Equivariant localization for AdS/CFT

We explain how equivariant localization may be applied to AdS/CFT to compute various BPS observables in gravity, such as central charges and conformal dimensions of chiral primary operators, without solving the supergravity equations. The key ingredient is that supersymmetric AdS solutions with an R-symmetry are equipped with a set of equivariantly closed forms. These may in turn be used to impose flux quantization and compute observables for supergravity solutions, using only topological information and the Berline--Vergne--Atiyah--Bott fixed point formula. We illustrate the formalism by considering $AdS_5\times M_6$ and $AdS_3\times M_8$ solutions of $D=11$ supergravity. As well as recovering results for many classes of well-known supergravity solutions, without using any knowledge of their explicit form, we also compute central charges for which explicit supergravity solutions have not been constructed.

The systematic analysis of the geometry that underlies the AdS/CFT correspondence in the supersymmetric context is an important ongoing investigation in holography.
One considers spacetimes of the form AdS × M in either D = 10 or D = 11 supergravity, where M is a compact manifold, together with a warped product metric and general fluxes, both of which are preserved by the symmetries of the AdS spacetime.
Demanding that some supersymmetry is preserved then leads to the existence of certain Killing spinors on M which, possibly combined with the equations of motion and/or the Bianchi identities for the fluxes, defines the supersymmetric geometry on M . The Killing spinors define a G-structure on M [1], and the geometry can be effectively studied by analyzing Killing spinor bilinears.
The first such analysis was carried out for the general class of AdS 5 ×M 6 solutions of D = 11 supergravity in [2]. The geometry on M 6 was precisely characterized and, moreover, was then used to construct infinite classes of explicit AdS 5 × M 6 solutions, dual to N = 1 SCFTs in d = 4. Subsequently, similar analyses have been carried out for many different cases in various dimensions. In some cases, one finds that the geometry is particularly tractable when one considers particular sub-cases where some of the fluxes are set to zero and/or additional restrictions on the Killing spinors are made. For example, substantial progress has been made for AdS 5 × SE 5 solutions with just five-form flux and AdS 4 × SE 7 solutions with just electric four-form flux, where SE 2n+1 is a Sasaki-Einstein space [3]. In particular, it has been shown that various physical quantities of the dual SCFTs can be computed by suitably going offshell and solving a novel variational problem [4,5]. More recently, starting with [6], there has been similar progress for AdS 3 × GK 7 solutions of type IIB with general five-form flux [7] and AdS 2 × GK 9 solutions of D = 11 with electric four-form flux [8], where GK 2n+1 is a GK geometry [9].
In this paper, and also in [10,11], we reveal an important general structure that has been overlooked in these various works for nearly twenty years. Specifically, provided that the preserved supersymmetry includes an R-symmetry, we show there is a natural equivariant calculus involving the Killing spinor bilinears. Furthermore, this calculus can be used to compute physical quantities of the dual SCFTs without knowing the full supergravity solution, and thus defines a canonical way to do various computations off-shell.
When the dual SCFT has an R-symmetry the geometry on M has a Killing vector ξ. It is then natural to introduce the equivariant exterior derivative d ξ ≡ d − ξ . This derivative acts on polyforms, i.e. sums of differential forms, and satisfies d 2 ξ = −L ξ , where L ξ is the Lie derivative with respect to ξ. In the case that the compact manifold M has even dimension, by explicit computation for several different set-ups, we show that various equivariantly closed polyforms Φ, satisfying d ξ Φ = 0, can be constructed from the subset of the Killing spinor bilinears which are invariant under the action of L ξ . Furthermore, we show that the integrals of the polyforms compute physical quantities of the dual SCFT, such as the central charge and flux integrals. The Berline-Vergne-Atiyah-Bott (BVAB) fixed point formula [12,13] can then be used to compute these physical quantities as a sum of contributions arising from the fixed point set of the action of ξ.
In this paper we will illustrate this new calculus for three classes of solutions in D = 11 supergravity. We first consider the class of AdS 5 × M 6 solutions of [2]. We show in detail how the new formalism can be used to obtain the central charge and scaling dimensions of certain chiral primaries for various different classes of examples of M 6 . In several classes, the explicit solutions are known and we can compare our results with the known answer, finding exact agreement. This detailed exercise reveals how the new equivariant calculus can be used and, in particular, we will see that the details depend on which case is being considered. We also carry out similar computations for classes of solutions which are not known in explicit form.
Our next class of examples consist of AdS 3 × M 8 solutions. A sub-class of geometries were first analysed in [14] and this was later extended to a general classification in [15]. Here we will display the equivariant polyforms for the sub-class considered in [14], for which it is known that there is a rich class of solutions. We certainly expect that our results can be generalized to the general class of [15], but we shall leave that to future work. We will again deploy the new calculus to recover some known results, as well as obtain some new results, focusing on cases where M 8 is an S 4 fibration over a four-dimensional base B 4 , associated with M5-branes wrapped on B 4 . In [11] we consider examples where the R-symmetry has isolated fixed points and show that one can obtain off-shell expressions for the central charge expressed in terms of 'gravitational blocks' [16], while here we consider a complementary class of examples.
Of course, this solution is explicitly known and one can immediately obtain the central charge of the dual field theory by a straightforward computation. However, it is illuminating to analyse it from the new point of view as it underscores the universality of the approach.
Before concluding this introduction we briefly compare our work with that of [17], which also discussed equivariant localization in holography. A detailed discussion of the so-called equivariant volume of symplectic toric orbifolds was made in [17] and, in particular, it was shown that various known off-shell holographic results for Sasaki-Einstein geometry and GK geometry could be recast in an elegant way in this language. Furthermore, by rephrasing certain off-shell field theory results it was suggested that, more generally, the equivariant volume should play a role in analysing certain classes of supergravity solutions associated with wrapped M5branes and wrapped D4-branes. However, the crucial step of how to go off-shell in a supergravity context was not given. By contrast, we emphasize that our approach does provide a canonical procedure for going off-shell, utilizing the structure of spinor bilinears, and furthermore this does not rely on the specific setting of symplectic toric orbifolds considered in [17]. Thus, our approach should be applicable more universally to all holographic geometries with an R-symmetry, as illustrated here and in various other examples in [10,11].
The plan of the rest of the paper is as follows. In section 2 we briefly review the localization formula of BVAB, as well as discuss some simple examples. In sections 3 and 4 we consider AdS 5 × M 6 and AdS 3 × M 8 solutions, respectively, and we conclude with some discussion in section 5. We also have three appendices. In appendix A we examine how spinors with a definite charge under a Killing vector behave near a fixed point, and how this is correlated with the chirality of the spinor. In appendix B we discuss the AdS 7 × S 4 solution using equivariant localization. Finally, in appendix C we prove some homology relations for manifolds which are the total space of evendimensional sphere bundles, focusing on S 2 and S 4 bundles, that we use in the main text.

