Rigid 6D supersymmetry and localization

We construct rigid supersymmetric theories for interacting vector and tensor multiplets on six-dimensional Riemannian spin manifolds. Analyzing the Killing spinor equations, we derive the constraints on these theories. To this end, we reformulate the conditions for supersymmetry as a set of necessary and sufficient conditions on the geometry. The formalism is illustrated with a number of examples, including manifolds that are hermitian, strong Kaehler with torsion. As an application, we show that the path integral of pure super Yang-Mills theory defined on a Calabi-Yau threefold M_6 localizes on stable holomorphic bundles over M_6.


Introduction
Recently, supersymmetric field theories on curved backgrounds have regained considerable attention, in particular for the application of localization techniques in the computation of indices, partition functions and Wilson loops, see e.g. [1,2,3,4,5]. A systematic approach to the study of rigid supersymmetry in four-dimensional curved space has been initiated in [6] and further developed in [7,8,9,10,11]. Holographic applications of these theories have been studied in [12,13] by embedding the curved space at the boundary of an asymptotically AdS space. A similar analysis of supersymmetric theories has been performed for curved five-dimensional spaces in [14,15] and in [16] for three-manifolds. The construction of these curved theories is based on the existence of an underlying off-shell supergravity in which the full off-shell supergravity multiplet is set to classical background values. The consistency of this limit requires the existence of solutions of the corresponding Killing spinor equations which in turn poses non-trivial constraints on the background fields. 1 In the present paper we will focus on rigid supersymmetric theories in six-dimensional Riemannian spin manifolds. We will study the coupling of off-shell Yang-Mills (YM) multiplets to a number of uncharged on-shell tensor multiplets. Such couplings have been constructed in 6D flat space-time of Minkowski signature [18]. In view of the applications mentioned above, we shall extend this model to curved background and in Euclidean signature. One way to construct these theories is to start from the Euclidean version of off-shell supergravity coupled to YM and tensor multiplets, and then take the rigid limit in which the fields in the supergravity multiplet are frozen in a manner consistent with supersymmetry. In particular, the vanishing of the fermionic fields in this multiplet would yield the Killing spinor equations. However, while the N = (1, 0) off-shell supergravity theory in 6D with Minkowskian signature is well known [19,20], thanks to the superconformal tensor calculus methods [19], the general couplings to Yang-Mills and multi-tensor multiplets is still missing. Extra complications may arise in the passage to Euclidean signature. This suggests an alternative approach in which we begin by formulating the Euclidean version of the Killing spinor equations with suitable background fields. We distinguish two cases depending on whether these equations descend from an off-shell theory with (θ = 0) or without (θ = 0) R-symmetry gauging. In the former case, additional Killing spinor equations have to be imposed. Next, we translate the model of [18] to Euclidean signature and then elevate the supersymmetry parameter to be Killing. Performing a Noether procedure which takes this into account then produces the full dependence of the couplings on the nontrivial background fields. Following this approach we construct the non-trivial supersymmetric tensor/vector couplings on a curved background.
Next, we systematically analyse the Killing spinor equations in six-dimensional Riemannian spin manifolds M 6 using G-structures and we derive the constraints for the existence of non-trivial solutions. We reformulate these constraints as a set of conditions on the geometry. The necessary and sufficient conditions for minimal supersymmetry are given in equations (5.27), (5.35) below as constraints on an SU(3) structure. In particular for a theory whose background descends from a supergravity with R-symmetry gauging (θ = 0), these constraints require M 6 to be hermitian, strong Kähler with torsion, but not necessarily conformally balanced. For θ = 0 on the other hand, the manifold M 6 is not necessarily hermitian; we also show that it cannot be (strictly) nearly Kähler.
In either case there is a crucial difference with the structures arising in supersymmetric heterotic compactifications: The Killing spinor equations (2.1), (2.3) below are of the same form as those for the vanishing of the gravitino and dilatino variation in the heterotic theory, provided we set the background field E • m to zero. For θ = 0, this would impose the additional condition that the manifold should be conformally balanced, which is indeed a necessary condition in supersymmetric heterotic compactifications. However as already mentioned for the rigid supersymmetric theories considered here the conformally balanced condition need not be imposed. Indeed as we will see in the following, a rigid (1,1)-supersymmetric theory can be defined on the round S 3 × S 3 which is hermitian, strong Kähler with torsion but not conformally balanced. For θ = 0, the manifolds need not to be hermitian, while hermiticity is a necessary condition in supersymmetric heterotic compactifications.
As a special case of our formalism we obtain pure super Yang-Mills (SYM) on a Calabi-Yau threefold M 6 by setting to zero all background fields except for the metric. Applying the localization procedure then yields the result that the SYM path integral localizes on stable holomorphic bundles on M 6 .
The rest of the paper is organized as follows: In section 2, we present the Killing spinor equations obtained from off-shell supergravity in the superconformal tensor calculus. In section 3, we construct the Lagrangian and supersymmetry transformation rules for Yang-Mills theory on curved Riemannian manifolds M 6 with background fields admitting solutions to the Killing spinor equations. Section 4 extends the construction to include a number of tensor multiplets interacting by Chern-Simons couplings with the vector fields. In section 5 we analyse the constraints to be satisfied by the background fields in order to admit non-trivial solutions. We reformulate the conditions for minimal supersymmetry as a set of necessary and sufficient conditions on a suitable SU(3) structure. Moreover, we examine the conditions for (1,1) and (1,2) extended supersymmetry. The formalism is illustrated with a number of examples, including Calabi-Yau (CY) threefolds, the round S 3 × S 3 and T 2 bundles over noncompact K3's. In section 6 we apply the localization procedure to SYM theory defined on a CY M 6 and show that the path integral localizes on stable holomorphic bundles over M 6 .

