Matrix models from localization of five-dimensional supersymmetric noncommutative U(1) gauge theory

We study localization of five-dimensional supersymmetric $U(1)$ gauge theory on $\mathbb{S}^3 \times \mathbb{R}_{\theta}^{2}$ where $\mathbb{R}_{\theta}^{2}$ is a noncommutative (NC) plane. The theory can be isomorphically mapped to three-dimensional supersymmetric $U(N \to \infty)$ gauge theory on $\mathbb{S}^3$ using the matrix representation on a separable Hilbert space on which NC fields linearly act. Therefore the NC space $\mathbb{R}_{\theta}^{2}$ allows for a flexible path to derive matrix models via localization from a higher-dimensional supersymmetric NC $U(1)$ gauge theory. The result shows a rich duality between NC $U(1)$ gauge theories and large $N$ matrix models in various dimensions.


Introduction
The existence of gravity introduces the gravitational constant G N into a physical theory. It is wellknown that the gravitational constant G N leads to a certain scale known as the Planck length L P = G N c 3 = 1.6 × 10 −33 cm in which spacetime coordinates become noncommutative (NC) operators obeying the commutation relation [y a , y b ] = iθ ab , (a, b = 1, · · · , 2n). (1.1) We are interested in the NC space with a constant symplectic matrix (θ ab ) = α ′ (I n ⊗ iσ 2 ) which is isomorphic to the Heisenberg algebra of an n-dimensional harmonic oscillator. The NC space (1.1) will be denoted by R 2n θ and l s ≡ √ α ′ is a typical length scale for the noncommutativity. Thus the NC space (1.1) is similar to the NC phase space in quantum mechanics obeying the commutation relation given by [x i , p j ] = i δ i j , (i, j = 1, · · · , n). (1.2) We will get an important insight from this similarity to understand the NC spacetime correctly. As we have learned from quantum mechanics, the NC phase space (1.2) introduces a complex vector space called the Hilbert space. This is also true for the NC space (1.1) since its mathematical structure is essentially the same as quantum mechanics. Therefore the NC space (1.1) admits a separable Hilbert space H on which any object O defined on R 2n θ linearly acts. In particular, NC fields become linear operators acting on the Hilbert space H. Since the NC space R 2n θ is isomorphic to the Heisenberg algebra of an n-dimensional harmonic oscillator, its Hilbert space is given by the Fock space and so has a countable basis, the representation of NC fields on the Hilbert space H is given by N × N matrices where N = dim(H) → ∞. Consequently, the NC space (1.1) brings about an interesting equivalence between a lower-dimensional large N gauge theory and a higher-dimensional NC U(1) gauge theory [1,2,3].
To illuminate the remarkable equivalence, let us consider a five-dimensional field theory defined on the NC space R 3 × R 2 θ with coordinates X M = (x m , y a ) where M = 1, 2, · · · , 5, m = 1, 2, 3 and a = 4, 5. Here R 3 is a usual commutative space while R 2 θ is a NC plane whose coordinates obey the commutation relation [y 4 , y 5 ] = iα ′ (1. 3) where α ′ = l 2 s is a constant parameter measuring the noncommutativity of the space R 2 θ . If we define annihilation and creation operators as The representation space of the Heisenberg algebra (1.5) is thus given by the Fock space H = {|n | n ∈ Z ≥0 }, (1.6) which is orthonormal, i.e., n|m = δ nm and complete, i.e., ∞ n=0 |n n| = I H , as is well-known from quantum mechanics.
The field theory we will consider is defined by dynamical fields on R 3 × R 2 θ which are elements of A θ (R 3 ) ≡ C ∞ (R 3 ) ⊗ A θ where A θ is a NC ⋆-algebra generated by the NC space (1.3). Consider two arbitrary fields Φ 1 (X) and Φ 2 (X) on R 3 × R 2 θ whose multiplication is defined by the star product See [4,5] for a review of NC field theories and matrix models. In quantum mechanics, physical observables are considered as operators acting on a Hilbert space. Similarly the dynamical variables Φ 1 (X) and Φ 2 (X) can be regarded as operators acting on the Hilbert space H. Thus one can represent the operators acting on the Fock space (1.6) as N × N matrices in End(H) ≡ A N (R 3 ) where N = dim(H) → ∞: |n n| Φ 1 (x, y)|m m| := ∞ n,m=0 |n n| Φ 2 (x, y)|m m| := ∞ n,m=0 (Φ 2 ) nm (x)|n m|, (1.8) where Φ 1 (x) and Φ 2 (x) are N × N matrices on R 3 . 1 Then one gets a natural composition rule for the products |n n| Φ 1 (x, y)|l l| Φ 2 (x, y)|m m| = ∞ n,l,m=0 (1.9) The above composition rule implies that the ordering in the NC algebra A θ (R 3 ) is compatible with the ordering in the matrix algebra A N (R 3 ) and so it is straightforward to translate multiplications of NC fields in A θ (R 3 ) into those of matrices in A N (R 3 ) using the matrix representation (1.8) without any ordering ambiguity. 1 Note that the eigenvalue n corresponds to the radius r 2 = y 2 4 + y 2 5 of the plane since the radial operator is given by r 2 = 2α ′ (a † a + 1 2 ) and r 2 |n = 2α ′ (n + 1 2 )|n . Therefore, for a well-localized NC field which rapidly decays at asymptotic regions, one may truncate the matrix representation of the NC field to a finite-dimensional space, i.e., N = dim(H) < ∞.
To formulate a gauge theory on R 3 × R 2 θ , it is necessary to dictate the gauge covariance under the NC star product (1.7). The covariant field strength of NC U(1) gauge fields A M (X) = ( A m , A a )(x, y) is then given by where the covariant derivative is defined by D M (X) = ∂ M − i A M (X). Note that the covariant derivative along R 2 θ can be written as an inner derivation, i.e., D a (X) = −i[ φ a (x, y), · ] ⋆ where φ a (x, y) ≡ p a + A a (x, y) with B ab = (θ −1 ) ab and p a = B ab y b . Now we will apply the matrix representation (1.8) to a five-dimensional NC U(1) gauge theory whose action is given by 2 Using the relations, 13) and the matrix representation (1.8), the above five-dimensional NC U(1) gauge theory is exactly mapped to the three-dimensional U(N → ∞) gauge theory on R 3 with a scalar triplet Φ A = (σ, φ 4 , φ 5 ), A = 0, 4, 5: (1.14) where g 2 5 = (2πα ′ )g 2 3 and η AB = diag(−1, 1, 1). Now all the dynamical fields in the action (1.15) are N × N matrices in the adjoint representation of U(N → ∞). The action (1.15) respects the SO(3) × SO(2, 1) global symmetry where SO(3) is the Lorentz symmetry group acting on R 3 and SO(2, 1) is the R-symmetry group acting on (x 0 , y 4 , y 5 ) (see footnote 2).
Let us summarize the isomorphic map from a five-dimensional NC U(1) gauge theory taking values in A θ (R 3 ) = C ∞ (R 3 ) ⊗ A θ to a three-dimensional large N gauge theory taking values in 2 Note that the kinetic term for the σ field has the unusual sign. It is to follow the convention in [6]. (See also the footnote 3 in [7].) A motivation for the wrong sign is to consider Euclidean Yang-Mills theory and yet work with physical fermions. This can be accomplished by making σ time-like. As a result, the N = 2 gauge theory on R 3 × R 2 θ , that we will consider later, can be reduced from a six-dimensional super Yang-Mills theory on R 3,1 × R 2 θ . The scalar field σ in the five-dimensional theory corresponds to the gauge field component compactified along the time-like direction. [1,2,3]: (1. 16) The conventional Coulomb branch of the large N gauge theory (1.15) is defined by for a = 4, 5. In this case the U(N) gauge symmetry is broken to U(1) N . It is important to perceive that, in the limit N → ∞, we have a new phase of the Coulomb branch in addition to the conventional Coulomb branch (1.17) [3,8]. The new vacuum is called the NC Coulomb branch and it is defined by Note that the NC Coulomb branch (1.18) saves the NC nature of matrices while the conventional vacuum (1.17) dismisses the property. We emphasize that the NC space R 2 θ in Eq. (1.18) arises as a vacuum solution of the large N gauge theory (1.15) when we take the limit N → ∞. Consequently, the three-dimensional large N gauge theory (1.15) in the NC Coulomb branch is exactly mapped to the five-dimensional NC U(1) gauge theory (1.14) and thus we verify their equivalence in a reverse way. If the conventional vacuum (1.17) were chosen, we would have failed to realize the equivalence. Indeed it turns out [3,8] that the NC Coulomb branch (1.18) is crucial to realize the large N duality which implies the emergent gravity from matrix models or large N gauge theories.
Recently a localization technique using fixed point theorems provides us a very powerful tool for the exact computation of the path integral both for topologically twisted supersymmetric theories and for more general rigid supersymmetric theories defined on curved spaces. See Refs. [9]- [26] for the collection of reviews of this subject. The power of localization is to reduce the dimensionality of the path integral using supersymmetries such that the path integral receives contribution from the locus of fixed points of supersymmetry. We will put a supersymmetric quantum field theory on S 3 × R 2 θ so that the path integral on S 3 is free of infrared divergences. Our aim is to exactly compute the expectation value O of a BPS observable in the quantum theory, which is defined by where Φ is the set of fields in the action. The usual partition function Z corresponds to the expectation value of the identity operator, i.e., Z = I . We are interested in supersymmetric field theories with a supercharge Q which obeys Q 2 = B with B a linear combination of bosonic charges conserved by the theory. We will assume that the BPS observable O as well as the action S[Φ] is preserved by the supercharge Q, i.e., 20) and the fermionic symmetry generated by Q is free of anomaly. Then we can use the freedom to deform the path integral of a supersymmetric quantum field theory by adding a Q-exact term to the classical action because where t is a non-negative real parameter and P [Φ] is a fermionic functional invariant under B. This means that In the end, the path integral (1.19) is localized to the locus F Q = {Φ|S loc [Φ] = 0} which is BPS field configurations annihilated by the supercharge Q. Depending on the spacetime dependence of the field configuration in the localization locus F Q , we may be left with the path integral of lowerdimensional field theory or, in favorable cases, F Q consists of constant field configurations with a finite-dimensional integral of a zero-dimensional quantum field theory such as matrix models [9]- [26]. Since we will consider a supersymmetric field theory on S 3 × R 2 θ , we will have the supersymmetric version of the equality in Eqs. (1.14) and (1.15). Although two theories are defined in different dimensions with different gauge groups, they are mathematically equivalent to each other. Therefore, we can apply the localization to either a five-dimensional supersymmetric NC U(1) gauge theory or a three-dimensional supersymmetric U(N → ∞) gauge theory. On the one hand, we can first apply the localization to the five-dimensional theory to obtain a two-dimensional NC gauge theory and then consider the matrix representation of the resulting NC gauge theory to yield a zero-dimensional matrix model, as depicted in Fig. 1. On the other hand, we can first implement the matrix representation to the five-dimensional theory to get a three-dimensional large N gauge theory and then apply the localization to the large N gauge theory on S 3 to derive a zero-dimensional matrix model. Both routes should end in an identical zero-dimensional matrix model. The aim of this paper is to verify the flowchart outlined in Fig. 1 using the localization technique and the matrix representation of NC field theories. This paper is organized as follows. In section 2, we construct a five-dimensional N = 2 supersymmetric NC U(1) gauge theory on S 3 × R 2 θ . Using the matrix representation (1.8), the five-dimensional Figure 1: Flowchart for zero-dimensional matrix model supersymmetric NC U(1) gauge theory is isomorphically mapped to a three-dimensional supersymmetric U(N → ∞) gauge theory on S 3 . In section 3 we perform the localization of the five-dimensional supersymmetric NC U(1) gauge theory on S 3 × R 2 θ , which results in a two-dimensional NC U(1) gauge theory. The matrix representation of the two-dimensional NC U(1) gauge theory leads to a zero-dimensional matrix model at the localization locus. We thus explore the red arrows in Fig. 1 to derive a zero-dimensional matrix model via the localization of a five-dimensional NC U(1) gauge theory.
In section 4 we follow the blue arrows in Fig. 1 to get the zero-dimensional matrix model via the localization of a three-dimensional U(N → ∞) gauge theory. Using the fact that a large N gauge theory admits a NC Coulomb branch in the limit N → ∞ and any N × N (Hermitian) matrix can be regarded as the matrix representation of a higher-dimensional NC field, the localization of the three-dimensional large N gauge theory can be easily done by mapping the problem to the one of a five-dimensional NC U(1) gauge theory.
