Equivariant localization and holography

We discuss the theory of equivariant localization focussing on applications relevant for holography. We consider geometries comprising compact and non-compact toric orbifolds, as well as more general non-compact toric Calabi-Yau singularities. A key object in our constructions is the equivariant volume, for which we describe two methods of evaluation: the Berline-Vergne fixed-point formula and the Molien-Weyl formula, supplemented by the Jeffrey-Kirwan prescription. We present two applications in supersymmetric field theories. Firstly, we describe a method for integrating the anomaly polynomial of SCFTs on compact toric orbifolds. Secondly, we discuss equivariant orbifold indices that are expected to play a key role in the computation of supersymmetric partition functions. In the context of supergravity, we propose that the equivariant volume can be used to characterise universally the geometry of a large class of supersymmetric solutions. As an illustration, we employ equivariant localization to prove the factorization in gravitational blocks of various supergravity free energies, recovering previous results as well as obtaining generalizations.


Introduction
The theory of equivariant co-homology is a rich mathematical subject, with numerous applications in diverse areas of geometry and mathematical physics. A remarkable consequence of this theory is that quite generally, on manifolds endowed with a Hamiltonian action of a Lie group, there exist localization formulas that express certain integrals in terms of contributions arising from the fixed point sets of the group action, thus simplifying enormously their evaluation. A prime example of this feature is the classic result of Duistermaat-Heckmann [1]. More generally, such integrals involve equivariant characteristic classes and may arise in the evaluation of equivariant indices of transversally elliptic operators [2,3]. See for instance the review articles [4,5], or [6] for a discussion perhaps more accessible to physicists. In this paper we will need an extension of these results to the setup of orbifolds [7], including spindles and other orbifolds with singularities in co-dimension less than two, that are not so often considered in the physics literature but have recently received attention in the context of holography.
A very general framework in which the ideas of equivariant localization are realized is that of symplectic toric geometry. In this case one starts with a symplectic manifold (M 2m , ω) in dimension 2m, equipped with an effective Hamiltonian action of the real torus T m . The image of the associated moment maps is a convex rational polytope P ⊂ Q m [8].
Together with the angular coordinates on the torus, φ i ∼ φ i +2π, the moment maps y i can be used as "symplectic coordinates", endowing the manifold with a natural coordinate system (y i , φ i ) in which any metric can then be written in a canonical form in terms of combinatorial data of the polytope [9]. In the toric setting the fixed points sets of the T m action are always isolated singularities, so that the localization formulas take the form of sums over fixed points. The applications of the localization theorems in this context range from the quantization of symplectic manifolds to algorithms for computing the volumes of polytopes and counting integral points. See e. g. [10,11]. The extensions to symplectic toric orbifolds was discussed in [12] and that to non-compact toric cones in [13].
In theoretical physics equivariant localization came to the fore in the work of Nekrasov as a technique for calculating partition functions counting instantons in supersymmetric field theories [14]. See e.g. [15] for a review. Applications of toric geometry motivated by the AdS/CFT correspondence were first discussed in [16] and further developed in [17], where the volume functional of toric Sasakian manifolds was shown to be extremized by Sasaki-Einstein metrics. The extension to the more general equivariant setting and the relation to fixed point theorems and the index-character of the associated Calabi-Yau cone singularities was explored in [18]. From the viewpoint of holography, these results can be used to compute the volume and other properties of Sasaki-Einstein manifolds, without explicit knowledge of the metric, from which in turn one can infer properties of the dual field theories. Subsequent developments in geometry include results about: toric Sasaki-Einstein metrics [19,20], Kähler-Einstein metrics [21], extremal Sasaki metrics [22,23], conformally Kähler Einstein-Maxwell metrics [24].
Following on the steps of [17,18] we take holography as a motivation for uncovering precise mathematical relationships between geometry and supersymmetric field theories. In particular, we provide new evidence that toric geometry and equivariant localization are well-suited mathematical frameworks for this purpose. In the context of supergravity our aim is to develop a universal approach to study the geometry underlying supersymmetric solutions based on extremization problems analogous to those formulated in [17] and [25]. We will argue that the functionals to be extremized can be calculated in each case using the technique of equivariant localization, generalizing the results appeared in [18,[26][27][28]. 1 We will consider, in particular, the novel supergravity constructions featuring compact spaces with conical singularities, like spindles and other orbifolds . These solutions imply that we should work in the orbifold setting from the outset and indeed from our results we will obtain a direct localization proof of the factorization in gravitational blocks [54] recently discussed in the literature [35,40,51,55].
In the context of supersymmetric field theory our motivation is that of extending to the orbifold realm some tools that are well-established for field theories compactified on smooth manifolds. Specifically, working with orbifold equivariant co-homology, we wish to put on a sounder mathematical footing the technique of integration of the anomaly polynomials of even-dimensional SCFTs compactified on orbifolds. Furthermore, following [56], where a new index for three-dimensional N = 2 theories was computed exploiting the spindle index-character, we present a discussion of equivariant orbifold indices, that we expect to be key building blocks for computing supersymmetric partition functions of SQFTs on orbifolds, extending Pestun's [57] approach to supersymmetric localization. When the underlying space is (the resolution of) a non-compact Calabi-Yau singularity the same objects have been employed previously to compute Hilbert series of the moduli spaces of supersymmetric field theories [58][59][60]. We expect that new insights can be gained from the study of equivariant indices and their relation to the equivariant volume. We now summarize the structure of this paper.
In section 2 we recall the symplectic geometry description of compact toric orbifolds, following [9,12,61,62]. We introduce the equivariant volume of symplectic toric orbifolds, that is our main object of interest. We discuss two alternative methods for evaluating this, namely the Berline-Vergne fixed point formula and the Molien-Weyl integral formula, that exploits the presentations of the toric orbifolds as symplectic quotients, based on the Lerman and Tolman's generalization [12] of Delzant's construction [8]. Although the topics covered in this section are mainly not original, we consider an extension to non-compact toric orbifolds, which leads to localization formulas for odd-dimensional orbifolds, arising as the base of complex cones thus generalizing the results of [18,63] for the Sasakian volume. Aspects of the equivariant volume of non-compact symplectic toric manifolds were recently studied in [64,65]. In this section we also discuss the equivariant orbifold index of the Dolbeault complex, twisted by a holomorphic line orbi-bundle. In the toric setting, the geometric interpretation of this object is that of counting integer lattice points inside a convex integral polytope (or polyhedral cone, in the non-compact case), corresponding to sections of the line bundle. It can then be thought of as the quantum (or K-theoretical) version of the equivariant volume, which is recovered in a limit in which the lattice spacing goes to zero. Besides this close relationship with the equivariant volume, equivariant orbifold indices are expected to provide fundamental building blocks for the construction of supersymmetric partition functions defined on orbifolds [56], with applications to black hole microstrate counting. We therefore present some examples of these indices in appendix C.
In section 3 we discuss in detail a number of examples of toric orbifolds and their associated equivariant volumes. We start with the complex projective line WP 1 [n 1 ,n 2 ] , also known as the spindle. This is the simplest compact toric orbifold, which arises in complex dimension one. It has a prominent role in several recent supergravity constructions corresponding to various wrapped branes [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49]. Moving to complex dimension two, we consider generic compact toric orbifolds, described by a rational convex polyhedron with an arbitrary number of vertices. We then consider in more detail triangles, namely weighted projective spaces WP 2 [n 1 ,n 2 ,n 3 ] and their quotients by discrete groups, as well as quadrilaterals. Examples of supergravity solutions comprising Hermitian (non-Kähler) metrics on quadrilateral toric orbifolds are discussed in [50][51][52][53]. As a warm-up for section 5, we also describe the pecularities of the non-compact case in a few explicit examples.
Section 4 concerns the application of equivariant localization to the calculation of the anomaly polynomial of 4d and 6d SCFTs compactified on various orbifolds. Using this approach we prove the localized form of the anomaly polynomial for theories compactified on the spindle [33,[35][36][37], as well as on general four dimensional toric orbifolds, for which one example was considered in [50]. Our approach leads to a uniform derivation for SCFTs compactified on different orbifolds and explains the localized form of the integrated anomaly polynomials that was previous observed in examples.
Finally, in section 5 we discuss the application of equivariant localization in the context of supergravity. Firstly, we show that the results of [18] on the localized form of the Sasakian volume are immediately recovered from the equivariant volume of the associated non-compact Calabi-Yau singularity. Furthermore, we prove that also the master volume introduced in [66], in the context of GK geometry [67], can be extracted from the equivariant volume and as a corollary we prove the localized form of the gravitational block formulas for D3 and M5 branes wrapped on the spindle, reproducing the results of [55] in the toric case. We then propose analogous constructions for other branes wrapped on the spindle from which, in each case, we can extract the localized form of the gravitational blocks, which was anticipated in [40]. This leads us to propose that equivariant localization is the common thread of the geometry of supersymmetric supergravity solutions, at least in setups with a holographic interpretation. In particular, we believe that the equivariant volume should be a key object for setting up extremal problems characterising supersymmetric geometries, in different supergravity theories.
In section 6 we discuss our findings and indicate some directions for future work. The paper contains three appendices with some complementary material.

