Holographic duals of M5-branes on an irregularly punctured sphere

We provide explicit holographic duals of M5-branes wrapped on a sphere with one irregular puncture and one regular puncture of arbitrary type. The solutions generalise the solutions corresponding to M5-branes wrapped on a disc recently constructed by Bah-Bonetti-Minasian-Nardoni by allowing for a general choice of regular puncture. We show that the central charges, flavour central charges and conformal dimensions of BPS operators match with a class of Argyres-Douglas theory.


Introduction
Compactifying SCFTs on compact manifolds has been a fruitful avenue for constructing new SCFTs. Given a parent SCFT with a holographic dual, it is natural to consider the holographic dual of the compactified theory. The earliest examples of such dual pairs were constructed in [1], and studied the compactification of both 4d N = 4 super-Yang-Mills and the 6d N = (2, 0) theory on a Riemann surface with genus g > 1. Since then this avenue of research has been extended in multiple directions. In this work we will primarily be interested in 4d N = 2 theories that can be obtained from M-theory. Theories of class S are 4d N = 2 SCFTs which arise from wrapping the 6d N = (2, 0) theory living on M5-branes on a punctured Riemann surface with gravity duals classified in [2,3] 1 . Extensions to 4d N = 1 theories from wrapped M5-branes have been studied in [6][7][8][9].
The prototypical setup for studying branes wrapped on various compact cycles is to consider embedding the cycle in some larger geometry, for example a Calabi-Yau or G2 manifold and taking the metric on the compact space to be the one with constant curvature.
Given that we typically view these types of solutions as the IR fixed point of an RG flow interpolating between dimensions, or rather the near-horizon of some black object, one can motivate this ansatz, at least for Riemann surfaces, from known uniformization theorems [10,11]. Under some assumptions, they state that one can take the UV solution to have an arbitrary metric on the Riemann surface, and along the RG flow this is washed out leaving just the constant curvature metric at the IR fixed point. Key assumptions for these theories are that supersymmetry is realised by a topological twist and that the metrics are smooth.
Recently, solutions with non-constant curvature metrics have been studied in string and M-theory, thus evading the uniformization theorems. These solutions are known as spindles [12][13][14][15][16][17][18][19][20][21][22] and discs [18,[23][24][25][26][27][28][29] and can be viewed as the horizon of accelerating black objects. 2 Spindles are the orbifold WCP 1 n − ,n + , i.e. a two-sphere with conical deficits at both poles. Discs on the other hand have the topology of (unsurprisingly) a disc, with an orbifold singularity at the centre and a boundary on which the metric is locally a cylinder but singular. The two types of solutions are intimately related and have been shown to be different global completions of the same local solutions [18,25]. Apart from having metrics of non-constant curvature one of the interesting features of these geometries is the method in which supersymmetry is preserved. The mechanism is not the usual topological twist but instead requires a mixing 1 The type IIA picture is given in [4,5]. 2 For discussion on accelerating black holes not embedded in string or M-theory see for example [30][31][32] and references within.
between the parent R-symmetry and the isometry direction of the compactification space. For spindles there are two different types of twist, the anti-twist or the topological topological twist [20], while for discs a different mechanism involving a holonomy for a gauge field on the boundary of the disc allows for the preservation of supersymmetry. See also [33,34] which consider 4d orbifolds with non-constant curvature.
In this work we will be interested in extensions of disc solutions, specifically for M5-branes wrapped on a disc. In [23,24] (BBMN) the first disc solution was presented and a dual field theory was proposed. The dual field theory is a 4d N = 2 SCFT of Argyres-Douglas type and constitutes the first holographic dual for such a theory. The dual field theory was shown to be (IN ,k , Y l ), which is the theory on a stack of M5-branes of A-type wrapped on a twice punctured sphere, with one irregular puncture of type I at one pole and a regular puncture of a particular type at the other pole, we will review this nomenclature in section 4. The goal of the present paper is to extend to more general Argyres-Douglas theories. In particular we will provide holographic duals for the SCFTs with arbitrary regular puncture and fixed type I irregular puncture. We provide evidence for the proposal by matching various observables on the two sides.
The outline of the paper is as follows. In section 2 we begin by reviewing the disc solution in [23,24]. Our presentation uses a different set of coordinates which are better adapted for later. We embed the solution into the classification of N = 2 AdS 5 solutions in M-theory in [2,35] studying the equivalent electrostatic problem. The electrostatic problem has some new and novel features, in particular unusual boundary conditions for the potential. In section 3 we generalise the disc solution building on the original solution. We show that the new solutions are well-defined supergravity solutions, performing a regularity analysis and flux quantisation. We proceed to compute various observables of the gravity solution with which to match to the field theory analysis in section 4. We provide a dictionary between the field theory and gravity solution with which we can compare the observables on the two sides and find perfect agreement in the holographic limit. We relegate some material to two appendices.
The first shows how to obtain AdS 7 and the Maldacena-Nunez solutions from our starting point, whilst the second performs an independent check of the observable computations by using anomaly inflow.

M5-branes on a disc
In [24] an AdS 5 × Σ solution in 7d gauged supergravity was found where Σ is topologically a disc. The boundary of the disc corresponds to a singularity of the overall metric, while the centre of the disc has a conical singularity. Upon uplifting the solution to 11d supergravity on an S 4 one obtains a 1/2 BPS AdS 5 solution, and one can give a physical interpretation of the singularities; the boundary of the disc arises due to a stack of smeared M5-branes whereas the conical deficit is due to the presence of a monopole. In [24] they conjectured that the solution was holographically dual to an Argyres-Douglas theory with one regular and one irregular puncture. In the following we will review the solution, albeit from a different parametrisation, before embedding the solution into the classification of N = 2 AdS 5 solutions of 11d supergravity constructed in [2,35]. We use the reformulation of the solution in terms of an electrostatics problem for a single potential, determining it for the disc solution before studying its properties and flux quantisation within the electrostatics reformulation.

