More on BPS States in N=4 Supersymmetric Yang-Mills Theory on R x S3

We perform a systematic analysis on supersymmetric states in N=4 supersymmetric Yang-Mills theory (SYM) on R x S^3. We find a new set of 1/16 BPS equations and determine the precise configuration of the supersymmetric states by solving all 1/16 BPS equations when they are valued in Cartan subalgebra of a gauge group and the fermionic fields vanish. We also determine the number of supersymmetries preserved by the supersymmetric states varying the parameters of the BPS solutions. As a byproduct we present the complete set of such supersymmetric states in N=8 SYM on R x S^2 by carrying out dimensional reduction.

1 Introduction N = 4 supersymmetric Yang-Mills theory (SYM) in four dimensions has been a fundamental theoretical tool to extract important lessons of duality. In particular N = 4 SYM enjoys S-duality [1,2], which is a generalization of the electro-magnetic duality [3,4], and also exhibits a dual description of type IIB supergravity on AdS 5 × S 5 under a particular limit [5]. For precise investigation of these kinds of strong-weak duality, supersymmetry often plays important roles.
Since maximally supersymmetric completion of Yang-Mills theory gives rise to full quantum conformal invariance in the flat space [6,7,8,9], N = 4 SYM can be studied by mapping the system onto R × S 3 by a conformal transformation, which makes the system free from infra-red divergence and discretizes the spectrum of the system with the mass gap of order the inverse radius of the three sphere. In particular discrete spectrum of the supersymmetric (or BPS) states of the theory has attracted a great deal of attention in the context of AdS 5 /CFT 4 duality, which can be tested by confirming match of supersymmetric spectra in both sides [10,11].
One aspect in the study of BPS spectrum is counting BPS states. (Other works related to N = 4 SYM on R × S 3 are found, for instance, in [12,13,14,15].) In particular an important tool to study BPS spectrum is a (superconformal) index [16], which encodes BPS spectrum as a form of polynomial. A superconformal index of the N = 4 SYM has been computed exactly in arbitrary SU(N) gauge group [17]. (See also [18,19,20].) Under the large N limit with the charges kept finite the index of the N = 4 SYM precisely matches that of multi-particle states of supergravity multiplet in type IIB supergravity on AdS 5 × S 5 [17]. This result is reasonable in the sense that other supersymmetric objects such as (dual) BPS giant gravitons [21,22,23] and BPS black holes [24,25,26] in the dual geometry have much larger charges, which are of order N, N 2 respectively. Counting of supersymmetric states with such large charges in the N = 4 SYM has also been done as a purely mathematical problem by recasting BPS states as cohomology of a supercharge [27]. As a result, an index of 1/8 BPS (dual) giant gravitons [28,29] has been reproduced from the N = 4 SYM, whereas that of 1/16 supersymmetric black holes has not. 1/16 BPS states are less understood and expected to have a key to make more progress in AdS 5 /CFT 4 duality. (See [30] for a recent study in this direction.) Under the circumference this paper is aimed at carrying out a systematic analysis on 1/16 BPS states of the N = 4 SYM. The goal of this paper is to clarify basic information on the BPS states of the N = 4 SYM such as the 1/16 BPS equations, the BPS configuration and preserved supersymmetries by setting assumptions that the fermionic fields vanish and that the fields are valued in Cartan subalgebra of a gauge group.
The rest of this paper is organized in the following way. In §2 we review basics of N = 4 SYM explaining our convention. In §3 we carry out a systematic analysis on supersymmetric states of the N = 4 SYM by setting the fermionic fields to zero. In §3.1 we study the spherical symmetric BPS states for warm-up. In §3.2 we study supersymmetric states with angular momenta, which contain 1/16 BPS states. In §3. 3 we count the number of preserved supersymmetries of the BPS states determined in §3.2. In §4 we derive similar information of BPS states in N = 8 SYM on R × S 2 by performing dimensional reduction. §5 is dedicated to discussion and future works. Appendix contains our convention (A), and basic information on S 3 (B) including construction of the scalar spherical harmonics on S 3 (B.1) and reduction of information of S 3 to that of S 2 (B.2).
2 N = 4 SYM on R × S 3 In this section we review basics of N = 4 SYM on R × S 3 so as to be self-contained. The N = 4 SYM consists of six real scalars, gauge field and their supersymmetric partners, all of which are valued in the adjoint representation of a gauge group. 1 This theory has P SU(2, 2|4) global symmetry, and especially SO(6) R ≃ SU(4) R R-symmetry. The fermionic fields are in the anti-fundamental representation of SU(4) R , which we denote by λ A , and the six scalar fields form the anti-symmetric representation of SU(4) R , which is denoted by X AB satisfying where A, B = 1, 2, 3, 4. The action of N = 4 SYM on R × S 3 is given by where g is a gauge coupling, r is the radius of the three sphere, d 3 Ω = 1 8 dθ sin θdφdψ is the volume form of the unit three sphere, D µ is the covariant derivative with respect to gauge and space index, which acts on the fields as follows.
where ω µ,νρ is the one form connection of the space-time. See Appendix B for more details. For later convenience we present equations of motion and conserved charges in the bosonic part. These can be computed in the standard way. Equations of motion for the gauge field and complex scalar fields in the bosonic part are In this paper we do not need to specify a gauge group.

