The Coulomb branch of N=4 SYM and dilatonic scions in supergravity

We find a parametrically light dilaton in special confining theories in three dimensions. Their duals form what we call a scion of solutions to the supergravity associated with the large-N limit of the Coulomb branch of the N=4 Super-Yang-Mills (SYM) theory. The supergravity description contains one scalar with bulk mass that saturates the Breitenlohner-Freedman unitarity bound. The new solutions are defined within supergravity, they break supersymmetry and scale invariance, and one dimension is compactified on a shrinking circle, yet they are completely regular. An approximate dilaton appears in the spectrum of background fluctuations (or composite states in the confining theory), and becomes parametrically light along a metastable portion of the scion of new supergravity solutions, in close proximity of a tachyonic instability. A first-order phase transition separates stable backgrounds, for which the approximate dilaton is not parametrically light, from metastable and unstable backgrounds, for which the dilaton becomes parametrically light, and eventually tachyonic.

In Refs. [1] and [2], we proposed a mechanism giving rise to an approximate dilaton in the spectrum of composite states of special classes of confining theories, in four dimensions, that admit a higher-dimensional gravity dual. We provided two explicit, calculable realisations of this mechanism, which generalises the ideas proposed in Ref. [3] (and further critically discussed in Refs. [4][5][6]).
We considered non-AdS gravity backgrounds in proximity of classical instabilities-generalising the proximity to the Breitenlohner-Freedman (BF) unitarity bound [7] in AdS space. An approximate dilaton (a scalar particle coupling to the trace of the energy-momentum tensor) emerges along special branches of supergravity solutions. Portions of the branches yield stable solutions, while complementary ones describe metastable or even unstable solutions. Moving in parameter space along the branch of solutions, the dilaton has finite mass for stable solutions, becomes parametrically light for metastable ones, and tachyonic for the unstable ones.
The gravity duals of the confining theories studied in Refs. [1,2] exhibit large departures from AdS geometry. What renders one of the scalar fluctuations parametrically light along the metastable branch is the interplay between the presence of a vacuum expectation value (VEV) breaking spontaneously scale invariance, the explicit breaking due to relevant deformations, and the effects of the nearby instability. The resulting scalar is an approximate dilaton, in the sense that it couples as expected to the trace of the energy-momentum tensor of the dual field theory; this is demonstrated by the failure of the probe approximation (which ignores the fluctuations of the trace of metric [131]) to reproduce correctly the mass spectrum. We refer the reader to the original publications for the details, and we defer commenting on potential phenomenological applications to extensions of the standard model and to Higgs physics [132,133].
The purpose of this paper is to exhibit a third example of this mechanism, but realised in a lower dimensional theory, in backgrounds that in the far UV approach an AdS geometry, with bulk scalar mass close to the BF bound. This hence highlights differences and similarities with other proposals for the origin of the dilaton. We study a particular truncation of the N = 8 maximal supergravity theory in D = 5 dimensions, which is (loosely) associated with the Coulomb branch of N = 4 super-Yang-Mills (SYM) theories. By introducing a relevant deformation, and compactifying one dimension on a circle, we build a scion 1 of gravity backgrounds yielding a light dilaton and admitting an interpretation in terms of a field theory in three dimensions that confines. The scion provides a one-parameter family generalisation of the gravity background occasionally denoted in the literature as QCD 3 , and for which the spectrum of fluctuations is known [60].
The paper is organised as follows. We summarise in Sect. II the main features of the Coulomb branch, as well as the consistent truncation of maximal supergravity in D = 5 dimensions, its reduction to D = 4 dimensions, the lift to D = 10 type IIB supergravity, and 1 The dual of the Coulomb branch is sourced by a discrete distribution of displaced D3 branes; conversely the pure supergravity action and lift we borrow from the literature [134] leads to singular supersymmetric solutions. In view of this loose relation between supergravity theory and Coulomb branch, we refer to our new solutions, obtained by elaborating on the gravity theory, as forming a scion, rather than a branch, as a way to emphasize their hybrid nature.
the prescription for Wilson loops. Sect. III summarises the classes of solutions we investigate in this paper: we present the UV and IR expansions, then compute curvature invariants and Wilson loops to characterise the solutions. We compute the spectrum of fluctuations for the solutions in Sect. IV, in the appropriate number of dimensions. In Sect. V we compare the free energies, to discuss the stability of the solutions. After the conclusions in Sect. VI, we supplement the material with Appendix A, summarising known results for the supersymmetric solutions, and Appendix B, exhibiting asymptotic expansions of the fluctuations used in computing the spectra.

