D4-branes wrapped on a topological disk

Employing the method applied to M5-branes recently by Bah, Bonetti, Minasian and Nardoni, we study D4-branes wrapped on a disk with a non-trivial holonomy at the boundary. In $F(4)$ gauged supergravity in six dimensions, we find supersymmetric $AdS_4$ solutions and uplift the solutions to massive type IIA supergravity. We calculate the holographic free energy of dual three-dimensional superconformal field theories.


Introduction
To our understanding of the AdS/CFT correspondence, [1], topological twisting has been essential in field theory, [2,3,4], and also in supergravity, [5]. Recently, new examples of the AdS/CFT correspondence beyond the topological twisting have been proposed. The first kind of such AdS solutions are from branes wrapped on a spindle which is topologically a two-sphere with orbifold singularities at the poles. There are AdS solutions from D3-branes, [6,7,8], M2-branes, [9,10], and M5-branes, [11], wrapped on a spindle. The solutions from D3-branes were previously found in [12,13,14,15,16] in various contexts, but the interpretation as a spindle solution and the AdS/CFT correspondence are newly proposed.
The second kind of such AdS solutions are from branes wrapped on a topological disk which is a disk with non-trivial U (1) holonomies at the boundary. The AdS 5 solutions from M5-branes wrapped on a topological disk was found in [17,18]. The dual field theory was proposed to be the Argyres-Douglas theory, [19], from 6d N = (2, 0) theories on a sphere with irregular punctures. The construction was soon applied to AdS 3 solutions from D3-branes and M5-branes in [20] and AdS 2 solutions from M2-branes in [21]. See also [22] and [23] for spindle and disk solutions from D3-and M2-branes. To recapitulate, topological dics and spindle are not manifolds with constant curvature and the supersymmetry is not realized by topological twist.
In this paper, we construct the D4-D8 brane system wrapped on a topological disk. Fivedimensional superconformal field theories were first discovered in [24,25] and their gravity dual was proposed to be the D4-D8 brane system, [26], and found to be supersymmetric AdS 6 × w S 4 solution, [27], of massive type IIA supergravity, [28]. The solution is also realized as the supersymmetric fixed point, [29], of F (4) gauged supergravity in six dimensions, [30].
From topological twisting in F (4) gauged supergravity, D4-D8-brane system wrapped on supersymmetric two-and three-cycles was studied in [31,32] and also from massive type IIA supergravity in [33]. The D4-D8-brane system wrapped on a supersymmetric four-cycle provides the horizon geometry of supersymmetric AdS 6 black holes, [34], and the Bekenstein-Hawking entropy was shown to match the field theory calculation of topologically twisted index, [35,36]. See [37] also for non-supersymmetric solutions. So far, solutions were obtained from D4-D8branes wrapped on supersymmetric cycles with constant curvature.
In this paper, we study D4-D8-branes wrapped on a disk with non-trivial holonomy at the boundary. In particular, we construct supersymmetric AdS 4 solutions of F (4) gauged supergravity and uplift the solutions to massive type IIA supergravity. The dual field theories are 3d N = 1 SCFTs that arise from the twist compactification of 5d N = 1 U Sp(2N ) superconformal gauge theories, [24,25], on a topological disk. We calculate the holographic free energy of dual 3d superconformal field theories.
In section 2, we review F (4) gauged supergravity in six dimensions. In section 3, we construct supersymmetric AdS 4 solutions and uplift the solutions to massive type IIA supergravity. In section 4, we conclude and discuss some open questions. The equations of motion are relegated in appendix A. In appendix B we derive the metric of D4-D8-branes smeared over four directions.
Note added: Some time after this work, the spindle solutions from D4-branes are obtained in [38,39]. In appendix C, we show that the topological disk solution we obtain, in fact, matches the local solution of spindle in [38].

F (4) gauged supergravity in six dimensions
We review SU (2) × U (1)-gauged N = 4 supergravity in six dimensions [30]. The bosonic field content consists of the metric, g µν , a real scalar, ϕ, an SU (2) gauge field, A I µ , I = 1, 2, 3, a U (1) gauge field, A µ , and a two-form gauge potential, B µν . The fermionic field content is gravitinos, ψ µi , and dilatinos, χ i , i = 1, 2. The field strengths are defined by The bosonic Lagrangian is given by where g is the SU (2) gauge coupling constant and m is the mass of the two-form gauge potential. The supersymmetry transformations of the fermionic fields are where T I , I = 1, 2, 3, are the SU (2) left-invariant one-forms, Mostly minus signature is employed. For g > 0 and m > 0 the theory admits a supersymmetric AdS 6 fixed point when g = 3m. At the supersymmetric AdS 6 fixed point, all the fields are vanishing except the AdS 6 metric.

Supersymmetry equations
We consider the background, with the gauge fields, and the scalar field, ϕ = ϕ(r). The gamma matrices are given by where α = 0, 1, 2, 3 are four-dimensional flat indices and the hatted indices are flat indiced for the corresponding coordinates. ρ α are four-dimensional gamma matrices with {ρ α , ρ β } = 2η αβ and σ 1,2,3 are the Pauli matrices. The four-and six-dimensional chirality matrices are defined to be, respectively, The spinor is given by where ϑ is a Killing spinor on AdS 4 and η = η(r, θ). The Killing spinors satisfy where s = ±1.
The supersymmetry equations are obtained by setting the supersymmetry variations of the fermionic fields to zero. From the supersymmetry variations, we obtain where the first three and the last equations are from the spin-3/2 and spin-1/2 field variations, respectively. By multiplying suitable functions and gamma matrices and adding the last equation to the first three equations, we obtain The spinor is supposed to have a charge under the U (1) θ isometry, where n is a constant. It shows up in the supersymmetry equations in the form of where α 0 is a constant. We also define We solve the equation of motion for the gauge fields and obtain where b is a constant. Employing the expressions we discussed in (3.8) beside the second equation, we finally obtain the supersymmetry equations, The supersymmetry equations are in the form of M (i) η = 0, i = 1, 2, 3, where M (i) are three 2 × 2 matrices, as we follow [17,18], (3.14) We rearrange the matrices to introduce 2 × 2 matrices, from the column vectors of From the vanishing of A ij , B ij and C ij , necessary conditions for non-trivial solutions are obtained. From A ii = 0, we find (3.21)

Supersymmetric solutions
From the second equation of (3.18), we obtain Then, from the third equation of (3.17) with (3.22), we obtain From the third equation of (3.18), we find an expression for Also from (3.12), we find another expression for √ g 1 √ g 2 , Equating (3.24) and (3.25), we find an ordinary differential equation for A θ and it gives where C is a constant. From (3.11), we find Then, from (3.24) or (3.25), we obtain Therefore, we have determined all functions in terms of the scalar field, ϕ(r), and its derivative.
The solution satisfies all the supersymmetry equations in (3.17) to (3.21) and the equations motion which we present in appendix A. We can determine the scalar field by fixing the ambiguity in reparametrization of r due to the covariance of the supersymmetry equations, where r > 0. Finally, let us summarize the solution. The metric is given by The gauge field is given by The metric can also be written as Now we consider the range of r for regular solutions, i.e., the metric functions are positive definite and the scalar fields are real. We find regular solutions when we have where and r 1 is determined from h(r 1 ) = 0. We plot a representative solution with s = −1, b = 0.1, C = 1 and g = 3m = 3 in Figure 1. The metric on the space spanned by Σ(r, θ) in (3.30) has a topology of disk with the origin at r = r 1 and the boundary at r = 0. Near r → 0 the AdS 4 warp factor vanishes and it is a curvature singularity of the metric, This singularity is resolved when the solution is uplifted to massive type IIA supergravity. Approaching r = r 1 , the metric becomes to be where we introduced a new parametrization of coordinate, ρ 2 = r 1 − r. The function, E(b; g, m), is given by Then, the ρ-θ surface is locally an R 2 /Z l orbifold if we set , (3.40) where l = 1, 2, 3, . . . .
Employing the Gauss-Bonnet theorem, we calculate the Euler characteristic of Σ, the r-θ surface, from (3.30). The boundary at r = 0 is a geodesic and thus has vanishing geodesic curvature. The only contribution to the Euler characteristic is 41) where 0 < θ < 2π. This result is natural for a disk in an R 2 /Z l orbifold centered at r = r 1 with g = 3m.

