Index calculation by means of harmonic expansion

We review derivation of superconformal indices by means of supersymmetric localization and spherical harmonic expansion for 3d N=2, 4d N=1, and 6d N=(1,0) supersymmetric gauge theories. We demonstrate calculation of indices for vector multiplets in each dimensions by analysing energy eigenmodes in S^pxR. For the 6d index we consider the perturbative contribution only. We put focus on technical details of harmonic expansion rather than physical applications.


Introduction
Indices are special type of partition functions in supersymmetric theories defined in such a way that the contributions of bosonic and fermionic states partially cancel each other and only modes satisfying some BPS conditions make non-trivial contributions. Thanks to the supersymmetry, indices are protected from uncontrollable quantum corrections, and it is possible to calculate them exactly even in strongly coupled theories. They are powerful tools to analyze dynamics of supersymmetric theories and have been used to test various dualities.
There are variety of indices. The simplest one is the Witten index [1], which is defined by where H and F are the Hamiltonian and the fermion number, respectively. The trace is taken over all gauge invariant states. This gives an integer, which is the difference between the numbers of bosonic and fermionic vacua. To calculate I W the system is put in a torus to avoid IR divergence. We can generalize I W by replacing the background space with another compact manifold M (or a non-compact manifold with appropriate boundary conditions). If we take M to be S p , the index gives BPS spectrum of the theory in S p × R. We can map the background to R p+1 by a Weyl transformation, and for superconformal theories the index can be associated with local BPS operators via the state-operator correspondence. This kind of indices, which we will focus on in this paper, are called superconformal indices [2][3][4]. The purpose of this paper is to demonstrate derivation of the superconformal indices for 3d, 4d, and 6d supersymmetric theories using harmonic expansion. For a given p-dimensional manifold M , the index is defined and calculated as follows. We consider a p + 1 dimensional supersymmetric field theory defined on M × R. We suppose the theory has k commuting bosonic conserved charges including the Hamiltonian H and other k − 1 charges F i (i = 1, . . . , k − 1). Let Q be a supercharge, and Q be another supercharge such that {Q, Q} = H + c i F i . The index is defined by I(e −β , e −γi ) = tr[(−1) F e −βH−γiFi ]. (2) The operator βH + γ i F i in the exponent must commute with Q. Therefore, only k − 1 variables in (β, γ i ) are independent. It is easy to show that the deformation of the Hamiltonian by a Q-exact term does not change (2). If we take V = Q, the deformation is equivalent to the shift of parameters (β, γ i ) → (β, γ i ) + t(1, c i ). This means that the index depends on the parameters through γ i − βc i , and the index is a function of k − 2 variables. If we choose V so that the deformation term contains quadratic terms of fields, the theory becomes weakly coupled when we take t → ∞ limit. This enables us to calculate the index exactly from the information of eigenmodes of the deformed Hamiltonian in the weak coupling limit.
Let Φ be fields in the theory. We expand the solution of the free field equations by eigenmodes of charges (H, F i , t I ) labeled by n, where t I are Cartan generators of the gauge group. Let (ω n , f i,n , t I,n ) be the eigenvalues of a mode n. The solution is given by where τ is a Euclidean time and x = (x 1 , . . . , x p ) are coordinates in M . The frequencies ω n are always non-zero in the theories we consider in the following sections except for gauge degrees of freedom, which have to be dealt with separately. The spectrum of multi-particle states in a free theory is uniquely determined by the spectrum of single-particle states. It is convenient to define the single-particle index in a similar way to (2) by I sp (e −β , e −γi , e iaI ) = tr[(−1) F e −βH−γiFi+iaI tI ] = n (−) F e −βωn−γifi,n+iaI tI,n sign ωn , where the trace is taken over all single-particle states including charged states. The exponent, sign ω n , represents the fact that negative frequency modes correspond to anti-particles, which carry opposite quantum numbers to those of particles corresponding to positive frequency modes. If the field is complex we have to take account of both positive and negative frequency modes, while for a real field we need to include only positive (or negative) frequency modes. The index for multi-particle states is given by I mp (e −β , e −γi , e iaI ) = Pexp I sp (e −β , e −γi , e iaI ), (6) where Pexp is the plethystic exponential defined by Pexp f (x, y, . . .) = exp ∞ n=1 1 n f (x n , y n , . . .) .
This multi-particle index includes the contribution of charged states as well as gauge invariant states. To obtain the index for physical states, we need to pick up the contribution of 2/26 gauge invariant states by integrating over a I . To determine the measure of this integral let us rewrite (2) in the path integral form where S is the supersymmetric action of the theory defined on the background M × S 1 , and S ′ = tδ Q V is a Q-exact deformation term corresponding to the deformation in (3). The parameters β and γ i in (3) are taken in the path integral formulation as the period of S 1 and the Wilson lines associated with the global symmetries, respectively. In the weak coupling limit the saddle point approximation gives the exact answer. For a gauge theory we have to fix the gauge symmetry and carefully take the associated ghost factor into account. Let α be weights in the adjoint representation of the gauge group G including both roots and Cartan generators, and t α be the corresponding generators. Let A be a fluctuation of the gauge potential around a saddle point. It belongs to the adjoint representation and is expanded as where α = 0 means the sum is taken over only roots, and I labels the Cartan generators. We divide the gauge field in M × S 1 into two parts, the x-independent part A 0 and the xdependent part A ′ ≡ A − A 0 . The gauge fixing for A ′ will be done in the following sections. We here focus only on A 0 = A 0 τ (τ )dτ . The path-integral of A 0 essentially gives the integral over the Wilson line u = P exp 1 S A 0 ∈ G. We take the static gauge where a I are τ -independent constant, which is identified with the chemical potentials for t I in (5). In this gauge the integral over u can be rewritten as where r = rank G and |W | is the number of elements of the Weyl group. α(a) is defined by [a, t α ] = α(a)t α for a Cartan element a of the gauge algebra. As well as the non-zero mode contribution (6), the measure factor in (11) is also written in the form of the plethystic exponential of the single-particle index We can interpret this as the contribution of ghost constant modes. After all, we obtain the following final expression for the index In the rest of the paper we calculate the single particle index I sp for vector multiplets in 3, 4, and 6 dimensional theories. For 6d theories we only take account of the perturbative contribution. 3/26