AdS 5 × Σ solutions of 7d gauged supergravity
In this section we will study the AdS 5 × Σ solution of 7d U(1) 2 gauged supergravity originally found in [24] using the conventions there for the gauged supergravity theory. As pointed out in [18,25] one can obtain disc solutions as different global completion of the same local solutions from which one may construct spindle solutions. In the following we will use the parametrisation of the spindle solutions in [36] specialised to the disc solution. 3 The solution is ds 2 7 = wP (w) where ds 2 (AdS 5 ) is the unit radius metric on AdS 5 , satisfying R µν = −4g µν and the functions take the form The solution depends on two real constants, s i . In order to obtain a disc the functions f (w) and P (w) must have a common root. Given the form of the polynomials, it is clear that this must necessarily be at w = 0 and therefore in order to obtain a disc we set, without loss of generality, s 2 = 0. In fact this limit leads to an enhancement of supersymmetry, with the solution preserving N = 2 supersymmetry, rather than N = 1. We will show this later by explicitly embedding the solution into the N = 2 classification of AdS 5 solutions of 11d supergravity in [2,35]. This local metric also admits other interesting limits. One may recover both pure AdS 7 and the Maldacena-Nunez solution [1] by taking different limits of this local metric as we show in appendix A.

Regularity in 7d
Let us begin by considering the regime of the s 1 which gives rise to a well-defined metric. We require f (w) to admit two roots, one at 0 and the second positive. The latter condition is necessary for the metric to have the correct signature, since there are terms proportional to w appearing in the metric. The roots of f (w) are In order for there to be a real positive root and for the scalars to be positive we require This leads to two positive roots and we take the domain of the line interval parametrised by We must now check how the metric degenerates at the end-points.
First consider the end-point at w = w − . Since P (w − ) = 0 we need only consider the metric on Σ at this end-point. Expanding the metric on Σ around w = w − we find where w − − w = r 2 . Fixing the period of z to be the space is the orbifold R 2 /Z l .
Let us now consider the end-point at w = 0. Expanding the 7d metric around w = 0 we find ds 2 7 w 3/5 w where we performed the change of coordinate w = r 2 . Clearly this is singular, as one can verify by computing the Ricci scalar, or any other curvature invariant. In addition, the scalars also have a singular behaviour, One should contrast this singular behaviour with the singular behaviour of the 4d and 5d solutions for M2-branes [18,27] and D3-branes [25,26] on discs. One notes that in addition to the singular metric only a single scalar diverges in each of these cases with the other scalars tending to zero. These scalars describe the stretching and squashing of the sphere in the uplifted theory, when written in embedding coordinates adapted to the U(1) n symmetry. In the M2-branes and D3-branes cases the sphere diverges along one direction and shrinks in the remaining directions. In contrast, here we have three scalars parametrising the squashing; two of which diverge and only one vanishes. We will see later, using the uplifted solution, that the behaviour of M5-branes on a disc is somewhat different to that of the M2-brane and D3-brane cases.

Magnetic charge and holonomy
The metric is supported by a single magneticaly charged gauge field. The magnetic charge is defined to be which for the solution at hand is Note that due to the orbifold it is not necessary that Q is integer but rather the weaker condition l Q ∈ Z. As such let us define Since the disc also has a boundary one can define the holonomy of the gauge field on the boundary. One should choose a gauge for the gauge field so that it is globally well-defined on the disc. Since the circle shrinks at the centre of the disc we must require that the gauge field vanishes there. This uniquely fixes the gauge and the globally well-defined gauge field is The holonomy of the gauge field along the boundary is then which gives minus the total magnetic charge threading through the disc.

Euler Characteristic
The final observable that we can compute is the Euler characteristic of the disc. We have where in the going to the second line we have used that the geodesic curvature of the boundary of the disc is 0, it is locally a cylinder there, and hence the boundary contribution vanishes.
As an aside one may express everything in terms of the orbifold weight l and the integer magnetic charge p defined in (2.12). The roots in terms of these integer parameters are , (2.17) and the period satisfies It is useful for later to introduce the 2π-periodic coordinateẑ aŝ

Dictionary to compare with BBMN
To translate between the parametrisation given above and the solution appearing in [24] one should perform the following identifications: This puts the metric into the form which is as given in [24]. Note that the gauge field and scalar are equivalent as well after the above redefinition. By using this dictionary one finds that the regularity analysis performed above agrees with the equivalent analysis performed in [24].
This concludes our review of the 7d solution and we turn our attention to the uplift of the solution to 11d supergravity on an S 4 .

11d uplift and regularity
In the previous section we have studied the regularity of the 7d solution. We have seen that the solution exhibits two distinct singular behaviours; one at the centre of the disc and one along the boundary. In this section we will study the 11d uplift of the solution, focussing in particular on the singular regimes from the 7d solution.
Using the uplifting formula in [37] the metric is with Note that X 0 = X 2 for the disc. Given this symmetry it is useful to parametrise the µ I as 24) and to define Note thatΩ vanishes at (w = 0, µ = 1) but is otherwise positive definite. Next definê then the metric takes the form Note that the singularity at w = 0 of the 7d metric persists in the uplifted solution. This is in contrast to the behaviour of the M2-brane and D3-brane disc solutions discussed in [18,27] and [25,26] respectively, where the line w = 0 is no longer singular only the point (w = 0, µ = 1).
In order to interpret the solution it is useful to observe that it can be written in the form 1]. Away from the boundary of the rectangle the metric is smooth and the fibers are non-shrinking. Along the boundary various fibers shrink, see figure 1. 4 Since the behaviour of the various edges of the rectangle will play a prominent role later we will study this in detail. The results we find are in agreement with the results in [24] and the reader familiar with the analysis there may skip to the next section safely. We will first study the degeneration along the sides away from the vertices.
Consider first the degeneration at µ = 0. We can see that the φ 1 circle shrinks smoothly giving R 2 if φ 1 has period 2π. This is of course the expected behaviour given the S 4 origin.

