D4-branes wrapped on a spindle

We construct supersymmetric solutions of D = 6 gauged supergravity, where is a two-dimensional orbifold known as a spindle. These uplift to solutions of massive type IIA supergravity using a general prescription, that we describe. We argue that these solutions correspond to the near-horizon limit of a system of Nf D8-branes, together with N D4-branes wrapped on a spindle, embedded as a holomorphic curve inside a Calabi-Yau three-fold. The dual field theories are d = 3, N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2 SCFTs that arise from a twisted compactification of the d = 5, N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 1 USp(2N) gauge theory. We show that the holographic free energy associated to these solutions is reproduced by extremizing an off- shell free energy, that we conjecture to arise in the large N limit of the localized partition function of the d = 5 theories on . We formulate a universal proposal for a class of off-shell free energies, whose extremization reproduces all previous results for branes wrapped on spindles, as well as on genus g Riemann surfaces Σg. We further illustrate this proposal discussing D4-branes wrapped on , for which we present a supersymmetric solution of D = 6 gauged supergravity along with the associated entropy function.


Introduction
A plethora of examples of AdS/CFT dualities have been constructed following the idea of [3] of wrapping branes on supersymmetric cycles. On the field theory side, these constructions realise supersymmetric lower-dimensional theories as "twisted" compactifications of the theories living on the branes. Here the twisting refers to the coupling of the field theory to a background R-symmetry gauge field that gets identified with a connection on the tangent bundle of the manifolds on which the theory is compactified, so that supersymmetry can be preserved simply taking constant spinors. This is referred to as a topological twist. On the gravity side, one generically expects to find supersymmetric solutions incorporating the backreaction of the large number of branes wrapped, and when the dual theory is a SCFT, the solutions will comprise an AdS factor. Focussing on compactifications on two-dimensional manifolds, these constructions have been realised for M2, D3, D4 and M5-branes wrapping constant curvature Riemann surfaces, which include the round two-sphere as the genus g = 0 case. The references presenting these solutions, along with a discussion of the field theory duals, are summarised in the first row of Table 1.
These solutions usually have been constructed in some U(1) d gauged supergravity in D = p + 2 dimensions, where p is the world-volume dimension of the brane, and then lifted to D = 10 or D = 11 supergravities, which is a necessary step in order to compare gravity computations with calculations performed in the dual field theory. An exception to this is the solution corresponding to D4-branes wrapped on Σ g , that was obtained directly in massive type IIA supergravity [2]. Below we will show that, in fact, that solution can also be obtained in a D = 6, U(1) 2 gauged supergravity and uplifted to massive type IIA, provided we take due care of the flux quantization conditions. The case of M2-branes, corresponding to supersymmetric AdS 2 × Σ g solutions, is particularly interesting, because on general grounds it corresponds to the near-horizon limit of BPS black holes in AdS 4 . In this case, one can also add rotation to the AdS 2 × S 2 solutions. The black holes are interpreted as "flows" across dimensions, with the AdS 4 conformal boundary representing the parent three-dimensional SCFT in the UV and the AdS 2 nearhorizon region corresponding to the one-dimensional IR theory. Such flows have also been constructed for higher-dimensional AdS solutions, although usually they are known only numerically.
The solution presented in [8] opened up a new, unexpected, direction of exploration in the landscape of AdS/CFT constructions. This comprises a supersymmetric AdS 3 × Σ background of minimal D = 5 gauged supergravity, where Σ = WCP 1 [n − ,n + ] is a weighted projective space, also known as a spindle. This uplifts to an AdS 3 ×M 7 solution of type IIB supergravity and it has been argued to be dual to a class of d = 4, N = 1 SCFTs compactified on the spindle with a novel type of twist, different from the topological twist, that was later dubbed "anti-twist". A similar construction, for AdS 2 ×Σ solutions of minimal D = 4 gauged supergravity, was presented in [7]. These have been later extended to spindle solutions of STU gauged supergravities in D = 5 [10,11] and D = 4 [12,13], respectively. A supersymmetric AdS 5 × Σ solution corresponding to M5-branes wrapped on the spindle was constructed in [9] and, differently from the previous constructions, it realises supersymmetry by means of a "topologically topological twist". Namely, the background R-symmetry gauge field is identified with a connection on the tangent bundle of the spindle, as for the topological twist, but the corresponding local curvatures are not equal. It turns out that these local solutions comprising spindles contain, as interesting degenerate limits, solutions corresponding to branes wrapped on disks or Riemann surfaces with non-constant curvature [13][14][15][16][17][18].
In this paper we will construct an AdS 4 × Σ solution, corresponding to D4-branes wrapped on the spindle, thus filling the outstanding entry in Table 1. We will show that our construction realises the topologically topological twist, as for the M5-brane solution in [9], with which it shares some similarities. We will first present the solution in a D = 6 gauged supergravity model and then we will discuss how to uplift this to a globally consistent solution in massive type IIA supergravity. We will elucidate the global structure of the Killing spinors, identifying the precise bundles of which they are sections and showing how they differ from the Killing spinors of the previous constructions for M2 [7] and D3-branes [8]. Our solution completes the panorama of the "basic" branes wrapped on spindles.
While for the SCFTs compactified on Riemann surfaces with the standard topological twist various supersymmetric partition functions have been computed and studied in the large N limit, for compactifications on spindles similar results are not yet available. For theories in d = 4 and d = 6 this lack of knowledge can be bypassed employing the recipe of [8] for extracting the trial central charge of the (d − 2)-dimensional theories from the anomaly polynomials of the parent theories. In d = 3 an entropy function was obtained in [19], from the on-shell gravitational action of the suitably regularised black hole solutions, employing the method of [20]. Extremizing this reproduces the entropy associated to the AdS 2 × Σ solution of [7]. An extension of this entropy function was conjectured in [12] and shown to reproduce correctly the entropy of multi-charge spindle solutions. Taking inspiration from that, in this paper we will propose a conjectural offshell free energy, whose extremization will, remarkably, reproduce the gravitational free energy associated to our solutions.
In the last part of the paper we will propose a universal class of off-shell free energies for various branes wrapped on spindles, analogous to the entropy functions, to which these reduce in d = 3. Specifically, we conjecture that for a large class of SCFTs in dimensions d = 3, 4, 5, 6, possessing large N gravity duals, when these are compactified on a spindle Σ, the exact superconformal R-symmetry of the SCFTs in dimension d − 2 is determined extremizing the following off-shell free energies where the variables ϕ i , ǫ and the magnetic fluxes n i satisfy the constraints The form (1.1) is suggested by the idea of gluing universal contributions called gravitational blocks, advocated in [21] for the entropy functions of SCFTs compactified on different manifolds. The "building blocks" are the functions F d above, which have different interpretations in the different dimensions d, being proportional to either the central charge or the sphere partition function of the SCFTs. They are also related to the prepotentials of the various gauged supergravities in dimension D = d + 1. Their precise form will be given later, see Table 2. The sign σ = ±1 labels the different twists that may occur on spindles. The sign σ = +1 corresponds to the topologically topological twist, which includes the standard topological twist as a special case, while the sign σ = −1 corresponds to the anti-twist, realised by M2 [7] and D3-branes [8]. The sign ± depends on the gluing, in the language of [21], and we shall comment below on its relation to the sign of σ. For example, in d = 3, taking n + = n − = 1 in the above formulas leads to the entropy functions for the supersymmetric black holes with AdS 2 × S 2 near-horizon geometry [21]. In this case, for σ = +1 we must take F − and this reduces to the entropy function [22] of the supersymmetric AdS 4 black holes with a topological twist [4]. On the contrary, for σ = −1 we must take F + and this reduces to the entropy function [23] for the supersymmetric rotating Kerr-Newmann AdS 4 black holes [24].
More generally, we will provide evidence that in D = 4, 6 the gluing sign ± coincides with −σ, while in D = 5, 7 they appear to be independent. In D = 4, the fact that −σ coincides with the sign ± may be understood as follows. The AdS/CFT correspondence implies that, in the large N limit, the free energies F ± should be identified with the appropriately regularised gravitational on-shell action of the dual supergravity solutions. In [25] it has been proved, in the context of minimal gauged supergravity, that the onshell action of any (Euclidean) supersymmetric solution takes the form of a sum over contributions from fixed points of the canonical Killing vector field, defined as a bilinear in the Killing spinors of the solution. The relative sign of these contributions is determined by the chirality of the Killing spinors at the fixed points, and in all the known supergravity solutions comprising spindle (including S 2 as a special case) we have that the chiralities at the north and south poles of the spindles are the same for the topologically topological twist and opposite for the anti-twist. A general proof of this fact is given in [26].
Our proposal reproduces all the previously known results for AdS × Σ solutions, including the AdS × S 2 solutions as special cases 1 . Moreover, in d = 5, taking σ = +1, corresponding to the topologically topological twist, we will show that the extremization of the function F − (ϕ i , ǫ; n i , n + , n − , +1) in (1.1) precisely reproduces the gravitational S 3 free energy of our solution (see eq. (3.71)). We then conjecture that this should arise in the large N limit of the localized partition function on S 3 × Σ, with the topologically topological twist. To add weight to our proposal, we will also discuss D4-branes wrapped on the four-dimensional orbifold Σ × Σ g . The effective field theory obtained from the twisted compactification of the d = 5 SCFT is expected to be superconformal, at least in some ranges of the magnetic fluxes. We will discuss the corresponding supersymmetric AdS 2 ×Σ×Σ g solutions of D = 6 gauged supergravity and show that the entropy function constructed from the "spindly" gravitational blocks correctly reproduces the geometric entropy.
The rest of the paper is organised as follows. In section 2 we discuss the uplift of solutions of a D = 6, U(1) 2 gauged supergravity model to massive type IIA. As a warmup, we illustrate this obtaining the (global) solutions of [27] and [2] from known solutions in D = 6. In section 3 we construct new supersymmetric AdS 4 × Σ solutions and discuss global properties of these both in D = 6 and D = 10. In section 4 we discuss aspects of the field theory duals of these solutions. In particular, we conjecture a field-theoretic large N off-shell free energy and show that extremizing this reproduces the holographic free energy associated to our solutions. In section 5 we discuss how our proposal fits in a general scheme of off-shell free energies for field theories compactified on spindles (as well as on genus g Riemann surfaces), comprising and extending entropy functions and trial central charges, previously discussed in the literature. In section 6 we begin investigating D4-branes wrapped on four-dimensional orbifolds, focussing on a class of supersymmetric AdS 2 × Σ × Σ g solutions. We conclude with a discussion in section 7. Appendix A contains technical details useful for comparing known solutions, in different conventions. In appendix B we demonstrate how the Killing spinors of the AdS 4 × Σ solutions encapsulate the OSp(2|4) superalgebra of the dual SCFTs.
2 Uplift of D = 6 solutions to massive type IIA In this paper we discuss solutions of a D = 6 gauged supergravity with gauge group U(1) 2 , comprising two gauge fields A 1 , A 2 , a two-form B and two real scalar fields ϕ = (ϕ 1 , ϕ 2 ). These can be uplifted locally to solutions of massive type IIA supergravity by means of the consistent truncation formulas presented in [28]. However, we will see that globally the solutions uplifted through this ansatz are incompatible with quantization of the fluxes and need to be supplemented by an additional parameter that arises in D = 10.