The equivariant localization theorem
The key mathematical result we use in this paper is the equivariant localization formula of Berline-Vergne-Atiyah-Bott (BVAB) [12,13]. We begin by introducing some notation, reviewing the formula 1 and illustrating how it can be used with some simple examples.
We consider compact Riemannian manifolds (M, g) of dimension 2n, with a U (1) isometric action generated by a Killing vector ξ. On the space of polyforms, which are formal sums of differential forms of various degrees, one introduces the equivariant differential d ξ ≡ d − ξ . (2.1) This differential squares to minus the Lie derivative along ξ: d 2 ξ = −L ξ . Thus, on the space of invariant polyforms {Φ}, with L ξ Φ = 0, the differential above is nilpotent and we can construct an equivariant cohomology. In the special case that ξ generates a (locally) free U (1) action on M , where all the orbits of ξ are circles, this cohomology is simply the de Rham cohomology of the quotient M/U (1). However, we are interested in the case that ξ does not act freely, and there is a non-trivial fixed The integral of a polyform is defined to be the integral of its top component. We also define the inverse of a polyform via the geometric series where Φ a is understood to be a wedge product of a copies of Φ, and note this series necessarily truncates.
The BVAB theorem expresses the integral of an equivariantly closed form Φ on M in terms of contributions from the fixed point set F of the group action. The connected components of F have even codimension, so the normal bundle N to F in M is an even-dimensional orientable bundle. Then the following relation holds 3) Here f : F → M is the embedding of the fixed point locus, e ξ (N ) is the equivariant Euler form of the normal bundle, and d F ∈ N is the order of the orbifold structure group of F (in cases where F and M have orbifold singularities).
To be more concrete, consider a connected component of F with codimension 2k.
Then, ξ generates a linear isometric action on the rank-2k normal bundle, so there are local coordinates in which it has the form Here (x 2i , x 2i−1 ) are Cartesian coordinates on the ith two-plane R 2 ⊂ R 2k in a normal space, with φ i corresponding polar angles with period ∆φ i = 2π. For generic 2 ϵ i , N splits into a sum of complex line bundles N ∼ = L 1 ⊕ · · · ⊕ L k , with L i rotated by the equivariant Euler classes of each summand L i , given by the sum of the Chern class of the bundle (representing the ordinary curvature), and the infinitesimal action generated by ξ to obtain where R denotes a curvature two-form. In the last step we used the restriction of the Lie derivative to the normal bundle 6) and the definition of the Pfaffian of a skew-symmetric matrix in the form above is The inverse of the equivariant Euler form (2.5) can then be found using the geometric series (2.2) and we have In general, the integral of an equivariantly closed form on a 2n-dimensional manifold can receive contributions from connected components of the fixed point set of various dimensions. Putting these points together, we can therefore write the BAVB formula in the following form where the normal space to a subspace F 2n−2k is R 2k /Γ, where Γ ⊂ SO(2k) is the finite group of order d F 2n−2k defining the orbifold structure (which is the trivial group in the case of smooth manifolds). Notice that in writing (2.9) we have suppressed the dependence of both the weights ϵ i and the line bundles L i on the particular connected component of F . If we fix a particular connected component, and (continuing to abuse the notation) also refer to that as f : F → M , then a more precise notation would be ϵ i (F → M ) and L i (F → M ), where both are associated to the normal bundle We shall occasionally use this notation for clarity. We remark that entirely analogous formulae to ( The metric on the round two-sphere with unit radius can be written where ϑ ∈ [0, π] and φ is a periodic coordinate with ∆φ = 2π. We consider the Killing vector ξ = b ∂ φ , where b is a non-zero constant, which has isolated fixed points at the north and south poles ϑ = 0, π, respectively. Here |b| is also the weight of the linearization of the action on the R 2 normal to the poles, but the sign at the two poles must be opposite in order to have a consistent orientation on the entire two-sphere. In particular, ξ rotates the plane tangent to S 2 at the north pole counter-clockwise, so the weight of the isometric linear action (2.6) is ϵ N ≡ ϵ(N → S 2 ) = +b at the north pole N and ϵ S ≡ ϵ(S → S 2 ) = −b at the south pole S. The integral of 1 b vol can be computed by first completing the volume two-form to a equivariantly closed polyform: with d ξ Φ = 0. We then use the BVAB formula (2.9), which in this case receives contributions only from the north and south poles, to get Notice that we can replace Φ 0 → Φ 0 + c, for an arbitrary constant c, without spoiling the condition d ξ Φ = 0. However, this constant "gauge" freedom drops out of the final integral.
One can thus compute the volume of the four-sphere by applying the BVAB formula (2.9). Since the fixed point set is the two poles, there are two contributions and we obtain 16) where ϵ N i and ϵ S i are the weights of the linear isometric action on the normal space to the north pole N and south pole S, respectively. From (2.14), we find that at the north pole ϵ N 1 ϵ N 2 = b 1 b 2 , but at the south pole, because of the change of orientation on the normal R 4 necessary for the consistency of the orientation of S 4 , ϵ S 1 ϵ S 2 = −b 1 b 2 . It is worth emphasizing that it is only the orientation of the entire R 4 that is fixed, not that of the individual summands in R 4 = R 2 ⊕ R 2 . Equation (2.16) then gives the expected result Again, notice if we replace Φ 0 → Φ 0 + c with an arbitrary constant c, we maintain d ξ Φ = 0 and c simply drops out of the final integral.

CP 2 example
Our final example, which displays more structure, is CP 2 . The standard Fubini-Study metric is where σ i are left-invariant one-forms on SU (2) = S 3 , with dσ i = − 1 2 ϵ ijk σ j ∧ σ k , and ϑ ∈ [0, π/2]. There is an isometry generated by the Reeb vector of the Hopf fibration on S 3 and using Euler angle coordinates on S 3 , we can write ξ = ∂ ψ , where ψ ∈ [0, 4π]. The square norm of ξ is ∥ξ∥ 2 = 1 4 sin 2 ϑ cos 2 ϑ, which vanishes at ϑ = 0, π/2. At ϑ = 0 there is an isolated fixed point, where the entire S 3 collapses and regularity of the metric is guaranteed by the periodicity of ψ being 4π (as on S 3 ). On the other hand, at ϑ = π 2 the two-sphere fixed by ξ, and with volume form proportional to σ 1 ∧ σ 2 , has finite size and radius 1 2 . One then proves that the polyform Φ Here we have explicitly included a real constant c in Φ 0 ; we will see that the integral of Φ is independent of c, but in a slightly less trivial way than the previous two examples.
In this case, the BVAB formula (2.9) receives contributions from the isolated fixed point and the fixed two-sphere, which are a nut and a bolt, in the terminology of [19].
The contribution from the former is where ϵ 1 ϵ 2 = 1 4 . On the other hand, the contribution from the bolt is where ϵ = 1 2 is the weight of the linear action on the normal bundle L to the CP 1 = S 2 bolt. Just as in the previous cases, we have fixed a natural orientation, and in this case it is such that the first Chern number of the normal bundle to the CP 1 bolt is +1. Therefore, overall we have which matches the volume computed from the metric (2.18), and is independent of c, as it should be.

AdS 5 × M 6 solutions
In this section we consider supersymmetric AdS 5 × M 6 solutions of D = 11 supergravity, generically dual to N = 1 SCFTs in d = 4, as first analysed in [2]. We begin by constructing a set of equivariantly closed forms, and examine some general properties of these forms when restricted to fixed point sets under the R-symmetry Killing vector ξ. In the subsequent subsections we then apply this to a wide variety of examples, including the holographic duals to M5-branes wrapped on Riemann surfaces Σ g [20] (which includes the Maldacena-Núñez solutions [21] as special cases) and spindles [22], together with all the explicit families of solutions found in [2]. As well as recovering results for known supergravity solutions using this new technology, crucially, without using the explicit forms of those solutions, we also compute (offshell) central charges and other observables in cases for which solutions have not yet been constructed.

Equivariantly closed forms
The D = 11 metric takes the warped product form where we take AdS 5 to have unit radius, and will assume that M 6 is compact without boundary. In order to preserve the symmetries of AdS 5 the four-form flux G and warp factor function λ are defined on M 6 . Note that in this paper we find it convenient to absorb the overall length scale of the D = 11 metric into the warp factor e 2λ . The flux quantization condition is for any four-cycle, C 4 , and ℓ p is the D = 11 Planck length.
Supersymmetry requires the existence of a Dirac Killing spinor, ϵ, on M 6 . From this one can construct the following real bilinear forms 3 1 =εϵ , sin ζ ≡ −iεγ 7 ϵ , K ≡εγ (1) where γ 7 ≡ γ 123456 . In particular from the Killing spinor equation one finds thatεϵ is a constant, which we have normalized to 1, and the vector field ξ dual to the one-form bilinear ξ ♭ is Killing. The factor of 1 3 has been introduced into the definition in (3.3) so that the R-charge of the Killing spinor under ξ is q = 1 2 : The Killing spinor is globally defined and so too are all of the bilinears in (3.3). We can introduce a local coordinate ψ so that ξ = ∂ ψ , and define the function y ≡ 1 2 e 3λ sin ζ , where dy = e 3λ K , (3.5) which was also used as a canonical coordinate in [2]. Next we have the following contractions [2]: The first two equations imply that L ξ λ = 0 = L ξ G, showing that ξ generates a symmetry of the full solution. Moreover, these equations can be used to establish that the polyform is equivariantly closed under d ξ . Using the further equations from [2], likewise we have the equivariantly closed form Both of these will be used later when quantizing the flux G. We also note from [2] that we have where the Hodge star is with respect to the metric ds 2 (M 6 ), and there is also an equivariantly closed from involving * G: Note that Φ * G and Φ Y are related via: It is next useful to introduce the local SU (2) structure defined by a Dirac spinor in six dimensions. This consists of a real two-form J, a complex two-form Ω (which will play no role in what follows), and two orthonormal one-forms e 5 and e 6 . In terms of the spinor bilinears already introduced, we have [2] K = cos ζ e 5 , ξ ♭ = 1 3 cos ζ e 6 , Y = J − sin ζ e 5 ∧ e 6 , Y ′ = − sin ζ J + e 5 ∧ e 6 . (3.13) The volume form on M 6 is vol = 1 2 J ∧ J ∧ e 5 ∧ e 6 , and taking the Hodge dual one finds that * Y = J ∧ e 5 ∧ e 6 − 1 2 sin ζ J ∧ J . (3.14) The further bilinear equation 15) then implies that the following polyform is equivariantly closed: This is particularly significant, since the a central charge for such a solution is [23] a = 1 2(2π) 6 ℓ 9 and this then localizes. The final bilinear equation we shall need from [23] is which will again be helpful for imposing flux quantization on G. For later use, notice that since the bilinears are globally defined on M 6 , the left hand side is globally exact.

