(1,0) gauge theories on the six-sphere

We construct gauge theories with a vector-multiplet and hypermultiplets of $(1,0)$ supersymmetry on the six-sphere. The gauge coupling on the sphere depends on the polar angle. This has a natural explanation in terms of the tensor branch of $(1,0)$ theories on the six-sphere. For the vector-multiplet we give an off-shell formulation for all supersymmetries. For hypermultiplets we give an off-shell formulation for one supersymmetry. We show that the path integral for the vector-multiplet localizes to solutions of the Hermitian-Yang-Mills equation, which is a generalization of the (anti-)self duality condition to higher dimensions. For the hypermultiplet, the path integral localizes to configurations where the field strengths of two complex scalars are related by an almost complex structure.


Introduction and summary
The study of supersymmetric theories on curved spaces has led to insights into the dynamics of strongly coupled theories. Several exact results for partition functions and other supersymmetric observables have been obtained using supersymmetric localization. One of the simplest curved spaces that admits global supersymmetries is a d-dimensional sphere, S d . Since S d is conformally flat there is a canonical way to put a superconformal field theory (SCFT) on it. For non-conformal theories, radius of S d serves as an IR regulator which preserves supersymmetry. Following the seminal work of Pestun [1], theories on S d for d ≤ 7 and with various number of supersymmetries have been studied in [2][3][4][5][6][7] (see [8] for a review). All known results for supersymmetric partition functions on S d can be expressed as analytic functions of d [9]. This analytic continuation is consistent with the decompactification limit of corresponding theories [10]. In [11], partition functions for theories with four and eight supersymmetries were obtained, treating the dimension of S d as an analytic parameter.
A generic feature of theories that have been studied using localization is the existence of a Coulomb branch. Exact results remain elusive for theories with minimal supersymmetry in four and six dimensions. For S 4 , the localization computation has not been possible due to a technical difficulty [12,13] no suitable localization term is known. For S 6 there was no known construction with eight supercharges.
The case of supersymmetric theories on S 6 is intriguing. Based on the approach of [14] a limited analysis of the off-shell 6D supergravity was carried out in [15]. It did not find S 6 to be a supersymmetric background. Group theoretic arguments of [16] also seem to suggest that theories on S 6 with eight supersymmetries do not exist: there is no supergroup which contains the isometry group of S 6 and only eight supercharges. An implicit assumption in these arguments is that the bosonic symmetry group of the theory contains the full isometry group of the sphere. However, it is possible that the Lagrangian explicitly breaks part of the sphere isometry. This can be done by adding terms in the Lagrangian that depend on the position on the sphere. Such constructions, though somewhat exotic, are not unfamiliar in supergravity and field theory. From the perspective of supergravity one can have a supersymmetric solution with non-trivial values of background fields that do not necessarily preserve the isometries of the metric. From the perspective of field theory such situations arise by promoting the parameters to be space dependent. For example, deformations of N = 4 SYM in flat space with varying gauge coupling and theta-angle were considered in [17,18].
Theories with eight supercharges and space dependent coupling on certain four-manifolds were constructed in [19].
Another motivation to look for the construction of (1, 0) theories on S 6 is their relation to 6D (1, 0) SCFTs [20]. These SCFTs have a tensor branch of vacua on which the low energy theory is the (1, 0) super Yang-Mills (SYM). The gauge coupling in the IR theory is related to the VEV of the scalar in the tensor multiplet. Being an SCFT, it can be naturally put on S 6 while preserving all supersymmetries. The IR theory on the tensor branch on S 6 will then be a conventional supersymmetric gauge theory on S 6 . The scalar field gets a conformal mass term on S 6 and a constant non-zero value is no longer a solution of equations of motions (EoMs). It is conceivable that the VEV on S 6 gives a supersymmetric theory with non-constant coupling.
Let us demonstrate this point in a little more detail for the case of 4D N = 4 SYM. In flat space the theory has a supersymmetric Coulomb branch of vacua parameterized by the VEVs of the scalars. In the notation of [1] the conformally coupled theory on S 4 of radius r is Clearly a non-zero constant value of the scalar fields does not satisfy the above constraints. The simplest solution to the EoM is φ I = c I 1 + β 2 x 2 , where c I is constant, β = 1 2r , and x µ are stereographic coordinates on S 4 . This solution breaks half of the 32 supersymmetries. Broken supersymmetries correspond to special conformal supersymmetries in the flat space while the Poincaré supersymmetries are preserved. This analysis is closely related to the existence of half-BPS scalar field configurations of maximally supersymmetric theories on AdS spacetime [21].
Let us now turn to 6D theories. The scalar in the tensor multiplet couples to the vectormultiplet in the following way where . . . denote terms irrelevant to our discussion. On the tensor branch this interaction gives rise the effective gauge coupling of the theory (1.4) On S 6 , however one has a conformal mass term 3 r 2 φ 2 for the scalar field in (1.3). In its presence a constant φ with other fields vanishing is not a solution of the EoMs. The simplest solution is φ ∝ 1 + β 2 x 2 2 . The effective theory on the tensor branch on S 6 has a position dependent coupling.
It remains to be shown that such a solution preserves some amount of supersymmetry on the sphere.
We shall see in this paper that the heuristic picture presented above emerges from a more rigorous analysis. In this paper we construct theories on S 6 with features described above, i.e., a non-constant coupling and (1, 0) supersymmetry. We first construct Lagrangian for the (1,0) vectormultiplet. We start from a flat-space theory and then deform the Lagrangian and the supersymmetry transformations to obtain sufficient conditions to put this theory on a curved space while preserving supersymmetry. Indeed we find that by allowing the coupling to depend on the polar angle, we can construct a supersymmetric theory on S 6 . We find the following profile for the effective coupling.
where g YM is a constant parameter with the dimension of length. The coupling is zero at the south pole (x → ∞) and smoothly varies to a non-zero value g YM at the north pole (x → 0). The position dependence of the coupling is the same as argued above. The resulting Lagrangian is invariant only under an SO(6) ⊂ SO(7) isometry group of S 6 , which leaves the polar angle fixed. We then carry out a similar analysis for hypermultiplets and construct a supersymmetric Lagrangian on S 6 .
With an eye towards application of localization we give an off-shell formulation of these theories.
For the vector-multiplet, we start from an off-shell formulation on R 6 for all supersymmetries and obtain an off-shell formulation on S 6 for all supersymmetries. For hypermultiplets, however, we give an off-shell formulation only for a particular supercharge. This involves introducing pure spinor-like objects as is familiar from off-shell formulation of higher dimensional SYM [22].
Using localization, we show that the path integral for the vector-multiplet localizes onto so- however, diverges at the south pole which leads to a divergent classical action.
The rest of this paper is organized as follows: In section 2 we start by introducing necessary notation and our approach in 4D. We then explicitly construct the action and off-shell supersymmetry transformations for the (1,0) vector-multiplet in section 3. In section 4 we do the same for interacting hypermultplets. In section 5 we apply the localization procedure to these theories and obtain the localization locus. We present our conclusions and discuss furhter issues in section 6.
Appendices contain our conventions and technical details of numerous computations.