SU(4) R R-symmetry charge is
The energy is where |A| 2 = AA † . The angular momentum is The action (2.2) is invariant under the following supersymmetry transformation rule.
where ∇ µ is the covariant derivative for spin and ǫ A is a supersymmetry parameter valued in Grassmann number, which is given by a conformal Killing spinor on R × S 3 This can be solved as follows. ǫ A is a constant spinor. On the other hand, for ǫ where ∇ S 2 µ is the spin covariant derivative on S 2 . More explicitly, the covariant derivatives act on a spinor as follows.
The solution of Killing spinor equation is A is a constant spinor.

BPS states in the N = 4 SYM
In this section we perform a systematic analysis on BPS states of N = 4 SYM on R × S 3 . That is we study a condition of the fields such that the supersymmetry variation given by (2.10) vanishes for a certain Killing spinor. In this paper we consider a situation where the gaugino fields vanish. In this case we have only to study conditions of the bosonic fields for the supersymmetry transformation of gaugino to vanish. By using the Killing spinor equation (2.11) the supersymmetry variation of gaugino can be written as There are two kinds of BPS states on the sphere: one is spherically symmetric, the other is not. This separation is achieved automatically by studying whether a preserved Killing spinor of a BPS state is projected or not by a certain projective operator, which breaks spherical symmetry. In this paper we perform projection for a constant spinor in Killing spinors (2.13), (2.16) by using γ 0 .

