${\cal N}=(2,2)$ AdS$_3$ from D3-branes wrapped on Riemann surfaces

We construct $\mathcal{N}=(2,2)$ supersymmetric AdS$_3$ solutions of type IIB supergravity, dual to twisted compactifications of 4d $\mathcal{N}=4$ super-Yang--Mills on Riemann surfaces. We consider both theories with a regular topological twist, and a twist involving the isometry group of the Riemann surface. These solutions are interpreted as the near-horizon of black strings asymptoting to AdS$_5\times \text{S}^5$. As evidence for the proposed duality we compute the central charge of the gravity solutions and show that it agrees with the field theory result.


Introduction
Since the seminal work of [2], studying the low energy dynamics of branes wrapped on compact manifolds has been a successful research direction. From the field-theoretic point of view one may generate families of lower-dimensional field theories from a higher-dimensional parent theory living on the worldvolume of the branes in flat spacetime. By placing the branes on a non-trivially curved manifold and requiring the preservation of some supersymmetry, one must perform a twist of the field theory. Geometrically, this is realised by studying the embedding of a calibrated cycle, on which the brane is wrapped, into a special holonomy manifold. The different ways of embedding the compactification manifold give rise to the different families of lower-dimensional field theories.
For a large enough number of branes, and small enough curvatures, one may use the holographic correspondence. Since the original work of [2], which studied the holographic duals of 4d N = 4 super-Yang-Mills (SYM) and the 6d N = (2, 0) theory compactified on a Riemann surface, there have been many generalisations. The theories of class S study the twisted compactification of the 6d (2, 0) SCFT living on M5-branes on a punctured Riemann surface [3] preserving N = 2 supersymmetry in four dimensions. These were later extended to theories preserving N = 1 supersymmetry in four dimensions in [4][5][6][7] and the holographic dual of Argyres-Douglas theories in [8,9]. Theories of class R study the compactification of the 6d (2, 0) A N −1 theory on hyperbolic three-manifolds, [10]. Whilst the theories in [11] arise from compactifying the D4-D8 bound state on a Riemann surface. In addition, the work of [12] studies various compactifications of branes on Riemann surfaces with punctures using gauged supergravity, and [1,[13][14][15] consider field theories wrapped on spindles.
In this work we will restrict to studying the holographic dual of a stack of D3-branes compactified on a Riemann surface with a twist preserving N = (2, 2) supersymmetry and flowing to a two-dimensional SCFT. 4 The UV theory is 4d N = 4 SYM which flows to the 2d SCFT after being placed on the Riemann surface. Via the AdS/CFT correspondence the two theories, the UV and IR, admit AdS duals. For the UV theory this is of course AdS 5 × S 5 whilst for the IR one obtains the AdS 3 solutions which form the content of this work. The full flow solution is a five-dimensional asymptotically AdS 5 black string with AdS 3 × Σ 2 near-horizon geometry. 5 4 Recently constraints on the elliptic genera of 2d N = (2, 2) SCFTs admitting holographic duals have been investigated in [16][17][18]. Though we will not compute the elliptic genus of the field theory duals of our solutions it would be interesting to see where these theories lie in the landscape mapped out there. 5 Static black string solutions in 5d STU supergravity have been found in [19] for example.
The microscopic origin of the Bekenstein-Hawking entropy of black objects is one of the fundamental problems in theoretical physics. Certain thermodynamic quantities of black objects may be computed in gravity in either the asymptotic or near-horizon limit without loss of information. One such quantity is the Bekenstein-Hawking entropy. The fact that one only needs the near-horizon to compute the Bekenstein-Hawking entropy and not the full black string solution is a considerable simplification to the problem. One can then identify the microstates of the black string as the SCFT dual to the AdS 3 near-horizon theory. In this paper we will take this latter approach and interest ourselves in the near-horizon of black strings preserving N = (2, 2) supersymmetry and not the full black string solution.
Solutions of this type were classified in [20], whilst AdS 3 solutions preserving other amounts of supersymmetry have been studied in . The conditions for preserving supersymmetry and satisfying the equations of motion, which we review in section 2, are very similar to the conditions for other AdS solutions which are identified as wrapped brane solutions. In particular the form of the solutions are similar to the AdS 5 solutions arising from M5-branes on a Riemann surface [7] and AdS 4 solutions in massive type IIA obtained from the D4-D8 bound state on a Riemann surface, [11,63]. We will reduce the conditions of [20] to a Monge-Amperé like equation for a potential. We will present two distinct classes of solutions. The first has a constant curvature metric on the Riemann surface and we find that the solution is a deformation of the Maldacena-Nuñez solution [2]. The benefit of our formulation is that the inclusion of punctures has a natural interpretation in this description: they are interpreted as sources to the Monge-Amperé equation. The second class of solutions we study has a non-constant curvature metric on the Riemann surface wrapped by the D3-branes. Topologically the space is a disk with a Z k orbifold singularity at the centre and smeared D3-branes on the boundary.
The paper is organised as follows. In section 2 we will review the construction of AdS 3 solutions in type IIB preserving N = (2, 2) supersymmetry before reducing the conditions under the assumption of the existence of a flavour symmetry. We next consider solutions where the branes wrap a constant curvature Riemann surface in section 3. We begin by reviewing the field theory in section 3.1 before constructing a simple holographic dual in section 3.2 and analysing the regularity and central charge of the solution in section 3.3. We generalise this solution further in section 3.4 before computing additional observables of the theory and comparing to the field theory results. In section 4 we consider the holographic duals of 4d N = 4 SYM on a topological disk equipped with a non-constant curvature Riemann surface, analysing the solutions in detail. Some technical material is relegated to two appendices.