Warm-up in four dimensions
Before considering S 6 , it is instructive to do the analysis in a familiar four-dimensional setting.
Curved four-manifolds admitting supersymmetric gauge theories have been studied in detail using supergravity techniques (see for example [23,24] where D is an auxiliary field. The supersymmetry transformations are given by In Minkowskian signature ψ 1 (ξ 1 ) is related to ψ 2 ξ 2 by complex conjugation but in Euclidean signature they are a priori independent.
Let us introduce the following useful notation, 3) The indices i, j, ... take values 1 and 2. They are raised and lowered by the antisymmetric matrix ε ij defined as The auxiliary field can also be expressed as the 2 × 2 matrix With this notation, the Lagrangian and the supersymmetry tranformations can be written in a compact form. To put the theory on a curved four-manifold M 4 we start by covariantizing the Lagrangian and the supersymmetry transformations.
The change in L M 4 under a supersymmetry transformation is (see appendix B.1 for derivation) The action of two consecutive supersymmetry transformations on fields is given by (see appendix B.2) where v µ is the vector field v µ ≡ ξ i Γ µ ξ i . The first term in δ 2 A µ is a Lie derivative along the vector v µ while the second term is a gauge transformation w.r.t parameter −A ν v ν . To complete the construction, we need to show that the r.h.s of eq. (2.7) vanishes up to total derivatives and δ 2 generates a bosonic symmetry of the theory.
There are two simple ways to satisfy both of these conditions. One is to assume that the supersymmetry parameter is a Killing spinor (KS) This ensures that δL M 4 = 0 and v µ is a covariantly constant vector. All but the first term in Another way to satisfy both conditions is to let the supersymmetry parameter be a conformal Killing spinor (CKS), in which case δL M 4 and the last term in δ 2 D ij vanish due to a numerical accident, i.e., Γ ρ Γ µν Γ ρ = 0 in four dimensions. The second term in δ 2 D ij does not vanish or take the form of a bosonic symmetry for arbitraryξ i . However, forξ i ∝ ξ i 2 , Using the Fierz identity in eq. (A.10) one can explicitly show that the second term in δ 2 ψ i becomes This along with the first term forms the spinorial Lie derivative along the vector field v µ . The conditionξ i ∝ ξ i also ensures that v µ is a Killing vector field and hence the supersymmetry algebra closes onto bosonic symmetries of the theory.
Since S 4 admits non trivial CKS, we conclude that a minimally supersymmetric theory can 2 The proportionality constant is β for S 4 and is fixed by the integrability condition.
be defined on S 4 . The Lagrangian we constructed is the same as the one found in [12] by the dimensional reduction of 5D SYM.

