More on BPS States in N=4 Supersymmetric Yang-Mills Theory on R x S3
aa r X i v : . [ h e p - t h ] S e p TIFR/TH/14-19June 2014
More on BPS States in N = 4 SupersymmetricYang-Mills Theory on R × S Department of Theoretical Physics, Tata Institute of Fundamental Research,Homi Bhabha Road, Mumbai 400005, India
E-mail: yokoyama(at)theory.tifr.res.in
Abstract
We perform a systematic analysis on supersymmetric states in N = 4 su-persymmetric Yang-Mills theory (SYM) on R × S . We find a new set of 1 / /
16 BPS equations when they are valued in Cartan subal-gebra of a gauge group and the fermionic fields vanish. We also determine thenumber of supersymmetries preserved by the supersymmetric states varyingthe parameters of the BPS solutions. As a byproduct we present the completeset of such supersymmetric states in N = 8 SYM on R × S by carrying outdimensional reduction. Introduction N = 4 supersymmetric Yang-Mills theory (SYM) in four dimensions has been a funda-mental theoretical tool to extract important lessons of duality. In particular N = 4 SYMenjoys S-duality [1, 2], which is a generalization of the electro-magnetic duality [3, 4], andalso exhibits a dual description of type IIB supergravity on AdS × S under a particularlimit [5]. For precise investigation of these kinds of strong-weak duality, supersymmetryoften plays important roles.Since maximally supersymmetric completion of Yang-Mills theory gives rise to fullquantum conformal invariance in the flat space [6, 7, 8, 9], N = 4 SYM can be studied bymapping the system onto R × S by a conformal transformation, which makes the systemfree from infra-red divergence and discretizes the spectrum of the system with the massgap of order the inverse radius of the three sphere. In particular discrete spectrum of thesupersymmetric (or BPS) states of the theory has attracted a great deal of attention in thecontext of AdS /CFT duality, which can be tested by confirming match of supersymmetricspectra in both sides [10, 11].One aspect in the study of BPS spectrum is counting BPS states. (Other works relatedto N = 4 SYM on R × S are found, for instance, in [12, 13, 14, 15].) In particular animportant tool to study BPS spectrum is a (superconformal) index [16], which encodesBPS spectrum as a form of polynomial. A superconformal index of the N = 4 SYMhas 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 preciselymatches that of multi-particle states of supergravity multiplet in type IIB supergravity on AdS × S [17]. This result is reasonable in the sense that other supersymmetric objectssuch as (dual) BPS giant gravitons [21, 22, 23] and BPS black holes [24, 25, 26] in the dualgeometry have much larger charges, which are of order N , N respectively. Counting ofsupersymmetric states with such large charges in the N = 4 SYM has also been done asa purely mathematical problem by recasting BPS states as cohomology of a supercharge[27]. As a result, an index of 1 / N = 4 SYM, whereas that of 1 /
16 supersymmetric black holes has not. 1 / /CFT duality. (See [30] for a recent study in this direction.)Under the circumference this paper is aimed at carrying out a systematic analysis on1 /
16 BPS states of the N = 4 SYM. The goal of this paper is to clarify basic information onthe BPS states of the N = 4 SYM such as the 1 /