The equivariant volume 2.1 The symplectic geometry description of toric orbifolds
In this section we review the geometry of symplectic toric orbifolds following the general formalism developed in [9,12,61,62]. We emphasise that although it appears that we are relying on symplectic geometry, we will be interested in computing topological quantities that do not depend on the existence of an integrable symplectic structures and it should be possible to reformulate our computations entirely in terms of complex geometry. In particular, our results are applicable also to situations in which one is interested in metrics that are not compatible with a symplectic structure, such as the Hermitian (non-Kähler) metrics on toric orbifolds constructed in [50][51][52][53].
We consider a symplectic toric orbifold (M 2m , ω) in dimension 2m equipped with an effective Hamiltonian action of the real torus T m = R m /2πZ m . We introduce symplectic coordinates (y i , φ i ) with i = 1, . . . , m where φ i are angular coordinates on the the torus, φ i ∼ φ i + 2π. In terms of these coordinates the symplectic form is given by 2 ω = dy i ∧ dφ i . (2.1) By a generalization of Delzant's theorem [8], compact symplectic toric orbifolds are classified by labelled polytopes which are the image of M 2m under the moment maps y i associated with the toric action [12]. The image of M 2m is simply obtained by forgetting the angular coordinates φ i and it is a rational simple convex polytope P in R m . We can describe it by introducing a set of linear functions 3 where v a are vectors in R m . The convex polytope is the subset of R m defined by The linear equations l a (y) = 0 (2.4) define the facets F a of the polytope. We denote with d the number of facets of P.
The condition that P be rational is equivalent to the fact that the vectors v a have integer entries. The condition that P be simple requires that each vertex p lies at the intersection of precisely m facets l a 1 (p) = l a 2 (p) = · · · = l am (p) = 0 , (2.5) and the corresponding m vectors {v a 1 , . . . , v am } are a basis for R m . The polytope comes equipped with a label for each facet, a positive integer n a such that the structure group of every point in the inverse image of F a is Z na . This construction exhibits M 2m as a torus fibration over the polytope P and generalises the familiar construction for toric varieties. As in the latter case, the torus fibration is non-degenerate in the interior of P. A particular one-cycle in T m , determined by the vector v a , collapses at the facet F a . Thus each facet F a defines a symplectic subspace of M 2m of real codimension two. In the complex case this becomes a divisor 4 which we will denote D a . Similarly, at the intersection of q facets, more cycles in T m degenerates and we have a symplectic subspace of M 2m of real codimension 2q. The intersection of m facets is a vertex of the polytope and it corresponds to a fixed point of the T m action. We denote with n the number of such fixed points. In particular, we can give an alternative definition of the polytope P as the convex hull of the images of the fixed points of M 2m .
In the context of toric varieties, the vectors v a define what is called the fan of M 2m . We will use the same terminology but the reader should be aware that we are dealing with a generalization of the concept which allows for more general type of orbifold singularities.
For toric varieties, the vectors v a are primitive, while in the case of generic symplectic toric orbifolds they are not. We can always define for each v a a primitive vectorv a and a positive integer n a such that v a = n av a . 5 The integer n a is precisely the label of the facet F a defined above. In particular, each symplectic divisor D a has a local Z na singularity. This cannot happen for toric varieties which are normal and have no singularities of complex co-dimension less than two. Notice also that the local singularity at the fixed point given by the intersection of the m facets F a i , with i = 1 . . . m, has order d = | det(v a 1 , . . . , v am )|. M 2m is a smooth manifold if and only if all the labels n a are one and for each vertex | det(v a 1 , . . . , v am )| = 1.
We want to equip M 2m with a compatible complex structure. Any T m -invariant Kähler metric on M 2m is of the form [61] where G ij is determined by a symplectic potential G(y) as and G ij = (G −1 ) ij is the inverse matrix. Holomorphic coordinates are given by The existence of a symplectic potential is equivalent to the integrability of the complex structure.
The canonical metric on M 2m is given by [9] G(y) = 1 2 d a=1 l a log l a , (2.9) so that (2.10) Notice that G ij has poles at the facets but the metric is smooth up to orbifold singularities. The inverse matrix G ij has rank m − 1 at the facets indicating that a one cycle in T m is degenerate. With a change of coordinates we obtain a smooth orbifold metric on F a . The structure of poles of (2.9) is compatible with the degeneration of T m at the faces of P and it is precisely designed to obtain an orbifold metric on M 2m . The most general Kähler metric on M 2m is discussed in [62] and it is obtained by replacing the symplectic potential G(y) for the canonical metric with G(y) + h(y), where h(y) is smooth on the whole P. The topological quantities we will discuss in this paper do not depend on the metric and we can safely set h(y) = 0. In our applications to holography we will encounter metrics that are not Kähler and not even symplectic, but the underlying spaces are in fact symplectic toric orbifolds and we can therefore use the symplectic coordinates and the canonical metric to compute topological quantities that ultimately will not depend on the metric.
Each facet F a defines a T m -invariant divisor D a and an associated line bundle L a . These objects will be important for our construction so we will spend some time discussing their properties. The first Chern class of L a has been explicitly computed in [9,62] where with [α] we denote the co-homology class of the differential form α. An explicit representative is given by 7 (2.14) Notice that µ i a can be seen as a moment map for the torus action We can relate the Chern classes of the divisors to the Kähler form ω as follows. From we obtain Using the degeneration of G ij at the facets, one can check that a G ij v a j dφ i is a welldefined one-form, so that the last term on the right hand side of (2.19) is exact. Therefore we obtain the important relation (see Theorem 6.3 in [9]) (2.20) 6 We believe that there is a minus sign error in equation (6.17) in [9] and we have therefore changed the sign here. 7 Writing as in (2.8), and, therefore, for a torus invariant function f that only depends on y With some abuse of language, we will often denote with c 1 (L) an explicit representative of the corresponding co-homology class.
Notice that this equation holds only in co-homology. We see that the parameters λ a defining the shape of the polytopes through (2.3) are parameterizing the Kähler moduli of the symplectic orbifold.
There are actually only d − m independent line bundles. Indeed, using (2.13) and (2.16), we find m relations among the Chern classes In this paper we will also consider non-compact cases, in particular non-compact Calabi-Yau cones. In this case, the polytope is replaced by a rational convex polyhedral cone. Calabi-Yau cones have typically singularities worse than orbifold. This happens when more than m facets intersect at a vertex. To use the general formalism of this section we will perform a partial resolution to have only orbifold singularities. In applications to holography, we will also encounter polytopes and polyhedral cones that are not convex. We will obtain results by performing a suitable extrapolation from the convex case.

The definition of the equivariant volume
In this section we define the equivariant volume. In order to simplify the exposition, in this and the next two sections we assume that M 2m is compact. We will discuss the subtleties and necessary modifications for the non-compact case in section 2.5.
We want to work equivariantly in the T m action on M 2m , which is generated by the m vector fields ∂ φ i . Correspondingly we introduce m equivariant parameters ǫ i , with i = 1, . . . m and the vector field ξ = ǫ i ∂ φ i . We can introduce a Hamiltonian H = ǫ i y i for this vector field and define an equivariant Kähler form We similarly introduce equivariant Chern classes for the line bundles L a using the moment maps µ a . All these forms are equivariantly closed which is sometimes referred in the literature to as "symplectic volume" or "equivariant symplectic volume". We can write the equivariant volume as Notice also that the two integrands in (2.29) are not equal but they differ by an equivariantly exact form by (2.20) and the integrals are therefore equal. Notice that the last inequality strictly holds only if M 2m is compact. We will return to this point in section 2.5. A geometrical interpretation of the equivariant volume of compact orbifolds is that it is the generating functional for the integrals of the equivariant Chern classes The equivariant intersection numbers are polynomials in ǫ i and they are topological in nature. Notice that the integrand in (2.31) is a formal linear combination of forms of various degree and the integral selects the piece of degree 2m. In particular, the equivariant intersection numbers are different from zero for all p ≥ m. The expression in (2.27) can be easily reduced to an integral over the polytope P by performing the angular integrations Thus we have another interpretation of the equivariant volume as the volume of the polytope P with measure e −H . Integrals of this type and their relation to equivariant localization are discussed in [11]. The equivariant volume satisfies some interesting identities. From the expression (2.29) e a λa(c 1 (La)+2πǫ i µ a i ) (2.33) and the identities (2.21) and (2.16) we find that the identity holds, for arbitrary β i ∈ R m . This formula reflects the fact that only d − m parameters λ a are geometrically independent. A closely related and useful identity can be obtained by taking derivatives of (2.33) and using again (2.21) and (2.16) In the next sections we review two different methods to evaluate V(λ a , ǫ i ), the fixed point and the Molien-Weyl formula. They are both discussed in the literature. Here we adapt them to our notations and we discuss the relations among them.

The fixed point formula
The equivariant volume can be computed using a fixed point formula. This can be obtained from (2.27) using the Duistermaat-Heckman theorem [1], the localization formula for equivariant co-homology [68,69], or a direct evaluation of (2.32) [10,11]. Here we use the localization approach and, for completeness, in Appendix A we report an explicit proof for m = 2.
Consider a symplectic toric orbifold M 2m with the properties discussed in section 2.1. The torus action T m acts on M 2m with n isolated fixed points corresponding to the vertices of the polytope P. Consider an equivariantly closed form α T on M 2m . The equivariant localization theorem for orbifolds applied to our situation states that where the sum is over the fixed points y A of the T m action, e T is the equivariant Euler class of the tangent bundle at y A and d A are the orders of the orbifold singularity at the fixed point y A . In particular, applying this theorem to the computation of the equivariant volume (2.29) gives To evaluate the localization formula we need to compute the restriction c T 1 (L a )| y A of the equivariant Chern classes of L a to the A-th fixed point. The fixed point y A is defined by m linear equations l a 1 = . . . = l am = 0 (2.39) for a choice of m intersecting facets associated with the vectors {v a 1 , . . . , v am }. The order of the orbifold singularity is given by We will denote the A-th fixed point also with the m-tuples of indices A = (a 1 , . . . , a m ) defining the vertex. In the neighbourhood of the fixed point A = (a 1 , . . . , a m ) up to regular pieces. This can be inverted to give where the vectors u a i A are constructed by taking the wedge product of the m − 1 vectors v a j with j = i. The sign ambiguity is determined by requiring u a i A · v a i = d A . The vectors u a i A have integer entries and satisfy They have a dual geometrical interpretation as the inward normal vectors to the facets of the cone (v a 1 , . . . , v am ) in the fan or, equivalently, as integer vectors along the m edges of the polytope P meeting at y A . Notice that the m-tuplet of vectors u a i A depends on the choice of vertex A.
The restriction of the moment maps (2.14) to the fixed points then gives and, therefore Notice that the restriction of c T 1 (L a ) to a fixed point is not zero only if the point belongs to the facet F a .
Finally the tangent bundle at a fixed point splits as a direct sum of the m line bundles L a i associated with A = (a 1 , . . . , a m ) and so we have [18] Now the fixed point formula gives where A runs over the n vertices of the polytope. This formula can be also obtained by computing the volume of the polytope P, as discussed, for example, in [11]. We note on passing that the identities (2.34), (2.35) and (2.36) can be also derived from the fixed point formula. For example,

The Molien-Weyl formula
The equivariant volume can be also computed using a Molien-Weyl integral formula following [64,65].
The orbifold M 2m can be realized as a symplectic quotient. Consider the space C d , where d is the number of vectors in the fan, and the subgroup K of T d of elements of the form e 2πiQ 1 , . . . , e 2πiQ d ∈ T d , Then M 2m is the symplectic reduction [12,62] M 2m = C d //K , (2.52) generalizing a familiar result in toric geometry. Notice that, in general, the group K has a continuous part as well as a discrete part of more complicated characterization. Consider first the case where there is no discrete part in the quotient and M 2m = C d //U(1) d−m . Using the results in [64,65], we can write a Molien-Weyl formula for the equivariant volume . (2.55) A particular contour of integration should be used depending on the direction of the vector of Kähler parameters t A . A prescription based on the Jeffrey-Kirwan (JK) residue [70] has been proposed in [64,65]. Formula (2.55) should be also divided by the order of the discrete group corresponding to the torsion part of K when this is present. We will discuss subtleties related to the choice of contour and discrete groups when illustrating examples. Notice that V M W depends on d − m Kähler parameters t A , which is the right number of geometrically inequivalent parameters for M 2m . It also depends on d equivariant parametersǭ a associated to the T d action on the ambient space, which is larger than the m parameters that we expect for M 2m . However, d − m equivariant parameters can be eliminated by shifting the integration variables. Indeed, up to an exponential factor, the Molien-Weyl integral is invariant under the shift for η A ∈ R.
We thus have two expressions for the equivariant volume, V(ǫ i , λ a ), depending on m equivariant parameters ǫ i and d Kähler parameters λ a but with the "gauge" invariance (2.34), and V M W (ǭ a , t A ), depending on d equivariant parametersǭ a and d − m Kähler parameters t A but with the "gauge" invariance (2.56). The two set of parameters can be related by comparing gauge invariant quantities. Based on examples, we find that and that the two expressions for the equivariant volume agree up to a multiplicative factor when expressed in terms of the over-redundant variables (λ a ,ǭ a ) which is also zero because of the identity (2.35) and the identification (2.57).