(1,0) vector-multiplet on S 6
We now perform an analysis, similar to that of the previous section, in 6D. We will focus on putting the (1,0) theory on S 6 . We shall see that the situation is different in this case as a mere covariantization does not give a supersymmetric theory on S 6 . One has to modify the Lagrangian and the supersymmetry transformations further.

Lagrangian and supersymmetry transformations
On-shell (1,0) vector-multiplet consists of a gauge field and a Weyl-fermion with four DoFs each. Off-shell, they have five and eight DoFs respectively. For off-shell supersymmetry we need three auxiliary fields. They fit nicely with the notation already introduced, where we impose To put the theory on a curved manifold M 6 , we proceed as in the case of four dimensions by covariantizing the flat-space theory. The variation of the Lagrangian under a supersymmetry transformation and two supersymmetry transformations of the fields are given by For a KS, the supersymmetry algebra closes off-shell and the Lagrangian is invariant. In this way 3 Usually, one would use three real auxiliary fields K1, K2, K3 in terms of which the auxiliary field contribution is manifestly positive-definite ( proportional to K 2 we can construct supersymmetric theories on CY3-folds as was done in [15]. For a CKS, δL M 6 does not vanish. This is essentially because Γ ρ Γ µν Γ ρ = 0 in 6D. A modification of the covariantized Lagrangian and (or) supersymmetry transformations is required to construct a supersymmetric theory.
To proceed, we specialize to S 6 and choose a set of CKSs. The round metric on S 6 is where indices with a hat are the flat indices. Indices on coordinates and their differentials will always be flat indices, i.e., x µ = xμ = x µ = xμ. A set of frame fields is defined by eμ as We choose spinor parameters ξ i to be where ǫ i is a constant spinor. This is a particular set of solutions of CKS equation with 4 We take the Lagrangian on S 6 to be the covariantized Lagrangian multiplied with a factor of e φ , where φ is a position dependent function on S 6 . Using the supersymmetry variation in eq. (3.2) and the total derivative term eq. (3.3) we get We can get rid of this last term by modifying the supersymmetry transformation of the auxiliary fields to This gives a Lagrangian on S 6 which is invariant under eight supersymmetry transformations generated by eight independent solutions of the CKS equation given in eq. (3.6). The overall factor of e φ captures the position dependence of the inverse squared coupling and it matches with the heuristic argument given in the introduction. 4 The other set is given by Γμxμ √ 1+β 2 x 2 ǫ i . In r → ∞ limit this corresponds to conformal supersymmetries of flat space.

Closure of the supersymmetry algebra
We now compute the action of two supersymmetry transformations on fields in our construction of (1,0) vector-multiplet on S 6 . For covariantized transformations, this is already given in eq. (3.2).
The supersymmetry transformation of the auxiliary field is modified from the covariantized version as in eq. (3.10). Only the variation of the gaugino and auxiliary fields is affected by this modification.

