Wilson loops on three-manifolds and their M2-brane duals

We compute the large N limit of Wilson loop expectation values for a broad class of N=2 supersymmetric gauge theories defined on a general class of background three-manifolds M_3, diffeomorphic to S^3. We find a simple closed formula which depends on the background geometry only through a certain supersymmetric Killing vector field. The supergravity dual of such a Wilson loop is an M2-brane wrapping the M-theory circle, together with a complex curve in a self-dual Einstein manifold M_4, whose conformal boundary is M_3. We show that the regularized action of this M2-brane also depends only on the supersymmetric Killing vector, precisely reproducing the large N field theory computation.


Introduction and summary
There has recently been considerable interest in defining and studying supersymmetric gauge theories on compact manifolds. This stems from the fact that certain observables may be computed exactly in such quantum field theories using localization. The first examples of such computations in the literature typically studied round sphere backgrounds, but more generally the observables also depend on the choice of background geometry, leading to a richer structure. Such exact computations may be used to test and explore non-perturbative dualities, and the focus of this paper will be the gauge/gravity duality.
In [1] the partition function Z of three-dimensional N = 2 supersymmetric gauge theories on a general class of background three-manifold geometries M 3 was computed exactly. In particular Z was shown to depend on the background geometry only through a certain supersymmetric Killing vector field K. 1 There are rich classes of N = 2 superconformal gauge theories which have a large N gravity dual in M-theory. For these theories one can compute the large N limit of the partition function using the matrix model saddle point technique of [3]. When M 3 is diffeomorphic to S 3 with the standard action of U(1) × U(1) on S 3 ⊂ R 2 ⊕ R 2 , and writing K = b 1 ∂ ϕ 1 + b 2 ∂ ϕ 2 in terms of the generators ∂ ϕ i of U(1) × U(1), one finds [1] the large N free energy where F round is the large N limit of the free energy on the round three-sphere, which scales as N 3/2 [4].
In [5] the field theory result (1.1) was reproduced in a dual computation in fourdimensional gauged supergravity. Here M 3 ∼ = S 3 arises as the conformal boundary of a self-dual Einstein four-manifold M 4 , where the supersymmetric Killing vector K also extends over M 4 . The asymptotically locally Euclidean AdS metric on M 4 is conformally Kähler, and supersymmetry requires one to turn on a graviphoton field A proportional to the Ricci one-form of this Kähler metric. A remarkable feature of the computation of the holographic free energy in [5] is that one does not need to know the form of the Einstein metric on M 4 explicitly -rather (1.1) is proven for an arbitrary such metric.
In [6] the vacuum expectation values of BPS Wilson loops on the round sphere were computed for a variety of gauge theories, and matched to regularized M2-brane actions in AdS 4 × Y 7 . Here the choice of internal space Y 7 determines the gauge theory on M 3 . The purpose of this paper is to extend these computations to general supersymmetric backgrounds M 3 = ∂M 4 . A Wilson loop is BPS if it wraps an orbit of K, and we will find that the large N Wilson loop VEV satisfies Here W round denotes the large N limit of the Wilson loop on the round sphere, whose logarithm scales as N 1/2 , and 2πℓ denotes the length of the orbit of K. Such orbits always close over the poles of S 3 , i.e. at the origins of each copy of R 2 in S 3 ⊂ R 2 ⊕ R 2 , where the lengths are then ℓ = 1/|b 1 | and ℓ = 1/|b 2 |, respectively. For these Wilson loops (1.2) becomes a function of b 1 /b 2 , exactly as in (1.1). The supergravity dual configurations are given by M2-branes wrapping a supersymmetric copy of the Mtheory circle in Y 7 [6] and a complex curve Σ 2 ⊂ M 4 , with boundary ∂Σ 2 ⊂ M 3 being the Wilson line. Identifying the logarithm of the VEV with minus the holographically renormalized M2-brane action, we also prove that (1.2) holds in general, thus verifying the matching of this observable in AdS/CFT in a very broad (infinite-dimensional) class of backgrounds.
The outline of the rest of this paper is as follows. In section 2 we review the geometry of M 3 , the definition of the BPS Wilson loop and how it may be computed using localization techniques in the large N limit to find (1.2). Section 3 analyses supersymmetric M2-branes in M 4 × Y 7 backgrounds in M-theory and we also derive the formula (1.2) in supergravity. Since our arguments are for general backgrounds they are somewhat implicit; in section 4 we therefore look at some explicit toric self-dual Einstein spaces, to exemplify our general formulae. We conclude in section 5 with a brief discussion.
2 Wilson loops in N = 2 gauge theories on M 3 The field theories of interest have UV descriptions as N = 2 Chern-Simons gauge theories coupled to matter on M 3 , where M 3 is a supersymmetric three-manifold. We begin this section by reviewing the geometry of M 3 , and then define the BPS Wilson loops of interest. These have been studied on particular squashed sphere backgrounds in [7], [8] (see also [9], [10]), and the extension to the general backgrounds of [1,11] is straightforward. After explaining how the Wilson loop VEVs localize in the matrix model, we then take the large N limit to derive (1.2).