µ = 1 degeneration
At µ = 1 we see that the S 2 shrinks smoothly giving R 3 . As before this is the expected result given the S 4 origin.  To properly understand the degeneration at w = w − we should rewrite the metric in the form of the S 1 z circle fibered over the S 1 φ 1 circle. It is also convenient to perform a gauge transformation of the gauge field whilst performing the rewriting, δA 1 = n z dz. The choice of the constant n z we will make is such that the Killing spinor on the disc is independent of the angular coordinate. This is achieved by taking n z = − 1 2 . The S 1 z × S 1 φ 1 part of the metric, ignoring the overall warp-factor w 1/3Ω1/3 , after the rewriting takes the form Note that R φ 1 vanishes at both µ = 0 and w = w − whilst R z only vanishes at the point (w = w − , µ = 0). Note also that the function L is piecewise constant on the two-edges: µ = 0 and w = w − . Along µ = 0 it vanishes, whilst along w = w − it is a non-zero constant. Since Ldφ 1 defines a connection for the fibration the physical parameter iŝ and we findL(w − , µ) = −l. This signifies the presence of a monopole at (w = w − , µ = 0).
To see this more clearly let us take the simultaneous limit towards this point. We change coordinates to and then take the r → 0 limit. The S 2 metric remains of finite size and the remaining 4d part of the internal space becomes This is the metric on R 4 /Z l , and is due to the presence of a monopole.
The final edge of the rectangle is the one along w = 0. Series expanding along w = 0 the metric reads Away from µ = 1 this is the metric on an M5-brane wrapping AdS 5 × S 1 z , located at the tip of R 3 and smeared along the two directions spanned by µ and φ 1 .

LLM reformulation
In the previous section we have studied the uplifted solution. In this section we will extend this analysis by showing that the solution can be embedded into the classification of N = 2 preserving AdS 5 solutions of 11d supergravity of [2,35]. 5 Ultimately we want to consider the equivalent electrostatic description of the problem [2], but for ease of exposition we will present the intermediate steps. We will follow the conventions of [2] in the following. Any N = 2 AdS 5 solution of 11d supergravity takes the following form: ds 2 11 = e 2λ 4ds 2 (AdS 5 ) + y 2 e −6λ ds 2 (S 2 ) + ds 2 4 , (2.37) , (2.40) As before, the metric on AdS 5 is the unit radius one. The potential D, which determines the full solution is a solution of the (infinite) Toda equation After a little rewriting the metric takes the form ds 2 11 = e 2λ 4ds 2 (AdS 5 ) + w 2 (1 − µ 2 )e −6λ ds 2 (S 2 ) (θ,φ 2 ) + ds 2 4 , (2.44) From the coefficient of the two-sphere we can then identify Since the solution has an enhancement of symmetry compared to the general classification, there is an additional U(1) symmetry, we define the polar coordinates We now want to identify the radial coordinate r and potential D in terms of w, µ. We find that they are given by , (2.48) with H an arbitrary function of one variable with continuous first derivative. 6 We can fix the function to be of the form H(x) = x α , with α a constant. There are different choices one could make for α, for example if one takes α = 1 2 one finds that the potential D is independent of the coordinate µ. We will instead make the seemingly crazy choice α = −γ −1 , see equation (2.18) for the definition of γ. This turns out to be useful because the metric takes the canonical LLM form upon making the change of coordinates One can check that the potential satisfies the Toda equation (2.42) as it should.

Electrostatics reformulation
In the previous section we have rewritten the solution in terms of the classification of N = 2 AdS 5 solutions of 11d supergravity, determined by a potential satisfying the Toda equation.
For solutions with two U(1)-isometries, like our solution, there is a formulation one can use by performing a Bäcklund transform [2]. Rather than being determined by a potential satisfying the Toda equation, the solution is now determined by a potential satisfying the 3d cylindrical Laplace equation. In terms of this potential the problem can be interpreted as an electrostatic problem with a linear line-charge density λ as we will now review.
To perform the Bäcklund transform we follow the conventions in [2] and introduce the new coordinates ρ, η defined via The fact that the function H is undetermined is due to the conformal symmetry of the solution in the Toda picture. 7 When w− = w+ the centre of the disc becomes H 2 rather than R 2 . The radial coordinate is This behaviour is rather different to the more general case that we will study here. Note that at the H 2 end-point the solution is of the form AdS5 × H 2 . As we show in appendix A.2, taking the limit carefully one obtains the Maldacena-Nunez solution.
The metric and flux after the coordinate transformation become The potential V satisfies the 3d cylindrical Laplace equation To every potential giving rise to a sensible geometry and satisfying the Laplace equation one can define a line-charge density The benefit of the electrostatic description is that the cylindrical Laplace equation is linear and therefore we may construct more general solutions using superposition of known solutions.
Let us now turn our attention to obtaining the new coordinates ρ, η and the potential V for the disc solution. The coordinate ρ is simple to extract in terms of w, µ and is given by where we have taken the positive root without loss of generality. To compute η note that the integrability of the coordinate change implies the two constraints which are independent of the potential V . We may now solve for η which gives This is defined up to the addition of a constant, however since this may always be absorbed by a coordinate transformation later we set this constant to zero. With ρ and η in hand we can now determine the potential V , which is also defined up to the addition of a constant, and again this constant is trivial. This indeed satisfies the cylindrically symmetric Laplace equation in 3d as it should.
We have now determined both the new coordinates ρ, η and the potential V for the electrostatic problem. However in our presentation above the potential is still written in terms of the original w, µ coordinates. To invert this we note that we may determine the w, µ coordinates in terms of ρ, η as where one should insert the expression for w into µ 2 . The final potential is and w should be understood to be the function of ρ, η, depending on the constants p, l given in (2.61). It is interesting to note that the potential can be broken into three pieces each of which are solutions of the 3d cylindrical Laplace equation on their own: The second term is the simplest, non-trivial solution to the cylindrical Laplace equation one can construct. Note that both pure AdS 7 and the Maldacena-Nunez solution have the same form of blocks, see appendix A. Of course this is expected given that both of these solutions can be obtained from the same local solution considered here as we show in appendix A.

