5D SYM on 3D Deformed Spheres

We reconsider the relation of superconformal indices of superconformal field theories of class S with five-dimensional N=2 supersymmetric Yang-Mills theory compactified on the product space of a round three-sphere and a Riemann surface. We formulate the five-dimensional theory in supersymmetric backgrounds preseving N=2 and N=1 supersymmetries and discuss a subtle point in the previous paper concerned with the partial twisting on the Riemann surface. We further compute the partition function by localization of the five-dimensional theory on a squashed three-sphere in N=2 and N=1 supersymmetric backgrounds and on an ellipsoid three-sphere in an N=1 supersymmetric background.


Introduction
we obtain the q-deformed Yang-Mills theory on Σ via localization? This was the original motivation in the previous paper [1,2].
Second, when replacing the round S 3 by a deformation of the S 3 , such as a squashed S 3 and an ellipsoid S 3 , as discussed in [21,22], whether will we obtain a deformation of the Schur index for the round S 3 , like the mixed Schur index in [20]?
We will make an attempt to answer both of the questions in this paper, which is organized as follows: in sections 2 and 3, we will begin with the construction of the five-dimensional supersymmetric Yang-Mills theory on a curved space, based on the idea of [23] that the fields of an off-shell supergravity multiplet are utilized as background fields to preserve supersymmetries of the field theory on a curved space. In fact, through the dimensional reduction of the six-dimensional N = (2, 0) conformal supergravity in [25], on-shell supersymmetry transformations and an on-shell action of the five-dimensional theory compactified on a curved space have been derived in [24], following the idea [23].
Therefore, sections 2, 3, and 4 are essentially devoted to a review of [24], up to a few points that we perform the dimensional reduction in the time direction of the six-dimensional theory, instead of the spatial direction as in [24]. And we obtain off-shell supersymmetry transformations and an off-shell action of the five-dimensional theory on a curved space in section 6, which are necessary to carry out localization.
In section 5, we will discuss the partial twistings mentioned above -the N = 1 twisting and the N = 2 twisting -in more details, in the language of the background gauge field of the R-symmetry group, and we will decribe the supersymmetric background on a round S 3 in [1,2] in terms of supergravity background fields for the N = 1 twisting in subsection 5.1, and give a supersymmetric background on the round S 3 for the N = 2 twisting in subsection 5.2.
We will proceed to consider two supersymmetric backgrounds on a squashed S 3 -the analog of the background in [21] and of the one in [22] -in subsections 5.3 and 5.4, respectively. Especially, for the former, we will give supersymmetry backgrounds for both of the twisitings.
In subsection 5.5, we will discuss a supersymmetric background for the N = 1 twisting on an ellipsoid S 3 , in an analogous way to [21].
After the discussions about the off-shell formulation of the five-dimensional theory in section 6, as mentioned above, we will explain our localization method in depth in section 7. We will compute the partition functions by localization on the round and squashed S 3 's in section 8 for the background in section 5.3 and that on the ellipsoid S 3 in section 9 for the background in section 5.5.
However, the computation of the partition function on the squashed S 3 for the background in section 5.4 somewhat doesn't seem straightforward to be done by localization, and we will leave it as an open question. Finally, section 10 is devoted to the summary and discuusions of this paper.
Appendix A is a simple collection of our conventions about the (anti-)symmetrization of various indices and about differential forms, used in this paper, and the gamma matrices of the Lorentz groups in five and six dimensions are shown in our repsentation in appendix B. The R-symmetry group of the six-and five-dimensional theories are commonly Spin(5) R ≃ Sp(2) R and the associating gamma matrices in our represenation are given in appendix C. The spinors in the theories are symplectic Majorana-Weyl spinors and in appendix D, our convetions about those spinors are explained.
After the dimensional reduction of the conformal supergravity, supersymmetry transforms of the fermionic fields in the supergravity multiplet (the Weyl multiplet) yield supersymmetry conditions on the background fields to preserve supersymmetries on the curved background. Besides the supersymmetry condition derived from the gravitino field, there is another supersymmetry condition from the fermionic auxiliuary field in the Weyl multiplet and it is too long to write down explicitly in the text. Therefore, the explicit form of the supersymmetry condition is written in appendix E.
In appendix F, a few formulas which we think are useful to verify the invariance of the actions in sections 3 and 4 under the supersymmetry transformations are given.
In appendix G, Killing spinors and metrics are discussed on the round, squashed, and ellipsoid S 3 , following [21,22].
Appendix H explains the difference among the notations used in [25], in [24], and in this paper, and further the difference between the notations used here and in the previous paper [1].

Euclidean 5D N = SYM in SUGRA Backgrounds
In this section, the dimensional reduction along the time direction will be performed for the sixdimensional N = (2, 0) conformal supergravity derived in [25]. This section, the sections 3 and 4 are essentially a review of [24], but the spatial dimensional reduction was carried out there.
In subsection 2.1, we will recapitulate the main results of [25], which we will need in this paper about the supergravity mulitplet called the Weyl multiplet in the conformal tensor calculus.
In subsection 2.2, we will discuss the dimensional reduction of the Weyl multiplet, which play roles of supersymmetric background fields to retain supersymmetreis of the five-dimensional Yang-Mills theory on a curved space. Subsection 2.3 is just a small digression about the relaton of Killing spinors with Killing vectors.

Weyl Multiplet in 6D N = (2, 0) Conformal Supergravity
In this paper, following [24], we will carry out dimensional reduction of the six-dimensional N = (2, 0) supergravity in [25] to obtain a five-dimensional Euclidean maximally supersymmetric Yang-Mills theory in supergravity backgrounds. It has been discussed in [23] that the supergravity backgrounds provide a systematic method for supersymmetric compactifications of supersymmetric field theories. The construction of the supergravity in [25] is based on the conformal tensor calculus. (See the textbook [26] for the conformal tensor calculus and references therein.) In this approach, one starts with a gauge field theory by gauging the six-dimensional N = (2, 0) superconformal symmetry group OSp(2, 6|4), whose bosonic part consists of the conformal group SO(2, 6) and the R-symmetry group Spin (5). The symmetry group OSp(2, 6|4) is generated by P a : translation, D : translation, M ab : Lorentz, K a : special conformal, R IJ : R−symmetry, Q α : supersymmetry, S α : conformal supersymmetry, whose corresponding gauge fields are shown in Table 1.
The fermionic fields ψ µ α and φ µ α are the gauge fields of the supersymmetry and the conformal supersymmetry, respectively. They are symplectic Majorana-Weyl spinors of positive and negative chirality, respectively. See Appendix D for our conventions about symplectic Majorana-Weyl spinors.  A straightforward manner of gauging translations doesn't lead to general coordinate transformations which is indispensable to a theory of gravity. To gain general coordinate transformations from translations in the conformal tensor calculus approach, auxiliary fields 4 in Table 2 are introduced and the transformation laws of the gauge fields are deformed by imposing some constraints on the gauge field strengths and the auxiliary fields such that the resulting transformation laws give a closed algebra, as explained in [26].
Furthermore, one requires the invertibility of the gauge field E a µ of translations to solve the constraints, which allows us to regard it as the sechsbein. Solving the constraints makes the gauge fermionic field χ αβ γ Γ 7 χ αβ γ = χ αβ γ , χ αβ γ = −χ βα γ , Ω αβ χ αβ γ = χ γβ γ = 0, fields Ω µ ab , f a µ , and φ α µ dependent fields given in terms of the other gauge fields and the auxiliuary fields. In fact, they are given by where the ellipses · · · denote the contributions from the fermionic fields. One can see that the spin connection Ω µ ab is a generalization of the Levi-Civita spin connection ω µ ab satisfying and R µ a (Ω) is the Ricci tensor R µ a (Ω) = Θ ν b R νµ ba (Ω), of the curvature tensor of the spin connection Ω a b , where Θ ν b denotes the inverse of the sechsbein E a µ , i.e., coframe. After the deformation, one finds a closed algebra with the (covariant) general coordinate transformations. The remaining independent gauge fields and auxiliary fields form a multiplet called the Weyl multiplet including the graviton, the gravitini and the others. We show the resulting bosonic transformations of the independent gauge fields, except for the (covariant) general coordinate transformations, where Λ D , Λ ab , Λ K a , and Λ R IJ , are the parameters of dilatation, the Lorentz, special conformal, and R-symmetry transformations, respectively, and under the first four transformations, the auxiliuary fields transform as δT αβ µνρ = Λ D T αβ µνρ , δM αβ γδ = 2Λ D M αβ γδ , δχ αβ γ = 3 2 Λ D χ αβ γ + 1 4 Λ ab Γ ab χ αβ γ .
Under the R-symmetry transformations, they transform in the representations shown in the Table  2, respectively. The resulting supersymmetry (Q-) transformations and superconformal (S-) transformations on the gauge fields and the auxiliuary fields are given by with (trace) denoting necessary terms to give the same irreducible representations of the R-symmetry group as the fields on the left hand sides. The parameter ǫ α of a supersymmetry transformation and η α of a superconformal transformation are symplectic Majorana-Weyl spinors of positive and negative chirality, respectively; The operation T denotes transpose, and so (ǫ α ) T and η α T are the tranposes of ǫ α and η α , respectively. The curvature R α ab (Q) is the field strength of the supersymmetry gauge field (graitini) ψ α µ , whose exact form can be seen in [25], but it will not be necessary in this paper.
Here the covariant derivatives of ǫ α and T αβ abc are given by Here, the field strength of the R-symmetry gauge field V µ IJ is given by

