Cohomological Localization of $\mathcal N = 2$ Gauge Theories with Matter

We construct a large class of gauge theories with extended supersymmetry on four-dimensional manifolds with a Killing vector field and isolated fixed points. We extend previous results limited to super Yang-Mills theory to general $\mathcal{N}=2$ gauge theories including hypermultiplets. We present a general framework encompassing equivariant Donaldson-Witten theory and Pestun's theory on $S^4$ as two particular cases. This is achieved by expressing fields in cohomological variables, whose features are dictated by supersymmetry and require a generalized notion of self-duality for two-forms and of chirality for spinors. Finally, we implement localization techniques to compute the exact partition function of the cohomological theories we built up and write the explicit result for manifolds with diverse topologies.


Introduction and summary
The analysis of N = 2 gauge theories has led to numerous advances in our understanding of the dynamics of quantum field theories at strong coupling. Moreover, it has proven to be a fruitful arena to develop interesting connections between physics and mathematics. Two broad classes of developments in the study of N = 2 gauge theories, namely the equivariant topological twisting and Pestun's supersymmetric localization on the four-sphere, were unified in a single framework in [1]. That analysis only applied to N = 2 super Yang-Mills theories; hence, here we consider its extension to theories that include both vector and hypermultiplets.
The first class of developments mentioned above, revolves around the topological twisting of N = 2 gauge theories. Witten showed in [2] that a certain twisting of N = 2 super Yang-Mills theory results in a four dimensional topological field theory whose correlators are the Donaldson invariants [3]. This work was later generalized by [4][5][6][7][8][9][10][11][12], who constructed twisted versions of N = 2 gauge theories with vector and hypermultiplets. In particular, for the purposes of our work, we are interested in the equivariant version of the topological theory of Witten, which can be defined on manifolds admitting a torus T 2 action, and was studied in [13][14][15][16]. When specifying to the omega background, its partition function was calculated in the seminal work of Nekrasov [17,18].
For more general non-compact toric manifolds M, the partition function is a product of Nekrasov partition functions for each of the fixed points of the torus action on M [19], see also [20][21][22]. The extension of this analysis to compact toric manifolds has been considered both recently [23,24] and in earlier works [25][26][27].
The second class of developments is related to [28], where Pestun placed an N = 2 gauge theory on a round four-sphere preserving all eight supercharges. He was able to show that the partition function of this theory and certain supersymmetric Wilson loop observables, can be computed exactly using localization techniques. His work was later generalized to squashed four-spheres in [29,30]. These advances led to the realization that localization techniques are a powerful tool to derive exact results in supersymmetric field theories in different dimensions (for a summary of these developments see for instance the review [31]).
The localization result for the partition function of Pestun's theory on the four-sphere is fairly simple. It consists of an integral over a real parameter of the product of two factors, each of which can be associated by equivariant localization to one of the poles of the sphere. These are the fixed points of a U (1) action generated by squaring the supercharge used for localization.
One factor includes a Nekrasov partition function for instantons, the other a partition function for anti-instantons. Hence, the final result is structurally similar to the form of the partition functions stemming from equivariant topological twisting, which only involve Nekrasov partition functions for instantons. This relation was explained in [1]  their supersymmetric observables can depend on. These results were formulated in a more rigorous mathematical framework in [32].
