Localizing Wrapped M5-branes and Gravitational Blocks

We consider $d=2$, $\mathcal{N}=(0,2)$ SCFTs that can arise from M5-branes wrapping four-dimensional, complex, toric manifolds and orbifolds. We use equivariant localization to compute the off-shell central charge of the dual supergravity solutions, obtaining a result which can be written as a sum of gravitational blocks and precisely agrees with a field theory computation using anomaly polynomials and $c$-extremization.


INTRODUCTION
Supersymmetric wrapped branes continue to provide a fertile arena for exploring the AdS/CFT correspondence.They give rise to rich classes of novel SCFTs in various spacetime dimensions, and they also provide a concrete framework for obtaining a microstate interpretation of the Bekenstein-Hawking entropy for asymptotically AdS black holes.In addition, the supersymmetric AdS solutions of supergravity associated with wrapped branes give rise to novel geometric structures which are of interest in their own right.
In a recent paper [1], a new calculus was introduced for supersymmetric solutions of supergravity that have an Rsymmetry.For several general classes of such solutions, it was shown there exists a set of equivariantly closed differential forms which can be constructed from Killing spinor bilinears.Furthermore, various BPS observables can then be computed using localization via the Berline-Vergne-Atiyah-Bott (BVAB) fixed point formula [2,3], without solving the supergravity equations of motion.
Here we want to further develop these tools for a general class of AdS 3 solutions of D = 11 supergravity that arise from M5-branes wrapping four-dimensional manifolds and orbifolds.The preserved supersymmetry is such that the AdS 3 solutions are dual to d = 2, N = (0, 2) SCFTs and, in particular, they have an R-symmetry.
More precisely, we focus on the class of supersymmetric AdS 3 × M 8 solutions of D = 11 supergravity considered in [4] and further analysed in [5].We construct a set of equivariantly closed forms and show that they can be used to compute the central charge of the dual SCFT, as well as the conformal dimensions of operators in the SCFT that are dual to supersymmetric wrapped probe M2-branes.To illustrate the formalism, we focus on examples where M 8 is an S 4 fibration over a toric B 4 , which are associated with M5-branes wrapping B 4 .Focusing on toric B 4 is of interest since we can both compare with some known and conjectured field theory results, as well as obtain results which provide new field theory predictions.As we shall see the localization results are remarkably simple for these examples because the fixed points of the R-symmetry, which is linear combination of the U (1) 2 action on S 4 and the U (1) 2 action on B 4 , are a set of isolated points on M 8 .We can use the BVAB formula to implement flux quantization as well as obtain an off-shell expression for the central charge.After extremizing over the undetermined R-symmetry data, we then obtain an on-shell expression for the central charge.As explained in more detail in [6], it is important to emphasize that this will give the correct central charge, without solving the supergravity equations, just assuming the supergravity solution actually exists, or equivalently, that the low energy limit of M5-branes wrapped on the specific toric B 4 does indeed flow to a SCFT in the IR, in the large N limit.The formalism therefore provides a geometric, off-shell version of c-extremization [7,8].
The off-shell expression for the central charge that we derive takes the form of a sum of gravitational blocks [9].The terminology 'gravitational block' is perhaps somewhat confusing, as in many works it is not referring to a computation in gravity, but rather is a conjecture that should arise for some unspecified gravity computation.More precisely various off-shell expressions for BPS quantities have been proposed which either can be derived in field theory, for example using anomaly polynomials, or alternatively have been noted to give the correct on-shell result for some specific, explicitly known supergravity solutions.For the former, invoking AdS/CFT, there is then an expectation that there will be a corresponding offshell computation within gravity that leads to the same off-shell result for the central charge.However, it is not at all clear, in general, how one should go off-shell on the gravity side.That being said, in the setting of GK geometry, for Sasaki-Einstein fibrations over spindles, this was recently achieved in [10].The results of this paper, as well as [1,6] indicate that the equivariant calculus of [1] provides a universal way of deriving gravitational blocks within a gravitational context.Moreover, the new results make it clear that the origin of gravitational blocks is when a trial R-symmetry has isolated fixed points on the space that the brane is wrapping.
In [6] we provide some further details of the equivariant calculus for the general class of AdS 3 solutions of arXiv:2308.10933v1[hep-th] 21 Aug 2023 D = 11 supergravity discussed here.In addition, we will also analyse other examples of wrapped M5-branes, where the R-symmetry fixed point set no longer consists of isolated points and, in particular, gravitational blocks are not relevant.
We consider supersymmetric solutions of D = 11 supergravity of the form where λ, F and f are a function, a four-form and a oneform on M 8 , respectively.In addition, ds 2 (AdS 3 ) is the metric on a unit radius AdS 3 and vol(AdS 3 ) is the corresponding volume form.The Bianchi identity implies d(e 3λ f ) = 0, and it is convenient to introduce a function a 0 , locally defined in general, via e 3λ f = da 0 .We assume that the preserved supersymmetry is such that the dual d = 2 SCFTs have N = (0, 2) supersymmetry.We will focus on the class of solutions classified in [4]: following the conventions of [5], there is then a complex spinor ϵ on M 8 , with εϵ = 1 = εc ϵ as well as εc γ 9 ϵ = 0, where γ 9 ≡ γ 1 . . .γ 8 .There is an R-symmetry Killing vector ξ, with a dual one-form ξ ♭ which can be constructed as a bilinear: We have introduced the notation γ (r) = 1 r! γ µ1•••µr dx µ1 ∧ • • • ∧ dx µr , and have normalized ξ so that L ξ ϵ = i 2 ϵ.We also define a scalar, two two-forms and a four-form bilinear and introduce the locally defined function y, given by y = 1 2 (e 3λ sin α − a 0 ).These ingredients can be used to define the following polyforms on where * denotes the Hodge dual and vol 8 is the volume form on M 8 .A key result [11] is that the differential and algebraic conditions satisfied by the above bilinears, along with the Bianchi identity and equation of motion for the four-form, imply these polyforms are equivariantly closed: d ξ Φ = d ξ Φ F = d ξ Φ * F = 0. Thus, we can compute their integrals on closed cycles using the BVAB formula.In particular, the integral of Φ F on a fourcycle Γ 4 represents the flux of the four-form of elevendimensional supergravity, which (in the large N limit) should be quantized as where ℓ p is the Planck length.By computing the effective three-dimensional Newton constant, one can show that the integral of Φ is proportional to the trial central charge