16 BPS equations, the BPS configurationand preserved supersymmetries by setting assumptions that the fermionic fields vanish andthat the fields are valued in Cartan subalgebra of a gauge group.The rest of this paper is organized in the following way. In § N = 4 SYM explaining our convention. In § N = 4 SYM by setting the fermionic fields to zero. In § § /
16 BPS states. In § § § N = 8 SYM on R × S by performing dimensional reduction. § S (B) including construction of the scalar spherical harmonics on S (B.1) and reduction of information of S to that of S (B.2). N = 4 SYM on R × S N = 4 SYM on R × S so as to be self-contained.The N = 4 SYM consists of six real scalars, gauge field and their supersymmetric part-ners, all of which are valued in the adjoint representation of a gauge group. This theoryhas
P SU (2 , |
4) global symmetry, and especially SO (6) R ≃ SU (4) R R-symmetry. Thefermionic 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 isdenoted by X AB satisfying X AB = − X BA , ( X AB ) † = 12 ε ABCD X CD =: X AB , (2.1)where A, B = 1 , , ,
4. The action of N = 4 SYM on R × S is given by S = 1 g Z dtr d Ω Tr (cid:20) − F µν F µν − D µ X AB D µ X AB + 14 [ X AB , X CD ][ X AB , X CD ] − r X AB X AB + i ( λ A ) † γ µ D µ λ A + λ A [ X AB , λ B ] + ( λ A ) † [ X AB , ( λ B ) † ] (cid:21) (2.2)where g is a gauge coupling, r is the radius of the three sphere, d Ω = dθ sin θdφdψ is thevolume form of the unit three sphere, D µ is the covariant derivative with respect to gaugeand space index, which acts on the fields as follows. D µ X AB = ∂ µ X AB + i [ A µ , X AB ] (2.3) D µ λ A = ∂ µ λ A + 14 ω µ,νρ γ νρ λ A + i [ A µ , λ A ] (2.4)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 thebosonic part. These can be computed in the standard way. Equations of motion for thegauge field and complex scalar fields in the bosonic part are D µ F µν − i [ X AB , D ν X AB ] = 0 , (2.5) D µ D µ X AB + [ X CD , [ X AB , X CD ]] − r X AB = 0 . (2.6) In this paper we do not need to specify a gauge group. U (4) R R-symmetry charge is R C A = i Z d Ω 1 g Tr (cid:0) − X CB D X AB + D X CB X AB (cid:1) . (2.7)The energy is H = Z d Ω 1 g Tr (cid:18)
12 ( F i ) + 14 ( F ij ) + 12 | D X AB | + 12 | D i X AB | + 14 | [ X AB , X CD ] | + 12 r | X AB | (cid:19) (2.8)where | A | = AA † . The angular momentum is J i = Z d Ω 1 g Tr (cid:18) F µ F iµ + 12 ( D X AB D i X AB + D i X AB D X AB ) (cid:19) (2.9)where i = θ, φ, ψ .The action (2.2) is invariant under the following supersymmetry transformation rule.∆ ǫ A µ = i (( ǫ A ) ∗ γ µ λ A + ǫ A γ µ ( λ A ) † ) , ∆ ǫ X AB = i ( − ε ABCD ǫ C λ D − ( ǫ A ) ∗ ( λ B ) † + ( ǫ B ) ∗ ( λ A ) † ) , (2.10)∆ ǫ λ A = 12 F µν γ µν ǫ A − D µ X AB γ µ ( ǫ B ) ∗ − i [ X AB , X BC ] ǫ C − X AB γ µ ∇ µ ( ǫ B ) ∗ , where ∇ µ is the covariant derivative for spin and ǫ A is a supersymmetry parameter valuedin Grassmann number, which is given by a conformal Killing spinor on R × S ∇ µ ǫ ( ± ) A = ± i r γ µ γ t ǫ ( ± ) A . (2.11)Let us solve this Killing spinor equation briefly. For ǫ (+) A , (2.11) reduces to ∂ t ǫ (+) A = i r ǫ (+) A , ∂ ψ ǫ (+) A = i γ t ǫ (+) A , ∂ θ ǫ (+) A = 0 , ∂ φ ǫ (+) A = 0 . (2.12)This can be solved as follows. ǫ (+) A = e i r t e − i γ ψ η (+) A (2.13)where η (+) A is a constant spinor. On the other hand, for ǫ ( − ) A , (2.11) becomes ∂ t ǫ ( − ) A = − i r ǫ ( − ) A , ∇ S θ ǫ ( − ) A = − ir γ θt ǫ ( − ) A , ∇ S φ ǫ ( − ) A = − ir γ φt ǫ ( − ) A , ∂ ψ ǫ ( − ) A = 0 , (2.14)where ∇ S µ is the spin covariant derivative on S . More explicitly, the covariant derivativesact on a spinor as follows. ∇ θ ǫ ( − ) A = ∂ θ ǫ ( − ) A , ∇ φ ǫ ( − ) A = ∂ φ ǫ ( − ) A + 12 cos θγ ǫ ( − ) A . (2.15)The solution of Killing spinor equation is ǫ ( − ) A = e − i r t e − i γ θ e − γ φ η ( − ) A (2.16)where η ( − ) A is a constant spinor. 3 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 .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 thegaugino fields vanish. In this case we have only to study conditions of the bosonic fieldsfor the supersymmetry transformation of gaugino to vanish. By using the Killing spinorequation (2.11) the supersymmetry variation of gaugino can be written as∆ ǫ ( ± ) λ A = 12 F µν γ µν ǫ ( ± ) A − i [ X AB , X BC ] ǫ ( ± ) C + ( − D X AB ± X AB ir ) γ t ( ǫ ( ± ) B ) ∗ − D i X AB γ i ( ǫ ( ± ) B ) ∗ (3.1)where i = θ, φ, ψ .There are two kinds of BPS states on the sphere: one is spherically symmetric, theother is not. This separation is achieved automatically by studying whether a preservedKilling spinor of a BPS state is projected or not by a certain projective operator, whichbreaks spherical symmetry. In this paper we perform projection for a constant spinor inKilling spinors (2.13), (2.16) by using γ . In this subsection we study spherically symmetric BPS solutions for warm-up. For thispurpose we study a BPS condition which preserves at least one Killing spinor without anyprojection. As such a preserved Killing spinor we first choose ǫ (+) , where = 1 , , ,
4. Weuse German letters , , , as symbols representing 1 , , , σ such that i = σ ( i ), where i = , , , , i = 1 , , ,
4. For the supersymmetryvariation of gaugino to vanish with nonzero ǫ (+) , the fields are required to satisfy F µν = 0 , [ X AB , X B ] = 0 , − D X A + X A ir = 0 , D i X A = 0 (3.2)for A = , , , , since γ µ ǫ (+) and ( γ µ ǫ (+) ) ∗ are linearly independent for all µ . This is the1/8 BPS condition which preserves a Killing spinor ǫ (+) . 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 ofthe gauge group. In this case these become F µν = 0 , ∂ X A = X A ir , ∂ i X A = 0 . (3.3)This can be easily solved as X A = x (+) A e i tr (3.4)4here x A are integral constants valued in complex number. Other components are deter-mined by using the relation (2.1) as X = X † = x (+) † e − i tr , X = − X † = − x (+) † e − i tr , X = − X † = − x (+) † e − i tr . (3.5)This is a general 1/8 BPS solution which preserves a Killing spinor ǫ (+) .Let us study a case for this BPS solution to preserve another Killing spinor. One willsoon notice that ǫ ( − ) is broken unless all fields are trivial. So let us set it to zero to studya nontrivial BPS solution. Under the BPS condition (3.2) the supersymmetry variation ofgaugino becomes∆ ǫ ( ± ) λ =( X ir ± X ir ) γ t ( ǫ ( ± ) ) ∗ + ( X ir ± X ir ) γ t ( ǫ ( ± ) ) ∗ , ∆ ǫ ( ± ) λ =( X ir ± X ir ) γ t ( ǫ ( ± ) ) ∗ + ( X ir ± X ir ) γ t ( ǫ ( ± ) ) ∗ , ∆ ǫ ( ± ) λ =( X ir ± X ir ) γ t ( ǫ ( ± ) ) ∗ + ( X ir ± X ir ) γ t ( ǫ ( ± ) ) ∗ , ∆ ǫ ( ± ) λ =( − X ir ± X ir ) γ t ( ǫ ( ± ) ) ∗ + ( − X ir ± X ir ) γ t ( ǫ ( ± ) ) ∗ + ( − X ir ± X ir ) γ t ( ǫ ( ± ) ) ∗ . (3.6)In order for these to vanish, it is required to satisfy( ǫ (+) ) ∗ = 0 or ( ǫ (+) ) ∗ = 0 , X = X = 0 , (3.7)( ǫ (+) ) ∗ = 0 or ( ǫ (+) ) ∗ = 0 , X = X = 0 , (3.8)( ǫ (+) ) ∗ = 0 or ( ǫ (+) ) ∗ = 0 , X = X = 0 , (3.9)( ǫ ( − ) ) ∗ = 0 or ( ǫ ( − ) ) ∗ = 0 , X = 0 , (3.10)( ǫ ( − ) ) ∗ = 0 or ( ǫ ( − ) ) ∗ = 0 , X = 0 , (3.11)( ǫ ( − ) ) ∗ = 0 or ( ǫ ( − ) ) ∗ = 0 , X = 0 , (3.12)where we used the relation (2.1). Therefore the 1/8 BPS solution has enhanced supersym-metry in the following situation.1. One complex scalar field is trivial: X = 0. In this case, the Killing spinor ǫ ( − ) isalso preserved and the solution (3.4) becomes 1/4 BPS.2. Two scalar fields are trivial: X = X = 0. In this case, the Killing spinors ǫ ( − ) , ǫ ( − ) , ǫ (+) are also conserved and the solution (3.4) becomes 1/2 BPS.In the same way we can obtain a BPS condition and solution which preserve ǫ ( − ) . TheBPS condition to conserve ǫ ( − ) is F µν = 0 , [ X AB , X B ] = 0 , D X A − X A ir = 0 , D i X A = 0 . (3.13)5hen the fields take values in Cartan subalgebra of the gauge group, this reduces to F µν = 0 , ∂ X A + X A ir = 0 , ∂ i X A = 0 . (3.14)The equations of the complex scalar field can be easily solved as X A = x ( − ) A e − i tr . (3.15)This is a general 1/8 BPS solution which preserves ǫ ( − ) . This BPS solution has enhancedsupersymmetry in the following situation.1. One complex scalar field is trivial: X = 0. In this case, the Killing spinor ǫ (+) isalso conserved, and the solution becomes 1/4 BPS.2. Two scalar fields are trivial: X = X = 0. In this case, the Killing spinors ǫ (+) , ǫ (+) , 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. X A = x ( ± ) A e ± i tr for A = , , . ǫ ( ± ) . 4( BPS) X A = x ( ± ) A e ± i tr for A = , , ǫ ( ∓ ) , ǫ ( ± ) . 8 X = 0. ( BPS) X = x ( ± ) e ± i tr , ǫ ( ∓ ) , ǫ ( ∓ ) , ǫ ( ± ) , ǫ ( ± ) . 16 X = X = 0. ( BPS) X = X = X = 0. ǫ ( ± ) , ǫ ( ± ) , ǫ ( ± ) , ǫ ( ± ) . 32(Unique vacuum)Table 1: We list spherically symmetric BPS solutions, preserved Killing spinors and itsnumber. Signature in multi-column has to be chosen with order respected except the lastcolumn.Let us compute conserved charges (2.8), (2.9), (2.7) under the BPS conditions. H = 1 g Z d Ω 4 r Tr | X A | , (3.16) P i =0 , (3.17) R = Z d Ω 2 g ∓ r Tr | X A | , (3.18)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 P A =1 R AA = 0. We can easily see the BPS relation of conservedcharge as rH = ∓ R = ± X A = R AA . (3.19)6 .2 Non-spherical BPS states ǫ (+) In this subsection we study supersymmetric states with angular momenta. To this endwe study supersymmetric states which preserves a Killing spinor projected by a certainoperator which breaks spherical symmetry. We first choose ǫ (+) as such a preserved Killingspinor. We carry out projection for a constant spinor η (+) such that γ η (+) = iw (+) η (+) (3.20)where w (+) = ± For this projected constant spinor, the Killing spinor ǫ (+) given by(2.13) is evaluated as ǫ (+) = e i r t e i w (+)2 ψ η (+) . (3.21)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 := ˆ L = − i∂ φ , and J ψ := ˆ R = i∂ ψ , respectively, where ˆ L i , ˆ R i are defined in Appendix B. Thecharge assignments are summarized in Table 2. Therefore the BPS condition of charges H J φ J ψ R R R R ǫ (+) − r − w (+) − − −