Temporally Dimensional Reduction of the Weyl Multiplet
In this subsection, the dimensional reduction of the Weyl multiplet along the time direction will be considered in the same way as the dimensional reduction along one spatial direction was performed in [24], where the strategy in [27] was followed.
The latter condition makes the dilaton field α covariant constant [27]; which will be convenient for the calculations below. The partial gauge fixing conditions are summarized as We will use b µ as shorthand for α −1 ∂ µ α and b a = θ a µ b µ . Therefore, under the gauge fixing condition, one has the dependent gauge field Ω µ ab in (1) Among them, after the dimensional reduction, the component often appears in the covariant derivatives, and we refer to it as (Ω c ) ab . The auxiliuary fields V a IJ , T αβ abc are decomposed into five-dimensional fields S IJ , V a IJ , t I ab by with ε 12345 = ε 12345 = 1. Note that the gauge field A µ IJ is given by Let us remove the underline from M αβ γδ to denote its reduced one as M αβ γδ . It is sometimes convenient to replace the spinor indices α, β of M αβ γδ by the vector indices I, J as The field M IJ is in the reprensentation 14 of the Spin(5) R group and enjoys the symmetry properties The time component of the gravitini is set to zero by the gauge fixing condition (4); ψ α t = 0, and we will denote the remaining components ψ α µ (µ = 1, · · · , 5) simply as since it is of positive chirality, and our convetion of the chirality is found in Appendix D.
Since the auxiliuary spinor χ αβ γ is also of positive chirality, we will take with the convenient coefficient 15/16 in [24].
The parameters ǫ α and η α of supersymmetry and conformal supersymmetry transformations are of positive and negative chirality, respectively, and we will take The gauge fixing condition (4) is changed under the supersymmetry (Q-) tranformation (3). In particular, the zeroth component of the gravitino transforms under the supersymmetry (Q) and the conformal supersymmetry (S) as under the gauge fixing condition (4). However, combining the supersymmetry (Q-) and the conformal supersymmetry (S-) transformations, one can find that one linear combination of them leaves the condition ψ α 0 = 0 unchanged. For any ǫ α , one can see that the conformal supersymmetry transformation with the parameter compensates for the deviation (5) from the gauge fixing condition on the gravitini. Among the other gauge fixing conditions in (4), the condition E a t = 0 remains unchanged under the supersymmetry (Q-) and the conformal supersymmetry (S-) transformaions. But, the remaining gauge fixing conditions b 0 = 0 and b µ = α −1 ∂ µ α are changed under those transformations. However, the deviations can be canceled by the special conformal (K-) transformations with appropriate paramemters Λ K a . Note here that E a t and ψ α 0 are left invariant under the special conformal (K-) transformations. Thus, one may define a supersymmetry transformation in the reduced fivedimensional theory as the linear combination of supersymmetry (Q-), conformal supersymmetry (S-), and special conformal (K-) transformations.
Following the ideas in [23], we are seeking for supersymmetric backgrounds of the reduced theory to obtain supersymmetric compactifications of the N = 2 supersymmetric Yang-Mills theory in five dimensions. Since we would like to consider bosonic backgrounds, we will turn off background spinor fields, and we will find the supersymmetric bosonic backgrounds leaving the spinor fields ψ α µ , χ αβ γ unchanged under some of supersymmetry transformations in the reduced theory. From the supersymmetry transformation of the gravitini with the covariant derivative of the supersymmetry parameter one can see that the supersymmetric bosonic backgrounds should obey Under a supersymmetry transformation, the auxiliuary spinor χ αβ γ transforms as with t αβ ab = t I ab (ρ I Ω −1 ) αβ , and S α β = (1/2)S IJ (ρ IJ ) α β , where the ellipse · · · denotes the necessary terms 5 to leave the right hand side in the representation 16 of the Spin(5) R symmetry, since χ αβ γ is in the reprensentation 16. Here, the two covariant derivatives are given by Therefore, the other condition for the supersymmetric backgrounds is that the right hand side of (8) should vanish. The explicit form (92) of the supersymmetry condition is given in Appendix E, because the equation is very lengthy to write it here. Thus, (7) gives the Killing spinor equation, and supersymmetric backgrounds have to allow the existence of the solutions (the Killing spinors) to the equation. One may interpret that (92) determines the background field M αβ γδ , which will appear in the mass term of the scalar fields in the five-dimensional N = 2 supersymmetric theroy, as will be seen below.
The vector field ξ a obeys the conformal Killing vector equation with the covariant derivative ∇ µ ξ a ≡ ∂ µ ξ a + ω µ a b ξ b , which is related to the previous covariant derivative as In fact, the equation D a ξ b + D b ξ a = 0 leads to which gives the conformal Killing vector equation (9).

Tensor Multiplet in the Supergravity Theory
To the conformal supergravity, tensor multiplets can be added as matters, and after the dimensional reduction, they give rise to N = 2 gauge multiplets in five dimensions. It therefore yields a fivedimensional N = 2 supersymmetric Abelian theory in the supergravity background. It is the topic of this section. A tensor multiplet (B µν , φ αβ , χ α ) of the N = (2, 0) supergravity is listed in Table 3, and the field strength of the two-form B is given by The transformation rules and the equations of motion of the tensor multiplet were derived in [25].
tensor multiplet symmetries Spin(5) R weight bosonic fields Under a fermionic transformation (supersymmetry+ conformal supersymmetry), the tensor multiplet transforms as where H ± = (1/2) (H ± * H). (See the definition of the Hodge dual * in Appendix A.) The covariant derivative of the scalar field φ αβ is The equations of motion of the tensor multiplet are given by with the covariant derivatives

Dimensional Reduction of the Tensor Multiplet
From the six-dimensional Minkowski space to the five-dimensional Euclidean space, the dimensional reduction of the tensor multiplet gives rise to the five-dimensional abelian gauge multiplet (A µ , φ I , χ α ), The remaining components B ab are described by A µ and φ I through the equation (11) of motion of H − , which is reduced to Since the components H ab0 reduce to the field strength F µν of A µ , one can see that the components H abc are reduced as We have previously seen that a six-dimensional supersymmetry transformation with a transformation parameter ǫ α combined with the superconformal transformation with η α in (6) is reduced to a five-dimensional supersymmetry transformation. Substituting the parameter η α in (6) into the fermionic transformation rules in (10) of the tensor multiplet, one can see that their reduction gives the supersymmetry transformation of the abelian gauge multiplet, with the covariant derivative of φ I , The reduction of the external derivative of the equation (11) d * H + φ αβ T αβ = 0, yields the equation of motion of the gauge field A µ , where The equations (12,13) of motion are reduced into where the covariant derivative of χ α with the spin connection Ω µ ab = ω µ ab + (e a µ θ νb − e b µ θ νa )b ν , and the scalar curvature From the equations of motion (15,16), one obtains the bosonic part of the action of the abelian gauge multiplet with and the fermionic part One can verify that the total action L = L F + L B is left invariant under the supersymmetry transformation (14). However, it is a lengthy calculation to verify the supersymmetry invariance of the action L. Although we do not intend to pause for a detailed demonstration of it, we will discuss a supersymmetry transformaion of the mass term of the scalar fields φ I in the action in Appendix F, which we think is one of the keys to verify the supersymmetry invariance of the action.

The Generalization for a Non-Abelian Gauge Group
The reduced theory of the six-dimensional tensor multiplet gives rise to the abelian gauge theory in five dimensions. We will extend the abelian gauge multiplet (A µ , φ I , χ α ) to the adjoint representation of a non-abelian gauge group G and replace the partial derivatives by covariant ones; We will henceforth denote the covariant derivatives as For the non-abelian extension of the supersymmetry transformations (14) and the equations of motion (15,16), there are two conditions to be satisfied. In the flat limit where all the backgrounds go to zero, they should be reduced to the ones in the N = 2 supersymmetric Yang-Mills theory on a flat space, and in the abelian limit g → 0, the extension has to go back to (14,15,16). Our ansatz for the non-abelian extension of the supersymmetry tranformations is with the field strength of the non-abelian gauge field A µ In the abelian gauge theory, the algebra of the supersymmetry transformaions (14) is closed onshell, and in the flat limit of the non-abelian gauge theory, it is also closed on-shell. Therefore, in order to see the closure of the algebra of the supersymmetry transformations (19), we make an ansatz for the equation of motion of the spinor χ α , The supersymmetry transforms of D µ φ I and (F ab − 2φ I t I ab ) may be useful to see that the algebra of the supersymmetry transformations is closed on-shell; Using the equation of motion (20) and the Killing spinor equation (7), one can verify that the algebra of the supersymmetry transformations (19) is closed on-shell.
with the Killing vector ξ a = (η · γ a ǫ), where the parameters are given by (See Appendix D for the abbreviation (η · ρ I 1 ···In γ a 1 ···am ǫ) ), and the covariant derivative of φ I (η · ρ I ǫ) is Since we have seen that the supersymmetry transformation (19) gives an on-shell closed algebra with the equation of motion (20), we will proceed with (19) and (20) to obtain the non-abelian extension of the action (17,18).
A simple calculation shows that the equation of motion (20) may be derived from the fermionic part of the non-abelian action where the symbol tr denotes a trace in the adjoint representation of the gauge group G.
In the abelian limit g → 0, the non-abelian action should go to (17), -more precisely, the abelian action of the |G| abelian gauge multiplets with |G| denoting the dimension of the adjoint representation of G -, and in the flat limit, we must regain the familiar non-abelian action in the N = 2 supersymmetric Yang-Mills theory. It therefore seems natural to take the ansatz where In order to examine the supersymmetry invariance of the sum S F + S (0) B , one needs to perform a similar calculation to what is done for the abelian action L. The calculation may be painful, especially in the mass term of the scalar fields φ i , of which the details is shown in Appendix F.
However, it turns out that the variation of the sum S F + S B under the supersymmetry transformation (19) doesn't vanish at the order O(g). Therefore, in order to obtain a supersymmetric action, as discussed in [24], one needs the additional term to cancel the supersymmetry variation of S

Supersymmetric Backgrounds
In this section, we will discuss the supersymmetric solutions to the Killing spinor equation (7) and the condition (92) from the spinor variation δχ αβ γ , which gives rise to supersymmetric backgrounds for the five-dimensional supersymmetric Yang-Mills theory.
In this paper, we will make an assumption which is satisfied by the background in the previous papers [1,2], as will be seen below. In [1,2], we have considered the product space of a round S 3 and a Riemann surface Σ. In this paper, we are especially interested in supersymmetric backgrounds for deformed 3-spheres -a squashed and an ellipsoid S 3 . We will find supersymmetric backgrounds on the product spaces of those 3-spheres and Σ, which turn out to satisfy the assumption (26). It is convenient under the assumption (26) to decompose the supersymmetry parameter ǫ α as in the representation with ρ 5 = diag.(+1 2 , −1 2 ). While the Killing spinor equation (7) in a generic background gives a differential equation of ǫα and εα coupled to each other, the assumption (26) splits them into with t ab ≡ t 5 ab , where the covariant derivatives are defined by We will further make an ansatz for the Killing spinors, with two-dimensional spinors ǫ,ǫ on the S 3 and constant two-dimensional spinors on Σ, obeying that τ 2 ζ ± = ±ζ ± , with the Pauli matrix τ 2 . Note that they satisfy For later convenience, let us consider the commutation relation of the covariant derivatives acting on εα, which by definition gives and acting γ ab on this, one obtains On the other hand, using the Killing spinor equation (28) twice for γ ab D a D b εα and equating it and the right hand side of (31), one finds that Decompsing the fields χ α , φ I in the representation with ρ 5 = diag.(+1 2 , −1 2 ) as one can see that the supersymmetry transformation under the assumption (26) becomes The equations of motion of the spinors ψα, λα under the assumption (26) give with the covariant derivatives