In section 5 we appeal to the mathematical identity depicted in Fig. 1, in particular, the equivalence between a higher-dimensional NC U(1) gauge theory and a lower-dimensional large N gauge theory. This implies [3,8] that the five-dimensional NC U(1) gauge theory describes a five-dimensional gravity according to the large N duality or gauge/gravity duality. We discuss a physical implication of the localization in Fig. 1 from the point of view of the emergent gravity.
We include five appendices, containing our notation and conventions, some details on the supersymmetric transformations, the harmonic analysis on S 3 , and the Clebsch-Gordan coefficients for the irreducible representation of the tensor products j ⊗ 1 and j ⊗ 1 2 .
2 Three-dimensional large N gauge theory from five-dimensional NC U (1) gauge theory The vector multiplet in the five-dimensional N = 2 supersymmetric Yang-Mills (SYM) theory consists of an N = 1 vector multiplet and an N = 1 hypermultiplet in the adjoint representation of gauge group g. We start with the N = 2 SYM theory on the NC space R 3 × R 2 θ with the gauge group g = U(1) ⋆ . We will follow Refs. [27,28,29] for the supersymmetric actions with minor modifications. Later we will put the theory on S 3 × R 2 θ to carry out the localization, which will require some additional terms in the action and a modification of supersymmetry transformations. For a notational simplicity, we will omit the hat symbol to indicate five-dimensional NC fields and implicitly assume the star product (1.7) for the multiplication between NC fields. We hope it does not cause much confusion with three-dimensional large N matrices. The vector multiplet in the five-dimensional N = 1 SYM theory consists of a gauge field A M , a real scalar field σ and a doublet of spinor fields Ψα where the doublet of SU(2) R-symmetry is labeled by the indicesα,β = 1, 2. The spinor field obeys the symplectic Majorana condition To realize an off-shell supersymmetry, we also introduce an auxiliary field Dαβ in the adjoint representation of SU(2) R R-symmetry: Thus the auxiliary field may be represented by Dαβ = D m (σ m )αβ where D m (m = 1, 2, 3) is the triplet of real scalar fields. The N = 1 vector multiplet describes the supersymmetric version of the five-dimensional NC U(1) gauge theory (1.11). The N = 1 hypermultiplet consists of complex scalar fields Hα, a spinor field Θ = Θ 1 + iΘ 2 and two auxiliary fields F α (α = 1, 2). They are also in the adjoint representation of the gauge group g so that the hypermultiplet is combined with the N = 1 vector multiplet to form a vector multiplet in the N = 2 theory. The action for the N = 1 vector multiplet is given by The above action is invariant under the following supersymmetric transformations  (1.15) except the auxiliary term. The matrix representation for the fivedimensional spinors Ψα(x, y) is basically the same as the bosonic fields and it is denoted by the same symbol Ψα(x) that are now N × N matrices over R 3 . But, in order to get a three-dimensional picture after the matrix representation, it is convenient to represent the symplectic Majorana spinors Ψα(x) in terms of three-dimensional spinors: 3 √ where two-dimensional Weyl spinors are the eigenvectors of iΓ 4 Γ 5 , i.e., iΓ 4 Γ 5 ζ ± = ±ζ ± . Then the conjugate spinors are given by where the barred spinors are defined by λ ≡ λ † = (λ * ) T , etc. Using this result, we can get the three-dimensional supersymmetric large N gauge theory whose action takes the form 10) 3 The multiplication factor √ 2πθ is to match the physical mass dimension of five-and three-dimensional spinors since where we have introduced the complex coordinates for R 2 ∼ = C given by 4 z = y 4 + iy 5 , z = y 4 − iy 5 (2.11) and the complex scalar field defined by The auxiliary fields Dαβ(x, y) are also represented by matrices dαβ(x) ≡ √ 2πθDαβ(x). Since the chirality condition, iΓ 4 Γ 5 ζ ± = ±ζ ± , has been imposed on the spinor ζ ± , the SO(2, 1) global symmetry now reduces to U(1).
Since the three-dimensional large N gauge theory (2.10) has been obtained from the matrix representation of the five-dimensional N = 1 vector multiplet without any loss of supersymmetry, the theory (2.10) must preserve N = 2 supersymmetries in three dimensions. Using the matrix representation (1.8) again, it should be straightforward to derive the supersymmetry transformations for the three-dimensional large N gauge theory (2.10) from the five-dimensional ones in Eq. (2.4) by taking the supersymmetry transformation parameters For localization, we will eventually put the N = 2 theory on S 3 , which is a maximally supersymmetric background, i.e., preserves four supercharges [30]. However we must select the supercharges that will be used to localize. Therefore we want to focus on the N = 1 sector in the N = 2 supersymmetries which will be identified with a localizing supercharge on S 3 . For that reason, we consider the supersymmetry transformation parameters given by (2.14) Then the supersymmetry transformations generated by the above parameter are given by [28] δA m = ig 3 (ǫγ m λ − λγ m ǫ), 4 In terms of complex coordinates, the commutator for the star product (1.7) is then given by (2.16) The N = 1 hypermultiplet is described by the action given by where Hα is the complex conjugate of Hα and Θ = Θ † . It is invariant under the following supersymmetry transformations where Λ α are symplectic Majorana spinors and Λ α = (Λ β ) T C 5 ε βα . Like the vector multiplet, the action (2.17) is invariant under the above supersymmetric transformations but it is not necessary to use the Fierz identity for fermionic cubic terms. We can similarly apply the matrix representation (1.8) to the hypermultiplet. For this purpose, let us represent the complex scalar fields Hα(x, y) and spinor Θ(x, y) in terms of three-dimensional fields in the adjoint representation of U(N → ∞): 5 Hα(x, y) → hα(x), We also denote the spinors in Eq. (2.7) with a compact notation: Using this matrix representation, it is straightforward to get the three-dimensional action for the hypermultiplet given by is the matrix representation of the auxiliary fields F α (x, y). The supersymmetry transformations for the three-dimensional hypermultiplet will be obtained by the matrix representation of the five-dimensional ones in Eq. (2.18). Having in mind a localizing supersymmetry on S 3 , let us consider the supersymmetry transformation parameters given by which are related to the spinors Σα in Eq. (2.14) by Λα = Γ 5 Σα. For this reason, we will use the spinor indexα for the spinors in Eq. (2.23). The supersymmetry transformations generated by the spinors Σα and Λα are easily deduced from Eq. (2.18), which are given by Using the matrix representation (1.8), we have obtained two mathematically equivalent theories, that are defined in different dimensions with different gauge groups. Although the two theories superficially look quite different, they should be physically equivalent to each other. To explore the physical implications of the equivalence, one may apply localization techniques to each of them to compute, for example, partition functions and some correlators exactly. On the one hand, one can first apply the localization to the five-dimensional theory to obtain a two-dimensional NC gauge theory and then take the matrix representation to the two-dimensional NC gauge theory to yield a zero-dimensional matrix model, as depicted in Fig. 1. On the other hand, one can first apply the matrix representation to the five-dimensional theory to have a three-dimensional large N gauge theory and then consider the localization of the large N gauge theory to get a zero-dimensional matrix model. Both routes should end up with an identical zero-dimensional matrix model. In the end, the localization will verify a rich duality between NC U(1) gauge theories and large N matrix models in various dimensions as outlined in Fig. 1.
To carry out the localization, we put the theory on S 3 × R 2 θ . Let r be the radius of S 3 . For this purpose, it is convenient to represent five-dimensional fields as the form of three-dimensional fields in which extra coordinates y a are regarded as parameters living in R 2 θ . For the vector multiplet on S 3 × R 2 θ , we take the following representation [27,28] One may notice that the pattern of the above decomposition is equal to the three-dimensional large N matrices in the action (2.10). This replica is not accidental because the matrix representation of the NC fields in Eq. (2.25) will give rise to the three-dimensional large N gauge theory on S 3 whose action is precisely equal to Eq. (2.10). Indeed the corresponding five-dimensional action after the decomposition (2.25) can be written as the same form as Eq. (2.10) with simple replacements Let us consider the supersymmetry transformation parameters as Eq. (2.14) and take ǫ to be a Killing spinor on S 3 obeying the following equation where the covariant derivative acting on a spinor is given by Then the covariant derivative acting on a spinor Ψ in the adjoint representation of g = U(1) ⋆ is defined by For a gauge singlet which does not depend on (y 4 , y 5 ), it reduces to Eq. (2.27). The supersymmetry transformations generated by the Killing spinor ǫ obeying Eq. (2.26) are also given by Eq. (2.15) with the replacement g 3 → g 5 . However, after imposing the condition (2.26), the supersymmetry transformations will no longer be closed because the covariant derivative now acts on the spinor ǫ nontrivially. Fortunately, to achieve a closed algebra, it is enough to modify the transformation law only for the auxiliary fields by adding We verify the closed algebra in appendix C. The result is essentially the same as the one in Refs. [27,28] although two-dimensional surface in our case is a NC space.
Using the matrix representation of the NC fields in Eq. (2.25), the five-dimensional U(1) gauge theory on S 3 × R 2 θ can easily be transformed to a three-dimensional large N gauge theory on S 3 . 6 From now on, we distinguish the curved space indices, µ, ν, · · · = 1, 2, 3 from the flat space ones m, n, · · · = 1, 2, 3.
The dreibein on S 3 is denoted by e m = e m µ dx µ and obeys the structure equation de m = 1 r ε mnp e n ∧ e p . The metric and spin connections are given by ds 2 = e m ⊗ e m = e m µ e m ν dx µ ⊗ dx ν and ω µ = 1 4 ω mn µ γ mn = i 2r γ µ , respectively.
The three-dimensional action on S 3 is obtained from the result in Eq. (2.10) on R 3 with an obvious replacement, The supersymmetry transformations of large N matrices on S 3 can easily be deduced from the five-dimensional ones using the matrix representation in a similar way. Moreover, the closedness of the three-dimensional supersymmetric algebra simply results from the five-dimensional one.
The modified supersymmetry transformations generated by the spinor Σα obeying Eq. (2.26) will be denoted by ∆ ǫ = δ + δ ′ where δ ′ -transformation is given by Eq. (2.29). Since the supersymmetry transformation parameter ǫ obeys the nontrivial Killing spinor equation (2.26), the action (2.3) is no longer invariant under the ∆ ǫ -transformations. Indeed its variation reduces to In order to preserve the ∆ ǫ -supersymmetry, it is necessary to add an extra action such that its supersymmetric transformation cancels the variation (2.30). It turns out [27,28] that the extra action is given by We also put the hypermultiplet on S 3 × R θ for the localization. The strategy is the same as the vector multiplet. In order to achieve a closed algebra of the supersymmetry on S 3 × R θ generated by the Killing spinor obeying the condition (2.26), it is necessary to modify the supersymmetry transformations in Eq. (2.18) by simply adding additional transformations where the spinors Σα and Λα are given by Eqs. (2.14) and (2.23), respectively. It is straightforward (though a bit tedious) to verify that the modified supersymmetry transformation ∆ Σ = δ + δ ′ leads to a closed supersymmetry algebra on the fields in the hypermultiplet. For example, one can show that Note that the adjoint scalar field σ in Eq. (2.37) can be absorbed into a local gauge transformation parameter [31]. Since the supersymmetry transformations are now generated by the Killing spinor ǫ obeying Eq. (2.26), there are extra contributions from the derivative of the Killing spinor given by 38) and the modified transformations introduced in Eqs. (2.29) and (2.36). They are combined to get the total variation of the action (2.17) generated by the supersymmetry transformation ∆ ǫ = δ + δ ′ and denoted by ∆ ǫ S 5H = δS 5H + δ ′ S 5H , which turns out to be non-vanishing. Therefore, as in the vector multiplet, it is necessary to add a compensating action given by HαHα . (2.39) Then the total action is invariant under the supersymmetry transformations, i.e., ∆ ǫ (S 5H + S M 5H ) = 0.