Localization and fixed point sets
Integrals of the equivariantly closed forms Φ G , Φ Y , Φ defined in (3.7), (3.9), (3.16) will localize to the fixed point set F of ξ. On the other hand, from (3.13) we have implying that Here we have introduced the notation · | F to denote evaluation on (a component of) the fixed point set F . The sign in (3.20) is correlated with the chirality of the spinor at that fixed point, since correspondingly −iγ 7 ϵ | F = ±ϵ. Since from (3.13) also K | F = 0, we deduce from (3.5) that both y and hence e 3λ are locally constant on F , with e 3λ F = ±2y, and hence constant on each connected component. Introducing the rescaled two-form 4 ω ≡ e 6λ J , (3.21) we conclude from (3.13), (3.14) that From (3.8) it follows that dω | F = 0. Thus ω defines a cohomology class on each connected component of F , which will again be useful in what follows. Finally, notice that the left-hand side of (3.18) is globally exact, so on a closed four-dimensional fixed point set F 4 we find  Using the formula for Φ G in (3.7) we may write a similar formula for the integral of G over a ξ-invariant submanifold C 4 and hence impose Dirac quantization. It is helpful to consider two separate cases. The first is when the entire C 4 is a fixed by the action of the Killing vector and then from (3.23) we have (3.25) The second case is when the fixed point set consists of either or both of C 2 , C 0 and then (3.26) Here C 2 , C 0 ⊂ C 4 are the fixed submanifolds inside C 4 , with L 1 being the normal bundle of C 2 in C 4 . Note in both cases that the integer N C 4 only depends on the homology class [C 4 ] ∈ H 4 (M 6 , Z), since G is closed.
We can also compute other observables using the same techniques. One example is the conformal dimension ∆ of chiral primary operators in the dual N = 1 SCFT, corresponding to M2-branes wrapping supersymmetric two-cycles Σ 2 ⊂ M 6 . Such M2-branes are calibrated by Y ′ , i.e. vol| Σ 2 = Y ′ | Σ 2 , and their conformal dimension may be computed in the gravity dual via [23] ∆( Since e 3λ Y ′ is the top-form of the restriction to Σ 2 of the four-form flux polyform Φ G in (3.7), this integral can also be evaluated by localization when Σ 2 is ξ-invariant.
We compute Here Σ 0 ⊂ Σ 2 are the fixed points inside Σ 2 and, as for the flux integral, the first term is only present (and by itself) when the entire Σ 2 is fixed, while the second term is present (and by itself) when the fixed point set is Σ 0 ⊂ Σ 2 .

M5-branes wrapped on Σ g
As a first example of this new technology we consider the supergravity solutions constructed in [20]. These describe the near-horizon limits of N M5-branes wrapped over a Riemann surface Σ g inside a local Calabi-Yau three-fold. The latter is the total space of the bundle The near-horizon limit of the wrapped M5-branes is then an S 4 bundle over Σ g : where the two copies of C i are twisted using the line bundles O(−p i ), i = 1, 2, respectively. There are two special cases: when p 1 = p 2 = −(g − 1) and when p 1 = 0, p 2 = −2(g − 1) corresponding to the supergravity solutions of [21] when g > 1 and dual to N = 1 and N = 2 SCFTs in d = 4, respectively.
We now show how to compute the central charge of such solutions, making just one more assumption. We first introduce vector fields ∂ φ i rotating the C i above, and hence acting on the S 4 fibre over the Riemann surface. Our extra assumption is that ∂ φ i are Killing vectors in the full solution. We then write the R-symmetry vector field as As explained in appendices A and B, with an appropriate choice of sign conventions, regularity of the spinor at the north pole of the S 4 requires the sum of the weights there to satisfy b 1 + b 2 = 1, and then by continuity this fixes the sum to be 1 everywhere. 6 We can thus write The parameter ε may be found for the (on-shell) solutions in [20], but for now we leave it arbitrary.
Following a similar argument in [10], we begin by choosing an arbitrary point on Σ g , and consider a linearly embedded S 2 i ⊂ C i ⊕ R ⊂ R 5 in the fibre over it. We choose the submanifold S 2 i to be invariant under the action of ξ. The homology class of this S 2 i is trivial, so it follows using localization for Φ Y in (3.9) that where the N and S subscripts refer to the north and south poles in the fibre sphere S 4 , respectively. To obtain (3.33), we used the fact that the weights at these north and south poles are where here notice that i = 1, 2 labels different submanifolds S 2 i , with the normal spaces at each pole of those two-spheres being R 2 . Note also the change of relative orientation of those spaces, as discussed in the S 2 example in section 2. Equation (3.33) immediately implies that |y N | = |y S |.
Next consider flux quantization of G through a copy of the fibre S 4 , again at an arbitrary point on Σ g . In the notation of section 3.2 we have C 4 = S 4 , and the fixed points C 0 under ξ are precisely the north and south poles N , S of the S 4 . Denoting the flux as N = N S 4 from (3.26) we have This is the same as in the S 4 example considered in section 2. For N > 0 we must then have y N = −y S > 0, which allows us to solve for y N : Notice that a priori the value of y N could have depended on the point chosen on the Riemann surface. The fact it does not reflects the fact that copies of S 4 over different points on Σ g are homologous, and the flux quantization condition (3.34) depends only on the homology class.
There are two other four-cycles of interest, namely the total spaces of the S 2 i bundles over Σ g , with the fibres S 2 i as defined in (3.33). Denoting these four-manifolds by C (i) 4 , as explained in appendix C we have the following homology relations where the factors of p 2 , p 1 arise from the first Chern classes of the normal bundles of Here Σ N g , Σ S g are the fixed copies of the base Riemann surface, at the north and south poles of the fibre sphere S 2 i . In our conventions for the assignment of north and south pole labels, these have normal bundles 4 , i = 1, 2, which give the respective factors of p i and −p i in (3.37). The corresponding weights are ϵ N i = b i and ϵ S i = −b i , respectively (c.f. the discussion for the S 2 example in section 2). From (3.36) we also have , and integrating the closed form e 6λ Y over both cycles (which are also fixed point sets) immediately implies that Σ N g ω = Σ S g ω. The above equations are then easily solved to find With all of this in hand, we may simply write down the central charge using (3.24). This reads for the product of weights on the normal spaces R 4 = R 2 ⊕ R 2 for each of Σ N g , Σ S g , respectively. As discussed for the simple S 4 example in section 2, there is no invariant way to assign orientations to each R 2 factor. Correspondingly, the normal bundle to Σ N g can be taken to be either of the upper or lower signs in , while the normal bundle to Σ S g may be taken to be either of The expression in (3.40) immediately simplifies to Here in the second equality we have used (3.29), (3.32).
This expression precisely agrees with the off-shell trial a-function in the dual field theory, as a function of the parameter ε. Specifically, compare (3.41) to equation (2.16) of [20], where z there = 1 + p 2 g−1 . The on-shell central charge is obtained by extremizing over the undetermined parameter ε and the result agrees with that obtained from the explicit supergravity solutions in [20].
For example, for the special case when p 1 = p 2 , on-shell we find ε = 0, so b 1 = b 2 = 1 2 , and a central charge which agrees with the explicit supergravity solutions dual to N = 1 SCFTs of [21], which only exist when g > 1. Similarly, for the special case when p 1 = 0, on-shell we find ε = 1/3, so b 1 = 2 3 , b 2 = 1 3 , and a central charge in agreement with the supergravity solutions dual to N = 2 SCFTs of [21], which again only exist when g > 1.
The fact that we obtain the off-shell central charge via localization arises from the fact that the ingredients entering the localization calculations are only a subset of the full supersymmetry conditions. If all of the supersymmetry conditions are imposed we automatically impose the equations of motion [2]. However, if only a subset are imposed then there are some remaining variations to consider. Remarkably, as we show in section 3.6, these remaining variations are associated with varying the offshell central charge.
We now consider M2-branes wrapped on various calibrated cycles and compute the dimension of the dual chiral primaries using (3.28). We first consider the cycles Σ N g , Σ S g . From (3.13) we can infer that these will be calibrated cycles provided that , which can be argued to be the case for the explicitly known solutions of [20]. Choosing an appropriate orientation for Σ N g , (3.28) then gives Similarly, with an an appropriate orientation ∆(Σ N g ) = ∆(Σ S g ). These results agree with the off-shell expressions obtained in field theory in equation (5.18) of [20]. Furthermore, after substituting the value of ε that extremizes a in (3.41) we find exact agreement with the result (4.10) computed in [20] using the explicit supergravity solution.
Another possibility, which has not been previously considered, is to wrap M2branes on the invariant, homologically trivial, submanifolds S 2 i , defined just above (3.33). Now since ξ is tangent to S 2 i , the S 2 i will be calibrated by Y ′ provided that the volume form on S 2 i is, up to sign, given by e 5 ∧ e 6 . To see this we recall (3.13) and that e 6 J = 0 so that Y ′ | S 2 i = e 5 ∧ e 6 | S 2 i . Choosing a suitable orientation to get a positive result, we can then compute ∆(S 2 i ) using localization via (3.28). We have fixed points at the north and south poles of S 2 i , and find where, as in (3.33), we used ϵ N i = b i , ϵ S i = −b i and also (3.35). It would be interesting to verify the calibration condition using the results of appendix D of [20] for the explicit supergravity solutions, and also to identify the operators in the dual field theory proposed in [20].