Killing spinor equations I
Our starting point is the off-shell formulation of six-dimensional supergravity obtained from superconformal tensor calculus with the dilaton Weyl multiplet coupled to a linear multiplet after particular gauge fixing [19,20,21]. The resulting bosonic fields including the space-time metric will constitute the background fields in the rigidly supersymmetric field theories to be constructed in this paper. As discussed in the introduction, when deriving rigid supersymmetric theories from supergravity one has to impose the vanishing of the supersymmetry variation of all fermionic fields from the supergravity multiplet. This defines the Killing spinor and poses constraints on the bosonic background fields. In our conventions (cf. appendix A) and switching from Minkowsi to Euclidean signature, the Killing spinor equations imposing the vanishing of the gravitino fields' variation, read with covariant spinor derivatives etc., a background vector field V • m , and a closed three-form mn] . In passing from Minkowski to Euclidean signature, we have replaced the symplectic Majorana-Weyl spinors by complex Weyl spinors ǫ, ζ, of positive chirality, thus reducing the manifest R-symmetry from Sp(1) to U(1). In particular, unlike in Minkowski signature, the spinors ǫ and ζ are no longer related by complex conjugation. If the Killing spinor equations (2.1) descend from a supergravity theory in which the R-symmetry is gauged, there are additional Killing spinor equations to be taken into account. Namely, imposing the vanishing of the remaining fermion variations in that case furthermore implies where L • denotes the background dilaton and E • m is the Hodge dual of the background five-form field strength 2 In the following we will construct rigidly supersymmetric field theories on backgrounds that allow for non-trivial solutions of the Killing spinor equations (2.1), (2.3). In slight abuse of standard notation, we will refer to a background as N = (p, q) supersymmetric when equations (2.1), (2.3) admit p independent solutions ǫ and q independent solutionsζ. 3 Flat space with vanishing background fields would thus correspond to N = (4, 4). Note the unlike the analogous structures in four dimensions [22,6] the Killing spinor equations (2.1), (2.3) for ǫ andζ are entirely decoupled. For the study of minimal supersymmetry we can thus consistently setζ = 0. While we will give a full-fledged analysis of the Killing spinor equations in section 5 below, let us finish this section by deriving a few immediate consequences for backgrounds that are at least N = (1, 1) supersymmetric. Straightforward computation gives rise to the relations for the Killing spinor bilinears In particular, the first equation shows that ξ m is a Killing vector of the background 2 Since we are in curved background geometry, the completely antisymmetric ε mnklpq is understood to be defined as a tensor, i.e. it carries an implicit factor of √ g • −1 .
metric. Furthermore, integrability of the Killing spinor equations (2.1) yields Contracting both equations with γ n and summing them up implies Together with (2.5) we obtain and anticipating the result of the full analysis that the background field strength V • mn vanishes (cf. equation (5.15) below), shows that the scalar combination ξ H is in fact constant.