Localization of five-dimensional NC U (1) gauge theory
In this section we will compute the partition function of five-dimensional N = 2 supersymmetric NC U(1) gauge theory on S 3 × R 2 θ by using the localization method. For the localization of fivedimensional quantum field theories, see Refs. [23,24] and references therein. As we pointed out in footnote 2, the adjoint scalar field σ in the vector multiplet has a wrong sign. In order to define the path integral properly, it needs to be analytically continued by replacing σ by iσ.
To carry out the localization procedure, we need to identify the Grassmann-odd symmetry, denoted by δ Q . It is required that δ Q is a symmetry of path integral (i.e. δ Q is not anomalous and preserves the action) and obey δ 2 Q = L B where L B is a Grassmann-even symmetry that could be a combination of Lorentz, R-symmetry, and gauge transformations. We apply the same twisting procedure as [32] by considering a global symmetry group H = SU(2) 1 × SU(2) 2 × SU(2) R where SU(2) 1 × SU(2) 2 is the rotational symmetry on S 3 and SU(2) R is the global symmetry of the N = 2 theory. We embed the Lorentz group K into H as After the twisting, we get a scalar supercharge Q which is thus Lorentz invariant (in the K sense). It is enough to have one scalar supercharge Q for localization and Q is regarded as a BRST operator.