Spherical Harmonics
In this section, we define spherical harmonics on S p in preparation for the analysis in the following sections.

Local frame on S p
Let f a (a = −1, 0, 1, . . . , p − 1) be p + 1 unit vectors that form an ortho-normal basis in R p+1 . We denote the position vector by y, and the ortho-normal coordinates y a are defined by y = f a y a , or, equivalently, y a = f a · y. The unit sphere S p is defined by y a y a ≡ y · y = 1.
Let T ab be the generators of the rotation group G ≡ SO(p + 1) that act on f a as where ρ V cd (T ab ) := δ ac δ bd − δ ad δ bc are representation matrices for the vector representation. For an anti-symmetric tensor λ ab we denote the corresponding G generator 1 2 λ ab T ab by λ. It acts on a vector v = f a v a as S p can be given as the coset G/H, where H = SO(p) is the subgroup of G that does not move a specific vector n ∈ S p . We choose n to be n = f −1 . S p = G/H means that there exists a projection map π : G → S p defined by In other words, G is an H fibration over S p . This fiber bundle is called the frame bundle of S p . Let g be a section of the frame bundle. Namely, g is a map from S p to G satisfying y = g(y)n. With this section we can define a local basis ξ (y) i at every point y ∈ S p by ξ (y) i = g(y)f i (i = 0, 1, . . . , p − 1).
We call g(y) a frame section. The vielbein 1-form e i and the spin connection 1-form ω ij are defined by the relations where κ i is the extrinsic curvature 1-form, which we are not interested in. Two equations in (18) are equivalent to and κ i = −e i . (19) is used in the following to rewrite covariant derivatives of harmonics in an algebraic form. A change of the frame section by g ′ (y) = g(y)h(y), h(y) ∈ H reproduces the local frame rotation 4/26

Spherical harmonics
A scalar harmonic Y 0 : S p → R with angular momentum ℓ is given by a homogeneous polynomial of the orthonormal coordinates y a = y · f a of order ℓ.
The coefficients c a1···aℓ are components of a totally symmetric traceless tensor satisfying c bba3···aℓ = 0. We can rewrite (21) as where is a traceless symmetric ℓ-tensor, and is an H-invariant ℓ-tensor. (22) is a scalar function on S p because it is invariant under the change of the frame section g(y) → g ′ (y) = g(y)h(y).
Next, let us consider vector harmonics. An arbitrary R p+1 -vector function S p → R p+1 that in general has both normal and tangential components to S p can be expanded by a set of vector functions of the form A vector harmonic Y 1 : S p → T S p can be obtained by projecting away the normal component from (25). Its components are given by where F and N i are defined by It is easy to generalize the above construction of the scalar and the vector harmonics to general spins. Let S and R be a spin and an angular momentum, which are representations of H and G, respectively. We suppose these are irreducible. Let V R and V R be the representation space of R and its dual space, respectively. Although R is an irreducible representation of G, it may be reducible as an H representation. For the existence of the S harmonics with angular momentum R S must appear in the H-irreducible decomposition of R. In other words, there must be an H-invariant subspace V S ⊂ V R and its dual space V S ⊂ V R associated with the spin representation S.
Let E µ ∈ V R (µ = 1, . . . , dim R) be basis vectors of V R , and E µ ∈ V R be the dual vectors satisfying ( E µ , E ν ) = δ µν . G acts on these vectors as where ρ R µν (g) = ( E µ , gE ν ) is the representation matrix for R. We also introduce basis vectors E α ∈ V S and E α ∈ V S that are transformed by H in a similar way to (28).