Three-dimensional background geometry
The manifold M 3 belongs to a general class of "real" supersymmetric backgrounds, with two supercharges related to one another by charge conjugation [11]. If χ denotes the Killing spinor on M 3 then there is an associated Killing vector field This Killing vector is nowhere zero and therefore defines a foliation of the threemanifold. This foliation is transversely holomorphic with local complex coordinate z. In terms of these coordinates the background metric may be written as 2 where dz is a local one-form and w (0) (z,z) is a function. We introduce an orthonormal frame for the three-metric ds 2 M 3 : and will use indices i, j, k = 1, 2, 3 for this frame.
It is important to stress here that arbitrary choices for φ (0) and w (0) (subject to M 3 being smooth) lead to supersymmetric backgrounds. The corresponding Killing spinor equation for χ may be found in [1,11]. Choosing the three-dimensional gamma matrices, in the frame (2.3), to be simply the Pauli matrices, one finds that the Killing spinor solution is where χ 0 is a constant and α(ψ, z,z) is a phase. The latter will play an important role later.
In much of what follows, and as in [1], we will assume that M 3 ∼ = S 3 with a toric structure, so that we have a U(1) × U(1) symmetry. If we realize M 3 ∼ = S 3 ⊂ R 2 ⊕ R 2 then we may write where ϕ 1 , ϕ 2 are standard 2π-period coordinates on U(1) × U(1).

The Wilson loop
In N = 2 supersymmetric gauge theories the gauge field A i is part of a vector multiplet that also contains two real scalars σ and D and a two-component spinor λ, all of which are in the adjoint representation of the gauge group G. The BPS Wilson loop in a representation R of G is given by where x i (s) parametrizes the worldline γ ⊂ M 3 of the Wilson loop and the path ordering operator has been denoted by P. For a Chern-Simons theory the gauge multiplet has a kinetic term described by the supersymmetric Chern-Simons action where k denotes the Chern-Simons coupling and vol 3 is the Riemannian volume form on M 3 .
The full set of supersymmetry transformations for a vector multiplet and matter multiplet may be found in [1]. For our purposes we need note only that localization of the path integral, discussed in the next section, requires one to choose a Killing spinor, namely χ in (2.4). We then need the following two supersymmetry transformations (3) is the one-form dual to the supersymmetric Killing vector K = ∂ ψ . Thus the Wilson loop (2.6) is indeed a BPS operator provided one takes γ to be an orbit of K. Notice that the topology of M 3 has not been used in this subsection, and thus any Wilson loop wrapped along an orbit of K is BPS, regardless of the topology of M 3 .