Localization of vector multiplet
According to our twisting, we will define the BRST transformation δ Q ≡ ǫQ by setting ǫ to zero and replacing the Grassmann-odd parameter ǫ by a Grassmann-even parameter in the supersymmetric transformations (2.15). Then δ Q is anticommuting with twisted spinors although all fields have integer spins with respect to K. The corresponding BRST transformations for the vector multiplet are then given by It is straightforward to check that the above BRST transformations are nilpotent, i.e., δ 2 We deform the action (3.2) by adding a BRST Q-exact term such that the total classical action is given by It may be useful to use the Fierz identity: where t is a non-negative real parameter. The explicit form for the Q-exact Lagrangian is given by where k µ = ǫγ µ ǫ and ǫǫ = 1. Note that the second line of L Q 5V can be recast as up to total derivatives. Since the Lagrangian (3.5) is BRST-exact, the modified action (3.4) with a parameter t leads to the same partition function as the undeformed one as was explained in Eq. (1.21). To be precise, the partition function Z(t) for the modified action is t-independent, i.e., dZ(t) dt = 0. Therefore we can calculate the partition function in the large t limit. In this limit, especially t → ∞, the fixed point is given by a solution obeying and fermions = 0. Since the deformation term (3.5) is positive semi-definite, the solution of Eq. (3.7) constitutes the localization locus F Q given by and obey the condition [φ z , σ] = iD z σ = 0. Then the classical action at the locus F Q is given by 8 where g 2 = g 5 2πr 3/2 is a two-dimensional gauge coupling constant. After the matrix representation (1.8), the classical action is mapped to the zero-dimensional matrix model where g = g 2 √ 2πα ′ is a coupling constant of the matrix model. Recall that the matrix σ must be subject to the condition [φ, σ] = 0. Now we compute the one-loop determinant coming from quadratic fluctuations of the fields about the fixed points in (3.8). For that purpose, the gauge-fixing procedure is also necessary for the computation of the path integral. We take the usual gauge-fixing term given by There remains the residual gauge symmetry that acts on where the gauge transformation parameter ω(z, z) is constant along the S 3 . Thus the gauge symmetry is the redundancy of the background σ and φ z , but not of the fluctuations. Using this freedom, we can put the background as 9 For this gauge-fixing of the background fields, we will add another gauge-fixing term [28] given by The path integral of the ghost fields gives the one-loop determinant Since we are interested in the large t limit and perform the path integral over the fluctuations around the fixed points defined by Eq. (3.13), let us expand the fields Φ about the saddle point configuration in Eq. (3.13) and rescale the fluctuation fields as and take the limit t → ∞. Φ 0 denotes the background at the fixed points in which Φ 0 = − i 2α ′ z, i 2α ′ z, and σ 0 for φ z , φ z , and σ, respectively, and Φ 0 = 0, otherwise. Taking t to be large then allows us to keep only the quadratic terms in the Lagrangian (3.5): where all fields indicate the fluctuations δΦ in (3.16). Although the multiplication between fields has originally been defined by the star product (1.7), we can ignore the star product for the quadratic terms since all nontrivial star products in this case are total derivatives and thus can be dropped. Hence we will regard the fluctuations in the Lagrangian (3.17) as fields on S 3 × C. Integral over the auxiliary fields D, F and F have already been performed, which contributes trivial constant terms to the partition function. We will ignore an overall constant of the partition function.
In order to calculate the one-loop determinant of U(1) gauge fields, we first proceed with separating the gauge field into a divergenceless and pure divergent part: where D µ B µ = 0. Then the delta function constraint from Eq. (3.11) becomes δ( 0 φ). One can see that the longitudinal mode in (3.18) can be absorbed into the complex scalar field with the form Thus the longitudinal mode A L µ = −∂ µ φ appears only in the gauge-fixing term in Eq. (3.11) and so we can integrate over φ using the delta function, which picks up a Jacobian factor of det −1/2 0 [33]. 10 The integral of the ghosts in Eq. (3.11) contributes a factor of det 0 . Therefore the one-loop determinant from b and φ as well as the ghosts c and c contributes a factor det To evaluate the path integral of the bosonic fields, it is more convenient to use a differential form notation. For that purpose, let us introduce the dreibeins e m = e m µ dx µ ∈ Γ(T * S 3 ) (m = 1, 2, 3) on S 3 obeying the structure equation de m = 1 r ε mnp e n ∧ e p . See appendix D for the differential geometry on S 3 . The dual frame basis is given by l m = e µ m ∂ µ ∈ Γ(T S 3 ) that is left-invariant vector fields on S 3 satisfying the following commutation relation where k m = ǫγ m ǫ is the Killing vector field. Using this form notation, the quadratic bosonic action on S 3 can be written as We expand a scalar field Φ(x, y) in the basis of the scalar spherical harmonics S m j (x) (j = 0, 1 2 , 1, 3 2 , · · · ; −j ≤ m = (m 1 , m 2 ) ≤ j) on S 3 as The harmonics S m j (x) belong to the (j, j) representation of SO(4) = SU(2) L × SU(2) R . Let us use the ket notation for the spherical harmonics (3.28) They obey the following properties: In order to calculate the determinant of the operators in the quadratic Lagrangian (3.25), we will take the same strategy as section 3.5 in Ref. [33]. We identify l m = e µ m ∂ µ with 2i r L m , where L m are operators in the su(2) Lie algebra. We also choose the Killing vector k µ as δ 3 µ [28,33] and thus k = k µ ∂ µ = 2i r L 3 . Since we will use the SU(2) L -invariant frame on S 3 (when thinking of S 3 as SU(2) and letting it act on itself), the angular momentum operators L m are acting on the SU(2) L index m in Eq. (3.28) and it is given by where L ± = L 1 ± iL 2 . In order to have a similar expansion for the one-form B = e m B m , we need the other set of basis, which can be constructed by considering a tensor product of the scalar spherical harmonics with the dreibeins. In particular, the dreibeins e m on S 3 are taken as the eigenstate of the spin operators − → S · − → S and S 3 where (S m ) ln = iε lmn is the spin-1 representation of SU (2). In terms of the spin-1 basis |s = 1, s z (s z = −1, 0, 1), they are given by and satisfy where S ± = S 1 ± iS 2 . The spin-1 basis |s = 1, s z transforms as the (1, 0) representation of SU(2) L × SU(2) R . Therefore the tensor product may be decomposed into the following irreducible representations, (j, j) ⊗ (1, 0) = (j + 1, j) ⊕ (j − 1, j) ⊕ (j, j), (3.33) where the last representation is a gradient of the scalar spherical harmonics. A general vector field on S 3 is then expanded as a combination of gradients of the scalar spherical harmonics plus a set of vector spherical harmonics V m j± (x) which are in the representation (j ± 1, j) of SU(2) L × SU(2) R [34,35]. The vector spherical harmonics in the decomposition (3.33) are then given by where j, m 1 ; 1, s|k, m are the Clebsch-Gordan coefficients of the spin-j representation and the spin-1 representation of the SU(2) group into the spin-k representation, with k = (j − 1), j, (j + 1).
See appendix E for the Clebsch-Gordan coefficients. Let us denote the vector spherical harmonics in (3.34) by V m k,j ≡ V k,m;j,m 2 . They have the following properties * dV m j+1,j = 2(j+1) and form an orthonormal basis It is not difficult to derive the following formulae  For example, for a real scalar field Φ(x, y), the reality condition in the harmonic expansion (3.27) is equivalent to the . This kind of the reality condition has to be incorporated into the sum (3.41) and the corresponding one-loop determinant. However, the reality condition may be easily implemented by relaxing the condition and then taking the square-root of the final one-loop determinant. We will take this strategy for simplicity since the one-loop determinant resulting from bosonic and fermionic fluctuations will eventually be cancelled each other.
Since the one-form gauge fields B ± have a different allowed range for j and m, special treatments are needed when m is close to ±j, ±(j + 1) and j = 0. The action S Q B;j, m of the modes for j ≥ 1, |m| ≤ j − 1, |m ′ | ≤ j, is given by after the scale transformation of gauge fields Note that the shifting form (3.45) was suggested in the last line of Eq. (3.26). The above scale transformation is just for convenience and it will be recovered later through the Jacobian factor in the path integral measure. The action (3.42) can then be written as (3.48) The crossing terms of B m j±1,j and σ m j can be diagonalized by shifting the scalar field σ m j as The modes for the bosonic fluctuations after the above shifting may be arranged into the following form Then the one-loop determinant from these modes yields 12 where detD is the functional determinant of the operator D since the expansion coefficients in Eqs.
Here we have incorporated the Jacobian factor for the change of variable (3.46). After some similar algebra, the action S Q B;j, m of the modes for j ≥ 1/2, m = −j, |m ′ | ≤ j, can be read as and the one for j ≥ 1/2, m = j, |m ′ | ≤ j as Therefore the one-loop determinant from these sector with j ≥ 1/2, m = ±j, |m ′ | ≤ j is given by Since the action S Q B;j,±(j+1),m ′ of the modes with j ≥ 0 is coming from the contribution of B + only, we directly calculate the action without shifting scalar fields and a scale transformation and it is then given by Finally, the action S Q B;j=0,0,0 contains remaining modes with j = 0 and it is simple as with the factor up to an overall constant. This result is essentially the same as Ref. [28] with the root α = 0.
Next we consider the fermions in the quadratic Lagrangian (3.17). Note that P − ≡ 1 2 (1 − γ µ k µ ) in the fermionic Lagrangian is a projection operator, i.e., P 2 − = P − and thus projects out some component of a spinor; in our case with k µ = δ 3 µ , the upper component. Fortunately the problem reduces to computing spin-orbit coupling in quantum mechanics by identifying l m = e µ m ∂ µ and γ m = e m µ γ µ with 2i r L m and 2S m , respectively [33]. Since ∇ µ = ∂ µ + i 2r γ µ , we have the relation In order to get the harmonic expansion of the spinors λ and ψ on S 3 , it is useful to introduce the eigenspinors θ k,ms;j,m 2 (x) = θ m k,j (x) of the operator D ≡ γ m ∇ m by [28] θ m k,j (x) = and form an orthonormal basis where the complex conjugate obeys the usual relation θ m k,j (x) And it is easy to verify that We will use the shorthand notation for the coefficient modes such that λ j± 1 2 ,m+ 1 2 ;j,m ′ → λ j± 1 2 and ψ j± 1 2 ,m+ 1 2 ;j,m ′ → ψ j± 1 2 whenever such a notation is enough. Since the Lagrangian (3.17) is also quadratic in fermionic fields, it is straightforward to evaluate the action for the mode expansion (3.64), which can be read as In order to implement the Gaussian integration for the action (3.65), we shift the spinor modes for The corresponding action for these modes is then given by Thus the above fermionic fluctuations lead to the one-loop determinant (3.67) Here we have considered the fact that λ j± 1 2 and ψ j± 1 2 are complex spinors. Cf. footnote 11. For the remaining fermionic modes with j ≥ 0, m = −(j + 1), j, |m ′ | ≤ j, the corresponding action reads as which leads to the one-loop determinant given by Combining all the contributions from the fermionic fluctuations yields the one-loop determinant (3.70) Note that the one-loop determinant (3.70) from the fermionic fluctuations exactly cancels the one (3.58) from the bosonic fluctuations. This result is somewhat expected [28] since our result corresponds to the five-dimensional N = 2 Yang-Mills theory on S 3 ×R 2 since the noncommutativity can be ignored at the quadratic order. However the classical action (3.9) at the localization locus needs not be quadratic and thus the noncommutative structure between background fields must be kept. The matrix representation of the background fields consequently gives rise to a zero-dimensional matrix model with the action (3.10) subject to a perplexing constraint [φ, σ] = 0. Thus we have explored the red arrows in Fig. 1 to derive a large N matrix model from a five-dimensional NC U(1) gauge theory.