The D = 6 gauged supergravity
The D = 6 supergravity model of interest can also be obtained as a sub-sector of an extension of Romans F (4) gauged supergravity [29], coupled to three vector multiplets [30]. The bosonic part of the action reads 2 where F i = dA i and the scalar fields ϕ are parameterised as The scalar potential is with g the gauge coupling and m the mass parameter, and where for later convenience we defined X 0 = (X 1 X 2 ) −3/2 . Here we consistently set B = 0 because in the first part of the paper we will restrict to configurations with F 1 ∧ F 2 = 0. We will restore the two-form B in section 6.1. It is worth mentioning that locally the ratio m/g can be set to any non-zero value rescaling the scalar fields X i and the field strengths F i . In particular, m can be absorbed in the coupling constant transforming and defining the new gauge coupling asg = mg 3 l 1/4 , with l a positive constant. The action (2.1) keeps the same form, with the scalar potential becoming However, for the time being we will keep both parameters m and g. A solution to the equations of motion of the model is supersymmetric if and only if it satisfies also the following set of Killing spinor equations [30]: These follow from setting to zero the supersymmetry variations of the fermionic fields of the theory with three vector multiplets [30], that do not vanish automatically in the sub-truncation that we are considering. Here (σ 3 ) A B is the usual third Pauli matrix and {Γ a , Γ b } = 2η ab . The SU(2) indices A, B are raised and lowered as ǫ A = ε AB ǫ B and ǫ A = ǫ B ε BA , where ε AB = −ε BA and its inverse matrix ε AB is defined such that ε AB ε AC = δ B C . The supersymmetry parameter ǫ A is an eight-component symplectic-Majorana spinor, hence it satisfies the condition where B 6 is related to the six-dimensional charge conjugation matrix C 6 by B 6 = i C 6 Γ 0 .

Improved uplift to massive type IIA
Any solution to the equations of motion of this theory can be embedded in massive type IIA supergravity by means of the dimensional reduction ansatz presented in [28] 3 , provided the gauge coupling and mass parameter are related as m = 2g/3. The metric in the string frame and the dilaton are given by where ds 2 6 is the six-dimensional metric and we defined the one-forms σ i ≡ dφ i − gA i . The angular coordinates φ 1 , φ 2 have canonical 2π periodicities, and the warp factor is The coordinates µ a , with a = 0, 1, 2, satisfy the constraint µ 2 a = 1, which can be solved for example defining µ 0 = sin ξ , µ 1 = cos ξ sin η , µ 2 = cos ξ cos η , (2.14) and taking η ∈ [0, π/2], ξ ∈ (0, π/2], where the range of ξ arises from the necessity of having µ 0 > 0. At any point in the six-dimensional space-time, the metric inside the square brackets in (2.11) parameterises a four-dimensional hemisphere, that we will denote by S 4 . This metric is in general squashed and it reduces to the metric on "half the round four-sphere" when X 1 = X 2 = 1. The only non-vanishing fields of the RR sector are the ten-dimensional Romans mass and the four-form flux. This is conveniently written in terms of its Hodge dual as where vol(M 6 ) is the volume form of the six-dimensional space and The Hodge star operator ⋆ 10 in (2.16) is computed using the string frame metric (2.11), while ⋆ 6 is defined using the six-dimensional metric ds 2 6 . Provided the equations of motion of the six-dimensional supergravity hold, the above field configuration solves the equations of motion of massive type IIA supergravity, whose action in the string frame reads 4 , (2.18) 4 Here B n (2) denotes the wedge product of B (2) with itself n times, divided by n!.
where the field strengths are defined from the NS two-form B (2) and the RR potentials C (1) and C (3) as (2.19) It will be important to notice that the equations of motion of massive type IIA are invariant if the fields are transformed as with n = 2, 4, where λ is any strictly positive constant. However, this scaling symmetry holds only at the classical level in supergravity and it is broken upon imposing the Dirac quantization conditions on the fluxes. As we will discuss momentarily, this additional parameter will be crucial for ensuring that six-dimensional solutions yield globally regular solutions in D = 10, in particular that the fluxes are correctly quantized. Notice that the reduction ansatz of [28] applies only after setting m = 2g/3 in the six-dimensional theory and it implies that the Romans mass of the ten-dimensional theory is fixed in terms of the gauge coupling constant g as in (2.15). It is natural to suspect that there may exist a more general truncation ansatz that relates the six-dimensional mass parameter m to the ten-dimensional parameter λ, so that the Romans mass F (0) is an independent parameter. This would be a mechanism analogous to the ten-dimensional origin of dyonic four-dimensional supergravity discussed in [32]. We leave this interesting question for the future and proceed to discuss different globally regular solutions of massive type IIA originating in D = 6.
In summary, our strategy will be as follows. We construct the solutions in D = 6 and after setting m = 2g/3 we uplift these to local solutions in D = 10, using the formulas in [28]. Then we introduce the parameter λ and proceed to quantize the fluxes, finding globally consistent solutions of massive type IIA supergravity.

The AdS 6 solution and its uplift
The equations of motion following from the action (2.1) admit the well-known supersymmetric vacuum with constant scalars X 1 = X 2 = 3m 2g 1/4 , vanishing gauge fields A 1 = A 2 = 0 and metric where ds 2 AdS 6 is the metric on AdS 6 with unit radius. We now set m = 2g/3 and uplift this solution to massive type IIA using the formulas in [28]. After introducing the parameter λ using the local scaling symmetry (2.20), we obtain ds 2 s.f. = λ 2 (sin ξ) −1/3 L 2 AdS 6 ds 2 AdS 6 + 4 9 dξ 2 + cos 2 ξ ds 2 S 3 , e Φ = λ 2 (sin ξ) −5/6 , where ds 2 S 3 denotes the metric on a unit radius round three-sphere ds 2 S 3 = dη 2 + sin 2 η dφ 2 1 + cos 2 η dφ 2 2 , (2.23) and vol(S 3 ) its associated volume form. The quantization conditions of the (non-zero) fluxes in massive type IIA read where ℓ s is the string length. In the solution (2.22) these imply that 5 It is clear from the second equation that setting λ = 1 leads to an inconsistent relation between the integers N and n 0 . This problem arises because without introducing λ there is only one free dimensionless parameter (gℓ s ) and two conditions to impose. Thus the scaling symmetry (2.20) plays a crucial role in making the uplifted solution globally consistent. After imposing (2.25) the uplifted solution (2.22) can be matched with the solution of [27] identifying with n 0 = 8 − N f , where N f is the number of D8-branes, and Q 4 is related to the number of D4-branes N by Note that the constant C is a trivial parameter which can be set to any non-zero value redefining Q 4 → C 2/3 Q 4 . Although both the six-dimensional vacuum and the tendimensional solution of [27] are well-known, clarifying their relationship will allow us to discuss the six-dimensional origin of more interesting solutions in the following. As discussed in [33], the effective six-dimensional Newton constant that should be proportional to the large N limit of the S 5 free energy of the dual field theory is divergent, due to the singularity of the ten-dimensional solution on the boundary of the hemisphere S 4 , i.e. for ξ → 0. This problem was circumvented in [33] by calculating the holographic entanglement entropy across a three-sphere and then extracting from this the free energy F S 5 . We refer the reader to [33] for the details and here, for completeness, we only quote the result