Localization in the matrix model
The VEV of the BPS Wilson loop (2.6) is, by definition, obtained by inserting W into the path integral for the theory on M 3 . The computation of this is greatly simplified by the fact that this path integral localizes onto supersymmetric configurations of fields. This is by now a fairly standard computation, and we shall simply summarize the main steps, referring the reader to [1,3,12,13,14] for further details. In particular the localization of the Wilson loop was explained in detail in [6] for the round S 3 case. This section generalizes that discussion to a generic supersymmetric manifold M 3 ∼ = S 3 .
The central idea is that the path integral, with W inserted, is invariant under the supersymmetry variation δ corresponding to the Killing spinor χ. We have written two of the supersymmetry variations in (2.8), and the variations of other fields (including fields in the chiral matter multiplets) may be found on the curved background M 3 in [1].
Crucially, δ 2 = 0 is nilpotent. There is then a form of fixed point theorem that implies that the only net contributions to this path integral come from field configurations that are invariant under δ [15].
For the N = 2 supersymmetric Chern-Simons-matter theories of interest, one finds that the δ-invariant configurations on M 3 ∼ = S 3 are particularly simple: where the function h = 1 2 * (e 1 (3) ∧ de 1 (3) ), and with all fields in the matter multiplet set identically to zero [1]. Here we may diagonalize σ by a gauge transformation. The exact localized partition function then takes the saddle point form [1] Here the integral is over the Cartan of the gauge group, k denotes the Chern-Simons level, the first product is over positive roots α ∈ ∆ + of the gauge group, and the second product is over weights ρ in the weight space decomposition for a chiral matter field in an arbitrary representation R matter of the gauge group. We have also defined the R-charge of the matter field is denoted r, and s β (z) denotes the double sine function.
In this set-up, the VEV of the BPS Wilson loop (2.6) reduces to Notice the integrand is the same as that for the partition function (2.11), with an additional insertion of Tr R (e 2πℓσ ) arising from the Wilson loop operator. Note also that we have normalized the VEV relative to the partition function Z, so that 1 = 1, as is usual in quantum field theory. We have also defined γ ds = 2πℓ (2.14) so that ℓ parametrizes the length of the Wilson line. More precisely, the integral (2.14) is well-defined only for a closed orbit of the Killing vector These have been studied in the present context in [8]. If on the other hand b 1 /b 2 is irrational, then the only closed orbits are at the two "poles" of where ∂ ϕ 1 = 0 and ∂ ϕ 2 = 0, respectively. Over these poles γ ds = 2π/|b 2 |, 2π/|b 1 |, respectively. Wherever the loop is located, we denote its length γ ds by 2πℓ as above.
For a U(N) gauge group we may write σ = diag( λ 1 2π , . . . λ N 2π ), thus parametrizing 2πσ by its eigenvalues λ i . Localization has then reduced the partition function Z and the Wilson loop VEV to finite-dimensional integrals (2.11), (2.13) over these eigenvalues, but in practice the formulae are difficult to evaluate explicitly. For comparison to the dual supergravity results we must take the N → ∞ limit, where the number of eigenvalues, and hence integrals, tends to infinity. One can then attempt to compute this limit using a saddle point approximation of the integral. In [3] a simple ansatz for the large N limit of the saddle point eigenvalue distribution was introduced. One seeks saddle points with eigenvalues of the form with x i and y i real and assumed to be O(1) in a large N expansion. In the large N limit the real part is assumed to become dense. Ordering the eigenvalues so that the x i are strictly increasing, the real part becomes a continuous variable x, with density ρ(x), while y i becomes a continuous function of x, y(x).
, with x supported on some interval [x min , x max ], and to apply the saddle point method one then extremizes One then finally also extremizes over the choice of interval, by varying with respect to x min , x max , to obtain the saddle point eigenvalue distribution ρ(x), y(x).
As it turns out, if one caries out the large N limit with the ansatz (2.15), one finds a very simple relation between the round sphere results F round and log W round and their squashed counterparts (with arbitrary b 1 and b 2 ) F and log W . To obtain this result for F , one may first relabel σ as |b 2 |σ in (2.11). The partition function then takes the same form as that in [16], where the large N limit was computed in detail. In particular in the latter reference it was shown that in the large N limit F [ρ(x), y(x)] is simply a rescaling of the round sphere result by a factor (βQ) 3 /2 3 β 2 , provided one also rescales the Chern-Simons coupling k as k → (2/βQ) 2 · k. This then leads to the large N result (1.1).
The same logic may be applied to the calculation of the Wilson loop. For the class of N = 2 supersymmetric Chern-Simons theories coupled to matter on the round threesphere studied in [6], x max is always proportional to 1/ √ k. According to the above prescription, the result for x max on a general background M 3 is given by rescaling the round sphere result by |b 2 | · (βQ/2) = (|b 1 | + |b 2 |)/2. Here the factor of |b 2 | comes from the relabelling σ → |b 2 |σ, while the factor of βQ/2 comes from the rescaling of the Chern-Simons coupling. Thus where x round max determines the supremum of the support of ρ(x) for the field theory on the round three-sphere. For the theories studied in [6], the eigenvalue density is always a continuous piecewise linear function supported on [x min , x max ]. Using this fact, the large N limit of the Wilson loop (2.13) in the fundamental representation may be easily computed with a saddle point approximation, and is Here recall that the length γ ds is in general 2πℓ. The round three-sphere Wilson loop in particular is obtained by setting b 1 = b 2 = 1 and ℓ = 1 and is, as shown in [6], We thus obtain This is the field theory result for the VEV of a supersymmetric Wilson loop on a general supersymmetric manifold M 3 ∼ = S 3 . In the next section we will look at the M2-brane dual to this Wilson loop, and show quite generally that the holographic dual computaton of the VEV agrees with (2.20).

Dual M2-branes
In this section we analyse the supersymmetric M2-brane probes that are relevant for computing the holographic dual of the Wilson loop VEV (2.20). The dual solution is constructed in four-dimensional gauged supergravity [5], and we begin by summarizing the geometry of these solutions. We then look at the eleven-dimensional uplift, and finally we compute the regularized action of the M2-brane.