The N = 1 SUSY Background in the Previous Paper
We start with the background in the previous paper [1,2], where the compactification on the product space of a unit round S 3 and a Riemann surface, S 3 × Σ was considered, and we will reinterpret it as a supersymmetric background in terms of A µ i j , S ij , G ab , t ab ≡ t 5 ab . See Appendix H.1 for the differences of the old notations used in [1] from the ones in this paper.
The background in [1] can be read in the notations of this paper as in the Lorentz frame t ab , G ab , where we have replaced the unit radius of the S 3 by r.
On the Riemann surface Σ with local coordinates (x 4 , x 5 ), the twisting is required to preserve supersymmetries by turning on the background gauge field A i j as with the spin connection ω 45 on the surface Σ. This together with S 12 = S 34 break the Spin(5) R R-symmetry group to SU(2) l × U(1) r ⊂ SU(2) l × SU(2) r , when regarding the subgroup Spin(4) of the Spin(5) R as SU(2) l × SU(2) r . We refer to it as the N = 1 twisting, following [16]. The supersymmetry condition (92) determines the background M IJ where the scalar curvature R(Σ) is derived from the spin connection ω 45 , and substituting these into (24) gives 6 The Killing spinor equation (28) in the background (34) is identical to the one in [1], with the ansatz (29). The scalar curvature R(Ω) on the S 3 × Σ is given by for the round S 3 of radius r. Since the gauge field A i j is minus the half of the spin connection ω 45 on the surface Σ, the field strength of A ij results in The equation (32) identically holds for the curvatures and the background fields, and it is consistent with the existence of the Killing spinor εα. In fact, as explained in [1,2] and in Appendix G, the Killing spinor is given by with ǫ 0 a constant spinor on the S 3 , which is consistent with our ansatz (29). For the other supersymmetry paramter ǫα, the Killing spinor equation (27) in the same background gives Note that S 12 = S 34 obeys S ij σ ij = 0. With A 12 = A 34 , we have A ij σ ij = 0, and the twisting of the background A ij have no effects inside the covariant derivative D a ǫα. In a generic Riemann surface Σ, we don't have a solution to the above Killing spinor equation. In fact, the calculation of γ ab D a D b ǫα shows that the scalar curvature R(Σ) is an obstacle to the existence of a Killing spinor for ǫα.
We can see from (34), (35) that the background breaks the Spin(5) R group of the R-symmetry into SU(2) l ×U(1) r , which is a subgroup of SU(2) l ×SU(2) r ≃ Spin(4) R ⊂ Spin(5) R . The symmetry breaking is caused by the twisting A 12 = A 34 (and also S 12 = S 34 ). As we have seen just above, the twisting only retains the half of the supersymmetries. Therefore, it is consistent with the fact that the SU(2) l symmetry doesn't give rise to the SU(2) R R-symmetry in four-dimensional N = 2 supersymmetric theories [6,16].
The background (34) is not a unique solution 7 to yield an N = 1 supersymmetric background on the round S 3 . Even under the ansatz with only non-zero components G 45 and t 45 , there exists a Killing spinor for εα, if which can be read from the Killing spinor equation (28). They may therefore be parametrized by S; The other supersymmetric condition (92) gives one more constraint -the backgrounds are constant on Σ, and determines the remaining background M IJ , for i, j = 1, · · · , 4. The scalar mass parameters M B IJ are given by When S = 1/r, it certainly retains the mass term of the scalar σ in the previous papers [1,2]. 7 It has been pointed out in [28] in the context of five-dimensional N = 1 supersymmetric theories.
5.2 N = 2 SUSY Backgrounds on the Round S 3 × Σ While the background in [1,2] preserves half of the supersymmetries, we will find a new supersymmetric background preserving both of εα and ǫα on the S 3 × Σ.
Taking the breaking of the R-symmetry group Spin(5) R into account, we will turn on A 12 and S 12 = S only, and it would break the Spin(5) R group down to U(1) R × SU(2) R . We could instead turn on A 34 or S 34 only, but it is just a matter of convention. We refer to this partial twisting as the N = 2 twisting.
Since we have the covariant derivatives with the ansatz (29), in order to cancel the spin connection ω 45 by A 12 in both of the covariant derivatives, the chirality of ǫα on the surface Σ should be the same as the one of εα; iγ 45 εα = (τ 3 )αβǫβ. Therefore, the twisting works for both of εα and ǫα. When we turn on the components G 45 and t 45 only, the Killing spinor equations (27), (28) become For a = 4, 5, the Killing spinor equation is satisfied with ǫα and εα constant on Σ, if With the ansatz (29), the Killing spinors on the round S 3 (See Appendix G.1) are lifted to and the comparison of this with the above Killing spinor equations for a = 1, 2, 3 leads to Depending upon the sign, there are two solutions; We will call the former background type B and the latter type A, respectively. Let us begin with the type A background; In the background, since the Killing spinor equation (7) is reduced into one obtains the solution to them, with ǫ 0 andǫ 0 constant spinors and U the mapping of the 3-sphere to the SU(2) group given in Appendix G and with C 3 the three-dimensional charge conjugation matrix explained in Appendix B.
The supersymmetry condition (92) determines the background M IJ ; which gives rise to the masses M B IJ of the scalar fields φ I , Turning on the field t 45 = t I=5 45 breaks the Spin(5) R symmetry group into Spin(4) R and with the twisting by A 12 = −ω 45 into U(1) × U(1). Thus, the background doesn't repsect the R-symmetry of the four-dimensional N = 2 conformal algebra, but it retains the N = 2 supersymmetry.
Let us move on to the type B background; It gives rise to the Killing spinor equation and one gives the same constant solution for the both ǫα and εα; with ǫ 0 andǫ 0 constant spinors as above.
The supersymmetric condition (92) is obeyed by the background, if the background fields M IJ satisfy which surely respects the R-symmetry group U(1) R ×SU(2) R . The scalars σ, φ 3 , φ 4 remain massless, while the remaining φ 1 , φ 2 , are lifted by a half of the scalar curvature R(Σ); Thus, they respect the remaining R-symmetry group Turning on either of A 12 or A 34 without t 45 = 0 breaks the R-symmetry group SO(5) R into SO(2) × SO(3)≃ U(1) R × SU(2) R , which can be identified with the R-symmetry group of the N = 2 superconformal group, if the theory flows into an infrared fixed point. On the other hand, as in the previous papers [1,2], turning on both A 12 and A 34 such that A 12 = A 34 , the SO(5) R group is broken to SU(2) l × U(1) r , which is the subgroup of SU(2) l × SU(2) r ≃ SO(4) ⊂ SO(5) R . The subgroup SU(2) l cannot be identified to the R-symmetry group SU(2) R , because the above results shows that such a background preserves only half of the supersymmetries 8 . This is consistent with the result in [6,16].

A Squashed 3-Sphere with constant Killing spinors
A squashed 3-sphere is a deformation of a round S 3 , and regarding it as a circle fibration over a round 2-sphere, i.e., the Hopf fibration, the radius of the fiber differs from the radius of the base. See Appendix G for more details. In [21], three-dimensional supersymmetric field theories on the squashed 3-sphere has beed discussed, and we will make use of their construction for the five-dimensional theory.
The constant solution εα on the round solves the differential equation where ω ab is the spin connection of the squashed S 3 with the fiber radiusr and the base radius r. See Appendix G for the squashed S 3 . We will begin with the N = 2 twisting by turning on A 12 only. A comparison of (37) with the Killing spinor equation (28) suggests that For the other supersymmetry parameter ǫα, it is easy to find a Killing spinor on the squashed S 3 , if we make the same ansatz as for εα; it is a constant spinor (ǫ 1 , One then see that it obeys the same differential equation (37), and thus the background (38) preserves the both Killing spinors εα, ǫα.
The other supersymmetry condition (92) determines the background fields M IJ , and plugging them into (24), one obtains the scalar masses M B IJ , 8 We thank Yuji Tachikawa for clarification on this point.
Let us proceed to the N = 1 twisting so that we will turn on the gauge field A ij of only one SU(2) subgroup of the Spin(5) R group by requiring that A 12 = A 34 , and then a comparison with the Killing spinor equation (7) identifies the background R-symmetry gauge field and for the other background fields, taking account of (92), one finds that for i, j = 1, · · · , 4. In the limitr → r, one regains the N = 1 supersymmetric background on the round S 3 in the previous subsection. It follows from (24) that

A Squashed 3-Sphere with non-constant Killing Spinors
Upon the Kaluza-Klein compactification on the time circle to the round S 3 ×Σ, the periodic boundary condition t → t + 2π was assumed in the previous papers [1,2]. The partition function is supposed to give the index of the six-dimensional theory. Let us generalize this by considering a slant boundary condition where ψ is the fiber coodinate in the Hopf fibration of the 3-sphere. See Appendix G for more details. It has been explained in [22] that the Kaluza-Klein reduction along this slant circle gives rise to a squashed S 3 . Changing the local coodinates (t, ψ) into (t,ψ) bỹ the ordinary reduction in thet direction will be carried out. Then, the mapping U(ψ, θ, φ) in (95) from the 3-sphere to the SU(2) group is given in terms of the new coordinates by and the vielbeinμ (0) in the new coordinates of the 3-sphere, is related to µ (0) in the original coordinates bỹ Under the change of coordinates, in the six-dimensional metric in addition to the trivial change of the base part in the Hopf fibration the last two terms are changed into Therefore, the slant boundary condition turns on the graviphoton field C and deforms the radius of the circle fiber, which results in a squashed 3-sphere.
Upon the reduction to five dimensions, one has the metric and the field strength of the graviphoton, Note thatα is the radius of the circle in thet direction, whiler is the radius of the fiber in the Hopf fibration of the squashed S 3 . If we started with a non-vanishing graviphoton field C in the round S 3 × Σ, we wouldn't gain a simple squashed 3-sphere. Therefore, let us consider the supersymmetric backgrounds in subsection 5.2, where we have C = 0 on a round S 3 × Σ. One can now see that the above change of coordinates leads the backgrounds in subsection 5.2 to supersymmetric backgrounds on the squashed S 3 × Σ. We will begin with the N = 1 supersymmetric background in 5.2 by turning on breaking the Spin(5) R symmetry down to SU(2) l × U(1) r . Besides the R-symmetry gauge field A i j , the only auxiliuary field t 45 is turned on in subsection 5.2.
Returning to six dimensions, the sechsbein a (a = 1, 2, 3), by a local Lorentz transformation, where From the six-dimensional view point, the background t 45 may be regarded as Recall that T αβ abc is anti-self under the Hodge duality. The field T αβ abc is transformed under the Lorentz transformation (40), and one obtains Therefore, on the squashed S 3 ×Σ, besides the R-symmetry gauge field A 12 = A 34 and the graviphoton field G, the auxiliuary fieldst Under the Lorentz transformation (40), a six-dimensional supersymmetry parameter ǫ α transforms as and recalling that it is of positive chirality, one can see that the five-dimensional spinor ǫ α transforms as The Lorentz transform (41) of the Killing spinors in the subsection 5.2 also gives the Killing spinors in the background on the squashed S 3 × Σ. In fact in the subsection 5.2, depending on the sign of the background t 45 = ±1/2r, one has the Killing spinors and they are transformed under the Lorentz transformation into One can thus see that the former never survive the Kaluza-Klein reduction, and the latter yields the Killing spinor on the squashed S 3 × Σ, which is the solution to which agrees with the Killing spinor equation (28) with the background obtained in this subsection. Let us turn to the remaining supersymmetry condition (92) from δ ǫ χ αβ γ = 0, which determines the auxiliuary field M IJ . Substituting the background fields into (92) and noticing that 9 one obtains In summary, we have found the supersymmetric background on the squashed S 3 × Σ, with the above scalar masses M B IJ .