Uplift to massive type IIA supergravity
We review the uplift formula of F (4) gauged supergravity to massive type IIA supergravity, [29]. We present the uplift formula in our conventions of [30]. 1 The non-trivial fields are the metric, the dilaton, and the four-form flux, respectively, We employ the metric and the volume form on the gauged three-sphere by 1 In contrast to [30], mostly plus signature is employed in [29]. The couplings and fields in [29] are related to the ones of [30] by where the tilded ones are of [29]. We will switch the solution to mostly plus signature, but keep following the normalization of [30].
where h I = σ I − gA I , (3.48) and σ I , I = 1, 2, 3, are the SU (2) left-invariant one-forms which satisfy A choice of the left-invariant one-forms is We also defined quantities, For our solutions, we have X = r. In particular, the metric can be written by h (s (gr 4 − 3m)) 4/3 dr 2 + where we have ∆ = r cos 2 ξ + r −3 sin 2 ξ .

Uplifted metric
The six-dimensional internal space of the uplifted metric is an S 1 θ × S 3 fibration over the 2d base space, B 2 , of (r, ξ). The 2d base space is a rectangle of (r, ξ) over [0, r 1 ) × 0, π 2 . See Figure 2. We explain the geometry of the internal space by three regions of the 2d base space, B 2 .
• Region I: The side of P 1 P 2 .
• Region II: The sides of P 2 P 3 and P 3 P 4 .
• Region III: The side of P 1 P 4 . Region I: On the side of ξ = 0, unlike the D3-, M2-, and M5-branes wrapped on a topological disk, there is no circle that shrinks.
Region II: Monopole We break (σ 3 − gA 3 ) 2 in S 3 and complete the square of dθ, [18,23], to obtain the metric of The metric functions are defined to be The function, L(r, ξ), is piecewise constant along the sides of r = r 1 and ξ = π 2 of the 2d base, B 2 , L r, π 2 = 0 , L (r 1 , ξ) = 1 CE(b; g, m) = l . (3.56) The jump in L at the corner, (r, ξ) = r 1 , π 2 , indicates the existence of a monopole source for the Dθ fibration. Due to the complexity of the expressions, we were not able to take the r → r 1 and ξ → π 2 limit to obtain the metric of the monopole.
Region III: Smeared D4-D8-branes The singularity at r → 0 in the warp factor of fivedimensional metric, (3.37), has been resolved in the uplifted metric, (3.52). On the other hand, there is a singularity at (r → 0, sin ξ → 0) and we consider this singularity. We first transform the metric from Einstein frame to string frame, ds 2 string = e Φ/2 ds 2 Einstein . Then, in the limit, the metric asymptotes to ds 2 string ≈ B 3 2/3 m 8/13 r −4/3 sin 2/3 ξ ds 2 AdS 4 + g 2 m 2 C 2 dθ 2 The metric implies the smeared D4-D8-brane sources. The D4-D8-branes are • extended along the AdS 4 and θ directions; • localized at the center of r direction; • smeared along the ξ andS 3 directions.
This matches the metric of D4-D8-branes smeared over four directions which can be obtained by following [40]. We present the derivation of metric for smeared D4-D8-branes in appendix B. 2 Lastly, we briefly present the comparison of our geometry with the geometry of wrapped M5-branes in [17,18]. The overall geometries are given by Wrapped D4-branes : where ∅ is empty and we denote the gauged coordinates with D, e.g., Dϕ. For each metric, we presented the factors in the same order so that the corresponding factors are easily found.