Spherically symmetric BPS states
In this subsection we study spherically symmetric BPS solutions for warm-up. For this purpose we study a BPS condition which preserves at least one Killing spinor without any projection. As such a preserved Killing spinor we first choose ǫ (+) 4 , where 4 = 1, 2, 3, 4. We use German letters 1, 2, 3, 4 as symbols representing 1, 2, 3, 4 in a way that there exists a permutation σ such that i = σ(i), where i = 1, 2, 3, 4, i = 1, 2, 3, 4. For the supersymmetry variation of gaugino to vanish with nonzero ǫ (+) 4 , the fields are required to satisfy and (γ µ ǫ (+) 4 ) * are linearly independent for all µ. This is the 1/8 BPS condition which preserves a Killing spinor ǫ (+) 4 . As asserted, the last equation in (3.2) constrains BPS states to be spherically symmetric.
Let us study this BPS condition when the fields take values in Cartan subalgebra of the gauge group. In this case these become This can be easily solved as where x A4 are integral constants valued in complex number. Other components are determined by using the relation (2.1) as This is a general 1/8 BPS solution which preserves a Killing spinor ǫ (+) 4 . Let us study a case for this BPS solution to preserve another Killing spinor. One will soon notice that ǫ is broken unless all fields are trivial. So let us set it to zero to study a nontrivial BPS solution. Under the BPS condition (3.2) the supersymmetry variation of gaugino becomes In order for these to vanish, it is required to satisfy (ǫ (+) 1 ) * = 0 or (ǫ (+) 1 ) * = 0, X 34 = X 24 = 0, (3.7) (ǫ (+) 2 ) * = 0 or (ǫ (+) 2 ) * = 0, X 34 = X 14 = 0, (3.8) where we used the relation (2.1). Therefore the 1/8 BPS solution has enhanced supersymmetry in the following situation.
1. One complex scalar field is trivial: X 14 = 0. In this case, the Killing spinor ǫ In the same way we can obtain a BPS condition and solution which preserve ǫ (−) 4 . The BPS condition to conserve ǫ When the fields take values in Cartan subalgebra of the gauge group, this reduces to 14) The equations of the complex scalar field can be easily solved as This is a general 1/8 BPS solution which preserves ǫ (−) 4 . This BPS solution has enhanced supersymmetry in the following situation.
2. Two scalar fields are trivial: X 14 = X 24 = 0. In this case, the Killing spinors ǫ are also unbroken, and the solution becomes 1/2 BPS.
We summarize the result in the following table.
BPS solutions. Preserved Killing spinors. Number of SUSY.
4 . 32 (Unique vacuum) Table 1: We list spherically symmetric BPS solutions, preserved Killing spinors and its number. Signature in multi-column has to be chosen with order respected except the last column.
where the upper sign is for the BPS condition given by (3.2) and the lower for that of (3.13). We can show that 4 A=1 R A A = 0. We can easily see the BPS relation of conserved charge as In this subsection we study supersymmetric states with angular momenta. To this end we study supersymmetric states which preserves a Killing spinor projected by a certain operator which breaks spherical symmetry. We first choose ǫ as such a preserved Killing spinor. We carry out projection for a constant spinor η given by (2.13) is evaluated as Let us fix charges of this Killing spinor. The energy, the angular momentum of φ direction, and that of ψ direction are evaluated as eigenvalues of the operators H := i∂ t , J φ :=L 3 = −i∂ φ , and J ψ :=R 3 = i∂ ψ , respectively, whereL i ,R i are defined in Appendix B. The charge assignments are summarized in Table 2. Therefore the BPS condition of charges Note that the relative factors match that obtained from psu(2, 2|4) superconformal algebra in the standard normalization [31,17].
In order to study a BPS condition preserving ǫ (+) 4 , it is convenient to divide the supersymmetric transformation of gaugino into two parts such that In terms of a Killing spinor the projection is given by Pǫ Let us extract the terms containing η (+) 4 from these. By using (3.20) and (3.21). we find where the ellipses represent the other terms which do not contain η 4 ) * are linearly independent under the projection (3.20). By using the fact that all the components of F µν are real, these can be simplified as This is the 1/16 BPS condition preserving ǫ projected by (3.20). This result is essentially the same as 1/16 BPS equations derived in [27], (4.9) and (4.10), in a different way by making the energy density complete square and looking for configurations to saturate the Bogomol'nyi bound.