An Ellipsoid 3-Sphere
As explained in [21], an ellipsoid 3-sphere is defined by the set of solutions ( which is solved by polar coordinates (φ, χ, θ) x 1 = r cos θ cos ϕ, x 2 = r cos θ sin ϕ, x 3 =r sin θ cos χ, x 4 =r sin θ sin χ, The metric is induced by embedding it into a flat R 4 , For more details, see [21] or Appendix G.5. The Killing spinors ǫ and ǫ c = C −1 3 ǫ * on the ellipsoid S 3 are given 10 in Appendix G.5. Using them, we form five-dimensional Killing spinors, ε 1 = ǫ ⊗ ζ + , ε 2 = ǫ c ⊗ ζ − , with the same ζ ± as before, and see that they obey where the R-symmetry gauge field A i j in the covariant derivative Dεα is given by with V the background U(1) gauge field on the ellisoid S 3 given in (103) of Appendix G.5, and with ω 45 the spin connection on the Riemann surface Σ. We will consider an N = 1 supersymmetric background by taking A 12 = A 34 , and break the For the other background fields S ij , G ab , and t ab , the Killing spinor equation (28) in the background satisfying where we have assumed that S = S 12 = S 34 , is reduced into (42). However, substituting them into the other supersymmetry condition (92), we find that the background is the only solution to (92), and that the background fields M IJ are given by It means that the scalar masses M B IJ are

The Off-Shell Formulation of the Reduced Theory
In order to implement the localization technique in calculating the partition function of the fivedimensional theory in a supergravity background, a supersymmetry transformation for the localization must be defined off-shell. To this end, the half of the supersymmetry transformations (19) will be extended off-shell by introducing auxiliuary fields. The supersymmetry parameters ǫα, εα are decoupled in the supersymmetry transformations, if the background fields obey the conditions The backgrounds we would like to consider in this paper, obey these conditions. Therefore, we will content ourselves with the construction of an off-shell formulation of the theory in this restricted type of backgrounds. In addition, the backgrounds in this paper also satisfy the condition b µ = 0, and we will add it to the above conditions. We would like to use one of the supersymmetry transformations for the localization. Since the supersymmetry parameters ǫα, εα are decoupled in the supersymmetry transformations in the restricted type of background, we will turn off the parameter ǫα and focus only on εα. It is then convenient to regard the N = 2 gauge multiplet as the sum of an N = 1 gauge multiplet (σ, A µ , λα) and N = 1 hypermultiplets (φ i , ψα).
We will introduce an auxiliuary field Dαβ in the adjoint representation of the SU(2) r subgroup of the Spin(5) R R-symmetry group; in (33), to which the supersymmetry tranformation (19) is reduced in the restricted type of backgrounds.
In a consistent way to the above replacement, the supersymmetry transformation of Dαβ is determined by using the equation of motion of λα, and one obtains The off-shell supersymmetry transformations (46) are closed into the other bosonic transformations. Using the Killing spinor equation (28) in this type of backgrounds, one obtains with the transformation parameters Let us proceed to the hypermultiplets (φ i , ψα). In writing the off-shell transformation for them, it will turn out that the spinor index notation of the scalars φαβ, will be convenient. In order to formulate an off-shell supersymmetry transformation 11 of the hypermultiplets, we will introduce auxiliuary fields Fαβ with the indexβ labeling a doublet of a new SU(2) flavor group, which is not a subgroup of the Spin(5) R R-symmetry group, following [29].
Further, we will also introduce different supersymmetry parameters εα from εα and ǫα, the former of which span the whole four-dimensional spinor space with εα.
The auxiliuary fields Fαβ and the new parameters εα are expected to play the role to impose the equation of motion of the spinors in the off-shell supersymmetry formulation. To this end, requiring that δ ǫ Fαβ be proportional to the equation of motion of the spinor ψα, one obtains an off-shell supersymmetry transformation In terms of the vector notation of the scalars φ i , it gives where the conditions are assumed; which will be necessary for the off-shell closure of the supersymmetry transformations on φαβ and ψα. Note that in terms of the spinor index notation, the covariant derivative D µ φαβ can be read as Making use of (50), one can verify that with the parameters in (48), where the transformation parameters of the other SU(2) subgroup of the Spin(5) R are given byΛ So far, we have seen that the supersymmetry transformations (49) on φαβ, ψα are closed off-shell, if we require the condition (50) on the supersymmetry parameters. However, we will see that the supersymmetry transformations (49) on the auxiliuary fields Fαβ are not automatically closed. In order to achieve an off-shell supersymmetry transformation, it seems that one has to require the supergravity backgrounds to obey additional conditions.
Let us look at the supersymmetry transformations on Fαβ. Using the condition (50), one obtains where the parameterΛαβ of the new SU(2) transformation is given by G ab ηβγ ab εγ −εβγ ab ηγ , and the last two terms on the right hand side suggest that the supersymmetry transformations fail to be closed off-shell, where the parameters (Θ ij )β˙γ, and ∆β˙γ are given by Therefore, in order to gain an off-shell closed algebra of the supersymmetry transformations, the conditions are required. Although the implications of the condition (53) have been unclear for the authors, the backgrounds in the section 5 satisfy the condition (53). So henceforth, we will assume that the backgrounds satisfy the condition (53).
Let us make sure that the supersymmetric backgrounds in section 5 obey the conditions (50) and (53). With the ansatz (29) and as explained in Appendix G, the supersymmetry parameters εα, εα are given by obeying that εα = γ 5 εα. It follows from this that and the condition (50) is satisfied by the supersymmetry parameters εα, εα.
Since the field S ij in all the N = 1 supersymmetric backgrounds in section 5 satisfies S ij σ ij = 0, and the field strength F ab ij satisfies F ab ij σ ij = 0, it is easy to see that they satisfy the former condition in (53). In the N = 2 background in subsection 5.3, the non-zero components of F ab ij are F 45 ij and F 12 ij , and G ab has only nonzero component G 45 . At the first sight, the nonzero F 45 ij and G 45 seems to give the contributions to the former condition of (53), but, since we have the formulā they yield no contributions to the condition. Furthermore, the nonzero F 12 ij appears on the left hand side of the condition with the term but, the conditions γ 3 εα = εα and γ 3 ηα = ηα reduceηαγ 3 εβ −εαγ 3 ηβ to the left hand side of the first condition in (50). Therefore, the former condition in (53) is obeyed also for the N = 2 supersymmetric background. The covariant derivatives D a t bc , D a G bc , and D a S ij are vanishing except for the background in subsection 5.4. However, ever for the background, D a t ab + D a G ab /(4α) = 0. Further, it is obvious that G ab G cd ε abcde = 0. Therefore, taking (54) into account, one can see that all the backgrounds in section 5 also satisfy the latter condition in (53). Now we can see that all the supersymmetry backgrounds in (53) allow the off-shell supersymmetry.
Let us proceed to the construction of an off-shell supersymmetric action. In order to perform the replacement (45) with Dαβ within the on-shell invariant action S = S F + S (0) (22)(23)(24)(25), we will add the term to the on-shell action S. For the hypermultiplets, the off-shell supersymmetry transformation of ψα has an additional term F taγ εγ, compared to the on-shell supersymmetry transformations. Therefore, under the offshell supersymmetry transformation, the on-shell fermionic action S F in (22) gains an additional term with the ellipsis denoting the other terms of the equation of motion for ψα, from the term in δ ǫ S F . Thus, it is necessary to add the term to cancel the additional term from δ ǫ S F . Finally, the construction of an off-shell action is achived as where the 'matter' Lagrangian L is given by with the 'mass' parameters

Localization and Twistings
In this section, let us proceed to compute the partition function of the theory by using the localization. Before going on, there is a subtle point that the kinectic terms of the fields σ, Dαβ, φ i , and Fαβ have the negative sign in the Lagrangian (57). In order to circumvent it, we would like to follow the same strategy for σ, Dαβ, and Fαβ as in [1,2]. Recall that the scalars φ i had the positive kinetic terms in [1,2], where the five-dimensional theory was obtained by the dimensional reduction from the six-dimensional maximally supersymmetric Yang-Mills theory.
To this end, we will perform the 'analytic continuation' for the scalars, For the auxiliuary fields Dαβ and Fαβ, let us carefully recall what we have done in the previous papers [1,2]. First, we have shifted D˙1˙1 as 12 and we then impose the reality condition In previous papers, we implicitly left the sign of the kinetic term of Fαβ negative. Since the integration over the auxiliuary fields Fαβ is trivial; there is no dependence on the vacuum expectation value of σ, we have just ignored the divergence from it. Therefore, we assumed that Although we don't have any rationale for the prescriptions, it seems to work well, and we will also follow the same prescriptions in this paper.
In order to carry out the localization, we will define a BRST transformation out of the supersymmetry transformation by setting both of ε˙2 and ε2 to be zero, following the strategy in [1,2]. Note that this is possible, because ε˙2 decouples from ε˙1 in the Killing spinor equation. This is also the case for εα. Furthermore, introducing bosonic Killing spinors ε andε, we take where Υ is a Grassmann odd number. For a generic field Φ, then we define the BRST transformation of Φ by Before the shift (58), it follows from (46) that the BRST transformation on the gauge multiplet is given by and from (49) that for the hypermultiplets, whereλα is an abbreviation for (λα) T C,ε is for ε T C.
The algebra of the supersymmetry transformations in (47), (51) and (52) may be used to check the nilpotency of the BRST transformation, assuming that (50) and (53) are satisfied. Substituting into (47-52), through the relation so thatη · ε and ξ a on the right hand sides of (47-52) are zero. Recalling that ε is chiral -iγ 45 ε = ε -on Σ, one can see that However, sinceη ·σ ij γ a4 ǫ andη ·σ ij γ a5 ǫ are not necessarily zero for a generic background, the BRST transformation isn't always nilpotent. But, for the backgrounds of our interest in this paper, since there are no fields carring the mixed components tangent to the 3-sphere and to the Riemann surface at the same time, one can find that it is nilpontent.
Let us now take the shift (58) into account. Although it never affects the nilpotency of the BRST transformation δ Q , it does affect δ Q λ˙1 and δ Q D˙1˙1, which is the BRST transform of a functional Ψ of the fields, with a parameter t. More explicitly, we will choose the regulator action to be Since the partition function Z never depends on the parameter t, one can take a large t limit, while leaving the value of Z intact. In the large t limit, the main contributions to Z comes from the fixed points of the fields given by δ Q λα = 0 and δ Q ψα = 0. Then, writing the fields Φ in terms of quantum fluctuationsΦ about one of the fixed points Φ 0 as and interating over the fluctuations to carry out the one-loop computation, one may compute the partition function Z exactly.
In order to carry out the localization, it is convenient to convert spinor and vector fields to scalar fields 13 on the 3-spheres, and then there is no need to introduce spinor or vector spherical harmonics on the three-spheres.
For the N = 1 hypermultiplet, Note that ǫ and ǫ c are the Killing spinors on each of the S 3 s and that they are linearly independent as two-component vectors, as discussed in Appendix G. The fields (χ, ξ, η, κ) and (χ,ξ,η,κ) are scalar fields on the three-spheres. For the N = 1 vector multiplet, for m = 1, 2, 3. The three-component vectors ǫ † τ m ǫ , ǫ c † τ m ǫ , and ǫ † τ m ǫ c are orthogonal among them, and are normalized by ǫ † ǫ = ǫ c † ǫ c = 1. See Appendix G in more detail. Let us recall that the background on the squashed S 3 in subsection 5.4 is not up to our mind here, because we will leave the calculation of the partition function for the background undone in this paper, as explained in Introduction.
In order to denote the scalar fields in the gauge multiplet, we use the same Greek letters χ, ξ, η, as for the ones in the hypermultiplet. But, we never mean that they are the same fields. What it really means is the shortage of the Greek letters we can assign to each of the fields. We will compute the one-loop contributions from the gauge multiplet and the hypermultiplet, separately. Therefore, we believe and hope that it wouldn't cause any confusion.