Localization of hypermultiplet
Using the same twisting as the vector multiplet, we can deduce the BRST transformations for the hypermultiplet given by The above BRST transformations are nilpotent, i.e., δ 2 Q = 0, as was expected. After a tedious calculation, the BRST transformation of the action (2.17) can be determined as (3.72) The above variation is exactly cancelled by the BRST transformation of the mass term (2.39). Thus the total action for the hypermultiplet is BRST invariant, i.e., δ Q (S 5H + S M 5H ) = 0. Similarly we deform the action S (inv) 5H ≡ S 5H + S M 5H by adding a BRST-exact term where the bosonic part of the Lagrangian L Q 5H is given by up to total derivatives and the fermionic part by Now the total classical action is defined by A fixed point of the BRST-exact action (3.73) is given by a solution to δ Q η = δ Q χ = 0 and δ Q η = δ Q χ = 0 in addition to fermions = 0. One gets H 1 = F 1 = 0 from the first condition and H 2 = F 2 = 0 from the second one. As a result, one finds no nontrivial background for the hypermultiplet and thus S 5H | F Q = 0.
Since we expand the fields in the hypermultiplet around the saddle point configuration as Eq. (3.16) and take the limit t → ∞, it is enough to keep the quadratic order of the fluctuations. After shifting the fields, the bosonic Lagrangian (3.74) reduces to a simple form: Note that we have already carried out the integration over the auxiliary fields F 1 and F 2 and their contributions to the partition function are simply an overall constant. Similarly the fermionic Lagrangian (3.75) also becomes a quadratic form: where S ± = 1 2 (γ 1 ± iγ 2 ) acts on the spinors S + ς − = ς + , S − ς + = ς − . One can show by a direct calculation using the mode expansion (3.79) below that the last term in Eq. (3.78) vanishes. A more easier way to see this is to consider a change of variable (cf. footnote 5), χ → C −1 3 χ * , χ → χ T C 3 , under which χk m γ n iγ m D n χ − 3 2r χ → χk m γ m iγ n D n χ + 1 2r χ after integration by parts and using the relation D n k m = − 1 r ε mnp k p .
The one-loop partition function for the hypermultiplet can be obtained in the same way as the vector multiplet by employing the harmonic expansions of the fluctuations 13 (3.80) The above mode expansion for the fermionic Lagrangian (3.78) leads to the action given by Therefore the one-loop determinant from these fermionic modes yields a factor Wrapping up the bosonic and fermionic contributions, one finds that the hypermultiplet contributes just a constant to the total partition function.