Let us solve these BPS conditions when the fields are valued in Cartan subalgebra of a gauge group. In this case these BPS equations boil down to and First let us solve the matter BPS equation. The equation of motion of the matter fields (2.6) suggests that solutions thereof are expanded by S 3 scalar spherical harmonics. We denote the scalar spherical harmonics by Y s,l 3 ,r 3 with non-negative half integer s and two half integers l 3 , r 3 whose moduli are bounded above by s. These quantum numbers are Cartan charges of the representation of the spherical harmonics of su (2) whereL i ,R i are differential operators generating su(2) L × su(2) R algebra. See Appendix B for more details. By using these operators the second equation in (3.30) can be written asR respectively, andR ± is given by (B.16). Therefore BPS solutions are expanded by the spherical harmonics of highest (or lowest) weight of su(2) R for w (+) = +1 (or −1).
where x s,l 3 A4 (t) are an unknown function of time, which is determined from the first equation in (3.30). Plugging the above into the first equation in (3.30) gives This equation is easily solved as where x s,l 3 A4 is an integral constant valued in complex number. 3 As a result we obtain This is a general 1/16 BPS solution of the complex scalar fields preserving the Killing spinor ǫ (+) 4 with projection (3.20). Note that this satisfies the equation of motion of the scalar field (2.6).
Let us move on to determining BPS solutions of gauge sector. For this purpose we substitute the BPS condition (3.29) into the equations of motion of the field strength (2.5) and one of the Bianchi identities, which are in our situation given by ∇ µ F µν = 0, ∇ 1 F 23 + ∇ 2 F 31 + ∇ 3 F 12 = 0, respectively. Under the BPS condition (3.29) the equations of motion reduce to the following three equations 40) and the Bianchi identity becomes After multiplying sin θ to both sides in (3.40), we can write it as A general solution of (3.41) and (3.42) is given by where A s,l 3 (t), B s,l 3 (t) are unknown functions of time. The reason is as follows. First we recall that Y s,l 3 ,w (+) s is annihilated by the operatorR w (+) . By dividing Y s,l 3 ,w (+) s into the real and imaginary part, Y s,l 3 ,w (+) s = u + iv, we can rewriteR w (+) Y s,l 3 ,w (+) s = 0 as This is equivalent to where A, B are arbitrary constants independent of θ, φ, ψ. This implies that the equations s,l 3 are integral constants taking real values with the range a (+) Plugging the explicit expression of the spherical harmonics given by (B.25) into these, we find the general 1/16 BPS solution of the field strength as where c s,l 3 is given by (B.26). 4 Other components of the field strength are determined by (3.29). The moduli space of the 1/16 BPS solution is given by s,l 3 . Note that we do not take into account the flux quantization condition here and hereafter, which would make the moduli space quantized in a certain manner.
Let us compute conserved charges (2.8), (2.9) and (2.7). We first simply them by using the BPS solution (3.29), (3.30). As a result the conserved charges are written in terms of X A4 , ∂ 2 X A4 , ∂ 3 X A4 , F 01 , F 23 . We present results by separating the matter sector and gauge sector. The conserved charges in the matter sector are simplified as where (c.c.) means the complex conjugation of the first term. The gauge field part is the following.
Note that at this stage we see the BPS relation of charges given by (3.22) in the gauge sector.
Let us compute conserved charges using the BPS solution (3.37), (3.48), (3.49). The matter part is the following. 5 In the matter sector we also find the BPS relation given by (3.22). The gauge field part is computed as follows.
(3.69) 5 For this computation we used formulas such that where |l 3 | < s. In the main text, we also used these formulas formally at |l 3 | = s.
Note that the energy and momenta become divergent when the modes with |l 3 | = s are nonzero.