The AdS 4 × Σ g solutions and their uplift
In this section we will discuss a class of supersymmetric solutions of the D = 6, U(1) 2 supergravity comprising an AdS 4 factor, that were constructed in [1] (see also [31]). We will use these to illustrate our procedure for uplifting to solutions of massive type IIA supergravity, showing that the uplifted solutions coincide with the solutions constructed in [2], by directly solving the supersymmetry conditions of [34] in ten dimensions. The solutions have the form of a product AdS 4 × Σ g , where Σ g is a genus g Riemann surface equipped with a constant curvature metric. As usual, we can distinguish three cases for the curvature κ = ±1, 0. When κ = +1 we have the round two-sphere with g = 0, while for κ = −1 locally we have the metric on the two-dimensional hyperbolic space H 2 , which can be quotiented to obtain a constant curvature Riemann surface with genus g > 1. In order to encompass both non-zero curvature cases 6 we denote by ds 2 Σg the metric on Σ g and define the one-form ω g such that dω g = vol(Σ g ). Explicitly, we can take In our conventions, the metric, scalars and gauge potentials take the form ds 2 6 = L 2 AdS 4 ds 2 AdS 4 + e 2G ds 2 Σg , where with p 1 , p 2 two constant parameters obeying the supersymmetry constraint 7 The fluxes of the gauge fields through Σ g are given by 8 where the quantization condition above arises from the requirement that gA i be welldefined connection one-forms on U(1) bundles over Σ g . The six-dimensional solution is therefore specified by the genus g and one integer, say p 1 (1 − g).
We now set m = 2g/3 and, after uplifting to massive type IIA using the formulas in [28], we introduce the parameter λ. The ten-dimensional metric and dilaton are while the Romans mass and four-form flux read We have σ i = dφ i − p i 2 ω g and we defined We have checked that the above configuration satisfies the ten-dimensional equations of motion and that dF (4) = 0. Imposing flux quantization (2.24) we obtain the relations (2.25), exactly as for the vacuum solution, showing that λ = 1 would be again inconsistent. We have therefore obtained a globally regular (modulo the ever-present singularity at ξ = 0) ten-dimensional solution with an AdS 4 factor, parameterised by the genus g and the integers n 0 , N, p 1 (1 − g). This is precisely the solution presented in [2], as we show in detail in appendix A.2.
Writing the ten-dimensional metric (2.34) in the form we have that ds 2 M 6 is the metric on the internal space M 6 , that is the total space of an S 4 bundle over Σ g , namely Recall that S 4 is a hemisphere of the four-sphere due to the fact that the warp factor is singular at µ 0 → 0, corresponding to the location of the O8-plane [27]. However, we can also think of the internal geometry as an S 4 bundle over Σ g , before the inclusion of the O8-plane. Either way, there is a U(1) × U(1) ⊂ SU(2) R × SU(2) F symmetry acting on S 4 and the gauge fields gA i are connections on the associated circle bundles, twisting these over Σ g , with Chern numbers P i . The solution can be interpreted as follows [2]: one starts with a geometry of the type R 1,2 × R × Y 6 , with Y 6 a local Calabi-Yau three-fold of the form where −gA i are Hermitian connections on the line bundles O(−p i ). Then (2.32) implies that the total space of Y 6 has vanishing first Chern class and hence is a Calabi-Yau threefold. At the origin of R there are an O8-plane and N f = 8 − n 0 coincident D8-branes, and in addition one wraps N D4-branes over the zero section Σ g of Y 6 . At low energies the effective theory on the D4/D8-system will be a d = 3, N = 2 field theory, obtained from the compactification on Σ g of the d = 5, N = 1 SCFT with gauge group USp(2N) [27], with the standard topological twist. The supergravity solution above strongly suggests that in the large N limit this is a SCFT and its S 3 free energy can be computed holographically, from the full ten-dimensional solution. Specifically, this can be obtained using the formula presented in [32] adapted to the string frame, and it reads [2] where the parameter z is related to our parameters as p 1 = κ + z, p 2 = κ − z. This expression has been reproduced exactly by a direct field theory computation, using the large N expansion of the localized partition function on S 3 × Σ g [35,36].
where ds 2 AdS 4 denotes the unit radius metric on AdS 4 and q 1 , q 2 are two real parameters. The real constants α i are pure gauge and we have included them as they will play a crucial role for understanding the global properties of the solution. These backgrounds can be obtained by doing an analytic continuation [37] of a class of six-dimensional BPS black holes [38]. A curvature singularity lies at y = 0, hence without loss of generality, in the subsequent analysis we will make sure that the globally regular solutions will be restricted to y > 0.
Before turning attention to the global structure of the solutions, we will demonstrate that they are supersymmetric by constructing the local form of the Killing spinors solving the equations (2.6) -(2.8). We employ the following orthonormal frame whereêâ,â = 0, . . . , 3, is the vierbein on AdS 4 , whose coordinates are denoted as xμ. Equation (2.6) then splits in the following equations where we definedh ≡ log(h 1 h 2 ). Equations (2.7) and (2.8) yield the same constraint From the third equation in (3.3) we see immediately that setting leads to Killing spinors independent of z and we will adopt this choice in the reminder of this section. We consider the specific decomposition of the gamma matrices where γâ are the (Lorentzian) gamma matrices in D = 4, γ 5 = i γ 0 γ 1 γ 2 γ 3 is the related chiral matrix and ρî,î = 1, 2, are the (Euclidean) gamma matrices in D = 2. For ρî we choose the following representation and we take B 2 = −σ 2 so that B 6 = B 4 ⊗ B 2 . The ansatz for the symplectic-Majorana Killing spinors ǫ A is where ϑ = ϑ(xμ) is a Majorana Killing spinor on AdS 4 and ϑ ± are its chiral components, i.e. γ 5 ϑ ± = ±ϑ ± . Thus we have ϑ * ± = B 4 ϑ ∓ and∇μϑ ± = 1 2 γμϑ ∓ . The spinors η A ± = η A ± (y) are two-component Dirac spinors defined on the spindle 9 .
Adopting the decomposition (3.6), the symplectic-Majorana condition (2.10) and the Killing spinor equations (3.3), (3.4) can be solved to give where ξ is a complex constant and we have defined 10) 9 Notice that η A ± are not chiral spinors, despite the notation suggests otherwise. The index A = 1, 2 is an internal index and we will see below that these four Dirac spinors are actually not all independent.
which satisfy F (y) = f 1 (y)f 2 (y). Notice that the four two-dimensional spinors above can be expressed in terms of just one of them, say η 1 + , by means of the relations Notice also that all the spinors never vanish, as it can be seen from their norm, given by We can now count the number of supersymmetries preserved by our AdS 4 ×Σ solution. ϑ is a Majorana spinor, hence it has four real degrees of freedom, while the spinors η A ± are fully determined by the complex constant ξ. Therefore, there are eight real independent Killing spinors, that is half the number of supersymmetries of the six-dimensional N = (1, 1) theory, hence the solution is 1/2-BPS. The eight Killing spinors correspond to four Poincaré supercharges Q and four superconformal supercharges S in the d = 3, N = 2 SCFTs. The precise identification of the spinors with the supercharges is discussed in appendix B. In particular, we show that ∂ z is part of the superconformal R-symmetry, namely the U(1) generator in the OSp(2|4) superalgebra.

Global analysis I: metric and magnetic fluxes
From now on we set m = 2g/3 without loss of generality. In order to have a well-defined metric on the spindle Σ, and positive scalars X i we need to take F > 0, h 1 > 0, h 2 > 0 in a closed interval not containing the curvature singularity in y = 0, thus without loss of generality we restrict to y > 0. Taking a look at the explicit form of F and its first derivative (3.14) from the expression of F ′ we see that there exist at most three turning points, hence at most four distinct real roots of F . Since the coefficient of y 6 in F (y) = 0 is positive and we restricted to y > 0, we need at least three positive roots and, to this end, Descartes' rule of signs implies the necessary conditions Without loss of generality we take q 1 > 0, q 2 < 0. Recalling that we have h 1 h 2 > 0 when F > 0. Moreover, for positive y and q 1 , h 1 > 0 and, accordingly, h 2 > 0 as well.
The conditions for Σ to be a spindle are obtained studying ds 2 Σ in the neighbourhood of the zeros of F . Denoting [y N , y S ] the range of the coordinate y, as the latter approaches one of the end-points of this interval, say y α , the metric becomes where we defined ̺ 2 = |y − y α |. As a consequence, ds 2 Σ is a smooth orbifold metric on the spindle if the following conditions hold where the minus sign in the second relation is due to the fact that F ′ (y S ) < 0. Here n ± are two co-prime integers and ∆z is the periodicity of the z coordinate. The Euler characteristic of metric (3.13) can be computed noticing that We then find where we employed F (y α ) = 0 and the following identity: (3.21) We now proceed to the quantization conditions for the magnetic fluxes of our AdS 4 ×Σ solution across the spindle. The integrated fluxes of F i are given by where the quantization of the p i arises from the requirement that gA i be well-defined connection one-forms on U(1) bundles over Σ (c.f. appendix A of [7]). In particular, the total flux reads where we made used the relation (3.21) and F (y α ) = 0. From (3.23) it follows that and therefore the two integers p i can be conveniently parameterised as where z is an appropriate rational number 10 . The situation is analogous to that obtained for M5-branes wrapped on the spindle [9] as here we also see that the "topologically topological twist" is realised by the solution, in contrast to the "anti-twist" that was first encountered for D3-branes and M2-branes wrapped on spindles. In [26] it is shown that very generally on the spindle only these two types of twists are possible. As explained in this reference, the occurrence of the twist case in our solution is correlated with the behavior of the Killing spinors at the north and south poles of the spindle, which we will discuss in section 3.4.