M5-BRANES WRAPPED ON B4
Within the above setup, we are interested in solutions that describe holographic duals to M5-branes wrapping a holomorphic four-cycle B 4 inside a Calabi-Yau four-fold.A local model for the Calabi-Yau is given by the sum of two line bundles which also guarantees the supersymmetry of a wrapped M5-brane.For the associated supergravity solutions (in the near horizon limit), we take M 8 to be an S 4 bundle over B 4 , Here we write S 4 ⊂ C 1 ⊕ C 2 ⊕ R, where the C i factors are twisted by the respective line bundles L i .We will assume that the solutions have U (1) 2 ⊂ SO(5) isometry of the S 4 , as well as the isometries of B 4 .In the following, we will first consider the B 4 base to be a complex toric surface and compute the corresponding trial central charge using equivariant localization.Later we will consider cases when the toric B 4 has orbifold singularities and we will then also slightly generalize the Calabi-Yau condition (7).Other classes of B 4 are considered in [6].
For the toric B 4 examples considered here, the Rsymmetry will only have isolated fixed points and as a consequence the BVAB formula takes a particularly simple form.On M 8 , and even-dimensional invariant submanifolds M 8−2k ⊂ M 8 , the integral of a general equivariantly closed polyform Φ is given by a sum of contributions from the fixed points x ℓ Here M can have orbifold singularities, where the normal space to the point x ℓ is R 2k /Γ ℓ and d ℓ is the order of the finite group Γ ℓ .On this normal space ξ generates a linear isometric action with weights ϵ ℓ i .In the sequel we impose one other condition on the class of solutions that we are considering; namely, that a certain flux integral threading the S 4 vanishes [12]:

SMOOTH TORIC BASE
The first family of solutions we focus on is when the base B 4 is a toric complex surface, with B 4 having U (1) 2 isometry.The R-symmetry Killing vector ξ on M 8 generically mixes the U (1) 2 ⊂ SO(5) isometry of the S 4 with the U (1) 2 of B 4 , and so we can write where b i , ε A are constants.Here ∂ φi rotate the two copies of We can now use the BVAB formula to compute the flux of Φ F through the S 4 cycle over any of the d fixed points on the base.Since these cycles are all in the same homology class [13], using (5) we have where N S 4 is the number of wrapped M5-branes.Here y a N/S denotes the value of the function y at the fixed point (N/S, a), and b a i are the weights of the action of ξ on the normal space R 4 = C ⊕ C to the fixed point in S 4 .We can similarly compute the flux of Φ * F through the same cycles which, recall from (10), we assumed to vanish.Utilizing the BVAB formula then immediately gives (y a N ) 2 = (y a S ) 2 .Thus, requiring that N S 4 ̸ = 0 we conclude that We can similarly evaluate the central charge, with contributions from the 2d fixed points given by Here the normal space to the fixed points in M 8 is R 8 = C ⊕4 , with b a i , ϵ a A being the associated weights of the action of ξ on those four copies of C.
It is remarkable how simply the key expression (14), as a sum of blocks, emerges from our formalism.In particular, we see that each block is related to the off-shell central charge for the d = 6, N = (0, 2) SCFT in the large N limit [14].To obtain our final off-shell result it remains to compute b a i , ϵ a A in terms of the R-symmetry vector (11), together with global topological data for M 8 .In fact we will be able to do this straightforwardly, utilizing various standard results in the toric geometry literature.