N = (2, 2) AdS 3 solutions
We now turn our attention to constructing the supergravity dual of the 2d N = (2, 2) SCFTs discussed in the introduction. We will reduce N = 4 SYM on a Riemann surface, breaking the SU(4) R R-symmetry to its U(1) 3 Cartan in order to perform a topological twist. As such, the resultant two-dimensional field theory should admit three U(1) symmetries: two will furnish the U(1) L ×U(1) R R-symmetry of the N = (2, 2) superconformal algebra whilst the third will be a flavour symmetry of the theory. Given these considerations, the metric of the gravity solution should take the form where I 1 , I 2 are two line intervals. Solutions of this form, supported by five-form flux were classified in [20]. 6 The metric is given by with AdS 3 having unit radius and L a constant. Supersymmetry fixes the internal metric to take the form ds 2 (X 7 ) = cos 2 ζ(dψ 1 + σ) 2 + sin 2 ζdψ 2 2 + sin 2 ζ 4y 2 cos 2 ζ dy 2 + sin 2 ζ y g (4) The coordinate y defines an almost-product structure, implying that the one-form σ has "legs" only along the four-dimensional base, though it may still depend on y non-trivially.

7)
6 For convenience we make a few redefinitions of the results in [20], in particular we redefine the warp factor via e −4∆ y = sin 2 ζ. 7 One can reduce the number of these conditions further by reinterpreting the conditions in terms of derivatives of the determinant of the metric, see [20]. This reformulation is reminiscent of the ones appearing for AdS5 solutions in M-theory [64] and AdS4 solutions in massive type IIA [63]. However, for our purposes the most convenient formulation of the conditions are the ones we present here.
The metric is supported by the five-form flux Supersymmetry implies that all equations of motion are satisfied provided all the torsion conditions given above hold. Contrast this with the N = (0, 2) case [23], where one must also impose the Maxwell equation for F (2) .
Consistent with these geometries preserving N = (2, 2) supersymmetry, which has Rsymmetry group U(1) L ×U(1) R , the metric admits two independent Killing vectors: The generator dual to the left-moving R-symmetry is ∂ ψ 1 −∂ ψ 2 whilst the right moving is dual Recall that the solution we are looking for should have three U(1) isometries.
The general form of the internal metric accounts for two of the three required, and therefore we require that the metric g (4) should admit a single Killing vector.

Embedding into the (0, 2) classification
Before we proceed, it is useful to study the embedding of the N = (2, 2) conditions into the classification of N = (0, 2) solutions [23]. We first want to isolate the right-moving R-symmetry vector field, to do this we make the change of coordinates A small rearrangement of terms after the coordinate transformation, and identifying the warp factor to be puts the solution into the form This is the same form for the metric in the N = (0, 2) classification [23]. 8 The one-form dual to the R-symmetry vector is (2.17) and the Kähler base Y 6 is defined above. The SU(3)-structure forms, in terms of the SU(2)structure ones are given by and it is simple to see that they satisfy the N = (0, 2) conditions, 19) provided the N = (2, 2) torsion conditions (2.4)-(2.10) hold. Moreover, since N = (2, 2) supersymmetry imposes the Maxwell equation for F (2) it follows that the N = (0, 2) master equation is also satisfied.

Canonical form of solutions with T 3 fibration
In appendix A we have reduced the torsion conditions (2.4)-(2.10) under the assumption that g (4) contains an additional flavour U(1) direction. The solution is determined by a potential Once a solution to (2.20) has been provided the metric is given by Dualising along the non-R-symmetry directions to type IIA and then uplifting to M-theory one will obtain an AdS 3 solution in the class discussed in [65]. It would be interesting to study the solutions discussed in the following sections from an M-theory perspective.
The form of the metric suggests that one should look for solutions with D3-branes wrapped on the (X 1 , X 2 ) directions; we will do just that in section 3. However, we should stress that these directions can lie in the co-dimensions of the wrapped D3-branes as we shall find in section 4. Our focus in this work are cases where (X 1 , X 2 ) span a Riemann surface of constant curvature. Going away from the constant curvature metric and including punctures on the Riemann surface is beyond the scope of this work, however the generalisation of our ansatz in this direction is obvious, though technically challenging.

D3-branes wrapping a constant curvature Riemann surface
In this section we will consider the compactification of 4d N = 4 SYM on a constant curvature Riemann surface Σ g , of genus g, preserving N = (2, 2) supersymmetry. We begin in section 3.1 by reviewing the construction of the 2d SCFT arising from the compactification. In particular, we will compute the central charge and the R-charges of baryonic operators with which we may compare to a dual gravity computation. Our gravity analysis begins with the reduction of the general N = (2, 2) supersymmetry conditions of (2.2) for a constant curvature Riemann surface. We find that there is no solution for g = 0, whilst the solution for g = 1 has an enhancement of supersymmetry to N = (4, 4) and is the double T-dual of the D1-D5 system. For g > 1 we find a family of solutions which generalise the Maldacena-Nuñez solution [2], breaking the SU(2) symmetry of their solution to U(1). We present the regularity analysis of these solutions before computing their central charge and R-charges, which match with the field theory analysis in section 3.1.

Twisted D3-branes
Let us consider compactifications of 4d N = 4 SYM on a Riemann surface. As is well known when the Riemann surface has non-trivial curvature it is necessary to perform a topological twist of the theory in order to preserve supersymmetry. 9 Let us begin by defining our conventions for 4d N = 4 SYM. For the moment we will use N = 4 representations, however as we will explain later, in order to compare to the R-charges computed in gravity it is useful to rewrite the theory in terms of N = 1 multiplets. 9 Recently, compactifications which twist with the isometry of the compactification space have been investigated, see for example [66].
As is well known 4d N = 4 SYM contains a single N = 4 gauge multiplet, which in N = 1 language consists of a single N = 1 gauge multiplet and three chiral superfields Φ i .
The bosonic symmetry group of the theory is Generically this does not preserve any supersymmetry unless the Riemann surface is the two-torus T 2 . Since we are interested in preserving (2, 2) supersymmetry in the resultant 2d SCFT we must perform a topological twist. We will twist with the Cartan of the SU(4) R R-symmetry, namely U(1) 3 . Under the Cartan the representations of SU(4) R that we need, decompose as (see for example [67]): and to extract out the R-charges, we will perform the analysis here as well. We turn on a background gauge field A along the generator T with T i the generator of the i'th U(1) in the U(1) 3 Cartan. Supersymmetry of the theory is preserved subject to a Killing spinor equation being satisfied. By turning on a background gauge field, the Killing spinor equation on the Riemann surface is given by The gauge field A has field-strength F = −κT dω, for κ = 0, with ω the spin connection on the . The constant s is ±1 for positive/negative chirality spinors respectively. In order for supersymmetry to be preserved we must tune the parameters a i such that the Killing spinor equation admits constant spinor solutions. This requires the contribution from the spin connection to be cancelled by the background gauge field.
Using the decomposition in (3.5) and denoting the spinors as m ± , m ∈ {1, 2, 3, 4} we see that constant spinors satisfy the Killing spinor equation provided for each of the spinors m ± respectively. N = (2, 2) supersymmetry is preserved if the a i are chosen such that two of the conditions above are satisfied, one for a positive chirality spinor and one for a negative chirality spinor.
Without loss of generality, we can choose these spinors to be 1 + and 2 − , which implies that the a i 's satisfy Recall that a 2d N = (2, 2) SCFT admits a U(1) L ×U(1) R R-symmetry. We may identify the right-moving R-symmetry by requiring that the positive chirality spinor has R-charge 1 and the negative chirality spinor has vanishing R-charge under it. Conversely under the leftmoving R-symmetry the positive chirality spinor has vanishing R-charge whilst the negative chirality spinor has R-charge 1. The most general right-moving U(1) R-symmetry with these properties is given by 10
whilst for the left-moving R-symmetry we have Note in addition that parametrises a flavour symmetry of the theory. For a theory which is conformal at the IR fixed point it is necessary that the 't Hooft anomaly 11 of a left-and right-moving current vanishes [68]. In addition at the fixed point the 't Hooft anomaly for a flavour and R-symmetry must vanish, therefore altogether we must impose Note that the first two are the conditions imposed by c-extremization, whilst the latter is an additional constraint which imposes N = (2, 2) supersymmetry. As an aside, if one naively performs c-extremization preserving N = (0, 2) supersymmetry and then tunes the parameters to preserve N = (2, 2) supersymmetry as in (3.9) one finds the non-sensical result that the central charge vanishes. By imposing the additional constraint that k LR = 0 we alleviate this problem.
In order to evaluate the 't Hooft anomalies we need to know the multiplicities of the 2d fermions. This can be computed by using the Riemann-Roch index theorem. The difference between the number of zero modes on Σ g is given by [2] n The constraints from (3.14) are satisfied provided which gives the following 't Hooft anomalies for the left-and right-moving R-symmetries Consequently the central charges are given by where we have used c R = 3k RR and c L = 3k LL . Note that the central charge is only well- We will see later that these are indeed reproduced by the gravity solution of section 3.4.5.

