M5-branes and D4-branes wrapped on a direct product of spindle and Riemann surface

We construct multi-charged $AdS_3\times{\Sigma}\times\Sigma_{\mathfrak{g}}$ and $AdS_2\times{\Sigma}\times\Sigma_{\mathfrak{g}}$ solutions from M5-branes and D4-branes wrapped on a direct product of spindle, ${\Sigma}$, and Riemann surface, $\Sigma_{\mathfrak{g}}$. Employing uplift formula, we obtain these solutions by uplifting the multi-charged $AdS_3\times{\Sigma}$ and $AdS_2\times{\Sigma}$ solutions to seven and six dimensions, respectively. We further uplift the solutions to eleven-dimensional and massive type IIA supergravity and calculate the holographic central charge and the Bekenstein-Hawking entropy, respectively. We perform the gravitational block calculations and, for the $AdS_3\times{\Sigma}\times\Sigma_{\mathfrak{g}}$ solutions, the result precisely matches the holographic central charge from the supergravity solutions.

An interesting generalization would be to find AdS solutions from branes wrapped on an orbifold of dimensions more than two.Four-dimensional orbifolds are natural place to look for such constructions and some solutions were found.First, by uplifting AdS 3 × Σ solutions, where Σ is a spindle, [6], or a disk, [21], AdS 3 × Σ × Σ g × S 4 solutions from M5-branes were obtained where Σ g is a Riemann surface of genus g.Also AdS 2 × Σ × Σ g solutions with spindle, Σ, from D4-branes were obtained, [13,12].More recently, performing and using a consistent truncation on a spindle, AdS 3 × Σ 1 ⋉ Σ 2 solutions from M5-branes wrapped on a spindle fibered over another spindle were found, [26].Also AdS 3 × Σ ⋉ Σ g solutions on a spindle fibered over Riemann surface were found, [26].
In this work, we fill in the gaps in the literature.First, we construct multi-charged AdS 3 × Σ × Σ g solutions from M5-branes.Employing the consistent truncation of [1], we obtain the solutions by uplifting the multi-charged AdS 3 × Σ solutions, [6], to seven-dimensional gauged supergravity.When the solutions are uplifted to eleven-dimensional supergravity, they precisely match the previously known AdS 3 × Σ × Σ g × S 4 solutions in [6] and [21], which were obtained by uplifting the AdS 3 × Σ solutions of five-dimensional gauged supergravity.However, it is the first time to construct the AdS 3 × Σ × Σ g solutions in seven-dimensional gauged supergravity.
Second, we construct multi-charged AdS 2 × Σ × Σ g solutions from D4-branes.Inspired by the consistent truncation in [27], we construct them by uplifting the multi-charged AdS 2 × Σ solutions, [14], to matter coupled F (4) gauged supergravity.Our solutions generalize the minimal AdS 2 ×Σ×Σ g solutions in [12] and also the solutions obtained in [13].We then uplift the solutions to massive type IIA supergravity to obtain AdS 2 × Σ × Σ g × S4 .
Finally, we perform the gravitational block calculations and, for the AdS 3 × Σ × Σ g solutions, the result precisely matches the holographic central charge obtained from the supergravity solutions.
In section 2, we construct AdS 3 × Σ × Σ g solutions from M5-branes.We uplift the solutions to eleven-dimensional supergravity and calculate the holographic central charge.In section 3, we construct AdS 2 × Σ × Σ g solutions from D4-branes.We uplift the solutions to massive type IIA supergravity and calculate the Bekenstein-Hawking entropy.In section 4, we present the gravitational block calculations.In section 5, we conclude.We present the equations of motion in appendix A and briefly review the consistent truncations of [1] in appendix B.

gauged supergravity in seven dimensions
We review U (1) 2 -gauged supergravity in seven dimensions, [28], in the conventions of [26].The bosonic field content is consist of the metric, two U (1) gauge fields, A 12 , A 34 , a three-form field, S 5 , and two scalar fields, λ 1 , λ 2 .The Lagrangian is given by where F 12 = dA 12 , F 34 = dA 34 and the scalar potential is 2) The equations of motion are presented in appendix A.