Summary of results. In the present work we extend the results of [1,32] to theories involving hypermultiplets. The inclusion of hypermultiplets generically requires the manifold to admit a spin structure. Nevertheless, this substantially enlarges the set of supersymmetric field theories which can be analyzed using localization techniques. We reformulate the theories we construct in terms of cohomological (twisted) fields. A generalization of the notion of self-duality for twoforms was found necessary in [1] to define the cohomological SYM theory. Similarly here we find that a generalization of a Weyl spinor is necessary to define cohomological variables for the hypermultiplets. The use of such spinors is forced upon us by supersymmetry, thus establishing an interesting relation between supersymmetry and the geometry of the manifolds where our theories live. Furthermore, we employ localization techniques to compute the partition function of the cohomological theories we constructed. Computations of path integrals via a purely cohomological formulation of supersymmetry appeared for theories defined on specific manifolds in 3d [33][34][35], 4d [25,32,[36][37][38][39], 5d [40][41][42][43][44], and 7d [45,46]. We refer to [31] and to references therein for an exhaustive bibliography. In this paper we aim to present a general formula for the partition function of N = 2 gauge theories with matter on any simply connected Riemannian spin manifold admitting an isometry with isolated fixed points as well as non-trivial fluxes. Technically, the computation of the relevant one-loop contributions descends from the index of a transversally elliptic differential operator of Dirac-type. Combining the result of this paper with those of [1], we find where the formula above holds for a manifold with p plus points and (l − p) minus fixed points.
In (1.1), Z inst , Z VM and Z HM respectively are the instantons contribution, the vector multiplet (a 0 , k i ) are Barnes double gamma functions [47]. With a specific choice of regularization (other choices are considered in section 5) we get for instance: where we have a product over hypermultiplet representations R, while the fugacity Φ 0 at a plus point is a combination of the Coulomb branch parameter a 0 and the function k + encoding the flux contribution: As we spell out in the main text, the expression forZ HM Outline of the paper. In section 2, we review how to place an N = 2 field theory on Euclidean four-manifolds preserving some supersymmetry. This can be accomplished by coupling the theory to a supersymmetric rigid supergravity background, whose properties we review. In particular, we show that the usual hypermultiplet action, quadratic in derivatives, when coupled to rigid supergravity, is δ-exact up to total derivatives.
In section 3, we consider a Riemannian spin four-manifold M that admits a Killing vector field with isolated fixed points. We review the arguments in [1] showing that M admits globally well defined Killing spinors that depend on a choice of a plus/minus label for each fixed point. In order to deal with off-shell hypermultiplets, we also need to show the existence of auxiliary Killing spinors satisfying certain properties. We prove that a smooth, globally well-defined choice for these auxiliary Killing spinors exists.
In section 4, we move to reformulate the gauged hypermultiplet in terms of cohomological fields.
For this we first define novel splits of the Dirac spinor bundle into two subbundles. These splits require the existence of a vector field with isolated fixed points and depend on the choice of ± at each fixed point. We also study the relation between these spinor bundles and the "flipping" sub-bundles of the bundle of two-forms studied in [1]. The appropriate cohomological fields for the hypermultiplet are elements of the novel spinor bundles we introduced. We explicitly prove that there is a smooth invertible map between cohomological variables and the usual hypermultiplet component fields. We also show how supersymmetry organizes the cohomological fields in different multiplets. Finally we rewrite the action for a gauged hypermultiplet in terms of the cohomological fields. In the reformulation of the theory in terms of cohomological variables, there is an important difference with [1], where only vector multiplets were considered. In the case of the hypermultiplet the twisted fields are spinors so that generically the four-manifold has to be spin. This is however too restrictive. For instance, depending on the flavor symmetry of the theory, it may be enough for the manifold to be spin c .
Finally in section 5, having constructed the cohomological theory, we use it to set up the localization computation of the partition function. As an application, we apply our formula to the specific case of the squashed four-sphere. We find perfect agreement with the localization result for N = 2 matter multiplets found in [29].
Outlook. A compelling direction to be explored in a future work would be applying the technology