The gaugino
For the gaugino we get We do the computation explicitly for i = 1. After using eq. (B.29) and CKSE we get Using the second Fierz identity in eq. (A.13) we can write After simplifying the gamma matrix structures appearing on the r.h.s 5 and expanding the contraction over the j index we get (3.14) The third term in eq. (3.12) can be written in the following way by using the second identity in eq. (A. 13).
Similarly, using Fierz identity the fourth term in eq. (3.12) becomes (3.16) 5 We use mathematica package FeynCalc [25,26] for gamma matrix manipulations Expanding the contraction over j index we get Finally, combining all the terms we see that A similar result holds for ψ 2 with 1 → 2 in the above equation.

Auxiliary fields
We have We now simplify above terms using CKS equation and Clifford algebra commutation relations. For terms involving the auxiliary fields we get For terms involving the gauge field we have These two terms precisely cancel each other. So we conclude that

The closure
The non-trivial part of two supersymmetry variations (i.e., the part without gauge transformation term) of a field Φ is where the second term is a Weyl transformation with weight Ω Φ . We have Ω A = 0, Ω ψ = 3 2 , Ω D = 2. We now show that the action is invariant under the bosonic transformation δ v . We note that v µ is a conformal Killing vector (CKV). Using the CKS equation and the gamma matrix identities, we find The transformation δ v acts only on dynamical fields and leaves the background fields invariant. In particular The Lie derivative of metric along v µ is Then the Weyl transformation of the metric is fixed to be We now examine the variation of the action S = d 6 x √ gL vec S 6 under δ v . We first consider the action of the Lie derivative. The factor √ ge φ transforms as a scalar density of weight 2 3 , i.e., Since ∂ µ v µ = 0, this terms essentially transforms as a scalar. Since all indices are contracted properly the rest of the terms in the Lagrangian also transform as a scalar, hence We next look at the action of the Weyl transformations with respect to a parameter Ω = 2v µ ∂ µ f . It is easier to work with a finite version of the infinitesimal Weyl transformation appearing in δ v . Under a Weyl transformation g µν → e 2Ω g µν and Φ → e −Ω Φ Ω Φ. The Weyl transformations of different terms in the Lagrangian are Note that the second term in the Weyl transformation for the fermion kinetic term is trivially zero.
So the Weyl transformation leaves the action invariant. We have shown that the supersymmetry algebra closes off-shell for the supersymmetry parameter ξ i . This conclusion is true more generally when one considers the anti-commutator of two supersymmetry variations w.r.t two different We have only focused on an abelian gauge group but all the analysis goes though for any gauge group by merely gauge-covariantizing different terms appearing in the action and supersymmetry transformations. To summarize, the Lagrangian 6 is invariant under supersymmetry transformations where D µ ≡ ∇ µ + A µ is the gauge-covariant derivative. We assume that the gauge indices are contracted using an invariant bilinear form. We also assume the real form of the gauge group so that generators are anti-hermitian. This completes the construction of (1,0) vector-multiplet on S 6 .

(1,0) hypermultiplet on S 6
We now construct the Lagrangian for (1,0) hypermultiplet on S 6 . We start by considering free, on-shell hypermultiplet in R 6 and then systematically modify the Lagrangian and supersymmetry transformations to obtain an interacting, off-shell hypermultiplet on S 6 .

Free hypermultiplet
The (1,0) hypermultiplet in 6D consists of four bosonic and fermionic degrees of freedom onshell. We denote scalars by 2 × 2 matrices φ iī and the fermions as χī. The fermion χī has opposite chirality as compared to the supersymmetry parameter ξ i . Both barred and unbarred indices are 6 Omitting the trace over gauge indices. separately raised and lowered by 2 × 2 antisymmetric matrices. The flat-space Lagrangian is invariant under supersymmetry transformations On S 6 the covariantized version of above Lagrangian and supersymmetry transformations gives which does not vanish in general. We modify the supersymmetry variation of the fermion to Under modified supersymmetry transformations First term vanishes if c = 4 3 . The second term can be cancelled by adding a mass term for scalars with a specific coefficient. It can be checked easily that the right term is 3 r 2 φ ij φ ij , i.e., the conformal mass term in 6D.