Vector multiplets
We will now construct rigidly supersymmetric gauge theories on a background that admits non-trivial solutions of the Killing spinor equations (2.1), (2.3). The resulting couplings are motivated by the coupling [21] of a single off-shell vector multiplet to the off-shell supergravity after integrating out the auxiliary fields. However, since the spinor properties, in particular their R-symmetry representation content change with the passage from Minkowski to Euclidean space, we will perform the full construction from scratch. Consider a set of vector multiplets {A r m , λ r , ν r } , labeled by an index r, described by a Lagrangian is the standard Yang-Mills field strength with structure constants f st r , and the spinor covariant derivatives are defined as in (2.2). B • mn is the background two-form potential, the background field E • m has been defined in (2.4) and satisfies ∇ m E • m = 0. The θ r denote a set of coupling constants which single out one of the vector fields and descend from an R-symmetry gauging of the underlying supergravity [21]. Accordingly, their gauge invariance requires an abelian factor within the Yang-Mills gauge group. (For θ r = 0 on the other hand there is no restriction on the gauge group.) Again, the spinors λ r and ν r are not related by complex conjugation, such that the variation of the vector fields and the Lagrangian are in fact complex, as usual in Euclidean supersymmetry, cf. [23,24,25], see also the discussion in section 6.
It is straightforward to check that the Lagrangian (3.1) is invariant under the supersymmetry transformations rules δA m r =ν r γ m ǫ −ζγ m λ r , provided the supersymmetry parameters satisfy the Killing spinor equations (2.1), (2.3). The latter equations are only required in case the θ r are non-vanishing. For θ r = 0, the only condition on the supersymmetry parameters are equations (2.1). Let us note that variation of the gauge connection in the Dirac term a priori gives rise to a quartic fermion term in the variation of (3.1) Fierzing in λ t ) (ν s shows that upon using that γ m γ kln γ m = 0. By virtue of antisymmetry of the structure constants, also the quartic variation (3.3) thus vanishes. It is instructive to work out the N = (1, 1) algebra of supersymmetry transformations (3.2) which takes the form 4 with parameters ξ n and ξ H defined in (2.6) and (2.9), respectively, and defining the standard Lie derivative on spinor fields Interestingly, we find that the algebra closes not only into the standard translations and gauge transformation but also into a global U(1) acting on the fermions with parameter ξ H that is constant due to (2.9). 5 4 Part of these and the following calculations have been facilitated by use of the computer algebra system Cadabra [26,27]. 5 As we will see in the following the full analysis of the Killing spinor equations shows that the constant ξ H vanishes in the case of backgrounds for which E • m = 0, but is generically non-vanishing. However, as the Lagrangian (3.1) possesses the global U (1) appearing under closure on the r.h.s. of (3.5), the form of the algebra in any case is consistent, whether or not ξ H is zero.
To summarize, the Lagrangian (3.1) defines non-abelian gauge theory on a curved Euclidean background with supersymmetry parameters defined as solutions of the Killing spinor equations (2.1), (2.3).

Tensor multiplets
We will now generalize the construction of the previous section to also include tensor multiplets. To this end, we will first derive the field equations for a set of free tensor multiplets on curved background and then proceed to introduce interactions with the vector multiplets. Since supersymmetry is expected to impose self-duality equations for the tensor fields that do not allow for a standard Lagrangian description, we will perform the entire construction on the level of the field equations. Moreover, since there is no off-shell formulation for self-dual tensor fields in six dimensions, their couplings on a curved background cannot directly be read off from an underlying supergravity but have to be constructed from scratch.
Let us consider n tensor multiplets {φ I , B I mn , χ I ,ψ I }, labelled by an index I, with supersymmetry transformations denotes the abelian tensor field strength which decomposes into its selfdual and antiselfdual part according to Closure of the supersymmetry algebra on the tensor fields gives rise to the selfduality equation relating the tensor field strength to the background three-form. Under the supersymmetry transformations (4.1), the self-duality equation transforms into the fermionic field equations provided the supersymmetry parameters ǫ,ζ satisfy the Killing spinor equations (2.1). In turn, these equations under supersymmetry transform into the scalar field equation where again (2.1) is required. For consistency, we may check the N = (1, 1) algebra of supersymmetry transformations (4.1) which close into with the same translation and U(1) parameters ξ m , ξ H , as in (3.5), and the additional tensor gauge parameter Λ m I ≡ −ξ n B nm I + ξ m φ I . Finally, we may couple the system of tensor multiplets to the non-abelian vector fields introduced in the previous section by introducing Chern-Simons interactions of the form In presence of d I rs the supersymmetry transformation rules (4.1) change to and similar forψ , which again can be verified by checking closure of the algebra. The field equations change into and can be checked to vary into each other under supersymmetry, provided the supersymmetry parameters ǫ,ζ satisfy the Killing spinor equations (2.1), (2.3). This generalizes the field equations found in flat space [18] for a tensor/vector system interacting by Chern-Simons couplings to curved background. In deriving this result, we had to use the gaugino field equation coming from (3.1). I.e. unlike in flat space where the YM sector can be kept off-shell in the presence of Chern-Simons couplings [18], the non-vanishing background field H • mnk relates the dynamics of the vector and tensor sector.
It would be interesting to further generalize the interactions by allowing for minimal couplings between vector and tensor fields with the latter charged under a non-abelian gauge group. The corresponding supersymmetric system in flat space has been constructed in [28,29] and allows for an action modulo the standard subtleties concerning self-dual three-forms. Replacing the constant supersymmetry parameters of that system by solutions to the Killing spinor equations (2.1) should lead to a deformed version of these tensor/vector interactions on curved space. It would also be interesting to analyze if the curved space models can be generalized to include couplings to hypermultiplets, by applying the same procedure to the flat space models of [30,31].