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Spin S harmonics with angular momentum R are given by where α = 1, . . . , dim S is a spin index, and µ = 1, . . . , dim R labels harmonics belonging to R. The harmonics are transformed under the local frame rotation g ′ (y) = g(y)h(y) by This means Y SR αµ have spin S. Under an isometry transformation y ′ = g −1 y by g ∈ G, Y SR αµ are transformed as where h(y) = g −1 (y)gg(g −1 y) ∈ H is the local frame rotation compensating the change of the local frame due to the isometry rotation. The relation (31) shows that the harmonics have angular momentum R. For a fixed spin S there are infinite number of representations R that contain S in their H-irreducible decomposition. We can show by using Peter-Weyl theorem that the collection of all Y SR αµ for such representations form a complete basis of spin S fields.

Covariant derivatives
The harmonics we defined above are eigenfunctions of the Laplacian on the sphere. This is easily shown by expressing the covariant derivatives in an algebraic form.
At the last step we used (19). This is equivalent to Note that (33) is again a harmonic, and has the general form (29) of harmonics with E α replaced by T i(−1) E α . The second derivative of Y SR αµ is obtained in the same way as By contracting indices i and j, we obtain where C SO(n) 2 (R) is the quadratic Casimir of an SO(n) representation R defined by 1 2 n a,b=1 It is also easy to obtain the curvature tensor R ij kl = δ ik δ jl − δ il δ kj from the anti-symmetric part of (34). 6/26

N = 1 superconformal index in 4d
In this section we demonstrate the calculation of the superconformal index of vector multiplets in a 4d N = 1 supersymmetric gauge theory. The index is defined in [2,3], and the relation to short and long multiplets of the superconformal algebra is investigated. The superconformal indices for extended supersymmetric theories are also defined in [3]. In particular, the index for the N = 4 supersymmetric Yang-Mills theory is calculated, and the agreement in the large N limit with the corresponding quantity in the gravity dual is confirmed. The N = 1 superconformal index for a general N = 1 supersymmetric gauge theory is derived in [5] using the Lagrangian of theories in S 3 × R constructed in [2]. The index is used in [5] to test Seiberg duality [6].
There are two kinds of multiplets of 4d N = 1 supersymmetry: vector multiplets and chiral multiplets. Due to limitations of space we only consider vector multiplets. A vector multiplet consists of a vector field A a , a gaugino field λ, and an auxiliary field D. All component fields belong to the adjoint representation of the gauge group.

S 3 harmonics
We consider a gauge theory in S 3 × R. In this section we label coordinates y a ∈ R 4 in a different way from the previous section. We use (y 4 , y 1 , y 2 , y 3 ) instead of (y −1 , y 0 , y 1 , y 2 ). For S 3 × R coordinates we use x a (a = 1, 2, 3, 4), x i (i = 1, 2, 3) for S 3 and x 4 for R. The theory has four commuting conserved charges: the Hamiltonian H = −∂/∂x 4 , a U (1) R charge R, and the angular momenta J 1 = 1 i T 12 and J 2 = 1 i T 34 . The Dirac matrices used in this section are where T D k are generators of the diagonal subgroup H = SU (2) D . A spin representation S is specified by s ∈ (1/2)Z ≥0 , while a G representation R is specified by the left and the right angular momenta j L , j R ∈ (1/2)Z ≥0 . The G representation R = (j L , j R ) must contain the H representation s in its H-irreducible decomposition. This requires the triangular inequality S 3 harmonics are labeled by six half integers: Y s(jL,jR) α(mL,mR) . When we do not need to explicitly show (j L , j R ) and (m L , m R ), we often omit them to simplify the expression. The eigenvalues of the Laplacian are given by We define We denote the operator by "rot" because this becomes the rotation for vector fields; β . We can calculate eigenvalues of this operator as where σ is given by

