Gravity duals of N = 2 superconformal field theories with no electrostatic description

We construct the first eleven-dimensional supergravity solutions, which are regular, have no smearing and possess only SO(2,4) x SO(3) x U(1)_R isometry. They are dual to four-dimensional field theories with N = 2 superconformal symmetry. We utilise the Toda frame of self-dual four-dimensional Euclidean metrics with SU(2) rotational symmetry. They are obtained by transforming the Atiyah--Hitchin instanton under SL(2,R) and are expressed in terms of theta functions. The absence of any extra U(1) symmetry, even asymptotically, renders inapplicable the electrostatic description of our solution.


Introduction
General solutions of eleven-dimensional supergravity as duals of N = 2 superconformal field theories were constructed in [1]. The metric and form fields are given by ds 2 11 = κ 2 /3 11 e 2λ 4ds 2 AdS 5 + z 2 e −6λ dΩ 2 2 + where the AdS 5 and S 2 have unit radii and , 2) Hence, these solutions boil down to determining the scalar function Ψ(x, y, z). It turns out that this obeys the continual Toda equation found in the context of continuum Lie algebras [2]. Demanding background regularity imposes appropriate boundary conditions given by z = 0 : e Ψ = finite = 0 , ∂ z Ψ = 0 .
Finding explicit solutions to this well posed mathematical problem has been so far possible only in two cases: 1. Whenever there in an extra U (1) symmetry of the background allowing one to map (1.3) with (1.4) to the Laplace equation [3] and the electrostatic analogue problem of a line charge distribution over an infinite conducting plane [1]. The specifics of the line charge distribution has an one to one correspondence with N = 2 quiver gauge theories [4]. Since this realisation a lot of works, notably [5][6][7], have appeared in the literature, all of them assuming, to the best of our knowledge, this extra U (1) isometry which essentially makes the problem tractable by elementary means.
2. When it is possible to find separable solutions. Due to the non-linearity of the continual Toda equation this method is quite limited in finding solutions in our context.
Defining the complex coordinate q = 1 2 (x + iy), a separable solution is the product of a quadratic polynomial in z and a function obeying Liouville's equation leading to e Ψ = c 3 |∂f | 2 (1 − c 3 |f | 2 ) 2 −z 2 + c 1 z + c 2 , (1.5) where the c i 's are real constants and f = f (q) is a locally univalent meromorphic function.
Evidently it is extremely important to find genuine solutions to (1.3), (1.4) since that would allow to explore novel aspects of these theories. It is the purpose of the present paper to find the first such solution in the literature that would fall outside the realm of electrostatic analogue description or (1.5).
The idea of our construction is the following: First we recall that (1.3) has appeared before in the context of four-dimensional self-dual Euclidean metrics with a rotational Killing vector. A particular solution corresponds to the Atiyah-Hitchin instanton metric [8,9]. This was originally formulated by utilising the SU (2) invariance of the metric and the Darboux-Halphen system [10] of differential equations arising from self-duality. What is directly useful for our purposes is not this formulation but rather the rewriting of the Atiyah-Hitchin metric in terms of the continual Toda potential which was found by an appropriately coordinate transformation in [11]. This provides a solution to (1.3), but the corresponding function Ψ does not satisfy the boundary conditions (1.4). We overcame this final obstacle by first using the extension of the construction of [11] to arbitrary solutions of the Darboux-Halphen system given in [12]. Then we transform the Atiyah-Hitchin instanton under the SL(2, R) covariance of the Darboux-Halphen system and by appropriately choosing parameters of the transformation we obtain the required solution.
The organisation of this paper is as follows: In Sec.

Gravitational instantons
The purpose of this section is to provide the necessary review materials for our later construction.