Four-dimensional supergravity dual
In [5] it was shown that supersymmetric three-manifolds M 3 of precisely the form de- Killing vector bilinear over M 4 , and the four-metric is then Einstein, has anti-self-dual Weyl tensor, and is conformal to a Kähler metric. Supersymmetry also requires one to turn on a specific graviphoton field A. After summarizing these solutions, and deriving some relevant formulae, we then use them to study the BPS M2-branes dual to the Wilson loops of the previous section.
The four-dimensional metric on the manifold M 4 takes the form and w = w(y, z,z) satisfies the Toda equation Supersymmetry requires that the graviphoton gauge field A takes the local form where the field strength F is defined by F = dA. Indeed, the metric (3.1) is conformal to the Kähler metric ds 2 Kahler = y 2 ds 2 M 4 , which is asymptotic to a cylinder R × M 3 near to the conformal boundary y = 0. The gauge field (3.4) is then 1 2 of the Ricci one-form for this Kähler metric. These solutions were referred to as self-dual solutions in [5], since the Weyl tensor is anti-self-dual 3 and F is anti-self-dual, i.e. * 4 F = −F . Moreover, the metric (3.1) is Einstein with negative cosmological constant. We shall use the following orthonormal frame for (3.1) As already mentioned, the solutions of interest are asymptotically locally Euclidean AdS (asymptotically hyperbolic), with the conformal boundary at y = 0. In particular imposing boundary conditions such that w(y, z,z) is analytic around y = 0, i.e.
w(y, z,z) = w (0) (z,z) + yw (1) (z,z) + 1 2 then setting r = 1/y the metric (3.1) expands to leading order as when r → ∞. Here we have also expanded the one-form φ tangent to M 3 In fact by expanding (3.2) one can show that φ (1) = 0. Equation (3.7) shows explicitly that the metric is asymptotically locally Euclidean AdS around y = 0, and moreover there is a natural choice of conformal class for the metric on the boundary M 3 given precisely by (2.2).
The four-dimensional geometry that we have just described, together with the gauge field A, form a supersymmetric solution to Euclidean gauged supergravity. There is correspondingly a Dirac spinor ǫ satisfying the Killing spinor equation of this theory. In the orthonormal frame (3.5) and using the gamma matrices with τ i the Pauli matrices, the Killing spinor ǫ is given by In particular the bulk spinor (3.10) precisely matches onto the boundary two-component So far we have not imposed the U(1) × U(1) symmetry we imposed on the boundary M 3 at the end of section 2.1. Doing so will simplify the subsequent discussion. Thus as in [5] we assume that the four-manifold M 4 is M 4 ∼ = B 4 ∼ = R 2 ⊕ R 2 and that the torus U(1) × U(1) acts in the standard way on R 2 ⊕ R 2 . The Killing vector K = ∂ ψ is then parametrized as again precisely as in (2.5) on the conformal boundary. It will be important to fix carefully the orientations here. Since the metrics are defined on a ball, diffeomorphic to R 4 ∼ = R 2 ⊕ R 2 with U(1) × U(1) acting in the obvious way, we choose ∂ ϕ i so that the orientations on R 2 induce the given orientation on R 4 (with respect to which the metric has anti-self-dual Weyl tensor). This fixes the relative signs of b 1 and b 2 . Given that K has no fixed points near the conformal boundary, we must also have b 1 and b 2 non-zero. Thus b 1 /b 2 ∈ R \ {0}, and its sign will be important in what follows.
In order to construct such backgrounds one can start with a toric (U(1) × U(1)invariant) self-dual Einstein metric on a ball, which is asymptotically locally Euclidean AdS. There are many examples of such metrics -we discuss the two simplest in section 4, but as explained in [5] the moduli space is in fact infinite-dimensional (each metric inducing a different conformal structure on the boundary M 3 ∼ = S 3 ). One can then choose a Killing vector (3.12), and then writing K = ∂ ψ the metric will necessarily take the form (3.1). Thus in particular the choice of K determines the conformal Kähler metric, which in turn determines the instanton gauge field A and Killing spinor ǫ. However, not all choices of K in (3.12) give non-singular gauge fields. While the metric (3.1) is smooth by assumption, the instanton F = dA and Killing spinor ǫ are singular where the conformal Kähler metric is singular. Regularity is in fact equivalent which notice is y 0 = ∞ when b 1 /b 2 = −1.