M5-branes wrapped on a spindle
In this section we consider N M5-branes wrapped over a spindle. The full supergravity solutions were constructed in [22], with M 6 being the total space of an S 4 orbibundle fibred over a spindle Σ = WCP 1 [n + ,n − ] . The latter is topologically a two-sphere, but with conical deficit angles 2π(1 − 1/n ± ) at the poles. The key difference, compared to the previous subsection, is that here the R-symmetry vector ξ generically also mixes with the spindle direction Σ. As a result the fixed point sets are all isolated and consequently the localization formulae are used rather differently than in the previous subsection. On the other hand, the final results of the two subsections should coincide after setting g = 0 and n + = n − = 1, respectively, so that Σ = S 2 , and we will see that this is indeed the case.
The physical set-up in this section is very similar to that in the last section: we are considering the near-horizon limit of N M5-branes wrapped on a spindle surface Σ = WCP 1 [n + ,n − ] inside a local Calabi-Yau three-fold. The latter is the total space of the bundle O(−p 1 )⊕O(−p 2 ) → Σ, where in order for the total space to be Calabi-Yau (giving a topological twist as in the known supergravity solution) we have (3.46) The near-horizon limit is then an S 4 orbibundle over Σ, where again S 4 ⊂ C 1 ⊕ C 2 ⊕ R = R 5 , and the two copies of C i are twisted using the complex line orbibundles O(−p i ), i = 1, 2, respectively. One technical difference in this case is that the fibres over the poles of the spindle base Σ are in general orbifolds S 4 /Z n ± , where the action of Z n ± is determined by the twisting parameters p i ∈ Z. This is discussed in detail in [24] for circle orbibundles (or their associated complex line orbibundles), and that discussion carries over straightforwardly for each of O(−p i ), with an induced action on the S 4 fibres. However, we will not need any of these details in what follows. We write the R-symmetry vector as where ∂ φ i rotate the two copies of C i , as before, while ∂ φ 3 is a lift of the vector field that rotates the spindle, where we use the construction of such a basis in [25]. We assume that ∂ φ i and ∂ φ 3 are Killing vectors of the full solution.
Consider first fixing one of the poles on Σ, say the plus pole with orbifold group Z n + , and consider a linearly embedded S 2 i ⊂ C i ⊕ R ⊂ R 5 in the covering space of the fibre over it. The same argument as in (3.33) then gives where the N and S subscripts again refer to the poles in the fibre sphere S 4 , and we have used where we shall determine the weights b + i below. This immediately implies that |y + N | = |y + S |. A similar argument using the minus pole of the spindle allows us to conclude |y ± N | = |y ± S |. We next consider flux quantization through the fibres S 4 /Z n ± over the poles of Σ. Similar to (3.34), we can use equation (3.26) to obtain and note that the orbifold factors of d = n ± follow since the north and south poles of S 4 /Z n ± are both orbifold loci. With N ± > 0, this fixes the signs to be y ± N = −y ± S > 0. Moreover, from the homology relation between these cycles we deduce which can be obtained from [25] and we are assuming that we are preserving supersymmetry via the twist viz. (3.46), as in the known supergravity solutions [22]. These equations imply Here there are 4 = 2 × 2 isolated fixed points: the two north and south poles N , S of the fibre spheres, over the two poles of the spindle. Notice that the weights of ∂ φ 3 on the tangent spaces to the spindle poles are precisely ∓1/n ± , since these are orbifold loci, and we have again included the factors of d = n ± . Using (3.51) this simplifies to Remarkably this expression takes a "gravitational block" form (see [16]), involving a difference of M5-brane anomaly polynomials 8 in the numerator, one associated to each ± pole of Σ. The appearance of gravitational blocks for this case is related 9 to the fact that ξ has isolated fixed points, arising from the mixing with the U (1) symmetry of the spindle as in (3.47). Indeed, by contrast, the expression (3.41) for the Riemann surface case, Σ g , displays a different structure. For the special case when Σ g ∼ = S 2 one can consider deforming the expression for the Killing vector (3.31) by allowing a mixing with the azimuthal symmetry of the sphere. However, this Abelian isometry sits inside the non-Abelian SO(3) isometry, and for the superconformal R-symmetry one expects (and finds) trivial mixing.
In order to evaluate (3.53) further we need to describe the fibration structure in more detail. The normal bundle to the M5-brane wrapped on Σ is N = N (Σ → (3.46). Compare to (3.29) in the case that g = 0 and n + = n − = 1, so that Σ = S 2 . The weights b ± i may then be computed using the results in [25]. We have ). We may then solve these constraints by introducing new variables ϕ i via with the constraint The central charge is then This derives the conjectured off-shell gravitational block formula in [26], where we have corrected the overall sign. In that reference it was shown extremizing a over the variables ϕ i (subject to (3.56)) gives the central charge of the explicit supergravity solutions constructed in [22]. Moreover, (3.57) agrees off-shell with the trial a-function in field theory, obtained by integrating the M5-brane anomaly polynomial over the spindle. As in the previous subsection, the reason that we have obtained the off-shell central charge via localization is because we have only imposed a subset of the supersymmetry conditions, and moreover, any remaining variations to go on-shell are associated with varying the central charge, as we explain in section 3.6.
It is instructive to see how (3.57) for the spindle case reduces to (3.41) for the S 2 case of the previous subsection in the appropriate limit. In this section we should first set n + = n − = 1, so that Σ = S 2 , and then in order to relate to the variables to those in section 3.3, we should identify with the same result for Σ S (up to orientation), ∆(Σ N ) = ∆(Σ S ). Evaluating this on the extremal values ϕ * i , ε * , one can verify the result agrees with that computed using the explicit supergravity solutions in [22], which provides strong evidence that the cycles are indeed calibrated. Notice also that (3.58) agrees with (3.44) in the appropriate limit, even though it was calculated using localization quite differently.
Finally, notice that in section 3.3 we needed to impose flux quantization of G through the four-cycles C (i) 4 in order to compute the central charge, while here we did not. On the other hand, we may compute the fluxes through the analogous cycles in the spindle solutions also using localization. Thus, let C (i) 4 be the total space of the S 2 i bundles over the spindle, i = 1, 2. Using (3.26) we can write down , (3.59) which using (3.51) gives where we have used (3.54) in the second equalities. One can check that this reduces to (3.38) in the case n + = n − = 1, exactly as it should, and (3.60) is a manifestation of the spindle generalization of the homology relations (3.36).