Hyper-Kähler geometries
Let us consider a hyperkähler manifold, which is a solution of Einstein vacuum equations with at least one isometry and associated Killing vector ξ = ξ µ ∂ µ . If where the fibering is defined along the action of the Killing vector ξ = ∂ ϕ , associated with this isometry. If ∂ ϕ is a translational Killing vector then we can always choose a coordinate system such that For this metric the self-duality (or anti-self duality) condition can be simply written as thus V −1 is the "electrostatic potential" of the translational Killing vector. The localised solutions of (2.4) read where (multi-)Eguchi-Hanson and (multi-)Taub-Nut correspond to ε = 0, 1 respectively.
If ∂ ϕ is a rotational Killing vector then we can always choose a coordinate system such that which is the Toda frame [14]. For this metric the self-duality (or anti-self duality) condition is the three-dimensional continual Toda (1.4) which here we write using complex coordinates 2q = x + iy and c.c.
The conformal transformations are symmetries of the Toda equation; i.e. q → F (q) and Ψ → Ψ − ln |∂F | 2 leaves invariant (2.7). Furthermore, the Kretschmann scalar of (2.2), (2.6) has leading singular behaviour Hence, regular solutions of four-dimensional Euclidean self-dual metrics require that ∂ z Ψ = 0. This is not in conflict with the boundary condition (1.4) since the latter refers to elevendimensional metrics. However, this implies that we cannot simply take over solutions to Toda equations appropriate for the four-dimensional metrics (2.2) and use them in (1.1) since the resulting background will be singular.

Review of Bianchi IX folliations and Toda frame
The way to find solutions to (2.7) is to make contact with another class of four-dimensional metrics of Euclidean Einstein gravity, namely metrics of the Bianchi IX type where a, b, c are functions of t and σ i are the left-invariant Maurer-Cartan forms of SU (2).
We choose our normalisation such that (2.10) We will use the explicit parametrisation where for later use we have also written the volume form of the three-sphere. The range of variables is The Bianchi IX foliation (2.9) is invariant under the left-action of the SU (2) algebra, generated by the Killing (right-invariant) fields This exhausts the isometry when a = b = c. When two metric coefficients are equal, an extra U (1) ⊂ SU (2) (right-action) survives. For a = b e.g. this U (1) is generated by ∂ ψ .
The Killing fields (2.13) can be rotational or translational depending on the self-duality conditions. As discussed in the literature in several instances [13,[15][16][17][18][19][20]], 1 integrating once the self-duality conditions for the curvature leads to the first order system of non-linear differential equationsȧ where the derivative is with respect to t.  [24]. For our purposes we shall focus on the case of the rotational Killing vector ξ 3 (λ = 1) for which we shall find its corresponding Toda frame. After a lengthy computation which is presented in full detail in App. A we find that This is still in terms of the original coordinates ϑ, ψ and t. The appropriate coordinate transformation is given by where we have defined 1 See also [21] for a general review of the subject and for an exhaustive classification see also [22,23].
and F (t, k) and E(t, k) are the incomplete elliptic integrals of the first and second kind, respectively which we write for completeness The only task remaining is to find appropriate solutions of the Darboux-Halphen system for the a, b and c and use them to construct the Toda potential Ψ from (2.15). The choice of these functions should be such that the boundary conditions (1.4) are satisfied.
This is a non-trivial task which we will undertake in the next section.

N = 2 SCFTs and continual Toda
In this section we will construct a family of solutions to the continual Toda equation which will respect the appropriate boundary conditions (1.4) for the eleven-dimensional solution (1.1). As we shall see this will be done by using the Atiyah-Hitchin instanton solution, followed by an appropriate SL(2, R) transformation which leaves the Darboux-Halphen system invariant.
We begin by first expressing various terms in the eleven-dimensional supergravity back- such that z = 1 2 (V + b nut ). Then we have that various combinations appearing in the background (1.1) and in (1.2) are conveniently expressed as In addition, we have that where the components of the metric γ ij in the (t, ϑ, ψ) coordinate system can be found in (A.1) as well as the frame components e ± in (A.8). Our solution for Ψ should satisfy (1.4) or, equivalently the boundary conditions 5) and the product V z be kept finite and different than zero.