II. THE MODEL
The N = 8 maximal supergravity in D = 5 dimensions [135][136][137] has played a central role in the history of gauge-gravity dualities. It descends from dimensional reduction of type IIB supergravity in D = 10 dimensions on the 5-sphere S 5 [138,139]. It has recently been established that this is a consistent truncation [140,141], and the full uplift back to type IIB is known [141][142][143] (see also Refs. [134,144,145]). The gauge symmetry is SO(6) ∼ SU (4)-capturing the isometries of S 5 , and the R-symmetry of the dual field theory, respectively. The field content includes 42 real scalars that match N = 4 field-theory operators on the basis of their transformation properties under SU (4): the complex singlet 1 C corresponds to the holomorphic gauge coupling, the symmetric 10 C to the fermion masses, and the real 20 to the matrix of squared masses for the scalars X i , with i = 1 , · · · , 6 (see e.g. Sect. 2.2.5 of Ref. [11], or the introduction of Ref. [146]).
One of the background solutions of D = 5 maximal supergravity lifts in type IIB to the AdS 5 × S 5 background geometry providing the weakly-coupled dual description of N = 4 SYM with SU (N ) gauge group, in the (decoupling) limit of large N and large 't Hooft coupling [8]. The supergravity solution is also the appropriate decoupling limit of the configuration sourced by a stack of N coincident D3 branes. Following Ref. [147] (see also Refs. [23,148,149]), we call Coulomb branch the space of inequivalent vacua of the N = 4 theory that preserve 16 supercharges. The space is so called because away from its SO(6)-invariant configuration the gauge group of the field theory is partially higgsed, and the massless gauge bosons mediate Coulomb interactions.
In the language of extended objects in D = 10 dimensions, the literature identified multi-centred D3-brane solutions [23,147] with the moduli space of N = 4 SYM, in the sense that points of the Coulomb branch are associated with distributions of the N D3 branes over R 6 (conveniently parametrised as a cone over the sphere S 5 ), accompanied by the higgsing of SU (N ). By taking N → ∞, while introducing a continuous distribution of D3 branes, one might hope to recover a supergravity description of the Coulomb branch still within maximal N = 8 supergravity in D = 5 dimensions. In fact, the resulting metrics satisfy the supergravity equations [147], but are singular (see for example the discussion after Eq. (3.12) of Ref. [150]). These solutions are captured by a consistent truncation [134,144,145] that retains only the 20 scalars, dual to the symmetric and traceless operator There are five subclasses of solutions that preserve SO(n) × SO(6 − n) subgroups of SO (6), with n = 1, · · · , 5 [147]. With abuse of language, the supersymmetric backgrounds of this type are referred to as the Coulomb branch, though such solutions are singular, and hence incomplete as gravity duals. We further restrict our attention to the n = 2 and the n = 4 cases [23]. The spectrum of the n = 2 case is quite peculiar: both the spin-0 and spin-2 spectra have a gap and a cut opening above a finite value [12,147,[151][152][153][154] (see also Refs. [154,155] for the spectra of vectors). The choices n = 2, 4 are convenient also because they are both captured by one of the subtruncations of the theory in Ref. [156]-which retains only two scalars, one in the 20 and the 10 C , respectively (see also the discussions in Refs. [157,158]). Setting to zero the latter of the two scalars reduces the field content to just one scalar (φ in our notation), and the lift to D = 10 dimensions is comparatively simple.
The gravity descriptions for n = 2, 4 are different; n is associated with the ball B n inside the internal space (including the radial direction) over which one distributes the N D3 branes, and is then reflected in the Ramond fluxes in supergravity. We will identify two distinct classes of solutions to the supergravity equations, distinguished by the negative or positive sign of φ at the end of space, which we associate with n = 2, 4, respectively (see also the discussion at the end of Section 2 in Ref. [134]). We display the supersymmetric solutions and summarise their known properties in Appendix A. We reconsider the system consisting of the scalar φ coupled to gravity in D = 5 dimensions, and describe more general classes of solutions with respect to the literature. These more general deformations break explicit supersymmetry and scale invariance, hence lifting the space of vacua, and modifying the spectrum of the theory. We focus on solutions that involve either of two possibilities.
• In Sects. III B and III C, we display singular domain wall solutions that generalise the supersymmetric ones while preserving Poincaré invariance in four dimensions. The dual field theory is deformed by mass terms, breaking supersymmetry, R-symmetry, and scale invariance. We compute the spectrum of fluctuations-which barring the singularity would be interpreted as bound states of the dual field theory in four dimensions-and the behaviour of the quark-antiquark potential between static sources, generalising the results of Ref. [23]. We discover one new special subclass of mildly singular solutions, that yield a long-distance potential E W ∝ 1/L 2 persisting up to infinite separation L.
• In Sect. III D, we identify background solutions for which one of the dimensions of the external spacetime is compactified on a circle, which shrinks smoothly to zero size at some finite value of the radial (holographic) direction. These are regular solutions, and the dual field theory yields linear confinement in three dimensions, as explicitly shown by the Wilson loops. We compute the spectrum of fluctuations, generalising the results of Ref. [60], and discover new features, such as the emergence of an approximate dilaton.
A. Sigma-model in D = 5 dimensions We denote with hatted symbols quantities characterising the theory in D = 5 dimensions. The action of the canonically normalised scalar φ coupled to gravity is the following (in the notation of Ref. [131]): Hereĝ 5 is the determinant of the metric,ĝMN its inverse, andR 5 the Ricci scalar, while V 5 is the potential. The Domain Wall (DW) solutions manifestly preserve Poincaré invariance in four dimensions. They can be obtained by adopting the following ansatz for the metric: By assumption, the only non-trivial functions determining the background are A(ρ) and φ(ρ), with no dependence on other coordinates. The resulting second-order equations of motion are the following: The conventions we are using [16] in writing the action in Eq. (2) are such that if the potential V 5 of the model can be written in terms of a superpotential W satisfying for the metric ansatz ds 2 DW , then the solutions to the first-order equations are also solutions to the second-order Eqs. (4)-(6). C. Reduction to D = 4 dimensions ng the notation in Ref. [61] (see also Ref. [50]), the reduction to D = 4 dimensions is obtained by adopting ing ansatz: angle 0  ⌘ < 2⇡ parametrises a circle, the four-dimensional metric takes the domain wall form ackground values of the new scalars (r) and the metric warp factor A(r), depend only on the the radial e r. ds that the action of the theory in D = 4 dimensions is sigma-model metric for the scalar fields ( , ) is G ab = diag 1, 3 2 , and the potential is itly verified that ake use of the change of variables @ r = e @ ⇢ . It is also convenient to simplify the notation by choosing this choice is purely conventional. ations of motion for the background functions can be written as follows: and is depicted in Fig. 1.
The first-order equations admit solutions that yield a departure from AdS 5 in the interior of the geometry, which may correspond to n = 2 (D3 branes distributed on B 2 ) or n = 4 (D3 branes on B 4 ). For small φ one finds that W − 3 2 − φ 2 + · · · , hence these solutions are interpreted in terms the VEV of an operator of dimension ∆ = 2 in the dual field theory. This saturates the BF bound and, with respect to Refs. [1,2], brings this study in closer contact with the arguments in Ref. [3].

B. Reduction to D = 4 dimensions
We want to model a confining dual field theory in three dimensions. Following Ref. [59], we therefore assume that one spatial dimension is a circle, the size of which may depend on the radial direction parametrised by ρ in the five-dimensional geometry, and we hence allow for the breaking of four-dimensional Poincaré invariance.
Regular background solutions in which the size of the circle shrinks smoothly to zero (at some finite value of the radial direction ρ = ρ o ) introduce a mass gap in the (lower-dimensional) dual field theory, and exhibit the physics of confinement-we discuss how in Sects. III C and III D.
We elect to describe the geometry by applying dimensional reduction of the gravity theory to four dimensions, with the introduction of a new dynamical scalar that encodes the size of the circle. In the remainder of this subsection we provide the technical details of the construction.
We reduce to D = 4 dimensions by adopting the following ansatz, as in Refs. [131] and [52]: where the angle 0 ≤ η < 2π parametrises a circle, the four-dimensional metric takes the domain wall form and the new background scalar χ(r) and warp factor A(r) depend only on the new radial coordinate r.
The action in D = 4 dimensions is where the sigma-model metric for the scalar fields Φ a = {φ, χ} is G ab = diag (1,3), and the potential is We explicitly verified that S 5 = dη S 4 + ∂S , where After the change of variables ∂ r = e −χ ∂ ρ , the equations of motion for the background read as follows: By combining Eqs. (17) and (18), we obtain which defines a conserved quantity along the flow in ρ.
C. Lift to type IIB in D = 10 dimensions We take the lift to type IIB supergravity in D = 10 dimensions from Ref. [134]. The dilaton/axion subsystem is trivial (see Sect. 3.2 of Ref. [134]), and there is no distinction between Einstein and string frames.
By making use of the equations of motions for the scalars φ and χ and for the function A, we find that R 10 = 0 identically. Yet, other invariants, such as the square of the Ricci and Riemann tensors, are non-trivial.