N = theories on four-manifolds
In this section we review the construction of rigid N = 2 supersymmetric fields theories on a curved four-manifold by coupling to background supergravity. Assuming the existence of appropriate Killing spinors, we write down the supersymmetry variations for fields in vector multiplets and hypermultiplets. We also present a δ-exact Lagrangian for a gauged hypermultiplet.

Review of N = 2 rigid supergravity
In order to couple a supersymmetric field theory defined in flat space to off-shell supergravity, we have to set the fermionic fields in the supergravity multiplet to zero and freeze the bosonic supergravity fields to fixed values. If this background is invariant under some supergravity variation, the resulting theory is supersymmetric [49]. Here we consider N = 2 theories with a conserved SU (2) R-current. Their supercurrent multiplet was studied by Sohnius [50] and the N = 2 Poincaré supergravity to which they couple is described in [51][52][53][54]. Rigid N = 2 supergravity backgrounds and the conditions they have to satisfy to preserve supersymmetry have been considered in [30,55,56].
In order to specify the supergravity background we need the following data: • A Riemannian manifold M equipped with a metric g and a spin structure 1 .
• An SU (2) R connection V µ i j . (Here and in the following i, j . . . are SU (2) R indices.) • Various other auxiliary fields: a one-form G µ , a two-form W µν , a scalar N , a closed two-form F µν , and a scalar S ij transforming as an SU (2) R triplet.
The supergravity variations are parametrized by a left-handed spinor ζ i α and a right-handed spinor χα i , both transforming in the fundamental representation of the SU (2) R R-symmetry. Here i is the SU (2) R index and α,α are spinor indices. For a brief review of the conventions we use, see Appendix A . Unless otherwise noted, we take these spinors to obey symplectic-Majorana reality conditions: Requiring the background to be invariant under the supergravity variation parametrized by ζ i α and χα i , one obtains two sets of Killing spinor equations. The first set is where R is the Ricci scalar and the spinors η i andη i are defined as: Here we use the notation W + = 1 2 W µν σ µν and W − = 1 2 W µνσ µν (similarly for F).
The spinors ζ i andχ i can be used to construct various bilinears. Restricting our attention to singlets of the SU (2) R-symmetry, we have the scalars 5) and the vector field The reality conditions (2.14) imply that v µ , s ands are real and that s ands are nowhere negative. The vector v and the scalars s,s satisfy ||v|| 2 = ss, hence they are not independent.
Using the Killing spinor equations we can show that v µ is a Killing vector and that s,s are constant along the orbits of v. We assume that both √ s and √s are smooth in a neighborhood of the fixed points.

Supersymmetric multiplets
Here we assume that the Killing spinor equations introduced in the previous subsection are satisfied in some supergravity background. We present the structure of the supersymmetry variations for vector multiplets and hypermultiplets coupled to this background.

Vector multiplet
The N = 2 vector multiplet contains a complex scalar field X, a gauge field A µ , two gauginos λ iα andλ iα that transform in the fundamental of SU (2) R and an auxiliary scalar field D ij transforming as a triplet of SU (2) R . All these fields (except A µ ) transform in the adjoint representation of the gauge group. The supersymmetry variations are given by: We used the shorthand notation F + = 1 2 F µν σ µν and F − = 1 2 F µνσ µν where F µν is the field strength for the gauge field A µ .
The square supersymmetry variation of a field Ψ in the vector multiplet results in a translation along the vector field v defined in (2.6), together with a gauge transformation and an SU (2) R transformation Here L v is the Lie derivative along v, and • denotes that Ψ is acted upon according to which SU (2) R representation it belongs. The gauge transformation parameter is and Λ (R) is a SU (2) R transformation parameter given by: (2.10)

Hypermultiplet
For the following we will embed the gauge group in Sp(k) and consider a hypermultiplet in the fundamental of this Sp(k). The hypermultiplet contains a scalar q ni (where the index n transforms under Sp(k) and runs over n = 1, . . . , 2k, while the index i transforms under the fundamental of SU (2) R ), and a pair of spinors ψ αn andψα n . Additionally there are auxiliary fields F nǐ that are necessary for the off-shell closure of the supersymmetry algebra. The F nǐ transform in the fundamental of an SU (2)Ř symmetry that is generically distinct from the SU (2) R-symmetry.
We take the Grassmann-even fields in the hypermultiplet to satisfy the following reality conditions: The supersymmetry variations of the hypermultiplet components are: where X n m = X α t α n m (and similarly for other vector multiplet components) and the derivative D µ is covariant with respect both to the gauge symmetry and the R-symmetry. Finally, we introduced the checked spinorsζǐ andχǐ, which need to satisfy the constraints: for the off-shell closure of the supersymmetry algebra. Unless otherwise noted, we assume thatζǐ andχǐ satisfy symplectic-Majorana reality conditions: Except when acting on the auxiliary fields F nǐ , the square supersymmetry variation of a field Ψ of the hypermultiplet, is given by: This includes a translation along v µ = 2χ iσµ ζ i , an SU (2) R transformation and a gauge transformation. The gauge transformation parameter Φ and the SU (2) R transformation parameter Λ (R) are as in (2.9) and (2.10). The squared supersymmetry variation of the auxiliary fields similarly includes SU (2)Ř transformations: HereV µ is a background connection for SU (2)Ř and the SU (2)Ř transformation parameter Λ (Ř) is given by: The derivativeĎ µ is covariant with respect to the background SU (2)Ř connectionV µ . Note that in (2.16), the connectionV µ cancels between the terms iv µV µ • Ψ and iΛ (Ř) • Ψ .