S 2 bundle over B 4
Our final set of examples use the same technology, but are also rather different in detail. The ansatz we make for the topology covers all of the supergravity solutions found in the original reference [2], but also many more cases for which explicit solutions have not been constructed.
We take M 6 to be the total space of an S 2 bundle over a base four-manifold B 4 : Here B 4 is a priori an arbitrary closed four-manifold, where the R-symmetry Killing vector ξ is assumed to rotate just the S 2 fibre and not act on B 4 . In [2] all solutions for which a certain natural almost complex structure on M 6 is integrable were found in closed form, and all have the structure (3.61), with, moreover, B 4 either Kähler-Einstein or the product of two Riemann surfaces with constant curvature metrics. In particular, for those cases B 4 is complex, and the S 2 bundle is that associated to the anti-canonical line bundle L over B 4 . More precisely, here we view S 2 ⊂ C ⊕ R = R 3 , and use L to fibre the copy of C over B 4 to construct the bundle in (3.61). We shall impose these conditions on the topology of M 6 from the outset in this section, but without assuming any metric on B 4 , and will see that not only do we reproduce all of the results for the explicit solutions found in [2], but we also compute closedform expressions for various BPS quantities for considerable generalizations of those solutions, assuming the latter exist.
The fixed point set of ξ consists of two copies of B 4 at the north and south poles of the S 2 fibre. We denote these by F 4 = B ± 4 (in contrast to the N and S notation used in the previous subsections). The normal bundles are then respectively Noting the discussion around equation (3.23), we observe that ω defines a cohomology class on each of B ± 4 . First pick representative two-manifolds Γ α ⊂ B 4 for a basis for the free part of H 2 (B 4 , Z), and denote the copies of these in B ± 4 by Γ ± α , respectively. We may then write with the real constants c ± α then parametrizing the cohomology classes. The [Γ ± α ] ∈ H 2 (M 6 , Z) in turn satisfy the homology relation, proved in appendix C (see (C.15)), where we have defined the Chern numbers where the weight of the R-symmetry vector on the ± poles of the fibre S 2 is ±ϵ, where in turn the R-charge of Killing spinor fixes ϵ = 1. 11 Again, we recall that y is necessarily constant on each copy of B ± 4 , and we have denoted those constants by y ± in (3.66).
Next we turn to flux quantization for G. Equation (3.23) immediately gives where I = (I αβ ) is the inverse of the intersection form for the four-manifold B 4 , and we introduced the notation ⟨·, ·⟩ for the bilinear form defined by I. The intersection form is a unimodular integer-valued symmetric matrix, and hence its inverse has the same property. Further background and discussion of this may be found in appendix C.
We also have four-cycles C that are the total spaces of the S 2 bundle over Γ α ⊂ B 4 11 The R-charge of the Killing spinor is q = 1 2 under ξ = ∂ ψ , but this is also precisely the charge of a spinor on the tangent space C to a pole of S 2 that is regular at the origin, where ξ = ∂ ψ rotates C with weight one. This fixes ψ to have period 2π hence ϵ = ±1, and in our conventions ϵ = +1.
in the base. Using (3.26) gives Substituting this into N ± in (3.67) gives Substituting the equations (3.66), (3.69) for c ± α , this simplifies to . (3.73) Turning to conformal dimensions of wrapped M2-branes, first applying (3.28) to On the other hand, for Σ 2 = Γ ± α we instead have, with appropriate orientations 12 , We now solve (3.70), subject to the constraint (3.71), and then substitute into which defines the flux number M ∈ Z. Assuming ⟨N, N ⟩ > 0, we then find that there are two solutions for y ± : These then give rise to the following two expressions for the central charge, respectively: (3.80) 12 As we will see, these overall signs give rise to ∆ > 0 in known explicit solutions.
Instead, wrapping Γ ± α , we compute The expressions for the central charge and conformal dimensions given in (3.79) and (3.80), (3.81) are new general results that go well beyond known explicit supergravity solutions. We conclude this subsection by briefly making some checks for some specific cases where explicit supergravity solutions are known.

B 4 = KE + 4 : simple class
An interesting family of explicit solutions found in [2] involves taking B 4 to be a positively curved Kähler-Einstein four-manifold KE 4 . The central charge of these solutions was computed in [23]. The fluxes N α for these solutions are not arbitrary, but, by assumption, are constrained to be proportional to the Chern numbers: for some constant k. Following the notation of [23], we can then write whereM is a topological invariant of KE 4 (and note thatM = M there in [23]), and hence we also have ⟨N, n⟩ = kM as well as ⟨N, N ⟩ = k 2M . To compare with [23] we further consider the class with N − = −N + , which corresponds to setting the flux number M = 0. In order to match the notation in [23] we define Substituting these fluxes into the general formulae above using (as below) the first branch, we obtain the central charge which agrees with the explicit result in [23]. Furthermore, from (3.80), (3.81) we compute ∆(S 2 fibre ) = which again agree with [23].

B 4 = S 2 × Σ g
We also consider the case B 4 = S 2 × Σ g when the base is a product of a two-sphere with a Riemann surface of genus g, for which explicit solutions were given in [2]. The (inverse) intersection matrix and Chern numbers are then given by We can now write The central charge then reads This result precisely coincides 13 with the large N limit of the central charge computed in [27], who obtained their result by consideration of anomaly inflow. Expressions for ∆(S 2 fibre ) and ∆(Γ ± α ) can similarly be obtained from from (3.80), (3.81), and these comprise new results for this class. Below we will write explicit expressions for these for a further sub-class associated with explicit supergravity solutions considered in [23].
Specifically, we next further restrict to the B 4 = S 2 × S 2 case, correspondingly setting g 2 = 0 and hence χ = 2. We also restrict to the sub-class with M = 0. We and find that the central charge can be written as This precisely coincides with the result computed from the explicit solution in [23].
The conformal dimensions of the chiral primaries associated to M2-branes wrapping supersymmetric cycles take the form 92) 13 We should identify N 1 = −N there which again match the results of [23] for p, q, N > 0. Finally, we connect with the Y p,q class of explicit solutions associated with B 4 = T 2 × S 2 . We now set g 2 = 1 and so χ = 0 and write The central charge is then given by which agrees with the central charge of the explicit supergravity solutions [23]. One can check that the expressions for the dimensions of the chiral primaries associated to M2-branes wrapping supersymmetric cycles also agree with those in [23]. This class of explicit solutions are the M-theory duals of the well-known AdS 5 × Y p,q Sasaki-Einstein solutions of type IIB string theory [28]. It is amusing to note that in this case the central charge may similarly be computed in type IIB without knowledge of the explicit supergravity solution, instead employing volume minimization [4]. This is in the same spirit as the present paper, but the details in M-theory and type IIB are very different.

Action calculation
For the general class of AdS 5 × M 6 solutions of D = 11 supergravity considered in [2], one can show that the D = 11 equations of motion, as given in [29], give rise to D = 6 equations of motion for the metric, the scalar λ and the four-form G. These can be written in the form R, and substitute into the action. Next, we eliminate the e 3λ G 2 terms from the resulting expression using the scalar equation of motion in (3.95). We then find that the remaining scalar terms combine into a total derivative and we obtain the following simple expression for the on-shell action Using the definitions (3.7) and (3.12), as well as (3.97), it is interesting to note that we can also write the on-shell action in terms of equivariant forms in an alternative way: 14 It is plausible that these are not extra assumptions, but we will not pursue that further here.
To conclude this section, as somewhat of an aside, we show that (3.97) can also be obtained via localization. We start by recalling (3.10) e 3λ * G = 3d(e 6λ ξ ♭ ) − 4e 6λ Y . (3.100) Since ξ ♭ smoothly goes to zero at a fixed point, the first term is globally exact. Thus, for closed M 6 we have where recall Φ Y = e 6λ Y + 1 3 y 2 . At a fixed point set we have sin ζ = ±1, Y ′ = − sin ζY = ∓Y and also 2y = e 3λ sin ζ = ±e 3λ . Localizing we then get possible contributions from four-forms, twoforms and zero-forms. Specifically: where we used the bilinear (3.18). We also have On the other hand Assuming the fixed point set is non-empty, this proves that where M 6 Φ = M 6 e 9λ vol computes the central charge. On the other hand, when the fixed point set is empty, localization says both integrals are just zero (which is a contradiction for the central charge, which is manifestly positive). This completes our localization proof of (3.97).