Appropriate boundary conditions
We will explore the boundary condition (3.5). In order to present the general analysis it is convenient to adopt an index notation. In particular, we will use a i = (a, b, c) and the directional cosines Then the Darboux-Halphen system, i.e. (2.14) for λ = 1, can be written compactly aṡ An alternative to the a k 's basis is to use the Ω k 's defined as where we also wrote the inverse relation. They satisfẏ which is the original Darboux system [25]. Then note that This rewriting is particularly convenient as it treats all three coordinate axes as equivalent.
We return to the determination of the behaviour of the various functions necessary to satisfy the boundary conditions. How this behaviour is actually achieved together with the full solution will be the subject of the next section.
The boundary conditions are easily described in terms of the z coordinate. However, in terms of the angles ϑ and ψ, equivalently the directional cosines, and the coordinate t the description is more involved. Since the solution for Ψ is written explicitly in terms of these coordinates, it is necessary to provide such a description as well. Let's denote with a star the constant variables for which the boundary condition is achieved, i.e. t * and n * k . Generically, if n * k = 0 (there is always such a directional cosine) the solution of the boundary conditions requires that a k → ∞ for some t → t * . From that we see that there are values for the Ω's which will be denoted by a star such that as t → t * they have the following specific limits These guarantees that e Ψ = finite = 0 and that V → ∞. Next, vanishing of z requires in addition that which already reveals that we must have that Ω * i Ω * j < 0. Let's also define for later convenience (3.14) Note that none of the α or β can become zero since in that case n * k = 0 which would have violated our hypothesis that n * k = 0. Then (3.13) is equivalent to the condition Keeping zV finite requires some extra care. We have that where we used that Then this gives the smooth limit as z → 0 Thus the last term in the eleven-dimensional metric (1.1) remains finite [7]. In the following, we will use the SL(2, R) transformations of the Atiyah-Hitchin instanton satisfying (3.12), as a solution-generating pattern for eleven-dimensional supergravity. The solutions, which will be regular, have no smearing and possess only SO

Punctures
The boundary conditions which were given in (1.4) are satisfied in the regions with no punctures. In the case of punctures, the S ϕ circle, associated to the U (1), shrinks in a This condition is related with the existence of a non-trivial 4-cycle which can support a non-trivial 4-form flux. More concretely, to have a non-trivial 4-cycle that support 4-flux, we look for solutions where S ϕ circle shrinks to zero size at a point z = z c , such that the 4-cycle is given by the product where g is the genus of the surface, z = z c corresponds to the zeros of the Toda potential and K T is the number of punctures of the surface. A rewriting of the integrant in terms of the quantities which were introduced in the App. A can be done with the use of (3.4).
We next apply this computation to the Toda potentials we have considered so far.
Backgrounds with no electrostatic description: In this case there is no extra U (1) symmetry. The 2-cycle is defined by the zeros of e Ψ , which were given by the four different cases in (A.13). Out of them the last one with a = b = c is not consistent with our boundary conditions and is dismissed. For all of the other three we obtain that ∆ = 0 and therefore a two-torus, i.e. g = 1, with no punctures.
Hence, there are no punctures and g = 1.
Maldacena-Núñez metric [26]: Using the Toda potential of (C.13) we find that the two circle is defined for z = z c = N and the energy reads where we have defined the integral by its principal value. Therefore this corresponds to a 2-sphere with no punctures.

Bianchi IX instantons with strict SU (2) isometry
In this section we explicitly construct solutions of the Darboux-Halphen system in terms of theta functions that indeed satisfy the appropriate boundary conditions.

