFT correspondence, the free energy of pure AdS 4 with an asymptotic boundary of S 3 is given by e.g., [50], where G 4 is the four-dimensional Newton's constant.This matches with the field theory free energy of the ABJM theory and the mass-deformed ABJM theory at large N , respectively, (2.12) Furthermore, the ratio below is universal, [55], (2.13) When the complex scalar field, (χ, ψ), is vanishing, the truncation reduces to the STU model with three scalar fields and four gauge fields.By setting all gauge fields equal and all scalar fields vanish, the truncation further reduces to minimal gauged supergravity.See appendix A for details.

AdS 2 ansatz
We consider the background with the gauge fields, where ds 2 AdS 2 is a unit radius metric on AdS 2 and V , f , h, and a α , α = 0, . . ., 3, are functions of the coordinate y only.The scalar fields, χ and λ i , i = 1, . . ., 3, are also functions of the coordinate y.In order to avoid partial differential equations from the equations of motion for gauge fields, we take the scalar field, ψ, to be ψ = ψz where ψ is a constant.This brings where B z is a function of the coordinate y.
We employ an orthonormal frame, where ēa is an orthonormal frame for ds 2 AdS 2 .The frame components of the field strengths are given by The equations of motion for gauge fields are combined and integrated to give the integrals of motion, and ) where E F i and E R i are constant.Among the six integrals of motion in (3.5) and (3.6),only three of them are independent, e.g., three in (3.6), and others can be obtained by combining the three independent ones.

BPS equations
We employ the gamma matrices, where Γ m are two-dimensional gamma matrices of mostly plus signature.The spinors are given by ϵ = ψ ⊗ χ , (3.9) and the two-dimensional spinor on AdS 2 satisfies where κ = ±1 fixes the chirality.The spinor, χ, is given by where the constant, s, is the gauge dependent charge under the action of azimuthal Killing vector, ∂ z .We consider the case of sin ξ ̸ = 0.The complete BPS equations are derived in appendix B and are given by with two constraints, The scalar-field dressed field strengths are given by We have checked that the BPS equations solve the equations of motion from the Lagrangian in (2.1) as presented in appendix A.

Integrals of motion
Following the observation made in [13], we find an integral of the BPS equations, where k is a constant.Hence, at the poles of the spindle solution, h = 0, we also have sin ξ = 0. From (3.12) and (3.13), we obtain and then the two constraints in (3.13) can be written as By employing the field strengths in (3.14), we can express the integrals of motion by and

Boundary conditions for spindle solutions
We fix the metric to be in conformal gauge, and the metric is given by where we have for the metric on a spindle, Σ.For the spindle solutions, there are two poles at y = y N,S with deficit angles of 2π 1 − 1 n N,S .The azimuthal angle, z, has a period which we set ∆z = 2π . (3.23)

Analysis of the BPS equations
Following the argument in section 3.3.1 of [13] we study the BPS equations to find the spindle solutions.At the poles of the spindle solution, y = y N,S , as we have k sin ξ → 0, we find cos ξ → ±1 if k ̸ = 0. Thus, we express cos ξ N,S = (−1) t N,S with t N,S ∈ {0, 1}.We choose y N < y S and y ∈ [y N , y S ].We assume that the deficit angles at the poles are 2π 1 with n N,S ≥ 1.Then we require the metric to have From the symmetry of BPS equations in (B.32) and (3.15), we further choose Then we find (k sin ξ) ′ | N > 0 and (k sin ξ) ′ | S < 0. Hence, we impose Two different classes of spindle solutions, the twist and the anti-twist, are known, [15].Spinors have the same chirality at the poles in the twist solutions and opposite chiralities in the anti-twist solutions as cos ξ| N,S = (−1) t N,S ; Twist: (t N , t S ) = (1, 1) or (0, 0) , Anti-Twist: (t N , t S ) = (1, 0) or (0, 1) .
As we find (k sin ξ) ′ = − cos ξ (s − B z ) from the BPS equation in (3.16), we obtain (−1) l N,S +t N,S +1 . (3.27) We consider the flux quantization for R-symmetry flux.From (2.7) we have F R = dB + d 1 2 (cosh (2χ) − 1) Dψ .At the poles, as χ = 0 unless Dψ = 0, the second term on the right hand side of F R does not contribute to the flux quantization.Then, we find the R-symmetry flux quantized to be We have Once again, as χ = 0 unless Dψ = 0 at the poles, we find ∂ χ B z = 0 at the poles.From the constraint in (3.17) we also find ∂ χ W = 0 at the poles.Hence, we have We further assume that the complex scalar field, (χ, ψ), is non-vanishing at the poles and we find χ| N , χ| Thus, we find that the flux charging the complex scalar field should vanish, Note that, from (3.30) and the second condition in (3.29), we find In order to find the values of the integrals of motion, E R i in (3.18), we introduce two quantities, and note that M (1) > 0. The integrals of motion, (3.18), are functions of V , λ i and cos ξ.We can eliminate V by using the first equation in (3.17) and cos ξ by (3.26).Then we find the integrals of motion to be with Finally, we can eliminate one of the scalar fields, λ i , say λ 3 , by the condition on the left hand side of (3.32).Hence, we have three independent integrals of motion in terms of two scalar fields, λ 1 and λ 2 .As the integrals of motion have identical values at the poles, we find three algebraic equations with four unknowns, (λ 1N , λ 1S , λ 2N , λ 2S ), Unlike the case for the Leigh-Strassler theory in [13], we are one equation short to solve for the values of all scalar fields at the poles. 1 We need to find an additional constraint to determine all the values of the fields at the poles and it will be found in the analysis of fluxes in the next subsection.