Multi-charged AdS 3 × Σ solutions
We review the AdS 3 × Σ solutions of U (1) 3 -gauged N = 2 supergravity in five dimensions, [6].These solution are obtained from D3-branes wrapped on a spindle, Σ.The metric, gauge fields and scalar fields read where I = 1, . . ., 3 and the functions are defined to be where K I and α are constant and satisfy the constraint, In the case of three distinct roots, 0 < y 1 < y 2 < y 3 , of cubic polynomial, P (y), the solution is positive and regular in y ∈ [y 1 , y 2 ].The spindle, Σ, is an orbifold, WCP 1  [n − ,n + ] , with conical deficit angles at y = y 1 , y 2 , [6].The spindle numbers, n − , n + , are arbitrary coprime positive integers.The Euler number of the spindle is given by where R Σ and vol Σ are the Ricci scalar and the volume form on the spindle.The magnetic flux through the spindle is given by and we demand p I ∈ Z.One can show that the R-symmetry flux is given by where the supersymmetry is realized by, [14], Anti-twist : (η 1 , η 2 ) = (+1, +1) , In minimal gauged supergravity, K 1 = K 2 = K 3 , only the anti-twist solutions are allowed.Otherwise, both anti-twist and twist are allowed.One can express ∆z, y 1 , y 2 , and the parameters, K I , α, in terms of the spindle numbers, n − , n + , p 1 , and p 2 , [6].The period of the coordinate, z, is given by ∆z In the special case of = X (2) , A (1) = A (2) , (2.10) expressions of y 1 , y 2 , and K 1 = K 2 are simpler, where we define q ≡ p 1 = p 2 .For the expression of α, we leave the readers to [6].For this special case, the AdS 3 × Σ solutions are also solutions of SU (2) × U (1)-gauged N = 4 supergravity in five dimensions, [29].The solutions can be uplifted to eleven-dimensional supergravity, [30], as it was done for a spindle, [6], and for a disk, [21].

Multi-charged AdS 3 × Σ × Σ g solutions
A consistent reduction of seven-dimensional maximal gauged supergravity, [31], on a Riemann surface was performed in [1].Empolying the consistent truncation, we uplift the AdS 3 × Σ solutions in section 2.2 with to U (1) 2 -gauged supergravity in seven dimension.We briefly summarize the uplift by consistent truncation in appendix B. As a result, we find the AdS 3 × Σ × Σ g solutions, where Σ g is a Riemann surface and we define and g 2 L 2 AdS 5 = 2 4/3 .The gauge coupling and the radius of asymptotic AdS 5 are fixed to be g = 2 2/3 and L AdS 5 = 1, respectively.
The flux quantization through the Riemann surface is given by where we find s 1 + s 2 = 2 (1 − g).Fluxes through the spindle are quantized to be where p 1 and p 3 are introduced in (2.6) and p i ∈ Z.The minus signs in the definition of n i are introduced for later convenience in the gravitational block calculations.By (2.7) the total flux is obtained to be where η 1 and η 2 are given in (2.8) and, thus, both twist and anti-twist solutions are allowed.

Uplift to eleven-dimensional supergravity
We review the uplift formula, [32], of U (1) 2 -gauged supergravity in seven dimensions to elevendimensional supergravity, [33], as presented in [26].The metric is given by where and L is a length scale.We employ the parametrizations of coordinates of internal four-sphere by with where The four-form flux is given by where and * 7 is a Hodge dual in seven dimensions.
We find a quantization condition of four-form flux through the internal four-sphere, where l p is the Planck length and N is the number of M5-branes wrapping Σ × Σ g .For the metric of the form, the central charge of dual two-dimensional conformal field theory is given by [34,35], and we follow [6], where the eleven-dimensional Newton's gravitational constant is G For the solutions, with (2.11), we find the holographic central charge to be where vol Σg = 4π (g − 1).This precisely matches the result obtained from the solutions by uplifting AdS 3 × Σ to eleven-dimensional supergravity, [6].

Matter coupled F (4) gauged supergravity
We review F (4) gauged supergravity, [36], coupled to a vector multiplet in six dimensions, [37,38], in the conventions of [12].The bosonic field content is consist of the metric, two U (1) gauge fields, A i , a two-form field, B, and two scalar fields, φ i , where i = 1, 2. We introduce a parametrization of the scalar fields, with The field strengths of the gauge fields and two-form field are, respectively, The action is given by where the scalar potential is and ε 012345 = +1.The norm of form fields are defined by The equations of motion are presented in appendix A.

Multi-charged AdS 2 × Σ solutions
We review the AdS 2 × Σ solutions of U (1) 4 -gauged N = 2 supergravity in four dimensions, [10,14].These solution are obtained from M2-branes wrapped on a spindle, Σ.The metric, gauge fields and scalar fields read where I = 1, . . ., 4 and the functions are defined to be H = (y + q 1 ) (y + q 2 ) (y + q 3 ) (y + q 4 ) , P = H − 4y 2 . (3.8) In the case of four distinct roots, y 0 < y 1 < y 2 < y 3 , of quartic polynomial, P (y), the solution is positive and regular in y ∈ [y 1 , y 2 ].The spindle, Σ, is an orbifold, WCP 1  [n 1 ,n 2 ] , with conical deficit angles at y = y 1 , y 2 , [10,14].The spindle numbers, n 1 , n 2 , are arbitrary coprime positive integers.The Euler number of the spindle is given by where R Σ and vol Σ are the Ricci scalar and the volume form on the spindle.The magnetic flux through the spindle is given by and we demand p I ∈ Z.One can show that the R-symmetry flux is given by where the supersymmetry is realized by, [14,15], Anti-twist : When parameters, q I , I = 1, . . ., 4, are all identical or identical in pairwise, only the anti-twist solutions are allowed.Otherwise, for all distinct or three identical with one distinct parameters, both the twist and anti-twist solutions are allowed.Unlike five-dimensional U (1) 3 -gauged supergravity which has a unique U (1) 2 subtruncation, there are two distinct U (1) 2 subtruncations from four-dimensional U (1) 4 -gauged supergravity, ST 2 model : and their permutations.