Global gauge for A
As remarked after equation (3.11), we will want to choose a gauge for A in which it is a global, smooth one-form on M 4 . The reason for this is that we will evaluate the Wess-Zumino term in the M2-brane action in section 3.4 by using Stokes' theorem for F = dA, which requires us to write A as a global one-form. This was also discussed to some extent in [5], but for the computation of the Wilson loop we need a little more information.
The key point is to recall that A is proportional to the Ricci one-form for the conformal Kähler metric ds 2 Kahler = y 2 ds 2 M 4 . When b 1 /b 2 > 0 the associated complex structure identifies M 4 ∼ = R 2 ⊕ R 2 ∼ = C 2 . The orientation in which the Weyl tensor is anti-self-dual is the same as the canonical orientation on C 2 . One can then introduce standard complex coordinates z i = ρ i e iψ i , i = 1, 2, on C 2 . The spinor ζ in (3.11), which is used to construct the Killing spinor (3.10), is the canonical spinor that exists on any Kähler manifold [5]. As such we have (3.14) Denoting the complex structure tensor by J we also have that J(V −1 ∂ y ) = ∂ ψ = K.
Since y is decreasing as we move away from the origin of C 2 , where recall that the origin is at y = y 0 > 0, this means that for b 1 > 0 and b 2 > 0 we must then identify ϕ i = −ψ i , where ϕ i are the coordinates on U(1) × U(1) in (3.12). This is because for r any radial coordinate on C 2 we have J(r∂ r ) = a 1 ∂ ψ 1 + a 2 ∂ ψ 2 where necessarily a 1 , a 2 > 0 (that is, the Reeb cone is the positive quadrant in R 2 -see, for example, [17]). On the other hand for b 1 < 0 and b 2 < 0 we instead have ϕ i = +ψ i , i = 1, 2.
The other non-singular case is b 1 /b 2 = −1. This is qualitatively different from the case b 1 /b 2 > 0 in the last paragraph, as here y 0 = ∞ (3.13). Moreover, the origin y = y 0 of M 4 ∼ = R 2 ⊕ R 2 is now identified with the point at infinity in C 2 , rather than the origin. One can see this from the conformal Kähler metric ds 2 Kahler = y 2 ds 2 M 4 , which is asymptotically Euclidean around y = y 0 . Thus now V −1 ∂ y has the correct orientation for a radial vector on C 2 , and we deduce that for b 1 < 0 and b 2 > 0 we have ϕ 1 = −ψ 1 , ϕ 2 = +ψ 2 , while for b 1 > 0 and b 2 < 0 we instead have ϕ 1 = +ψ 1 , ϕ 2 = −ψ 2 .
Putting all of the above together, we may compute the charge of the Killing spinor ǫ under the supersymmetric Killing vector K = ∂ ψ : Since in all cases the Kähler structure is defined on C 2 , the canonical bundle is trivial and one may indeed take A to be a global one-form on M 4 . We denote the restriction of this global A to the conformal boundary M 3 = ∂M 4 at y = 0 by A (0) . Then the formula (3.15) for the charge of ǫ under K = ∂ ψ means that This is the restriction of (3.4) to y = 0, together with a gauge transformation A → A + γdψ which accounts for the charge (3.15). One can show independently that (3.17) then defines a global one-form on M 3 ∼ = S 3 , which leads to another formula for γ that was derived in section 3.3 of [5], although we will not need this in the present paper.

Uplifting to D = 11 supergravity
In order to study the M2-branes dual to Wilson loops, we need to uplift the fourdimensional geometry to an eleven-dimensional supergravity solution. More precisely, we are interested in a class of N = 2 supersymmetric M 4 × Y 7 backgrounds of Mtheory in Euclidean signature. In Euclidean signature there are certain factors of i that appear relative to the uplifting formula in Lorentzian signature of [18]. Again, this will be important for correctly evaluating the M2-brane action.
The action of D = 11 supergravity in Euclidean signature is Here we have denoted by g 11 the eleven-dimensional metric, with associated Ricci scalar R, C is the three-form potential and ℓ p denotes the eleven-dimensional Planck length.
The equations of motion for the metric and C-field follow immediately: where we have defined G ≡ dC and A, B, C = 1, . . . , 11. It is also useful to define G 7 = i( * 11 G + i 2 C ∧ G) so that the equation of motion for G is simply dG 7 = 0. An ansatz that leads to a consistent truncation to four-dimensional gauged supergravity in Lorentz signature was given in [18]. Here the internal space Y 7 is taken to be any Sasaki-Einstein seven-manifold Y 7 with contact one-form η, transverse Kähler-Einstein six-metric ds 2 T with Kähler form ω T = dη/2, and with the seven-dimensional metric normalized so that Ric = 6g Y 7 . The consistent truncation ansatz in Euclidean signature then becomes As before, ds 2 M 4 is the four-dimensional gauged supergravity metric on M 4 with gauge field A, field-strength F = dA and volume form vol 4 . The radius R is , (3.21) where N is the number of units of flux Substituting the ansatz (3.20) into the equations of motion (3.19), we find the latter are equivalent to the metric g µν corresponding to ds 2 M 4 and F satisfying