Killing spinors
Supersymmetry transformations in a 4d conformally flat background are parameterized by a left-handed spinor ǫ and a right handed spinor ǫ satisfying the Killing spinor equations To calculate the superconformal index, we need to determine Killing spinors in S 3 × R. From (44) we can show ∆ S 3 ǫ = − 3 4 ǫ, (and the same equation for ǫ). Comparing this with (40), we find ǫ and ǫ must have (j L , j R ) = ( 1 2 , 0) or (j L , j R ) = (0, 1 2 ). There are four linearly independent Killing spinors for ǫ: We also have four Killing spinors for ǫ. By using the general formula (33) we obtain where γ iso is the chirality operator for the isometry group G = SO(4), and γ iso = +1 (−1) for Killing spinors belonging to R = ( 1 2 , 0) (R = (0, 1 2 )). From these equations, we find The equations in (44) with a = 4 determine the x 4 dependence of the Killing spinors; We have obtained eight linearly independent Killing spinors. (4 for ǫ and 4 for ǫ.) To perform localization, we choose the following specific ones with quantum numbers We denote the supercharges corresponding to ǫ and ǫ by Q and Q, respectively. Among linear combinations of four conserved charges H, J 1 , J 2 , and R, the following three commute with Q and Q.
We define the superconformal index by Because the spinors in (50) are SU (2) R singlet, the cancellation occurs between modes with the same SU (2) R quantum numbers. In the following analysis it is convenient to separate 8/26 SU (2) R quantum number from others. We define Cartan generators of SU (2) L and SU (2) R by and rewrite the index (52) by introducing z 1 = tx and z 2 = t/x as

Mode analysis
The transformation rules of 4d N = 1 vector multiplets are conformally invariant, and we can use the rules for the flat background except that the parameters ǫ and ǫ are Killing spinors satisfying (44); where To perform index calculation, we deform the action by adding Q-exact terms. A standard form of the action used in the literature is S ∼ δ Q ((δ Q ψ) † ψ), where ψ denotes fermions in the theory. An advantage of this action is that the bosonic part (δ Q ψ) † δ Q ψ is manifestly positive definite, and the path integral is well defined automatically. However, this has also a disadvantage that it contains spinor bilinear ǫ † γ i ǫ, which breaks the rotational symmetry G. To avoid this problem, we adopt another action where ǫ ′ is a Killing spinor such that ǫǫ ′ is a G-invariant scalar. For example we can use the Killing spinor obtained from ǫ by replacing Y The absence of ǫγ i ǫ ′ in the action is guaranteed by the algebra {δ(ǫ), δ(ǫ ′ )} = 0, and the spinors ǫ and ǫ ′ appear in the action only through ǫǫ ′ , and the Lagrangian is G-invariant; We fix the gauge symmetry by the gauge fixing function where primes indicate the ghost and the anti-ghost fields do not include the constant modes on S 3 . They are expanded by the scalar harmonics Y 0(j,j) (mL,mR) (j ≥ 1 2 ), and the path integral gives the factor −4j(j + 1) for each j and (m L , m R ). Note that V gh is not the full 4d divergence D a A a = D i A i + D 4 A 4 but the 3d divergence in S 3 . Therefore, the gauge fixing is partial, and the gauge transformation with parameter that depends only on x 4 is not fixed. The fixing of this remaining gauge symmetry has been already discussed in section 1, and gives the Jaccobian factor in (11).