Killing spinor equations II
So far, we have derived the couplings and field equations for vector and tensor multiplets on a curved background under the assumption that the supersymmetry parameters satisfy the Killing spinor equations (2.1), (2.3). In this section, we will analyze these Killing spinor equations and derive the resulting constraints for the background fields, in particular the background geometry. We obtain the necessary and sufficient conditions for the background geometry in order to admit at least one Killing spinor, i.e. for the theory to have (1,0) supersymmetry. In subsections 5.2, 5.3 we derive stronger conditions for the existence of several supercharges. As this section deals exclusively with the background structure we will for notational simplification omit the superscript ' • ' which so far has distinguished the background structures from the matter fields. I.e. (for this section only) we pass to H • → H, L • → L, etc.

(1,0) supersymmetry
The Killing spinor equation (2.1) can be thought of as a parallel-transport equation, ∇ ′ ǫ = 0, with respect to a metric-compatible connection ∇ ′ with torsion given by the background three-form H. Therefore the holonomy of ∇ ′ (in the spinor representation) is a subgroup of G, the stabiliser group of ǫ, and M 6 admits a G-structure [32]. Since ǫ transforms in the 4 of the structure group Spin(6) ∼ = SU(4) of the Riemannian spin manifold M 6 , the stabiliser is G = SU(3) as can be seen by the decomposition 4 → 3 ⊕ 1 under SU(4) → SU(3). Hence M 6 must admit an SU(3) structure.
The topological obstruction for an oriented Riemannian six-dimensional manifold M 6 to admit an SU(3) structure is that it should be spin (see e.g. [33]). Since in the present paper we are assuming that M 6 is spin, there is no additional topological condition imposed by the existence of an SU(3) structure on M 6 . However, as we will see in the following, supersymmetry imposes additional geometrical conditions in the form of constraints on the torsion classes of the SU(3) structure. In order to systematically analyze these constraints we will now reformulate the SU(3) structure in terms of certain globally defined forms on M 6 .
From equation (2.1) it can easily be seen that either ǫ vanishes identically, or it is nowhere-vanishing. Assuming that ǫ is not identically zero, let us parametrize: where η is a unimodular Weyl spinor of positive chirality, and A is a function on the six-dimensional manifold. For the purposes of this section we will assume that η is a commuting spinor: we have the freedom to make this choice since the Killing spinor equations are linear in ǫ.
The existence of the unimodular spinor η allows us to define the bilinears where our spinor conventions are explained in detail in appendix A. In particular, it follows from the properties of Weyl spinors in six Euclidian dimensions that J is a real two-form and Ω is a complex three-form which is imaginary self-dual. Using the Fierz identities it can further be shown that J m n (where we have raised one index with the metric) defines an almost complex structure: with respect to which Ω is (3, 0): are the projection operators onto the (1,0) and (0,1) parts. Moreover, it can be seen that The globally defined forms J, Ω subject to the above conditions can be seen to specify an SU(3) structure on M 6 [34]. The intrinsic torsion parametrizes the failure of the spinor η to be covariantly constant. In the case of an SU(3)-structure manifold M 6 , the intrinsic torsion decomposes into five modules (torsion classes) W 1 , . . . , W 5 : where W 1 is a complex scalar, W 2 is a complex (1,1)-traceless form, W 3 is a real traceless (2, 1) + (1, 2) form, W 4 is a real one-form and W 5 is a (1,0) form. In terms of SU (3) representations, Equivalently these torsion classes appear in the SU(3) decomposition of the exterior derivatives of J, Ω. Explicitly we have, Intuitively, the intrinsic torsion characterizes the geometry of M 6 and parameterizes the failure of the manifold to be of special holonomy -which can also be thought of as the failure of the closure of J, Ω. As a final ingredient before we proceed to the analysis of the Killing spinor equations, we will need the decomposition of the background three-form H from (2.1) with respect to the reduced structure group SU(3). Explicitly we have where the normalization above has been chosen so that

2W
(1,0) 4 (5.14) Moreover it follows that the background vector field is exact, 15) and the irreducible components of H are determined in terms of the torsion classes. Explicitly, in form notation: (5.16) For the remaining analysis, it will make a crucial difference whether the parameters θ r in the Lagrangian (3.1) are vanishing or not. Only in the latter case do we have to impose the Killing spinor equations (2.3) on the supersymmetry parameters. This leads to further restrictions on the background geometry, to be discussed in this section. The case θ r = 0 is examined separately in section 5.1.2.
Without loss of generality, we may assume in this section the background field L to be non-vanishing. Otherwise, it is straightforward to see that the Killing spinor equation (2.3) implies that also E m = 0 (taking into account that E m is real). For the Lagrangian (3.1) and the supersymmetry transformation rules (3.2), this amounts to setting θ r = 0, and brings us back to the situation discussed in the next section. With the field L non-zero, the second Killing spinor equation (2.3) implies that the scalar component of H must vanish: As a consequence of the above and (5.16) there is now an additional constraint on the torsion classes: Taking the above into account, it follows that the H field is given in terms of the torsions classes: It follows that the closure of H can be written equivalently as: Eq. (5.19) can also be written as: where we have used If W 4 is exact (which need not be true in general), W 4 = dφ, the manifold is called conformally balanced. In that case (5.23) can be written as: Finally the background field E m is given by:  Conditions (5.27) have the following geometric interpretation: The vanishing of W 1 , W 2 is equivalent to the condition that the almost complex structure is integrable, i.e. the condition that M 6 is hermitian. 6 The second line of (5.27) can be rephrased as the statement of proportionality between the Lee forms L Ω and L J of Ω and J respectively. More explicitly the Lee forms are defined by: 6 A manifold equipped with a Riemannian metric g mn and an almost complex structure J m n is called almost hermitian if the almost complex structure is an orthogonal transformation with respect to the metric: J m p J n q g pq = g mn . (5.28) If moreover the almost complex structure is integrable, the manifold is called hermitian.