Flux quantization
In order to properly quantize the flux fields in massive type IIA supergravity, we rescale the fields with a positive parameter, λ, [38], dŝ 2 string = λ 2 ds 2 string , eΦ = λ 2 e Φ ,B (2) = λ 2 B (2) , where the metric is in the string frame and n = 2, 4. Under the transformation the equations of motion are invariant. As there is only one free parameter, g = m/3, and two constraints from the quantizations of F (0) and F (4) , the additional parameter, λ, is required. For the convenience of notation, we will remove the hat on the fields from now on. The quantization condition on the Romans mass, F (0) = g 3λ 3 , is given by where n 0 = 8 − N f and N f is the number of D8-branes. We consider the flux quantization condition for the four-form flux. The integral of the fourform flux over any four-cycle in the internal space is an integer, see, e.g., [33], where l s is the string length. First, we consider the F (4) ξα 1 α 2 α 3 component of the four-form flux and we obtain (3.62) where volS3 = 2π 2 and N ∈ N is the number of D4-D8-branes wrapping the two-dimensional manifold, Σ. This integration contour corresponds to the interval, Q 1 Q 2 in Figure 2.
Second, we consider the F (4) rθα 3 ξ component of the four-form flux and we obtain (3.63) Plugging C from (3.40) and l s from (3.62), we obtain where K ∈ Z is another integer. From (3.60) and (3.62), we determine (3.65) By eliminating E(b) from the constraints, (3.40) and (3.64), and with the expression of r 1 (b) in (3.36), we obtain 3 and we also find Then, by plugging r 1 (b), (3.67), in (3.40) with the expression of E(b) in (3.39), we also find another expression for b, Finally, identifying (3.66) and (3.68), we can solve for C and then for b in terms of the quantum numbers, N and K, (3.69)

Holographic free energy
Now we calculate the holographic free energy of dual 3d SCFTs. Consider the metric of the form, in the Einstein frame. The formula for holographic free energy is given in e.g., [33], where the ten-dimensional metric is in the string frame, ds 2 string = e Φ/2 ds 2 Einstein . Employing the formula, we find where we have 0 < θ < 2π, 0 < ξ < π 2 , and volS3 = 2π 2 . For the solutions in (3.35) with r min = 0 and r max = r 1 , we obtain the holographic free energy, where we used (3.69). If we set K ∼ N , the free energy scales as F ∼ N 5/2 as the free energy of 5d SCFTs in [24,25]. Even though the uplifted solutions have singularities, we obtain a well-defined finite result for free energy.

Conclusions
Employing the method applied to M5-branes recently by [17,18], we constructed supersymmetric AdS 4 solutions from D4-D8-branes wrapped on a two-dimensional manifold with non-constant curvature. We uplifted the solutions to massive type IIA supergravity and calculated the holographic free energy of dual three-dimensional superconformal field theories. The first natural question would be to identify the three-dimensional superconformal field theory which is dual to the solution and match the free energy calculated from the field theory.
In this work, we only constructed a class of AdS 4 fixed points from D4-D8-branes on a nonconstant curvature manifold. The holographic RG flow from the AdS 6 fixed point dual to 5d superconformal field theories would enable us to understand more details of the solution.
Among twist compactifications of branes and their dual field theories, D4-D8-brane system is lesser understood and we look forward to seeing development to come in the future.

Acknowledgements
We thank Chris Couzens and Hyojoong Kim for very helpful discussions. We also thank an anonymous referee for very instructive suggestions and encouragement. This research was supported by the National Research Foundation of Korea under the grant NRF-2019R1I1A1A01060811.

A The equations of motion
In this appendix, we present the equations of motion of F (4) gauged supergravity,

B Smeared D4-D8-branes
In this appendix, we derive the metric of D4-D8-brane system smeared over four directions by following appendix B of [40]. The metric describing D4-branes in the worldvolume of D8-branes were constructed in [46]. In particular, for the D4-branes extending from x 0 to x 4 and the D8-branes along all directions beside x 9 , the metric is given by where the harmonic functions satisfy The solutions are given by where r 2 = 8 i=5 (x i ) 2 is the radial coordinate in the x 5 to x 8 directions and Q 4 and Q 8 are constant.

C Equivalence with the spindle
Although the spindle and disk solutions are physically distinct, the solutions originate from different global completions of common local solutions. In this appendix, we show that the solution of topological disk we obtained in (3.30) matches the local solution in [38], by simple change of a coordinate, (C.3): by identifying the r coordinate here with the scalar field, X(y), in [38].
(C.2) It is equivalent to the reduction of matter coupled F (4) to pure F (4) gauged supergravity. Now we perform a change of coordinate by identifying the scalar field, X(y), with the coordinate, r, in (3.1), X(y) = r .

(C.3)
Note that y is the spindle coordinate of [38] and r is the disk coordinate of ours. We further make identifications of parameters, Then the solution reduces to our solution obtained in (3.30) with the parameters, This shows that our disk solution matches the local solution of spindle in [38].