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The bosonic part of the Lagrangian including the gauge fixing term is where the D 2 term is neglected because the path integral of D gives just a constant. (V i , V gf ) and A a are related by with a differential operator D A . The energy eigenmodes are obtained by solving D A A = 0. This can be easily solved by harmonic expansion. We expand A and V as follows.
2 ), there are two components for each A and V , and the coefficients are related by The determinant of the matrix in (61) is This is canceled by the ghost factor, and does not correspond to any physical modes. When (j L , j R ) = (0, 0), a 3 mode does not exist, and we have only a 4 mode. This corresponds to the Wilson line integrated in (11). The (j + 1, j) mode f 1 and (j − 1, j) mode f 2 depend only on a 1 and a 2 , respectively. By using formula (42) we obtain These give the physical modes in Table 1. Table 1 Bosonic physical modes in S 3 × R are shown. We denote J L eigenvalues by m. J R eigenvalues always take values between −j and j, and we do not explicitly show them in the table.
For the gaugino λ we need to solve the Dirac equation D \ λ = 0 in S 3 × R. We expand λ and D \ λ by the harmonics as follows.
The corresponding eigenmodes are summarized in Table 2. Table 2 Fermionic physical spectrum in S 3 × R. We denote J L eigenvalues by m ′ .
Let us compare bosonic modes in Table 1 and fermionic modes in Table 2. If we replace m ′ in Table 2 where χ j (x 2 ) is the SU (2) character χ j (x 2 ) = x 2j + · · · + x −2j . The [λ2] modes with m ′ = j − 1/2 contribute to the index by Combining the bosonic contribution (66), the fermionic contribution (67), and the ghost contribution (12), we obtain the total single-particle index Refer to [5] for the result for chiral multiplets. Before ending this section, we comment on a close relation between the 4d superconformal index and the S 3 partition function of 3d supersymmetric field theories. The 3d partition function of N = 2 supersymmetric field theories on round S 3 is first calculated in [7] for canonical fields. It is generalized to non-canonical fields in [8,9], and squashed S 3 in [10]. (See also [11].) Once we obtain the formula for the 4d superconformal index, it is easy to obtain a general formula for the S 3 partition function by taking a small radius limit β → 0 [12][13][14]. 11/26 4. N = 2 superconformal index in 3d In this section we consider a 3d N = 2 supersymmetric theory. 3d superconformal index is defined in [4], and the relation to multiplets of the superconformal algebra is investigated. An important application of the index is a test of AdS 4 /CFT 3 correspondence. For example, the index of the ABJM model [15] is calculated in [16] for the perturbative sector and the agreement with the corresponding quantity on the gravity side is confirmed. The check including monopole contribution is done in [17], in which formula of the N = 2 superconformal index for canonical fields (without anomalous dimensions) is derived. The index for chiral multiplets with anomalous dimensions is derived in [18].
Just like the 4d N = 1 case, we have two kinds of multiplets: vector multiplets and chiral multiplets. Again we only consider a vector multiplet, which consists of a vector field A a , a scalar field σ, a gaugino field λ, and an auxiliary field D.
We use different labeling of the coordinates y a from section 2. Instead of (y −1 , y 0 , y 1 ), we use (y 3 , y 1 , y 2 ). For S 2 × R we use coordinates x a (a = 1, 2, 3), x i (i = 1, 2) for S 2 and x 3 for R. We use Pauli matrices as the Dirac matrices; γ a = σ a .
We consider an N = 2 supersymmetric theory in S 2 × R. There are three conserved charges: the Hamiltonian H = −∂/∂x 3 , the angular momentum J 3 = −i T 12 , and a U (1) R charge R.
An important difference of S 2 from S 3 discussed in the previous section is the existence of topologically non-trivial gauge field configurations, monopoles. Harmonics describing charged particles coupling to such monopole backgrounds are called monopole harmonics.

Monopole harmonics
Let us consider harmonics with angular momentum j ∈ 1 2 Z ≥0 . The representation space V j is spanned by satisfying The dual vectors are defined by E s = E −s = (E s ) * . All representations of H are singlets. Spin s harmonics are given by where ρ j is the SU (2) representation matrix for angular momentum j. Let us consider a field with U (1) electric charge q ∈ Z in the monopole background with magnetic charge m ∈ Z. We can take the gauge so that the gauge potential is given by A = (m/2)ω 12 , and then the covariant derivative for a spin s field becomes Therefore, we can treat a charged spin s particle in the magnetic flux as a particle with shifted spin Thus we do not have to introduce anything new for dealing with monopole backgrounds.
In the case of non-Abelian gauge theory, m = I m I t I becomes a Cartan element of the 12/26 gauge algebra, and the factor qm in (73) must be replaced by α(m) for the components of an adjoint field specified by the weight α.
It is convenient to define v ± for a general vector v by v ± ≡ v 1 ± iv 2 . For example, we define The basis E s with s = ±1 are given by E ±1 = f 1 ± if 2 . We should note that spin ±1 components of a vector v are E ±1 · v = v ∓ . The representation matrices (T 3± ) ss ′ ≡ ρ j ss ′ ( T 3± ) have non-vanishing components ( T 3± ) s±1,s , and satisfy From the general formula (33) and (35) we obtain

Killing spinors
The parameters of N = 2 supersymmetry are two-component spinors ǫ and ǫ. In a Euclidean space they are treated as independent spinors. They must satisfy the Killing spinor equations D µ ǫ = γ µ κ and D µ ǫ = γ µ κ. We can easily show that each component of ǫ and ǫ must be a spinor harmonic with j = 1 2 . A general form of such spinors are where m are summed over ± 1 2 . We can determine the x 3 dependence of the coefficients in the same way as the 4d case, and obtain eight linearly independent Killing spinors. We use the following specific ones for localization.
Among linear combinations of the three conserved charges H, J 3 , and R, the following two commute with Q and Q.
We define the superconformal index by where m I are the background monopole charges. 13/26