D. Rectangular Wilson loops
The expectation value of rectangular Wilson loops of sizes L and T in space and time, respectively, is computed using the standard holographic prescription [20,21] (see also Refs. [22][23][24]). Open strings, with extrema bound to the contour of the loop on the boundary of the space at ρ = +∞, explore the geometry down to the turning point ρ o in the holographic direction, and the problem reduces to a minimal surface one. The warp factor Ω 2 depends on θ, but we restrict attention to configurations with θ held fixed, and focus on the limiting cases θ = 0 and θ = π/2. Taking T → +∞, we obtain the effective potential between static quarks as a function of the separation L between end-points of the string. 2 Compared to Ref. [134], we have σ i (here)= 2σ i (Ref. [134]).
The calculation of the Wilson loop can proceed along the lines of the prescription in Refs. [22][23][24][25][26]. Starting from the elements of the metric in D = 10 dimensions, ds 2 = g tt dt 2 + g xx dx 2 + g ρρ dρ 2 + · · · , we introduce the functions F 2 (ρ, θ) ≡ −g tt g xx and G 2 (ρ, θ) ≡ −g tt g ρρ , and the convenient quantity where the dependence on (constant) θ is implicit. The separation between the end points of the string is and the profile of the string in the (ρ, x)-plane is The energy of the resulting configuration is As , also written explicitly as Eq. (33) is UV-divergent, requiring the introduction of ρ Λ as a UV cutoff, and to define the regulated E Λ (ρ o ) by restricting the range of integration. We define the following: where the integral extends all the way to the end of space ρ o , choose the case θ = 0 as a counterterm, and finally define the renormalised energy as In confining theories, at large separations L(ρ o ) the energy grows linearly and the string tension is given by A limiting configuration consists of two straight rods at distance L, both with fixed θ, extending from the boundary to the end of space, connected by a straight portion of string atρ o = ρ o . Its energy is configuration is indistinguishable from two disconnected ones, yielding screening in the dual theory-barring the caveats discussed in Ref. [160]. We set the normalisation κ = 1 from here on. There may be cases in which this procedure shows the emergence of a phase transition for the theory living on the probe [23] (see also the discussions in Refs. [26,161]).

III. BACKGROUND SOLUTIONS
We classify in this section the background solutions we are interested in. We present their UV and IR expansions, and discuss curvature invariants and Wilson loops.