(5.30)
On the other hand, taking (5.10) into account, it follows that hence the second line of (5.27) can be rewritten in terms of the Lee forms as: The third line of (5.27) is the condition that the Hermitian manifold M 6 is strong Kähler with torsion (SKT). One immediate consequence of the SKT condition is that M 6 cannot be special hermitian [35]. 7 The first two lines of (5.27) also appear as necessary conditions for supersymmetric heterotic compactifications [37,38,39]. This is not surprising given the fact that the Killing spinor equations (2.1), (2.3) are of the same form as those for the vanishing of the gravitino and dilatino variation in the heterotic theory, provided we set the background fields V , E to zero and we identify L = e −2φ , where φ is the dilaton. Crucially, however, this would impose the additional condition that the torsion class W 4 is exact: W 4 = dφ as follows from eq. (5.26); in particular the manifold is constrained to be conformally balanced. We stress again that for the rigid supersymmetric theories considered here the conformally balanced condition need not be imposed.

θ r = 0
With all parameters θ r vanishing, the background fields L and E m completely decouple form the Lagrangian (3.1) and supersymmmetry transformation rules (3.2) and equations (2.3) do no longer have to be imposed on the supersymmetry parameters. As a result, the constraints on the background geometry are relaxed w.r.t. the last section and in particular the torsion class W 1 is no longer zero, cf. (5.18). Equations (5.19), (5.23) for the H field now get modified to: and: A special hermitian six-manifold is a manifold which is both half-flat (and therefore can be lifted to a seven-dimensional manifold of G 2 holonomy [36]) and hermitian. Equivalently, a special hermitian six-manifold is a manifold which admits an SU (3) structure whose only non-zero torsion class is W 3 .
Finally, the necessary and sufficient conditions can be summarized as The fact that now W 1 is not necessarily zero implies that the almost complex structure is not integrable in general and the manifold M 6 is not necessarily hermitian. This is a major difference from the L = 0 case discussed above. The second line above has the same interpretation as in section 5.1.1 as the proportianality between the Lee forms of J and Ω. The last line in (5.35) is the condition of closure of H; for W 1 = 0 it reduces to the third line in (5.27).
As we will now show, the system (5.35) does not admit as solution any (strictly) nearly Kähler manifold. Nearly Kähler manifolds are SU(3)-structure manifolds whose only nonvanishing torsion class is W 1 . Equivalently, they can be defined as sixdimensional manifolds admitting a Killing spinor, ∇ m η = 1 4 W 1 γ m η c , cf. eq. (5.8), so that they are Einstein, R mn = 5 4 |W 1 | 2 g mn , and the metric cone over M 6 is a G 2holonomy manifold [40]. In the case of a nearly Kähler M 6 it can immediately be seen from eqs. (5.10) that W 1 is constant. Performing a constant phase redefinition of Ω we can take W 1 = iw, with w a real constant. Eqs. (5.10) then reduce to: while dReΩ = 0 follows from the above. Moreover the background field H is given by and it is not closed, unless w = 0 and M 6 is a Calabi-Yau.