The Darboux-Halphen system and its generic solutions
Our starting point is the four-dimensional self-dual gravitational instantons of the type (2.9) which, by utilising (3.8) we will write as where T = t/2. We will focus on the Darboux-Halphen system, (2.14) with λ = 1, because in the case at hand, the isometry group is strictly reduced to the left SU (2), generated by the Killings (2.13). These are rotational and will provide solutions of the continual Toda, potentially eligible for higher-dimensional supergravity embeddings, dual to superconformal field theories. As we will show, the required boundary conditions can indeed be reached and this is the central result of the present work.
The Omega's satisfy (3.9) without the overall 1/2 factor on the right hand side due to the rescaling on t as above. For convenience we repeat it here where now the derivative is with respect to T . Consider the same system in the complex The general solutions of this system have the following properties [10,27]: • The ω's are regular, univalued and holomorphic in a region with movable boundary i.e. a dense set of essential singularities). The location of this boundary accurately determines the solution.
• If ω i,0 (z) is a solution, then is another solution with singularity boundary moved according to z → az + b cz + d .
The fully anisotropic case is our main motivation here. In this case no algebraic first integrals exist and the general solution (see [10,27,28]) is expressed in terms of Concretely with E i (z) triplet 2 of holomorphic weight-2 modular forms of Γ(2). Again, the SL(2, C) acts as a permutation on ω's.
Since our purpose is the description of gravitational backgrounds, we are interested in real solutions of the real coordinate T are obtained following the general pattern as The reason for the choice iT as the argument is that we get converging real functions.
According to (4.3), new real solutions are generated as 3 Note that the entries of the SL(2, C) and SL(2, R) transformation matrices are related as a = A, d = D, c = −iC and b = iB, so that the determinant condition is preserved.
2 Notice their general transformations as generated by z → −1/z and z + 1: In addition, note that an SL(2, R) transformation on the Ω's never changes their relative position in the real line. This remains the same as in the seed solution. Also note that the Darboux system (4.2) does not allow a cross over behaviour for its solutions. The reason is that, if for some finite T two of the Ω's, initially different, become equal, then their derivatives at that point are equal as well. Hence these Ω's will subsequently remain equal.
A particular solution of the Darboux system (4.2) is the original Halphen solution [10].
In the present language, it corresponds to where we have used standard notation for the Theta functions and the quasimodular form of weight two. Details are given in App. B. This is also the solution found by Atiyah and Hitchin [8,9] as the Bianchi IX self-dual gravitational instanton relevant for describing the configuration space of two slowly moving BPS SU (2) Yang-Mills-Higgs monopoles [29,30].
This solution and the SL(2, R)-transformed ones, possess rotational Killing fields, given in (2.13), as opposed to the strict-SU (2)-isometry family of solutions of the Lagrange system found earlier in [31], where the same fields are translational.

General solution and boundary conditions
We now turn to the problem of finding solutions of the system (4.2), which satisfy the prescribed boundary condition that there exist some T = T * so that the corresponding values for the Ω i 's, i.e. Ω * i 's satisfy (3.12) and (3.15). It should be noticed at this point that the metric ansatz (4.1) assumes the product of the three Ω's be positive. In the opposite instance, it is enough to change the overall sign of the ansatz to recover a positive-definite metric. The domain of definition of the Euclidean time T is set by the roots or the poles of the Ω's, and T which is a root of Ω 3 . The existence of such a limiting value is consistently required from the higher-dimensional perspective, even though from the point of view of the four-dimensional gravitational instanton, a simple root of any Ω i is a genuine curvature singularity (see [31]). The original Halphen solution (4.7), according to (4.5), along the imaginary z-axis leads to real Ω's , These do not satisfy the boundary conditions (3.12) and (3.15). The reason is that none of the Ω's in that solution has any root (see Fig.1). This precise property makes the Atiyah-Hitchin instanton regular everywhere in the range 0 < T < ∞ (connecting the Taubian infinity with the bolt). We will need the behaviours for small and large T of the Halphen functions Ω i,H defined in (4.8). This is obtained from the well-known behaviours of the modular and quasimodular forms introduced above, along the imaginary axis τ = iT using the results of the App. B. We obtain that and that The reason for that is the argument of the various functions should have positive imaginary part for them to converge. Let us introduce for later convenience (4.12) There are two distinct generic behaviours for the Ω's, depending on the sign 4 of Z ∞ : : Ω's are defined on 2 disconnected sets T < T ∞ < T 0 < T ; : Ω's are defined on a single set T 0 < T < T ∞ (0 < Z < ∞).