Mode analysis
The supersymmetry transformation rules for 3d N = 2 vector multiplets in a conformally flat background are δA a = i(ǫγ a λ) − i(ǫγ a λ), δσ = (ǫλ) + (ǫλ), For localization we use the Q-exact Lagrangian where ǫ ′ is the Killing spinor obtained from ǫ by replacing Y and saddle points are given by V a = 0. This condition is solved, up to gauge transformation, by the monopole backgrounds where m ∈ Z is the monopole charge. These backgrounds induce the shift of the effective spin for a field associated with a weight α. We consider fluctuations around the saddle points.
We fix the gauge symmetry by the gauge fixing function The quadratic part of the Lagrangian including the gauge fixing term is We expand the bosonic fields by the monopole harmonics as follows.
Unlike the 4d case, harmonics in the four components all have the same G quantum numbers j and m, and can mix among them. When j ≥ |s 0 | + 1, all four components exist, while if 14/26 j is equal to or smaller than |s 0 | some of them are absent. When all components exist the coefficients are related by  and the determinant of the 4 × 4 matrix is For smaller j some of rows and columns are absent, and the determinant is given by The factor [j(j + 1) − s 2 0 ] is canceled by the factor arising from the ghost term L gh = c ′ D (0) i D i c ′ and does not correspond to any physical modes. Other two factors, ∂ 3 − j and ∂ 3 + j + 1 correspond to physical modes in Table 3. Table 3 Bosonic physical spectrum in S 2 × R. We denote J 3 eigenvalues by m.
The fermionic part of the Q-exact Lagrangian is L = λD λ λ with We expand the components of λ and D λ λ by When j ≥ |s 0 | + 1 2 , the two components are non vanishing, and the relation among the coefficients is The determinant of the matrix is When j = |s 0 | − 1 2 only one of the two components of λ and D λ λ exists, and the matrix element becomes ±(∂ 3 + j + 1). The physical modes are summarized in Table 4. 15/26 Table 4 Fermionic physical spectrum in S 2 × R. For convenience we use j ′ and j ′′ instead of j and m ′ instead of m.
Let us compare the bosonic and fermionic modes. By the replacement and the latter contribute to the index. For the ghost index (12), we must take account of the gauge symmetry breaking due to the monopole background. The ghost zero-modes are present only for unbroken symmetry, and the ghost index becomes By combining (97), (98), and (99), we obtain Refer to [18] for the chiral multiplet contribution.

N = (1, 0) superconformal index in 6d
The superconformal indices in 6d theories are defined in [4]. The derivation by harmonic expansion in this section is based on [19]. The method used in [19] is essentially the same as that in [20], in which the S 5 partition function is calculated by using CP 2 harmonics in [21]. See also [22][23][24] for the S 5 partition function.
Although it is known that instantons make non-perturbative contribution to the index, it seems in practice impossible to calculate it by harmonic expansion, and we are going to calculate only perturbative contribution to the index. 16