BPS states preserving
Next we study BPS states preserving a Killing spinor ǫ is projected in such a way that given by (2.16) is evaluated as where we used notation such that We determine charges of this Killing spinor (ǫ (−) 1 ) * to find out the BPS relation of charges in Table 3. 6 Therefore the BPS relation of charges associated with the Killing To determine a BPS condition preserving ǫ (−) 1 , we extract the terms containing η (−) 1 6 The reason why we determine charges of (ǫ 1 ) * has the negative energy, which has a corresponding supersymmetry charge, while ǫ has the positive energy, which corresponds to a special superconformal charge. We fix a BPS relation of charges for a particular supersymmetry charge. from (3.24), (3.25) as done previously. The results are as follows.
where the ellipses describe the other terms than η (−) 1 . In order for these to vanish with η nonzero, the fields are required to satisfy 1 ) * are linearly independent under the projection (3.70). Using the fact that all the components of F µν are real, we can simplify these BPS equations as follows.
where x s,r 3 A1 (t) represents time-dependence of the scalar field, which is determined from (3.77).
This equation is easily solved as where x s,r 3 A1 is an integral constant. As a result we obtain BPS solutions This is a general 1/16 BPS solution which preserves ǫ (−) 1 projected by (3.20). It is not difficult to check that this satisfies the equation of motion of the scalar field (2.6).
Let us move on to determining the BPS solution of the gauge sector combining the equations of motion (2.5) and Bianchi identity. In the current situation they are given by ∇ µ F µν = 0 and ∇ 1 F 23 + ∇ 2 F 31 + ∇ 3 F 12 = 0, respectively. Under the BPS conditions (3.75),(3.76) the equations of motion reduce to the following three equations We can solve (3.85) and (3.86) by noticing the fact that they can be rewritten asL w (−) G = 0, where G = sin θF 01 + i sin θF 23 . Therefore this solution is given by the spherical harmonics of the highest (or lowest) weight of su(2) L algebra.
is an unknown function of time, which can be easily determined from the other BPS equations (3.83), (3.84).
s,r 3 < 2π. Plugging this back into the above gives where c s,r 3 is given by (B.29). 7 Other components can be obtained from (3.75), (3.76). The moduli space of the 1/16 BPS solution is given by for s ≥ 0, |r 3 | ≤ s for x s,r 3 A4 , s ≥ 1 2 , |r 3 | ≤ s for a (+) s,r 3 . Here we do not take into account the flux quantization condition.
Let us compute conserved charges (2.8), (2.9), (2.7) of the BPS solution. For this end we first simplify them by using the BPS conditions (3.75), (3.76), (3.77), (3.78). The 7 We exclude the mode with s = 0. results of the matter part are the following.
The gauge sector is as follows. The matter parts are 8 which satisfies the BPS relation of charges given by (3.73). The gauge field part is the following.
Note that the modes with |r 3 | = s give divergent contribution to the energy and momenta.