BPS M2-branes
We are interested in calculating the action of M2-branes that are dual to Wilson loops of the dual gauge theory on M 3 . These M2-branes wrap Σ 2 × S 1 M , where the surface Σ 2 ⊂ M 4 has boundary given by the Wilson line ∂Σ 2 = S 1 ⊂ M 3 = ∂M 4 , and S 1 M ⊂ Y 7 is a copy of the M-theory circle. In particular we will show that submanifolds Σ 2 ⊂ M 4 parametrized by the radial direction y in M 4 and an orbit of the Killing vector K are complex with respect to the complex structure J of the conformal Kähler metric to ds 2 M 4 . The wrapped M2-brane is then supersymmetric. 5 Over the poles S 1 ⊂ M 3 ∼ = S 3 the topology of Σ 2 is a disc, where y ∈ (0, y 0 ] serves as a radial coordinate with the origin of the disc at y = y 0 > 0.
The action of the M2-brane is A supersymmetric M2-brane satisfies an appropriate κ-symmetry condition, which may be written as [19] Pǫ with i, j, k indices on the worldvolume. Here ǫ 11 is the eleven-dimensional Killing spinor for the background (3.20), which is constructed as a tensor product of the fourdimensional spinor ǫ and the Killing spinor on the internal space Y 7 . TheΓ M are eleven-dimensional gamma matrices, with X M describing the M2-brane embedding. One can analyse (3.25) precisely as the authors did in [6]. The upshot is that S 1 M ⊂ Y 7 must be a calibrated circle in Y 7 [6], while taking Σ 2 ⊂ M 4 to be a surface at constant z, parametrized by y and ψ, one finds (3.25) is equivalent to the projection condition Here we have used the orthonormal frame (3.5), and Γ 5 ≡ Γ 0 Γ 1 Γ 2 Γ 3 with Γ µ defined by (3.9) (in the orthonormal frame). Using the explicit form for ǫ in (3.10) it is trivial to see that (3.26) indeed holds. Moreover, Σ 2 is calibrated with respect to the Kähler form for the conformal Kähler metric, making it a complex curve.
Let us now calculate the action (3.24) for our M2-brane. Using the self-dual fourdimensional supergravity solution of section 3.1 and the uplift (3.20) the C-field is computed to be and dΓ = −3vol 4 . The area of the surface Σ 2 in M 4 is divergent, but can be regularized by subtracting the length of its boundary, i.e. the length of the S 1 in M δ 3 at y = δ → 0. Notice this is then a local boundary counterterm. If we denote by M δ 4 the manifold M 4 with boundary M δ 3 = {y = δ} (with 0 < δ < y 0 ), and similarly for Σ δ 2 etc, the action of the M2-brane is Here we have written vol S 1 M for the volume form on S 1 M induced from the metric g 7 , and similarly for vol Σ 2 and the metric g M 4 . Applying Stokes' theorem for the gauge field term F = dA we then compute 6 Recall here that 2πℓ denotes the length of the orbit of K, as in (2.14). The contribution of the M-theory circle S 1 M is exactly the same as for the AdS 4 × Y 7 backgrounds studied in [6], and is expressed in terms of the contact form η on Y 7 and the Dirac quantized number N of (3.22). The gauge field integral is easily computed, thanks to (3.17) Putting everything together, and using the formula (3.13) for y 0 , we have Using the round sphere result of [6] log W round gravity = (2π) 2 i.e. log W round QFT = log W round gravity holds to leading order at large N. Assuming this to be the case, equations (2.20) and (3.34) mean that we have shown very generally that in the large N limit log W QFT = log W gravity (3.35) where now the field theory is defined on a general class of background three-manifolds M 3 , with fillings M 4 in four-dimensional gauged supergravity.
We conclude this section with two further comments. Firstly, it is interesting to note that when the orbit of K is one of the poles of S 3 , where correspondingly ℓ = 1/|b 1 | or ℓ = 1/|b 2 | respectively, the Wilson loops are then functions only of |b 1 /b 2 |, just as for the free energy (1.1). Secondly, in the case that b 1 /b 2 = m/n is rational and the Wilson line wraps a generic orbit γ ⊂ T 2 ⊂ S 3 (i.e. not at either pole), then the curve Σ 2 ⊂ M 4 ∼ = C 2 wrapped by the dual M2-brane is the Brieskorn-Pham curve {z n 1 = z m 2 } ⊂ C 2 . This follows since supersymmetry pairs the orbit of K with its complexification in M 4 ∼ = C 2 , meaning that Σ 2 is swept out as a generic C * orbit of (z 1 , z 2 ) → (λ m z 1 , λ n z 2 ), with λ ∈ C * . The curve {z n 1 = z m 2 } adds the origin in C 2 at y = y 0 , which is a singular point when m, n > 1, although notice this does not affect our computation of the M2-brane action, which is finite. It is well-known that (m, n) torus knots in S 3 may be realized as links of the above Brieskorn-Pham curves, and it is interesting to see that this construction is realized as the holographic dual of the knot.