AdS 3 × M 8 Solutions
We now consider supersymmetric AdS 3 × M 8 solutions of D = 11 supergravity that are dual to N = (0, 2) SCFTs in d = 3, also discussed in [11]. These solutions have a canonical Killing vector that is dual to the R-symmetry of the SCFT. A general class of solutions associated 15 with M5-branes wrapping a holomorphic four-cycle inside a Calabi-Yau four-fold were classified in [14]. Moreover, infinite classes of explicit solutions were also constructed in [14], including the particular solution found in [30] that generalizes the Maldacena-Núñez construction. The classification results of [14] were later extended to the most general supersymmetric AdS 3 × M 8 solutions in [15]. In this section we will show that there are natural equivariant polyforms, given in [11], that can be constructed from Killing spinor bilinears. Furthermore, once again they can be used to compute the central charge of the dual SCFTs, without knowing the explicit solution. For simplicity we will focus on the general class of AdS 3 × M 8 solutions considered in [14], and discussed in section 4.3 of [15], but we strongly suspect that our results will have a simple generalization to the most general class of [15]. We will illustrate the formalism for a class of M 8 which are S 4 fibrations over a four-dimensional base B 4 associated with M5-branes wrapping B 4 . Here we will consider cases when the R-symmetry acts just on the S 4 fibre, while in [11] we consider cases where B 4 is toric and the R-symmetry acts on both S 4 and B 4 .
We follow the notation and conventions of [15], with some minimal notational changes to maintain some cohesion with the previous sections. The ansatz for the D = 11 metric and four-form are given by ds 2 = e 2λ ds 2 (AdS 3 ) + ds 2 (M 8 ) , where the Hodge star is with respect to the metric ds 2 (M 8 ). The flux quantization 15 The class may also describe more general kinds of wrapped M5-branes.
condition 16 is as in (3.2) and so for any four-cycle, C 4 , we have Having supersymmetric solutions that are dual to N = (0, 2) SCFTs in d = 3 requires that M 8 admits two Majorana spinors ϵ i satisfying Killing spinor equations given in [15]. It is convenient to define the complex spinor ϵ ≡ ϵ 1 + iϵ 2 . For the most general class of solutions, we can construct the following scalar bilinears where ζ is real and S is complex, and γ 9 ≡ γ 1···8 . We also have the following one-form where K, ξ ♭ are real and P is complex. The vector ξ, dual to the one-form ξ ♭ , is the Killing vector that is dual to the R-symmetry and it generates a symmetry of the full solution: L ξ λ = L ξ F = L ξ f = 0. The action of ξ on ϵ is given by and we notice that L ξ ζ = 0. In addition we recall the following bilinears that are also invariant under ξ: Further bilinears are defined in [15], but will not be needed in what follows.
We now restrict our considerations to a sub-class of solutions by demanding that the complex scalar and one-form bilinears which are charged under the action of ξ all vanish, which means S = P = 0 , (4.8) 16 We are interested in large N results and hence we are neglecting the extra Pontryagin class contribution of [31]; see also the discussion in sec 2.3 of [14]. 17 In comparing with [15] note that λ = ∆ there , m there = −1/2, ϵ i = χ there i and ξ ♭ = 1 2 L there . We have also written Ψ = Φ there to avoid confusion with our notation for polyforms. We also note that in this section, in contrast to section 3.1, we use Γ 012...10 = −1 as in [15]. and, as in [15], we now write ζ = sin α . (4.9) In this case, it is possible to write down the bilinears in terms of a local SU (3) structure on M 8 . Introducing an orthonormal frame e i , we define the tow form j = e 12 + e 34 + e 56 and the three form θ = (e 1 + ie 2 )(e 3 + ie 4 )(e 5 + ie 6 ). The bilinears we introduced above can then be written cos α e 7 , K = cos α e 8 , J = j + sin α e 78 , ω = sin α j + e 78 , These expressions allow one to easily obtain the expressions for the contraction of ξ on the bilinears.
The bilinears satisfy the differential conditions Additional conditions are given in [15].
With this set-up we can now construct the equivariant polyforms, as in [11]. To begin we impose the condition d(e 3λ f ) = 0 in (4.2) which allows us to introduce a function a 0 , which in general is only locally 18 defined, via Moreover, using ξ f = 0 ((2.49) of [15]) we see that L ξ a 0 = 0 is invariant under the action of the Killing vector. From (4.11) we can then write d(e 3λ sin α − a 0 ) = 2e 3λ K . Since d(e 3λ K) = 0, as follows from the first line of (4.11), we can introduce a local coordinate y so that y = 1 2 (e 3λ sin α − a 0 ), K = e −3λ dy . (4.14) We can now construct an equivariant form, Φ F , involving the four-form F : We can also construct another equivariant form, Φ * F , involving the four-form * F : We also find another equivariant form, with the four-form Ψ as the top form, given by which is in fact equivariantly cohomologous to Φ * F : and hence can be evaluated using localization. We now consider some properties of the fixed point set. Since ∥K∥ 2 = 4∥ξ ♭ ∥ 2 = cos 2 α, at fixed points we have α = ±π/2 and the Killing spinor is chiral/anti-chiral: Since dy = e 3λ K, at the fixed points y is constant and It will also be helpful to note from (4.10) we have e 6λ ω 2 F = e 6λ j 2 F .

(4.24)
We can also compute the conformal dimension ∆ of chiral primary operators in the dual N = (0, 2) SCFT, corresponding to M2-branes wrapping supersymmetric two-cycles Σ 2 ⊂ M 8 . By following similar arguments given in [23], we claim that such M2-branes are calibrated by ω, i.e. vol| Σ 2 = ω| Σ 2 , and their conformal dimension may be computed in the gravity dual via (4.25) By considering the restriction of the four-form flux polyform Φ F in (4.15) to Σ 2 , we see that when Σ 2 is ξ-invariant we can evaluate the integral in (4.25) by localization: Here Σ 0 ⊂ Σ 2 are the fixed points inside Σ 2 and the first term is only present (and by itself) when the entire Σ 2 is fixed, while the second term is present (and by itself) when the fixed point set is Σ 0 ⊂ Σ 2 .

M5-branes wrapped on B 4
We will consider a class of solutions that were explicitly found in [32] (generalizing those in [30]). These describe the near-horizon limit of M5-branes wrapping a complex four-manifold B 4 inside a Calabi-Yau four-fold. We assume the normal bundle of B 4 is a bundle of the form N = L 1 ⊕ L 2 → B 4 . The total space of the bundle is Calabi-Yau if [32] where t 1 , t 2 are the Chern roots 19 of the holomorphic tangent bundle T B 4 . Equation where S 4 ⊂ C 1 ⊕ C 2 ⊕ R and each C i is twisted by L i . The R-symmetry Killing vector ξ acts on the four-sphere with fixed points at the two poles. We write it as where we are explicitly assuming that ∂ φ i are Killing vectors of the supergravity solution. Also, we take ∆φ i = 2π and hence each ∂ φ i rotates C i with weight 1. Since the sum of the weights must be b 1 + b 2 = 1, as may be argued similarly to section 3.3, using appendices A and B, we can also write The fixed point set consists of two copies of B ± 4 over the two poles, denoted in this section by ±, with normal bundles N ± (which are isomorphic to each other, up to orientations). We denote by Γ α the two-cycles forming a basis of the free part of as well as 20 As in section 3.5, here I is the inverse of the intersection matrix for B 4 , and ⟨·, ·⟩ is the bilinear form defined by I.
The integrals of G through the four-cycles should be quantized, and we can evaluate the integrals using localization. For S 4 there are fixed points at the poles and we deduce 34) 20 We will shortly provide a consistency check of (4.33) using localization.
where we used ϵ + 1 ϵ for the products of weights on the R 4 normal spaces to the two poles inside S 4 . For C (iα) 4 , which have fixed point sets Γ ± α , we find associated fluxes Here we have used the weights ϵ + i = b i , ϵ − i = −b i on the R 2 spaces normal to the two poles of the S 2 i fibres inside C (iα) 4 (which are the copies Γ ± α of the two-submanifolds Γ α in B 4 ), and we have also defined (4.36) Recall from (4.23) that c ± α only depend on the homology class of Γ ± α , and that the normal bundles to Γ ± α inside C (iα) 4 have opposite orientations, correlated with the weights ϵ ± i . Notice we can solve for c + α − c − α : From (4.31) we have and hence Continuing, we now impose flux quantization through B ± 4 . To do so we recall from (4.11), (4.17) that we have Thus, where we used (4.10), (4.24) and also B ± 4 e 6λ ω 2 = ⟨c ± , c ± ⟩ . We now briefly pause to explain how we can obtain (4.43) using localization without requiring the homology relation (4.33), or equivalently how the homology relation is consistent with the orientation choices that we use in the localization formulae. The idea is to trivially extend the equivariant form Φ F in (4.15) to have top-form that is the zero six-form. We then integrate this six-form on the six-cycles , to trivially get zero, and then also evaluate it using the BVAB formula (2.9): Here, in the first line we used Continuing, we next consider integrals of Φ * F . Using localization for the first two, we find In the last line we are simply emphasizing that the integral of Φ * F 4 ≡ e 6λ * F − a 0 e 3λ F over B ± 4 will appear in various places below. The homology relation (4.31) then implies Write the left hand side as 1 2 (y + + y − )(c + α − c − α ) + 1 2 (y + − y − )(c + α + c − α ) and substitute (4.39). Then substituting (4.34) into the right hand side, we deduce that either The former possibility is inconsistent with N S 4 ̸ = 0 and so we deduce that To proceed, we can combine (4.41) and (4.43) to get an expression for the quantity (y + ) −1 We also get an expression for B + using (4.33) and the first line in (4.47). These can be solved for B ± 4 Φ * F 4 and after using (4.34) and also (4.49), we deduce If we now substitute this as well as (4.49) into (4.41) we also deduce that Notice that, in contrast to (3.76), in this case it is not possible to add an additional flux number M = (N + + N − )/2, which is set to vanish by the geometry.
We now turn to the central charge (4.21). We need to compute the integral M 8 Φ using localization. After some computation, the BVAB formula (2.9) gives where we used (4.45).
If we now make the additional assumption that things simply further. More physically, from the first equation in (4.47) we see that this is equivalent to saying that the four-form flux associated with Φ * F , through the S 4 fibre, is zero: In contrast, the four-form flux associated with Φ F , through the S 4 fibre, is N S 4 , the number of M5-branes, as in (4.34 This is an off-shell expression for the central charge, since it still depends on b i subject to the constraint b 1 + b 2 = 1. After substituting (4.30) and extremizing over ε, we find that the extremal value is given by and the extremal value of ε is We now consider M2-branes wrapped on various calibrated cycles and compute the dimension of the dual chiral primaries using (4.26). We first consider the invariant submanifolds Γ ± α located at the north and south poles of the S 4 fibre. From (4.10) we can infer that these will be calibrated cycles provided that ω| Γ ± α = ±j| Γ ± α . Choosing an appropriate orientation, (4.26) then gives the off-shell result where we used (4.36) and (4.49). Similarly, choosing a suitable orientation, we have ∆(Γ − α ) = ∆(Γ + α ). Another possibility, is to wrap M2-branes on the invariant, homologically trivial, submanifolds S 2 i in the S 4 fibre. Now since ξ is tangent to S 2 i , from (4.10) the S 2 i will be calibrated by ω provided that the volume form on S 2 i is, up to sign, given by e 7 ∧ e 8 . Choosing a suitable orientation to get a positive result, we can then compute ∆(S 2 i ) using localization. We have fixed points at the north and south poles of S 2 i , and find where we used ϵ + i = b i , ϵ − i = −b i and also (4.34). Note that this result dose not use the assumption (4.53).
As an example, we consider B 4 = KE − 4 , a Kähler-Einstein four-manifold with negative curvature. We choose a normalization of the KE − 4 metric so that the Kähler form satisfies J KE = −R = −2πc 1 (L), where L is the anti-canonical bundle. It is also useful to recall that the first Pontryagin number and the Euler number of the KE − 4 are given, respectively, in terms of the Chern roots t 1 , t 2 , by (4.60) We also have Now we can solve the Calabi-Yau condition (4.27) for the Chern roots with a twist by writing Here t 1 , t 2 are topological invariants of B 4 = KE − 4 , so that the variable z we have introduced then describes a one-parameter family of associated local Calabi-Yau four-folds, via the twisting of the normal bundle L 1 ⊕ L 2 over B 4 = KE − 4 . We compute the formulae Substituting into the central charge (4.55) and using (4.30), we obtain which agrees with the large N limit of the off-shell expression (6.6) of [32] provided we take N there = −N S 4 > 0. After extremizing over ε we get the on-shell result Explicit supergravity solutions with z = 0 were constructed in section 4.2 of [30], and extended to |z| < 1/3 in [32]. It would be interesting to check the predictions for the conformal dimensions of chiral primaries associated with wrapped M2-branes given in (4.58) and (4.59) with the explicit supergravity solutions.