Solution of the regularity conditions
In this subsection we elaborate on the conditions worked out in the global analysis above. Specifically, we will aim to derive expressions for y N , y S and ∆z in terms of the spindle parameters n ± and the flux parameter p 1 (or z). Imposing F (y α ) = 0 we obtain the sum and product of q 1 and q 2 in terms of the roots y N and y S as Plugging the first of the equations (3.26) into (3.27) and changing variables as we obtain the equation where we defined µ ± ≡ n + ± n − and µ ≡ µ − /µ + . From 0 < y N < y S it immediately follows that w > 0 and 0 < x < 1. This quadratic equation for w can be easily solved, giving the following expressions for the two roots in terms of the parameter µ and the new variable x (3.31) 10 We must take z < −1 to ensure that p 1 < 0 , p 2 > 0, as follows from (3.22) and the signs of q i . Moreover z must be chosen such that p 1 and p 2 are integers Notice that in order for y N , y S to be real, there are two alternative set of conditions, namely Below we will turn attention to the variable x and we will check which of these two conditions hold. From (3.22) and (3.18) we obtain the following useful relations Extracting ∆z from (3.34) and inserting it into (3.35), along with the expression for the roots y N , y S from (3.31), we obtain Recalling that x > 0, in order for this equation to have a real solution we need η = +1, hence any solution will have to satisfy the conditions (3.32). In particular, being x positive by construction, we need µ > 0 as well, hence µ − > 0 and n − < n + . Bearing these existence conditions in mind and recalling that p 1,2 = µ + (1±z)/2, from (3.36) we find that x must be a root of the following quartic equation From (3.35) we can then obtain the periodicity of z, namely and from (3.26) we can express q 1 and q 2 in terms of x as All the relevant quantities are now given in terms of a solution x of equation (3.37), with 0 < x < 1 and meeting the constraints (3.32). We are left with the analysis of the existence and uniqueness of such solution. Focussing on the second condition of (3.32), namely x < µ, we know that, by definition, µ < 1, thus this constraint reduces the range of existence of x to (0, µ). Recalling that z 2 > 1 and µ 2 < 1, we have Since P (x) is continuous, there must exist at least one zero in the interval (0, µ). Let us now prove by contradiction that the first condition of (3.32) holds. Assuming that multiplying this by the positive quantity x, and using equation (3.37) to get rid of the quartic term, we obtain (3.43) in contradiction with the starting hypothesis, which must be false. In this way, we showed that there exists at least one real solution to equation (3.37), lying inside the range 0 < x < 1.
We shall now prove that this root is unique inside the interval (0, µ). The first derivative of P (x) reads is strictly increasing and can thus have only one root inside the considered range. In the other case, we can focus on the zeros of the polynomial, multiply the condition P ′ (x) > 0 by x and remember that, at these points, P (x) = 0, ending up with This inequality is true since 8z 2 − 3 − 9µ 2 < 0 for hypothesis and µ − x > 0. Hence, in every zero of P (x) in the range (0, µ) the polynomial must be increasing, but being P (x) continuous the root must be unique.

Global analysis II: gauge fields and Killing spinors
We shall now discuss global properties of the Killing spinors and the gauge fields A i , following closely the exposition in [7]. Recall that the Killing spinors that we wrote in section 3.1 were obtained in the frame (3.2) and with the gauge fields in the gauge given by the expressions (3.1), subject to 46) which was motivated by the fact in this gauge they are independent of z. However, both the frame and the gauge fields are singular at the north and south poles of the spindle, where the azimuthal coordinate z is ill-defined. In order to shed light on the global properties of the spinors and of the gauge fields, we shall therefore cover the spindle by two patches U ± as usual, and check that spinors and gauge fields can be correctly glued across the equator, where they overlap, identifying the correct bundles of which they are sections and on which they are connections, respectively. We begin introducing the angular coordinate ϕ defined as with canonical 2π periodicity, so that the two gauge potentials in (3.1) read and we define the R-symmetry gauge field A R ≡ g(A 1 + A 2 ). The open sets U ± cover the two hemispheres containing the south and north poles, respectively. Specifically, we have y S ∈ U + ≃ R/Z n + and y N ∈ U − ≃ R 2 /Z n − . In these two patches independently we can perform the gauge transformations so that the transformed gauge fields A ± i are non-singular in their respective patches provided that where s 1 = +1, s 2 = −1. We then have that A − i | y=y N = 0 and A + i | y=y S = 0, implying that both gauge fields are non-singular at the poles, as required. The corresponding gauge transformations for the R-symmetry gauge field A R are given by so that on the overlap U − ∩ U + the gauge fields transform as meaning that gA i are connections on O(p i ) bundles and A R is a connection on the O(n + + n − ) bundle (which is the tangent bundle) on the spindle, respectively. From the covariant derivative (2.9) we see that the R-symmetry charge of ǫ 1 is 1/2, while that of ǫ 2 is −1/2. This implies that a gauge transformation A R → A R + Λ R dϕ acts on the Killing spinors as ǫ 1 → e iϕΛ R /2 ǫ 1 and ǫ 2 → e −iϕΛ R /2 ǫ 2 , with the spinors η 1 ± and η 2 ± behaving accordingly. We now move to the analysis of the global structure of the Killing spinors. Since the frame spanning the spindle {e 4 , e 5 } in (3.2) is again singular at the poles, we consider two distinct local frames in each of the two patches. We define ̺ ± the geodesic distance between y and each root contained in U ± , y S and y N respectively. We can thus write (cf. (3.17)) where the sign of e 4 in U + is due to the fact that approaching y S the coordinate y is increasing, while ̺ + is decreasing. In the patch U − we introduce the complex coordinate z − = ̺ − e iϕ/n − = x − + i y − , which is non-singular in y N and, thus, defines a smooth oneform dz − on the orbifold. This one-form, in turn, determines a non-singular frame, that can be obtained rotating the initial frame as follows: This is an SO(2) ∼ = U(1) rotation of the frame in the patch U − , which induces a transformation of the spinors given by the action of the exponential of the spinor representation of the infinitesimal version of the SO(2) frame rotation. Explicitly, this is a U(1) rotation of the components of the spinors by means of the matrix Performing in U − the frame rotation (3.54) and the gauge transformation (3.49) with gauge parameter Λ − R (3.51), the spinors undergo an R-symmetry rotation plus a U(1) rotation. The total action on, e.g., η 1 + is (3.56) The ϕ coordinate is not well-defined in y N , however since f 2 (y N ) = 0, the transformed spinor is smooth and well-defined at this pole of the spindle and, thus, in the whole patch U − . Of course the same is true for the spinors η 1 − and η 2 ± . A similar analysis can be performed in the patch U + defining the non-singular coordinate z + = −̺ + e −iϕ/n + , related to the initial frame by the rotation This transformation is analogous to the frame rotation in U − , but is performed in the opposite direction, and the same happens to the spinors. The corresponding spinor rotation and gauge transformation combine so that the spinor η 1 + transforms as 58) giving, again, a well-defined spinor in U + .
In conclusion, we have shown that the Killing spinors on the spindle are smooth and well-defined, in the appropriate orbifold sense, in line with all the previous constructions of supersymmetric solutions involving spindles. The spinor transition function in going from the path U + to the patch U − reads where the sign of the rotation in U + is reversed because we started with a non-singular spinor in the patch U + . This identifies the positive and negative chirality spin bundles S (±) on the spindle as the bundles O(∓ 1 2 (n + +n − )), respectively. Recalling that our spinors are also charged under A R and therefore they are sections of the bundles O(±(n + + n − )) 1/2 = O(± 1 2 (n + + n − )), we conclude that, for example as it was indeed obvious from the explicit transition functions obtained from passing from the expression in (3.56) to that in (3.58). Notice that all the Killing spinors have definite chirality at the north and south poles of the spindle, and this is the same at both poles. For example, at the poles the spinor η 1 which have both positive chirality. The other spinors behave similarly. As discussed in [26], this behaviour is indeed consistent with having a global topological twist.