Localization of three-dimensional large N gauge theory
The relationship between a lower-dimensional large N gauge theory and a higher-dimensional NC U(1) gauge theory in Fig. 1 is an exact mathematical identity. The identity in Fig. 1 is derived from the fact that the NC space (1.1) admits a separable Hilbert space and NC U(1) gauge fields become operators acting on the Hilbert space. Using the matrix representation (1.8) of NC fields, we have obtained a three-dimensional large N gauge theory described by the action (2.10) for the vector multiplet and the action (2.22) for the hypermultiplet. Now we will explore the blue arrows in Fig.  1 to illuminate how to derive the same large N matrix model starting from a three-dimensional large N gauge theory. See Refs. [15,16] and references therein for the localization of three-dimensional quantum field theories.
The BRST invariant theory for the vector multiplet in the three-dimensional large N gauge theory is given by the action (3.4) by applying the isomorphic map (1.16). The localization locus F Q is defined by where ǫ is a Killing spinor satisfying Eq. (2.26). The space of F Q consists of all possible solutions obeying the above conditions. It may be characterized by the BPS equations given by and D = F = F = 0. See, e.g., [36] for the localization at a Dirac monopole configuration. But we will consider a more simplified set of solutions satisfying A µ = ∂ µ σ = ∂ µ φ z = 0 and thus F Q is defined by the set of constants obeying Then the classical action at the locus F Q is given by the zero-dimensional matrix model (3.10).
The conventional choice of vacuum in the Coulomb branch of U(N) gauge theory is given by Eq. (1.17). It means that the locus F Q takes values in the Cartan subalgebra of the Lie algebra of g = U(N) such that Here H i (i = 1, · · · , N) are generators of the Cartan subalgebra of u(N) of rank N. In this case the U(N) gauge symmetry is broken to U(1) N . Note that the gauge group g in our case is U(N), in particular, in the limit N → ∞. In order to find all possible solutions defining the space of F Q , it is important to notice that the limit N → ∞ opens a new phase of the Coulomb branch, the so-called NC Coulomb branch [3,8]. For example it may be characterized by the vacuum (1.18) satisfying the Moyal-Heisenberg algebra. It should be emphasized that the NC Coulomb branch arises as a vacuum solution of the large N gauge theory (2.10) and it saves the NC nature of matrices while the conventional vacuum (4.4) dismisses the property.
To be specific, the locus F Q in the NC Coulomb branch is given by and It is then obvious that [φ z , σ] vac = 0 and [φ z , φ z ] vac = 1 2α ′ I N ×N . Of course, we have to take the limit N → ∞ to make sense of this NC vacuum. 14 One may observe that the NC vacuum (4.6) can be represented by the root system r of the Lie algebra su(N) as Therefore the NC Coulomb branch is in sharp contrast to the conventional vacuum (4.4) which takes values in the Cartan subalgebra of u(N) only. The localization of a large N gauge theory at the conventional Coulomb branch (4.4) has been discussed by many authors [33,36,37,38,39,40]. See also a review [35]. So we will focus on the localization at the NC Coulomb branch. Let us represent all possible deformations of the vacuum F Q 14 It might be remarked that √ 2α ′ φ z corresponds to a = x+ip √ 2 and √ 2α ′ φ z to a † = x−ip √ 2 and the operators x and p obey the Heisenberg algebra [x, p] = i . As is well-known from quantum mechanics, the representation space of the Heisenberg algebra is the infinite-dimensional Fock space H and so the representation φ z and φ z on the Hilbert space H requires infinite-dimensional matrices. by σ(x, z, z) = σ 0 where Φ(x, z, z) collectively represents remaining fields with the vanishing vacuum expectation value at F Q . The notation in Eq. (4.8) means the large N matrices such that, for example, In other words, the matrix representation of Φ(x, z, z) on the Fock space H can be expanded in the Chevalley basis (H i , E ±α ) of the Lie algebra u(N) in the limit N → ∞. With this notation, it is obvious that the localization of the three-dimensional large N gauge theory around the locus (4.3) is exactly parallel to the five-dimensional case and thus arrives at the results (3.58) and (3.70) for the one-loop determinant from the bosonic and fermionic fluctuations described by large N matrices in Eq. (4.8). So we confirm the duality in Fig. 1 to illustrate how to use a five-dimensional NC U(1) gauge theory for the localization of the vector multiplet in the three-dimensional large N gauge theory. We can apply the same idea to the hypermultiplet in a three-dimensional large N gauge theory. The BRST invariant theory for the hypermultiplet in the three-dimensional large N gauge theory is given by the action (3.76) by applying the isomorphic map (1.16). The localization locus F Q for the hypermultiplet is defined by Given the locus F Q of the vector multiplet, the solution for the above equations is trivial and it is given by H 1 = H 2 = F 1 = F 2 = 0. The fluctuations around the locus F Q are described by which we assume the same expansion (4.9) in terms of the Chevalley basis (H i , E ±α ) of the Lie algebra u(N). So we will eventually arrive at the result (3.80) and (3.82) for the one-loop determinant for the hypermultiplet described by the three-dimensional large N gauge theory. This result also confirms the duality in Fig. 1 between a five-dimensional NC U(1) gauge theory and a three-dimensional large N gauge theory.