Properties of the electrostatic setup
Having reformulated the disc solution in terms of an electrostatic problem we will now study the solution from this vantage point. The first task is to identify the range of the ρ and η coordinates. By inserting the boundary values of w, µ we find that the boundary of the (w, µ)-rectangle is identified with We find that the ranges of ρ and η are Note that the location of the smeared branes (irregular puncture) defines an ellipse in the (ρ, η) coordinates, see the last condition in (2.64). The focal point of the ellipse is at which, for the sign within the domain (+), is the location of the monopole! In figure 2 we have plotted the resultant domain in the (ρ, η) coordinates, colour coded to match the regions in figure 1 for the (w, µ) coordinates. The prototypical example of solutions in the literature of this electrostatic problem includes a non-compact domain, with both ρ and η non-compact, see for example [2,35]. The typical boundary condition imposed is thatV vanishes along η = 0, and we find that this is also true of the solutions discussed here. Some solutions with η compact have been found in the literature, see for example [4,5,38], where the domain is a rectangle. The additional boundary condition imposed in these works is thatV vanishes along a line η = η * for some η * > 0. Our compact domain is quite novel being a quarter of an ellipse. The inclusion of the smeared brane (irregular puncture) leads to a non-trivial boundary condition for the electrostatic problem, that is the boundary condition thatV vanishes on an ellipse. This sets the disc solution apart from previous examples in the literature.

Line Charge
Having determined the potential V it is a simple matter to obtain the line charge λ(η). We where we have used (2.17) and (2.18) to express the result in terms of the magnetic charge defined in (2.12) and the orbifold weight. Note that the change in slope at the monopole point is l, the orbifold weight. The reader familiar with the conditions in [2] may be uneasy that η does not take integer values at the monopole point and that the slope is not integer.
As we will explain later this still gives rise to a well-defined solution, in fact the constraints in [2] are too strong and not all constraints imposed there are needed for a well-defined solution of this type. As we will see the non-integer slope leads to operators in the dual field theory having non-integer scaling dimensions.
As an aside, note that there is a scaling symmetry of the solution. One may perform the for M and Ξ some constants, and retain a solution to the cylindrical Laplace equation. This transformation leads to a transformation of the line charge as and therefore one could use this freedom to make the line charge satisfy the conditions in [2]. Note that the parameter c is precisely the linear shift we could have performed earlier when obtaining the coordinate η. We will refrain from performing these rescalings for the time being.

Line charge kinks analysis
Let us now substantiate our claim that the following line charge and potential do give rise to a well-defined geometry. We will show that the monopole number is indeed l and that the flux is properly quantised. Since the analysis goes through allowing for an arbitrary number of kinks in the line charge and we will need this later, we perform the analysis allowing for which form semi-circles around the monopole in this 2d plane. Expanding the terms appearing in the metric around r = 0 we havė where in taking the derivative of λ around the monopole point we have and the sum is over monopoles higher along the η axis. In this limit the metric becomes where α is an additive constant which can be removed by a gauge transformation. This is then the metric on AdS 5 × S 2 × R 4 /|δλ(n)| and implies that we should take δλ(n) to be a negative integer. This imposes that the line charge density is convex and has integer changes in the slope at a location of the monopole.

Flux quantisation
Next consider flux quantisation. We may rewrite the three-form potential in the form for all integral four-cycles Σ.
We must first identify all integral four-cycles in the geometry. There are two types of four-cycle to consider depending on shrinking cycles in the geometry. The first type of cycle, which we denote by C a , are constructed by taking the cycle which shrinks along ρ = 0 and the two-sphere which shrinks along η = 0 which is topologically a four-sphere. Pictorially they are given by a line stretching from a point on the η axis to the ρ axis. For f kinks there are f + 1 such cycles that can be constructed by taking the shrinking cycle to be located at some point between two kinks n a (the end-points included).
The second class of four-cycle is topologically either S 2 × S 2 or S 4 . and is constructed by taking the shrinking β cycle along ρ = 0 between two kinks (or the end-points) and the round two-sphere, and will be denoted by D a . For the cycles involving the two end-points the four-cycle is topologically an S 4 and otherwise is topologically S 2 × S 2 .
Let us first consider the four-cycles C a , and let a = {0, ..., f }. We must identify the exact S 1 that is shrinking along ρ = 0, this depends where it is located along the η-axis. We may identify the 2π periodic coordinate which parametrises this shrinking S 1 by finding a Killing vector with vanishing norm as ρ → 0 which is normalised such that the surface gravity is 1.
The Killing vector satisfying these properties is 8 Note that since λ(η) is linear that this is independent of the coordinate η, but depends on the location along the ρ = 0 line due to the jumps in λ (η). We find where we take λ in the interval given by the index 'a'. This should be integer for all fourcycles. Note that since λ(0) = 0 the flux through the cycle C 0 is 0 always, whilst the cycle C f gives the total number of M5-branes wrapping the punctured sphere.
Consider now the second type of cycle D a . Let the locations of the monopole along the η axis be denoted by n a , with n 0 =0, and n f +1 the end-point of the η-axis. We have (2.79) To simplify the results let us take the line charge to be the union of lines of the schematic form λ = r a η + m a . (2.80) Then the quantisation conditions in general are equivalent to (2.82) Note that r 0 − r 1 = l and is integer. The quantisation conditions are satisfied if and p ∈ Z. Note that this is different from the quantisation condition considered in [2], where the l is not present. We emphasise that despite the line-charge not satisfying all the properties required in [2], in particular the integer slopes the solution is globally well-defined. In [2] for certain choices of line charge where it plateaus one is still forced into having integer slopes, however when the line charge does not plateau and either increases to infinity or has a zero away from η = 0 then this assumption is too restrictive. As we will see when considering the field theory, the non-integer slope leads to operators in the dual SCFT with fractional scaling dimensions.
Having understood the constraints for a well-defined solution let us consider the scal- Observe that the convex condition implies that each term in the sum on the left-hand-side is positive and consistency requires each term to be integer, thus the right-hand-side is also positive and integer. We can interpret this constraint as a partitioning of the positive integer −r f N f +1 and will be related to the construction of a Young diagram for the regular puncture in section 4. In [24] it was shown that this particular gravity solution is dual to a certain Argyres-Douglas theory. The goal of this paper is to generalise this construction.