Remarks on the non-compact case
Most of the previous results hold also for non-compact orbifolds, but with some important differences. We still define the equivariant volume as which we can also write as an integral over a polyhedron P V(λ a , ǫ i ) = P e −ǫ i y i dy 1 . . . dy m . (2.61) Since M 2m and P are non-compact, we need to assume that the integrals are convergent. This happens for the examples we will be interested in, where P is asymptotically a cone. In this case the Hamiltonian H = ǫ i y i acts as a convergence factor, at least if the vector ǫ lies inside the cone. The fixed point formula and the Molien formula hold under general conditions also for non-compact orbifolds, and we can even use them as an operative definition for the equivariant volume. For example, the fixed point formula only assumes that there are isolated fixed points in P and that there are no contributions from infinity. Notice that the identities (2.34), (2.35) and (2.36), which follow from the fixed point formula, are still valid.
However, the co-homological interpretation (2.30) fails. In particular, the two integrals in (2.29) are not equal in general. The second term in (2.19) is still exact, but it cannot be ignored in a integral over non-compact orbifolds. 8 As a result, the equivariant volume can be still formally expanded in power series of λ a but the coefficients cannot be straightforwardly interpreted as generalized intersection numbers as in (2.30). In particular, while in the compact case, the power series of λ a starts at order m and the coefficients are polynomials in ǫ i , in the non-compact setting the power series has all coefficients different from zero and these are in general rational function of ǫ i , as it follows from the fixed point formula. We will discussed explicit examples of non-compact orbifolds in sections 3.3 and 5.

Relation to the equivariant orbifold index
In this section we briefly discuss how the equivariant volume arises as a limit of an index character, following the logic in [18,64,65]. Examples of calculations of characters are provided in the appendix C as this is not the main theme of this paper. The equivariant index for the twisted Dolbeault complex on M 2m is defined by where q ∈ T m is an element of the torus and Λ a ∈ Z, a = 1, . . . , d are integers specifying a choice of line bundle. The trace is taken on the induced action on the co-homology. The index can be computed with the Hirzebruch-Riemann-Roch theorem. We start with the smooth case that is considerably simpler and we take all the labels to be one, the vectors v a to be primitive and all the orders d A = 1. The fixed point formula for the equivariant index is where the symbol q n is an abbreviation for q n 1 1 . . . q nm m . The geometrical interpretation of the equivariant index is generically to count integer points as discussed, for example, in [11]. The equivariant volume can be obtained by setting q i = e − ǫ i and taking the limit → 0. The limit of (2.64) is singular but exhibits the equivariant volume as the coefficient of the leading pole Analogously, the Molien-Weyl formula reads [64,65] where the contour is defined again through the JK prescription. Settingq a = e − ǭa , The Molien-Weyl formula (2.68) can be interpreted as an average over the complexified group U(1) d−m acting on the coordinates of the ambient space C d . The continuous average in (2.68) should be supplemented by a discrete average if the group K appearing in the symplectic reduction M 2m = C d //K has a torsion part. The formulas (2.64) and (2.68) are familiar in the context of non-compact conical Calabi-Yau singularities. For Λ a = T A = 0 they have been used to compute Hilbert series for the mesonic moduli space of the dual N = 1 superconformal theories [58,59]. For Λ a , T A = 0 they compute Hilbert series for the baryonic moduli space [60].
The relation between the two formulas is (2.70) The corresponding fixed point formulas for orbifolds are more complicated [7,71]. Each fixed point contribution is replaced by a discrete Molien formula implementing the quotient , is a generator of the local orbifold singularity Z d A . In the limit → 0 only the term with k = 0 contributes at the leading order and we recover the fixed point formula (2.48) as the coefficient of the most singular term (see (2.67)).
In the special case of non-trivial labels but no extra orbifold singularities we can use the formulas in Appendix C to resum (2.71) and we obtain where the floor function ⌊x⌋ denotes the integer part of x, v a = n av a and u a = n aû a have been decomposed into primitive vectorsv a ,û a and the label n a and we assume that d a,a+1 = det(v a ,v a+1 ) = 1. One can check with elementary methods that this formula computes indeed the number of integer points in the polytope (2.66).
3 Examples from geometry and holography

The spindle
The first example that we consider is the weighted projective line WP 1 [n + ,n − ] , namely the spindle Σ, which is at heart of the recent novel supergravity constructions. It can be defined as the set of pairs of complex numbers (z 1 , z 2 ) under the identification This is the most general toric symplectic orbifold in two real dimensions. It is the simplest example of toric orbifold that is not a toric variety, as its singular points occur in complex co-dimension one. In particular, the orbifold points, that are fixed under the U(1) action, are also the divisors and therefore d = n = 2. The image of Σ under the moment map is simply a segment P = I, exactly as for P 1 ≃ S 2 and the two facets F a are its end-points, defined by the linear equations where the one-dimensional non-primitive vectors are v 1 = n 1 , v 2 ≡ −n 2 , where we will also denote n 1 ≡ n + and n 2 ≡ n − . 9 The two labels are n + , n − ∈ N. Thus Σ is a circle fibration over the segment I, with the circle collapsing at the end-points, where are the two C/Z n + , C/Z n − singularities, see figure 1. In the following it will be convenient to define two rescaled Kähler parameters λ 1 = n +λ1 , λ 2 = n −λ2 , so that l 1 (y) = n + (y −λ 1 ) , l 2 (y) = −n − (y +λ 2 ) . (3.3) 9 We use this notation to facilitate comparison with the spindle literature in supergravity. Figure 1: The spindle Σ as a circle fibration over a segment.
We can then write explicitely the one-dimensional (canonical) metric (2.10) 4) and the related moment maps In the conventions of section 2.1, in which l a (y) ≥ 0, we have that 10 y ∈ [y 1 , , with the extrema of the interval corresponding to the north and south poles of Σ, respectively. We can then write all the relations discussed in section 2.1 explicitly in terms of the moment maps (3.5). The explicit representatives of the first Chern classes c 1 (L a ) associated to the line bundles L a defined by the extrema of the interval (the two divisors) are where φ ∼ φ + 2π is the azimutal coodinate, and denoting by ǫ the equivariant parameter, their equivariant versions are The equivariant Kähler form is where H = ǫy is the Hamiltonian for the vector field ξ = ǫ∂ φ . With this information, we can check various relations explicitely, without appealing to the fixed point theorem. For example, it is immediate to compute The co-homological relation (2.20) can also be verified explicitly, noting that where Φ(y) = (n + − n − )(y +λ 2 )(y −λ 1 ) is such that Φ(y a ) = 0, implying that Φ(y)dφ is an exact one-form on the spindle. Let us now turn to the fixed-point relations and the equivariant volume. From (3.5) it immediately follows that the restriction of the moment maps to the fixed points reads where u 1 = 1, u 2 = −1 and therefore Using these and the fact that the equivariant Euler class of the tangent bundle at a fixed point y a is one can reproduce the integrals (3.9) from the fixed point theorem (2.37). The equivariant volume can be easily computed either directly 15) or using the fixed point formula which of course gives the same result. From this it is immediate to compute the non-zero equivariant intersection numbers (2.31), generalizing (3.9) to s > 1.
We end this subsection noting that it is straightforward to check directly that the equivariant volume depends only on the co-homology class of [ω]. Specifically, if we use a different representative of [ω], such as where the last equality follows trivially from a change of variables and the fact that Φ(y a ) = 0.

Four dimensional toric orbifolds
We now move to compact four-dimensional orbifolds (m = 2), that have applications in holography [50][51][52], as well as in geometry [72][73][74]. The image of M 4 under the moment map is a rational simple convex polygon P in R 2 and its facets F a , defined by the linear equations l a (y) = 0, are the edges of the polygon, as shown in figure 2. Of course each vertex p a of the polygon lies at the intersection of precisely 2 edges, so that the simplicity condition is automatic. Moreover, the number of facets/edges d of P coincides with the number of vertices n, that are also the fixed points of the T 2 action. The condition that P be rational is equivalent to the fact that the d normals to the edges, v a , have integer entries; however, they do not need to be primitive. The highest common factor n a of the two entries of each of the v a is precisely the label associated to an edge/facet Each facet defines a symplectic subspace of M 4 of real dimension two, namely a divisor D a of complex dimension one, with a local C/Z na singularity. The vertices p a of P correspond precisely to points where two divisors D a , D a+1 intersect, with local singularity The equivariant volume and equivariant intersection numbers can be conveniently written in terms of the quantities where u i a are the inward normals to the cones (v a , v a+1 ) as discussed in section 2.3 and pictured in figure 2. We can rewrite these as where ǫ ≡ (ǫ 1 , ǫ 2 ). In writing these formulas, we have assumed that the vectors v a lie on the plane in anticlockwise order and we have identified cyclically v a+d ≡ v a . In particular, the equivariant Euler class of the tangent bundle at a fixed point y a reads The restriction to the fixed points of the equivariant Chern classes c T 1 (L a ) (2.46) can be written as so that the general formula for the equivariant intersection numbers reads (3.26) Below we report explicit formulas for intersection numbers up to p = 4. First of all, for p = 0, 1 the integrals vanish automatically, giving rise to identities that may also be verified with elementary algebra. Specifically, we have (3.28) For the double intersection numbers we reproduce the intersections matrix of divisors (see e.g. [51]), which is independent of the equivariant parameters ǫ 1 , ǫ 2 : where we used the identity to deal with the terms occurring in the self-intersection D a ·D a . Similarly, we can compute the integral of three equivariant first Chern classes, that read: where the diagonal term can be shown to be linear in ǫ i , using the identity For the integrals of four equivariant first Chern classes we have where the diagonal term can be shown to be quadratic in ǫ i , using the identity which is easily evaluated in examples. On the other hand, starting from the definition (2.27), the equivariant volume can also be written as in integral over the polygon P as that can be evaluated explicitely using Stokes' theorem. Notice that the dependence on the Kähler parameters λ a arises from the shape of P = P(λ a ). The details of this calculation are given in appendix A.
As a first example of equivariant volume for four-dimensional toric orbifolds we consider the weighted projective space WP 2 [N 1 ,N 2 ,N 3 ] defined as the set of triples of complex numbers (z 1 , z 2 , z 3 ) under the identification There are various presentations in terms of labelled polytopes [62] and it is interesting to consider some of them in order to understand better the role of the labels.
Consider first the fan where each of the three facets have non-trivial labels. We take the labels n a to be mutually coprime. The GLSM charges are given by 39) and the symplectic quotient description WP 2 where U(1) acts with charge Q, is just the definition (3.37) of the weighted projective space. Notice however that our choice of fan corresponds to "non-minimal" Q a = N a that are products of integers.
The equivariant volume can be computed with the fixed point formula (3.35) This expression is a rational function of ǫ i but when expanded in power series in λ a as where V (k) is the component of degree k in λ a , all singular terms cancel. The constant and linear terms vanish and the other V (k) (λ a , ǫ i ) are homogeneous polynomials of degree k − 2 in ǫ that encode the equivariant intersection numbers of the line bundles L a . In particular, V (2) (λ a , ǫ i ) coincides with the non-equivariant limit V(λ a , 0) = 1 2 n 1 n 2 λ 1 + n 2 n 3 λ 2 + n 1 n 3 λ 3 n 1 n 2 n 3 2 = 1 2 encoding the classical intersections numbers of the divisors D a . Since only one divisor is geometrically independent because of (2.22), this is a quadratic form of rank one. We can compare the result with the Molien-Weyl formula (2.55), where we use the charges Q. The JK prescription for a single integration is very simple. 12 It prescribes to take all residues associated with φ with positive charge for t > 0, and minus the residues for φ with negative charge for t < 0. Taking t > 0 and performing the residue computation, we obtain
with non-equivariant limit We see that each residue corresponds to a fixed point. It is easy to see that the two master volumes coincides under the identification (2.58) It is interesting to consider also the fan where now the GLSM charges are the "minimal" ones However in this case there is also a torsion part in the symplectic quotient description. Indeed the symplectic action on C 3 is given by For the minimal fan (3.47) with relatively prime n a , the relations have solution (3.52) One k i can be further gauged away by choosing α = integer/n 3 (assuming again that the n a are relatively prime). So we are left with a continuous U(1) action generated by α and an extra discrete group 13 As an exercise, the interested reader can check that the relation with an extra factor of N 3 . Finally, we can consider an example taken from [62] 14 . The fan is 57) and the GLSM charges are the "minimal" ones have solution and one k i can be further gauged away by choosing α = integer/(n 1 n 2 n 3 ). So we are left with a continuous U(1) action generated by α and an extra discrete group so that is a discrete group of order N 1 N 2 N 3 . We still find where the Molien-Weyl integral should be further divided by N 1 N 2 N 3 . The non-equivariant volume is 1 2 (3.64) 14 The reader should be aware that, in [62], the weighted projective space is denoted with WP 2 (N1,N2,N3) and the example we are discussing with WP 2 [N1,N2,N3] .