Counting of supersymmetries
In this subsection we count number of supersymmetries preserved by BPS solutions constructed in the previous subsections. Let us count the number of supersymmetries of the BPS solutions given by (3.37), (3.48), (3.49), which preserves at least η (+) 4 projected by 8 We note useful formulas where |r 3 | < s. We however used these formulas formally at |r 3 | = s in the main text.
(3.20). For convenience we write them down here again.
Then the form of the BPS solution gets another constraint such that  A are also conserved. We summarize this result in Table 4.

Parameter region.
Preserved Killing spinors. Number of SUSY.
Next we consider a case where x s,l 3 34 = 0 for some positive s and l 3 . This assumption restricts us to two case to study BPS solutions to preserve another Killing spinor as (i) η 3 . We study the BPS solution of the matter fields since the constraint of the field strength can be discussed in the same way as above. As in the above case, the matter BPS solution should also be of the form such as (3.114). Due to the assumption, matching of (3.111) and (3.114) requires the signature of the projection for η is preserved without any projection. Furthermore in order to preserve η (−) 2 in addition it is necessary for X 24 to vanish. In this case supersymmetry is enhanced so that η (+) 3 is also preserved. We summarize this result in Table 5.
Parameter region.
Preserved Killing spinors. Number of SUSY.

BPS states in
In this section we investigate supersymmetric states in N = 8 SYM on R × S 2 by carrying out dimensional reduction of N = 4 SYM on R × S 3 so that the Hopf fiber direction ψ is degenerated. We present basic results obtained by this dimensional reduction for this paper to be self-contained. From the metric of R × S 3 we obtain that of R × S 2 as ds 2 R×S 2 = −dt 2 + µ −2 (dθ 2 + sin 2 θdφ 2 ) (4.1) Parameter region.
Preserved Killing spinors. Number of SUSY.
where we set µ = 2 r , which is the inverse radius of S 2 . For more detail, see Appendix B.2. Dimensional reduction for fields can be achieved by truncating the fields to leave the zero modes of the Hopf fiber direction. ∂ ψ Φ = 0, or ∂ 3 Φ = 0, where Φ is any four dimensional field. Accordingly the four dimensional gauge field is separated into the three dimensional one and a scalar field in a way that where i = 0, 1, 2. Thus the gauge field strength become Note that the local Lorentz indices 0, 1, 2 and the global indices t, θ, φ are now transformed to each other by using the transition function of R × S 2 given by (B.34). Performing this dimensional reduction to the N = 4 SYM action given by (2.2) we obtain the action of N = 8 SYM on R × S 2 where we set d 2 Ω = dθ sin θdφ, λ A = ψ A , 1 µ , and D µ is the covariant derivative in terms of gauge and spin indices in the three dimension.
Equations of motion for the gauge field and the scalar fields in the bosonic part are Conserved charges of this theory can be obtained by dimensional reduction. SU(4) R R-symmetry charge of the bosonic part is The energy is The angular momentums are Under the truncation supersymmetries specified by ǫ

(+)
A are all broken, since they are dependent on the Hopf fiber direction ψ. The other supersymmetries specified by ǫ

(−)
A are all preserved, which we denote by ξ A . The determining equation of ξ A comes from (2.14), which is now given by 11) and the solution is where η A is a constant spinor. By using this the supersymmetry transformation rule of N = 8 SYM on R × S 2 is obtained as Note that due to this reduction the global symmetry reduces from P SU(2, 2|4) to P SU(2|4). BPS states of N = 8 SYM on R × S 2 can be studied in the same manner as the case of N = 4 SYM on R × S 3 in §3.2.2. Especially calculation in §3.2.2 can be applied to this case as it is by exchanging the fields from four dimension to three dimension as discussed above. For this reason we do not repeat the similar calculation, and we present only relevant results.
Let us study BPS states of N = 8 SYM on R × S 2 . For this purpose we study a BPS state which preserves a Killing spinor ǫ 1 whose constant spinor η 1 is projected in such a way that γ 0 η 1 = iwη 1 (4.14) where w = ±1 and 1 = 1, 2, 3, 4. The BPS relation of charges associated with the Killing spinor ǫ 1 is By performing the dimensional reduction to (3.74), we find The gauge field strength has to be determined by combining with the equation of motion (4.5), which is now given by ∂ µ f µν + µγ 0µν ∂ µ φ = 0. 9 Under the BPS condition (4.17) this becomes ∂ t f 01 = −wµ∂ φ f 01 , µ∂ 2 φ φ = tan θ∂ θ (sin θf 01 ). where x s A1 , a s , α s are integral constants with the range x s A1 ∈ C, a s ≥ 0, 0 ≤ α s < 2π for all A = 2, 3, 4; s ≥ 1 2 , and Y s,ws (θ, φ), c s are given in Appendix B.2. Thus the real adjoint scalar field φ is determined as a s s c s sin s θ cos(s(wφ − µt) + α s ), (4.23) where φ 0 is another integral constant taking real values, which parametrizes the vacua of the theory. 10 The other components of field strength can be obtained from (4.17). This is a set of general 1/8 BPS solutions of N = 8 SYM on R × S 2 . 11 Note that this result is consistent with the BPS solutions obtained in [32], where BPS solutions of the N = 8 SYM were obtained from those of N = 6 Chern-Simons (or ABJM) theory defined on R × S 2 by performing a particular scaling limit from a half BPS solution of ABJM. We present the values of conserved charges under the BPS solution (4.20), (4.21), (4.23).
Tr(a s ) 2   , (4.28) which satisfies the BPS relation of charges given by (4.15). The number of supersymmetries preserved by the 1/8 BPS solution at each point of moduli space is given in Table 7. In particular this theory has degenerate vacua, which are parametrized by vacuum expectation values of the real scalar field denoted by φ 0 . 12