Observables
Having quantised the fluxes and checked regularity let us turn our attention to the observables of the theory which we can compare to the field theory. One such observable is the 'a' central charge, which we study in the following section. A second is the scaling dimension of probe M2-branes, and the third observable that we will consider is the flavour central charges of the solution.

Central Charge
The leading order contribution to the a central charge for an AdS 5 solution of the form [39] ds 2 11 = e 2A 4ds 2 (AdS 5 ) + ds 2 (M 6 ) , (2.89) From the form of the metric in electrostatic coordinates we identify and therefore we have (2.92) Using the cylindrical Laplace equation we may rewrite this as a = 2 10 π 6 (2π p ) 9 ∂ ρ (V 2 )dη ∧ dρ . (2.93) It follows that we can integrate over ρ by defining the integration domain to go from ρ = 0 to the ellipse and gives a = 2 10 π 6 (2π p ) 9 where it is understood that the second term is evaluated on the ellipse by eliminating the ρ coordinate. However, the boundary condition along the ellipse requiresV to vanish and therefore the only contribution comes from the line charge term and we have 9 a = 2 10 π 6 (2π p ) 9 λ(η) 2 dη . (2.96) Note that this holds generally for a solution whose domain is fixed like the disc solution, i.e.
has a similar boundary structure to figure 2 where the ellipse may be replaced by a more complicated curve along whichV = 0. Carefully performing the integral we find a = 2 π 3 p 3 l 2 p 2 12(p + 1) , which upon using the quantisation condition (2.85) gives a = N 3 l 2 p 2 12(p + 1) . (2.98) This agrees with the result in [24] upon using the following dictionary between our variables

Scaling dimensions of probe M2-branes
One may wrap probe M2-branes around calibrated two-cycles in the geometry giving a BPS particle. With the metric in (2.88) the calibration condition on a 2d submanifold Σ 2 reads where X is the calibrated two-form which can be constructed as a spinor bilinear and will be given momentarily. The right-hand-side denotes the restriction of the volume form on M 6 to the 2d submanifold. For such a calibrated two-cycle the conformal dimension of the BPS particle is given by (2.101) The calibration two-form X was given in [24,35] in terms of the Toda frame and we refer the reader to [24] in particular for further details on its construction. Following [24] it is convenient to write the metric on the round two-sphere as We may transform this to the electrostatic description by the change of coordinates which gives the calibration two-form Note that this holds for any solution written in electrostatic coordinates following the conventions used in this paper. The calibrated two-cycle that we will consider in the following is the round two-sphere at fixed positions in M 6 . The calibration condition forces the cycle to be located at the positions of the kink of the line charge.
Let us check the calibration condition for the two-cycles. The calibration condition is This condition holds at the monopole point as one can verify from the expansion in (2.71), in fact this is the only point that it holds true for the potential we study. The conformal dimension of the BPS particle is then For the solution in BBMN we find which agrees with the result in [24] upon using the dictionary provided in equation (2.99). 10 There are some factors of 2 and signs different between the expression here and the one presented in [24] which are related to the different normalisation we employ and the different choices of volume form.

Flavour symmetries
To each flavour symmetry we can associate a flavour central charge. One can use anomaly inflow methods to compute a mixed U(1) R -flavour Chern-Simons term in AdS 5 . From [2] we have that the contribution to the flavour central charge due to the flavour group at the kink

Generalised regular puncture
In the previous section we have discussed how to rewrite the disc solution of [24] in terms of an electrostatic problem. We have obtained a potential V which depends on two integers, p and l. Since the cylindrical Laplace equation is linear we may sum different potentials and obtain a new solution. In particular we can sum up the potential of the previous section with the different potentials depending on different parameters p and l. We will interpret this as giving rise to a general Young diagram for the regular puncture. As we will show there are some constraints that the potential blocks must satisfy in order to give rise to a consistent solution. In the following sections we will analyse this in detail and compute the observables of the theory considered in the previous section.

Building block of the generalised regular puncture
In this section we will construct the general building block form of the potential which we will use in the remainder of the section. We saw earlier that the potential directly following from the disc solution can be more conveniently written by using the constant scaling symmetry of the electrostatic description. In order to make the discussion of the superposition of potentials simpler we will use this symmetry to construct the building block potential. Taking the potential in (2.62) and transforming it via V (ρ, η; p, l) → 2lnV ρ 2ln , η 2ln ; p, l = V(ρ, η; p, l, n) (3.1) we end up with the building block V(ρ, η; p, l, n) = l 2 Note that we have included an additional integer parameter n which places the kink at n rather than at 1. This parameter is trivial in the single kink case since it could be absorbed by redefining the flux quanta N , however, with multiple kinks this is no longer true and it becomes a bonafide parameter. The domain for the building block potential is a quarter of an ellipse satisfying ρ ∈ [0, n p(p + 2)] , η ∈ [0, n(p + 1)] , 1 ≥ η 2 n 2 (p + 1) 2 + ρ 2 n 2 p(p + 2) , (3.4) with focus at It is interesting to note that the level-sets of w are ellipses defined by with focus at n. The maximal value of the level set is which is of course the value of the root of the disc, and occurs at ρ = η = 0. In terms of α this corresponds to 2α = p and therefore we should take −1 ≤ 2α ≤ p.