Quadrilaterals
Let us now discuss the most general four-dimensional toric orbifold with four fixed points, that we refer to as quadrilaterals, following [73]. A sub-class of this family of orbifolds has been discussed in [51] and below we will discuss how one can retrieve those results. It is convenient to parameterize the data of this orbifold in term of the six vector products where v a , a = 1, . . . , 4 is the set of toric data, as in figure 3. Then a vector identity implies that these satisfy the following relation showing that there are five independent integers characterising a quadrilateral. One can show that the charges of the U(1) 2 action on C 4 can be written in a particularly symmetric form using this parameterization, and read Without loss of generality, one can use an SL(2, Z) transformation to set v 1 = (n − , 0), where n − ∈ Z. The remaining vectors solving the constraint are then easily determined and the full set reads where d 1,2 , d 1,3 , d 4,1 are integer multiples of n − and a ± ∈ Z such that which always exist by Bezout's lemma. The equivariant volume V(λ a , ǫ i ) is easily computed using the general fixed point formula (3.35), but we shall not write it explicitly as it does not have a compact expression. Expanding it in Taylor series in powers of λ a as in (3.41), one can check that V (0) (ǫ i ) = V (1) (λ a , ǫ i ) = 0, whereas the quadratic part is independent of ǫ i and reads where D ab is the intersection matrix of divisors (3.29), which reads (3.72) It may be useful to compare with the orbifold geometry studied in [51] (see also [50]), which may be viewed as spindle fibered over another spindle, and interpreted as a natural orbifold generalization of Hirzebruch surfaces. This is obtained setting and For these values the vectors of the fan reduce to those written in eq. (4.27) of [51] and the GLSM charges become We can then perform the U(1) 2 quotient of C 4 in two stages. The first quotient using weights Q 1 gives the line bundle O(−t) → Σ m + ,m − . The second quotient using weights Q 2 projectivize this bundle, giving a Σ n + ,n − → Σ m + ,m − bundle. Indeed for n + = n − = m + = m − = 1 this reduces exactly to the Hirzebruch surface F t , which is the most general family of toric manifolds with four fixed points. It is instructive to compare with the Molien-Weyl formula. We do it explicitly for the case of the Σ n + ,n − → Σ m + ,m − bundle. The Molien-Weyl integral reads .
(3.75) The multiplicative factor n − takes into account that the U(1) 2 action is not effective since acts in the same way on all points in C 4 . The Molien formula mods out twice by a discrete subgroup Z n − and we need to multiplicate by n − to obtain the right result. The poles in the integrand are associated with two-dimensional vectors Q a = (Q 1 a , Q 2 a ), a = 1, 2, 3, 4. The JK prescription instructs us to take the (simultaneous) residues at the poles Q a 1 and Q a 2 if the vector of Kähler parameters (t 1 , t 2 ) is contained in the cone (Q a 1 , Q a 2 ). When Figure 4: The JK prescription for the Σ n + ,n − → Σ m + ,m − bundle.
(t 1 , t 2 ) lies in the first quadrant, it is contained in four different cones (see figure 4). In the Kähler chamber t 1 > 0 and t 2 > 0 we then add four contributions. One can check that where the gauge invariant variables (2.57) are The non-equivariant volume is (3.79)

Non-compact examples
We consider now some non-compact examples where the polytope P is asymptotically a polyhedral cone. We compute the volume using the fixed point formula, assuming that there is a finite number of fixed points and no contributions from infinity. As anticipated in section 2.5, the equivariant volume is a rational function of ǫ i . As we will see, two of the singular terms for the Calabi-Yau examples can be identified with the Sasakian volume of [17,18] and the master volume introduced in [66]. This identification will be discussed in more detail in section 5.

C 2 /Z p
Our first non-compact example is C 2 /Z p , which is a conical orbifold singularity. The most general toric action is obtained using as fan the vectors The corresponding Z p action on (z 1 , z 2 ) ∈ C 2 is given by where the GLSM charges are pure torsion, namely a Q a v a ∈ Z 2 , (3.82) whose solution is, for p and q coprime, given by This is then the C 2 /Z p quotient with ω p = e 2πi p , which for |z 1 | 2 + |z 2 | 2 = 1 gives the Lens space L(p, q). The equivariant volume is obtained from the general formula (3.35), and is entirely encoded by the the contribution of the single fixed point with Z p singularity, namely Expanding (3.85) in powers of λ a we obtain where the constant and linear terms are non zero, as a difference with the compact case. They are also singular functions of ǫ. The constant piece is actually the Sasakian volume of the Lens space L(p, q) and the linear term is the corresponding master volume introduced in [66]. In particular, for q = 1 that corresponds to the "supersymmetric" Lens space L(p, 1), the linear term coincides precisely with (B.6) of [76].
Since there are no continuous GLSM charges, the Molien-Weyl formula is trivial where there are no Kähler parameters t and we divide by p for the order of the discrete group Z p . Nevertheless, it is still true that where the gauge invariant combinations are Let us now consider the asymptotically conical non-compact orbifold 15 O(−p) → Σ, where Σ = Σ n + ,n − is a spindle with Z n + , Z n − singularities at its poles. This can be thought of as a "blow-up" of the C 2 /Z p of the previous section, where the apex of the cone is replaced with the spindle Σ. In general, this is a partial resolution, but in the special case that n + = n − = 1, it is a complete resolution.
The GLSM charges are given by 90) where notice that the space is a Calabi-Yau if and only if p = n + + n − . In particular, only for p = 2 we have a complete resolution of the CY singularity C 2 /Z 2 . The vectors of the fan solving the condition can be taken to be which always exist by Bezout's lemma. Indeed, we see that the basis (3.92) is obtained from (3.80) by adding the vector (t, n − ), which is normal to the edge representing the spindle, as illustrated in figure 5. The equivariant volume is obtained from the general formula (3.35), and now it receives contributions from the two orbifold fixed points, with Z n − , Z n + singularities, namely 94) and expanding this in λ a we get which coincides with (3.86). Thus we see that up to linear order, the information about the (partial) resolution is washed out as there is no dependence on n − and n + , nor on the Kähler parameter λ 2 , corresponding to the compact divisor Σ. Specifically, the leading and sub-leading terms reproduce precisely the Sasakian and GMS master volume of the Lens space L(p, q). Notice that this behaviour is independent of the Calabi-Yau condition p = n + + n − . Another notable case is given by n + = n − = p = 1, which corresponds to the smooth non-compact space O(−1) → P 1 , namely the blow-up of C 2 at one point 16 . This is not a Calabi-Yau and its link is the round S 3 . We can also compare with the Molien-Weyl formula , (3.96) where, according to the JK prescirption, for t > 0 we take the residues for the terms with positive charge Q i , φ = −ǭ 1 /n + and φ = −ǭ 3 /n − . We find, as usual where the gauge invariant combinations are The corresponding polytope is a conical non-compact polyhedron with four facets all intersecting at the tip of the cone. Since the vectors v a lie on a plane, or, equivalently a Q a = 0, this defines a conical Calabi-Yau three-fold. In order to use the results of section 2 we need to resolve the conifold singularity. This can be done in a standard way by triangulating the fan as in figure 6. The fan is now the union of the two cones (v 1 , v 2 , v 4 ) and (v 2 , v 3 , v 4 ), each associated with a vertex of the polytope and therefore with a fixed point of the torus action. The original conical singularity has been replaced by a compact two cycle, which can be visualised as a circle fibration over the segment connecting the two vertices of the polytope. There are actually two small resolutions related by a flop. The second one is obtained by triangulating the fan in a different way, by adding the line v 1 − v 3 and considering the fan which is the union of the two cones (v 1 , v 2 , v 3 ) and (v 1 , v 3 , v 4 ). 16 In this case we have t − q = 1 and therefore we can take t = 1, q = 0, reproducing the obvious fan v 1 = (1, 0), v 2 = (1, 1), v 3 = (0, 1).
where A runs over the triangular cones of the resolution, d a 1 ,a 2 ,a 3 = | det(v a 1 , v a 2 , v a 3 )| is the order of the orbifold singularity at the fixed point and u a A are the inward normals to the faces of the cones as in figure 7.
At λ a = 0 we recover the Sasakian volume of the 5-dimensional base of the singular cone (cfr (7.30) in [18]) , (3.103) and the quadratic part is, up to a normalization, the master volume introduced in [66] 17 (3.104) Things work similarly for the second resolution (a 1 , a 2 , a 3 ) = (1, 2, 3) , a 2 , a 3 ) = (1, 3, 4) , which is different from (3.102). The difference between the equivariant volumes for the two resolutions starts at order three in the Kähler parameters . (3.107) We need to specify a prescription for the contour and the residues to take, which depends on the sign of t and the choice of resolution. The JK prescription requires to take the residues where the charge Q is positive for t > 0 (first resolution) and minus the residues where the charge Q is negative for t < 0 (second resolution). For example, for t > 0 we take the two residues φ = −ǭ 1 and φ = −ǭ 3 , obtaining while for t < 0 we take (minus) the residues associated with negative charge, φ =ǭ 2 and φ =ǭ 4 . It is easy to check that in both cases where the gauge invariant variables explicitly read t = −λ 1 +λ 2 −λ 3 +λ 4 , ǫ 1 =ǭ 1 +ǭ 2 +ǭ 3 +ǭ 4 , ǫ 2 =ǭ 2 +ǭ 3 , ǫ 3 =ǭ 3 +ǭ 4 . (3.110)