4.2.2
The regime Z ∞ > 0 We will first prove that in the regime Z ∞ > 0, condition (3.15) fails. Let us concentrate on the region T 0 < T (central symmetry allows to draw similar conclusions for the alternative set T < T ∞ ). Using the results (4.9) and (4.10) of App. B, we can study the behaviour of Ωs for large T and around T 0 .
Behaviour for T 0 T : Around T 0 , the argument of the Ω i,H vanishes. Using (4.9) and (4.6) we obtain It is not hard to interpolate between the large-T and the T 0 T regions, and obtain the global picture for Ω i (T ). On the one hand, we observe that the Ω 2|3,H are positive and monotonically decreasing. Using (4.6) we may compute that (4.15) Hence since T > T ∞ a decreasing and positive Ω 2|3,H implies that Ω 2|3 remains decreasing and positive. On the other hand, since Ω 1 (T ) is negative around T 0 T , there is a T * such that Ω 1 (T * ) = 0. We expect Ω 1 (T ) to increase, become positive at T * , and decrease towards zero at large T . The generic behaviour is presented 5 in Fig.2. Since Ω 2|3,H always stay non-negative there is no way we could satisfy the boundary condition (3.15) (with i = 2 and j = 3). 4 Note that Z∞ cannot vanish due to unimodularity of the trajectory. Furthermore T0 = T∞ + 1 AC and consequently Figure 2. Generic solution (4.6) for Z ∞ > 0 in the range T 0 < T (Ω 1 < Ω 3 < Ω 2 ).

The regime Z ∞ < 0
Before entering the technical details, let us motivate the reason why the regime Z ∞ < 0, is appropriate for our purposes. For this type of SL(2, R) transformations, T 0 and T ∞ are poles. The behaviour of the Ωs around T 0 has already been exhibited in (4.14). Around T ∞ , we proceed in a similar manner.
Behaviour for T T ∞ : Around T ∞ , the argument of the Ω i,H diverges. Using (4.10) and (4.6) we find (4.17) From the behaviours (4.14) and (4.17), together with the expression (4.6) and the known behaviours of Ω i,H , we can qualitatively describe the solutions at hand. The function Ω 3 (T ) interpolates between two simple poles. It is positive around T 0 and negative around T ∞ , thus there is a root at T * , i.e.
withΩ * 3 < 0. The function Ω 1 (T ) is finite and negative between its double pole at T 0 and its simple pole at T ∞ , whereas Ω 2 (T ) is finite and positive between its simple pole at T 0 and its double pole at T ∞ . The generic behaviour is depicted in Fig. 3
Using (4.6) with Ω i,0 = Ω i,H , the explicit expressions of Ω i,H given in (4.8), as well as condition (3.15) (for i = 1 and j = 2), the normalisation in (3.14) and the identity (B.4), we obtain that where This allows to determine Z * as an elliptic modulus, in terms of the complete elliptic integral of the first kind K(κ) introduced in (B.8) with the modulus and complementary modulus of the elliptic integrals given by where we have used the identities (B.13) and (B.14) involving the complete elliptic integral of the second kind E(κ). Since T * and Z * are related via (4.20), A, B, C, D are not arbitrary unimodular real numbers. They must satisfy The approximate expressions near the extreme values α = 0 and 2 read . The simplest possible choice is α = β = 1, because it is symmetric and algebraic which is generically not the case. Using (4.21) we find Z * = 1, whereas (4.25) leads to A + C = 0.

Concluding remarks
The symmetry. An example of this is the reconstruction of the Toda potential for the Eguchi-Hanson metric in [35] (see equation (4.11) and below in that paper). The potential for the Finally, it will be of much interest to explore the underlying physical information of our solution with regard to field theory and keeping in mind the above comments. A Toda frame of the Atiyah-Hitchin metric