Uplift to massive type IIA and holographic free energy
By means of the reduction ansatz of [28] we can uplift our six-dimensional AdS 4 × Σ background (3.1) to massive type IIA and subsequently introduce λ as in (2.20). The metric and dilaton read the Romans mass is while the four-form flux takes the form For convenience, we defined the functions The quantization of the fluxes proceeds as in (2.24) and it yields again the relations (2.25), which fix the parameters g and λ in terms of the integers N and n 0 . Therefore our ten-dimensional solution is characterised by five integers: the pair n 0 and N, determining the dual five-dimensional theory, n ± defining the spindle, and z related to the magnetic charges threading this.
The ten-dimensional geometry has a form analogous to that of the solutions in section 2.4. The internal six-dimensional space M 6 has a fibration structure with the twisting of the bundle specified by the connection one-forms gA i with Chern numbers The solution can then be interpreted as follows: one starts with a geometry of the type At low energies the effective theory on the D4/D8-system will be a d = 3, N = 2 field theory, obtained from the compactification on Σ of the d = 5, N = 1 SCFT with gauge group USp(2N) [27], with the "topologically topological twist". The supergravity solution above strongly suggests that in the large N limit this is a SCFT and its S 3 free energy can be computed holographically as before [32]. Specifically, we have Notice that the dependence of x on the parameter z and µ could be made explicit by writing out the solution to the quartic (3.37), however this is extremely cumbersome and we will refrain from doing so. Alternatively, one could think of x and µ as the two independent parameters, with z given in terms of these two by solving (3.37), which is a simple quadratic equation. In any case, in the next section we will reproduce the expression (3.71) analytically, starting from a conjectural large N free energy of the dual field theories.
Noticing that µ is a free "small" parameter 11 , it is useful to expand in series of µ near to µ → 0 (holding χ fixed), which formally corresponds to reducing to a spindle with equal conical deficits, and in particular it includes the two-sphere for n + = n − = 1. The root of the quartic equation (3.37) meeting the required constraints then has the following expansion where we have defined t ≡ √ 8z 2 + 1, with t > 3. Inserting this in the free energy (3.71) we obtain the expansion which, after setting χ = 2, at leading order in µ agrees with the free energy (2.42) for g = 0 and κ = 1 [2]. This suggests that it may be possible to recover the AdS 4 × S 2 solution in [2] by performing a suitable scaling limit of our solutions, but we will not attempt to do so here. For future reference, let us also record the expansion for ∆z, that reads

Field theory
We conjecture that the solutions we have constructed in section 3 are holographically dual to three-dimensional SCFTs obtained by compactifying on a spindle Σ the five-dimensional SCFTs dual to the solution of [27]. In the reminder of this section we will provide evidence for this by proposing an off-shell free energy whose extremization reproduces exactly the holographic free energy (3.71). This is an extension of the entropy functions that have been shown to provide an efficient method for reproducing the entropy of supersymmetric AdS black holes in various dimensions. A priori, this function should be derived from first principles, by computing (minus the logarithm of) the localized partition function of the d = 5 SCFT, placed on the background of S 3 × Σ, and then taking the large N limit. This strategy has been implemented in [35,36] for the background S 3 × Σ g and, indeed, it led to reproducing the holographic free energy (2.42) previously obtained in [2]. Instead, we will follow a short-cut inspired by the "gravitational blocks" advocated in [21]. We will infer from the supergravity description the main ingredients involved in the fieldtheoretic construction and we will propose a large N off-shell free energy on S 3 × Σ obtained by suitably gluing the S 5 free energy of the d = 5 theories. We will then show that extremizing it will reproduce exactly the holographic free energy (3.71).

d = 5 SCFTs dual to the AdS 6 solution
Let us begin by recalling the salient features of the five-dimensional theory that is holographically dual to the AdS 6 × S 4 background of massive type IIA, arising in the nearhorizon limit of N D4-branes and N f D8-branes, that we reviewed in section 2.3. This is an N = 1 gauge theory with gauge group USp(2N), coupled to N f massless hypermultiplets in the fundamental representation and one hypermultiplet in the antisymmetric representation of USp(2N) [39]. At low energies, this theory flows to an interacting SCFT, with global symmetry SU(2) R × SU(2) F × E N f +1 , where the first two factors are realised as symmetries of the AdS 6 × S 4 solution [40]. Placing this theory on a rigid S 5 background, one can compute the exact localized partition function Z S 5 and consider the associated free energy, namely as a good measure of the degrees of freedom of the theory. This was computed in [33], that also showed that in the large N limit it becomes and is reproduced by a holographic calculation in the solution of [27]. The S 5 free energy may also be "refined", promoting it to an off-shell free energy, regarded as a function of the fugacities for the U(1) × U(1) Cartan subgroup of SU(2) R × SU(2) F , which we will denote as ∆ i , with i = 1, 2. A priori, the R-symmetry can mix with any flavour symmetry and the ∆ i parameterise this mixing. For the present theory this is actually not necessary, as the R-symmetry is non-Abelian, nevertheless this will be useful in the sequel. We can then write with the fugacities obeying, in a canonical normalization, the R-symmetry constraint For later convenience we have defined the auxiliary function F (∆ i ). Extremizing F S 5 (∆ i ) gives ∆ * 1 = ∆ * 2 = 1 and inserting these values back one reproduces the initial free energy Let us now move to discussing compactifications of this theory to d = 3 dimensions and the corresponding off-shell free energies.

d = 3 SCFTs dual to the AdS 4 × Σ g solutions
We can obtain three-dimensional N = 2 theories by compactifying the above d = 5 SCFT on a Riemann surface Σ g of arbitrary genus g, performing the standard topological twist [2]. Specifically, we place the theory on Σ g and couple it to two background gauge fields A i for the U(1) × U(1) Cartan subgroup of SU(2) R × SU(2) F , with appropriately quantized magnetic fluxes 12 The topological twist implies that the R-symmetry gauge field A R = A 1 + A 2 is identified with a connection on the tangent bundle, thus and the Killing spinors become just constant. It is then convenient to parameterise the magnetic fluxes as (4.8) 12 Here and in the following we shall rename the background gauge fields as gA i → A i , which is more natural from the field theory point of view. The magnetic fluxes n i correspond precisely to the fluxes P i defined in (2.33). However, we denote these with different symbols to emphasise the fact that the P i were defined as integrals of supergravity fields, living in D = 6, while the n i are defined as integrals of background gauge fields, living in d = 5.
The exact R-symmetry of the d = 3 theory will be determined by extremizing the off-shell S 3 free energy [41], viewed as a function of the fugacities ∆ i . Equivalently, this quantity may be thought of as the off-shell free energy of the d = 5 theory on S 3 × Σ g . The latter quantity was computed in [35,36] using localization, and in the large N limit it was shown to reproduce the holographic free energy (2.42). Below we will show that it can also be reproduced by a formula obtained by gluing two gravitational blocks. We begin defining the following conjectural large N off-shell free energy 13 where F (∆ i ) is defined in (4.3) and with ∆ i satisfying (4.4).
Notice that in addition to the fugacities ∆ i of the parent d = 5 theory and the magnetic fluxes n i , (4.9) depends a priori also on the parameter ǫ, although we shall see below that the extremization equations automatically set ǫ = 0. At least in the case g = 0, ǫ may be interpreted as the fugacity associated to the U(1) J ⊂ SU(2) J rotational symmetry of the two-sphere. Extremizing (4.9) with respect to ǫ and ∆ i , subject to (4.4), we easily find the critical values For the two-sphere, the fact that ǫ * vanishes means that the R-symmetry of the compactified theory does not have a component along U(1) J , as expected. In any case, inserting the critical values in (4.9) we get which agrees with (2.42). Since ǫ * = 0, we could have started setting ǫ = 0 in (4.9), thus reducing to the known ǫ-independent off-shell free energy corresponding to the standard topological twist [31,35,36,43]. In particular, extremizing (4.13) reproduces (4.12) with ∆ * 1 given in (4.11). We notice that ∆ * 1 , ∆ * 2 match with the scaling dimensions of two particular 1/2-BPS operators, corresponding to D2-branes wrapped on calibrated surfaces, embedded in the internal six-dimensional geometries, and sitting at the center of AdS 4 , described in [2]. For g > 1 we can set z = 0 so that ∆ * 1 = ∆ * 2 = 1 and the free energy (4.12) reduces to the universal relation [44] F (∆ * i , 0; n 1 = n 2 = 1 − g) = 16π(g − 1) (4.14)