Coupling to a vector-multiplet
To include interactions with the gauge field we replace the derivatives with the gauge-covariant derivatives in the appropriate representation of the gauge group. Supersymmetry requires additional terms cupling the hypermultiplet to other fields in the vector-multiplet. The minimal terms which couple the gaugino ψ i and the auxiliary field D ij to the hypermultiplet are where we have suppressed the gauge group indices. We claim that the supersymmetric coupling of the hypermultiplet to the vector-multiplet is given by the following Lagrangian.
To prove our claim we explicitly compute the variation of the above Lagrangian under supersymmetry transformations. Due to the coupling with the gauge field, the variation of the first three terms is no longer zero. (4.8) The variation of the fourth term in L hyp S 6 gives (4.9) The third term in above equation can be modified using the Fierz identity on the second line of eq. (A.13).
Last equality follows due to the fact that ψ i is antisymmetric in the gauge indices but the bilinear χīΓ µνρ χī is symmetric. Hence we see that the combined supersymmetry variation of the first The supersymmetry variation of the term involving the coupling of auxiliary field with hypermultiplet bosons can similarly be computed.
First two terms come from variation of D ij and by doing an integration by parts to remove derivatives acting on the gaugino. Last term comes from the variation of hypermultiplet scalars. This cancels the supersymmetry variation of the first four terms and we conclude that the Lagrangian L hyp S 6 given in eq.

Off-shell Interacting hypermultiplet
So far we have only realized supersymmetry on-shell. For a particular choice of supersymmetry parameter we can realize the supersymmetry off-shell. To match eight off-shell fermionic degrees of freedom of hypermultiplet we need four bosonic auxiliary fields K ij . Supersymmetry can be realized off-shell by appropriately modifying the supersymmetry transformations and the Lagrangian. One adds K jī ν j to the supersymmetry transformation of χī. Here ν i is a spinor of negative chirality which is to be specified in terms of ξ i . This is similar to the use of pure spinors in realizing off-shell supersymmetry for maximally supersymmetric gauge theories. One adds 1 8 K ij K ij to the Lagrangian. The supersymmetry transformation of K ij is then obtained by requiring the modified Lagrangian to be invariant under supersymmetry transformations. This leads to We relegate the details of the computation of two supersymmetry transformations of fields to appendix B.3. It is not possible to close the supersymmetry algebra off-shell for arbitrary ν i . We where Ω φ = 2 and Ω Y = 3. The last term in two variations of the scalar (auxiliary) field is proportional to the EoM for auxiliary (scalar) field. It is straightforward to verify that the Lagrangian is invariant under such a transformation of scalar and auxiliary field. Spinors ν i , however, can be chosen so that this transformation is trivially zero.
The invariance of the action under L v follows just as in the case of the vector-multiplet. Under a Weyl transformation all but the scalar kinetic and mass terms get scaled by e −6Ω . This cancels against the transformation of √ g in the action. The kinetic term for the scalars change as the conformal mass term for the scalar receive extra modification because of the change of the scalar curvature under Weyl transformation The second terms above and in the transformation of the scalar kinetic term cancel. The third term combine to give a total derivative and we deduce that the action is invariant.

Localization of the path integral
In this section we apply the localization procedure for theories constructed in this paper. We

Vector-multiplet
To implement the localization procedure we choose a supercharge Q which generates supersymmetry transformations w.r.t a specific parameter ξ i as defined in eq. (3.6). We normalized ξ i (∝ ǫ i ) such that We next take the Q-exact Lagrangian to be QV where where Qψ i † is defined in terms of the holomorphic fields and the spinor parameter ξ i . We will define it in such a way that it coincides with the complex conjugation of Qψ i along the contour To proceed we assume Note that this is a consistent condition 7 . Supersymmetry transformations still depend on the two parameters ξ 1 and ξ 2 holomorphically. We define A straightforward computation then gives 8 where we have defined and ξ i = ε ij ξ j . We impose this condition and drop the superscript (i) . It is shown in appendix B.4 that I µ ν satisfies following identities.
Combining the two terms in eq. (5.5) we get the following simple form for the localization Lagrangian.
We next write the localization Lagrangian in a manifestly positive-definite form. The term involving auxiliary fields is already positive-definite as argued earlier. For the term involving the vector field, we decompose the 2-form field strength w.r.t to the almost complex structure as follows.
where F +(−) has six(eight) independent components. More explicitly (5.10) The localization Lagrangian can now be written as Hence the localization locus is given by everywhere expect at the south pole. (5.12) First two equations are called Hermitian Yang-Mills equations [27]. These are natural analogues of (anti)-self-duality condition in four dimensions which describe instanton configurations. For the case of six-sphere these are studied in [28]. We leave a detailed analysis of the localization locus and computation of partition function for future work.