A. UV expansions
All the solutions of interest have the same asymptotic UV expansion, and they all correspond to deformations of the same dual theory. We expand them for z = e −ρ 1.
The integration constant A U can be reabsorbed and set to zero, while χ U can be removed by a shift of radial coordinate ρ. φ 2 is associated with the VEV of the aforementioned dimension-2 operator of the dual field theory, and φ 2l with its (supersymmetry-breaking) coupling. The parameter χ 4 is associated with the VEV of a dimension-4 operator which triggers confinement. Domain wall (DW) solutions in D = 5 dimensions are recovered (locally) for A = A − χ = 2χ, yielding two constraints on the five parameters of a generic solution: We illustrate the behaviour of the singular DW solutions with the comprehensive catalogue in Fig. 2. We devote to them Sects. III B and III C (and Appendix A), before The results agree with Ref. [23]. In the case of the n = 2 solutions ( , A ), for both ✓ monotonically increasing function of L, the separation between the end points of the string. T end points converges. The numerical calculation shows that in both cases there is a maximum at that point the energy agrees with the energy of a configuration in which the string reache function of L, the minimum of E W describes a second-order phase transition, and the configu lower energy.
In the case (n = 4) characterised by ( + , A + ), there is a very major di↵erence between the t The case of ✓ = 0 yields a convergent result: L vanishes when the string approaches the e first-order phase transition: E W is multivalued as L changes, there are three branches, w which the string reaches the end of space and becomes tensionless dominating at large L, w L = L max to configurations that have a turning point at finite⇢ o .
In this case for ✓ = ⇡/2, again for the ( + , A + ) solutions, we find a linear potential at reachable because L diverges, but notice that this is not the minimum-energy configuration.
All of these results are in complete agreement with Figs. 1, 4, and 5 of Ref. [23]. By co Ref. [23], we generated the figures by making use of numerical solutions obtained by setting up by means of the asymptotic IR expansions, rather than relying of the exact solutions. We checking that our formalism and numerical strategy agrees with known results, before we p new solutions, in the subsequent sections.
Ignoring the inconsequential constants A I and ρ o , this one-parameter family of solutions is labelled by the free separation L converges to zero for strings with end points at ⇢ = +1, in the limit in which the turning point of the string configuration reaches the end of space, and at that point the string tension vanishes. But in these backgrounds the behaviour is di↵erent for the case ✓ = ⇡/2, for which lim ⇢!⇢o F 2 (⇢) = 2 > 0 is finite. The separation L diverges, and one recovers the linear potential with E W = L + · · · . The top-right and bottom-left panels of Fig. 6 illustrate this behaviour, for the representative choices o = 1 and o = +1, respectively.
Notice that when o < 1 2 q 3 2 log 4 3 (the supersymmetric case), the assumption of keeping ✓ fixed fails, as at show distances the configurations with ✓ = ⇡/2 have smaller energy that those with ✓ = 0, but at large separations L the converse is true.
In this case the separation L diverges for strings that touch the end of space in the geometry, but the potential vanishes, and so does the string tension. The bottom-right panel of Fig. 6 shows this behaviour explicitly, in which we find a Coulombic potential at arbitrary separations L.
We should remember that all these background solutions are singular. Yet the case of the special ( + , A + ) solutions is singled out by the mildness of the divergence, which manifests itself only in the square of the Riemann tensor. It is interesting to notice how this behaviour is accompanied by the emergence of a Coulombic potential for arbitrary values of L. This is, after all, what one would expect to happen along the Coulomb branch, as the name indicates. It is also encouraging to remember, as we saw in Sect. V B, that the spectrum of fluctautions in this limit did not contain a tachyon. E↵ectively, this special solution is the limiting case of the one for o > 1 2 q 3 2 log 4 3 , in which the phase transition leading to a screening potential is removed to L max ! +1. These arguments are very suggestive, yet they must be taken with caution. We will return to this discussion later in the text, when we will compute the free energy and compare it to other classes of solutions. separation L converges to zero for strings with end points at ⇢ = +1, in the limit in which the turning point of the string configuration reaches the end of space, and at that point the string tension vanishes. But in these backgrounds the behaviour is di↵erent for the case ✓ = ⇡/2, for which lim ⇢!⇢o F 2 (⇢) = 2 > 0 is finite. The separation L diverges, and one recovers the linear potential with E W = L + · · · . The top-right and bottom-left panels of Fig. 6 illustrate this behaviour, for the representative choices o = 1 and o = +1, respectively.
Notice that when o < 1 2 q 3 2 log 4 3 (the supersymmetric case), the assumption of keeping ✓ fixed fails, as at show distances the configurations with ✓ = ⇡/2 have smaller energy that those with ✓ = 0, but at large separations L the converse is true.
In this case the separation L diverges for strings that touch the end of space in the geometry, but the potential vanishes, and so does the string tension. The bottom-right panel of Fig. 6 shows this behaviour explicitly, in which we find a Coulombic potential at arbitrary separations L.
We should remember that all these background solutions are singular. Yet the case of the special ( + , A + ) solutions is singled out by the mildness of the divergence, which manifests itself only in the square of the Riemann tensor. It is interesting to notice how this behaviour is accompanied by the emergence of a Coulombic potential for arbitrary values of L. This is, after all, what one would expect to happen along the Coulomb branch, as the name indicates. It is also encouraging to remember, as we saw in Sect. V B, that the spectrum of fluctautions in this limit did not contain a tachyon. E↵ectively, this special solution is the limiting case of the one for o > 1 2 q 3 2 log 4 3 , in which the phase transition leading to a screening potential is removed to L max ! +1. These arguments are very suggestive, yet they must be taken with caution. We will return to this discussion later in the text, when we will compute the free energy and compare it to other classes of solutions. separation L converges to zero for strings with end points at ⇢ = +1, in the limit in which the turning point of the string configuration reaches the end of space, and at that point the string tension vanishes. But in these backgrounds the behaviour is di↵erent for the case ✓ = ⇡/2, for which lim ⇢!⇢o F 2 (⇢) = 2 > 0 is finite. The separation L diverges, and one recovers the linear potential with E W = L + · · · . The top-right and bottom-left panels of Fig. 6 illustrate this behaviour, for the representative choices o = 1 and o = +1, respectively.
Notice that when o < 1 2 q 3 2 log 4 3 (the supersymmetric case), the assumption of keeping ✓ fixed fails, as at show distances the configurations with ✓ = ⇡/2 have smaller energy that those with ✓ = 0, but at large separations L the converse is true.
In this case the separation L diverges for strings that touch the end of space in the geometry, but the potential vanishes, and so does the string tension. The bottom-right panel of Fig. 6 shows this behaviour explicitly, in which we find a Coulombic potential at arbitrary separations L.
We should remember that all these background solutions are singular. Yet the case of the special ( + , A + ) solutions is singled out by the mildness of the divergence, which manifests itself only in the square of the Riemann tensor. It is interesting to notice how this behaviour is accompanied by the emergence of a Coulombic potential for arbitrary values of L. This is, after all, what one would expect to happen along the Coulomb branch, as the name indicates. It is also encouraging to remember, as we saw in Sect. V B, that the spectrum of fluctautions in this limit did not contain a tachyon. E↵ectively, this special solution is the limiting case of the one for o > 1 2 q 3 2 log 4 3 , in which the phase transition leading to a screening potential is removed to L max ! +1. These arguments are very suggestive, yet they must be taken with caution. We will return to this discussion later in the text, when we will compute the free energy and compare it to other classes of solutions. separation L converges to zero for strings with end points at ⇢ = +1, in the limit in which the turning point of the string configuration reaches the end of space, and at that point the string tension vanishes. But in these backgrounds the behaviour is di↵erent for the case ✓ = ⇡/2, for which lim ⇢!⇢o F 2 (⇢) = 2 > 0 is finite. The separation L diverges, and one recovers the linear potential with E W = L + · · · . The top-right and bottom-left panels of Fig. 6 illustrate this behaviour, for the representative choices o = 1 and o = +1, respectively.
Notice that when o < 1 2 q 3 2 log 4 3 (the supersymmetric case), the assumption of keeping ✓ fixed fails, as at show distances the configurations with ✓ = ⇡/2 have smaller energy that those with ✓ = 0, but at large separations L the converse is true.
In this case the separation L diverges for strings that touch the end of space in the geometry, but the potential vanishes, and so does the string tension. The bottom-right panel of Fig. 6 shows this behaviour explicitly, in which we find a Coulombic potential at arbitrary separations L.
We should remember that all these background solutions are singular. Yet the case of the special ( + , A + ) solutions is singled out by the mildness of the divergence, which manifests itself only in the square of the Riemann tensor. It is interesting to notice how this behaviour is accompanied by the emergence of a Coulombic potential for arbitrary values of L. This is, after all, what one would expect to happen along the Coulomb branch, as the name indicates. It is also encouraging to remember, as we saw in Sect. V B, that the spectrum of fluctautions in this limit did not contain a tachyon. E↵ectively, this special solution is the limiting case of the one for o > 1 2 q 3 2 log 4 3 , in which the phase transition leading to a screening potential is removed to L max ! +1. These arguments are very suggestive, yet they must be taken with caution. We will return to this discussion later in the text, when we will compute the free energy and compare it to other classes of solutions.
A special limiting case (corresponding to φ o → −∞) of the (singular) negative DW solutions is represented by the thick (grey) line in Fig. 2, and is given by the IR expansions in Eqs. (A7) and (A8). It satisfies the firstorder equations, as it coincides with the supersymmetric solutions (φ 2 , A 2 ) described by Eqs. (A3) and (A4)-the solution corresponding to the (n = 2) case of D3 branes distributed on a disk (B 2 ).
The study of the Wilson loops is exemplified in Fig. 3. The top-left panel depicts the case of backgrounds (φ − , A − ) with φ o = −1. The string is tensionless at the end of space, as lim ρ→ρo F 2 (ρ) = 0 for both choices θ = 0, π/2. The separation L converges to zero for strings with end points at ρ = +∞, when the turning point of the string configuration reaches the end of space, yielding the description of a phase transition such that the Wilson loop mimics screening at large L.   The five-dimensional curvature invariants diverge. In D = 10 dimensions the divergence appears in R 10,MN RMN 10 = 405e −8 √ 2 3 φo sin 4 (2θ) 2048 sin 10 (θ) + · · · . (48) The limit θ → 0 is singular at the end of space, even in the case of the supersymmetric solution-see also Eq. (A12) and the discussion that follows it. The calculation of the Wilson loops is exemplified in the top-right and bottom-left panels in Fig. 3, for φ o = −1 and φ o = +1, respectively. For θ = 0, once more lim ρ→ρo F 2 (ρ) = 0. The separation L vanishes when the turning point of the string configuration reaches the end of space, as we found for (φ − , A − ). But in the case θ = π/2, we find that lim ρ→ρo F 2 (ρ) = σ 2 > 0 is finite. The separation L diverges, and one recovers the linear potential E W σL. When φ o < − 1 2 3 2 log 4 3 , the assumption of keeping θ fixed fails, as at small L the configurations with θ = π/2 have lower energy than those with θ = 0, while at large L the converse is true.