A simple ansatz for constant curvature Riemann surfaces
In this section we consider a simple ansatz for D3-branes wrapping Riemann surfaces of constant curvature. As we shall see only the cases of T 2 and H 2 give rise to sensible holographic duals, the former reproduces the D3-D3 near-horizon while the latter generalises the N = (2, 2) solution of [2]. Our starting point is the system presented in section 2.2.
In order to make the comparison with the field theory more manifest it is useful to first make the change of coordinates where m ± are constrained to satisfy This change of coordinates puts the defining equation into the form We will make the assumption that the Riemann surface is of constant curvature, as such we assume that the warp factor e 2A appearing in front of the Riemann surface takes the factorised where the potential A 0 satisfies 12 Given the definition of e 2A , after substituting this ansatz into (2.20), the potential D is required to take the form with c an integration constant and a ± constants satisfying In addition we have defined the scalarÃ 0 which satisfies and has solutionÃ With these definitions equation (3.22) reduces to which we must solve for I. Notice that this equation is left invariant when I is shifted by any linear function of (t + − t − ), so we can without loss of generality set such terms to zero. We will take a similar ansatz to that used in [11], namely where we define In general the constants are fixed as

33)
12 In order to generalise to include punctures on the Riemann surface one should add source terms to the right-hand side of equation (3.24).
with additional constraints that depend on the curvature of the Riemann surface, namely .
Notice that we have so that when m ± = 0 and likewise the right-hand side of this expression, (µ 0 , µ + , µ − ) define an embedding of a surface into a three-dimensional ambient space. The metric of the constant curvature solutions take a simple form in terms of these coordinates, we have in general that where we define the warp factor 37) and the fibered terms Clearly, positivity of M 5 demands c ± > 0 and this manifold has the appearance of a U(1) 3 preserving deformed five-sphere. However, as we will see shortly, the actual topology of this space depends on that of the Riemann surface. We shall study the distinct solutions in more detail in the next section.

Analysis of solutions
We saw in the previous section that the solution has the appearance of a U(1) 3 preserving deformed five-sphere, fibered over a Riemann surface, subject to the embedding condition (3.35). In this section we will study the regularity of the solutions for all choices of Riemann surface.

T 2 case
We begin by considering the case κ = 0, taking the solution of (3.34) without loss of generality. The condition (3.35) then informs us that we should fix and one finds that η ± are merely two independent linear combinations of (ψ 1 , φ), that we shall identify as η ± = dτ ± so that in other words the warp factor is constant and the internal five-manifold is S 3 × R 2 . Locally there is of course no difference between this and S 3 × T 2 , making the entire space locally AdS 3 × S 3 × T 4 -clearly then we have reproduced the D3-D3 near-horizon 13 , one can check that the flux is consistent with this. There are no further solutions with κ = 0.