Hypermultiplet
Localization analysis for the hypermultiplet can also be performed in a completely analogous manner. We choose a localization Lagrangian QV hyp ∝ Qχ i † Qχ i , with Q being the same supercharge as the one used in the case of vector-multiplet. Qχ i † is defined as where O µ = D µ + 4∂ µ f . This definition coincides with the complex conjugation along the contour We choose the spinors ν i as ν 1 = Cξ 2 , ν 2 = Cξ 1 , where C is the charge conjugation matrix. A straightforward computation gives the following The term involving auxiliary fields is positive definite. First two terms can also be written as a manifestly positive definite form by completing the square Auxiliary fields vanish at the localization locus and scalar fields satisfy O µ φ 12 − I µ ν O ν φ 11 = 0, everywhere except the south pole. (5.16) This is one complex and two real constraints. We do not attempt to solve for a general solution to this constraint here. A simple solution is O µ φ ij = 0. This corresponds to 17) where C ij are dimensionless constants valued in the appropriate representation of the Lie algebra.
This φ ij is actually a solution of the EoMs of the classical action. The value of the classical action, however, is divergent unless C ij vanish identically. Moreover one can explicitly check that this a UV divergence localized at the south pole where the value of the scalar field diverges. So the path integral for the hypermultiplet does not receive contribution from this simple solution of eq. (5.16).
We conclude this section by showing that for the fields satisfying eq. (5.16) the supersymmetry variation of hypermultiplet fermions is zero. We focus on δχ1, computation for δχ2 is analogous.
We argue this point by showing that all bilinears of δχ 1 with ξ 1 and ξ 1c ≡ Cξ 1 vanish. We first show this for ξ 1c . The even ranked bilinear vanish trivially as both δχ1 and ξ 1c have negative chirality.
A short computation gives which vanishes on the locus eq. (5.16).
For the rank-3 bilinear we get Using the second identity in eq. (5.7) we can write For ξ 1 , the odd-ranked bilinears are identically zero due to the opposite chirality. The zeroranked bilinear is proportional to the vector field v µ and hence vanishes identically without using the localization locus equations. For the rank-2 bilinear we use Fierz identities to reduce it to the bilinear that we have already computed.
In the first line we have proportional sign because ξ 1c ξ 1 = 1 1+β 2 x 2 . The second line follows from the Fierz identity on the third line of eq. (A.13). All bilinears on the second line involve ξ 1c and δχ1 and vanish as we have shown earlier.