Hypermultiplet Lagrangian
Having completed the coupling to rigid supergravity, we can write a supersymmetric Lagrangian L = L B + L F for the hypermultiplet: It turns out that (up to total derivatives) this Lagrangian is itself the supersymmetry variation of some Grassmann-odd V G : where V G is given by the following expression: The expression for V G in (2.20) is in agreement with and generalizes a similar one found in [29], to which it reduces if we assume that s +s is a constant (in that case F µν = 0).

Construction of Killing spinors
In the last section we have considered gauged hypermultiplets coupled to a rigid N = 2 supergravity background. We assumed that this background preserves supersymmetry, i.e. that there are nonzero solutions to the Killing spinor equations (2.2) and (2.3). A general class of backgrounds allowing solutions to these equations was studied in [1]. Here we will briefly review their main properties.
Consider a Euclidean orientable four-manifold M with metric g and a spin structure. It was shown in [1] that the Killing equations can be satisfied by spinors ζ i andχ i that are both nonvanishing provided that the metric admits a Killing vector v whose fixed points are isolated. In addition to v, the supergravity background is specified by a choice of a real, nowhere-negative scalar s on M that is constant along orbits of v. Moreover s has to approach a positive constant K at a subset of the fixed points of v and needs to go to zero as ||v|| 2 /K at the remaining fixed points.
At these fixed points of v, the scalars = ||v|| 2 /s approaches K. Hence, the fixed points of v are separated into a set wheres = 0 with s = K and a second set where s = 0 ands = K. The fixed points satisfyings = 0 are plus fixed points, whereas those satisfying s = 0 are minus fixed points.
This is consistent with [1].

Killing spinors
In this subsection, we employ the Killing vector v and the scalar s to construct spinors ζ i α andχ iα that satisfy the reality conditions (ζ iα ) * = ζ iα and (χα i ) * =χ iα and such that We cover the manifold with charts U k , such that every fixed point of v belongs to a single distinct chart and we make a choice of vielbein e a k in each chart. We can also assume that there are no overlaps between charts containing different fixed points of v. We consider the spinors ζ i andχ i given by the following expressions in every chart: These spinors satisfy the reality conditions and the constraints (3.1).
In going from chart to chart, ζ transforms under SU (2) l × Z 2 SU (2) R . For the form of ζ above to be valid in each chart, we have to undo the SU (2) l transformation by an appropriate SU (2) R transformation. The expression forχ will then also be valid in each chart because it is directly related to ζ.
Unfortunately of v with s = 0, we can act with a further SU (2) R transformation: As a consequence, in charts containing a fixed point where s = 0, the spinors are:

Auxiliary Killing spinors
An additional element we need to consider is the construction of smooth auxiliary spinorsζǐ α anď χα ı , since they are used for the off-shell extension of the supersymmetry transformations for the N = 2 hypermultiplet. These spinors satisfy the reality conditions: as well as the following constraints: (3.6) These spinors transform under SU (2) l × Z 2 SU (2)Ř and SU (2) r × Z 2 SU (2)Ř respectively. The SU (2)Ř bundle is not generally identified with the SU (2) R bundle. The constraints in (3.6) determineζ andχ uniquely up to local SU (2)Ř transformations.
The construction of these spinors parallels that of ζ and χ. On patches U k that do not contain fixed points where s = 0, we define the spinors as follows: It can be checked using the expressions for ζ i andχ i in (3.2), that these satisfy the constraints (3.6).
In this case in going from patch to patch, we have to undo SU (2) r transformations by appropriate On a patch U l that includes a fixed point of v with s = 0, we take instead: which also satisfy the constraints (3.6) with ζ i andχ i in (3.4). As before, when transitioning to one of these patches, there is an extra SU (2)Ř transformation given by: Hence, we have found smooth solutions to the constraints (3.6). There is a different construction of solutions to these constraints that is often used in the literature. This is given by: In this case SU (2)Ř is identified with SU (2) R ; however the resulting spinors are singular at the fixed points of v.