B
We can also consider the class of solutions with B 4 ∼ = Σ 1 × Σ 2 , where Σ α are Riemann surfaces with genus g α , as considered in [32], extending [34]. The solutions in [32,34] were constructed in D = 7 gauged supergravity and then uplifted to D = 11.
For this case we have (4.66) Moreover, we denote the degrees of the normal bundles L 1 and L 2 on Σ α as −p α and −q α , respectively. Thus, the integration of (4.27) on each of the Σ α gives (4.67) Using the same parametrization as [32], this can be solved by writing where κ α = 0, ±1 is the curvature of the Riemann surface Σ α and z α ∈ Q (consistent with p α , q α ∈ Z). Substituting into (4.55) then gives the off-shell central charge which matches the large N limit of the off-shell formula in [32]. The on-shell expression can be obtained by extremizing over ε (or directly from (4.56)). Note that explicit supergravity solutions were constructed in [32], extending [34], when at least one of the g α > 1. In particular, no solutions for a four-torus, when g 1 = g 2 = 1, are known to exist. Note a special case is when we set z 1 = z 2 = 0. In this case we have p α = q α = (1 − g α ) and, with g α > 1, for which explicit supergravity solutions are known [30], this can be viewed as a special case of the KE − 4 case considered above. Indeed, setting z 1 = z 2 = 0 and κ α = −1 in (4.69) we get precise agreement with (4.65) after setting z = 0, χ = 8(1 − g 1 )(1 − g 2 ) and P 1 = 0.
It would be interesting to check the predictions for the conformal dimensions of chiral primaries associated with wrapped M2-branes given in (4.58) and (4.59) with the explicit supergravity solutions.

Action calculation
Similar to section in 3.6, we now investigate the action. For the general class of AdS 3 × M 8 solutions of D = 11 supergravity considered in [15], one can show that the D = 11 equations of motion, as given in [29], give rise to the following D = 8 equations of motion. From the Einstein equations we get arising from the Bianchi identity and the equation of motion for G (and already seen in (4.2)). These can be obtained 21 by extremizing the eight-dimensional action Substituting this into the action, and then following the same steps as in section 3.6, we find that the partially on-shell action can be written in the form with Φ given in (4.20). Thus, once again, the partially on-shell action is precisely proportional to the central charge, and we again see that varying the central charge with respect to any undetermined coefficients after carrying out localization, is a necessary condition for putting the system on-shell. 21 One should substitute f = e −3λ da 0 and F = e −3λ dC 3 and then vary over a 0 , C 3 , as well as the metric and the scalar field.
Finally, note that if we multiply the scalar equation of motion in (4.69) by e 9λ , solve for e 9λ and then substitute into (4.73) we can write the on-shell action as Then, after using (4.72) and recalling the equivariant polyforms in (4.15), (4.16), we find that we can also write the on-shell action in the form This is the analogue of the expression that we presented in (3.99).

Discussion
In this paper, and in [10,11], we have introduced a general calculus that allows one to cases we have illustrated our formalism by recovering some results for known supergravity solutions, as well calculated new results for more general classes of solutions (providing they exist). There are also other classes of solutions that can be analysed in a straightforward way; for example, it would be interesting to consider M 8 which are S 2 bundles over B 6 and make contact with the explicit solutions in [14].
A remarkable feature of our formalism is that we derive expressions for BPS quantities, such as the central charge, which are generically off-shell. This allows one to directly compare with off-shell field theory results, where analogous expressions are obtained utilizing field theory extremization techniques. In the examples we considered, the off-shell central charges are functions of some undetermined weights of the action of the R-symmetry. We also demonstrated that imposing a subset of the supersymmetry conditions as well as a subset of the supergravity equations of motion, one obtains a partially on-shell action that is proportional to our expression for the central charge. Thus, extremizing our expressions for the off-shell central charge over the undetermined weights is associated with necessary conditions for putting the whole system on-shell. It would be very interesting to determine the precise class of geometries that we are extremizing over when we partially go on-shell, analogous to what has been achieved in a different setting for Sasaki-Einstein geometry in [4] and GK geometry in [6].
We anticipate some immediate generalizations of this work, some also discussed in [10]. Firstly, we expect there will be straightforward generalizations of all of the calculations in this paper to AdS ×M solutions of type IIA and type IIB supergravity when M has even dimension. Second, for AdS × M solutions of type II and D = 11 supergravity where M has odd dimension we expect that the construction of equivariant polyforms will be similarly straightforward. However, for this case the details of utilizing localization formulae will be different. There are known techniques to deal with localization in odd dimensions, and we expect that it will be possible to use them to compute physical observables in the dual SCFT. This odd-dimensional case may overlap with work on Sasaki-Einstein and GK geometry, and we expect to make contact with the recent work of [17], as well the extensive recent work on GK geometry starting with [6]. We plan to report on these topics soon.

Acknowledgements
This work was supported in part by STFC grants ST/T000791/1 and ST/T000864/1.
JPG is supported as a Visiting Fellow at the Perimeter Institute. PBG is supported by the Royal Society Grant RSWF/R3/183010.