Conclusions and outlook
In this paper we constructed theories on S 6 with (1,0) supersymmetry. We wrote down explicit Lagrangians for vector and hypermultiplets with off-shell supersymmetry. We also determined the localization locus for the path integral in these theories. We showed that the path integral for the vector-multiplet localizes onto solutions of Hermitian Yang-Mills equations. In the perturbative sector we showed that the path integral for the hypermultiplet localizes to field configurations where the field strength of two hypermultiplet scalars is related to each other. There are a number of issues left open for future research on which we comment now.
First one would like to compute the partition function and other supersymmetric observables for these theories. This requires a complete analysis of the localization loci derived above. One also needs to understand how to take into account the configurations localized at the south pole. It will be worthwhile to explore various approaches to this problem. One can modify the Lagrangian near the south pole to capture the contribution from the singular scalar field configurations discussed in previous section. Such singularities are generally related to operator insertions at the south pole. This is similar to the singular field configurations produced by surface operators in N = 4 SYM [29]. It is also possible to use a different supercharge for localization which may give a different and simpler localization locus. This may also be achieve by suitably modifying the localization term.
Computation of partition function in the non-perturbative sector is even more challenging. Nonperturbative configurations of gauge fields in higher dimensions are not well understood. We are not aware of any explicit solutions of HYM equations on S 6 . Any progress in this direction will be an important step towards understanding the structure of partition function of these theories.
A possible extension of this work is to construct supersymmetric theories with non-constant coupling on spheres of different dimensions. A simple example will be 6D (1, 1) theory which arises by taking the hypermultiplet in our construction to be in the adjoint representation of the gauge group. This would be a construction of the maximally supersymmetric theory on S 6 different than the one given in [7]. This hints at the existence of different formulations of maximally supersymmetric theories in other dimensions. One can consider the dimensional reduction of 10D SYM but allowing for a non-constant coupling. We report this analysis in a companion paper [30].
Another interesting issue is to extend our analysis to include tensor multiplets on S 6 . For consistent theories in 6D with different gauge groups one needs to include a certain number of tensor multiplets [31,32]. Due to the self-duality constraint on the 3-form field strength the Lagrangian for the tensor multiplet cannot be written in a simple manner. However it is reasonable to expect that the analysis of supersymmetry can be performed at the levels of EoMs and the correct form of supersymmetry transformations can be determined. This would be of significant importance because the detailed structure of the partition function depends on the supersymmetry transformations (á la localization) and the actual Lagrangian only appears in the weighting factor multiplying contributions to the path integral from different loci. We hope to explore these issues in future. Here we list our spinor conventions and useful properties of Clifford algebra in 4D and 6D Euclidean space. We start with some generalities about Clifford algebra d = 2n dimensions.

Acknowledgements
The charge conjugation matrix C and gamma matrices satisfy the following: From these one can derive following useful identities Covariantly transforming bilinears of spinors are defined as follow: Using this definition and anti-commutation relations of gamma matrices we can obtain the following identity: where σ = + (−) for commuting (anti-commuting) spinors. From this identity one can immediately derive the following.
For 4D and 6D, following special cases of above identities will be frequently used.
One can define a chirality matrix Γ in even dimensions as follows: For even n, the irreducible chiral representation of Spin(2n) is real and for odd n it is complex. A useful identity relating different clifford algebra elements is For two positive chirality spinors, we have the following identity: where the subscript denote the chirality of the spinor. For opposite chirality spinors we have an extra minus sign on the right hand sign.

A.2. Fierz Identities
For 4D we have the basic Fierz identity which can be derived by usual methods: From this we can obtain following four identities: (A.11) In 6D the basic Fierz identity takes the following form : From this we obtain following identities for 6D spinors (A.13)

A.3. Explicit gamma matrices and Killing Spinors
A set of explicit Gamma matrices in Euclidean 6D is given by 14) The We choose ǫ 2 and ν 2 to be complex conjugates of ǫ 1 and ν 1 respectively.
To carry out localization analysis we imposed another consistence condition on supersymmetry parameters, i.e., ξ i = iξ i . Spinor parameters which satisfy this requirement are specified by.

B. Technical details of some computations
In this appendix we give details of some computations whose results are used in the paper.

B.1. Covariantized supersymmetry variation of covariantized Lagrangian in 4D and 6D
Covariantized Lagrangian in both 4D and 6D has the following form: In 4D, ψ 1 has positive chirality while ψ 2 have negative chirality. In 6D, however, they both have positive chirality. The supersymmetry variation of the first and the last term in the Lagrangian is straightforward to evaluate: The supersymmetry variation of the kinetic term for fermion takes the following form: where we have kept a total derivative term for future reference. After using δψ i , the first term becomes Using Bianchi identity for the gauge field, the last term on the r.h.s can be written as Combining all these terms, the total change in the Lagrangian becomes The last equality follows by using the identity in eq. (A.6) for anti-commuting spinors. t.d denotes the total derivative term Using δψ i , the first term on the r.h. s becomes: so the total derivative term is Note that the results in eqs. (B.6) and (B.9) hold in 4D and 6D.