Summing up building blocks
We now want to sum an arbitrary number of the building block potentials V(ρ, η; p, l, n) with different integers p, l, n. We must take the l's integer and we choose to take the n's integer too for later simplicity, this can always be arranged by the quantisation condition. The p's are no longer constrained to be integer though. Let us index the different potentials by a subscript a with a = {1, ..., f + 1}, and order the n a so that 0 < n 1 < n 2 < ... < n f < n f +1 .
The largest of the n's, n f +1 is the end-point and we understand p f +1 = l f +1 = 0 and n 0 = 0.
Then, the general potential is and depends (naively) on 3f parameters in total. However, as can be seen from the form of the potential in (3.9) the final potential actually depends only on 2f + 1 parameters: the f l a 's, the f n a 's and the end-point n f +1 , or equivalently the slope (3.14) The resultant line charge from this general potential is Generically the slope r f is not integer, but becomes integer if f a=1 n a l a = mn f +1 for some integer m. The end-point n f +1 has been fixed by solving λ(n f +1 ) = 0 and assuming that there is no other positive 0 of the line charge. If the line charge has a zero at a positive value smaller than n f +1 we must cut off the line-charge there. Note by construction that the line charge is convex and has kinks at n a with change of slope l a . 11 The conditions arising from flux quantisation impose (this follows straightforwardly from section 2.4 so we do not repeat the analysis) for all a. We may solve all these conditions by defining We end up with 2f +1 independent quanta, the f changes of slope l a and the f +1 positions N a , N f +1 included. We will relate combinations of these quanta to the field theory parameters.
The introduction of the integer N is useful for understanding the holographic limit of the solution, however in comparing to the field theory it is more useful to work with the integers N a and N f +1 .
One may wonder whether it is possible to extract out the building blocks used to construct the potential. One can recover information about the location of the monopoles of the building blocks and the associated change of slope, however there is no way to recover the information about p. One can construct multiple choices of building block with fixed n a and l a but varying  Another interesting aspect of the superposition of these solutions is that one can find solutions where the kinks and change of slope at the kinks are identical but the location of the zero at n f +1 is changed. This hints that the information contained in the kinks labels a regular puncture whilst the location of the non-trivial zero contains irregular puncture data.
This will motivate the proposal for the holographic dictionary we present in section 4. We we are unable to give a closed form expression for this point. One must solve 21) subject to (3.20) for the f − 1 remaining α a 's and finally for the location ρ * .

Central charge
We can now compute the central charge of the solution using the general form from (2.96) where we have used the short-hand It is useful to split the various terms up, we may write the first term as whilst the third term gives Putting everything together we have (3.28)

Scaling dimensions of BPS probe M2-branes
Using the result in (2.107) the scaling dimensions of BPS probe M2-branes located at the kinks is given by It is convenient to rewrite this by first defining We will see that the combination A a − N a appears later in the field theory section and can be obtained by studying a Young tableau.

Flavour symmetries
We may also compute the flavour central charge due to the flavour groups arising at the location of the kinks. From (2.109) we have that the central charge at the a'th kink is which gives withN as given in (3.25). Note that this is twice the scaling dimensions of the BPS operators considered above.

Field theory dual
In [24] the dual field theory of the disc solution that we reviewed in section 2 was identified to be the 4d N = 2 theory where the Young diagram Y is a general partition ofN , not necessarily rectangular.
To keep this paper as self-contained as possible and to clarify the notation we will use, we first review the Argyres-Douglas theories with emphasis on the observables that we can match with the holographic solutions in section 4.1, the reader familiar with these theories and their notation may safely skip this section. We then proceed to explain how to read off the holographic dictionary between the gravity solutions in section 3, in particular from the data contained in the line charge in equation (3.15), and the parameters of the field theory.
We show that the leading order contributions to the central charges from gravity match the field theory results and in addition compare the scaling dimensions of certain BPS operators.

Review of Argyres-Douglas theories
Argyres-Douglas theories describe a set of 4d N = 2 SCFTs which admit fractional scaling dimensions of the Coulomb branch operators and dimensional coupling constants. They were first found in [41,42] as a point on the Coulomb branch of pure N = 2 SU(3) gauge theory and have been extended to many different constructions since [43][44][45][46][47][48]. They are intrinsically strongly coupled theories and because of the non-integer Coulomb branch operators the conformal fixed points cannot be described by a N = 2 Lagrangian gauge theory. 12 One can engineer Argyres-Douglas theories using geometric engineering, in particular there are constructions in both type IIB and in M-theory. We will be concerned with the class of theories which can be obtained by compactifying the 6d N = (2, 0) theory of g=ADE type on a punctured sphere. If the punctures of the sphere are all of regular type one obtains theories of class S, with integer scaling dimensions. If instead, one allows for irregular singularities one can obtain Argyres-Douglas theories, [46,48,54].
One can only obtain a 4d N = 2 SCFT with an irregular puncture by compactifying on a sphere and not on a higher order genus Riemann surface. The complex coordinate of the Riemann surface should transform non-trivially under the U(1) R R-symmetry in the presence an irregular puncture. As such, the puncture must be placed at a fixed point of this rotational symmetry. In the space of Riemann surfaces, only the sphere admits a U(1) action with fixed points. It follows that we may place an irregular puncture at one of the poles of the sphere and at most one regular puncture at the other. Any other configurations containing an irregular puncture, whether it be a different Riemann surface or with more than one regular puncture, will not give rise to a N = 2 SCFT [46]. 13 The possible irregular punctures were classified in [46] using Type II , IIk ,N : Type III , III Y l ,··· ,Y 1 : The final dots denote non-divergent terms and the matrices T are SU(N ) valued diagonal matrices. For type I and type II theories they haveN independent eigenvalues whilst for type III theories the degeneracy of the eigenvalues is encoded in the corresponding Young tableaux 13 Note that if there is no irregular puncture the complex coordinate of the Riemann surface does not transform under the U(1) R R-symmetry and therefore the above analysis does not apply. One can allow for an arbitrary number of regular punctures for any Riemann surface and obtain a theory of class S. Y i . The inclusion of a regular puncture with one of the above three irregular punctures gives rise to a type IV theory. These theories are labelled by the choice of irregular puncture P and regular puncture Y , and will be denoted (P, Y ). In this work we will be interested in type IV theories where the irregular puncture is of type I. The Seiberg-Witten curve takes the form where S is defined by the Newton polygon for the theory. The scaling dimension of x and z are fixed by taking the Seiberg-Witten differential x dz to have dimension 1, and therefore We refer the reader eager for more details on the classification of irregular punctures and Newton polygons, after this short and simplified review, to [46,48].