S 2 case
In this section we shall study the case κ > 0, i.e. the two-sphere case. When κ = 0, positivity of the Riemann surface factor in (3.36) demands that so when κ > 0 we must have 2 > 1, the symmetry of the solution means we can take this to be > 1 and so m − < 0 without loss of generality. Turning our attention to (3.35) we see that when the right-hand side is positive/negative it becomes an embedding equation for dS 2 /AdS 2 , both of which are non-compact and pseudo-Riemannian -however (µ ± , µ 0 ) do not appear in the definition of M 5 with the correct signs to give rise to sub-manifolds of these topologies and in fact M 5 remains positive in both these cases. One can parametrise without loss of generality, however in either case 0 ≤ r < ∞, and the upper limit is at infinite proper distance in the ten-dimensional space, so the solutions following from these tunings are unbounded. There is of course one final option, fix Here one can parameterise but once again 0 ≤ r < ∞, with the upper bound at infinite proper distance. Hence while solutions exist with S 2 , they do not represent good holographic duals to CFTs.

H 2 case
In this section we shall study the case κ < 0, where the Riemann surface is a compact quotient of H 2 . Positivity of the metric in this case requires that 2 < 1, meaning that m ± > 0 and so (3.35) embeds a two-sphere in R 3 . We thus fix without loss of generality which makes the warp factor become Λ = 1 + sin 2 α + cos 2 α cos(2β) , (3.46) which has no zeros as −1 < < 1. The internal five-manifold on the other hand can be written as which is topologically a five-sphere with 0 ≤ α < π 2 , 0 ≤ β < π 2 , 0 ≤ ψ 2 < 2π and the U(1) directions τ ± are defined as and have period 2π. Notice that when = 0 we have Dβ = dβ and the metric becomes that of the AdS 3 solution of [2]. The general solution is a parametric deformation breaking the U(1)×U(1)×SU(2) isometry to U(1) 3 , whilst preserving N = (2, 2) supersymmetry. One can show that the two form flux is given by

Flux quantisation
In order to compare to CFT quantities we must first quantise the flux. To this end it is helpful to know the internal part of the five-form flux explicitly, we find The relevant part of this is the first line. We should impose in units where g s = α = 1. The integral is non-trivial but can be performed exactly, ultimately we find that we must tune

Central charge
In this section we will study the central charge with which we may compare to the field theory results presented in section 3.1. Since the R-charges may be obtained from a specification of the more general solution we present later in section 3.4.5, we will suppress the calculation of them here. The Brown-Henneaux formula [70] c sugra = 3 2G For the H 2 solution we have where by dvol(S 5 ), we mean the volume form on the unit norm round five-sphere, integrating to π 3 . We thus find where we have used that the volume of the Riemann surface is given by where ρ is the Ricci form, which follows since the chosen metric on the Riemann surface is Einstein.

A more general solution
In this section we will generalise the previous solution by adding two additional free parameters. We generalise the previous solution by taking as ansatz for I where Note that the previous ansatz for I is recovered for r + = −r − = 1 and c ± = 2R ± . As before the solution is split into distinct cases depending on the genus. The universal sector fixes the constants a + = κ 4 (2 + r + + r − ) , m ± = 1 ± 2 , R ± = 0 (3.62) whilst the curvature dependent part satisfies Note that these reduce to (3.34) for r + = −r − = 1 and c ± = 2R ± as they should. Moreover, we have a similar constraint on the functions µ 0 , µ ± as in (3.35) given by As before, this defines an embedding equation of a surface into a three-dimensional ambient space. Note that the inclusion of the parameters r ± shift the coefficients of the functions µ ± so that it is no longer simply m ± as in (3.35). We may now assemble the full metric of the solution. We have From the presentation above we see that the metric looks locally like a U(1) 3 preserving deformed five-sphere. In addition, from the N = (0, 2) perspective of section 2.1, one can identify the one-form η as the one-form dual to the R-symmetry vector. This puts the metric on the five-sphere in the form of a U(1)-fibration over CP 2 equipped with a non-Einstein metric. The two-form flux is given by with

T 2 case
As in section 3.3.1 the T 2 example reduces to the AdS 3 × S 3 × T 4 solution, as such we shall suppress further analysis of this case.

S 2 case
Let us turn to the S 2 case. It is clear that we require R ± > 0 and therefore positivity of the warp factor of the Riemann surface, e 4ν implies that In each case the solutions will be non-compact and therefore they do not give rise to good holographic duals as in section 3.3.2.

H 2 case
Instead, let us consider the g > 1 case. Positivity of the warp factor requires Due to the symmetry of the solution we may assume without loss of generality that r + > r − ; if this is not the case then we may relabel as + ↔ −. We see that we must therefore require and consequently the embedding equation is that of a two-sphere in R 3 . We may then parametrise as giving rise to a compact internal manifold.