Fluxes
In appendix B we have obtained the expressions of field strengths in terms of the scalar fields, warp factors, the angle, ξ, and k, where we have Thus we find that the fluxes are solely given by the data at the poles, From (2.5) we define R-symmetry and two flavor symmetry fluxes by, respectively, By (3.40) we find and where we used (3.32).Thus we recover the R-symmetry flux quantization, (3.28), and vanishing of the flux of massive vector field, (3.31), respectively, Furthermore, we define the fluxes of two flavor symmetry vectors by where p F 1 and p F 2 are integers.From these two constraints, we should be able to find expressions of k and one of the scalar fields, λ i , which was not determined in (3.36). 2ummary of the constraints to determine all the boundary conditions: Let us summarize the constraints we have obtained to determine all the boundary conditions.By solving seven associated equations, the left hand side of (3.32), (3.36), (3.44), and (3.45), we can determine the values of the scalar fields, λ 1 , λ 2 , λ 3 , at the north and south poles and also the value of the constant, k, in terms of n N,S , t N,S , p F 1 , and p F 2 .Then the values of V at the poles are determined from the definition of M (1) in (3.33).This fixes all the boundary conditions except the hyper scalar field, χ, which will be freely chosen when constructing the solutions explicitly.However, the constraint equations are quite complicated and it appears to be not easy to solve them.
Even though we are not able to solve for the boundary conditions in terms of n N,S , t N,S , p F 1 , and p F 2 analytically, if we choose numerical values of n N,S , t N,S , p F 1 , and p F 2 , the constraints can be solved to determine all the boundary conditions.For instance, in the anti-twist class, for the choice of we find the boundary conditions to be In this way, without finding analytic expression of the Bekenstein-Hawking entropy, we can determine numerical value for each choice of n N,S , t N,S , p F 1 , and p F 2 .Furthermore, we will be able to construct the solutions explicitly numerically.

The Bekenstein-Hawking entropy
The AdS 2 × Σ solution would be the horizon of a presumed black hole which asymptotes to the N = 2 AdS 4 vacuum dual to the mass-deformed ABJM theory.We calculate the Bekenstein-Hawking entropy of the presumed black hole solution.
From the AdS/CFT dictionary, (2.11) and (2.13), for the four-dimensional Newton's constant, we have, 1 Then the two-dimensional Newton's constant is given by Employing the BPS equations, we find Hence, the Bekenstein-hawking entropy is expressed by the data at the poles, where we expressed the Bekenstein-Hawking entropy in terms of M (1) .
As we can determine the numerical values of the boundary conditions for each choice of n N,S , t N,S , p F 1 , and p F 2 , we can find the numerical value of the Bekenstein-Hawking entropy as well.For instance, for the choice of (3.46), the Bekenstein-Hawking entropy is given by Furthermore, when there is no flavor charges, p F 1 = p F 2 = 0, we perform a non-trivial check that the numerical value of the Bekenstein-Hawking entropy precisely matches the value obtained from the formula given in (4.7) for the solutions from minimal gauged supergravity.

Solving the BPS equations 4.1 Analytic solutions for minimal gauged supergravity via W
In minimal gauged supergravity associated with the Warner N = 2 AdS 4 vacuum, utilizing the class of AdS 2 × Σ solutions in [6], we find solutions in the anti-twist class to the BPS equations in (3.12), (3.13) and (3.14).The scalar fields take the values at the Warner N = 2 AdS 4 vacuum, The metric and the gauge field are given by and we have Note that for the overall factor in the metric, we have L 2 AdS 4 = 2 3 √ 3g 2 for the Warner vacuum from (2.10).The quartic function is given by and the constants are We set n S > n N .For the two middle roots of q(y), y ∈ [y N , y S ], we find The Bekenstein-Hawking entropy is calculated to give where we employed (2.10) and (2.11) and F mABJM 3 N3/2 , (2.12), is the free energy of mass-deformed ABJM theory.