WEIGHTS FROM TORIC GEOMETRY
We begin by recalling some key facts about complex toric four-manifolds B 4 .By definition these are complex manifolds equipped with a holomorphic (C * ) 2 action, which has a dense open orbit.There always exists a compatible Kähler metric, where U (1) 2 ⊂ (C * ) 2 is an isometry, but we emphasize that no metric data enters the fixed point formulae we use.
Such a B 4 has a distinguished set of a = 1, . . ., d toric divisors D a ⊂ B 4 .By definition these are complex twodimensional submanifolds, invariant under the U (1) 2 action, where the normal space to a given D a is rotated by the U (1) ⊂ U (1) 2 subgroup specified by a vector with components v a A ∈ Z, A = 1, 2. The set of v a A is referred to as the toric data for B 4 .The toric divisors may be ordered cyclically, with x a = D a ∩ D a+1 precisely giving the set of d points that are fixed under the U (1) 2 action, the index a understood to be defined modulo d.
If ∂ ψ A denote vector fields generating the U (1) 2 isometry, then we may write a Killing vector on B 4 as , and the weights ϵ a A of this Killing vector on C a A are given by the standard toric geometry formulae The internal space M 8 is in turn the total space of an S 4 bundle over B 4 .By definition the vector fields ∂ φi in (11) rotate the two copies of C i in S 4 ⊂ C 1 ⊕ C 2 ⊕ R, with weight 1, but to define ξ we must also choose a lifting of the ∂ ψ A to M 8 .This may be achieved by choosing a lifting to each line bundle L i → B 4 , making these equivariant line bundles.On the other hand, a basis of such equivariant line bundles L a is naturally provided by the toric divisors D a .The corresponding equivariant first Chern class c ξ 1 (L a ), when restricted to the fixed point x b ∈ B 4 , is given by the formula where the weights ϵ a A are given by (15).We may thus write L i = − d a=1 p a i L a , where p a i ∈ Z specify both the topology of L 1 ⊕ L 2 → B 4 , and also a choice of lifting of the U (1) 2 isometry of B 4 to the total space.From ( 16) the weights of ξ on the two complex line fibres are then Having determined explicit formulae ( 15), ( 17) for the weights of ξ at the fixed points, finally we must impose that ξ is an R-symmetry: that is, there is a Killing spinor ϵ satisfying L ξ ϵ = i 2 ϵ.This is where the Calabi-Yau condition ( 7) enters as a further set of constraints on our parameters.
For a toric complex manifold B 4 we have the standard toric geometry formula c 1 (T B 4 ) = The c ξ 1 (L a ) are precisely a set of generators for the equivariant cohomology of B 4 , with no relations, so the coefficients in (18) must all be zero: 2 i=1 p a i = 1.On the other hand, the resulting SU (4)-invariant chiral spinor on the Calabi-Yau four-fold satisfies a standard set of projection conditions γ 2j−1,2j ϵ = iϵ in an orthonormal frame, for each j = 1, 2, 3, 4. The original local Calabi-Yau geometry is embedded inside M 8 as the normal bundle of the north pole section, in our conventions for the labelling of north/south poles.As shown in the appendix, given the above projection conditions the charge of the spinor at the point x N a in this north pole section is then Here we have used (17), and then imposed the Calabi-Yau condition in the form (18). Thus, together we have the following constraints on our parameters: We will discuss a generalization of these constraints later.
Inserting the formulae ( 15), ( 17) into (14) gives our final 'gravitational block' formula for the trial central charge in supergravity, expressed in terms of the toric data v a A , p a i , and choice of R-symmetry vector field (11), subject to (20).Our resulting expression agrees precisely with the field theory formula in [15], obtained via equivariant localization of the M5-brane anomaly polynomial, and provides a striking confirmation of AdS/CFT.To get the on-shell result, we need to extremize over the choice of R-symmetry; in [6] we show that on the gravity side this is indeed a necessary condition for imposing the supergravity equations of motion.