(1,1) supersymmetry
In the case of vanishing θ r , we have seen that the Killing spinor equations (2.3) do no longer have to be imposed. The remaining equations (2.1) are related by simple transposition, i.e. any solution ǫ of the first equation in (2.1) defines a solution to the second equation by setting ζ ≡ ǫ. In our notation, this means that the resulting theory in fact is N = (1, 1) supersymmetric rather than just N = (1, 0). 8 In contrast, for non-vanishing θ r , imposing (1,1) supersymmetry in general imposes further constraints on the background geometry, to be discussed in the rest of this section. In this case, we have to look for necessary and sufficient conditions so that the Killing spinor eqs. (2.1), (2.3) are obeyed for a pair of not identically vanishing spinors ǫ, ζ. We have already seen that (1,0) supersymmetry imposes that the background gauge field is exact, cf. eq. (5.15). Hence we may gauge away V by a suitable redefinition of the dynamical fermions of the theory. Setting V = 0 in (5.15) then implies that A is constant. In this section we will take A = 0 without loss of generality so that, cf. eq. (5.1), ǫ = η . where we have set L = e −2φ . Hence the manifold is constrained to be conformally balanced. As a consequence it also follows that the scalar ξ H defined in (2.9) vanishes. More generally, any (commuting) spinor ζ of positive chirality can be written as: where K is a (1,0)-form. 9 It can then be seen taking the formulae of appendix B into account that the Killing spinor equations are equivalent to the following set of conditions: 42) in addition to the conditions derived in section 5.1.1. Note that the penultimate line above implies that the Lie derivative of the background three-form field vanishes, L K H = 0; the second line can also be written as L K L = 0; the third line is equivalent to K being Killing. It also follows from the above that the norm of K is constant, ∂ m |K| 2 = 0. Hence for K = 0 the triplet (K, J, Ω) defines on M 6 an SU(2) structure. Normalizing |K| 2 = 2 the SU(2) structure can be given equivalently as the triplet (K, j, ω), where [39,41]: is a complex (2,0) form and is a real (1,1) form obeying the compatibility conditions If desired, the last line in (5.42) may be expressed in terms of SU(2) torsion classes. Note again that choosing E m = 0 in the first line of (5.42) implies that the manifold is conformally balanced and the scalar ξ H defined in (2.9) vanishes.

Examples
There are many examples of six-dimensional manifolds obeying the necessary and sufficient conditions (5.27), (5.35). One obvious class is that of Calabi-Yau manifolds; in this case all torsion classes vanish. A compact non-Ricci-flat, non-conformally-balanced example with (1,1) supersymmetry is that of the round S 3 × S 3 . We will also provide non-compact examples based on the Iwasawa manifold (with (1,1) supersymmetry), as well as certain T 2 bundles over non-compact K3's (with (1,2) supersymmetry). As we will see, however, our rigid-supersymmetric theory cannot be defined on the round six-sphere, at least for θ r = 0. The latter conclusion is in accordance with the representation-theoretic classification of Euclidean supersymmetries [42], as pointed out recently in [17]. Note however that the classification of [42] only considers "spacetimes" whose isometries are simple Lie groups. In particular it does not cover the cases without any isometries, such as e.g. Calabi-Yau manifolds.

The round S 3 × S 3
This example is based on an SU(3) structure given in [43]. We identify S 3 ×S 3 with the group manifold SU(2)×SU (2). The orthonormal frame is given by the SU(2)-invariant one-forms e a , f a , a = 1, 2, 3, satisying The SU(3) structure is given by: so that a basis of (1,0)-forms is given by (e 1 + ie 2 ), (e 4 − ie 5 ), (e 3 + ie 6 ). The metric associated with the structure (5.49) is that of the round S 3 × S 3 : Moreover we obtain W 1 , W 2 = 0 (hence M 6 is hermitian), and which is closed, dH = 0 (equivalently ∂∂J = 0, hence M 6 is strong Kähler with torsion). Furthermore, setting L = 0, E m = 0, the round S 3 × S 3 satisfies the necessary and sufficient conditions (5.35) and thus provides a consistent background for a rigid (1,1)-supersymmetric theory. Note in particular that this example is not conformally balanced: indeed it follows from (5.51) that W 4 is not closed and therefore not exact.

The noncompact Iwasawa manifold
The Iwasawa manifold can be viewed as a compact quotient G/Γ where G is the threedimensional complex Heisenberg group defined as with group product given by ordinary matrix multiplication, and Γ acts as Γ : (5.55) By 'noncompact Iwasawa' we mean the group manifold G (which is homeomorphic to C 3 ), i.e. before taking the Γ quotient. The left invariant forms on G are given by dz 1 , dz 2 , dz 3 + z 1 dz 2 . We can choose the orthonormal frame of one-forms e a , a = 1, . . . , 6 so that e 1 + ie 2 = dz 1 , e 3 + ie 4 = dz 2 , e 5 + ie 6 = dz 3 + z 1 dz 2 .  Imposing closure, dH = 0, is equivalent to which can only be satisfied for the noncompact Iwasawa. This example is conformally balanced since W 4 is exact. Hence by setting the background field E m to zero the noncompact Iwasawa satisfies the necessary and sufficient conditions (5.27) and moreover we obtain a (1,1)-supersymmetric theory. Note that although the vector dual to the (1,0)-form K = e 5 + ie 6 is Killing (corresponding to isometries along the z 3 direction), K is not ∂-closed: ∂K = dz 1 ∧ dz 2 . Hence the conditions (5.46) for (1,2) supersymmetry are not satisfied.