Toric data and three-cycles
We now want to analyse the regularity of these generalised solutions. We saw earlier that regularity of the metric can be ensured by correctly identifying the embedding coordinates.
In this section we will take an alternative approach using the toric data of the five-sphere at fixed point on the Riemann surface. Note that the compact part of the metric in (3.65), without the Riemann surface, has been written as a U (1)  compute the R-charges of these operators and compare to the field theory. In the following we will consider just the κ = 0 case since the κ = 0 case reduces to the previously studied The form of the ansatz for the function I is reminiscent of the canonical potential for a symplectic toric manifold. Here, the µ 2 • 's play the role of the functions defining the edges of the polytope. Note that the metric has a singular like behaviour when any of µ 0 or µ ± vanish, the goal is to obtain constraints such that the degeneration is smooth. 15 To this end we construct Killing vectors which have zero norm on some locus of the solution. The degeneration locus is associated to the edges of a 2d polytope over which the U(1) 3 is fibered.
Requiring that the surface gravity of these Killing vectors is normalised to 1 on the respective degeneration surface, and giving the dual coordinate period 2π leads to a smooth degeneration and a regular metric. In the case at hand there are three Killing vectors, which, after the prescribed normalisation condition discussed above, are given by (3.74) Defining a new 2π-periodic coordinate for each of these three vectors via, k • = ∂ ψ• , we find that the metric is regular. In order to compute various integrals it is useful to perform a change of coordinates, These new 2π-periodic coordinates give a free action for the U(1) 3 torus action.
To extend this regularity analysis to the full manifold we now need to make sure that the fibration over the Riemann surface is well defined. Viewing the fibration as a gauging of the U(1) 3 torus action via with the A • gauge fields on the Riemann surface. The fibration is well-defined provided that the field-strength of the gauge fields A • over the Riemann surface have integral period.

Explicit computation gives
and therefore the fibration is well-defined since From the non-trivial gauge fields we can read off the topological twist of the dual field theory.
We see that the topological twist is performed by turning on a background gauge field of the Finally we may identify the (right-moving) R-symmetry of the solution from the expression for the one-form η. We find (3.80) Note in particular that the R-symmetry does not mix with the isometries of the Riemann surface. This is a distinction between the solutions discussed in this section and the ones in the following section 4.

Flux quantisation and observables
Having discussed the geometry of the solution and the various three-cycles let us first consider the quantisation of the five-form flux. We must ensure that this is an integer over all integral five-cycles in the geometry according to the quantisation condition (3.51). Since the geometry is a five-sphere fibered over a Riemann surface the only integral five-cycle is the five-cycle at fixed point on the Riemann surface. The internal part of the five-form is where the suppressed terms do not have support on the topological five-sphere. Integrating the flux over the five-sphere we find the quantisation condition We can now compute the R-charges of baryonic operators in the SCFT. These correspond to wrapping D3-branes on three-cycles on the five-sphere. The three-cycles, let us denote them by S • , are the ones obtained by going to the facet of the polytope, upon which a U (1) shrinks. 16 Following [20] (see also [71]) the R-charge is given by where the last inequality has been written using (0, 2) language. Explicit computation gives which after using the definition of N become Note that the R-charges are all positive and moreover they sum to 2N as expected [71], and find agreement upon identifying Finally, let us compute the central charge. As before we use the Brown-Henneaux formula as given in (3.56), and as expected from the field theory analysis, the result is Note in particular that the newly introduced parameters drop out of the central charge, despite not dropping out of the R-charges of the fields.

D3-branes wrapping a topological disk
In [1] a solution with D3-branes wrapping a spindle was obtained in 5d U(1) 3 gauged supergravity and then lifted to ten dimensions. The result locally coincides with a solution originally derived in [72], but the interpretation in terms of a spindle and CFT analysis is new. The solution supports multiple charges and generically preserves N = (0, 2) supersymmetry, however, as we shall establish in this section, for a certain tuning of these charges it experiences an enhancement to N = (2, 2), where the D3-branes now actually wrap a Riemann surface with the topology of a disk and includes additional source D3-branes. This solution can be embedded in the canonical form of section 2.2, interestingly the Riemann surface which the branes wrap is not that of the (X 1 , X 2 ) directions. Instead, as we establish in section 4.3, (X 1 , X 2 ) span a two-sphere in the co-dimensions of the wrapped D3-branes.