Observables
Central charges The main observable that we wish to compare to our gravity solutions is the leading order contribution to the 'a'-central charge. There is a "straightforward" way of obtaining the central charge from knowledge of the central charge of the maximal puncture theory and the regular puncture theory [55]. We will focus on the leading order terms for largeN ,k and suppress the subleading terms. As explained in [55] the central charges of the (IN ,k , Y ) theory are equal to the sum of four pieces: Here a Y and c Y are the standard contributions from the puncture Y , I ρY is the embedding index of SU (2)  The final, as of yet unspecified, contributions are from a Y and c Y . Following the conventions in [56] we have (dropping some obviously subleading terms) where n h (Y ) and n v (Y ) are the effective number of hypermultiplets and vector multiplets from the regular puncture. We have Note that in the holographic limit n h (Y ) = n v (Y ) since the last two terms of (4.13) are subleading. Moreover, the first terms in (4.12) are also subleading and can therefore be dropped too leaving only the summation term. Note in addition that the summation term contains terms with different scaling properties in the holographic limit.

BPS operator scaling dimensions
There in the Seiberg-Witten curve. We may compute their dimension by using the dimension of the coordinates x, z:

Setting up the holographic dictionary
We now want to understand how to map the gravity parameters to the field theory parameters introduced in the previous section. One may ask whether we can use the rules of [2] to construct a dual quiver theory. The short answer is that generically we cannot apply the rules to construct a quiver. Given that the (IN ,k , Y ) theories are generically non-Lagrangian this is somewhat reassuring since we wish to identify them as a dual pair. To see why application of the rules in [2] for constructing a quiver fails one should note the non-integer slopes of the line-charge for the solutions discussed here. In [2] they are taken to be integer, which is a requirement for constructing a dual quiver since the values of the line charge at the integer values of η become the ranks of the gauge groups in the quiver. We emphasise that having integer slopes for the line charge is not a requirement of a well-defined supergravity solution, only that the change of the slopes at the kinks is integer. For certain choices of the parameters we can make the slope r f integer, and therefore all the other slopes too, which allows for the construction of a dual quiver theory, however this is not a generic choice one can make and so we will not focus on this possibility.
Recall that the line charge depends on 2f + 1 parameters, the f changes of slopes l a , the f kink locations n a and the end-point n f +1 . The field theory we conjecture is dual also has 2f + 1 parameters: the integerk and the 2f parameters describing the Young diagram giving a partition ofN . We conjecture that the dictionary between the field theory and gravity solutions isk Note that this reduces to the identifications made for the disc solution in [24] as it should.

Matching of observables
We now want to match the central charge of the proposed field theory with the gravity result in (3.22) using the above dictionary. The large N limit is obtained by taking then a 's,N and k all large and of the same order. It is useful to note that the central charge coming from field theory may be written as where we have used our dictionary which givesk +N = N f +1 . It is helpful to compare the N f +1 coefficients. The first and last terms match on the nose with the gravity expressions. The second term is not as obvious but after a short but not particularly enlightening calculation, and removing subleading terms one finds that this term also matches between the field theory and gravity result. We conclude that the central charge of the gravity solution we discuss in One interesting point to consider is when one can draw a dual quiver using the prescription in [2]. As we explained earlier, their prescription for drawing a dual quiver given a line charge, relies on the slope of the line charge being integer. In our setup this follows if the slope r f in Thus, for these theories the scaling dimensions of the operators we consider are integer. The non-integer slope of the line charge is therefore essential for obtaining theories containing operators with non-integer scaling dimensions. A final check of the proposed duality is to study the quiver following from the prescription in [2] when we take the slope of the line charge to be integer. Constructing the quiver following the prescription in [2] one can show that the central charges match exactly with the ones presented in (4.6)-(4.7), including subleading terms. This gives added evidence that our conjectured duality is correct.