7.1
The N = 2 Twisting and the N = 1 Twisting As we have seen in section 5, the N = 2 twisting by turing on A 12 only gives rise to the N = 2 supersymmetric backgrounds on a round and a squashed S 3 , and on the other hand, the N = 1 twisting by turning on A 12 and A 34 with A 12 = A 34 gives rise to the N = 1 supersymmetric backgrounds on a round, a squashed and an ellipsoid S 3 . The difference between the N = 2 twisting and the N = 1 twisting has no effects on the BRST transformation of the N = 1 gauge multiplet, but affects the transformation of the N = 1 hypermultiplet. Therefore, the one-loop contributions from the N = 1 gauge multiplet don't depend on which twisting is done and yield the same results on an identical sphere.
Therefore, before proceeding to the one-loop calculations, let us see how the spin content of the two-dimensional fields in the hypermultiplet is changed upon each of the twistings. Then, we will see the spin content of the N = 1 gauge multiplet after the twistings, too.
The spin content of the two-dimensional fields on Σ from the hypermultiplet can be read from the covariant derivatives of the component fields of the hypermultiplet along the surface Σ with the local coordinates (x 4 , x 5 ).
The results are summarized in Table 4. The notation (k, l) in the table denotes a (k, l)-form for integers k, l. For half integers k, l, ( 1 2 , 0) denotes a Weyl spinor of positive chirality, and (0, 1 2 ) of negative chirality. Whichever k and l are integer or half-integer, the covariant derivative of a field Φ of (k, l) carries the spin connection ω 45 of Σ as On a squashed and an ellipsoid S 3 , as we have seen in subsection 5.3 and 5.5, we have also turned on the background field A i j along the S 3 . When the Killing spinors are reduced to the threedimensional ones ǫ and ǫ c on the spheres, we refer to the background R-symmetry field as V so that the covariant derivatives of ǫ and ǫ c are given by the N = 2 twisting the N = 1 twisting 5D fields scalars spin (k, l) charge q spin (k, l) charge q Then, on the squashed S 3 in subsection 5.3, we have for the N = 2 twisting, and for the N − 1 twisting, where | S 3 denotes the components along the S 3 .
On an ellipsoid S 3 in subsection 5.5, the N = 1 twisting causes along the S 3 the background field which is identical to V given in (103). When a two-dimensional field Φ has the covariant derivative we will say that the field Φ carries charge q under the background field V . The charges of the two-dimensional fields from the hypermultiplet are listed in Table 4. Let us turn to the two-dimensional fields on the three-spheres in the N = 1 gauge multiplet and see how the spin content of them is changed under the twisting. As mentioned before, both of the N = 2 and N = 1 twistings affect the spin content of them in the same way.
The two-dimensional fields from the N = 1 gauge multiplet are also charged under the gauge field V . But, the charges of them don't depend upon which twisting we perform. The charges under V which two-dimensional fields from the N = 1 gauge multiplet carry are also listed in Table 5.
2D fields ϕ χφχ ξ ηξη V 0 V + V − spin (k, l) (1, 0) (0, 1) (0, 0) (0, 0) (0, 0) charge q 0 2 0 -2 0 2 0 -2 0 2 -2 8 Localization on the Round and Squashed S 3 In this section, we will compute the partition function by localization for the backgrounds on the squashed S 3 discussed in subsection 5.3. In the round limitr → r, we will see that the previous results in [1,2] are regained for the N = 1 twisting, and we will obtain new results for the N = 2 twisting on the round S 3 in subsection 5.2 and on the squashed S 3 in subsection 5.3.
As mentioned before, in order to carry out localization, we need to find fixed points of the regulator action S Q , which are given by δ Q λα = 0 and δ Q ψα = 0. In the squashed S 3 background in subsection 5.3, the former conditions gives with the complex coordinate z = x 4 + ix 5 combining the local coordinates (x 4 , x 5 ) of Σ.

The first equation means that
A m = 0, e m ∂ m σ = 0 → σ = σ(z,z), and the second equation in turn implys that We will 'diagonalize' the scalar field σ at the fixed point by partial gauge fixing, where H i (i = 1, · · · , r) are the generators of the Cartan subalgebra of the gauge group G with r the rank of G. It then follows from D z σ = 0 that the gauge field A z takes values in the Cartan subalgebra, too, and that ∂ z σ i = 0, i.e., the solution σ i is a constant. As for the latter conditions δ Q ψα = 0, a simple examination shows that the solution to δ Q ψα = 0 is given byH = H = 0, for both of the N = 2 and N = 1 twistings. We will proceed to calculate the one-loop contributions about the fixed points of the regulator action S Q in the next two subsections.

One-Loop Contributions from the N = 1 Gauge Multiplet
The BRST transformations of the N = 1 gauge multiplet are the same for both of the N = 2 and N = 1 twistings. As discussed in section 7, we would like to reduce all the component fields in the gauge multiplet into scalar fields on the S 3 .
In particular, when we will convert the gauge field A m to V 0 and V ± , the field strength F mn = D m A n − D n A m + ig [A m , A n ] may be rewritten in terms of them as where the ellipsis stands for the gauge interaction terms, which gives no contributions to the partition function in the large t limit, and we will omit them. Note that the formulas (101) were used to derive these. Also omitting the gauge interaction terms, the field strength F mz is given in terms of this language by After the conversion, we can see that the BRST transformation of the bosonic fields is given by where we denote a fixed point of the scalar field σ as the same letter σ, and the fluctuation about this fixed point σ asσ. Henceforth, we will keep this notation until the end of this section. The BRST tranformation of the fermionic fields is given by Taking account of (V ± ) † = V ∓ , we deduce that Each of these fluctuations is in the adjoint representation of the gauge group G, whose Cartan generators we denote as H i (i = 1, · · · , r) with r the rank of G, and the remaining generators as E α with α a root of G. We assume that they obey and are normalized as Since the fluctuations have no interactions in the large t limit, the fluctuations taking values in the Cartan subalgebra are decoupled from the remaining sector, and they yield an overall constant to the partition function. We are interested in the dependence of the partition function on the value σ at one of the fixed points, and therefore we will focus on the remaining sector, where the fluctuations are expanded in terms of the basis {E α } α∈Λ with Λ the set of all the roots of G.
We then assume that (σ · α) = r i=1 σ i α i is non-zero for a generic (σ 1 , · · · , σ r ). It implys that the operator [σ, ·] acting on the sector we are interested in is invertible, and the following shifts are allowed to be done: which enables us to 'gauge away' the fluctuationσ in the above BRST transformation. In order to ensure this, we need to use (102) in Appendix G.3.
We would now like to contemplate the relation of the scalar σ with a parameter θ of the gauge transformation. In order to elucidate the discussion, we will refer to the value σ at one of the fixed points as σ 0 . Before 'diagonalizing' σ 0 , the scalar field σ is given by the sum where the fluctuaionσ is defined as the non-zero modes on the S 3 so that σ = with the scalar spherical harmonics ϕ l,m,m (l = 0, 1/2, 1, 3/2, · · · ; −l ≤ m,m ≤ l) on the S 3 . The fixed point σ 0 (z,z) therefore corresponds to the zero mode ϕ 0,0,0 on the S 3 . With the parameter θ of the gauge transformation, the scalar field σ is transformed infinitesimally as The parameter θ may be expanded in terms of the harmonics ϕ l,m,m , Since there is the correspondence betweenσ l,m,m and θ l,m,m for l = 0, the measure of the nonzero modes l=1/2 −l≤m,m≤l [dσ l,m,m ] in the path integral can be cancelled by the gauge degrees of freedom, l=1/2 −l≤m,m≤l [dθ l,m,m ], if the fluctuationσ never appear in the integrand of the path integral. This is indeed the case, as we have seen above in the large t limit.
Thus, we will setσ to zero in the BRST transformations, and let us proceed to the evaluation of the one-loop determinants in the partition function.
From the bosonic part of the gauge multiplet in the regulator action S Q , we can see that the auxiliuary fields D 1 2 and D 2 1 show up in the last term (δ Qχ ) † · δ Qχ ∼ |D 2 1 | 2 , and we will immediately integrated them out in the path integral. Furthermore, we will also integrate out the auxiliuary field D 1 1 , since it appears only in the first term (δ Q ξ) † · δ Q ξ ∼ |D 1 1 | 2 + · · · , with no D 1 1 in the ellipsis. Then, the sum of the first term and the second term is reduced to and the third and fourth terms are summed to yield which we will integrate by parts to obtain where ∆ 0 denotes the differential operator Since the three differential operators with d = e m ∂ m obey the Lie algebra of SU(2), we can regard ∆ 0 as 4 , which is potitive in the root sector expanded in the basis {E α }, and the operator ∆ 0 is invertible in the sector. Using the inverse of it, we will shift A z and Az in the above integrand to give after integrations by parts, with where we have defined the operator ∆ −2 by From the definition, it is obvious that and so the operator ∆ −2 may be rewritten as Therefore, with the same reason as for ∆ 0 , we can see that the operator ∆ −2 is invertible. To achieve the above expression (66) of the integrand, we have used the formulas (102), repeatly. In particular, from (102), we can deduce more customized formulas for this purpose, with the abbreviations, In the sum of (64) and (66), we will shift V ± as This shift is possible, because the operators acting on the root E α have no zero-modes for a generic (σ · α). Since the term in δ Q ξ, We obtain the integrand of the resulting sum of (64) and (66), after integrations by parts, where we will shift V 0 appropriately to eliminate the term D + V − . This is possible, since the operator K 0 is invertible in the same sense as explained above. The last term in the above integrand then gives Note that K 0 V 0 is pure imaginary, which may be ensured by using We now see that the resulting integrand is 'diagonalized', and it is a simple of matter to compute the one-loop determinants from the bosonic fields of the gauge multiplet, where the determinant Det (k,l) [D] for an operator D is defined by for a bosonic (k, l)-form field ϕ on Σ and its partner ϕ † , both of which are also scalar fields on the S 3 . We denote the one-loop contribution from V 0 as Z V,0 . Since V 0 is a real field; V † 0 = V 0 , some care is required to integrate over it. Upon expanding it in terms of the basis {E α } α∈Λ , we have and the reality condition implys that (V α 0 ) † = V −α 0 . Therefore, Z V,0 is given by the path integral up to the Cartan part, with Λ + the set of all the positive roots of Λ. Taking account of (69), we will integrate it to obtain where we defined the determinant Det (k,l) [D] for an operator D as for a bosonic (k, l)-form field ϕ α on Σ and its partner (ϕ α ) † , both of which are also scalar fields on the S 3 . They are just one components of an adjoint ϕ in the expansion Therefore, up to an overall constant including the Cartan part, Let us proceed to the one-loop contributions from the fermionic part of the gauge multiplet, of which the part in the regulator action S Q is Substituting (61) into this and integrating by parts, the first two terms become ξ η , and the remaining terms yield Integrating over ϕ, χ,φ, andχ to give the one-loop determinant , the latter terms in the integrand are reduced to after integration by parts. Summing this and the first two terms in the integrand results in where the operators D 1 , D 3 , and D 4 denote − , and the zero in the top right component of the matrix is seen from the calculation with help of (68). Integrating over ξ, η,ξ, andη, we obtain the one-loop determinants .
Thus, we compute the one-loop contributions from the fermionic fields of the gauge multiplet, where the last factor is easily evaluated as