Summary of [1]
The metric of the solution of [1] takes the form 17 1 3 x + 3K I , where µ I embed a unit radius two-sphere into R 3 and the functions of x are where [1] constrain K 1 + K 2 + K 3 = 0, which kills the O(x 2 ) term in H. The x direction can then be bounded between two real roots of the cubic polynomial P for which H = 0, and the surface becomes Σ 2 = WCP 1 [n + ,n − ] , which is an orbifold known as a spindle. Here n ± ∈ N, are related to the period of ϕ. When K 1 = K 2 = K 3 = 0 the solution of [13] is recovered. In this section we shall consider a different tuning of these parameters, not considered in [1]. N = (2, 2) tuning

An
We would like to extract an N = (2, 2) solution from the local solution of the previous section.
The first thing to appreciate is that K 1 +K 2 +K 3 = 0 is not a requirement for supersymmetry (at least in ten dimensions), so let us instead tune the parameters as The connection A 3 then becomes topologically trivial, {A 1 = A 2 , X 1 = X 2 = X −2 3 } and so the U(1) 3 isometry of the five-dimensional internal space gets enhanced to SU(2)×U (1) -similar to the N = (2, 2) solution of [2]. This solution experiences an enhancement of supersymmetry to N = (2, 2), as we prove in the following section, 4.3. The two functions defining the solution become For certain tunings of (K, x 0 ) the polynomial Q contains two real roots x = x ± namely Σ 2 exhibits spindle-like behaviour between these loci, however a physical metric should be real and positive which requires {Q > 0, x > x 0 , 3K + x > 0}, and Q is strictly negative for x − < x < x + . It is however possible to achieve a physical metric and to bound x to the 4x 0 ). At x = x 0 H and P both vanish, but P/H remains finite, so similar to the solution of [9], the topology of Σ 2 is now that of a disk with R 2 /Z k orbifold singularity at x = x − . Close to x = x − only P exhibits a zero, at this loci, so that we have a R 2 /Z k orbifold when the period of ϕ is ∆ϕ = for k ∈ N or a regular zero for k = 1. The behaviour close to x = x 0 in the full space needs more care to disentangle: first one should note that warp factor W can now be written as which provided µ 3 = 0, remains finite as x approaches x 0 . Similarly around x = x 0 when . We see then that the sub-metric spanned by (x, φ 3 + ϕ) vanishes as R 2 in polar coordinates. The behaviour as both µ 3 , (x − x 0 ) → 0 is a bit subtle, one can study it by changing coordinates to and then expanding about r = 0. To leading order the metric then becomes where (θ, φ 3 + ϕ) span the two-sphere and (β , φ 1 , φ 2 ) span the three-sphere (in topological joint coordinates). It is not hard to see that this behaviour is singular, it is in fact that of a partially localised stack of D3-branes with worldvolume (AdS 3 , ϕ) that are smeared over S 3 .
Thus we see that at generic points of the deformed two-sphere spanned by µ I the solution is bounded between a regular zero at x = x 0 and a R 2 /Z k orbifold at x = x − , with flavour D3-branes at the loci (x = x 0 , µ 3 = 0).

Embedding the topological disk solution into section 2.2
Above we have constructed a solution by taking a limit of the multi-charge solution of [72] and then analysed its regularity. In this section we prove that the solution does indeed preserve N = (2, 2) supersymmetry as claimed. We shall do so by embedding it into the general form of section 2.2. Much of this can be reverse-engineered by comparing (4.11) to (2.3) and (2.21). This is sufficient to establish how the coordinates are related, extract the values of (∂ 2 Θ D, ∂ y D) and to identify the Riemann surface of the local ansatz, which surprisingly turns out not to be Σ 2 . In fact it is not hard to show that the (X 1 , X 2 ) directions in (2.21) correspond to the two-sphere spanned by (β, τ 1 ) in (4.11), as such this is another example in the constant curvature ansatz, our first with a two-sphere. We must identify the coordinates of (2.21) as and decompose the potential defining the system as (4.14) so that the partial differential equation of (2.20) reduces to It is possible to extract the following derivative of I from the metric 18 where we introducẽ 18 We can also extract an expression for ∂y∂ΘI which is consistent with these. (4.17) The definitions (4.16) are self consistent (i.e. they give rise to the same ∂ 2 Θ ∂ y I) and are already enough to confirm that (4.15) is satisfied, which is an important consistency check. Further, though non trivial, the definition of ∂ y I in (4.16) can be integrated to give I up to a function of Θ which can then be fixed 19 by imposing consistency with the expression for ∂ 2 Θ I. The final expression is rather complicated but of closed form, we find it convenient to decompose it as I = I 0 + I + + I − + I Θ , where q ± ≡ q 1 ± q 2 and so on for p ± , l ± and m ± , with We have confirmed that (4.18) does indeed solve (4.15), so the solution preserves N = (2, 2) supersymmetry as claimed.