Discussion
We have done a systematic analysis of supersymmetric states in N = 4 SYM on R × S 3 by setting the gaugino fields to zero. As a result we have found two sets of 1/16 BPS conditions and we have solved them completely when the bosonic fields are valued in Cartan subalgebra of a gauge group. We have precisely counted the number of supersymmetries preserved by the BPS solutions varying the parameters of the solution. We have also obtained the most general 1/8 BPS solution of N = 8 SYM on R×S 2 with the precise number of supersymmetries under the same assumptions by performing dimensional reduction.
In this paper we have solved the 1/16 BPS solutions assuming that they are valued in Cartan subalgebra of a gauge group. It would be interesting to study more general BPS 11 If we include the non-normalizable mode, the general solution Parameter region.
Preserved Killing spinors. Number of SUSY. x s 12 , x s 13 , x s 14 , a s , φ 0 : generic. η 1 2 with γ 0 η 1 = iwη 1 . ( ( 1 4 BPS) x s 12 , x s 13 , x s 14 = 0 for ∀s > 0, η 1 . 4 a s = 0 for ∀s, r 3 . ( ( 1 2 BPS) x s 12 = 0 for ∀s > 0, solutions by relaxing this assumption. In this case one has to solve non-linear differential equations given by (3.28) or (3.74), which are much more complicated and technically much harder to solve. Therefore it will become important to reduce the problem to a simpler one by restricting attention to a special subsector as performed in [27]. An important problem is to clarify a relation between 1/16 BPS states in the N = 4 SYM and 1/16 BPS objects in type IIB supergravity on AdS 5 × S 5 . General 1/16 BPS (dual) giant gravitons have been constructed in [35] by using the same technique to construct a general 1/8 BPS giant graviton [36], where the configuration of 1/8 BPS giant gravitons of energy E is realized by the intersection of S 5 and the zero locus of a polynomial of the form n 1 +n 2 +n 3 =E/R c n 1 n 2 n 3 e −iEt z n 1 1 z n 2 2 z n 3 where R is the radius of AdS 5 , z 1 , z 2 , z 3 are coordinates of C 3 into which S 5 is embedded.
(See also [37,38,39,28,40,41,42,43].) Supersymmetric black holes have also been found in [25,24,26], which have turned out to be 1/16 BPS. Reproducing behaviors of these objects from the N = 4 SYM side is an important issue in AdS 5 /CFT 4 duality. Another interesting direction is to study the N = 4 SYM on R × S 3 /Z k by orbifolding the Hopf fiber direction so that the su(2) R algebra is broken and the global symmetry algebra of the system becomes psu(2|4). Study of a class of the theories with psu(2|4) symmetry is interesting because a family of BPS solutions with psu(2|4) symmetry (called bubbling geometries) and BPS objects in ten or eleven dimensional supergravity theories has extensively studied in [33,34,44,45]. It would be attracting to pursue the correspondence of BPS spectra of both sides more. (See [13,46,47,48,49] for the study of this direction.) It would also be fascinating to study BPS states in other supersymmetric gauge theories defined on R × S n as done in this paper, for example SYM on R × S 4 constructed recently in [50]. (The case of N = 6 Chern-Simons (ABJM) theory on R × S 2 was done in [32]. ) We leave these issues to future works.

B.2 Reduction to S 2
In this subsection we extract information on S 2 from the results obtained in the previous subsections by degenerating the Hopf fiber direction ψ of S 3 . The metric of R × S 2 is ds 2 R×S 2 = −dt 2 + 1 µ 2 (dθ 2 + sin 2 θdφ 2 ) (B.32) where µ is the inverse radius of S 2 , which is related to the S 3 radius by µ = 2 r . A local orthonormal frame is e 0 = dt, e 1 = µ −1 dθ, e 2 = µ −1 sin θdφ. Other components are zero. so(3) Killing actions on S 2 are given byL i with ∂ ψ eliminated. It is automatic thatL i restricted on S 2 form su(2) algebra. By using the operatorsL i the S 2 spherical harmonics are defined as root vectors of the su(2) algebra.
where s is non-negative half integer and |s 3 | ≤ s. The S 2 spherical harmonics can be obtained from S 3 spherical harmonics by taking a zero mode of ψ direction.