Conclusion
In this paper we have studied the holographic duals of the Argyres-Douglas theories with Y an arbitrary regular puncture generalising the work in [24]. We have shown that there is a perfect match between the field theory and gravity solutions in the holographic limit. It would be interesting to go beyond the leading order analysis conducted here and compute subleading corrections on the gravity side, for example to study the gravitational anomaly a − c and check the matching of the central charges more generally. Recently, 4d N = 2 SCFTs with a = c have been constructed in [58] with the holographic duals currently unknown. It is natural to wonder whether using similar singular gravity solutions as those in this paper one can construct the duals of these theories too.
One of the results of our analysis which deviates from folklore are the necessary constraints a line charge must satisfy to give rise to a well-defined N = 2 AdS 5 solution using [2]. We have seen that it is necessary for the integer condition of the slope to be relaxed if the line charge does not plateau. This can be reinterpreted as the T 2 fibration of the internal manifold being quasi-regular as opposed to regular when the slopes are all integer. This leads to M5-branes wrapping cycles in this torus and in the quasi-regular case the dual SCFT is non-Lagrangian and dual operators have fractional dimensions. The gravity analysis then gives a concrete condition on when one can construct a dual quiver and additionally by using the rules in [2] how to do this. It seems worthwhile to see whether relaxing this condition in previously studied solutions in the literature gives rise to interesting solutions which have been missed previously.
There remain a number of interesting avenues to pursue. For example it remains to construct the holographic duals of spheres with the other types of irregular puncture. We have constructed the holographic dual for the most general type IV theory using a type I irregular puncture however a holographic dual for the theories using the other irregular punctures is currently missing. In a similar vein it would be interesting to construct the holographic duals of the Argyres-Douglas theories constructed from the 6d N = (2, 0) D-series theory, one may obtain some inspiration from [59]. Further constructing the SCFT duals for the other disc compactifications is an outstanding problem. For the D3 discs, [25,26] it is tempting to conjecture that this is dual to N = 4 SYM on a punctured sphere which preserves N = (2,2) in 2d. Similarly the field theory dual for M2-branes on a disc [18,27] should correspond to ABJM on a punctured sphere, and for the D4-D8 system [28] to 5d N = 1 USp(2N ) theory on a punctured sphere. In a similar spirit identifying the dual field theories for compactifications on a spindle still remains an open problem.
where η = 1 2 cos θ cosh 2ζ, r = sin θ sinh 2 ζ. This can be obtained by taking a suitable limit of the general potential we give in (3.2).

A.2 Maldacena-Nunez solution
We noted in the main text that there is a second special point in s 1 parameter space with s 2 = 0 fixed, namely s 1 = − 1 4 . At this point the roots of f (w) are w = 0 twice and w = 1 2 twice. We noted earlier that since w = 1 2 is now a double root the metric at this end-point looks locally like H 2 rather than R 2 /Z l as at a generic value − 1 4 < s 1 < 0. Indeed, as we will show momentarily by taking a suitable scaling limit to this point in parameter space we find the Maldacena-Nunez solution.
First, let us take µ = sin θ and perform the rescalings 14 whilst also performing the coordinate shift which removes a singular term from the gauge field. Expanding around λ = 0 the metric becomes ds 2 11 = 1 2 (1 + cos 2 θ) 1/3 4ds 2 (AdS 5 ) + 2dθ 2 + 2 cos 2 θ 1 + cos 2 θ ds 2 (S 2 ) + 4 sin 2 θ 1 + cos 2 θ (dφ 1 + y −1 dx) 2 which is precisely the metric in [1], with the coordinates used there. We therefore find that the same metric can be globally completed to obtain three seemingly distinct solutions, disc solutions, Maldacena-Nunez solution and pure AdS 7 . It is interesting to note that the Maldacena-Nunez solution preserves supersymmetry via a topological twist whilst the disc solution does not involve a topological twist but an altogether different mechanism, in particular the Killing spinors of the disc are not independent of the disc coordinates as they would be for a topological twist.
The electrostatic potential for the Maldacena-Nunez solution is where η = 1+y 2 4y cos θ, r = y−1 y+1 . This can also be obtained from the general potential we provide in (3.2) after performing the limit in (A.3).

B Anomaly inflow
In this section we will study the global symmetries and 't Hooft anomalies of the dual SCFT by using anomaly inflow methods [60][61][62]. This section is an extension of the computations performed in [24] to account for the more general flavour symmetry that our solutions exhibit, as such we will present the bare bones computation when there is no risk of confusion. 15 The ultimate goal of this section is to give an independent derivation of the observables studied in the gravity theory and to understand the breaking of a u(1) isometry algebra by a Stückelberg mechanism as observed in [24].
The 6d anomaly polynomial of the 4d SCFT, in the holographic limit, is given by with the integration over the M 6 fibers.

B.1 Constructing E 4
We now want to construct E 4 for the background in the main text. We will gauge all the symmetries in M 6 . As in [24] one finds that there is a spontaneous breaking of the continuous symmetries of M 6 by a Stückelberg mechanism. We will review this quickly for completeness, but refer the reader there for further details. The four-form takes the form We may gauge the continuous isometries of M 6 , in total we have the SU (2)  and thus this acts as an obstruction to constructing E 4 as above. As explained in [24] this requires the introduction of an axion which leads to the spontaneous symmetry breaking of the gauge field A β .
Following [24] one introduces an axion α with field strengthf which satisfies df = The Bianchi identity for the axion field strength is As shown in [24] this leads to the U(1) gauge field A 1 becoming massive via a Stückelberg mechanism leading to the symmetry being spontaneously broken.
We now want to include the contributions of the flavour symmetries. Recall that at the locations of the kinks of the line-charge the metric is locally the orbifold R 4 /Z la and leads to an SU(l a ) flavour symmetry. The orbifold leads to l a − 1 resolution two-cycles on which we may wrap the three-form potential C 3 leading to l a − 1 abelian gauge fields for each kink. Let the resolution two-cycles be denoted by ω a,i with i ∈ {1, .., l a − 1} and a ∈ {1, .., f }. These should be understood to be localised at the kink locations and therefore the intersection of any two of these two-cycles located at different kinks is zero. The intersection pairing then gives R 4 /Z l a ω a,i ∧ ω a,j = −C

B.2 Anomaly polynomial from anomaly inflow
We can now insert the ansatz for E 4 into the 12-form anomaly polynomial and integrate over M 6 to obtain the anomaly polynomial for the 4d theory. To proceed we need a few results that may be extracted from the literature. The Bott-Cattaneo formula [63] gives The final result is due to non-perturbative M2-brane states which enhance the U(1) la−1 symmetry to the full non-abelian symmetry SU(l a ) as opposed to its Cartan.
Plugging all these ingredients into (B.2) we find in agreement with the result we used from [2].