Examples
Our derivation of the formula (3.34) was necessarily somewhat indirect, as we have shown that it holds for a very general (infinite-dimensional) class of solutions. In particular we didn't need to use the explicit form of the solution to the Toda equation (3.3). In this section we illustrate our general results by discussing two explicit families of solutions, where all quantities in the previous section may be written down in closed form. We will focus on the four-dimensional part of the M2-brane calculation, in particular showing how the factor ℓ(|b 1 | + |b 2 |)/2 in (3.34) arises explicitly in these cases. In order to do so we will use the results of the previous section that allow us to Here we cut off Σ 2 at y = δ, and (4.2) is then understood to be the limit δ → 0. We compute (4.2) directly in the examples, confirming that (3.34) indeed holds in these cases.

(4.3)
Here q is a radial variable with q ∈ [0, ∞), so that the origin of M 4 ∼ = R 4 is at q = 0 while the conformal boundary is at q = ∞. The coordinate ϑ ∈ [0, π 2 ], with the endpoints being the two axes of R 2 ⊕ R 2 ∼ = R 4 .
Of course the metric (4.3) is conformally flat, which leads to a trivial graviphoton A = 0. However, we may instead pick a general supersymmetric Killing vector K = b 1 ∂ ϕ 1 + b 2 ∂ ϕ 2 . This leads to a family of conformal Kähler structures on C 2 , where the explicit formulae for the conformal factor y and the metric function w(y, z,z) may be found in [5]. In particular one calculates the local gauge field given by (3.4) to be which is a non-trivial instanton on Euclidean AdS 4 . In fact this solution was first found in [16] using very different methods. One can check that the field strength Writing A as a global one-form and restricting to the conformal boundary at q = ∞ we obtain In particular notice this is well-defined at both poles ϑ = 0 and ϑ = π/2. The submanifold Σ 2 is parametrized by the radial direction q in AdS 4 and the S 1 wrapping ϕ 1 or ϕ 2 when ϑ = 0 or ϑ = π/2, respectively.
We now turn to computing (4.2). Notice that the dependence on b 1 and b 2 arises only via the gauge field A, and not from the metric. Indeed, we compute and ∂Σ 2 The overall factors of sign(b 1 ), sign(b 2 ) for ϑ = 0, π/2 arise because the orientation of ∂Σ 2 is determined by K. Equation (4.2) immediately gives for all regular cases that In particular using the variable ℓ introduced previously, which is given by ℓ = 1/|b 1 | and 1/|b 2 | for the ϑ = 0 pole and ϑ = π/2 pole, respectively, we obtain for both poles and all regular cases that as expected.

Taub-NUT-AdS 4
The Taub-NUT-AdS 4 metrics are a one-parameter family of self-dual Einstein metrics on the four-ball, and have been studied in detail in [20], [21]. The metric may be written where Ω(r) ≡ (r ∓ s) 2 [1 + (r ∓ s)(r ± 3s)] , (4.11) and σ 1 , σ 2 , σ 3 are left-invariant one-forms on SU(2) ≃ S 3 . The latter may be written in terms of Euler angle variables as σ 1 + iσ 2 = e −iς (dθ + i sin θdϕ) , σ 3 = dς + cos θdϕ . Here ς has period 4π, while θ ∈ [0, π] with ϕ having period 2π. The radial coordinate r lies in the range r ∈ [s, ∞), with the origin of the ball ∼ = R 4 being at r = s. The parameter s > 0 is referred to as the squashing parameter, with s = 1 2 being the Euclidean AdS 4 metric studied in the previous section. The metric is asymptotically locally Euclidean AdS as r → ∞, with so that the conformal boundary at r = ∞ is a biaxially squashed S 3 .
While the Taub-NUT-AdS metric (4.10) has SU(2) × U(1) isometry, a generic choice of the Killing vector breaks the symmetry of the full solution to U(1) × U(1). In particular, this symmetry is broken by the corresponding instanton A. On the other hand, in [20], [21] the SU(2)×U(1) symmetry of the metric was also imposed on the gauge field, which results in two subfamilies of the above solutions, which are 1/4 BPS and 1/2 BPS, respectively. In each case this effectively fixes the Killing vector K (or rather the parameter b 1 /b 2 ) as a function of the squashing parameter s.