T 2 bundles over noncompact K3
This example is based on the work of [44] (see also [45]). In that reference it was shown that six-dimensional manifolds with SU(3) structure can be constructed as T 2 fibrations over K3 surfaces. The metric on the total space is given by where ds 2 (K3) is the Ricci-flat metric on K3, φ is a function on K3, and α, β are local one-forms on K3 which satisfy dα = ω P , dβ = ω Q with The complex SU(3) structure on the total space is given by where ω, j are the holomorphic (2, 0) form, the Kähler form on the K3 base 10 respectively. In particular ω, j are both closed. Moreover we will assume that ω P , ω Q are anti self-dual ⋆ω P,Q = −ω P,Q , so that 11 By computing dJ, dΩ and comparing with (5.10) it can then be seen that the torsion classes are given by (5.68) From the above and (5.23) we find that the H field is given by The closure of H can then be seen to be equivalent to where ∇ 2 K3 is the (unwarped) Laplacian on K3, and we have defined |ω| 2 = ω * mn ω mn /2. Similarly to the previous example, the above equation can only be solved for noncompact K3's.
Since W 4 is exact we can set the background field E m to zero, hence this example satisfies the necessary and sufficient conditions (5.27) and possesses (1,1) supersymmetry. Moreover let us define the one form K by where z = x + iy, which is (1,0) with respect to the complex structure associated with (5.65). Clearly the vector dual to K is Killing for the metric in (5.63). Furthermore L K φ = 0 since φ only depends on the coordinates of K3 and not on the T 2 coordinate z. We also have dK = ω P + iω Q ∈ H 1,1 (K3), hence ∂K = (dK) 2,0 = 0. Finally from (5.65) we find Therefore all the conditions in (5.46) are satisfied and we obtain a rigid (1,2)-supersymmetric theory.
The round sphere S 6 Whether or not the six-dimensional sphere admits a complex structure is still an open question. It is known however that the six-dimensional sphere does not admit an orthogonal complex structure, i.e. one that obeys eq. (5.28) with respect to the standard 'round' metric [46]. In other words the round sphere S 6 is not a hermitian manifold. This implies that S 6 violates the first of the necessary and sufficient conditions in (5.27) therefore our rigid-supersymmetric theory cannot be defined on S 6 , at least for θ r = 0.

Euclidian 6D SYM and localization
In this section we shall set the parameters θ r and all other background fields to zero, except for the metric of the curved six-dimensional space M 6 . Then the analysis of section 5 implies that all torsion classes vanish and M 6 is a Calabi-Yau threefold 12 . Equivalently, this can be seen by the fact that imposing (6.1), the first Killing spinor equation (2.1) reduces to the condition that the spinor ǫ is covariantly constant, while eq. (2.3) is automatically satisfied. Moreover, as follows from the analysis of section 5.2, this example possesses (1,1) supersymmetry since θ r = 0. The upshot is that Euclidean rigid (1,1)-supersymmetric nonabelian YM theory can be consistently defined on CY threefolds.

Localization
In order to apply the localization procedure, the theory must be invariant under the action of a fermionic operator Q which is nilpotent, Q 2 = 0 (or more generally squares to a symmetry of the theory). Deforming the action by a Q-exact term, leaves invariant the expectation values of Q-closed operators. Hence we may take the limit of zero coupling constant, e 2 → 0, upon which the theory localizes to the set Σ of critical points of Q · U [47]. In this limit the path integral can be performed by restricting S to Σ and computing a one-loop determinant describing the fluctuations normal to Σ. This procedure has been carried out in detail in e.g. [48] for the case of SYM on the round S 4 . The action (3.1) is only invariant on-shell. In order to construct the fermionic operator Q of the previous paragraph we introduce auxiliary scalar bosonic fields K r . The operator Q is then defined by its action on the fields: where η is a commuting unimodular Weyl spinor, as in section 5, which is covariantly constant: Note that by virtue of the fact that η is commuting Q defined in (6.3) is indeed fermionic, i.e. transforms commuting to anticommuting fields and vice-versa. It can be checked that the operator Q defined in (6.3) is nilpotent off-shell, i.e. Q 2 = 0 on all fields without using the equations of motion. Furthermore it can be checked that Q leaves the following Lagrangian invariant up to a total derivative: where L off reduces to pure on-shell SYM upon eliminating the auxiliary fields by their equations of motion. Explicitly we have: The sign difference between the F 2 term in (3.1) and that in (6.5) above comes from the fact that the spinor parameter η in (6.3) is commuting whereas ǫ in (3.2) is anticommuting. From (6.5) we see that for convergence of the path integral the fields A r m , K r should be real. On the other hand the Q-transformations of A r m , K r given in (6.3) are complex. This does not affect the localization argument [48]: the action should be thought of as an analytic functional in the space of complexified fields, with the path integral understood as integration over the real subspace thereof.
We now deform the Lagrangian by a Q-exact term: where U is given by U = λ † Q · λ . (6.8) A straightforward computation using (6.3) then gives where we have taken the definition (5.3) into account. Note that since we are in the case where M 6 is a CY threefold and the spinor η is covariantly constant, J mn is a Kähler form and J m n is a complex structure. According to the localization procedure, the theory will localize, in the e 2 → 0 limit, to the set Σ of critical points of Q · U. To determine Σ we note that the right-hand side of (6.9) can be expressed as a sum of squares: where we have taken into account (5.4) and the identity: which can be proven e.g. by fierzing, taking into account the definition (5.3) of J as a spinor bilinear. To bring (6.10) to a more familiar form, it is useful to project on the (2,0), (0,2) and (1,1) parts (with respect to the complex structure J m n ) using the projectors defined in (5.6). Explicitly we expand: where Hence the set of critical points of Q · U is given by: In conclusion: the path integral of the SYM theory (6.5) defined on a CY threefold M 6 localizes on stable holomorphic bundles over M 6 . Note however that we have restricted our discussion to essentially classical considerations. A quantum-mechanical treatment taking into account the path-integral measure will not be attempted here. 13 More generally, for complex auxiliary fields K r we find (6.14) As we have already mentioned, for convergence of the euclidean path integral we must take the auxiliary fields to be real.