Special positive DW solutions
A limiting case of the positive DW solutions is depicted by the short-dashed (purple) line in Fig. 2. The IR expansions are: The only parameters are the inconsequential A I and ρ o . The five-dimensional curvature invariants diverge, but the lift to D = 10 dimensions yields Yet, these solutions are singular as well, as illustrated by the simultaneous limits ρ → ρ o and θ → 0 of the square of the Riemann tensor: lim ρ→ρo (R 10,MNRŜ ) 2 = 9(15 + 10 sin 2 (θ) + 7 sin 4 (θ)) 5 sin 6 (θ) .
These solutions are the limiting case φ o → +∞ of the (φ + , A + ) general class, and the bottom-right panel in Fig. 3 shows a peculiarly interesting behaviour for the quark-antiquark potential. The separation L is unbounded, the potential vanishes for L → +∞, and so does the string tension. We find the potential E W −e 1/6 /L at short L, and E W −e 6 /L 2 at large L (for A I = 0). In the case of confining solutions with = 0, the spectrum has been compu truncating completely the tower of excitations of . This special solution is so language) in the literature, and is a well known example in which the spectrum units of the lightest spin-2 excitation, they report (T 3 in Table 4  The results of our numerical study of the spectrum of fluctuations of the gravi The masses of scalars (blue disks) and tensors (red circles) are plotted as a fu display the result of applying the probe approximation to the treatment of the s tensor masses, as well as the masses of half the scalars (the second, fourth, six, with Ref. [58]. Yet, we notice that the truncation adopted in Ref. [58] misses t can be decoupled only for = 0-in this case, the probe approximation is accura states, but only within a narrow range around = 0, as can be seen in the Fig. The main feature emerging from the study are that the spectrum is positive large and negative values of I , we find the emergence of a tachyonic scalar instability. This is qualitatively similar to what we found in the case of DW po probe approximation yields to a dramatic result: the probe approximation alwa spectrum, even at the qualitative level, yielding an unphysical proliferation of t

C. Rectangular Wilson loops for confining so
The study of the rectangular Wilson loop (in D = 3 dimensions) in the cas out by fixing the coordinates ⌘ and ✓, and allowing the two sides of the recta non-compact space-like directions. The results are illustrated by Fig. 8. We with o = 1 (in which case we saw that the spectrum of fluctuations is posi case we found the presence of a tachyon in the spectrum). In all cases, the shor way to the linear potential typical of confinement, and L is unbounded.
We can compute the string tension, and we find (for A I = 0 = I ) and find:  In the case of confining solutions with = 0, the spectrum has been computed in Ref. [58], although only after truncating completely the tower of excitations of . This special solution is sometimes called QCD 3 (with abuse of language) in the literature, and is a well known example in which the spectrum of bound states can be computed. In units of the lightest spin-2 excitation, they report (T 3 in Table 4 of Ref. [58]) the spectrum of mass of the tensors to be For the scalars (S 3 in Table 4 The results of our numerical study of the spectrum of fluctuations of the gravity background are displayed in Fig. 7. The masses of scalars (blue disks) and tensors (red circles) are plotted as a function of the parameter I . We also display the result of applying the probe approximation to the treatment of the scalars. We notice that for I = 0 the tensor masses, as well as the masses of half the scalars (the second, fourth, six, eight, . . . ) are in excellent agreement with Ref. [58]. Yet, we notice that the truncation adopted in Ref. [58] misses the lightest of the scalar states, which can be decoupled only for = 0-in this case, the probe approximation is accurate for the first, third, fifth, . . . , scalar states, but only within a narrow range around = 0, as can be seen in the Fig. 7.
The main feature emerging from the study are that the spectrum is positive definite only for I > ⇤ I ' 0.5. For large and negative values of I , we find the emergence of a tachyonic scalar state, signaling the appearence of an instability. This is qualitatively similar to what we found in the case of DW positive solution. Comparison with the probe approximation yields to a dramatic result: the probe approximation always fails to capture the features of the spectrum, even at the qualitative level, yielding an unphysical proliferation of tachyonic states.

C. Rectangular Wilson loops for confining solutions
The study of the rectangular Wilson loop (in D = 3 dimensions) in the case of the confining solutions is carried out by fixing the coordinates ⌘ and ✓, and allowing the two sides of the rectangle to be along time and one of the non-compact space-like directions. The results are illustrated by Fig. 8. We considered two representative choices with o = 1 (in which case we saw that the spectrum of fluctuations is positive definite) and I = +1 (in which case we found the presence of a tachyon in the spectrum). In all cases, the short-distance Coulombic behaviour gives way to the linear potential typical of confinement, and L is unbounded.
We can compute the string tension, and we find (for A I = 0 = I ) and find: While the results of the study of the Wilson loops for the DW solutions are very suggestive, with the emergence of screening, confining, several types of Coulombic potentials and phase transitions, they must be all taken with caution; all the background solutions discussed so far (and in Appendix A) are singular. Hence, such solutions cannot be considered as complete gravity duals of field theories, but they provide only approximate descriptions that may miss important long-distance details.

D. Confining solutions
The solutions of this class are completely regular, and dual to confining, three-dimensional field theories. Here we present their IR expansions, discuss the gravitational invariants, and compute the Wilson loops. The expansion in proximity of the end of space ρ o , is The gravity invariants in five dimensions are finite, and when restricted to the (ρ, η) plane, the metric reduces to We fix χ I = 0 in order to avoid a conical singularity. The integration constant A I is trivial and can be reabsorbed by a rescaling of the three Minkowski directions. The constant φ I characterises this one-parameter family of solutions. The curvature invariants of the lift to D = 10 dimensions are regular, for all choices of 0 ≤ θ ≤ π/2. In the study of the rectangular Wilson loop (in three dimensions) we fix θ, and allow the two sides of the rectangle to align with time and one non-compact space-like direction. The static potential E W (L) is illustrated by Fig. 4, for two representative choices with φ o ± 1. The short-distance Coulombic behaviour gives way to the linear potential typical of confinement, and L is unbounded. For this reason, with some abuse of language, we call these regular solutions confining. We can compute the string tension, and we find The configuration with θ = 0 has lower energy in the case where φ I > 0, and vice versa.