Let us evaluate the determinant
.
For four operators A, B, C, and D, we have the formula (see for example, [30]) for an invertible D. If there is another differential operator D ′ satisfying the relation we then obtain the formula When we regard , using (68), we find the operator , and the determinant gives On the other hand, we also have the formula (see for example, [30]) for an invertible A. If there is another differential operator A ′ satisfying the relation we then obtain the formula

If we then identify
+ , using (68), we can find the operator and therefore, the same determinant have another expression For a (k, l)-form fermionic field v on Σ of charge 2, which is also a scalar on the S 3 , and its hermitian conjugate v * , integration by part is used to deduce which implys that Similarly, we can see that and that Using these, we may rewrite the determinant which also implys the formula .
Using (70) twice in a bit tricky way, we obtain Z 1−loop With the same reasoning as the argument about integration by parts in the integrand, it is easy to see that Using this, we will combine the one-loop contributions , and Z FP from the gauge multiplet to yield where we have made use of the invariance under α → −α for all the roots α ∈ Λ. The determinant Det (k,l) can be evaluated by using the basis {ϕ l,m,m ⊗ v ⊗ E α , ϕ l,m,m ⊗ v ⊗ H i }, for v running over all the basis vectors of Ω (k,l) (Σ), the set of all (k, l)-forms on Σ, upon regarding Ω (k,l) (Σ) as a linear space. Here, ϕ l,m,m (l = 0, 1/2, 1, 3/2, · · · ; −l ≤ m,m ≤ l) denote the scalar spherical harmonics on the S 3 , and through the relations (65) of the differential operators with the generators of the Lie algebra of SU(2), they provide the representations of the SU(2) algebra; L 3 ϕ l,m,m = mϕ l,m,m , L ± ϕ l,m,m = (l ∓ m)(l ± m + 1)ϕ l,m±1,m .
As explained in [31], the Hodge decomposition implys that for the space Ω k,l (Σ) of all the (k, l)forms on the Riemann surface Σ, where H p (Σ) is the space of all the harmonic p-forms on Σ. It follows from this that for a constant D, 15 Since the contributions from the basis vectors ϕ l,m,m ⊗ v ⊗ H i to the determinants are constant factors to the partition function, we will omit them.
with b i (Σ) = dim H i (Σ) the i-th Betti number, and with the Euler number χ(Σ) of the surface Σ; where we have used the Hodge duality; b 0 (Σ) = b 2 (Σ).
Taking account of this, we can reduce where we have replaced l by n = 2l (n = 0, 1, 2, · · · ). From the formula together with the zeta regularization, it follows that In the round limitr → r, Z V is in agreement with the previous result in [1].

One-Loop Contributions from the N = 1 Hypermultiplet
Let us proceed to compute the one-loop contributions from the hypermultiplet by localization. Since the BRST transfromation in the N = 1 twisting differs from the one in the N = 2 twisting, we will discuss them separately in the next two subsubsections. However, the BRST transformations of the scalar fields H, H † of the hypermultiplet are common in both the N = 1 and the N = 2 twistings. From (60), the BRST transformation of the scalar fields is reduced to

The N = 1 Twisting
Let us begin with the N = 1 twisting to calculate the one-loop contributions from the hypermultiplet to the partition function by localization. The BRST transformation of the fermions in the hypermultiplet is given by and its hermitian conjugate by Using the BRST transformation of the auxiliuary fields F 1 2 , F 2 2 , where we have omitted the terms g (σ i )α˙γ φ i , λ˙γ on the right hand sides of both the equations, because they vanish in the large t limit, we find that The system of (H,H † , χ, ξ,η,κ, F 1 2 ) is identical to the one of (H, H † ,χ,ξ, η, κ, F 2 2 ). If the former contributes the one-loop determinant Z 1−loop H to the partition function, both of the systems contribute (Z 1−loop H ) 2 . Therefore, we will focus on the former system only. From the fermionic part of the systerm (H,H † , χ, ξ,η,κ, F 1 2 ) of the regulator action S Q , where we have performed integration by parts, the one-loop determinant from the fermions of the system (H,H † , χ, ξ,η,κ, F 1 2 ) can be read as where the determinant Det (0, 1 2 ) is defined such that the path integral over a fermionic (k, l)-form ψ on Σ and its partner λ with a differential operator D yields .
For four differential operators D 1 , · · · , D 4 , we have the formula for an invertible D 1 . If there is another differential operator D ′ 1 satisfying the relation we obtain the formula In our case, we have which both act on the spinorκ of negative chirality on Σ and of charge q = 1. Using (99) in Appendix G.3, we can find the operator D ′ 1 , Therefore, we find that where the differential operator ∆ N =1 H,B denotes Note that The differential operator D ′ 1 in the determinant Det ( 1 2 ,0) [D ′ 1 ] on the most right hand side doesn't depend on the chirality of χ, and therefore, . It means that the ratio of the determinants is unity; and we obtain In the bosonic part of the system (H,H † , χ, ξ,η,κ, F 1 2 ) of the regulator action S Q , we will shift the auxiliuary fields Fαβ so that we can trivially integrate them out. We will integrate the remaining part of the action by parts to obtain and see that the one-loop determinant from the bosonic fields of the system (H,H † , χ, ξ,η,κ) is given by Therefore, the contributions from the hypermultiplet to the partition function are trivial; In the round limitr → r, the contributions from the hypermultiplet reproduce the previous results about the hypermultiplet in [1].

The N = 2 Twisting
Let us proceed to the N = 2 twising. In contrast to the N = 1 twisting, the system of the fluctuations (H,H † , χ, ξ,η,κ, F 1 2 ) yields the different contribution to the partition function from the one from (H, H † ,χ,ξ, η, κ, F 2 2 ), and we will treat them separately below. The BRST transformation of the fermions in the hypermultiplet is given by and its hermitian conjugate by Using the BRST transformation of the auxiliuary fields F 1 2 , F 2 2 , where we have omitted the terms g (σ i )α˙γ φ i , λ˙γ on the right hand sides of both the equations, because they vanish in the large t limit, we find that From the fermionic part of the systerm (H,H † , χ, ξ,η,κ, F 1 2 ) of the regulator action S Q , where we have performed integration by parts, the one-loop determinant from the fermions of the system (H,H † , χ, ξ,η,κ, F 1 2 ) can be read as with the determinant Det (0,0) defined in the previous subsection 8.2.1. As in the N = 1 twisting in subsection 8.2.1, upon computing the one-loop determinant (75), we may identify the differential operators D 1 and D 3 with respectively, and using (99) in Appendix G.3, we obtain the operator D ′ 1 , Using this relation, we can compute the one-loop determinant , where the differential operator ∆ N =2 H,B denotes In the bosonic part of the system (H,H † , χ, ξ,η,κ) of the regulator action S Q , we will shift the auxiliuary fields Fαβ so that we can trivially integrate them out. We will integrate the remaining part of the action by parts to obtain with the differential operator∆ N =2 H,B given by Therefore, we can read the one-loop determinant from the bosonic fields of the system (H,H † , χ, ξ,η,κ, F 1 2 ) as Let us move onto the fermionic part of the systerm (H, H † ,χ,ξ, η, κ, F 2 2 ) of the regulator action S Q , upto an integration by parts. It gives rise to the one-loop determinant with the determinant Det (1,0) defined in the previous subsection 8.2.1. As we have done just above, identifying the differential operators D 1 and D 3 with and using (99) in Appendix G.3, the operator D ′ 1 is found to be It follows from this relation that the one-loop determinant is computed to givẽ with∆ N =2 H,B given just above. In the bosonic part of the system (H, H  † ,χ,ξ, η, κ, F 2 2 ) of the regulator action S Q , we can trivially integrate Fαβ out in the same way as above. We will integrate the remaining part of the action by parts to obtain with the same∆ N =2 H,B as above. It immediately gives the one-loop determinant from the bosonic fields of the system (H, H † ,χ,ξ, η, κ, F 2 2 ),Z Combining the one-loop determinants from both the systems, we obtain .
Each of the fluctuations is in the adjoint representation of the gauge group G, whose Cartan generators we denote as H i (i = 1, · · · , r) with r the rank of G, and the remaining generators as E α with α a root of G. We assume that they obey and are normalized as As explained in [31], the Hodge decomposition implys that for the space Ω k,l (Σ) of all the (k, l)forms on the Riemann surface Σ, where H p (Σ) is the space of all the harmonic p-forms on Σ. It follows from this that for a constant D, Betti number, and with the Euler number χ(Σ) of the surface Σ; where we have used the Hodge duality; b 0 (Σ) = b 2 (Σ). The one-loop determinant Det (k,l) in Z 1−loop H is defined over the space with the basis 16 In terms of the basis {ϕ l,m,m ⊗ E α ⊗ v}, we obtain , up to an overall constant, where α is a root of the Lie algebra of the gauge group, and Λ is the set of all the roots of it. By the hermitian conjugation, we can see that and in a simialr way to above, we can compute upto a constant factor. After replacing spin l by n = 2l = 0, 1, 2, · · · , and shifting n → n − 1, we find that , 16 More precisely, the basis consists of {ϕ l,m,m ⊗ E α ⊗ v} and {ϕ l,m,m ⊗ H i ⊗ v}, with v ∈ Ω k,l (Σ). However, the Cartan part of the Lie algebra of G contributes a constant to the determinant, and we will omit them in computing the partition function.
where s b (x) is a double sine function; In particular, when b = 1, it is reduced to For more details on double sine functions, see [32,21,33,34]. The remaining factor in the one-loop contribution Z 1−loop H is computed in a similar way to yield .
Therefore, wrapping up all the factors, we obtain . To this end, we will derive the formula by using the zeta regularization, In this regularization, we can prove the above formula as follows: where we have used the formula We can make use of (77) to see the round limitr → r of Z 1−loop .
In summary, we have seen that the one-loop constribution in the N = 2 twisting on the squashed S 3 is given by 9 Localization on the ellipsoid S 3 We will calculate the partition function by localization on the ellipsoid S 3 in the background discussed in subsection 5.5. The calculations we will carry out are quite parallel to what we have done for the round and squashed S 3 's in the previous section. All we have to do is to replace the background gauge field V by the one in (103), andr/r 2 by 1/f . The fixed points discussed in the begining of subsection 9.1 are the same as for the background on the ellipsoid S 3 .
Therefore, we will briefly explain the calculations of the one-loop contributions from the N = 1 gauge multiplet and the N = 1 hypermultiplet, separately in the next two subsections.