Hopf fibration
In this section we consider a 6d N = (1, 0) supersymmetric theory in S 5 × R. As in the 3d and 4d cases, we need to choose a particular supercharge for localization. The choice breaks the rotational symmetry G = SO (6). Unlike the 3d and 4d cases it seems impossible to construct Q-exact Lagrangian that respect the full SO(6) symmetry. The Lagrangian we use respects only the subgroup G ′ = SU (3) × U (1). Fortunately, this is transitive, and harmonic expansion is still efficient. Because of this symmetry breaking, it is natural to regard S 5 as a Hopf fibration over CP 2 . In this subsection we discuss the Hopf fibration of general odd-dimensional spheres, and define a coordinate system convenient for the following analysis.
Let us consider S 2r+1 with unit radius defined as the subset of R 2r+2 by y a y a = 1. In this subsection we use indices a, b, . . . = −1, 0, 1, . . . , 2r, i, j, . . . = 0, 1, . . . , 2r, and m, n, . . . = 1, . . . , 2r. The first step to define the Hopf fibration is to specify a complex structure in R 2r+2 . Let I ab be the complex structure in R 2r+2 with non-vanishing components and let U (1) I be the subgroup of G ≡ SO(2r + 2) generated by I = (1/2)I ab T ab . We define Hopf fibration with U (1) I orbits. y ∈ S 2r+1 can be written as where 0 ≤ ψ < 2π is a coordinate along fibers, and y 0 (θ) is a representative in the fiber that is specified by CP 2 coordinates θ = (θ 1 , θ 2 , θ 3 , θ 4 ). The S 2r+1 metric is where V and ds 2 CP r are defined by ds 2 CP r is the Fubini-Study metric of CP r . A convenient choice of the local frame is given by a frame section of the form g(θ, ψ) = e −ψ I g 0 (θ), g 0 (θ) ∈ SU (r + 1), where SU (r + 1) is the special unitary group that rotates holomorphic vectors f 2k−1 + if 2k (k = 0, . . . , r). We call this a unitary frame. Because ∂ ψ y(θ, ψ) = ξ (y) 0 the 0 direction points the fiber direction. The vielbein e 0 is given by and the others e m (m = 1, 2, . . . , 2r) are the pull-back of the vielbein of the base CP r . The exterior derivative of V is At the second equality we use |y| = 1. Clearly dV satisfies (∂ ψ , dV ) = 0, and can be regarded as the pull-back of a two-form in the base CP r . The unitary transformation g(θ, ψ) keeps the complex structure intact, and the components of dV in (107) in the unitary frame is 17/26 essentially the same as I ab except that (−1) direction is absent in S 2r+1 . Namely, dV is given by where I mn is an anti-symmetric 2r × 2r matrix with non-vanishing components This is nothing but the complex structure in the base CP r . From the torsionless condition, we obtain the spin connection where ω CP r ≡ (1/2)ω CP r mn T mn is the spin connection of CP r . Let us read off the vielbein e i and the spin connection ω ij according to (19) from the Maurer-Cartan form Because g ∈ U (r + 1), SO(2r + 2) generators T i(−1) , whose coefficients are identified with the vielbein e i , always appear in the Maurer-Cartan form through the combinations Therefore, the components of ω ij corresponding to I and T n0 are written in terms of e i . We obtain ω = −e 0 I + e m I mn T n0 + ω SU (r) + r + 1 where ω SU (r) is the SU (r) part of the spin connection cting on the holomorphic vectors f 2k−1 + if 2k (k = 1, . . . , r). From (110) and (113), we obtain Under the subgroup G ′ a G representation R is decomposed into ⊕ i (R ′ i , q i ) where R ′ i and q i are SU (r + 1) representations and U (1) I charges. Correspondingly, S 2r+1 harmonics are expanded into Kaluza-Klein modes by By applying the covariant derivative to this expansion, we obtain ∇ is the covariant derivative on CP r defined by where Q V ≡ 1 i I.
To write down the Killing spinors in S 5 × R, we first define Killing spinors in S 5 . There are two spinor representations of G = SO (6). Let V (L) and V (R) be representation spaces for positive and negative chirality, respectively. We introduce basis vectors E (L) µ (µ = 1, 2, 3, 4) for V (L) and E (R) µ (μ =1,2,3,4) for V (R) . Both these spinor representations are irreducible also as H-representations, and we can identify V S with V (L) and V (R) . According to general prescription, we can define 8 linearly independent Killing spinors on S 5 : where E α are dual basis vectors for V S . By using the general formula (33) and the explicit representation of the Dirac matrices, we obtain where Γ iso is the chirality operator of the isometry group G = SO (6). Under the subgroup αµ and ε (R) αμ splits into a singlet and a triplet. This can be seen from the explicit form of I for the spinor representations: We can see that ε αμ belong respectively to the following G ′ representations.
For the index calculation we use the SU (3) singlet Killing spinors We can show that these ε I are ∇-constant.
This property drastically simplify the following calculation. 19/26 We define the 6d Killing spinors ǫ I in S 5 × R by using S 5 Killing spinors ε I by There are five conserved charges: the Hamiltonian H = −∂ 6 , the angular momenta J 1 = −i T 12 , J 2 = −i T 34 , J 3 = −i T 65 , and the SU (2) R Cartan generator R 3 . The supercharges Q I corresponding to ǫ I carry charges (H, The following four commute with Q ≡ Q 1 + Q 2 .
We define the superconformal index Because the supercharges Q I are SU (3) singlets, we can treat the SU (3) as a "flavor" symmetry, and it is convenient to separate SU (3) fugacities from z i . We use variables x and y i defined by z i = xy i (y 1 y 2 y 3 = 1) instead of z i . The index is rewritten in terms of these variables by where

CP 2 harmonics
According to the general construction in section 2, the scalar spherical harmonics in S 5 are characterized by rank k symmetric traceless tensors in R 6 . Let R k denote the tensor representation. Under G ′ = SU (3) × U (1) R k is decomposed into k + 1 representations R k,q (q = k, k − 2, . . . , −k) where R k,q is the symmetric traceless part of the product (3, +1) . The corresponding scalar harmonics are given by The following relations hold.
We normalize Y 0(k,q) µ by S 5 |Y 0(k,q) For simplicity of expressions we omit the G ′ indices µ and (k, q) in the following. 20/26 By using the scalar harmonics and the Killing spinors ε I we can define a complete set of spinor harmonics Spinor harmonics with different G ′ quantum numbers are orthogonal. The inner products of two harmonics with the same G ′ quantum numbers are Among four harmonics Y and Y 1 2 (2) exist when k = q = 0. A complete set of vector harmonics can be defined by where X mn are the tensors with the components mn (a = 1, 2, 3) can be defined also as the following bi-linears of the Killing spinors.
These are ∇-constant. The inner products of two vector harmonics with the same G ′ quantum numbers are