IV. MASS SPECTRA AND PROBE APPROXIMATION
The spectrum of small fluctuations of a sigma-model coupled to gravity of the form of Eqs. (2) and (13) in generic D dimensions can be interpreted in terms of the spectrum of bound states of the strongly-coupled dual field theory, by applying the dictionary of gauge-gravity dualities. We adopt the gauge-invariant formalism described in detail in Refs. [15][16][17][18][19]. Due to the divergences in the deep IR and far UV, we introduce two unphysical boundaries ρ 1 < ρ < ρ 2 in the radial direction-the physical results are recovered in the limits ρ 2 → +∞ and ρ 1 → ρ o . The calculation involves fluctuating solutions for which the metric has the DW form in D dimensions. The confining solutions assume the DW form in the dimensionally reduced (D = 4) formulation of the theory. For the confining solutions, it is also understood that in the following equations (59 -62) appearing in this section of the paper, A is to be replaced by A.
The tensorial fluctuations e µ ν are gauge-invariant, obey the equations of motion The notation follows the conventions of Ref. [19]. The sigma-model metric being trivial, the covariant derivative simplifies to V a |c ≡ D c V a = ∂ c (G ab ∂ b V ) , and the background-covariant derivative to D ρ a a = ∂ ρ a a .
The probe approximation is defined according to the prescription tested in Ref. [131], and we use it as a diagnostic tool to identify particles coupled to the trace of the energy momentum tensor, because of their mixing with h. The probe approximation ignores the fluctuation h, in the definition of a a , yielding variables p a that satisfy subject to Dirichlet boundary conditions p a ρi = 0.
The fluctuation h is interpreted as the bulk field coupled to the dilatation operator in the dual field theory. If the approximation of ignoring h captures correctly the spectrum, then the associated scalar particle is not a dilaton. Conversely, the probe approximation either completely misses, or fails to capture the correct mass of, an  113) and (114). o ! 1 the backgrounds are approaching the susy solutions in Eqs. (90) and (91), which also coincides with Eqs. (94) (95). The numerical calculations are performed with finite cuto↵s ⇢1 = 10 6 and ⇢2 = 0, but we checked explicitly that se choices are close enough to the physical limits ⇢1 ! ⇢o = 0 and ⇢2 ! +1 that the results do not display any discernible idue spurious dependence on the physical cuto↵s, given the numerical precision.
ablished. In the case of the background of this mode that respect four-dimensional Poincaré invariance, this is ver the case, due to the singularity in the background. Yet, it is instructive to perform the calculation, applying the es of gauge-gravity dualities, hence generalising the results for the (singular) supersymmetric solutions of Sec. IV B d references therein) to non-supersymmetric solutions, for which we already ascertained the singularities are no rse. The result of the numerical study of the fluctuations for the ( , A ) soluitions (corresponding to the n = 2 coset ng the Coulomb branch) is displayed in Fig. 4, as a function of the parameter o characterising the 1-parameter ily of solutions. We find it convenient to normalise the mass M of the spin-0 (displayed as the blue disks) as well spin-2 states (the red circles) so that the lightest tensor mode has unit mass. For any finite value of o the spectrum is characterised by an unremarkable discrete sequence of state, and by the stence of tachyonic state, which signals a fatal instability in the background solutions. Only in the strict limit ! 1, corresponding to the supersymmetric solution, the tachyon becomes exactly massless. In the same limit, spectrum shows the presence of a gap, followed by a continuum, in all the channels, for M 2 > 1. This result roduces precisely the results we already quoted from the literature, confirming that the gauge-invariant formalism adopted, and the choices of boundary conditions we impose, are such to correctly identify the poles of the relevant oint correlation functions int eh gauge-gravity prescription. By comparing the gauge-invariant spin-0 fluctuations to the probe approximation (the black diamonds in Fig. 4), e observes a huge discrepancy, both qualitative and quantitative. Clearly, the probe approximation fails most pletely to capture the lightest (tachyonic) state at all values of o , so that we can establish that the dilaton ponent is always important in such spin-0 state. For large and positive values of o , the probe approximation tures well all other excited sclalar states, and hence yields a good identification of the dilaton with the tachyon. xing e↵ects become prevalent for negative o , and such identification becomes more obscure. approximate dilaton. We tested these ideas on a large selection of examples in Ref. [131].
In order to improve the convergence of the numerical computation of the spectrum, we make use of the UV expansions for the fluctuations given in Appendix B, setting up the boundary conditions such that only the sub-leading modes are retained. This is the customary prescription, as well as the one selected by the boundary conditions in Eq. (61), in the limit in which we remove the UV regulator (boundary) at ρ 2 .
We start the analysis from the DW solutions. The result of the numerical study of the fluctuations for the (φ − , A − ) solutions is displayed in Fig. 5, as a function of the parameter φ o characterising this one-parameter family. We find it convenient to normalise the masses M of the spin-0 (blue disks) and spin-2 states (red circles) so that the lightest tensor mode has unit mass.
For any finite value of φ o the spectrum is characterised by an unremarkable discrete sequence of states, and by the existence of a tachyon, which signals a fatal instability in the background solutions. Only in the strict limit φ o → −∞ does the tachyon become exactly massless. In the same limit, the spectrum degenerates to a gapped continuum, in all the channels, for M 2 > 1 (in units of the lightest tensor mode). This result reproduces the results quoted from the literature in Appendix A, for the n = 2 case, confirming that the gauge-invariant formalism we adopt, the choices of boundary conditions we impose, and the numerical strategy we deploy combine Sect. V A 2, and particularly in Sect. V A 3, that solutions of this type have a milder singularit encouraging result. We shall return to it when appropriate, later in the text. Once again, the spectrum of the supersymmetric backgrounds is in splendid agreement wit further confirms that our numerical strategy is reliable. The spectrum for the case of nagetic always a tachyon, followed by a light scalar state and a densely packed sequence of heavy ex o ! 1, the spectrum agrees again with the case of the supersymmetric solution with ( addition of a tachyon. Indeed, this superficially surprising feature can be explained by star in Fig. 3, from which one can see that it is possible to choose boundary conditions for the ( yield a trajectory that can be made to approximate for arbitrarily long interval the special ( of ( , A ) with o ! 1. Yet, in the case of ( , A ), this limit is not well defined, as e depart from the limiting case, and (⇢) becomes positive (and divergent) close enough to the Once more, the comparison with the probe approximation is instructive: the tachyonic stat the probe approximations, which produces an arbitrary number of such negative-mass states, g value of o .