d = 3 SCFTs dual to the AdS 4 × Σ solutions
We now consider the compactification of the d = 5, N = 1 SCFT on a spindle, that we expect to be dual to the solutions we constructed in section 3. In particular, we perform a global topological twist, which means that we place the theory on Σ and couple it to two background gauge fields A i for the U(1) × U(1) Cartan subgroup of SU(2) R × SU(2) F , with appropriately quantized magnetic fluxes As for the standard topological twist, the R-symmetry gauge field A R = A 1 + A 2 becomes a connection on the tangent bundle, thus but crucially, this does not imply that the metric on the spindle has constant curvature, nor that the spinors are chiral and constant. Specifically, the rigid Killing spinors are expected to behave precisely as the spinors η A ± arising in the supergravity solution. As before in the paper, we will continue to parameterise the magnetic fluxes as Taking inspiration from the entropy function proposed in [12], we conjecture that the large N off-shell free energy for these theories is given by where F (∆ i ) is defined in (4.3) and The ∆ i are the fugacities parameterising the R-symmetry within the U(1) × U(1) ⊂ SU(2) R × SU(2) F global symmetries of the d = 5 theory, and are therefore still subject to the constraint (4.4), while ǫ is an equivariant parameter for the spindle, that is a fugacity for the U(1) J rotational symmetry. In general, we expect that this will parameterise a non-trivial mixing of the R-symmetry of the parent theory, with the U(1) J of the spindle. The off-shell free energy (4.18) bares a close resemblance to the off-shell central charge for D3-branes wrapped on spindle in [10] and we shall elaborate on this in the next section. Employing the parametrisation (4.17) we see that upon redefining χǫ =ǫ, the off-shell free energy (4.18) becomes F = χ · f (∆ i ,ǫ; µ, z), implying that the free energy at the critical point must be a function of µ and z only, with an overall factor of χ. Notice that setting n + = n − = 1 the present setup reduces to the case of twisted compactification on S 2 , that is the case g = 0 discussed in the previous subsection.
In order to implement the constraint (4.4) it is useful to introduce a Lagrange multiplier and consider the extremization of the following function S(∆ i , ǫ, Λ; n i , n + , n − ) = F (∆ i , ǫ; n i , n + , n − ) + Λ(∆ 1 + ∆ 2 − 2) , (4.20) that is analogous to the entropy functions studied in the literature. The corresponding extremality equations read These are four equations for the variables ∆ 1 , ∆ 2 , ǫ, Λ and, in order to solve these, it is convenient to process them further. Noticing that (4.18) is homogeneous of degree two in ∆ i and ǫ, by Euler's theorem we have The extremization equations for S written as then immediately imply Λ = −F . We can therefore eliminate Λ from the system and write the remaining two independent equations as where one has to use also the constraint ∆ 1 + ∆ 2 = 2. Notice that taking ǫ → 0 the second equation reduces to the first equality in (4.13).
After some work, we determined the critical values , (4.25) in terms of the parameter x, that is the unique root in the interval (0, 1) of the quartic (3.37) that we introduced in the discussion of the gravitational solution. Inserting these values back into (4.18) we obtain which, remarkably, agrees exactly with the gravitational free energy (3.71)! To arrive at the solution (4.25), we first solved the extremality equations perturbatively in µ around µ = 0, obtaining agreement with the expansion (3.73), up to high powers of µ. We then noticed that the result for ǫ * could be rewritten as which is a universal relation holding in all previous spindle solutions [7][8][9][10][11][12][13]. Using this, we then obtained the result for ∆ * 1 in (4.25). As discussed in section 3, in the limit n + = n − = 1 (4.26) reproduces the free energy (4.12) for g = 0, corresponding to the compactification of the d = 5 theory on the two-sphere, with the standard topological twist. Moreover, expanding ∆ * 1 in (4.25) in series of µ around µ = 0 we find that 28) again in agreement with the two-sphere value given in (4.11). It would be interesting to reproduce ∆ * 1 , ∆ * 2 by computing the scaling dimensions of some supersymmetric probe D2-branes wrapped on calibrated two-cycles in the ten-dimensional geometry (3.62).

Gravitational blocks for branes on spindles
The off-shell free energy that we discussed in the previous section may be regarded as a particular instance of a general class of off-shell free energies F ± , for twisted compactifications of d-dimensional theories on the spindle Σ. Below we will state our conjecture and we will then illustrate how it encapsulates and generalises various extremal functions discussed in the literature. From the constructions of M2, D3 and M5-branes wrapped on a spindle and the results we discussed so far in this paper, it has emerged that supersymmetry on a spindle can be preserved in two different ways, that can be referred to as twist and anti-twist. These are characterised by two types of background R-symmetry gauge field A R , with fluxes given by where σ = +1 for the twist and σ = −1 for the anti-twist. The sign σ = +1 corresponds to the choice we made in section 4.3, and that was also realised for M5-branes wrapped on the spindle in [9]. The sign σ = −1 has been realised by the supergravity solutions for D3branes [8,10,11] and M2-branes [7,12,13]. Below we will treat both cases simultaneously, with the understanding that not all cases may have a counterpart as gravity solutions.
In [26] it is proved that these are the only two possible twists preserving supersymmetry on the spindle. A large class of SCFTs in different dimensions are expected to be characterised by supersymmetric partition functions Z M , where M are rigid geometries comprising a metric on an appropriate curved space, the background gauge fields for the R-symmetry and possibly other flavour symmetries. The associated free energy is defined by and it is regarded as a function of the fugacities ∆ i associated to the Cartan subgroup of the continuous global symmetry group of the theory. For theories with an Abelian Rsymmetry, that we are interested in, the fugacities obey a constraint that we can always normalise to be where d is the rank of the global symmetry group, of which the Abelian R-symmetry is part. In general, these are complicated matrix models, which however simplify drastically in special limits, such as the large N limit or Cardy-like limits, reducing to simple local functions of the fugacities ∆ i . For example, in four dimensions the partition function corresponding to M = S 1 × S 3 is the (refined) superconformal index and in either limits its logarithm is related to the trial central charge a 4 (∆ i ) of the theory. In all SCFTs possessing an Abelian R-symmetry, it has been either proved or conjectured that the exact superconformal R-symmetry is determined by extremizing these quantities. We conjecture that for a general class of d-dimensional SCFTs compactified on the spindle, with either the twist or the anti-twist, the exact R-symmetry is determined by extremizing the following off-shell free energies 5) and the magnetic fluxes through the spindle, n i , satisfy the constraint The building blocks are the functions F d (∆ i ) summarised in Table 2. They are proportional to: the S 3 off-shell free energy of the ABJM theory, the trial central charge of the N = 4 SYM theory, the S 5 off-shell free energy of the d = 5, N = 1 SCFT and the trial central charge of the d = 6, (2, 0) SCFT, respectively. In the first two cases, it is straightforward to replace these with the corresponding quantities for more general d = 3, N = 2 theories and d = 4, N = 1 theories, but in this paper we will not pursue this.
In the last row we summarised the relations between the gluing sign ± and the sign σ, characterising the type of twist.
We expect that the form (5.4) arises, in the large N limit, from a fixed point formula, with the "blocks" F d (∆ i ) evaluated at the north and south poles of the spindle. The superscripts in F ± refer to the relative choice of sign ± in (5.4), corresponding to the type of gluing of the contributions of the two hemispheres of the spindle [21]. We will give circumstantial evidence that in D = 4, 6 the type of gluing is correlated with the type of twist, specifically that the gluing sign is −σ. On the other hand, in D = 5, 7 the results of the explicit supergravity solutions are all reproduced by the minus gluing sign.
In the examples that we discuss below, we will explain which choice of twist and gluing is relevant, but a more systematic understanding of these choices is clearly desirable.
The variable ǫ is a fugacity associated to the U(1) J rotational symmetry of the spindle and the significance of the fact that at the critical point of (5.4) this takes a non-zero value is that the R-symmetry of the parent d-dimensional theory mixes with U(1) J to give the exact superconformal R-symmetry of the (d − 2)-dimensional theory arising in the IR, when this flows to an SCFT. The constants r i are arbitrary, but subject to the constraint and parameterise the ambiguities of defining flavour symmetries [10]. In the previous section we picked the most symmetric choice, corresponding to r i = d 2 = 1. However, it is simple to show that the functions (5.4) evaluated at the critical point are independent of the choice of r i . Introducing a new set of variables defined as the off-shell free energies simply read where, from now on, we will omit n + , n − , σ from the arguments of the function, in order not to clutter the subsequent formulas. The variables ϕ i , ǫ satisfy the constraint inherited from (5.3) and (5.7). Since (5.9) does not depend on the constants r i , it follows that the critical values ϕ * i , ǫ * and F ± (ϕ * i , ǫ * ; n i ) do not depend on the r i either. Our proposal unifies previous proposals concerning entropy functions [12,21,45,46] and central charges [10], extending these to compactifications of d-dimensional theories on spindles with both twist and anti-twist. Below we shall illustrate this, recovering known results and discussing some generalisations. The constrained extremization problem can be carried out introducing a Lagrange multiplier. Defining and using the fact that (5.4) is homogeneous of degree 14 h in ϕ i and ǫ, by means of Euler's theorem we have that Λ = − h 2 F ± , and we can therefore eliminate Λ from the system.
The resulting extremization equations can be written as (j = 1, . . . , d) Notice that the last equation can be replaced by (5.13)

M2-branes
Supergravity solutions describing M2-branes wrapped on the spindle were first constructed in [7] in minimal D = 4 gauged supergravity and generalised to U(1) 2 gauged supergravity in [12,13]. They realise the anti-twist, σ = −1. The corresponding dual field theory is the ABJM model compactified on the spindle, with two background gauge fields with magnetic fluxes n 1 + n 2 = n + + σn − 2n + n − , (5.14) for σ = −1. However, it is straightforward to carry out the extremization leaving the twist unspecified. Picking the plus gluing sign in (5.9) gives with ϕ i , ǫ satisfying the constraint For σ = −1 this is exactly the entropy function proposed in [12], in the case of vanishing electric charges and angular momentum. Performing the extremization of (5.15), subject to (5.16), we get and inserting these values back in (5.15) we find which coincides with the entropy in [12] upon setting σ = −1. In minimal D = 4 gauged supergravity the entropy function (5.15), for ϕ 1 = ϕ 2 and n 1 = n 2 , was derived in [19], and its plus gluing sign is consistent with the fact that it indeed arises as the sum of contributions from the north and south poles of the spindle, which are the fixed points of the canonical Killing vector field [25]. The precise map between the variables here and those used in [12] is as follows 19) and using this, the constraint (5.16) becomes Note that with vanishing angular momentum and electric charges the entropy function is purely real (or purely imaginary) and therefore the critical points are purely real.
For the other type of gluing we obtain (for either choice of σ) the function whose extremization, however, leads to a degenerate result. The entropy function for the general four-charge model, for either type of gluing and either type of twist, reads ± (ϕ 1 − n 1 ǫ) (ϕ 2 − n 2 ǫ) (ϕ 3 − n 3 ǫ) (ϕ 4 − n 4 ǫ) .