A Regularity of spinors
In this appendix we examine how spinors with a definite charge under a Killing vector behave near a fixed point, and how this is correlated with the chirality of the spinor.
The charge q of the spinor with respect to a Killing vector is defined, as usual, via the Lie derivative: We consider a spinor which satisfies a differential condition of the general form 22 where M is a sum of forms. At a fixed point, as we make explicit below, we have ξ a (M · γ) a ϵ = 0 so that ξ a ∇ a ϵ = 0 and hence We first consider two-component Dirac spinors on R 2 . A basis for the Clifford algebra is given by γ 1 = σ 1 , γ 2 = σ 2 and we define γ * = −iγ 1 γ 2 = σ 3 . Chiral spinors are defined via γ * ϵ ± = ±ϵ ± .
We use Cartesian coordinates (x, y) and orthonormal frame e a = (dx, dy), both of which are of course globally well-defined on R 2 . We also have standard polar coordinates (r, φ), with ∆φ = 2π. We consider the Killing vector ξ = x∂ y − y∂ x = ∂ φ which generates an SO(2) isometry and has a fixed point at the origin. At the fixed point, since ξ x = ξ y = 0, we have ξ a (M · γ) a ϵ = 0. We also have dξ ♭ = 2dx ∧ dy and hence at the fixed point from (A.3) we deduce Our arguments also apply if we have ∇ a ϵ = (M · γ) a ϵ + (N · γ) a ϵ c .
We thus conclude that |q| = 1 2 and furthermore that the sign of q is correlated with the chirality of the spinor. This is sensible, since the charge labels the irreducible representation of U (1) = SO (2) in which the spinor transforms; but here the SO (2) is also the "Lorentz" group, so the charge labels the irreducible representations of the "Lorentz" group, whose elements are the chiral spinors.
We consider the Killing vector ξ to be a linear combination of rotations in the two planes: We compute, in the Cartesian frame, The norm squared of the Killing vector is ∥ξ∥ 2 = b 2 1 r 2 1 +b 2 2 r 2 2 . Thus, when both b i ̸ = 0 the norm vanishes at the origin of R 4 , and the fixed point set is a nut. If one of the b i = 0, then the fixed point set of ξ is a two-plane bolt.
A basis for the Clifford algebra is We define the chirality operator γ * ≡ γ 1234 = −σ 3 ⊗ σ 3 so that chiral spinors are given, in the Cartesian frame, by We now compute the Lie derivative along ξ. Notice that both for a nut and a bolt we have ξ a (M · γ) a ϵ = 0. Thus at a fixed point, from (A.3) we deduce We thus conclude |q| = 1 2 |b 1 ± b 2 | and there is a further correlation with the chirality of the spinor. For example, suppose the spinor has charge q = +1/2, as we assume in the bulk of the paper, with If we have a nut, i.e. both b i ̸ = 0, then we see that the spinor must either have

B.1 Equivariantly closed forms
The metric is a warped product where AdS 7 has unit radius, and compatibility with the AdS symmetries implies that λ is a function on M 4 and the four-form is given by for some constant G.
We can decompose the D = 11 Clifford algebra Cliff(10, 1) ∼ = Cliff(6, 1)⊗Cliff(4, 0) by writing Γ a = ρ a ⊗ γ 5 , where ρ a generate Cliff(6, 1) with ρ 0123456 = +1, the Hermitian γ m generate Cliff (4,0) and γ 5 ≡ γ 1234 with γ 2 5 = 1. We decompose the eleven-dimensional spinor ϵ 11 by writing where ϵ is a spinor on M 4 and ψ is a Killing spinor on AdS 7 satisfying D α ψ = − 1 2 ρ α ψ. This ansatz implies that ϵ satisfies the following equations on M 4 : We can now construct various bilinears using the spinor ϵ and consider the algebraic and differential equations that follow from (B.5). It is not difficult to deduce that λ must be constant, so we set e λ ≡ L. This then fixes the flux to be The first equation in (B.5) then becomes the Killing spinor equation ∇ m ϵ = γ m γ 5 ϵ and hence we deduce [35] that M 4 must be S 4 (or a quotient thereof) and ds 2 (M 4 ) = To proceed we first note that (B.5) implies ϵϵ is a constant and we normalize so that ϵϵ = 1. We define the following differential forms on M 4 : where γ (r) ≡ 1 r! γ µ 1 ···µr dx µ 1 ∧ · · · ∧ dx µr . From (B.5) it follows that the vector ξ, dual to the one-form ξ ♭ , is a Killing vector and we have chosen a normalization so that ϵ has charge 1 2 under the action of ξ: The square norm of the Killing vector is given by Thus, the fixed points of ξ correspond to sin ζ = ±1 and we observe that at such a fixed point the spinor is chiral/anti-chiral, satisfying γ 5 ϵ = ±ϵ.
We can also show that which implies that the polyform, extending the four-form G, is equivariantly closed: d ξ Φ G = 0. This allows us to use the equivariant localization formula to compute the flux quantization of the four-form.
The holographic central charge of the dual field theory is given by [36,37]  Here, we have written Y ′ = * Y to be consistent with the results in other dimensions.

B.2 Localization
With these polyforms, we are able to evaluate the central charge of the field theory dual to any M 4 using localization formulae after fixing the topology of M 4 . For this case we know that M 4 = S 4 , with a round metric, is the only solution, so we proceed with just the assumption that topologically M 4 = S 4 and that ξ is a Killing vector field of the form where φ i are the polar angles on C i if we view S 4 ⊂ C 1 ⊕ C 2 ⊕ R. The weights b i are arbitrary, and the fixed points of ξ are at the two poles, which we label N and S. We observe that ϵ N 1 ϵ N 2 = b 1 b 2 and ϵ S 1 ϵ S 2 = −b 1 b 2 , with the change of relative orientation at the poles as in the example considered in the S 4 example in section 2.
Quantization of the flux through S 4 can be imposed by applying the localization formula to (B.11): With the constraint (B.17), the result for the central charge in (B.16) matches the large N limit of the off-shell trial central charge of the six-dimensional (2, 0) theory, with mixing due to the U (1) 2 maximal torus in the SO(5) R-symmetry. Carrying out the extremization over the weights b i subject to (B.17), we deduce that the extremal on-shell weights are b 1 = b 2 = 1 2 and the central charge is a = 16/7N 3 . Note 24 also that at the extremal values b 1 b 2 > 0 and so N > 0. We have thus recovered the results that can alternatively be obtained from direct evaluation in the Freund-Rubin solution. We emphasize that our new derivation did not require the explicit metric on S 4 and, moreover, since we only utilised a subset of the information contained in the Killing spinors, we obtained an off-shell expression for the central charge.

C Homology relations from Poincaré duals
In the main text we have used a number of homology relations for manifolds which are the total spaces of even-dimensional sphere bundles. These may be proven using the explicit differential form construction of Poincaré duals in [33], together with a simple additional ingredient. We mainly focus on the total spaces of S 2 bundles over a four-manifold B 4 , as in section 3.5, to illustrate the general method. We then briefly indicate the extension to S 4 bundles that appear in sections 3.3 and 4.1.
Consider a six-manifold M 6 which is the total space of an S 2 bundle over a fourmanifold B 4 , with projection map π : M 6 → B 4 . More specifically, we are interested in unit sphere bundles inside the R 3 bundle L ⊕ R → B 4 , where L is a complex line bundle. Choosing coordinates (z, x) on the R 3 = C ⊕ R fibre, the unit sphere is where we have introduced the intersection form which is an integer-valued unimodular symmetric matrix. For example, we may then I αβ c α c β , (C.8) where we have introduced 26 (C.9) 26 Notice we can introduce a dual basisΨ α satisfying B4Ψ α ∧ Ψ β = δ αβ , so that ω = b2 α=1 c αΨα andΨ α = b2 β=1 Q −1 αβ Ψ β .
With this notation in hand, we may then expand where L i are two complex line bundles. Choosing coordinates (z 1 , z 2 , x) on the R 5 = C 1 ⊕ C 2 ⊕ R fibre, the unit sphere is |z 1 | 2 + |z 2 | 2 + |x| 2 = 1, with the north and south pole sections B N , B S at {z 1 = z 2 = 0, x = ∓1}, respectively.
In section 3.3 we consider the case where B = Σ g is a Riemann surface, and defined four-cycles C (i) 4 as the total spaces of S 2 i bundles over Σ g . Specifically, C 4 is defined as the locus {z 2 = 0}, whose fibre is then a unit sphere inside C 1 ⊕R ⊂ R 5 . Its normal bundle inside M is L 2 , with Poincaré dual PD C  where n 1α , n 2α are the coefficients of the expansion of c 1 (L 1,2 ) on Γ α , as in (C.14). Furthermore, we consider the four-cycles C