Flux quantisation and central charge
We now want to turn our attention to computing some observables of the dual CFT. We should first quantise the flux on M 5 , the relevant part is where . . . either vanishes on M 5 or does so when it is integrated over it. We then have We should also quantise the connections of the fibrations such that we have a well defined orbifold fibration, this demands that we impose that the field strengths F I = dA I satisfy 20 , for M ∈ N, which fixes Given the constraints on (K, x 0 ) discussed below (4.5) the positive branch is valid for M ≥ 1 and the negative one for M ≥ 2, so they are equivalent -to be concrete we take the former.
We can compute the holographic central charge using (3.56), we find after substituting for L and 3K + x 0 this takes the form where again we expect the central charges of the dual CFT to be c L = c R = c sugra . Clearly we have a deviation from the behaviour derived in section 3.1, but this should be no surprise as the D3-branes are now wrapping a topological disk which does not have constant curvature.

Discussion
In The dual field theory computation in this case is somewhat subtle. One cannot simply specialise the field theory result in [1] to our current case. Concretely, one finds that after c-extremization has been performed, despite the trial central charge having a well defined 20 Recall dA3 = 0. extremal point, the central charge is identically 0. Clearly this is inconsistent. This is in fact the same problem one would encounter if one naively specialises the general N = (0, 2) twist of 4d N = 4 SYM in [68] to a N = (2, 2) twist in the constant curvature Riemann surface case. The problem arises because c-extremization mixes the holomorphic (right-moving) and anti-holomorphic (left-moving) R-symmetries. We leave recovering the supergravity result from a field theory computation to the future.

A Reducing the conditions
We have seen that the gravity solution is fixed by determining a four-dimensional metric satisfying the torsion conditions (2.4)-(2.10). Our goal in this section is to obtain the reduced conditions after substituting in an ansatz for the four-dimensional base. We will take the most general 4d metric admitting a single U(1) isometry as used in [7] to study AdS 5 solutions in M-theory. Explicitly, the metric is where V I and V R have legs only along the Riemann surface parametrised by the coordinates (x 1 , x 2 ). The vector ∂ φ is taken to be a Killing vector and thus the three scalars A, B, C are independent of φ, though they may depend on the other three coordinates and y. We will use conventions in which the Hodge dual on the Riemann surface satisfies and the volume form is given by In addition, let us define the one-forms and the twisted exterior derivative on the Riemann surfacê With the above metric ansatz the SU(2)-structure forms are

A.1 Reducing the conditions
Let us reduce the torsion conditions (2.4)-(2.10) on the ansatz above. We begin with the conditions for the holomorphic volume form before moving on to the Kähler form.
Reducing (2.9): ∂ y Ω equation From equation (2.9) we obtain the conditions Since the first condition implies that the scalar C is independent of the y coordinate it follows that it may be removed by a change of coordinates. We will therefore set C = 0 in the following. Moreover we may solve the final condition by introducing the scalar Λ: First note that equation (A.20) implies that the twisted differential operatord 2 is nilpotent. Moreover, it follows that V R is locally exact with respect to this twisted exterior derivative and we may write it (locally) as for some function Γ. To proceed it is useful to make the change of coordinates [7] X i (x) = x i , −Γ(X, Θ, y) = θ .
With this change of coordinates the derivatives are given by whilst the twisted differential becomeŝ With this change of coordinates (and dropping the superscript X on d X 2 from now on) the one-form η θ takes the simple form where we have defined Since V 0 is independent of both θ and y it follows that for a suitable X dependent gauge choice for Γ we may, without loss of generality set V 0 = 0. It then follows that (A.18) is equivalent to The ten-dimensional metric becomes and we have defined where the reason for this labelling of the one-forms will become apparent soon. If we make the change of coordinates specified by dθ = GdΘ + Gdy , (A. 39) under which the partial derivatives transform as ∂ Θ = G∂θ , ∂ y → ∂ y + G∂θ , (A. 40) the functions G and G are eliminated from both the metric and the remaining conditions to solve. We may therefore set G to any non-zero constant, set G = 0 and revert back to the previous set of coordinates. We fix G = 1 4 so that the condition from setting G = 0 implies the integrability condition The flux is given by with J as given in (A.6).