(5.22)
The magnetic fluxes obey n 1 + n 2 + n 3 + n 4 = n + + σn − n + n − , (5.23) and the variables ϕ i , ǫ satisfy the constraint For n + = n − = 1 these reduce to the entropy functions proposed in [21]. In particular, for σ = +1, F − reduces to the entropy function of the supersymmetric AdS 4 black holes with a topological twist [4], whereas for σ = −1, F + reduces to the entropy function for the supersymmetric rotating Kerr-Newmann AdS 4 black holes [24]. More generally, it is natural to expect that the correct gluing for either type of twists is −σ, as reported in Table 2. It would be nice to corroborate this proposal showing that extremizing the entropy functions F −σ in (5.22) reproduces the entropy of the AdS 2 × Σ solutions of STU gauged supergravity.

(5.29)
Setting σ = −1 this reduces to the a 2 central charge obtained in [10,11]. On the other hand, setting σ = +1 we reproduce also the central charge for D3-branes wrapped on the spindle with the twist, for which supergravity solutions were recently presented in [26]. Further setting n + = n − = 1, (5.29) reduces to the result for D3-branes wrapped on S 2 with the standard topological twist [5]. Equivalently, we can extremize over the variables ∆ i , ǫ, with ∆ 1 + ∆ 2 + ∆ 3 = 2 obtaining the same result as in (5.29) for the critical central charge. However, the variables ∆ i are affected by the ambiguity related to the choice of the constants r i . Notice that for the other type of gluing we obtain (for either choice of σ) the function F + (ϕ i , ǫ; n i ) = −3N 2 (n 1 ϕ 2 n 3 + ϕ 1 n 2 n 3 + n 1 n 2 ϕ 3 ) ǫ + ϕ 1 ϕ 2 ϕ 3 ǫ . (5.30) It may be possible that this corresponds to a different type of twisted compactification of D3-branes on the spindle with a corresponding class of supergravity constructions.

(5.35)
Inserting the values of ϕ * i and ǫ * back in (5.33) we get 36) which reduces to the results of the previous section for σ = +1.
We expect the choice of minus gluing sign to be correlated to the fact that in our solutions the Killing spinors on the spindle have the same chirality at the north and south poles. It would be interesting to investigate the extremization of F + and to find out whether for σ = −1 it has a physical critical point, corresponding to AdS 4 × Σ solutions with the anti-twist, yet to be constructed.

(5.42)
Setting σ = +1 this reduces to the a 4 central charge obtained in [9] integrating the M5brane anomaly polynomial on the spindle. However, in [9] the extremization was carried out over the variables ∆ i , subject to ∆ 1 + ∆ 2 = 2, and it was found that the critical values are given by ∆ * 1 = ∆ * 2 = 1, while the critical ǫ * coincides with the one given in (5.40), up to a convention-dependent factor, specifically, ǫ * here = − 1 2 ǫ * there . One can check that the variables ∆ i utilised in [9] correspond (for σ = +1) to the choice of constants r 1 = 2 − r 2 given by It would be interesting to find out whether the critical points for σ = −1 correspond to AdS 5 × Σ solutions with the anti-twist, yet to be constructed. We note that, as for the case of AdS 4 × Σ g solutions, the extremization of the offshell free energy (5.38) reproduces also the central charge 15 of d = 4 SCFTs dual to the AdS 5 × Σ g solutions [6]. As anticipated in footnote 1, in this case the variables ϕ i and the magnetic fluxes are subject to the constraints ϕ 1 + ϕ 2 = 2 , n 1 + n 2 = 2(1 − g) , (5.44) and as usual the latter may be parameterised as The free energy (5.38) is extremized by 46) to which corresponds the critical value [6] Since ǫ * = 0, we could have started setting ǫ = 0 in (5.38), thus reducing to the known ǫ-independent off-shell free energy corresponding to the standard topological twist. In particular, extremizing (5.48) reproduces (5.47). For g > 1 we can set z = 0 so that ϕ * 1 = ϕ * 2 = 1 and the free energy (5.48) reduces to the universal relation [44] F − (ϕ * i , 0; n 1 = n 2 = 1 − g) = Notice that for the other type of gluing we obtain (for either choice of σ) the function F + (ϕ i , ǫ; n i ) = − 9 128 N 3 n 2 1 n 2 2 ǫ 3 + n 2 1 ϕ 2 2 + n 2 2 ϕ 2 1 + 4n 1 n 2 ϕ 1 ϕ 2 ǫ + It may be possible that this corresponds to a different type of twisted compactification of M5-branes on the spindle with a corresponding class of supergravity constructions.
6 D4-branes wrapped on Σ × Σ g In this section we will begin investigating constructions corresponding to D4-branes wrapped on orbifolds of dimension higher than two, focussing on a class of explicit solutions that can be easily obtained uplifting to massive type IIA supergravity the AdS 2 solutions of four-dimensional minimal gauged supergravity of [7]. Generically, for fourdimensional orbifolds M 4 , the field theories are d = 1 SCQMs obtained from twisted compactifications of the d = 5, N = 1 USp(2N) gauge theory. The off-shell free energies are in this case entropy functions, whose extremization determines the entropy of supersymmetric AdS 6 black holes with AdS 2 × M 4 near-horizon geometry. As we have discussed, we expect that these will take the form of a sum of gravitational blocks over the set of fixed points of the canonical Killing vector, which includes also the non-orbifold geometries as special cases. In particular, the entropy function for the product of two constant-curvature Riemann surfaces Σ g 1 × Σ g 2 with the standard topological twist may be recovered considering 16 subject to the constraints Extremizing (6.1) with respect to ǫ 1 and ǫ 2 sets ǫ 1 = ǫ 2 = 0 so that the entropy function reduces to [43] lim Extremizing this with respect to ϕ i reproduces the entropy of an associated class of AdS 2 × Σ g 1 × Σ g 2 supersymmetric solutions [31]. Below we will discuss the case of M 4 = Σ × Σ g , leaving M 4 = Σ 1 × Σ 2 and more general orbifolds for future work. Analogous solutions, corresponding to M5-branes wrapped on Σ × Σ g , were presented in [11].

AdS 2 × Σ × Σ g solutions
A class of supersymmetric AdS 2 × Σ × Σ g backgrounds may be easily obtained by lifting the AdS 2 ×Σ solutions of four-dimensional minimal gauged supergravity constructed in [7] to D = 6 matter-coupled gauged supergravity, using the consistent truncation presented in [46]. In our conventions, the bosonic part of the action reads where F i = dA i , H = dB and the scalar fields X i with the scalar potential V are given by (2.2) and (2.3), respectively. This model is the complete version of the truncation 16 The overall factor of − 1 4 may be fixed by splitting the compactification on Σ g1 × Σ g2 in two steps, first reducing from d = 5 to d = 3 and then from d = 3 to d = 1. This factor is then consistent with the rules summarised in Table 2.
presented in section 2.1, with non-vanishing two-form B. Below for simplicity we will restrict our attention to the static solution, which in the notation of [7] corresponds to setting j = 0. The six-dimensional solution then reads ds 2 6 = e −2C L 2 AdS 4 y 2 4 ds 2 AdS 2 + y 2 q(y) dy 2 + q(y) 4y 2 dz 2 + e 2C ds 2 Σg , where the function q(y) and the constant a are given by 17 The constants k 2 and k 8 are those appearing in (2.31), namely while C reads As usual, the parameter g is fixed in terms of m, while L AdS 4 is related to m by eq. (4.24) of [46], Notice that formally taking a = 0 the solution (6.5) reduces precisely to the AdS 4 × Σ g solution discussed in section 2.4. Let us now consider the quantization of the fluxes. For the fluxes through the Riemann surface we have with s 1 + s 2 = 2(1 − g). Recalling that [7] 1 y 1 − 1 y 2 ∆z 2π = n 2 + + n 2 − n + n − (n + + n − ) , (6.11) 17 In order to be consistent with our notation we have exchanged n + and n − with respect to [7].
where y 1 , y 2 are the two relevant roots of q(y), the fluxes through the spindle are given by (6.12) where p i ∈ Z. We then note that where ∆ * 1 was given in (4.11), thus we have 14) that will be important in the field theory extremization. The constraint ∆ * 1 + ∆ * 2 = 2 implies showing that there is an anti-twist over the spindle. Note that from the relations (6.14) we obtain which is a non-trivial Diophantine equation. One example of solution is given by the set of values κ = −1, z = 6, n + − n − = 6, p 1 = 5 and we have checked that other combinations exist. Moreover, when κ = −1 it is possible to smoothly take the limit z → 0, in which case the equation reduces to n + − n − = 2p 1 .
We can now uplift the solution (6.5) to massive type IIA using the recipe described in section 2.2. For simplicity, we will only present the relevant ingredients for the computation of the entropy, namely the metric and the dilaton, that read the effective two-dimensional Newton constant is given by 20) and therefore the entropy reads where A h is the area of the horizon of the four-dimensional black hole with L AdS 4 = 1, specifically [7] A h = 1 2 (y 2 − y 1 )∆z = π −(n + + n − ) + √ 2 n 2 + + n 2 − n + n − . (6.22) Both the Romans mass F (0) and the part of the four-form flux F (4) along the fourhemisphere S 4 remain unaltered with respect to (2.36) and (2.37), therefore the quantization of the fluxes in ten dimensions is unchanged, giving the relations (2.25). The final expression of the entropy is