Mode analysis
A vector multiplet consists of a vector field A a , a gaugino λ I and a SU (3) triplet auxiliary field D IJ . The transformation laws are δA a = −(ǫ I Γ a λ I ), 21/26 For index calculation we use the Q-exact action where τ 3 = diag(1, −1) is the matrix acting on SU (2) R doublets. The auxiliary fields D IJ does not give any physical modes. The gauge field terms in (141) with the gauge fixing term (D i A i ) 2 added can be rewritten as For the saddle points all terms in (142) must vanish. For the first term to vanish the gauge field in CP 2 must be anti-self-dual. This allows instanton configurations. However, it is difficult to calculate the instanton contribution to the index by means of harmonic expansion.
Here we focus only on the perturbative sector with zero instanton number. Then the saddle point is given by the trivial gauge configuration A a = 0 up to gauge transformations. Let us represent the fluctuation of the gauge potential as a column vector A and expand it by six basis vectors Y 1,...,6 . Their explicit forms are We can write the Lagrangian in the form L = A † DA with a certain differential operator D. It is straightforward to calculate the determinant of the differential operator D in each subspace with specific G ′ quantum numbers (k, q) and µ. For |q| ≤ k − 2, all six basis vectors are linearly independent, and the determinant is The factor k 2 (k + 4) 2 is canceled by the ghost factor, and the other factors correspond to physical modes. For |q| = k, the six vectors Y 1,...,6 are not linearly independent, and some factors in (144) are absent. Careful analysis gives the spectrum in Table 5. The gaugino terms in (141) take the form λ I D λ λ I with the differential operator  Table 5 Bosonic physical modes in the representation R k,q .
Comparing Table 5 and Table 6, all quantum numbers match except the ranges of q. Let us focus on the positive frequency modes because A a are real and λ I are symplectic Majorana. Only difference of the bosonic and the fermionic spectra is that there are q = k modes in 23 e iα x k χ (k,k) (y) − α ∞ k=0 e iα x k+3 χ (k,k) (y), where χ (k,q) (y) is the SU (3) character of the representation R k.q . By combining this with (12) we obtain where we used the formula ∞ k=0 x k χ (k,k) (y) = 1 for the SU (3) characters. Refer to [19] for index calculation for hypermultiplets by means of harmonic expansion.

Concluding Remarks
We reviewed how we calculate 3d, 4d, and 6d superconformal indices by using supersymmetric localization and harmonic expansion. We deformed the theories by introducing Q-exact terms, and derived the indices by means of mode analysis in S p × R backgrounds. The method of harmonic expansion works effectively when the deformed theory has G = SO(p + 1) rotational symmetry. This is the case in 3d and 4d. Although it is not the case in 6d and the deformed Lagrangian respect only the subgroup G ′ = U (3) ∈ SO(6), G ′ is still transitive and harmonic expansion is usufull.
In the case of 5d, however, this method does not work. Let Q be a supercharge that we use for localization. In general Q 2 is a linear combination of a G generator T and generators of internal symmetries. Unlike the S 5 case we discussed in section 5, in which T generates shifts along Hopf fibers, S 4 rotations generated by T always have (at least) two fixed points. These fixed points are often called North and South poles, and the existence of such special points clearly shows that the symmetry of the deformed theory is not transitive on S 4 . For this reason it is not practical to use harmonic expansion for the index calculation. This is also the case for the S 4 partition function of 4d theories.
The 5d superconformal index and the S 4 partition function have been calculated by using more sophisticated technique, called equivariant localization. The S 4 partition function is calculated in the seminal work by Pestun [26], and the result is extended to squashed S 4 in [27]. The 5d superconformal index is derived in [28,29]. All these works employ the method of equivariant localization.
In the method using equivariant localization, the existence of fixed points is not a trouble but what is required. The partition function and the index are given as the product of the contribution of each fixed point. This also make it possible to includes the instanton contribution as the contribution from fixed points, each of which is given by the Nekrasov partition function [30,31].
Unfortunately, we have no space to discuss these issues, and the interested reader is reffered to the original works cited above. 24/26