C. Rectangular Wilson loops for DW solutions
The calculation of the Wilson loops yields behaviours that generalise those we found fo solutions. We exemplify them in Fig. 6, where we focus on four illustrative examples: th ( , A ) with o = 1, the ( + , A + ) solution with o = 1, the ( + , A + ) solution with o ( + , A + ) solution corresponding to o ! +1. In both cases, we assume that the angle ✓ be fix the value ✓ = 0 or ✓ = ⇡/2.  Once more the spectrum of the supersymmetric background is in agreement with the literature, which further confirms that our numerical strategy is reliable.
The spectrum for φ o < − 1 2 3 2 log 4 3 always contains a tachyon, followed by a light scalar state and a densely packed sequence of heavy excitations. In the limit φ o → −∞, the spectrum agrees with the case of the supersymmetric solution (φ 2 , A 2 ), except for the addition of a tachyon. Indeed, this superficially surprising feature can be explained by close examination of the stream plot in Fig. 2, from which one can see that it is possible to choose boundary conditions for the (φ + , A + ) solutions that yield a trajectory approaching the special (supersymmetric) limit (φ 2 , A 2 ) with φ o → −∞. Yet, eventually all the φ + (ρ) solutions turn positive (and divergent), close enough to the end of the space; the tachyon emerges as an unavoidable consequence of the intrinsic instability of these flows.
The comparison with the probe approximation is instructive: the tachyon is never captured by the probe approximation, which rather produces an arbitrary number of negative-mass-squared states, depending on φ o . Conversely, in the region of large and positive φ o the probe approximation highlights that an infinite number of scalars mix with the dilaton.
The spectrum of confining solutions with φ = 0 has been computed in Ref. [60], although only after trun-cating the tower of excitations of φ. This background is sometimes called QCD 3 (with abuse of language) in the literature. In units of the lightest spin-2 excitation the spectrum of mass of the tensors (T 3 in Table 4 of Ref. [60]) is reported to be M 2 = 1 , 1.73 , 2.44 , .3.15 , 3.86 , 4.56 , · · · , (63) and for the scalars (S 3 in Table 4  We extend the numerical study to the whole oneparameter scion of solutions characterised by φ o , and retain both fluctuations of φ and χ. The resulting spectrum is displayed in Fig. 7. The masses of scalars (blue disks) and tensors (red circles) are plotted as a function of φ I . We also display the result of applying the probe approximation to the treatment of the scalars (black diamonds). For φ I = 0, the tensor masses, as well as the masses of half the scalars (the second, fourth, six, eight, . . . ) are in excellent agreement with Ref. [60], confirming for the third time the robustness of our procedure. Yet, the truncation adopted in Ref. [60] misses the lightest of the scalar states, which can be decoupled only for φ = 0-in this case, the probe approximation is accurate for the first, third, fifth, . . . , scalar states, but only within a narrow range around φ = 0.
The main feature that emerges is that the spectrum is positive definite only for φ I > φ * I −0.52. For large and negative values of φ I , we find the emergence of a tachyon, signaling the appearance of an instability. The probe approximation fails to capture the features of the spectrum, even at the qualitative level, yielding an unphysical proliferation of tachyons.
In summary, in the case of the confining solutions, and for φ I > φ * I −0.52, the solutions are regular (the curvature invariants computed in D = 5 and D = 10 dimensions are all finite) and smooth (there is no conical singularity at the end of space), the spectrum is positive definite, and the calculation of the Wilson loop via the dual gravity prescription leads to the linear potential expected in a confining field theory (in three dimensions). All of these properties are preserved all the way along the scion of confining solutions until the critical value φ * I , in proximity of which the lightest scalar separates from the rest of the spectrum, and becomes arbitrarily light, before turning into a tachyon. This light state, as shown by the probe approximation, has a substantial overlap with the dilaton, and couples to the trace of the energymomentum tensor of the dual confining theory.

V. FREE ENERGY AND STABILITY ANALYSIS
Because we regulate the theory by introducing two boundaries ρ 1 and ρ 2 in the radial direction of the geometry, the complete action in D = 5 dimensions must include also boundary-localised terms: The Gibbons-Hawking-York (GHY) term is proportional to the extrinsic curvature K, and λ i are boundary potentials. We choose λ 1 = − ∂ ρ A| ρ=ρ1 , and, as in Ref. [12] (see also Eq. (3.66) of Ref. [153]), where z 2 ≡ e −ρ2 , and the freedom in the choice of k reflects the scheme-dependence of the result. The explicit appearance of the term containing the unphysical constant k in this result is a peculiarity of this model, distinguishing it from those in Refs. [1,2]. It is due to the mass of the scalar field corresponding to the deforming field theory operator exactly saturating the BF unitarity bound. In this sense, this model is a more direct realisation of the ideas exposed in Ref. [3], where the proximity to the BF bound is the starting point of the analysis. As we shall see shortly, though, our results here are qualitatively similar to those in Refs. [1,2].
The need for counter-terms that are quadratic in φ, and their scheme dependence, imply that the concavity theorems do not hold for the free energy of this system. The free energy F and its density F are defined as and by using the equations of motion, supplemented by the observation that Eq. (20) defines a conserved quantity along the radial direction ρ, we find We can now use the UV expansions, take the e −ρ2 → 0 limit, and arrive at For the DW solutions, further simplifications yield Along the lines of Ref. [1], we find it convenient to define a scale Λ as follows [162]: While this is not a unique choice, its simplicity and universality gives it a practical value for our applications. In the one-parameter branch of confining solutions until the critical value I , in proximity of which t separates from the rest of the spectrum, and becomes arbitrarily light, before turning into a tachyon. as shown by the calculation of the spectrum in probe approximation, has a substantial overlap with ------------- the calculation of the free energy density, we set k = Λ, as this quantity scales with dilatations in the same way as z −1 .
We display in Fig. 8 the result of the calculation of FΛ −4 as a function of the source φ 2l Λ −2 , for the three classes of negative DW, positive DW and confining solutions. For negative values of φ 2l Λ −2 , as we saw the regular confining solutions have positive definite spectrum (as φ I > φ * I ), and furthermore their free energy is the lowest among the solutions we considered. When φ 2l Λ −2 is positive, but small, we still find regular confining solutions, but the lightest scalar state has lower mass, which vanishes when φ I = φ * I −0.52, after which it turns tachyonic. There is hence a regime of parameter space in which the lightest scalar has suppressed mass. But these solutions are metastable: the positive DW solutions (despite being singular) have lower free energy when φ 2l Λ −2 > φ 2l Λ −2 0.13, the critical value identified by the crossing in Fig. 8 (corresponding to φ I −0.067)

13
Once more, as in the models in Refs. [1,2], we find that the lightest scalar can be identified with a parametrically light dilaton along the metastable solutions. In the stable solutions the lightest scalar is still an approximate dilaton, but not parametrically light.