A.1 Construction of the Toda frame
In this appendix we shall revisit the derivation of the Toda frame for the Atiyah-Hitchin metric as this was presented in [11] and generalised in [12] for the general solution of the Darboux-Halphen first order (DH) system. We shall present the derivation done in [12], adopted however in our conventions. The key point of this computation is the repeated use of (2.14) for the Darboux-Halphen system, i.e. λ = 1.
Rewriting of the metric (2.9) in the Geroch formulation with a Killing vector ξ 3 where g and γ are the metrics depicted in the line elements d 2 and ds 2 respectively. Ricci flatness of the four-dimensional metric introduces the notion of the nut potential [13] δ( where the metric γ ij is used to define the superscripts and the covariant derivative. Using the latter and (2.14) we find the on-shell field Combining the nut potential and V we can built the fields Using the latter we can define the quantity where the strict inequality or equality is the criterion for a Killing vector to be rotational or translational, respectively [14]. In our case ∂S + 2 = 4, so ∂ ϕ is a rotational Killing vector 6 . This enables us to define a coordinate z as which will play the rôle of z in the Toda frame (2.6).
Straightforward calculation shows that the zeros of e Ψ occur at: that is in four different cases. We note that at these points there is a coordinate singularity of the metric g in the Toda frame (2.6), since which vanishes at the zeros of the Toda potential and its derivative.
Our next task is to compute q,q and complete the coordinate transformation. For this purpose we shall perform the coordinate transformation Using the latter, we find cosh p (cosh p + cos ϑ sinh p)) .
It is easy to check, that for this choice we retrieve the ϑ + component of (A. 16). This enables us to write a system of first order partial differential equations for W (t, p), namely: which can be easily proved through (2.14) to be compatible, i.e. ∂ ∂t , ∂ ∂p W (t, p) = 0. Integrating the second one, we find where ξ(t) is an arbitrary function of time and F (t, k), E(t, k) are incomplete elliptic integrals of the first and second type To specify this function, we shall compute the partial derivative of W (t, p) with respect to t, use (2.14) and compare it with the first equation of (A.20): Finally, we mention that the explicit solution of (4.2) was not used for the integration of q = q(t, ϑ, p) in contrast with the derivation which was performed in App. C of [12].

A.2 Verification of the solution
So, we have the desired coordinate transformation y α = (q,q, z) and the Toda potential Ψ as functions of x i = (t, ϑ, ψ). The scope of this section is to prove that Ψ satisfies the continuoual Toda equation (2.7). Taking into account that the expressions are transcendental, we shall use the chain rule However, the expressions we have, involve the new coordinates y α as functions of the old ones x i and so, we have to compute the Jacobian matrix of the coordinate transformation 7 whose expression can be found with the use of (A.6),(A.7) and (2.14). Inverting this matrix we can compute the derivatives with respect to y α as follows Using the latter and Eqs. (A.10),(2.14) we find a simple expression for the partial derivatives of Ψ with respect to z Note that the det γ transforms as it should: i.e. a scalar density of weight (hq, hq) = (1, 1); which is in agreement with (2.6). To compute the partial derivatives with respect to q,q of Ψ, it is convenient to consider the product and the ratio of e 2f and e 2f as in [12] Π = e 2Ψ = e 2f +2f , R = e 2f −2f .
(A. 28) To proceed, it is helpful to rewrite the partial derivatives with respect to q andq as where Q,Q should be considered as first order differential operators acting on the right and obeying Leibniz rule. Using these operators, we can write the second partial derivative of Ψ with respect to q andq as follows Commuting of the partial derivatives, provides the following constraint 8 which is satisfied after a lengthy but straightforward computation. Thus, we can write the second partial derivative of Ψ with respect to q andq in a symmetric way Finally, plugging (A.27) and (A.33) in the continual Toda equation (2.7), we find after a lengthy but straightforward computation, that it is indeed satisfied.

B Modular forms and elliptic integrals
We collect here some conventions for the modular forms and theta functions used in the main text. General results and properties of these objects can be found in [33,34].
Introducing q = e 2iπτ , τ ∈ C, we first define 8 We use the standard notations for commutators and anticommutators In addition note the Jacobi theta functions.
These have many remarkable properties. We quote here the Jacobi identity and Their transformation properties under a Möbius transformation are and E 2 (τ + 1) = E 2 (τ ) , The last relation can be used to provide a proof of (B.5).
The Jacobi functions and the weight-two quasimodular form are related to the complete elliptic integrals of the first and second kind, defined respectively as special values of the incomplete integrals (A.22) which satisfy the Legendre relation Setting τ as the elliptic modulus =⇒ κ 2 + κ 2 = 1 (B.11) and K(κ) = π 2 θ 2 3 (τ ) , (B.12) or equivalently Notice finally the useful inequality .