Twisted supersymmetry
In this section we rewrite the component fields for the vector multiplet and hypermultiplet in terms of twisted variables. The case of the vector multiplet was considered in [1]. The construction of these twisted variables is intimately connected with the geometry of M. In particular, for the vector multiplet it relies on a novel decomposition of the bundle of two-forms on M which we review. For the hypermultiplet we will introduce a corresponding decomposition of spinors on M.

Flipping projectors
Consider the bundle of two-forms Λ 2 (M). On an orientable Euclidean manifold with metric g, we can split Λ 2 (M) in two orthogonal sub-bundles, consisting of self-dual and anti self-dual twoforms. Self-dual two-forms play an essential role in Donaldson-Witten theory and its equivariant extension [2,13]. As shown in [1], on a manifold equipped with a vector field v with isolated fixed points, one can introduce a different splitting of the bundle of two-forms. This splitting arises naturally in the study of the N = 2 theories on M we presented in section 2.
Let v be a real vector field on M (with isolated fixed points) and for what follows we will take it to be the Killing vector. The s ands , which are the spinor bilinears in (2.5), determine the ± fixed points. We can define projectors P ± on the bundle of two-forms as follows: where we use 1 to indicate the identity operator throughout the paper. Here, κ = g(v) is the one-form dual to the Killing vector v. The objects P ± are well defined projectors as they satisfy (P ± ) 2 = P ± , P + P − = 0 and P + + P − = 1. Especially, P + induces a split of the bundle of two-forms into two orthogonal subbundles. The image of P + consists of two-forms that are self dual at the fixed points of v wheres = 0, and anti-self dual at those fixed points where s = 0. We refer to P + as a flipping projector.
Projectors Z ± ,Z ± analogous to P ± can be defined for spinors on M. Consider a Dirac spinor Ψ on M (see appendix A for a summary of our conventions). Its left-handed and right-handed components are ψ α andψα. Given two such spinors Ψ 1,2 we can define the SO(4) invariant product Ψ 2 Ψ 1 = ψ 2 ψ 1 +ψ 2ψ1 . The projectors L = 1 2 (1+γ 5 ) and R = 1 2 (1−γ 5 ) on the left-and right-handed components of Ψ are compatible with the product of two spinors, that is: If a vector field v with isolated fixed points related to s ands exists on M, we can define a new projector acting on Dirac spinors We have Z 2 + = Z + andΨ 2 Z + Ψ 1 = Z + Ψ 2 Ψ 1 , the projector Z + is then compatible with the inner product. Starting from Z + we can find more projectors The image of Z + (orZ + ) comprises spinors that are left-handed at the plus fixed points of v and right-handed at the minus fixed points. There is a direct relation between the Killing spinors ζ i andχ i and the projectors we introduced. Indeed we can construct a Dirac spinor z i which satisfies Z + z i = z i as a consequence of (3.2) or (3.4). This is a first indication of the strict relation between supersymmetry and the flipping bundles constructed here. Similarly we construct a Dirac spinoržǐ out of the auxiliary spinorsζǐ,χǐ: which, using (3.7), is seen to satisfyZ −žǐ =žǐ. Let Ψ 1,2 be spinors such that Z + Ψ 1,2 = Ψ 1,2 . We can construct the two-form which satisfies P + ω = ω with P + defined above in (4.1). This establishes a relation between the bundle of two-forms in the image of P + and the spinor bundle in the image of Z + . In the same way, starting from two spinors such thatZ − Ψ 1,2 = Ψ 1,2 the two-formΨ 2 γ µν Ψ 1 is in the image of P − = 1 − P + (similar relations can be found using Z − orZ + ).

Cohomological fields
Here we briefly review the rewriting of the vector multiplet in terms of cohomological (or twisted) fields introduced in [1]. We then proceed to constructing the appropriate twisted fields for the case of the hypermultiplet.