Multi-charged AdS 2 × Σ × Σ g solutions
A consistent reduction of matter coupled F (4) gauged supergravity on a Riemann surface was performed in [27].Inspired by the consistent truncation in [27], the AdS 3 × Σ × Σ g solutions in (2.13), and the minimal AdS 2 ×Σ×Σ g solutions in [12], we construct the AdS 2 ×Σ×Σ g solutions.However, only the T 3 model is obtained from the truncation of F (4) gauged supergravity and not the ST 2 model.Thus, we only find solutions by uplifting multi-charged AdS 2 × Σ solutions in section 3.2 with to six dimensions.After some trial and error we find the solutions to be where we define H = (y + q 1 ) 3 (y + q 4 ) , P = H − 4y 2 , ( and There are parameters, κ = 0, ±1, for the curvature of Riemann surface, and, z, which define

.18)
If we set q 1 = q 2 = q 3 = q 4 , it reduces to the minimal AdS 2 × Σ × Σ g solutions in [12].For our solutions, in order to satisfy the equations of motion, we find that we should choose and we find k 2 = k 4 = k −1 8 = 3.Then the solutions are given by where we have Notice that the components of F 1 on the Riemann surface is turned off by the choice of (3.19).
The flux quantization through the Riemann surface is given by where we find s 1 + s 2 = 2 (1 − g).Fluxes through the spindle are quantized to be where p 1 and p 4 are introduced in (3.10) and p i ∈ Z.By (3.11) the total flux is obtained to be and both the twist and anti-twist solutions are allowed. 1

Uplift to massive type IIA supergravity
We review the uplift formula, [39], of matter coupled F (4) gauged supergravity to massive type IIA supergravity, [40], presented in [12].Although the uplift formula is only given for vanishing of two-form field, B, in F (4) gauged supergravity, it correctly reproduces the metric, the dilaton and the internal four-sphere part of four-form flux.The metric in the string frame and the dilaton field are where the function, ∆, is defined by and the one-forms are σ i = dϕ i − gA i .The angular coordinates, ϕ 1 , ϕ 2 , have canonical periodicities of 2π.We employ the parametrization of coordinates, where 2 a=0 µ 2 a = 1 and η ∈ [0, π/2], ξ ∈ (0, π/2].The internal space is a squashed fourhemisphere which has a singularity on the boundary, ξ → 0. The four-form flux is given by where the function, U , is defined by and * 6 is a Hodge dual in six dimensions.The Romans mass is given by The positive constant, λ, is introduced from the scaling symmetry of the theory.It plays an important role to have regular solutions with proper flux quantizations, [12].The uplift formula implies m = 2g/3.The relevant part of the four-form flux for flux quantization is the component on the internal four-sphere, We impose quantization conditions on the fluxes, where l s is the string length.For the solutions, these imply that where we have n 0 = 8 − N f and N f is the number of D8-branes.These results are identical to the case of minimal AdS 2 × Σ × Σ g solutions in [12].
For the metric of the form in the string frame, the Bekenstein-Hawking entropy is by, [34,35], and in [12], For the solutions, we obtain the Bekenstein-Hawking entropy to be where the area of the horizon of black hole, multi-charged AdS 2 × Σ, in (3.7) is and y 1 and y 2 are two relevant roots of P (y).The free energy of 5d U Sp(2N ) gauge theory on S 3 × Σ g is given by, [41,42,12], By comparing (3.39) with (3.36), we find the Bekenstein-Hawking entropy to be and, for κ = −1 and z = 1, (3.19), we obtain2 Although formally the Bekenstein-Hawking entropy is in the identical expression of the one for minimal AdS 2 × Σ × Σ g solutions in [12], note that the black holes that give the area, A h , are different: it was minimal AdS 2 × Σ in [12], but now it is multi-charged AdS 2 × Σ, [14].We refer [15] for the explicit expression of A h for the multi-charged solutions.