Integrating the anomaly polynomial on orbifolds
In this section we will consider field theories associated to D3 and M5 branes and we will show how the formalism of equivariant integration can be employed to determine the anomaly polynomials of lower dimensional theories, obtained compactifying the original theories on orbifolds. For theories compactified on the spindle these results were obtained in [33,[35][36][37], but the equivariant formalism allows for uniform derivations for various branes wrapped on different orbifolds. In the case of theories compactified on manifolds the method of integration of the anomaly polynomial is well-known and is reviewed in [77], to which we refer for more details. In this reference is also discussed the extension in which background gauge fields for the isometries of the compactification manifolds are turned on, and the relation to equivariant integration is spelled out. In a SCFT in (even) dimension d, the anomaly polynomial can be thought of as a formal (d + 2)-form on an auxiliary (d + 2)-dimensional space Z d+2 that is the total space of a M p fibration over a and integrating it on M p gives a (d + 2 − p)-form, that is the anomaly polynomial of a (d − p)-dimensional SCFT. In [33] an extension of this construction to the case that M 2 = Σ is a spindle has been proposed and succesfully matched to the dual supergravity solution. More generally, here we will assume that M p (and hence, generically, also Z d+2 ) is an orbifold. The anomaly polynomial is a linear combination of characteristic classes of vector bundles on Z d+2 which (using the splitting principle) can always be decomposed in terms of first Chern classes of holomorphic line (orbi-)bundles on Z d+2 . Specifically, for any U(1) symmetry acting on M p , the latter can be fibered over B d+2−p by gauging this U(1) with a connection on a line bundle J over B d+2−p , denoted A J . When M 2m is a toric orbifold this consists in replacing the occurrences of each term dφ i with dφ i + A J i , where J i are auxiliary line bundles with c 1 (J i ) = [F J i ] /2π ∈ H 2 (B d+2−p , Z) and F J i = dA J i . This recipe is formally equivalent to consider equivariant first Chern classes, as follows. Starting from the line bundles L a with equivariant first Chern classes c T 1 (L a ) discussed in section 2.1, one can make the replacement One can then use these c 1 (L a ) as a basis to parameterise the characteristic classes appearing in the anomaly polynomials.

D3 branes on the spindle
For 4-dimensional SCFTs the anomaly polynomial is a 6-form defined on an six-dimensional orbifold Z 6 that we take to be the total space of a Σ fibration over an auxiliary space B 4 , Neglecting terms that are sub-leading in the large N limit the six-form anomaly polynomial is given by where c IJK are the cubic 't Hooft anomaly coefficients c IJK =Tr(F I F J F K ) and F I denote the generators of global U(1) symmetries. The c 1 (F I ) are formal first Chern classes associated to these U(1) I symmetries that can be decomposed as 18 where p a I ∈ Z are "fluxes stuck at the fixed points", L a are the line bundles discussed in section 3.1 and c 1 (F 2d R ) ∈ H 2 (B 4 , Z) is the first Chern class of the 2d R-symmetry line bundle.
The anomaly polynomial of the two-dimensional theory is obtained integrating the anomaly polynomial of the four-dimensional theory on the spindle, and is a four-form on B 4 . Substituting (4.5) in (4.4) it is immediate to see that this is exactly equivalent to the equivariant integral on the spindle where we have expressed the resulting four-form as a function of the variables ∆ I and the equivariant paremeter ǫ. This can also be understood as allowing the trial 2d R-symmetry to mix with the U(1) symmetry of the spindle, formally setting c 1 (J ) = ǫc 1 (F 2d R ) (thus "undoing" the replacement (4.2)), and extracting the coefficient of the four-form A 2d = F (∆ I , ǫ)c 1 (F 2d R ) 2 . Expanding the integrand we obtain 19 with the equivariant intersection numbers whose non-zero values are given in (3.17). Alternatively, the same result can be obtained by evaluating the equivariant integral (4.7) using the fixed point theorem (2.37), explaining the observation made in [35]. Specifically, we get where, in terms of the trial 4d central charge a 4d (∆ I ) = 9 32 I,J,K c IJK ∆ I ∆ J ∆ K , (4.11) 18 To avoid clumsiness in the formulas, we sistematically use the Einstein's notation for the indices a, b, . . . in this section. 19 We use that c IJK are totally symmetric to write (4.8).
we have (4.12) Notice that the terms singular in ǫ (cubic in the ∆ I ) cancels out, as expected.
The expression above depends on the p a I , but it can be rewritten in terms of "physical fluxes" n I , defined as the integrals of the flavour line bundles [77] namely (4.14) Recall that in general supersymmetry can be preserved on the spindle by coupling with a background R-symmetry gauge field with first Chern class where σ a = ±1, corresponding to either twist or anti-twist [41]. Since the supercharge should couple to 2c 1 (F 2d R ) + c 1 (E R ) we must have and we can introduce new variables In terms of these variables the gravitational blocks depend only on the physical fluxes, namely yielding in agreement with [40]. Notice that (4.21) is much simpler than the analogous expression in eq. (5.9) of [35] and is therefore the most convenient form to be used in the extremization. The reason is that in the variables ϕ I the expression is manifestly independent of the redundant parameters r I 0 used in the construction in [35], as we now explain. In this reference one starts from the background gauge fields A I = ρ I (y)dφ inherited from the supergravity solution, where y, φ are coordinates on the spindle. This is lifted to Z 6 by the gauging procedure dϕ → dϕ + A J , which leads to the connection one-forms A I = ρ I (y)(dφ + A J ), where dA J = 2πc 1 (J ), so that Comparing with (4.5) leads us to identify ρ I (y) precisely with our moment maps, namely ρ I (y) = −2πp a I µ a . (4.23) In our notation 20 , the functions ρ I (y) satisfy [35] ρ I (y 1 ) = 1 2 where I r I 0 = 2, while using (3.12) we have implying that we must identify (4.26) Summing over I we then get consistently with σ 1 = −1, σ 2 = 1, corresponding to the anti-twist [35]. This shows that the constants r I 0 parameterise the redundancy in the relation between the physical fluxes n I and the p a I . However, notice that (4.26) implies that r I 0 ∈ Q.

M5 branes on the spindle
The anomaly polynomial of M5 brane 6-dimensional SCFTs is an eight-form defined on an eight-dimensional orbifold Z 8 that we take to be the total space of a Σ fibration over an auxiliary space B 6 , Σ ֒→ Z 8 → B 6 . (4.28) Neglecting terms that are sub-leading in the large N limit, the eight-form anomaly polynomial is given by where p 2 (R) is the second Pontryagin class of the SO(5) R normal bundle to the M5brane in the eleven-dimensional spacetime. The F I , I = 1, 2 are the generators of U(1) 1 × U(1) 2 ⊂ SO(5) R global symmetries that are preserved when the M5-brane is compactified on the spindle [37]. The c 1 (F I ) are the first Chern classes of the line bundles on Z 8 associated to these U(1) I symmetries and are decomposed as The anomaly polynomial of the four-dimensional theory is obtained integrating the anomaly polynomial the six-dimensional theory on the spindle, and is a six-form on B 6 . We now substitute (4.30) in (4.29), where c 1 (L a ) is as in eq. (4.5) and c 1 (F 4d R ) ∈ H 2 (B 6 , Z) is the first Chern class of the 4d R-symmetry line bundle. Formally setting c 1 (J ) = ǫc 1 (F 4d R ) and extracting the coefficient of the six-form which may be expanded as where the non-zero equivariant intersection numbers are given in (3.17). Alternatively, the same result can be obtained evaluating the equivariant integral (4.32) using the fixed point theorem (2.37). Specifically, we get where, in terms of the "trial 6d central charge" 21 Notice that the terms singular in ǫ (quartic in the ∆ I ) cancel out, as expected. The rest of the discussion proceeds exactly as for the D3 branes in the previous subsection. In particular, supersymmetry implies that the constraints (4.17) hold and the physical fluxes n I obey (4.18). In terms of the variables ϕ I (4.19) the fixed point contributions read F a (ϕ I , ǫ) = (−1) a 1 ǫ a 6d (ϕ I − (−1) a n I 2 ǫ) , (4.37) which give the very simple expression F (ϕ I , ǫ) = − N 3 48 (ϕ 1 n 2 + ϕ 2 n 1 )(4ϕ 1 ϕ 2 + n 1 n 2 ǫ 2 ) , (4.38) in agreement with [40]. Let us briefly compare the results above with the corresponding calculation in [37]. Again, the functions ρ I (y) used in [37] should be identified with our moment maps as ρ I (y) = −2πp a I µ a , and the resulting p a I are easily obtained from the expressions in eq. (A.2) of [37]. However, due to the supergravity coordinates and the specific gauge used in [37], these are quadratic irrational functions of the spindle parameters and the physical fluxes and do not depend on any free parameter. One can check that the constraint (4.17) on the p a I is respected, with σ 1 = −1 and σ 2 = −1, consistently with the fact that the discussion in [37] concerns the twist case 22 . Notice that the form of (4.38) is much simpler than the corresponding function computed [37]. In particular, in (4.38) there are no linear and cubic terms in ǫ.

M5 branes on 4d orbifolds
We now consider M5-branes compactified on a four-dimensional toric orbifold 23 as discussed in section 3.2. The eight-form anomaly polynomial (4.29) is defined on an eightdimensional orbifold Z 8 that we take to be the total space of a M 4 fibration over an auxiliary space B 4 , 39) and integrating it on M 4 gives a four-form, that is the anomaly polynomial of a 2d SCFT: The c 1 (F I ) now decompose as where J 1 , J 2 are the auxiliary line bundles with c 1 (J i ) ∈ H 2 (Z 4 , Z). The L a , a = 1, . . . , d are the line bundles discussed in section 3.2 and c 1 (F 2d R ) ∈ H 2 (Z 4 , Z) is the first Chern class of the 2d R-symmetry line bundle.
Substituting (4.41) in (4.29) and setting c 1 (J i ) = ǫ i c 1 (F 2d R ), leads to the equivariant integral , (4.43) 22 We identify their P i with our −n I . 23 The M5 brane anomaly polynomial in an example of this type of compactification was studied in [50]. 24 Notice that there is no summation on a in the expressions inside the parenthesis.
in agreement with the formula conjectured in [51]. In terms of the gravitational blocks we have However, recall that this relation cannot be inverted and therefore it is not manifest that (4.43) depends only on the physical fluxes. That this is true was proved in [51], as we now recall, translating the arguments to our current notation 25  where λ I ∈ R 2 are arbitrary two-dimensional constant vectors 26 , leaves the physical fluxes q a I invariant. In order to discuss how this transformation affects (4.43) we will assume 27 , following [51], that supersymmetry requires the background R-symmetry gauge field to have first Chern class where σ a = ±1, generalizing the twist and anti-twist for the spindle [41]. This corresponds to the following constraints where ǫ = (ǫ 1 , ǫ 2 ) and W ∈ R 2 is a two-dimensional constant vector transforming as under (4.48). Performing this transformation in (4.43) implies that the variables ∆ a I change as Recall that in this paper we denote the non-primitive, "long", vectors of the fan by v a , the corresponding divisors as D a , and their intersection matrix as D ab . While in [51] these were denoted byv a , D a , andD a,b , respectively. Similarly, (p a I ) here = (m a p a I ) there and (q a I ) here = (q a I /m a ) there . 26 Notice that these λ I have nothing to do with the Kähler parameters λ a appearing elsewhere. 27 This holds in all known examples of supergravity solution corresponding to M5 branes wrapped on four-dimensional toric orbifolds [50][51][52][53]. It should be possible to prove this, along the lines of the analysis in [41]. thus F (∆ I , ǫ i ) → F (∆ I , ǫ i ) ,∆ I ≡ ∆ I − det(λ I , ǫ) , (4.53) where∆ I obey the same constraint as the ∆ I , namelỹ This is exactly the same function as the initial one, completing the proof.