C Extra Killing vector and electrostatics
We will consider solutions which have an additional U (1) isometry. Not much is really new material but we believe that the presentation is novel. It can be read independently from the rest of the paper.
In the case of backgrounds with an extra Killing vector field ∂ β and an associated Then, thanks to r∂ r (r∂ r ln r) = r δ(r) = 0 and using (1.3), the new function Ψ(r, z) satisfies which is the continual Toda equation (1.3) in cylindrical polar coordinates with azimuthal symmetry.
Here we present some necessary formulae for the case corresponding to an isometry along the polar angle in the x-y plane, Eq. (C.2). We let where Φ = Φ(ρ, η) satisfies the scalar Laplacian in the cylindrical coordinates (ρ, η) Given the above change of variable for ρ = ρ(r, z) one may compute the other variable η = η(r, z) from the exact differential In addition the potential Φ is computed form the following exact differential The boundary is at z = 0, which implies the boundary condition and a charge density λ(η) along the η-axis. We may take Φ(ρ, 0) = 0 and the boundary condition can be satisfied by considering an image line with opposite charge below the plane, i.e. λ(−η) = −λ(η). The above define an electrostatic problem. Due to the line density the right hand side of the Laplace equation (C.4) should be replaced by λ(η) δ(ρ) ρ .
From Gauss' law applied on a cylinder with small radius and small height which does not cross the η = 0 plane we have that for the potential and the radial component of the electric field Φ λ ln ρ + · · · =⇒ E ρ = −∂ ρ Φ − λ ρ + · · · . (C.8) If we may easily compute the behaviour of the potential Φ near ρ = 0, then this is a practical way to read off the charge line density λ(η). Another way is to write the second of the above as λ(η) ≡ ρ∂ ρ Φ ρ=0 = z(ρ = 0, η) . (C.9) Therefore if we know the explicit form of the coordinate change from (r, z) → (ρ, η), it might be more practical to use the above formulae. Returning to the computation of the potential Ψ and assuming that the space is free of boundaries we have that Φ(ρ, η) = − 1 2 ∞ 0 dη λ(η )G(ρ, η; η ) , (C. 10) where the Green's function is G(ρ, η; η ) = 1 ρ 2 + (η − η ) 2 − 1 ρ 2 + (η + η ) 2 (C.11) and satisfies the standard Green's equation with Dirichlet boundary condition The electrostatics picture of the Toda equation implies a smearing procedure has occurred so that an extra U (1) isometry develops. That introduces the limitation that the length-scales for which a solution is trustworthy should be larger than the smearing length, i.e. ρ sm . In addition the charge line density should obey some consistency conditions so to avoid coordinate singularities [4]. Finally, these profiles only tell us about the behaviour of the solution around ρ = 0. At larger distances, say for ρ > ρ U (1) , it should be replaced by a solution of the continual Toda without a U (1) β isometry. Thus the electrostatics picture is valid when ρ sm ρ ρ U (1) and for specific forms of the line density. An exception is the Maldacena-Núñez solution [26] which is an unsmeared solution.

C.1 Maldacena-Núñez solution
The first paradigm we are going to tackle is the Toda potential of the Maldacena-Núñez solution [26] e Ψ = 4 N 2 − z 2 One can invert the above expressions but the resulting expressions for r(ρ, η) and z(ρ, η) are not very illuminating. The electrostatic potential can be computed using (C.6). One finds that Φ = z − N tanh −1 z N + 1 + r 2 1 − r 2 z ln r . (C.16) 9 The corresponding Gibbons-Hawking four-dimensional geometry has an anti-self-dual Riemann twoform. It is a Ricci flat space whose Kretschmann scalar reads R κλµν R κλµν = 24N 4 z 6 , so that the geometry is singular at z = 0. The Toda potential for the regular Eguchi-Hanson metric has been computed in [35]. The expression (C.13) is an analytic continuation of equation (3.10) in [35].
Moreover, its charge density at ρ = 0 which corresponds to r = 0 or z = N reads λ(η) = η , 0 η N , N , η N . (C. 17) This line density can be also computed by expanding Φ near r = 0 and near z = N , corresponding to expanding near ρ = 0. In addition, for consistency one may verify that this electrostatic potential can be directly evaluated by using (C.10).
(C. 23) Again, this line density can be also computed by expanding Φ near ζ = 0 and near ϑ = 0, corresponding to an expansion near ρ = 0. In addition, this electrostatic potential can be directly evaluated by using (C.10) with the above line density.