Numerical solutions for
In section 3.3, although we were not able to find the analytic expressions of the boundary conditions, we were able to determine the numerical values of the boundary conditions for each choice of n N,S , t N,S , p F 1 , and p F 2 .Employing these results for the boundary conditions, we can numerically construct AdS 2 × Σ solutions in the anti-twist class by solving the BPS equations. 3 In order to solve the BPS equations numerically, we start the integration at y = y N and we choose y N = 0.At the poles we have sin ξ = 0. We scan over the initial value of χ at y = y N in search of a solution for which we have sin ξ = 0 in a finite range, i.e., at y = y S .If we find compact spindle solution, our boundary conditions guarantee the fluxes to be properly quantized.
We numerically perform the Bekenstein-Hawking entropy integral in section 3.3.3and the result matches the Bekenstein-Hawking entropy in (3.51) with the numerical accuracy of order 10 −6 .We present a representative solution in figure 1 for the choice in (3.46) in the range of y = [y N , y S ] ≈ [0, 4.9433467].The scalar field, χ, takes the values, χ| N ≈ 0.455 and χ| S ≈ 0.447928, at the poles.Note that h vanishes at the poles.
There appears to be constraints on the parameter space of n N,S , t N,S , p F 1 , and p F 2 .However, without the analytic expressions of the boundary conditions, it is not easy to specify the constraints.

Conclusions
In this paper, we constructed supersymmetric AdS 2 ×Σ solutions which is presumably asymptotic to the Warner N = 2 SU (3) × U (1) R AdS 4 vacuum dual to the mass-deformed ABJM theory, where Σ is a spindle.We derived the complete BPS equations required to construct the solution.
We found the flux quantization through the spindle from the analysis of the BPS equations.We found all the necessary algebraic constraints to determine the boundary conditions to construct solutions explicitly.Due to the complexity of the constraints, even though we were not able to solve for the boundary conditions analytically, we were able to determine numerical values of the boundary conditions for each choice of n N,S , t N,S , p F 1 , and p F 2 .Employing this, we explicitly constructed the solutions in the anti-twist class numerically and calculated the numerical values of Bekenstein-Hawking entropy.
As we discussed in the introduction, from a local solution of spindle, disk solutions can be obtained by distinct global completion.In our case, it is required to have A 1 = A 2 = A 3 , A 0 = 0, and λ 1 = λ 2 = −λ 3 for the disk solutions, [9,24].However, then, from (2.5), the flavor vector fields are trivial and, from (2.9), we do not have the non-trivial vacuum dual to the mass-deformed ABJM theory.
It would be most interesting to determine the boundary conditions analytically and find analytic expression of the Bekenstein-Hawking entropy.The analytic expressions of the boundary condition would enable us to study the parameter space of the solutions and also to see if the solutions in the twist class are allowed or not.
If the analytic expression of the Bekenstein-Hawking entropy is available, it would be inter-esting to reproduce it from the field theory calculations as in [56,57] or from the gravitational blocks by generalizing the results of [11,31,32,33] beyond the GK geometry.
The scalar 56-bein in the symmetric gauge is given by where we define ϕ ijkl = 1 24 ϵ ijklmnpq ϕ mnpq , ϕ ijkl = (ϕ ijkl ) * , (A.4) which are completely antisymmetric complex self-dual scalar fields.Then the U (1) 2 -invariant ϕ ijkl are given by where we introduced the scalar fields and they corresponds to the scalar fields, (λ i , φ i ), from three N = 2 vector multiplets and, (χ r , ψ r ), from the universal hypermultiplet where i = 1, 2, 3 and r = 1, 2. They parametrize the special Kähler and quarternionic Kähler manifolds, respectively, In the following, we will restrict ourselves to the case of The bosonic Lagrangian of the truncation is given by where we define We present the equations of motion from the Lagrangian in (A.8).The Einstein equations are with the energy-momentum tensors by where X denotes a scalar field.The Maxwell equations are and the scalar field equations are given by e 2(λ 1 −λ 2 +λ 3 ) F 2 µν F 2µν + 1 4 e 2(λ 1 +λ 2 −λ 3 ) F 3 µν F 3µν = 0 , (A.27)

B Supersymmetry variations B.1 Derivation of the BPS equations
Following the arguments in [13], we derive the BPS equations required to construct the spindle solutions.We consider the background with the gauge fields, From the y direction, the spin-3/2 field variation, (A.20), reduces to and, thus, the superpotential, W , is monotonic along the BPS flow if the sign of f sin ξ does not change.
We find an integral of the BPS equations, where k is a constant.Employing this to eliminate h, we find the BPS equations to be The frame is invariant under this transformation.We fix h ≥ 0 by this symmetry in the main text.

Figure 1 :
Figure 1: A representative AdS 2 × Σ solution in the anti-twist class for n N = 8, n S = 1, p F 1 = 1 and p F 2 = 2 in the range of y = [y N , y S ] ≈ [0, 4.9433467].The metric functions, e V (Blue) and h (Orange), are in the left.The scalar fields, λ 1 (Blue), λ 2 (Orange), λ 3 (Green), and χ (Red) are in the right.Note that h vanishes at the poles.