Vector multiplet
The vector multiplet comprises a complex scalar X, the gauge field A, an auxiliary scalar D ij and gauginos λ i ,λ i . There is an invertible map between these component fields and cohomological (twisted) fields. We present the details of the map in Appendix C. The twisted fields arrange themselves into various multiplets. There is one long multiplet made of the gauge field A, a scalar φ, and a Grassmann one-form Ψ both in the adjoint of the gauge group. Supersymmetry acts on these fields as follows: Here F is the field strength of A and ι v denotes contraction with the vector field v.
The rest of the twisted fields arrange in two short multiplets. One is formed by a scalar ϕ and a Grassmann scalar η and the second is formed by a Grassmann two-form χ satisfying P + χ = χ and a second two-form H also satisfying P + H = H. All these fields are in the adjoint of the gauge group. Supersymmetry acts as follows: With canonical reality conditions on the scalars X * =X, the field ϕ is real. On the other hand, the reality properties of φ involve a non-trivial shift depending on (s,s, ϕ): (4.10)

Hypermultiplet
The hypermultiplet cohomological fields are fermions q, b, c, h transforming in the fundamental of The fields q and c are related to the component fields of Section 2.3 by the following map: where we show explicitly their left-and right-handed components. The index n transforms in the fundamental of Sp(k).
There are two remaining cohomological fields. The first b n is given by: The last field h is related to the variation of b under supersymmetry h = −i δb: (4.14) In this formula T is a combination of supergravity background fields and derivatives of Killing spinor bilinears, where b is a one-form satisfying ι v b = 0, which parametrizes remaining freedom in choosing the supergravity background (see Appendix B).
The map from the hypermultiplet components to the cohomological variables has a smooth inverse. For the scalar q ni , this is where z is the Dirac spinor built out of the Killing spinors defined in (4.5). The map from b, c to the ordinary component fields ψ,ψ is 17) and the one for the ordinary auxiliary field is Equations (2.11) and (B.1) imply the following reality conditions on Grassmann-even spinors: The twisted fields for the hypermultiplet we defined above are spinors; consequently, a spin structure is necessary in order to define them. This is different from the case of the vector multiplet whose twisted fields can be defined provided the manifold is orientable and admits a U The cohomological fields we introduced split into two separate multiplets under supersymmetry: where, just like in (28) of [1], the supersymmetry algebra encodes gauge transformations G Φ with respect to the field Φ = (iι v A + φ). Here A and φ are fields in the twisted vector multiplet (see Appendix C). As in section 2.3, the object G Φ acts on fields according to their representation, e.g.
Few comments are in order: • It follows from (4.8) that G Φ is δ-closed, which ensures the closure of the algebra (4.20).
Indeed, δ acts on the cohomological fields as an equivariant differential, namely: • The action of supersymmetry commutes with the projectors Z ± ,Z ± . This is because the vector field that enters in their definition is the same as the Killing vector that appears in the supersymmetry variations (in general it is sufficient for the two vector fields to commute).
• The gauge field A and the scalar field φ always appear in the combination Φ. When discussing localization we will see that, after gauge fixing, Φ will be a Coulomb branch modulus appearing in one-loop determinants. If Φ is part of a dynamical vector multiplet, it will be integrated over. Φ could also arise from a background vector multiplet, in which case it will remain as a free parameter in supersymmetric observables. For instance it can be identified with a flavour fugacity.

Hypermultiplet Lagrangian in cohomological fields
We have shown in section 2.3 that the usual quadratic Lagrangian for a hypermultiplet coupled to a rigid N = 2 supergravity background is the δ variation of V G in (2.20). In terms of the cohomological fields (q, b, c, h), the deformation term V G reads as follows: where, in the expression above, Sp(k) gauge/flavor indices are contracted as Ψ 1 Ψ 2 = (Ψ 1 ) m Ψ m 2 . Here we used the short-hand notation: By taking a δ variation of V G , we obtain the Lagrangian for the hypermultiplet L = L B + L F : (4.25) and
In terms of Killing spinor bilinears, this corresponds to respectively setting eithers = 0 and s = constant ors = constant and s = 0. The Killing vector v vanishes in both cases, yielding a non-equivariant differential δ satisfying δ 2 = 0. In turn, the hypermultiplet cohomological complex fulfills δh = δc = 0, while Dirac spinors become pure Weyl spinors: Consequently, the deformation term V G becomes (4.28) The objects V G andṼ G are deformation terms reproducing the features of topologically twisted hypermultiplets, see for instance [57] and references therein.