Discussion
We emphasize that a NC space realizes a remarkable duality between a higher-dimensional NC U(1) gauge theory and a lower-dimensional large N gauge theory [3,41]. This duality is simply derived from a very elementary fact that the NC space (1.1) denoted by R 2n θ is equivalent to the Heisenberg algebra of an n-dimensional harmonic oscillator. A well-known property from quantum mechanics is that the NC space R 2n θ admits a separable Hilbert space and NC U(1) gauge fields become operators acting on the Hilbert space. The matrix representation of dynamical operators on the Hilbert space immediately leads to the picture depicted in Fig. 1. Therefore the relationship between a lowerdimensional large N gauge theory and a higher-dimensional NC U(1) gauge theory in the figure is an exact mathematical identity. In this correspondence, the dynamical variables in the lower-dimensional large N gauge theory take values in A N (S 3 ) = C ∞ (S 3 ) ⊗ A N and those in the higher-dimensional NC U(1) gauge theory take values in A θ (S 3 ) = C ∞ (S 3 ) ⊗ A θ . We have applied the localization technique to this correspondence. The result reveals a rich duality between NC U(1) gauge theories and large N matrix models in various dimensions, as clearly summarized in Fig. 1.
We note that both A N (S 3 ) and in H is a Lie algebra homomorphism. However there is another important lesson that we have learned from quantum mechanics. For example, the momentum (position) operator in the Heisenberg algebra (1.2) can be represented by a differential operator in position (momentum) space, i.e., p i = −i ∂ ∂x i or x i = i ∂ ∂p i . Recall that we have used the leftinvariant vector fields in Eq. (D.7) to represent the SU(2) Lie algebra. More generally a NC algebra A θ always has a representation in terms of a differential (graded) Lie algebra D and the map A θ → D is also a Lie algebra homomorphism [5]. To be specific, let us apply the Lie algebra homomorphism A θ → D to the dynamical variables in Fig. 1. We will get a set of differential operators derived from the five-dimensional NC U(1) gauge theory on S 3 × R 2 θ , which is isomorphically mapped to a three-dimensional large N gauge theory through the matrix representation as shown in Fig. 1. An interesting problem is to identify the theory described by the set of differential operators. It turns out [3] that the theory in a classical limit describes a five-dimensional gravity whose asymptotic (vacuum) geometry corresponds to S 3 × R 2 and the relationship between the five-dimensional gravity and the three-dimensional large N gauge theory is the well-known gauge/gravity duality or large N duality.
Therefore the localization of a higher-dimensional NC U(1) gauge theory and a lower-dimensional large N gauge theory in Fig. 1 can be interpreted as a localization of a five-dimensional gravity Figure 2: Localization for large N duality emergent from the gauge theory. A configuration at the localization locus F Q will be mapped to a BPS geometry according to the correspondence A θ → D. This means that there exists an isomorphic map from the NC U(1) gauge theory to the Einstein gravity which completes the large N duality [41]. In our case, Eq. (4.3) corresponds to a vacuum geometry S 3 × R 2 . As we pointed out in section 4, the locus is characterized by the BPS equations (4.2) whose solution is, in general, nontrivial, e.g. U(N) instantons on S 3 such as Nahm monopoles and U(1) N monopoles in R × S 2 [36]. Of course, putting instantons on a compact space is highly nontrivial. Nevertheless solutions exist, e.g., [42]. It is known [43] that NC U(1) instantons on R 2n θ are equivalent to n-dimensional Calabi-Yau manifolds in a commutative limit. Therefore it will be interesting to consider a nontrivial locus such as BPS solutions and study their emergent geometry around the locus from the geometric point of view.
Our localization scheme outlined in Fig. 1 may be directly applied to a localization problem in the AdS/CFT correspondence [20]. The AdS 5 space has a boundary R × S 3 in global coordinates. Hence one may consider the N = 4 U(N) SYM theory on R × S 3 [34,44] to study the AdS 5 /CF T 4 duality. The localization technique provides us a powerful tool for a nonperturbative analysis of the large N duality [6]. The N = 4 SYM theory has six adjoint scalar fields and the AdS/CFT duality typically considers the N → ∞ limit of U(N) gauge group. Therefore one can consider a vacuum in the NC Coulomb branch by turning on vacuum expectation values of the adjoint scalar fields such that the vacuum moduli obey the Heisenberg algebra (1.18) with rank(B) = 6. As we illustrated in section 4, the fluctuations around the NC Coulomb branch (1.18) are described by a ten-dimensional N = 1 supersymmetric NC U(1) gauge theory [8]. Although these two theories are defined in different dimensions with different gauge groups, they are mathematically equivalent to each other as depicted in Fig. 2. In this paper we have shown that the localization of a large N gauge theory at the NC Coulomb branch is equivalent to the localization of a higher-dimensional NC U(1) gauge theory. The corresponding picture for the AdS/CFT correspondence has been summarized in Fig. 2. As we remarked in section 1, the NC field theory representation of a lower-dimensional large N gauge theory in the NC Coulomb branch will provide us a powerful machinery to identify gravitational variables dual to large N matrices [3]. Hence one may study a nonperturbative aspect of the AdS/CFT correspondence using the localization technique along the flowchart in Fig. 2. We think that Fig. 2 will be a straightforward generalization of Fig. 1. We hope to address this interesting problem in the near future.