1/4 BPS solution:
The supersymmetric Killing vector for this solution is K = − 1 2s ∂ ς and we have (4.14) Here ς = ϕ 1 − ϕ 2 , ϕ = ϕ 1 + ϕ 2 is the change of angular coordinates. The boundary gauge field A (0) is [21] which is a global one-form on M 3 ∼ = S 3 . We may now take the surface Σ 2 wrapped by the M2-brane to be any S 1 orbit of the Hopf Killing vector ∂ ς (at any point on the base S 2 = S 3 /U(1) ς ), together with the radial direction r. This is supersymmetric, and the regularized volume of Σ 2 is while the gauge field integral is This leads to where ℓ = 4s is the length of K divided by 2π.

1/2 BPS solution:
The Taub-NUT-AdS metric (4.10) also admits a 1/2 BPS solution [20], [21]. There are thus two linearly independent Killing spinors, and an appropriate linear combination preserves U(1) × U(1) symmetry, leading to the Killing vector The boundary gauge field is This time we take the Wilson loop to wrap one of the two poles θ = 0, θ = π. These are both copies of S 1 , and Σ 2 is again formed by adding the radial direction r. The boundary gauge field is The regularized volume is again 8πs 2 , which then gives (4.23) In both cases we indeed have where ℓ = 1/|b 1 |, ℓ = 1/|b 2 | for the two poles.

Discussion
In this paper we have derived the formula (1.2), (1.3) for the expectation values of large N BPS Wilson loops, in both gauge theory and in supergravity. In particular the gauge theories are defined on a general class of supersymmetric backgrounds M 3 ∼ = S 3 , which in the supergravity dual arise as the conformal boundaries of self-dual solutions to gauged supergravity. A key feature of the gravity calculation is that we are able to evaluate the regularized M2-brane action, that is identified with the Wilson loop VEV, without using the explicit form of the metric and graviphoton field. This seems to be a general feature of such computations of BPS quantities in AdS/CFT, and allows us to verify the correspondence for these observables in a very broad class of solutions.
The results described in this paper lead to a number of questions, and possible future directions to pursue. First, in supergravity we have restricted to self-dual solutions, while more generally there are also non-self-dual solutions to gauged supergravity. A local study of these solutions appears in [22], while global asymptotically locally Euclidean AdS solutions were constructed in [21]. Presumably the methods we have used extend to this general class of solutions. In particular the Wilson loop was computed for a charged topological black hole background in [23], and successfully compared to a field theory calculation. The non-self-dual solutions in [21] all have the feature that the bulk M 4 has non-trivial topology, which in turn leads to issues in interpreting them holographically (and in particular uplifting to eleven dimensions restricts the choice of Y 7 , implying the solutions are only relevant to specific gauge theories on M 3 ). It would be interesting to try to calculate Wilson loops in such examples, and compare to a dual field theory computation. Indeed, in [21] it was argued that in appropriate circumstances W = 0 in supergravity, via a similar mechanism to that in [24]. Typically here the boundary M 3 in such examples has a non-trivial fundamental group, as in the large N gauge theory computation in [25], and there is indeed evidence that if the R-symmetry gauge field on M 3 has non-trivial topology then the large N Wilson loop VEV vanishes also in the gauge theory. 7 Finally, it is now clear that similar results should also hold in higher dimensions. A very similar formula to (1.2), (1.3) was found to hold for certain supersymmetric squashed five-sphere conformal boundaries and their gravity duals in [27], [28], and was conjectured to hold for general backgrounds in those references. It would also be interesting to compute Wilson loops in the general class 7 The matrix model behaviour is then much more complicated, and seems difficult to analyse analytically, but very roughly speaking the Wilson loop VEV averages to zero due to the sum p−1 k=0 ω k p = 0, where ω p is a primitive pth root of unity. This arises from the fact that the dominant contribution to the Wilson loop at large N comes from a non-trivial flat connection, with ω k p related to its holonomies [26]. of S 1 × S 3 Hopf surface geometries in [29].