VI. CONCLUSION AND OUTLOOK
We studied new classes of background solutions of maximal supergravity in D = 5 dimensions, truncated to retain only one scalar field. This is the theory related to the dual of the Coulomb branch of N = 4 SYM. We focused on solutions that are regular, have positive definite spectrum, and can be interpreted as the gravity dual of confining field theories in three dimensions. We found evidence that the lightest scalar state is an approximate dilaton, and can be made parametrically light, in a region of parameter space in which these new regular solutions are metastable.
The study confirms, in a lower-dimensional simple setting, for a well known example of gauge-gravity duality related to the study of N = 4 SYM, the qualitative features that emerged in the models in Refs. [1,2]. We notice the emergence of a first-order phase transition separating the metastable from the stable portions of the parameter space of the new confining solutions. As in Refs. [4][5][6], the approximate dilaton is not parametrically light in the stable solutions, confirming this generic feature also in confining theories in three dimensions.
Further exploration of the catalogue of supergravity theories will possibly help to understand whether the aforementioned results are universal or model dependent.
Of particular interest would be to ascertain whether it is possible, and under what conditions, to find systems for which the phase transition is weak enough to render the dilaton parametrically light already in the stable region of parameter space, in proximity of the phase transition itself. It would also be interesting to see whether systems exist for which the phase transition is of second order.
Both solutions exhibit a naked singularity in D = 5 dimensions, which softens in D = 10 dimensions. For the n = 2 solutions (φ 2 , A 2 ) we find R 10,MN RMN 10 = 135 4 cos 2 (θ)(ρ − ρ o ) 3 + · · · .(A11) For the (n = 4) solutions given by (φ 4 , A 4 ) the behaviour of this invariant is milder: R 10,MN RMN 10 = 5 sin 4 (2θ) 8 sin 10 (θ) + · · · . (A12) The singularity at the equator of S 5 displayed by the curvature invariant in Eqs. (A11) and (A12) at the end of space signals the incompleteness of the supergravity description in both supersymmetric cases. The Wilson loops for the supersymmetric solutions have been computed in Ref. [23]. We display our result in Fig. 9, as a test of our procedure. In the case of the n = 2 solutions (φ 2 , A 2 ), for both θ = 0, π/2 we find a monotonic potential, and a maximum value of L = L max . In the case n = 4 of (φ 4 , A 4 ), there is a very major difference between the two cases with θ = 0, π/2. The case of θ = 0 displays the features expected by a first order phase transition, with long-distance screening. Conversely, for where is the digamma function, q the four-momentum in Euclidean signature, and a depends on the momentum as a = 1 2 + 1 2 We notice the presence of a massless pole, besides the gap and the cut. We borrow also the tensorial correlation function from Section 3.3 of Ref. [153]: where  is some constant that is of no concern to us, and where a has been define in Eq. (111). The tensorial spectrum has the same properties and cut as the scalar one, except for the absence of a massless state. The other supersymmetric solutions, with n = 4, denoted by ( + , A + ) in Eqs. (92) and (93), correspond to swopping the global symmetries in the cosets. The resulting discrete spectrum is for example described in Eq. (26) of Ref. [149]. With j = 0, 1, · · · , the spectrum is given by M / p j(j + 1)/2 ' 0, 1, 1.7, 2.5, 3.2, · · · . 1 The spectrum of tensors can be found for example in Eq. (45) of Ref. [151], according to which it agrees '. . . to an extremely good approximation . . . ', but for the fact that there is no zero mode. four-momentum in Euclidean signature, and a depends on the momentum as a = 1 2 + 1 2 le, besides the gap and the cut. on function from Section 3.3 of Ref. [153]: θ = π/2 we find a linear potential at asymptotically large L, which is unbounded; this configuration has higher energy than the θ = 0 one. For the n = 2 solutions (φ 2 , A 2 ) in Eqs. (A3) and (A4), the 2-point function of the operator O dual to the scalar φ can be found in Section 3.3 of Ref. [153] (see also Eq. (8.6) of Ref. [12]): O(q)O(−q) = 16 3q 2 − 4 ψ(a(q) + 1) − ψ(1) ,(A13) and for the tensors where ψ is the digamma function, q the four-momentum in Euclidean signature, κ is a constant, and a is a(q) ≡ − 1 2 The scalar correlator displays a massless pole, a gap and a continuum cut, the tensor differs by the absence of the massless state. The n = 4 solutions (φ 4 , A 4 ) in Eqs. (A5) and (A6), have a discrete spectrum, for example described in Eq. (26) of Ref. [147]. With j = 0, 1, · · · , the spectrum is given by M ∝ j(j + 1)/2 0, 1, 1.7, 2.5, 3.2, · · · . The spectrum of tensors can be found for example in Eq. (45) of Ref. [151], according to which it agrees with that of the scalars '. . . to an extremely good approximation . . . ', but for the fact that there is no zero mode.
The results of studying the Wilson loops agree with Figs. 1, 4, and 5 of Ref. [23]. We relied on numerical solutions guided by the asymptotic IR expansions, rather than using the exact solutions as in Ref. [23]. Our numerical study of the spectrum yields numerical results in splendid agreement with pre-existing calculations, as can be seen in Figs. 5 and 6. These results and their agreement with earlier studies of the supersymmetric solutions confirm the robustness of our formalism and numerical strategy.

Appendix B: Expansions for the fluctuations
In the numerical calculation of the spectra, we used the asymptotic expansions of the gauge-invariant fluctuations, as a way to optimise the decoupling of spurious cutoff effects present at finite ρ 2 . In the case of DW solutions, we find that we can expand the physical fluctuations as follows: In these expressions, a φ 2l , a φ 2 , (e 0 ) µ ν , and (e 4 ) µ ν are the integration constants governing the solutions of the secondorder linearised equations, half of which are determined by the boundary conditions. In the probe approximation, the expansion for the scalars p φ is of the same form, up to O(z 4 ): The confining solutions do not satisfy the DW conditions. For the scalars we find where a φ 2l , a φ 2 , a χ 0 , and a χ 4 are the free parameters. For the probe approximation of the scalars, the free parameters are p φ 2l , p φ 2 , p χ 0 , and p χ 4 , and as a result of mixing in the second derivative of the potential in Eq. (62) we have: p χ = p χ 2l log(z)z 2 + p χ 2 z 2 + O(z 4 ) .
For the tensor fluctuations we find