A.2 Charge conjugation matrices
The five-dimensional charge conjugation matrix C 5 obeys where T denotes the transpose of a matrix. It is related to the three-dimensional charge conjugation matrix C 3 = iσ 2 by and thus C 3 satisfies the relation

A.3 Fermion bilinears
Symplectic Majorana spinors satisfy the following transposition property of fermion bilinears: The raising and lowering of SU(2) R indices are defined by Then the following relation is deduced: which should not be confused with the first one in (A.10). Our convention for the SU(2) R invariant tensors, εαβ and εαβ, is given by and thus εα˙γε˙γβ = δβ α .

A.4 Lie algebra g
The gauge group for NC U(1) gauge theories is U(1) ⋆ whose element is given by where λ = λ(X) ∈ A θ . We consider the matrix representation (1.8) of the NC gauge parameter λ ∈ A θ which leads to a gauge transformation parameter in U(N → ∞) gauge theory. In this way, we get the gauge group U(N) for large N gauge theories with the limit N → ∞, i.e., U(1) ⋆ → U(N) by e iλ ⋆ → e iΛ where Λ(x) = N 2 a=1 λ a (x)T a . The Lie algebra generators in u(N) are split into su(N) generators T a (a = 1, · · · , N 2 − 1) and a u(1) generator T N 2 = 1 √ N I. The su(N) generators are normalized as Tr(T a T b ) = δ ab and obey the commutation relation It is convenient to introduce the Chevalley basis (H i , E ±α ) for a simple Lie algebra, i.e. su(N), obeying the relations 16) where i = 1, · · · , N − 1 and α ∈ r is an element of the root system r of the Lie algebra su(N).

A.5 Integral on NC space
For the star product (1.7), the integral is invariant under cyclic permutations of the smooth functions f i [4]. In particular, the following useful relations are deduced from this property: (A.19) Note that the above cyclic permutations have been derived from the assumption that the functions f i appropriately behave, i.e., rapidly decay, at asymptotic infinities so that total derivative terms can be dropped. Thus one may worry about the first term in (1.11) since F M N does not decay to zero but approaches to a constant value at infinity. Fortunately, constant terms do not introduce any trouble for the cyclic permutation of the integral because they are immune from the star product and so they can be placed outside the integral. This property can also be understood using the matrix representation (1.8). In the matrix representation, the integral (A.17) is transformed into the trace over matrices, i.e., (A.20) Therefore the cyclic property of the integral (A.17) corresponds to the cyclic permutation of the matrix trace, e.g., for N × N matrices f 1,2,3 (x) over R 3 or S 3 . Note that the background B-field is mapped to the identity matrix (see A.4) and so it can freely escape from the trace. Hence the previous argument is confirmed.

B Vanishing cubic terms in supersymmetric transformations
This appendix is to check the supersymmetric invariance of five-dimensional NC U(1) gauge theory, in particular, the vanishing of the fermionic cubic terms in supersymmetric variations [45].
As in the commutative case, after cancellation of all the quadratic terms, we are left with the cubic terms of Ψ field: In order to show the vanishing of the cubic terms in (B.1), we need the Fierz identity for gamma matrices which leads to the identity for arbitrary symplectic Majorana spinors ǫ, ζ, η and φ. First note that, using the identity Ψα(ΣβΨβ)Ψα := −(Ψα) T (Ψα) T (ΣβΨβ) = −ΨαΨα(ΣβΨβ), we get

C Closed supersymmetric algebra
In this appendix, we present a detailed result for the closedness of the supersymmetry algebra on S 3 × R 2 θ for the vector multiplet. The modified supersymmetry transformations generated by the spinor ǫ obeying Eq. (2.26) will be denoted by ∆ ǫ = δ ǫ + δ ′ ǫ where δ ǫ -transformations are given by Eq. (2.15) with the replacement g 3 → g 5 . The result on S 3 is exactly the same as the five-dimensional case if g 5 is replaced by g 3 .
First, the vector multiplet satisfies the following closed algebra given by where ζ µ = ǫγ µ η − ηγ µ ǫ, ζ = ǫη − ηǫ (C. 8) and the covariant derivative D µ contains gauge and spin connections. In order to get the above result, we have used at several places the three-dimensional Fierz identity, (ǫψ)φ = − 1 2 ψ(ǫφ) − 1 2 γ µ ψ(ǫγ µ φ) (C.9) for complex spinors ǫ, ψ and φ. It is useful to recall that the transformation parameters ǫ and η do not depend on the NC coordinates y a ∈ R 2 θ , so they are immune from the star product that is implicitly assumed for all multiplications. It is easy to see that [∆ η , ∆ ǫ ] acts as an even symmetry of the theory since it can be written as a sum of a translation generated by the parameters ζ µ , a gauge transformation by ρ = ζ µ A µ + ζσ, a Lorentz transformation by κ µν = 1 r ε µνλ ζ λ and a U(1) transformation in SU(2) R symmetry by υ = ζ r [31]. Thus it verifies that the modified supersymmetry transformations ∆ ǫ form a closed algebra even off-shell.

D Harmonic analysis on S 3
Any element of SU(2) can be written in the form g = α β −β α , |α| 2 + |β| 2 = 1. See the appendix in Ref. [35] for their explicit coordinate representations. We can use the MC forms to analyze the differential geometry of S 3 . The dreibein of S 3 is proportional to ω m , and we write e m = r 2 ω m µ dx µ = The spin connection ω m n is introduced via the torsion-free condition de m + ω m n ∧ e n = 0.
The curvature tensor is given by R m n = dω m n + ω m p ∧ ω p n = 1 r 2 e m ∧ e n , or, equivalently, R mnmn = 1 r 2 (no sum). Thus the Ricci tensor and the Ricci scalar are given by R mn = 2 r 2 δ mn and R = 6 r 2 , respectively. The scalar Laplacian on S 3 can be written in local coordinates as 10) or equivalently 0 = −g µν ∂ µ ∂ ν + g µν Γ λ µν ∂ λ . The Peter-Weyl theorem says that any square-integrable function on S 3 ∼ = SU(2) can be written as a linear combination of the spherical harmonics in Eq. (3.28) as was illustrated in (3.27).
The dreibeins e m are taken as the eigenstate of the spin operators − → S · − → S and S 3 where (S m ) ln = iε lmn is the spin-1 representation of SU (2). The vector spherical harmonics on S 3 are then constructed by considering a tensor product of the scalar spherical harmonics with the spin-1 basis |s = 1, s z (s z = −1, 0, 1). The space of one-forms on S 3 can be decomposed using the vector spherical harmonics in Eq. Using the dreibein, we can define locally inertial gamma matrices as γ m = E µ m γ µ which satisfy the relations {γ m , γ n } = 2δ mn , [γ m , γ n ] = 2iε mnp γ p .
The covariant derivative acting on a spinor is defined by 14) It follows that the Dirac operator is Since the total angular momentum is defined by − → S corresponds to spin s = 1 2 and − → L to j, the possible eigenvalues of − → J are j ± 1/2. Thus the eigenvalues of the Dirac operator (D.15) are equal to 2 r j ± 1 2 j ± 1 2 + 1 − j(j + 1) = The eigenvectors of the Dirac operator are given by the spinor spherical harmonics introduced in Eq. (3.60).

E Clebsch-Gordan coefficients
We reproduce the Clebsch-Gordan coefficients for the products j ⊗ 1 and j ⊗ 1 2 in Ref. [28] for reader's convenience.