Cohomological localization
In this section we perform supersymmetric localization in the cohomological language described previously. The localization argument is the standard one: let Z hyper be the hypermultiplet partition function, which is a path integral with functional weights e −S hyper . We then make the replacement

BPS Locus
On the BPS locus, the fermionic fields b, c and their supersymmetric variations are vanishing. In particular, this yields h = 0 and Since the field Φ is complex and the Killing vector v is real, by imposing reality conditions on the above equations we end up with a trivial BPS locus: q = 0.

One-loop determinant from index theorem
The deformation term we employ is By construction, V loc leads to a positive definite deformation Lagrangian L loc = δV loc . Explicitly, The deformation term V loc can be recast in the quadratic form with entries At plus fixed points we have while at minus fixed points we find Consequently, the symbol σ [D 10 ] coincides with that of a chiral Dirac operator whenevers = 0 or s = 0, ensuring ellipticity of D 10 . On the other hand, implying that D 10 is not elliptic in the patch where s =s, because σ [D 10 ]| s=s = 0 for p µ = v µ = 0.
The operator D 10 is then transversally elliptic with respect to the Killing vector v, as the symbol σ [D 10 ] is everywhere invertible for any p µ = 0 such that p · v = 0.
The hypermultiplet contribution to the partition function is encoded into the index of the transversally elliptic operator D 10 [28,29]: ind(D 10 )(t) = x : x=x where t ∈ R, while x is the image of the coordinates x under the torus action induced by H and the sum is over the fixed points x = x. In a neighborhood of a plus fixed point, the metric becomes flat and the manifold can be parametrized by a pair of complex coordinates (z 1 , z 2 ). In (5.10), bosons q contributes to ind(D 10 )(t) with a plus sign, whereas fermions b contributes with a minus sign.
The Killing vector v in a neighborhood ofs = 0 reads phases q i with i = 1, 2: In a neighborhood of a plus point, the denominator entering the index formula (5.10) is where q i = q −1 i is the complex conjugate/inverse of the phase q i . We now need the action of H upon the spinors q and b. We then embed U (1) × U (1) into SU (2) + × SU (2) − ∼ Spin(4) to see how spinors transform under H. We define (5.14) Then, coordinates z and spinors Ψ = {q, b} transform as with g ∈ SU (2) + × SU (2) − being the spinor representation of the torus action L v . At a plus point, q is left-handed and b is right-handed: q = L q and b = R b. Then, the action of L v upon q, b is The operator H also contains G Φ , which acts non-trivially on q, b. Thus, for a hypermultiplet in the representation R of the gauge group, the index formula (5.10) yields ind(D 10 )| plus point = √ q 1 q 2 This index translates into a functional determinant by expanding ind(D 10 )| plus point in powers of q 1 , q 2 and converting the corresponding series into an infinite product. The latter requires regularization, which is a delicate matter. For instance, in cases linked to five-dimensional manifolds, the regularization was established in [37,[42][43][44]58]. Here, we will examine diverse regularizations.
We define: The difference between the two regularizations is Here, ρ is a sum over weights in the representation R and Φ 0 is with a 0 being a Coulomb branch moduli and k + ( 2 ) parametrizing the flux contribution at the plus fixed point. As in [1,29] (5.23) These are 1-loop contributions to the hypermultiplet partition function at a plus fixed point. We used Barnes multiple zeta and gamma functions [47,59], to regularize the infinite product that gives Z HM (a 0 , k + ). For instance, in the case of squashed S 4 worked out in [29], the regularization chosen for the plus point contribution is ++: In a neighborhood of a minus point, the denominator entering the index formula (5.10) is formally the same as the one computed at the plus point: At s = 0, there is a chirality flip with respect to the plus fixed point ats = 0. Indeed, q is right-handed and b is left-handed: q = R q and b = L b. Therefore, By taking into account the action of G Φ at a minus fixed point, the index formula (5.10) provides Again, we have four possible series expansions:  Using the results of [1], we can write down the complete partition function for an equivariantly twisted N = 2 gauge theory coupled to matter with p (respectively (l − p)) plus (minus) fixed points, giving formula (1.1).