Gravitational blocks
In this section, we briefly review the off-shell quantities from gluing gravitational blocks, [43], and show that extremization of off-shell quantity correctly reproduces the Bekenstein-Hawking entropy, central charge, and free energy, depending on the dimensionality, [12].Then apply the gravitational block calculations to the solutions we constructed in the previous sections.Depending on the dimensionality, the Bekenstein-Hawking entropy, central charge, and free energy are obtained by extremizing the off-shell quantity, [12], where F d are the gravitational blocks, [43].We also define quantities, and where σ = +1 and σ = −1 for twist and anti-twist solutions, respectively.The expressions of gravitational blocks are and the fugacities of dual field theories are normalized by The off-shell quantity can be written by where the variables satisfy the constraint, which originates from (4.6) and (4.8).

M5-branes wrapped on Σ × Σ g
For the AdS 3 × Σ × Σ g solutions, there is standard topological twist on Σ g for the magnetic charges, s i , and anti-twist on Σ for n i .Then the off-shell central charge is given by with the constraints, For the calculations, we employ Extremizing it with respect to ϵ 2 gives ϵ 2 = 0 and renaming ϵ 1 → ϵ, we find the off-shell central charge expressed by

.14)
We have started with the d = 6 gravitational blocks, F 6 , and we observe the d = 4 structure, F − 4 , naturally emerges.See section 5.2 of [12] for the calculations of d = 4 gravitational blocks.From the d = 4 point of view, the s 1 term of S(φ i , ϵ; n i , s i ) in (4.14) is the off-shell central charge for n 1 ̸ = n 2 = n 3 and the s 2 terms is for n 1 = n 3 ̸ = n 2 .Thus, extremization gives disparate results for each term.However, for the solution, as we have the solution chooses the s 1 term in the off-shell central charge.Extremizing this we find the values, .
(4.16) Then the off-shell central charge gives which precisely matches the holographic central charge from the supergravity solutions, (2.27), with σ = −1.

D4-branes wrapped on Σ × Σ g
For the AdS 2 × Σ × Σ g solutions, there is standard topological twist on Σ g for the magnetic charges, s i , and anti-twist on Σ for n i .Then the entropy function is given by with the constraints, For the calculations, we employ Extremizing it with respect to ϵ 2 gives ϵ 2 = 0 and renaming ϵ 1 → ϵ, we find the entropy function expressed by We have started with the d = 5 gravitational blocks, F 5 , and we observe the d = 3 structure naturally emerges.See section 5.1 of [12] for the calculations of d = 3 gravitational blocks.
From the d = 3 point of view, the s 1 term of S(φ i , ϵ; n i , s i ) in (4.21) is the entropy function for n 1 ̸ = n 2 = n 3 = n 4 and the s 2 terms is for n 1 = n 2 = n 3 ̸ = n 4 .Thus, extremization gives disparate results for each term.However, for the solution, as we have the solution chooses the s 1 term in the entropy function.However, in this case, the algebraic equations appearing in the extremization procedure are quite complicated and we do not pursue it further here.

Conclusions
In this work, we have constructed multi-charged AdS 3 × Σ × Σ g and AdS 2 × Σ × Σ g solutions from M5-branes and D4-branes.We have uplifted the solutions to eleven-dimensional and massive type IIA supergravity, respectively.We have also studied their spindle properties and calculated the holographic central charge and the Bekenstein-Hawking entropy, respectively.Although we have only considered the AdS 2,3 × Σ × Σ g solutions for spindle, Σ, the local form of our solutions naturally allows solutions for disk, Σ, by different global completion.However, the AdS 3 × Σ × Σ g solution for disk, Σ, was already constructed and studied in [21].Thus, it would be interesting to analyze the AdS 2 × Σ × Σ g solutions for disk, Σ, from the solutions we have constructed.
Unlike the minimal AdS 2 × Σ × Σ g solutions in [12] where z is a free parameter, only z = 1 is allowed for our multi-charged AdS 2 × Σ × Σ g solutions, (3.19).We would like to understand why it is required to fix the parameter for the solutions and if there are more general multi-charged solutions with additional parameters.

. 4 )
and the constants, b d , will be given later.The relative sign for gluing gravitational blocks in (4.1) is −σ for d = 3, 5 and − for d = 4, 6.The twist conditions on the magnetic flux through the spindle, n i , is given byd i=1 n i = n + + σn − n + n − ,(4.5)where n + and n − are the orbifold numbers of spindle and d is the rank of global symmetry group of dual field theory, i.e., d = 4 for d = 3, d = 3 for d = 4, and d = 2 for d = 5, 6.The constants are constrained by d i=1 r i = 2 , (4.6) and they parametrize the ambiguities of defining the flavor symmetries.The U (1) R-symmetry flux gives 1 2π Σ