Conclusions
In this paper, we have analyzed the conditions for rigid supersymmetry in six dimensions and translated them as a set of necessary and sufficient conditions on the geometry of the six-dimensional manifold. In particular, the background three-form flux H • kmn is given in terms of the torsion classes. Given the existence of Killing spinors on these background structures, we have constructed the explicit Lagrangians and field equations for the interactions of non-abelian vector-and tensor-multiplets on curved space. We have worked out a number of explicit examples, including CY threefolds, the round S 3 × S 3 , as well as noncompact examples based on the Iwasawa manifold and T 2 bundles over noncompact K3's. Finally, we have applied the localization procedure to pure SYM theory defined on a CY threefold M 6 and shown that the path integral localizes on stable holomorphic bundles over M 6 .
A number of interesting questions remains to be further analyzed. We have fully analyzed the conditions for rigid backgrounds allowing for the minimal amount of supersymmetries, i.e. N = (1, 0) and N = (1, 1) in our notation, and briefly discussed the consequences of N = (1, 2). The existence of more Killing spinors will impose further constraints on the background fields and geometry, just as in the four-dimensional case [6]. With increasing number of supercharges, the structure of the supersymmetry algebra becomes more and more constraining. Let us note that the algebra displayed in (3.5), and (4.7), is soft in the sense that the closure involves dependence on fields, including the background ones. Taking the flat space limit in which the dynamical fields and the background 2-form potential vanishes, yields the rigid superalgebra {Q α , Q β } = (Cγ µ ) αβ P µ , (7.1) which is the Euclidean Poincaré superalgebra in 6D [49]. If we set only the dynamical fields equal to zero, the interpretation of the resulting superalgebra depends on the details of the background fields, such as the Killing spinors and and isometries they may support. In this case, a superalgebra in lower dimensions which also contains internal symmetry generators may be expected. In particular, the analogs of the Poincaré, AdS and conformal superalgebras for space-times in arbitrary dimensions and signatures have been systematically constructed in [50]. Another interesting open question concerns possible anomalies of our model. In Minkowski space-time and with all background fields set to zero, the model reduces to that of [18]. In that model there are gauge anomalies due to the coupling of gauge fields to chiral fermions. For certain gauge groups these anomalies can be cancelled by Green-Schwarz mechanism involving the addition of a one-loop counterterm of the form B ∧ F ∧ F . Supersymmetrization of the model in the presence of this counterterm [51] leads to -corrections to the Yang-Mills field equation, which however is anomalous, as its divergence does not vanish, as expected. This also implies an anomaly term in the closure of the superalgebra. Nonetheless, as shown in [51], introducing a constant α ′ in front of the YM interactions in the tensor multiplet field equations, and then interchanging the role of α ′ with leads to a dual model such that (a) it is anomaly free upon treating the resulting -terms as one-loop counterterm corrections, and (b) the tensor multiplet equations are free, while they arise as source in the field equations of the Yang-Mills multiplet. While the Euclideanization of 6D space-time is not expected to upset these results, the consequences of retaining nontrivial background fields from the context of the anomalies are not clear, and remains an i! nteresting open problem. Nonetheless, a direct application of the Noether procedure in the dual formulation that uses the Killing spinor equations is expected to yield a formulation of these models in curved background. It is also worth noting that the tensor-YM systemà la [18] as well as its dual formulation, both in 6D Minkowski spacetime, were also obtained from different global limits of the (on-shell) heterotic supergravity compactified on K3, [52].
It should be straightforward to extend the localization results presented here to non-Calabi-Yau manifolds. This would entail modifying the analysis of section 6 to include non-vanishing background fields besides the metric. We should stress however that our treatment of localization was essentailly limited to classical considerations. A fully quantum-mechanical treatment should include the consideration of the path-integral measure and potential anomalies.
In the above we have taken into account the identity which can easily be shown by direct calculation.
The following identity is useful in deriving the formulae in section 5.3: where V is an arbitrary (1,0)-form and W is any three-form obeying W ∧ J = 0.