Entropy function
As before, we can deduce the main ingredients of the field theory construction from the supergravity solution. In particular, the magnetic fluxes s i show that there is a (standard) topological twist on Σ g , while from the form of the n i we see that there is an anti-twist on Σ. Then applying our conjecture twice, we obtain the following entropy function subject to the constraints Extremizing this with respect to ǫ 2 sets ǫ 2 = 0 and after renaming ǫ 1 → ǫ we obtain where c ≡ √ 2π 5 which has to be extremized subject to the constraints (6.25). So far we have not used the additional input (6.14) given by the supergravity solution discussed above. This strongly suggests that the extremization of (6.26), without imposing (6.14), will give the entropy of a more general supergravity solution 18 . However, below we will proceed enforcing (6.14). Doing so, it is convenient to introduce the rescaled variables subject to the constraint 29) in terms of which the entropy function reads Extremizing this we find the critical values where the sign ambiguity η = ±1 arises by solving the equations over the complex numbers 19 . Inserting these values back into the entropy function, we compute where we need to pick η = −1 in order to get a positive entropy that agrees with the one computed from the ten-dimensional supergravity solution. It should be straightforward to incorporate electric charge and rotation (along the spindle), by promoting the extremization of (6.30) to the Legendre transform. The entropy obtained in this way should match that of the supergravity solution obtained by uplifting to massive type IIA the rotating AdS 2 × Σ solution of [7] 20 .

Discussion
The work of [8] opened the way to a novel class of examples of the AdS/CFT correspondence, by constructing a supergravity solution with an AdS 3 factor, comprising the two-dimensional orbifold Σ, known as the spindle. This was interpreted as the nearhorizon limit of D3-branes wrapped on the spindle with the corresponding field theory 18 The recent paper [47] discusses AdS 2 × Σ × Σ g supergravity solutions and it would be interesting to understand the relationship to our work. 19 Generically, in the presence of rotation, we are forced to work with the complex numbers [19,20] and we therefore continue to do so also in the static case. 20 In the rotating solution, setting n + = n − = 1 will reduce to a rotating AdS 2 × S 2 ǫ × Σ g solution in D = 6, whose entropy function has been recently discussed in [42]. duals being a class of d = 4, N = 1 SCFTs compactified on the spindle, with a new type of supersymmetry-preserving twist, that was dubbed anti-twist. Following this, over the past year various extensions, including analogous constructions for M2 and M5-branes, have appeared, realising either the anti-twist [8], or a global generalization [9] of the standard topological twist. These constructions have left out the notable class of solutions describing D4-branes wrapped on the spindle, that we constructed in this paper. We found these AdS 4 × Σ solutions in D = 6 gauged supergravity and then uplifted them to massive type IIA supergravity. The type of twist realised in our solutions is the one previously found in [9] for M5-branes, which was referred to as a global topological twist. Differently from the standard topological twist, the R-symmetry gauge field does not cancel the spin connection, despite the fact that its integrated flux is equal to the Euler characteristic of the spindle. As a consequence the Killing spinors are then non-trivial sections (in particular, they are not simply constant) of the same bundles that occur in standard topological twist.
There are several aspects of our solution that may be interesting to investigate in the future. For example, it may be instructive to cast it in the form of the classification of supersymmetric AdS 4 solutions of massive type IIA supergravity [34], or to study supersymmetric probe D-branes in our background in order to extract further information about the dual field theories. Another question that arises from our work is whether there exists a more general consistent truncation of massive type IIA supergravity to D = 6, analogous to the dyonic consistent truncation to D = 4 supergravity found in [32]. It is also intriguing to investigate whether there exist supergravity solutions corresponding to D4-branes wrapped on the spindle with the anti-twist. Finally, as an important step towards improving control on the dual field theories, it would be worthwhile computing an appropriately regularised on-shell action, that should prove the validity of our conjectural off-shell free energy (4.18). To do this, one may need to know the full "black two-brane" solution, interpolating between AdS 6 asymptotically and AdS 4 × Σ in the near-horizon, similarly to [19]. However, it may be possible to employ the strategy of [25] to prove that the supergravity (Euclidean) on-shell action localises at the poles of the spindle, which are the fixed points for the canonical Killing vector field associated to any (Euclidean) supersymmetric solution of the theory [48].
There is also a number of assorted interesting questions in the field theory side. The most direct one is to study the five-dimensional SCFTs in the background of S 3 × Σ, computing the localized partition function from first principles, and then showing that in the large N limit the associated free energy reduces to (4.18). This is indeed an open problem for SCFTs compactified on spindles in different dimension.
It is compelling that the five-dimensional off-shell free energy that we conjectured here and the entropy function that was conjectured in [12] fit into a broader conjecture for the free energies (i.e. minus the logarithm of partition functions) of SCFTs in various dimensions, namely (5.9). These extend the idea of gravitational blocks put forward in [21] in two directions. Firstly, from compactifications on smooth manifolds, to the realm of compactifications on orbifolds. It is remarkable that compactifications on spindles can be incorporated by a simple modification of the constraint obeyed by the fugacities (see eq. (5.10)). This depends on the type of twist performed and it includes the standard topological twist and the no-twist as special cases. Secondly, we have pointed out that the form (5.9) should hold also for observables of higher-dimensional theories, beyond the entropy, associated to AdS 2 solutions dual d = 1 SCFTs.
Using the AdS/CFT correspondence, the free energies (5.9) are generically expected to arise as gravitational (Euclidean) on-shell actions, and their structure is suggested by the fact that for supersymmetric solutions, this form should arise from summing contributions at fixed points of the canonical Killing vector field, defined as a bilinear in the Killing spinors of the solution [25,49]. An immediate issue that would be nice to clarify is a better justification of the ± signs in (5.9) and their relationship with the type of twist. For example, while for field theories in odd dimensions we have proposed that the gluing sign should be −σ, for field theories in even dimensions, the explicit examples indicate that one should always pick the minus sign. It would interesting to find out whether the functions that have not been considered so far, may have critical points that correspond to new gravitational objects.
It is quite clear that the structure of (5.9) will extend to compactifications of SCFTs on higher-dimensional 21 orbifolds and in general we expect that the functions to extremize will take the form of a sum of contributions from fixed points of the canonical Killing vector on the compactification orbifold M. Assuming this is toric, for concreteness, there will be as many equivariant parameters ǫ I as the rank of the torus acting on M. One of the simplest examples of this type of construction is given by twisted compactifications of the d = 5 SCFT on M = Σ × Σ g , that we discussed in section 6. We have shown that the corresponding entropy function can be obtained iterating our conjecture for the off-shell free energies twice and extremizing this reproduces the entropy of a corresponding class of AdS 2 × Σ × Σ g supergravity solutions. More generally, we expect that there should exist solutions of the type AdS 2 ×Σ 1 ×Σ 2 in D = 6 supergravity and of the type AdS 3 ×Σ 1 ×Σ 2 in D = 7 supergravity, for which the corresponding entropy function and trial central charge can be obtained following the rules we proposed in this paper. Interestingly, the former case would be identified with the near-horizon limit of a novel class of supersymmetric black holes in AdS 6 , that may be accelerating. Work along these lines is underway and we hope to report in the near future.
As we continue to navigate the landscape of supergravity solutions corresponding to branes wrapping orbifolds, it will be revealing to employ the approach developed in [50][51][52][53], adapting it to situations with orbifold singularities. While this is already well developed for solutions arising from M2 and D3-branes, it is tantalising to think that an analogous geometric approach may be concocted for studying AdS 5 solutions of D = 11 supergravity or other backgrounds with an AdS factor.
A More details on the AdS 4 × Σ g solutions A.1 Relation with the Lagrangian of [1] Here we make contact between the D = 6 Lagrangian (2.1), that we use in the paper, and the Lagrangian used in [1], given explicitly in [54]. In order to minimise confusion we have relabelled some of the quantities in [54] as follows ϕ 1,2 →φ 1,2 , F 3,6 → 1 2 F 3,6 , m → m 1 , (A.1) where the 1/2 factor is due to the unusual definition of the field strengths as, e.g., . Consistency between the equations of motion and the supersymmetry equations requiresφ 1 = 0, and in this case the D = 6 Lagrangian in [54] reads where the scalar potential iŝ