One-Loop Contributions from the N = 1 Gauge Multiplet
For the BRST transformations of the N = 1 gauge multiplet, as discussed in section 7 and done in previous section 8, we will reduce all the component fields in the gauge multiplet into scalar fields on the S 3 .
As seen in section 8, upon converting the gauge field A m to V 0 and V ± , the field strength F mn and F mz are given, up to the gauge interactions, by where we have used (105), and we will omit the gauge interactions, as before, since they have no effects on the partition function in the large t limit.
The BRST transformation of the bosonic fields is given by where we denote a fixed point of the scalar field σ as the same letter σ, and the fluctuation about this fixed point σ asσ, as we have done in the previous section. Henceforth, we will keep this notation until the end of this section. The BRST tranformation of the fermionic fields is given by and furthermore, we find that As we have discussed in the previous section 8, assuming that (σ · α) = r i=1 σ i α i is non-zero for a generic (σ 1 , · · · , σ r ), we can see that the operator [σ, ·] acting on the sector with the basis {E α } we are interested in is invertible, and we will 'gauge away' the fluctuationσ by the shifts in the BRST transformation in the large t limit, where we used (107) in Appendix G.5.
Using the remaining gauge transformations, we will 'diagonalize' the value of the scalar σ at one of the fixed points. The latter results in the Fadeev-Popov determinant with the Fadeev-Popov ghost c α (z,z),c α (z,z) (α ∈ Λ), which are scalar fields on Σ.
This gauge-fixing procedure is quite the same as for the squash S 3 in section 8, and we will set σ to zero in the BRST transformations.
The bosonic part (63) of the gauge multiplet in the regulator action S Q , after integrating out the auxiliuary fields Dαβ and integrating by parts, is reduced to the sum of and where ∆ 0 denotes the differential operator which is potitive and so invertible in the root sector expanded in the basis {E α }.
As we have done for the squashed S 3 in the previous section 9.1, using with the abbreviations, derived from (107) in Appendix G.5, we will shift A z and Az in the latter integrand to give after integrations by parts, with where we have defined the operator ∆ −2 by which is also invertible in the sector we are interested in. Here, we have made use of which is also deduced from (107). When we shift V ± as for a generic (σ · α), the term in δ Q ξ, where we have used the formula which follow from (80) and of (107), together with (106) in Apeendix G.5. Therefore, the integrand of the sum of (79) and (81), after integrations by parts, becomes where we have shifted V 0 appropriately to eliminate the term D + V − , as before. Integrating over the remaining fluctuations, we obtain the one-loop determinants from the bosonic fields of the gauge multiplet, where Z V,0 denotes the one-loop contribution from V 0 . Taking account of the fact that V 0 is a real field; V † 0 = V 0 , we find that Therefore, up to an overall constant including the Cartan part, The computation of the one-loop contributions from the fermionic part of the gauge multiplet in the regulator action S Q is also parallel to that for the squashed S 3 .
Integrating by parts, the fermionic part is reduced to the sum of Integrating over ϕ, χ,φ, andχ gives the one-loop determinant and leaves the integrand after intergation by parts, where the operators D 1 , D 3 , and D 4 denote Integrating over the remaining ξ, η,ξ, andη, and combining the resulting determinant with (82), we obtain the one-loop contributions from the fermionic fields of the gauge multiplet, with Det (0,0) D 1 0 D 2 D 4 evaluated to give

The determinant
, noting the relation is evaluated to yield Since the determinant of an operator is the same as the one of its ajoint operator, it follows that Using them, (83) may be rewritten as For the determinant , using the formula and the relation of the determinant of an operator with that of the adjoint operator, .
we can see that it is reduced to Substituting (84) and (85)  , we obtain With the same argument about the adjoint operators, we can show that and using this, the one-loop contributions , and Z FP from the gauge multiplet are summaried to give The determinant Det (k,l) can be evaluated by using the basis {h n,m,k ⊗ v ⊗ E α , h n,m,k ⊗ v ⊗ H i }, for v running over all the basis vectors of Ω (k,l) (Σ), the set of all (k, l)-forms on Σ, upon regarding Ω (k,l) (Σ) as a linear space. Here, h n,m,k (n, m = 0, 1, 2, · · · ; k = 0, 1, · · · , n + m) denote scalar spherical harmonics on the S 3 (See Appedix G.5 for more details) and obey On the basis 17 {ϕ l,m,m ⊗ v ⊗ E α }, the determinants in Z 1−loop V are computed to give where the determinant det (k,l) is defined over the space Ω (k,l) (Σ). Taking account of (72), we can simplify Z V , . 17 We will again omit the contributions from the basis vectors ϕ l,m,m ⊗ v ⊗ H i to the determinants, as done for the squashed S 3 . Furthermore, from the formula it follows that In the round limitr → r, Z 1−loop V recovers the result for the round S 3 in section 8.

One-Loop Contributions from the N = 1 Hypermultiplet
Let us turn to compute the one-loop contributions from the hypermultiplet by localization.
Since the BRST transformations of the scalar fields H, H † of the hypermultiplet are independent of the background fields, they are the same as for the round and squashed S 3 's; where we have omitted the terms g (σ i )α˙γ φ i , λ˙γ on the right hand sides of both the equations, because their contributions vanish in the large t limit, we find that The system of (H,H † , χ, ξ,η,κ, F 1 2 ) is identical to the one of (H, H † ,χ,ξ, η, κ, F 2 2 ), as we have seen for the squashed S 3 case in subsection 8.2.1. If the former contributes the one-loop determinant Z 1−loop H to the partition function, both of the systems contribute (Z 1−loop H ) 2 . Therefore, we will focus on the former system only.
From the fermionic part of the systerm (H,H † , χ, ξ,η,κ, F 1 2 ) of the regulator action S Q , As we have done in subsection 8.2.1, for four differential operators D 1 , · · · , D 4 , we have the formula for an invertible D 1 . In the above case, we have which both act on the spinorκ of negative chirality on Σ and of charge q = 1. Using (107) in Appendix G.5, we can identify the operator D ′ 1 ,

and obtain
Det (k,l) We thus find that With the same reason as in subsection 8.2.1, we can show that and it follows from this that In the bosonic part of the system (H,H † , χ, ξ,η,κ, F 1 2 ) of the regulator action S Q , we will immediately integrate the auxiliuary fields Fαβ out, and integrate the remaining part of the action by parts to obtain and see that the one-loop determinant from the bosonic fields of the system (H,H † , χ, ξ,η,κ) is given by Therefore, the contributions from the hypermultiplet to the partition function are trivial; In the round limitr → r, the contributions from the hypermultiplet reproduce the previous results on the round S 3 about the hypermultiplet in section 8.

Summary and Discussions
In this paper, we have seen the effects caused by changing the twisting and by deforming a round 3-sphere to a squashed and an ellipsoid 3-spheres on the partition function on the round S 3 , which was computed in the previous paper [1,2]. We have discussed the two kinds of twistings -the N = 1 twisting and the N = 2 twising, the former of which breaks the Spin(5) R symmetry group to U(1) r × SU(2) l , which is the subgroup of SU(2) r × SU(2) l ≃ Spin(4) ⊂ Spin(5) R , while the latter breaks the Spin(5) R to U(1) R × SU(2) R , which is the subgroup Spin(2) R × Spin(3) R of the Spin(5) R . In the N = 1 twisiting, the only supersymmetry transformation with the parameter εα is preserved, and in the N = 2 twisting, the ones with both εα and ǫα are available.
The change of the twisting affects on the spin content of the N = 1 hypermultiplet, when the N = 2 gauge multiplet is viewed as the sum of the N = 1 gauge multiplet and the N = 1 hypermultiplet.
In all the cases we discussed in this paper, the classical action, S cl which is the value of the off-shell action (56) at one σ i of the fixed-points, can be compactly written down 18 as in the zero-area limit of the Riemann surface Σ, where 1/f is replaced byr/ 2 r for the squashed S 3 and by 1/r for the round S 3 . Recall that σ i is a constant at the fixed point, and notice that the scalar curvature R(Σ) of Σ disappears in the mass parameter M σ . The integration in the prefactor can be easily done to give where the intergers m i are the 'monopole' charges, which will be explained below for, just for brevity, the gauge group G = SU(2).
In the N = 1 twisting, we have seen that the one-loop contributions from the hypermultiplet are trivial to the partition functions.
Furthermore, on the squashed S 3 , the one-loop determinants from the N = 1 gauge multiplet remains the same as on the round S 3 , when we replace r byr. In fact, the partition function Z N =1 squashed for the squashed S 3 , is reduced to the one for the round S 3 , in the round limitr → r.
More specifically, let us take the gauge group G to be SU (2), and then the generators with our normalization, implying that the positive root α = √ 2. In our convention, we have where m, which was refered to above as the 'monopole charge', runs over all the integers. Substituting it to the classical action S cl , we see that the path integral gains the contributions only from the configurations for n ∈ Z.
Therefore, the partition function on the squashed S 3 is turned to with q = e −g 2 /(2πr) , where the character χ n (q) is defined by Especially when we consider the round S 3 and replacer by r in the above Z N =1 squashed , the result in the previous paper [1] is recovered; with q = e −g 2 /(2πr) , and we can see that it is consistent with the superconformal index computed in [3]. Using the 2(g − 1) structure constants and 3(g − 1) propagators in [3] to compute the index for the surface Σ of genus 19 g (therefore, χ(Σ) = 2 − 2g) with no punctures, one obtains the above Z N =1 round up to an factor 20 . In the review article [35], it has been elucidated 21 that the discrepancy is attributed to the difference of the renormalization prescriptions used here and there, and that it can be improved by the requirement of the S-duality (a.k.a. the bootstrap).
From the point of view of the number of supersymmetries, this result seems puzzling. The partition function Z N =1 round computed under the N = 1 twisting is supposed to be the index of a four-dimensional N = 1 supersymmetric theory, while the index in [3] was computed for a fourdimensional N = 2 superconformal theory. In [20], the superconformal index of of N = 1 class S fixed points has been calculated in four dimensions. Among their results, the mixed Schur index carrys two fugacities p and q in their notations. When we take p = q, the index takes the same form as the Schur index of the N = 2 fixed points given in [3].
The partition function Z N =1 squashed on the squashed S 3 is essentially the same as Z N =1 round on the round S 3 . However, the partition function Z N =1 ellipsoid on the ellipsoid S 3 is deformed from the one on the round S 3 .
This is a similar situation to the three-dimensional case in [21]. As we have done just before, taking the gauge group G = SU(2) and summing over the monopole charge m, we can see that the only configurations n, for n ∈ Z contribute to the partition function, and therefore the summation of n over intergers yields where q = e −g 2 /(2πr) and p = e −g 2 /(2πr) . It is also consistent with the mixed Schur index 22 of N = 1 rank one class S fixed points in [20], up to an factor from the renoramlization mentioned above. Let us turn to the N = 2 twisting. On the round S 3 , we have seen that the hypermultiplet contributes the same one-loop determinants to the partition fucntion as the N = 1 gauge multiplet does. Deforming the round S 3 to the squashed S 3 , we have observed that the one-loop contributions from the hypermultiplet are deformed by the deformation paramter of the S 3 .
In fact, in the partition function Z N =2 squashed on the squashed S 3 we have seen that the one-loop contributions . Therefore, for the gauge group G = SU(2), upon summing the magnetic charge m over intergers, we obtain , with q = e −g 2 /(2πr) , where the double sine functions may be rewritten as and in the round limitr → r, they reduce to with q = e −g 2 /(2πr) . When recognizing χ n (q) as the q-deformed number 23 [n] q , we may regard the square root of the double sine functions as a deformation of [n] q . 22 More specifically, our result corresponds to the case with l 1 = l 2 in their notations of [20], and to the N = 1 twist in [16], as may be seen from the background R-symmetry gauge field. 23 This definition of the q-deformed number [n] q slightly differs from the one in [1] by the factor 1/(q We thus find that the partition function Z N =2 round on the round S 3 is given for G = SU(2) by This result suggests that the partition function Z N =2 round does not corresponds to the Schur limit of the superconformal index discussed in [3]. We expect that it gives another simple limit of the superconformal index of N = 2 rank one class S fixed points, where the index can be calculated by the two-dimensional q-deformed Yang-Mills theory but with the measure 2 sinh πrg (σ · α)