Non-compact Calabi-Yau singularities
The geometry of many type II and M theory solutions arising from branes can be modelled on singular Calabi-Yau cones. In this section we consider the equivariant volume of partial resolutions of non-compact (asymptotically conical) Calabi-Yau singularities and provide many applications to holography. We assume that X is the (partial) resolution of a Calabi-Yau conical singularity, as in some of the examples in section 3.3. This means that the convex rational polyhedron is asymptotically a cone. More precisely, we assume that the large y i approximation of the polyhedron is a non-compact cone with a single vertex in y = 0 and d ′ facets. Notice that only the non-compact facets of P are relevant for the asymptotic behaviour and, in general d ′ ≤ d.
The vectors v a k , k = 1, . . . , d ′ , define the fan of the singular cone X sing of which X is a resolution. The fan of X sing consists of a single cone and it corresponds to a singularity which is, in general, not of orbifold type. The resolution replaces the singularity of P ′ with compact cycles of real dimension 2m − 2, corresponding to the bounded facets of P, as well as sub-cycles of smaller dimensions. 28 We assume that P has only orbifold singularities. From the dual point of view, the fan of X is the union of m-dimensional cones, each corresponding to a vertex of P and therefore to a fixed point. Each cone is specified by a choice of m adjacent vectors (v a 1 , . . . , v am ). We use the notation A = (a 1 , . . . , a m ) to identify the set of such cones and we assume that there are n of them.
Using the results of section 2.3, the equivariant volume is given by

GK geometry and the GMS master volume
In this section we explore some general properties of the equivariant volume for noncompact Calabi-Yau singularities and its relation with other volumes appearing in the literature in similar contexts, like the the Sasakian volume of [17,18] and the master volume introduced in [66]. Consider the formal expansion where V (k) (λ a , ǫ i ) is homogeneous of degree k in λ a . We start by observing some general properties of the homogeneous quantities V (k) (λ a , ǫ i ). From (2.35), by matching degrees in λ a , we have By considering the case i = 1 and the fact that, for Calabi-Yaus, v a 1 = 1, we also have d a=1 We can combine (5.7) and (5. It follows from this equations that V (k) are invariant under the gauge transformation which allows to eliminate m − 1 29 unphysical λ a that do not correspond to non-trivial co-homology classes.
In the compact case, the terms with k < m in the expansion (5.6) are identically zero. Indeed they can be written as in (2.31) and they are integrals of forms of degree less than 2m. As we already discussed in section 2.5, in the non-compact case this is not true and also the terms with k < m are not vanishing. They are rational functions of ǫ encoding some interesting geometrical information that we will now elucidate.
It is important to observe that V (k) (λ a , ǫ i ) with k < m only depends on the d ′ Kähler parameters λ a associated with the singular fan and the non-compact directions. The Kähler parameter λ a associated with the compact directions start contributing at order m in the expansion. This can be understood as follows. The compact Kähler parameters are associated with the bounded facets of the original polytope. We can always modify the polytope by adding new facets and making it compact. We can also assume that the bounded facets of the original polytope are not modified by this operation. The compact Kähler parameter enters in the fixed point formula only in the terms associated with the vertices of the bounded facets and, therefore, their contribution to V for the old and new polytope is the same. Since the new polytope is compact, this contribution must start at order m.
The term of zero degree is simply and it computes the equivariant volume of the singular Calabi-Yau cone X sing , or equivalently, the regularized volume of the polyhedron P ′ . At λ a = 0, the metric of X sing becomes conical ds 2 (X) = dr 2 + r 2 ds 2 (Y ) .

(5.12)
A choice of Kähler metric exhibits X sing as a cone over a Sasakian manifold Y of real dimension 2m − 1. As shows in [17,18] there is family of Sasakian metrics parameterized by the Reeb vector ξ = ǫ i ∂ φ i . They correspond to different choices of the radial coordinate , (5.14) was derived indeed in [18]. The term of degree m − 1 is also interesting. It coincides, up to a numerical factor, with the master volume introduced in [66]. To define the master volume, one foliates the 2m − 1 base Y as ds 2 (Y ) = η 2 + ds 2 2m−2 , where η is the dual one form to ξ (i ξ η = 1) and the metric ds 2 2m−2 is conformally Kähler with Kähler form ω B . We turn on the d ′ Kähler parameters λ a associated with the noncompact facets of the polyhedron by letting ω B vary [66] [ where c a are the co-homology classes that uplift to c 1 (L a ) on the Calabi-Yau X sing . Notice that [dη] = 2π a c a [66]. The master volume is then defined as (5.17) For example, for m = 3, the master volume reads [66] 18) where the vectors v a with a = 1, . . . , d ′ runs over the fan of the singular cone X sing .
Notice that, in the original approach of [66], the master volume is defined by considering conical non-Kähler metrics on X. In our approach instead, we give up the conical condition on X and we use a Kähler metric. Despite the difference of approaches, we can recover the master volume as the term of degree m − 1 in the equivariant volume where we stress that setting λ a = 0 and multiplying by λ a 2 . . . λ am , we can also rewrite (5.19) as where the analogy with (5.17) is manifest. As observed in [66], the master volume reduces to the Sasakian volume when all the Kähler parameters are equal. We can understand this statement in our formalism as follows The master volume plays an important role in supergravity solutions based on GK geometry [25,66]. In this context the so-called supersymmetric action plays an even more important role. It is the object that needs to be extremized in order to find a solution of the equation of motions. Using (5.8) we find

Gravitational blocks from the equivariant volume
In this section we use the equivariant volume to study type II and M theory branes compactified on a spindle. Supersymmetry is preserved either with a topological twist or an antitwist [41]. From the field theory point of view, we consider the case where a superconformal field theory compactified on a spindle flows in the IR to a superconformal quantum mechanics or to a two-dimensional superconformal field theory. The corresponding supergravity solutions describing the IR limit have a geometry AdS 2 × Z or AdS 3 × Z, respectively, and can be interpreted as the near-horizon geometry of black holes or black strings. The local geometry of such brane systems can be modelled in terms of CY m-folds. We can describe many D-branes and M-branes configurations in terms of a formal fibration CY m ֒→ CY m+1 → Σ , (5.26) where the CY m encode the geometry and the information about the original higherdimensional CFT. It is often useful to define an off-shell free energy F (∆ I , ǫ), or extremal function 30 , depending on chemical potentials ∆ I for the continuous global symmetries of the higherdimensional CFT and an equivariant parameter ǫ for the rotation along the spindle, whose extremization gives the entropy of the black hole or the central charge of the two-dimensional CFT. Extremal functions of known black holes and black strings can be expressed in terms of gravitational blocks [54]. The general form of the extremal functions for branes compactified on a spindle in this context was proposed in [40]. The characteristic form of the off-shell free energy F is given by a gluing where the block F encodes some universal properties of the higher-dimensional SCFT and it is related to the geometry of CY m , and the gluing depends on the details of the fibration (5.26). For D3 and M5 branes, F is related to the central charge of the higher-dimensional CFT. For other types of branes, F is the sphere-free energy of the higher-dimensional CFT at large N [40,54].
In this section we focus on the gravitational interpretation of (5.27). In the case of M2 and D3 branes, the off-shell free energy can be expressed in terms of the supersymmetric action in the formalism of GK geometry and the decomposition (5.27) was explicitly proved in [55] from the gravitational point of view. We will show how to recover and reinterpret this results in terms of the equivariant volume. In the case of D2, D4 and M5 brane systems, we cannot apply the GK formalism but we will propose a possible and intriguing extension.
where σ = ±1. This is a fibration over a spindle WP 1 [n + ,n − ] specified by the vectors w ± . Notice that the (m + 1)-dimensional geometry is still a Calabi-Yau if the first component of the vectors w ± is one, w ±1 = 1, which we will assume. As shown in [55], this geometry explicitly appears in the gravity solutions of M2 and D3-branes compactified on a spindle with where a − n + + a + n − = 1 and σ = ±1 and p is a (m − 1)-dimensional vector. In this context, supersymmetry is preserved with a twist (σ = 1) or an anti-twist (σ = −1). Notice also that, in the anti-twist case, the toric diagram is not convex and it does not strictly define a toric geometry. We will nevertheless proceed also in this case, considering it as an extrapolation from the twist case. The fixed point formula is where A runs over the polyhedral cones of a resolution and ǫ (m+1) = (ǫ 0 , ǫ 1 , . . . , ǫ m ). For ease of notation, we drop the label A from the normal vectors U a . We can choose a resolution for the CY m by subdividing the m-dimensional fan. We then obtain a resolution of the CY m+1 by considering polyhedra where V + and V − are added to the m-dimensional cones (V a 1 , . . . , V am ). Let's assume also d a 1 ,...,am = 1 for the CY m . The inward normals to the tetrahedra are 31) where u a i are the inward normals to the CY m cones, we identified a m+1 = ± and σ = −1 is obtained by analytic continuation.
The contribution of the tetrahedra with vertex V + is where ǫ (m) = (ǫ 1 , . . . , ǫ m ), and we used d a 1 ,...,a m+1 = n + and the identity among normals i u a i = (1, 0, . . . , 0). 31 The contribution of the tetrahedra of V − is obtained by replacing w + with w − , n + with −σn − and λ 4 ≡ λ + with a new variable λ − . The signs in U a i are compensated by d a 1 ,a 2 ,a 3 ,a 4 = −(−σn − ) (we work for positive σ and analitically continue the result), which also brings an overall extra sign.
The final result is and we see that V CY m+1 can be obtained by gluing two copies of V CYm .