B.2. Two covariantized supersymmetry variations in 4D and 6D
Now we compute the change in the fields under two successive covariantized supersymmetry transformations. We will find the following two relations useful in our computations: It follows from the second relation that ξ i Γ µνρ ξ i = 0.
(1) The gauge field Using the supersymmetry transformation of gauge field and fermion given in eq. (2.6) one gets Using eq. (B.10), this becomes Since we did not assume any specific form of the supersymmetry parameter, the above relation holds in general, as long as the supersymmetry transformations of gauge field and the fermion have the form given in eq. (2.6).
(2) The gaugino Using the covriantized supersymmetry transformations of eq. (2.6), we get the following: For simplicity, let us do the computation for i = 1. The i = 2 case is completely analogous. We get: Let us now specialize to 4D and 6D separately to simplify the above expression.

4D
In this case, the first term on the second line in eq. (B.14) is zero and we can write: Now we use Fierz identities of eq. (A.11) to see that The second equality on the first line follows from eq. (A.6) because ξ 1 is a commuting spinor. So For ψ 2 we get a similar expression with 1 → 2 in the above equation.
If the supersymmetry parameters are Killing spinors then the second term on the r.h.s of δ 2 ψ 1 vanishes. Moreover one can show that v µ is a Killing vector in this case. Hence δ 2 ψ i is equal to the Lie derivative of the spinor ψ i along a Killing vector.
If ξ i satisfy CKS equation, ∇ µ ξ i = Γ µξ i = βΓ µ ξ i , then we need to analyze the second term appearing on the r.h.s of δ 2 ψ 1 further. This term can be written as Now we use 4D Fierz identities to simplify above terms. We have So these two terms cancel each other. The next two terms become: For CKS, the vector field v µ ≡ ξ j Γ µ ξ j is a Killing vector field and it satisfies the following: From this we can write The term involving ξ 2 vanishes due to the first Fierz identity in eq. (A.11). Similar conclusions hold for δ 2 ψ 2 . Combining everything we conclude that for CKS in four dimensions

6D
Now we start wit the general form given in eq. (B.14). Using the second Fierz identity in eq. (A.13), we have where the last equality follows due to the first Fierz identity in eq. (A.13).
Using the third identity in eq. (A.13), we get where the second equality follows by using Clifford algebra commutation relations and noting that ξ 1 Γ µ ξ 1 = 0. Hence we see that the second and the third term in δ 2 ψ 1 as given in B.14 cancel each other. Other terms appearing there can also be simplified in a similar fashion. For the fourth term, upon using Fierz identity and after some algebra we get 2ξ 2 ξ 1 / ∇ψ 1 = 1 2 v µ Γ µ / ∇ψ 1 − 1 4 ξ 1 Γ µνρ ξ 2 Γ µν ∇ ρ ψ 1 .

(B.27)
And the last term in δ 2 ψ 1 takes the following form after using Fierz identity and Clifford algebra commutation relations.

(3) Auxiliary field
For auxiliary fields, using covariantized supersymmetry transformations, one gets The first term in the above equation actually vanishes due to Bianchi identity and i ↔ j antisymmetry of ξ i Γ µ ξ j .
The third term in δ 2 D ij just becomes So the full action of two supersymmetry transformation on D ij becomes In 4D, the last two terms vanish for all KS and CKS. In 6D, they vanish for KS but not for CKS.

B.3. Two supersymmetry variations of the hypermultiplet
Here we give details of the computation of two supersymmetry transformations acting on fields in the hypermultiplet.

Scalars
We have

(B.35)
These terms can be brought in the desired form by using Fierz identities and relations satisfied by ν i s.

B.4. Almost complex structures
Here we prove the identities of eq. (5.7) for I µν ≡ iǫ † Γ µν ǫ, where ǫ is a constant spinor. Using Fierz identity on the third line of A.13 we can write: The second equality is obtained by using standard gamma matrix relations and the definition of I µ ν . The same Fierz identity also gives ǫ † Γ σδ ǫǫ † Γ σδ ǫ = −6. Using this on easily obtains To prove the second identity we start by using eq. (A.8) to write ε µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 I µ 3 µ 4 I µ 5 µ 6 = 2iǫ † Γ µ 3 µ 4 ǫǫ † Γ µ 1 µ 2 µ 3 µ 4 ǫ where the second equality follows by expressing the rank four gamma matrix in terms of lower rank matrices. The third equality follows by using Fierz identity.