Appendix A Our Conventions of (Anti-)Symmetrization of Indices and Differential Forms
The convention of the antisymmetrization and symmetrization 24 may be seen from However, for the six-dimensional gamma matrice (for the definition of them, see the next appendix), we define For the five-dimensional gamma matrices, γ a 1 ···an is defined in the same way.
On the reduction along the time direction from the six-dimensional Minkowski space to the five-dimensional Euclidean space, where the gamma matrices are five 4 × 4 hermitian matrices γ a (a = 1, · · · , 5) satisfying γ a , γ b = 2δ ab 1 4 , γ 1 · · · γ 5 = 1 4 , with 1 4 the 4 × 4 unit matrix, we define for a = 1, · · · , 5, with the Pauli matrices τ 1 , τ 2 , τ 3 . The property (89) is reduced to with ε 12345 = ε 12345 = 1. The six-dimensional charge conjugate matrix C is related to the five-dimensional charge conjugation matrix C by and one can see that the charge conjugation matrix C enjoys the properties It follows from them that A more explicit form of the five-dimensional gamma matrices γ µ takes γ a = τ a ⊗ τ 2 , (a = 1, 2, 3) C Gamma Matices of the R-Symmetry Group Spin(5) R We give the explicit form of the gamma matrices of the R-symmetry group Spin(5) R with the Pauli matrices τ 1 , τ 2 , τ 3 , satisfying that where I, J run from 1 to 5. We use them to define with ε 12345 = 1.

D Symplectic Majorana-Weyl Spinors
For a six-dimensional Dirac spinor ψ, we define ψ = ψ † Γ 0 . In six-dimensional Minkowski space, the symplectic Majorana condition on an even number of spinors can be imposed. In our case, all the spinors in the Weyl multiplet and the tensor multiplet of the supergravity carry the spinor indices of the Spin(5) R symmetry group, and the dimension of the spinor representation is four -an even number. Let us take one of such spinors, say ψ α , and it obeys the symplectic Majorana condition and the other spinors in the multiplets obey the same condition.
In the Minkowski space, the Weyl condition Γ 7 ψ = ±ψ and the symplectic Majorana condition can be imposed on spinors at the same time. In fact, all the spinors of the multiplets are symplectic Majorana-Weyl spinors, and also so are the parameters of the supersymmetry and the conformal supersymmetry transformations, as explained in the text.
For the spinors εα, ηα, the Fierz transformation εαηβ = εαη˙γCε˙γβ = − 1 4 ηβ εα 1 4 + ηβγ a εα γ a − 1 2 ηβγ ab εα γ ab , F The SUSY Transform of the Mass Term of the Scalars When the interested readers attempt to ensure the supersymmetry invariance of the actions L and S in sections 3 and 4, respectively, it may be convenient to show how the mass term M B IJ φ I φ J in the actions transforms under a supersymmetry transformation 25 .
The two last terms on the right hand side of (93) depend on M I J and R(Ω). If they are given in terms of the backgrounds S IJ , G ab , t I ab , they may cancel the supersymmetry variation of the other terms in the actions. In fact, this is the case, if one uses the supersymmetry condition (92) and the Killing spinor equation (7), as will seen below.
Using the supersymmetry condition (92), the term − i 4 tr φ Iχ α · 4 15 M I J (Ωρ J ) αβ ǫ β on the right hand side of (93) can be straightforwardly replaced by terms depending on the backgrounds S IJ , G ab , t I ab . The commutation relation of the covariant derivatives gives and on the other hand, using the Killing spinor equation (7), one obtains D a G bc γ ab γ c ǫ α − 1 8α D a G bc γ ad γ d bc ǫ α + 1 2 D a t I bc (ρ I ) α β γ ad γ d bc ǫ β +S IJ (ρ IJ ) α β γ a D a ǫ β − 1 2α G bc γ ab γ c D a ǫ α − 1 8α G bc γ ad γ d bc D a ǫ α + 1 2 t I bc (ρ I ) α β γ ad γ d bc D a ǫ β .

G.1 Killing Spinors on a Round 3-Sphere
The Killing spinor equation on a round 3-sphere is given by When ǫ satisfies the Killing equation, the spinor C −1 3 ǫ * gives another solution to the equation. One solution to the equation is a constant spinor ǫ 0 ; dǫ 0 = 0.
A spherical harmonics f n corresponding to one of the highest weight states defined by is obtained by f n = c n e inχ sin n θ, with c n = n + 1 2π 2 , where n is an integer required by the periodicity under χ → χ + 2π and is non-negative required by the normalizability of f n ; n ∈ Z ≥0 . It has the eigenvalues (l = n/2, m = n/2,m = n/2) for L 2 =L 2 , L 3 andL 3 , L 2 f n =L 2 f n = n 2 In the text, we assume that the Killing spinors εα obey iγ 45 εα = (τ 3 )αβεβ, or more explicitly, we make the ansatz ε˙1 = ǫ ⊗ ζ + , ε˙2 = C −1 3 ǫ * ⊗ ζ − .

G.7 Auxiliuary SU (2) Flavor Spinors
In order to construct the off-shell supersymmetry of the five-dimensional theory, besides the supersymmetry parameter εα, we need to introduce an additional supersymmetry parameter εα (α = 1, 2), with the indexα of an additional SU(2) flavor group which are not an subgroup of the Spin(5) R R-symmetry group.
Since the two-dimensional spinors ζ ± are linearly independent, these supersymmetry parameters εα, εα form the basis of the four-dimensional linear space of five-dimensional spinors.
The covariant derivative D µ εα is defined by where we assume that the gauge fieldǍ µ ij takes the same as A µ ij ;Ǎ µ ij = A µ ij .

H Dictionary among Notations
In this paper, we follow the same procedure as in [24] to obtain five-dimensional N = 2 supersymmetric Yang-Mills theory in the supergravity background, but our notations are different from the ones in [24]. The procedure is based on the dimensional reduction of the off-shell supergravity in [25], where different notations from [24] and ours however, are used. Therefore, the list of the diffrent notations among [25,24] and ours may be convenient for the readers. We use the common Lorentz metric with the signature (−, +, · · · , +). However, the indices of Lorentz vectors are different; for a Lorentz vector V a in our notations, it is V a in [25] and V a in [24]. Note here that the underline indicates that it is a six-dimensional one in our paper. That's why the above V carries the underline V in ours, but not in [25,24]. However, the underline carried by the Lorentz indices or the coordinate frame indices means that they are six-dimensional in [24] and ours, but not in [25].
the Killing spinor equation is reduced to the one in the previous paper The Killing spinor in the previous paper has the additional property Γ 45 Σα = −i (τ 3 )αβΣβ.
The equations of motion are also reduced to the ones in the previous paper.
In order to lift the on-shell supersymmetry transformation to the off-shell one, introducing the auxiliuary field Dαβ to replace Dαβ = iσ (τ 3 )αβ + 2ig Hα , Hβ − 1 2 δαβ H˙γ , H˙γ , and using the equations of motion of the spinors with the background, the supersymmetry transformation of Dαβ reproduces the previous one It yields the off-shell supersymmetry of the vector multiplet (v M , λα, σ, Dαβ) in the previous paper. For the self-dual S ij ; ε ij kl S kl = 2S ij , the term in δΞ, can be rewritten by using the formulas S ij σ k 1α φ k σ ij αβ Σβ = 3 4 S ij σ ij αβ HαΣβ.
It facilitates the computation to verify that the off-shell supersymmetry transformation of the hypermultiplet (Hα, Ξ, F Hα ) with the auxiliuary field F Hα gives rise to the one in the previous paper. In fact, by modifying the on-shell supersymmetry transformation as δΞ → δΞ + F HαΣ α , δF Hα = iΣ α (the e.o.m. of Ξ), one can lift it to the off-shell supersymmetry transformation.
In the previous paper, we have given the action 27  27 The mass term of Hα in the action L didn't include the curvature term R(Σ)HαHα in the previous paper [1].
where the term ω c.s. denotes It enables us to compare it easily with the action S in (22)(23)(24)(25), and one can see that the substitution of the background on the round 3-sphere in subsection 5.1 into the action S exactly reproduces the above action.