The case of D3 branes
The case of D3 and M2 branes can be described in terms of GK geometry [66]. We consider a system of branes sitting at the tip of a conical toric Calabi-Yau three-fold singularity (CY m ) with Sasaki-Einstein base Y 2m−1 and further compactified on a spindle. The dual supergravity solution has an AdS 2 × Z 9 and AdS 3 × Z 7 near horizon geometry, for M2 and D3 branes respectively. The internal manifolds Z 2m+1 are obtained by fibering the Sasaki-Einstein base Y 2m−1 over the spindle. Supersymmetry requires that the cone C(Z 2m+1 ) is topologically a CY m+1 , although the supergravity metric is not Ricci-flat. The supergravity solution can be described in terms of a family of backgrounds that depends on the equivariant parameters ǫ i and the Kähler parameters λ α of the CY m+1 [66]. To avoid confusion, we will use Greek letters to label the vectors in the m + 1-dimensional fan and the associated Kähler parameters. The conditions for supersymmetry can be compactly written as Let e i be the canonical basis in R 3 . Then i u ai · e 2,3 = 0. For example i u ai · e 2 = det(e 2 , v 2 , v 3 ) + det(e 2 , v 3 , v 1 ) + det( On the other hand, i u ai · e 1 = det(e 1 , v 2 , v 3 ) + det(e 1 , v 3 , v 1 ) + det(e 1 , v 1 , v 2 ) = 1 since it is the sum of the areas of three triangular cones obtained by triangulating (v 1 , v 2 , v 3 ) with the insertion of e 1 (it lies in the same plane). The sum of the three areas is the area of the original cone d 1,2,3 = 1.
where S SUSY is the supersymmetric action (5.24) of the CY m+1 Let us now specialize to the case of D3-branes where m = 3. The solution AdS 3 ×Z 7 is dual to a two-dimensional CFT. In this context, the Killing vector ξ = m i=1 ǫ i ∂ φ i is interpreted as the R-symmetry of the dual CFT and the extremization of the supersymmetric action is the geometrical dual of c-extremization [78] in two-dimensional CFTs [25,66].
More explicitly, the on-shell value of the supersymmetric action, up to a normalization coefficient, is the exact central charge of the two-dimensional CFT, while the off-shell value of S SUSY as a function of ǫ i after imposing (5.35) equals the charge c as a function of a trial R-symmetry. This has been proved explicitly in [79] for the case of a compactification on S 2 , or on a Riemann surface, and in [55] for a spindle. More precisely, S SUSY as a function of ǫ i coincides with the trial c-function after the baryonic directions in the trial R-symmetry have been extremized. 33 It is also shown in [55] how to write the supersymmetric action as the gluing of two gravitational blocks [54]. We now recover this result in our formalism.
We can extract the supersymmetric action from the gluing formula (5.33) where Notice that the redefinitions are homogeneous in λ α . This means that the previous identity can be easily truncated at a given order in λ α . Taking the quadratic piece and using (5.19) and (5.25) we obtain since the supersymmetric action is the quadratic part of V CY 4 and the 3d master volume is the quadratic part of V CY 3 . We thus recover the factorization in gravitational blocks derived in [55] (see for example (7.27) and (7.31) in that paper). We can be more explicit in the special case of S 5 and the dual N = 4 SYM. This example is discussed in [55] in a different gauge. The vectors are v 1 = (1, 0, 0) , v 2 = (1, 1, 0) , v 3 = (1, 0, 1) , (5.41) and the equivariant and master volume read We parameterize the fluxes M α as where the constraint (5.37) further requires We will be cavalier about the normalization of the fluxes in the following, but it is clear than N should be proportional to the number of colors of the dual theory. We can use the gauge freedom (5.10) to set λ 1 = λ 2 = λ 3 = 0. The master volume evaluated at the two poles in this gauge reads and we can find λ ± by solving the α = ± components of the second equation in (5.35) (recall that only two of them are independent) the extremization in [66] must be done under the condition ǫ 1 = 2/(m − 3) = 2 which correponds to We then see that ∆ I , with I = 1, 2, 3, parameterize the R-charges of the three chiral fields of N = 4 SYM. The supersymmetric action (5.40) is then obtained by gluing blocks (5.51) We can see that this result reproduces the anomaly polynomial computed in section 4. In doing so, we need to be careful that there is some ambiguity in identifying the chemical potentials associated with the R-symmetry in (5.47). Any redefinition ∆ I → ∆ I + δ I ǫ with 3 I=1 δ I = 0 would respect the constraint (5.48) and would lead to a potentially good choice of R-symmetry chemical potential. To avoid this ambiguity, we can compare the objects that are invariant under this redefinition It is then easy to see that the same relations hold for the anomaly as written in (4.10) with σ 1 = −1 and σ 2 = −σ.
Notice that the supersymmetric action scales like N 2 as expected for N = 4 SYM. This can be understood from the fact that S SUSY | CY 4 is quadratic in λ α and, from (5.35), λ α scales linearly with N.

The case of M2 branes
We consider now the case of M2 sitting at the tip of a conical toric CY 4 with Sasaki-Einstein base Y 7 and further compactified on a spindle. The dual M theory solutions correspond to 4d black holes and have an AdS 2 × Z 9 near horizon geometry. We can again describe the system in terms of GK geometry. The supersymmetric action provides an entropy functional for the black hole. The construction is the gravitational dual of Iextremization [81] and has been applied to the case of a compactification on S 2 in [82][83][84] 34 and in the case of a compactification on the spindle in [55,85].
From the gluing formula (5.33) we find where Taking the piece cubic in λ we recover the result in [55] S SUSY | CY 5 = 2π To simplify the discussion we restrict to the case Z 7 = S 7 which is dual to the ABJM theory with k = 1 compactified on the spindle. The vectors are v 1 = (1, 0, 0, 0) , v 2 = (1, 1, 0, 0) , v 3 = (1, 0, 1, 0) , v 4 = (1, 0, 0, 1) , (5.56) and the equivariant volume is We parameterize the fluxes M α as where the constraint (5.37) further requires 59) and N will be related to the number of colors of the dual theory. We use again the gauge freedom (5.10) to set λ 1 = λ 2 = λ 3 = λ 4 = 0 and we determine λ ± solving (5.35). This time, the master volume evaluated at the two poles in this gauge reads and we can find λ ± by solving the α = ± components of equation (5.35) (recall that only two of them are independent) There is a sign ambiguity in solving (5.61) that we fix to match the gravitational entropy function result and the explicit geometric analysis in [55]. 35 Define The extremization in [66] must be done under the condition ǫ 1 = 2/(m − 3) = 1 which corresponds to We then see that ∆ I , with I = 1, 2, 3, 4, parameterize the R-charges of the four chiral fields of ABJM. The supersymmetric action (5.40) is then given by gluing blocks 35 We see from (5.61) should be positive. We then choose the solution λ + < 0, λ − > 0 which is consistent with [55] -for example, one can compare with (7.16) and (7.26) in that reference.
with the appropriate funnction for the ABJM theory [54] and This formula matches the entropy function found in [40,55]. 36 Notice that the entropy function scales like N 3/2 as expected for the ABJM theory. The reason is that S SUSY | CY 5 is cubic in λ α and, from (5.35), λ α scales as √ N . Notice also that the gluing (5.64) involves a different relative sign for the twist and anti-twist. This is typical of M2 and D4 branes and it is not present for D3 and M5 branes [40].

The case of M5 branes
The case of M5 branes compactified on a spindle has no known description in terms of GK geometry. However, a stack of M5 branes in flat space in M theory probes a transverse geometry that is C 2 × R suggesting that the relevant CY 2 is simply C 2 . In the associated supergravity solution AdS 7 × S 4 , the base S 3 of the cone C 2 is fibred over a real direction to give the S 4 . Notice that there is a supersymmetric generalization where the M5 branes probes a C 2 /Z p × R and the CY 2 could be replaced by C 2 /Z p .
We now consider the CY 3 consisting of C 2 fibered over the spindle with fan From the gluing formula (5.33) we have  (5.71) and the equivariant volume of C 2 can be read from (3.85) with p = q = 1 In analogy with S SUSY in the context of GK geometry, we expect to be able to extract an off-shell free energy F , for the spindly M5 branes AdS 5 solutions found in [37] from the equivariant volume of C 2 fibred over the spindle. We now show that this is the case by an appropriate generalization of the equations (5.35). A scaling argument for N suggests that the correct generalization is where the flux equation uses the quadratic piece but the extremal function is the cubic piece of the equivariant volume. In this way, λ α is linear in N and the extremal function F cubic in N, as expected for M5 branes. Here and in the following sub-sections we will not be concerned with precise normalizations. Following the same logic as in previous sections, we solve the constraint on the fluxes with n 1 + n 2 = − 1 n + − 1 σn − and define with ∆ 1 + ∆ 2 = 2. The chemical potentials ∆ 1 and ∆ 2 can be associated with the Cartan subgroup of the SO(5) R-symmetry of the (2, 0) theory. We find that, after solving for λ ± in the gauge λ 1 = λ 2 = 0, the extremal function is given by gluing two blocks with the appropriate function for the (2, 0) theory [54] F (∆ 1 , and We see that we have reproduced the (2, 0) anomaly integrated on the spindle (4.34) with σ 1 = −1 and σ 2 = −σ.

The case of D4 branes
We now turn to the case of D4 branes compactified on a spinde. The massive type IIA supergravity solution associated with a system of D4 and D8 branes is topologically AdS 6 times an hemisphere S 4 [86] suggesting that the relevant CY 2 is again C 2 . Supergravity solutions with an AdS 4 factor corresponding to compactifications of the dual 5d SCFT on a spindle have been found in [40,42]. It si intriguing to observe that we can again extract an off-shell free energy for these spindly solutions from a generalization of the equations (5.35) valid for GK geometry. We consider as before the equivariant volume (5.70) of C 2 fibred over the spindle. This time we take so that λ α ∼ N 1/2 and F ∼ N 5/2 as expected for D4 branes. We use the same definitions (5.74) and (5.75) of the previous section. After solving for λ ± in the gauge λ 1 = λ 2 = 0, the extremal function is given by 37 where for the 5d SCFT [54] and We have reproduced the entropy function in [40,42]. 38

The case of D2 branes
We finally consider the case of D2 branes compactified on a spinde. The massive type IIA supergravity solution associated with a system of D2-branes is topologically AdS 4 × X 6 , where the internal manifold is a Sasaki-Einstein five-manifold foliated over a segment 37 We made a choice of determination for the fractional power that is similar to the M2 brane case and correctly reproduce the supergravity result. 38 To compare with (5.33) in [40] one needs to identify n here [ 87,88]. Supergravity solutions with an AdS 2 factor corresponding to compactifications of the dual 3d SCFT on a spindle have been found in [46]. We expect that the corresponding extremal function is related to the equivariant volume of a CY 3 fibred over the spindle. A scaling argument suggests to use so that λ α ∼ N 1/3 and F ∼ N 5/3 as expected for D2 branes. We can verify that this is the case for the solution associated with C 3 where the three-dimensional SCFT is a pure N = 2 Chern-Simons theory with the same matter content and superpotential of the maximal supersymmetric Yang-Mills theory [87] that we indicate as D2 k . We have where V C 3 is given in (5.42). The fluxes can be written as in (5.43). Since the matter content of the theory is the same as for N = 4 SYM we can still use the parameterization with ∆ 1 + ∆ 2 + ∆ 3 = 2, and interpret the ∆ I , with I = 1, 2, 3, as the R-charges of the three chiral fields of the theory. We can use the gauge freedom (5.10) to set λ 1 = λ 2 = λ 3 = 0 and we can find λ ± by solving the first equation in (5.85). The extremal function (5.85) is then obtained by gluing two blocks where for the D2 k theory and This is the natural expectation for the extremal function for a D2 brane theory. It would be interesting to have a more complete analysis in supergravity to compare with.