Example: hypermultiplets on squashed S 4
As an example, let us apply (1.1) to the specific case of hypermultiplets on a squashed four-sphere S 4 ,˜ studied in [29]. This manifold possesses a plus and a minus fixed point, respectively dubbed north and south pole. In fact, S 4 ,˜ is a four-dimensional ellipsoid embedded in R 5 according to where , are the lengths of the ellipsoid axis, while r is its radius. If = = r, one recovers the round four-sphere S 4 , whose isometry group is SO (5). For arbitrary , , the group SO(5) is broken to SO(2) × SO (2), which is a real torus action rotating (x 1 , x 2 ) and (x 3 , x 4 ). At the fixed points, the equivariant parameters Indeed, we can make contact with the previous subsection by setting z 1 = x 1 +ix 2 and z 2 = x 3 +ix 4 , so that at the north pole of S 4 ,˜ the torus action becomes a complex torus action U (1) × U (1) generated by a Killing vector (5.11) with (+) 1 ,˜ we have vanishing fluxes, then k + = k − = 0, and for a hypermultiplet in the representation R of the gauge group we find Z HM Analogously, at the south pole we have a U (1) × U (1) generated by a Killing vector (5.26) with Combining the two contributions we obtain a Υ-function depending on the Coulomb branch parameter a 0 and on the lengths of the ellipsoid axis ,˜ : which matches [29]. We recall that the definition of the Υ-function is

Example: hypermultiplets on S 2 × S 2
As another example, we apply (1.1) to hypermultiplets defined on the product of two spheres of radii −1 1 , −1 2 , which we denote by S 2 1 × S 2 2 . Gauge theories on manifolds with such a topology were studied e.g. in [25], as well as in [37] by dimensional reduction from five-dimensional toric

A.2 Differential geometry
Greek letters µ, ν, . . . are used to denote curved indices and Latin letters a, b, . . . to denote frame indices. Let e a µ be the orthonormal vielbein corresponding to the metric g µν . We denote the Levi-Civita connection by ∇ µ . The corresponding spin connection is (A.14) The Riemann tensor is The Ricci tensor is R µν = R µρν ρ , and R = R µ µ is the Ricci scalar. In these conventions a round sphere has negative Ricci scalar.
(A. 16) The Lie derivative of spinors along a Killing vector v is given by [61] L

B Supergravity background solutions
In this appendix, we write down explicitly the supergravity background on which the Killing spinor equations are solved by the Killing spinor constructed in section 3.1. This is a summary of the results of [1].
The background supergravity fields are the metric g µν , a SU (2) R connection V µ i j , a one form G µ , a two-form W µν , a scalar N , a closed two-form F µν , and a scalar S ij transforming as an SU (2) R triplet. The Killing spinors do not determine the supergravity background completely. In the formulas below, this freedom is parametrized by two one-forms. One is G µ and is arbitrary, the other is denoted b µ and satisfies ı v b = 0. The reality conditions for the Killing spinors (2.14) are compatible with the following behavior of the background supergravity fields under complex conjugation: We will make use of various spinor bilinears. Besides the SU (2) R singlets s,s and v defined in (2.5) and (2.6) and used extensively throughout the paper, we introduce v (ij) We also define the combination Θ ij µν = 2 s +s s 2 +s 2 Θ ij µν + Θ ij µν .

(B.3)
The two-form W µν and the SU (2) R connection (V µ ) ij are given by: The graviphoton field strength F µν is: Here K is the constant defined in section 3 and such that (s +s − K)/(s 2s2 ) is smooth at the fixed points of v. The scalars S ij are: All the fields (C.1) are differential forms in the adjoint of the gauge group. They are all singlets under the SU (2) R symmetry. It follows from the definitions above that the two-forms χ µν and H µν are in the image of the projector P + .