OTHER OBSERVABLES
As also shown in [1,6], other physical observables may similarly be computed using equivariant localization in supergravity.
For example, consider the four-cycles Γ a i ⊂ M 8 that are the total spaces of S 2 i bundles over the toric divisors D a ⊂ B 4 , where S 2 i ⊂ C i ⊕ R is a linearly embedded twosphere within the fibre S 4 .There are four fixed points, namely the copies of x a , x a−1 at the north and south poles of the fibre S 2 i .In fact x a , x a−1 are also precisely the poles of D a ∼ = S 2 a , with ϵ a 2 and ϵ a−1 1 = −ϵ a 2 being the weights on the tangent space, respectively.Picking for instance i = 1, using localization we compute the flux 2  4 and similarly N a 2 = −q a 1 N S 4 .To get the third line in (21) we substituted (13), (17) and used (15).To get the final line we evaluated the determinants in (15), subject to det(v a−1 , v a ) = det(v a , v a+1 ) = 1, which follow from B 4 being smooth.In (21) q a i ∈ Z is (minus) the integral of the first Chern class of L i through the divisor D a where ) is the intersection matrix of the toric divisors A derivation of (21), using only algebraic topology, can be found in an appendix to [6].We can also compute the dimension of chiral primary operators dual to M2-branes wrapping submanifolds calibrated by ω.These are obtained by applying the BVAB formula to Φ F restricted to Σ 2 [6].Consider the (homologically trivial) S 2 i just considered over a fixed point x a ; if it is calibrated by ω then One can also similarly consider the divisor D a at, say, the north pole of the S 4 , and if this is calibrated by ω then (with an appropriate orientation choice) The same expression holds for the divisor D a at the south pole of S 4 , up to an overall sign related to the choice of orientation.Here we introduced the intersection of three equivariant Chern classes (see e.g.[16] for a review) Notice that the first terms in (25), as well as (24), are independent of ε A , and indeed match the analogous formulas in the absence of U (1) 2 symmetry on B 4 discussed in [6].In contrast, this is not true of D abc , and in fact ∆(D a ) is linear in ϵ a A .

ORBIFOLDS AND ANTI-TWISTS
The above discussion has some immediate generalizations, and this allows us to make a connection with the results of [16,17].
First, we may replace B 4 by a complex toric orbifold.Here the fixed points x a of the U (1) 2 action are now orbifold points, with tangent space C 2 /Z da , where more generally d a = det(v a , v a+1 ).The latter is positive, with appropriately oriented cyclic ordering of the toric divisors, and is equal to 1 when B 4 is a smooth manifold.
Appropriate factors of 1/d a then enter fixed point formulae, as also explained in [1,6].In particular, the S 4 a cycles over the fixed points x a on B 4 do not belong to the same homology class, but instead the classes ) are all equal.Correspondingly the fluxes through the cycles are not (12) However, from the above remarks the combinations d a N S 4 a are necessarily all equal, and we label these by N S 4 , so that ( 13) is not modified.The same orbifold order appears in the BVAB formula used to compute (14), which generalizes to This result provides a gravitational derivation of the gravitational block conjecture for M5-branes wrapped on toric four-orbifolds in [17].Similarly, the expressions (15) for the weights receive a factor of 1/d a , whereas ( 17) is formally unchanged.The expression (21) for the flux N a i is also formally unchanged, but if B 4 is not smooth the intersection matrix D ab reads with a similar expression for the generalization of (26) that may be found in [16].Second, we may relax the Calabi-Yau condition (7), and in particular (20).This is motivated by the socalled anti-twist, discovered in [18] as a novel way to preserve supersymmetry for D3-branes wrapped on a twodimensional orbifold known as a spindle, but which has since been generalized to many other setups.In particular the solutions [19], describing M5-branes wrapping orbifolds B 4 that are the total spaces of spindles fibred over spindles, have been further studied in [16,17,20,21], where it was proposed to relax the second condition in (20) to Here σ a = ±1 is a priori chosen freely for each toric divisor.Again using (17) one can check that the following identity now holds, for each a = 1, . . ., d: From the discussion in the appendix, we may interpret this as a necessary condition for the spinor ϵ to have R-charge 1  2 , but where the projection conditions on the spinor at different fixed points now depend on the choice of σ a .In particular, the chirality of the spinor at x N a is determined by the sign of σ a σ a+1 .This change of chirality at different fixed points is understood in detail for the spindle [22], and indeed the motivation for introducing σ a in [17] was precisely that this describes the known supergravity solutions where B 4 is a spindle fibred over another spindle [19].
It would be interesting to understand better what global constraints there are on the choice of projection conditions in a general setup, but we leave this for future work.
with weight 1, and ∂ ψ A are a lift of the generators of the torus isometry of B 4 to M 8 .For generic b i , ε A , the fixed points of the action of ξ on M 8 , where ∥ξ∥ = 0, are isolated, as noted above.Concretely, the U (1) 2 action on the S 4 has two fixed points, at the north and south pole.If we take the U (1) 2 isometry on B 4 to have d isolated fixed points we then have a total of 2d fixed points on M 8 .These are labelled by (N/S, a), where N/S refers to the north or south pole of S 4 , and a = 1, . . ., d labels the isolated fixed points on B 4 .

d a=1 c 1 ( 2 i=1
L a ).Since also by definition L i = − d a=1 p a i L a , imposing the equivariant version of equation (7) gives d a=1