Discussion
In this paper we provided several applications of equivariant localization to quantum field theory and holography. In particular, we have shown that the equivariant volume, a basic and well-studied geometrical object in symplectic geometry, is at the heart of many constructions characterising supersymmetric geometries. It generalises at once the Sasakian volume of [17,18] and the master volume introduced in [66] that have been proven very useful in the study of superconformal field theories dual to branes sitting at Calabi-Yau singularities, and generalizations thereof. The corresponding quantum (or K-theoretical) version, the equivariant index-character, is analogously crucial for studying the Hilbert series of the moduli spaces of the corresponding SCFTs and we expect that it still has many surprises to unveil. More specifically, we proposed that the equivariant volume should be the key object for all the extremal problems characterising supersymmetric geometries with a holographic interpretation, in different supergravity theories. We showed in this paper that all the extremization problems associated with compactifications on the spindle of M2 and D3 brane at Calabi-Yau singularities as well as M5, D4 and D2 brane configurations in flat space, can be re-expressed in terms of the equivariant volume. In the case of M2 and D3 branes it was known before that the extremization problem can be expressed in terms of the master volume [66] and we showed how this object is encoded in the equivariant volume of the associated fibered Calabi-Yaus. In the M5, D4 and D2 branes there is not yet an analogous of the construction in [66], but we showed that the relevant extremal functions can be written in terms of the equivariant volume. We leave for future work the analysis of more complicated situations, like M5 and D4 branes compactified on a four-dimensional orbifolds M 4 or D2 branes in massive type IIA associated with a generic Sasaki-Einstein five-manifold foliated over a segment [88]. These examples are more complicated but we are confident that the corresponding extremal functions and extremization problems can be written and formulated solely in terms of the equivariant volume.
In this paper we discussed the factorization (5.27) of the extremal functions associated to spindly black objects in terms of gravitational blocks both from the gravitational and the field theory point of views. On the geometry side, in all cases the factorization is an immediate corollary of the fixed-point localization formula for the equivariant volume, applied to the relevant non-compact geometry -see eq. (5.33). For D3 and M5 branes, F and F m are related to the central charge of the lower and higher-dimensional SCFT, respectively, and the gluing (5.27) is nothing else that equivariant localization applied to the computation of the higher-dimensional anomaly polynomial that we discussed in section 4. For other types of branes, F m is the sphere-free energy of the higher-dimensional SCFT at large N [40,54], and (5.27) is typically related to the large N factorization properties of SCFT partition functions of theories with a holographic dual [89][90][91]. In this context, the factorization becomes visible after taking the large N limit of the SCFT free energy, which is usually expressed in terms of a matrix model. The partition function of three-dimensional N = 2 SQFTs compactified on a spindle times a circle was derived in [56], generalizing at once the superconformal index and the topologically twisted index. The large N limit of this spindle index is currently under investigation [92] and we expect to reproduce the factorization (5.27), thus closing the circle.
The derivation of partition functions of SQFTs compactified on orbifolds is another arena where the results of this paper could be useful. Indeed, quantum field theory localization relies on the use of the equivariant index theorem [93] and the building blocks of this construction will involve generalizations of the index-character discussed in section 2.6 and appendix C. More precisely, the index-character enters as a basic building block for partition functions on M × S 1 where supersymmetry is preserved with a topological twist on M, while it needs to be generalized in the case of general σ a as in (4.49). In particular, it would be interesting to generalize the five-dimensional indices of [90,94,95] to the case of a five-dimensional SCFT defined on M 4 × S 1 and apply the result to reproduce the entropy function for the supergravity solutions associated with D4 branes compactified on a four-dimensional orbifold [51], as well as recover the anomaly polynomial results for M5 branes compactified on a four-dimensional orbifold discussed in section 4.3.
and, using Stoke's theorem, we have where a facet F a of the polytope is defined by the linear equation We can introduce a coordinate 40 s ∈ [s min a , s max a ] on each facet F a , writing The extrema of the interval can be determined by intersecting F a with F a−1 and F a+1 , respectively, and read s min where v, w ≡ det(v, w). Plugging (A.5) into ν and integrating we have Recalling that and using the vector identities we obtain the compact expression Rearranging the contributions of the vertices in the sum where we used another other standard vector identity v a+1 , ǫ ǫ · v a − v a , ǫ v a+1 · ǫ = − v a , v a+1 ǫ · ǫ , (A.14) we obtain the localization formula (3.35).
B Direct proof of the master volume formula for CY 3 We consider a singular Calabi-Yau cone with fan specified by the vectors v a = (1, w a ), with a = 1, . . . , d ′ and w a ∈ Z 2 . The projection on the plane with first coordinate equal to one is a convex polytope called toric diagram in the physics literature. We label the vectors v a along the toric diagram in anticlockwise order and we identify vectors cyclically, v d ′ +1 ≡ v 1 . In order to apply the fixed point formula we need to resolve the singularity. This can be done by triangulating the toric diagram. We add new vectors v α = (1,w α ), α = d ′ + 1, . . . , d withw α lying inside the toric diagram until it becomes the union of triangles.
The master volume V is a function of the λ a associated with the external vectors v a = (1, w a ) only. The equivariant volume V is a function of all λ and can be obtained from the fixed point formula (5.5), where the sum is extended to all triangles in which the toric diagram has been partitioned. The computation is fortunately local. Every particular external λ a enters in just two fixed point contributions, associated with the two triangles in figure 8, wherev is an internal point. Denoting v, w, u ≡ det(v, w, u), and taking into account the orientations of the inward normals u a A , we can write V = d 2 I e −(λ v a−1 ,v a ,ǫ +λ a−1 v a ,v,ǫ +λa v,v a−1 ,ǫ )/d I v a−1 , v a , ǫ v a ,v, ǫ v, v a−1 , ǫ + d 2 II e −(λa v a+1 ,v,ǫ +λ v a ,v a+1 ,ǫ +λ a+1 v,v a ,ǫ )/d II v a+1 ,v, ǫ v a , v a+1 , ǫ v, v a , ǫ + . . .
(B.1) whereλ is the Kälher parameter associated with the internal pointv, d I = | v a−1 , v a ,v | and d II = | v a+1 , v a ,v | are the orders of the local singularities and the dots refer to terms independent of λ a . We can then compute (B.2) thus recovering the master volume expression (5.18).

C The equivariant index
In this appendix we present some simple examples of character indices. It is an interesting topic, with many applications for example to localization computations [56], but, since it is not the main theme of this paper we will be brief.

C.1 The equivariant index for the spindle
We consider the spindle Σ = WP 1 [n + ,n − ] . As in section 3.1 we take v 1 = n + , v 2 = −n − and GLSM charges Q = (n − , n + ). We take n + and n − relatively prime.
The character computes the equivariant index Z(q, Λ a ) = where q is the weight for the T action. We recall that not all the T-invariant divisors are inequivalent. In particular, for the spindle, (2.22) implies n + D + = n − D − . The fixed point formula (2.71) has contribution from two poles of the spindle. In each contribution we need to add an average over the local singularity Z d A Z(Λ a , q) = 1 n + n + −1 k=0 e −2πikΛ + /n + q −Λ + /n + 1 − e 2πik/n + q 1/n + + Setting q = e −ǫ and Λ ± = −λ 1,2 / and taking the limit → 0 we recover the equivariant volume (3.16) as the coefficient of the leading pole as in (2.67). Only the terms with k = 0 contribute to the limit. The index can been resummed using [56] 1 k where ω k = e 2πi k , u is a complex number, α is an integer number and the floor function denotes the integer part. This formula can be proved by expanding both sides in power series of u. The result is Notice that, although there were fractional powers of q in (C.2), the final formula contains only integer powers. We see that, for Λ ± > 0, Z(Λ a , q) = q This can be checked by computing explicitly the residue related toq i . One obtain two sums over z k − =q −1/n − 1 e 2πik − /n − and z k + =q −1/n + 2 e 2πik + /n + for k ± = 0, . . . n ± − 1 that match the fixed point formula. 41 On the other hand, we can also evaluate the integral in a simpler way. We define the grand-canonical partition function (C.11) where we evaluated the integral by deforming the contour circling z k ± into a contour around z = w, which is the only other singularity of the integrand. From this expression we see that Z M W (T,q a ) for T > 0 counts the monomials in (q 1 ,q 2 ) of charge T under the rescaling with weights Q = (n − , n + ). These are precisely the holomorphic sections For simplicity, we only consider the case of the "non-minimal" fan v 1 = (n 3 , n 3 ) , v 2 = (−n 1 , 0) , v 3 = (0, −n 2 ) , (C.13) and Q = (N 1 , N 2 , N 3 ) = (n 1 n 2 , n 2 n 3 , n 1 n 3 ). The orbifold is M 4 = WP 2 [N 1 ,N 2 ,N 3 ] if the n a are coprime. There are only labels and not extra orbifold singularities in the sense that the corresponding primitive vectorsv a , obtained by dividing v a by the label n a , satisfŷ d a,a+1 = det(v a ,v a+1 ) = 1.
The fixed point formula (2.71) requires averaging over the local orbifold singularity. For example, we can consider the contribution from the fixed point associated with the cone (v 2 , v 3 ). The local data are d 2,3 = n 1 n 2 and u 1 = (−n 2 , 0) and u 2 = (0, −n 1 ). The local orbifold action is determined by J 2 n 1 = s, J 3 n 2 = p where s, p are integers: (e 2πiJ 2 , e 2πiJ 3 ) = (e 2πi s n 1 , e 2πi p n 2 ) . (C.14) The fixed point contribution is (1 − 1/q 1 )(1 − 1/q 2 ) + , (C. 16) in agreement with the general formula (2.73). One can check that This can be proved with elementary methods. By expanding in power series (C.15) in 1/q 1 and 1/q 2 we obtain all the integer points in an infinite cone, as in figure 9. By ( Λ2 n1 , Λ3 n2 ) Figure 9: The integer points counted by (C.15).
similarly expanding the other power series in (C.16) for |q i | > 1 and |q 2 /q 1 | > 1, we find three competing contributions which cancel except for the points inside the polytope as depicted in figure 10. The Molien-Weyl formula gives instead .
(C. 19) This can be explicitly evaluated by taking the residues at z Naq a = 1 and one recovers formula (2.70): , T = n 1 n 2 Λ 1 + n 2 n 3 Λ 2 + n 3 n 1 Λ 3 . (C.21) One can check that the three set of residues at z Naq a = 1 precisely correspond to the three fixed point contributions. For example, the residue associated withq 1 reads reproducing the contribution (C.15). In the sum we replaced k with n 3 k using the fact that the n i are coprime. We also used the identity can be also computed deforming the contour to encircle z = w which gives .
(C. 25) We see that Z M W (T,q a ) for T > 0 counts the monomials in (q 1 ,q 2 ,q 3 ) of charge T under the rescaling with weights Q = (N 1 , N 2 , N 3 ). These are precisely the holomorphic sections In the presence ofd a,a+1 = det(v a ,v a+1 ) = 1 we would need more general formulas for resumming the averages over the orbifold action and we will not discuss this case.