14 34
Table 2: Charges of the Killing spinor ǫ (+) are shown.associated with the Killing spinor ǫ (+) is rH = − R + 2 w (+) J ψ . (3.22)Note that the relative factors match that obtained from psu (2 , |
4) superconformal algebrain the standard normalization [31, 17].In order to study a BPS condition preserving ǫ (+) , it is convenient to divide the super-symmetric transformation of gaugino into two parts such that∆ ǫ ( ± ) λ C =∆ ( ± ) λ C + ¯∆ ( ± ) λ C (3.23)where ∆ ( ± ) λ A = 12 F µν γ µν ǫ ( ± ) A − i [ X AB , X BC ] ǫ ( ± ) C , (3.24)¯∆ ( ± ) λ A =( − D X AB ± X AB ir ) γ t ( ǫ ( ± ) B ) ∗ − D i X AB γ i ( ǫ ( ± ) B ) ∗ . (3.25) In terms of a Killing spinor the projection is given by P ǫ (+) = iw (+) ǫ (+) where P = U γ U − with ǫ (+) = U η (+) . η (+) from these. By using (3.20) and (3.21). we find∆ (+) λ A = e iµ t e i w (+)2 ψ (cid:20) iw (+) δ A ( − F + iF + iw (+) F − w (+) F ) γ η (+) + { iw (+) δ A ( iF + F ) + ( − i [ X AB , X B ]) } η (+) (cid:21) + · · · (3.26)¯∆ (+) λ A = e − iµ t e i − w (+)2 ψ (cid:20) (cid:18) ( − D X A + X A ir )( iw (+) ) − iD X A (cid:19) ( η (+) ) ∗ + ( − D X A − iD X A w (+) ) γ ( η (+) ) ∗ (cid:21) + · · · (3.27)where the ellipses represent the other terms which do not contain η (+) . In order for ∆ ǫ ( ± ) λ C to vanish with η (+) nonzero, it is necessary to satisfy − F + iF + iw (+) F − w (+) F = 0 ,iw (+) δ A ( iF + F ) − i [ X AB , X B ] = 0 , ( − D X A + X A ir )( iw (+) ) − iD X A = 0 , − D X A − iD X A w (+) = 0 , for arbitrary A , since η (+) , γ η (+) , ( η (+) ) ∗ , γ ( η (+) ) ∗ are linearly independent under the pro-jection (3.20). By using the fact that all the components of F µν are real, these can besimplified as F = − w (+) F , F = − w (+) F ,w (+) δ A ( iF + F ) − X AB , X B ] = 0 , ( D − ir − w (+) D ) X A = 0 , ( D + iw (+) D ) X A = 0 . (3.28)This is the 1/16 BPS condition preserving ǫ (+) projected by (3.20). This result is essentiallythe same as 1/16 BPS equations derived in [27], (4.9) and (4.10), in a different way bymaking the energy density complete square and looking for configurations to saturate theBogomol’nyi bound.Let us solve these BPS conditions when the fields are valued in Cartan subalgebra of agauge group. In this case these BPS equations boil down to F = F = 0 , F = − w (+) F , F = − w (+) F , (3.29)and ( ∂ − ir − w (+) ∂ ) X A = 0 , ( ∂ + iw (+) ∂ ) X A = 0 (3.30)8or A = , , . Note that by restricting our interest to the Cartan part the gauge sectorand the matter one decouple so that we can solve each sector independently. 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 scalar spherical harmonics. Wedenote the scalar spherical harmonics by Y s,l ,r with non-negative half integer s and twohalf integers l , r whose moduli are bounded above by s . These quantum numbers areCartan charges of the representation of the spherical harmonics of su (2) L × su (2) R actingon S . ˆ L Y s,l ,r = ˆ R Y s,l ,r = s ( s + 1) Y s,l ,r , (3.31)ˆ L Y s,l ,r = l Y s,l ,r , ˆ R Y s,l ,r = r Y s,l ,r , (3.32)where ˆ L i , ˆ R i are differential operators generating su (2) L × su (2) R algebra. See AppendixB for more details. By using these operators the second equation in (3.30) can be writtenas ˆ R w (+) X A = 0 (3.33)where ˆ R w (+) = ˆ R ± when w (+) = ± R ± is given by (B.16). ThereforeBPS solutions are expanded by the spherical harmonics of highest (or lowest) weight of su (2) R for w (+) = +1 (or − X A = X s ≥ , | l |≤ s x s,l A ( t ) Y s,l ,w (+) s ( θ, φ, ψ ) (3.34)where x s,l A ( t ) are unknown functions of time, which is determined from the first equationin (3.30). Plugging the above into the first equation in (3.30) gives ∂ x s,l A ( t ) = ir x s,l A ( t ) − r w (+) ( − iw (+) s ) x s,l A ( t ) . (3.35)This equation is easily solved as x s,l A ( t ) = x s,l A e i s +1 r t (3.36)where x s,l A is an integral constant valued in complex number. As a result we obtain X A = X s ≥ , | l |≤ s x s,l A e i s +1 r t Y s,l ,w (+) s ( θ, φ, ψ ) . (3.37)This is a general 1 /
16 BPS solution of the complex scalar fields preserving the Killingspinor ǫ (+) with projection (3.20). Note that this satisfies the equation of motion of thescalar field (2.6). This is not the case in the non-abelian sector [27]. More rigorously or mathematically speaking, x s,l A is a constant valued in Cartan subalgebra of agauge group over the complex number. But we often abuse such a variable as a coefficient of a Cartangenerator by which the variable is expanded. ∇ µ F µν = 0 , ∇ F + ∇ F + ∇ F = 0 , respectively. Under the BPS condition (3.29) the equationsof motion reduce to the following three equations ∂ t F + w (+) ∂ F = 0 , (3.38) ∂ t F + w (+) ∂ F = 0 , (3.39) ∂ ( w (+) F ) − ∂ F + 2 r cot θ ( w (+) F ) = 0 , (3.40)and the Bianchi identity becomes ∂ (sin θF ) + w (+) ∂ (sin θF ) = 0 . (3.41)After multiplying sin θ to both sides in (3.40), we can write it as ∂ (sin θF ) − w (+) ∂ (sin θF ) = 0 . (3.42)A general solution of (3.41) and (3.42) is given bysin θF = X s ≥ , | l |≤ s ( B s,l ( t )Re[ Y s,l ,w (+) s ( θ, φ, ψ )] − A s,l ( t )Im[ Y s,l ,w (+) s ( θ, φ, ψ )])sin θF = X s ≥ , | l |≤ s ( A s,l ( t )Re[ Y s,l ,w (+) s ( θ, φ, ψ )] + B s,l ( t )Im[ Y s,l ,w (+) s ( θ, φ, ψ )]) (3.43)where A s,l ( t ) , B s,l ( t ) are unknown functions of time. The reason is as follows. First werecall that Y s,l ,w (+) s is annihilated by the operator ˆ R w (+) . By dividing Y s,l ,w (+) s into thereal and imaginary part, Y s,l ,w (+) s = u + iv , we can rewrite ˆ R w (+) Y s,l ,w (+) s = 0 as ∂ u − w (+) ∂ v = 0 , ∂ v + w (+) ∂ u = 0 . (3.44)This is equivalent to ∂ ( Au + Bv ) + w (+) ∂ ( Bu − Av ) = 0 , ∂ ( Bu − Av ) − w (+) ∂ ( Au + Bv ) = 0 (3.45)where A, B are arbitrary constants independent of θ, φ, ψ . This implies that the equations(3.41) and (3.42) are solved by sin θF = Au + Bv, sin θF = Bu − Av . Superposing allmodes labeled by l, l gives a general solution (3.43), as asserted.The time dependence of the field strength is determined by (3.38), (3.39). The resultis sin θF = X s ≥ , | l |≤ s a (+) s,l Re[ Y s,l ,w (+) s ( θ, φ, ψ − w (+) r t + α (+) s,l )] (3.46)sin θF = X s ≥ , | l |≤ s a (+) s,l Im[ Y s,l ,w (+) s ( θ, φ, ψ − w (+) r t + α (+) s,l )] (3.47)10here a (+) s,l , α (+) s,l are integral constants taking real values with the range a (+) s,l ≥ , ≤ α (+) s,l < π. Plugging the explicit expression of the spherical harmonics given by (B.25) intothese, we find the general 1/16 BPS solution of the field strength as F = X s ≥ , | l |≤ s a (+) s,l c s,l tan w (+) l (cid:18) θ (cid:19) sin s − θ cos ( l φ + s ( 2 r t − w (+) ψ ) + α (+) s,l ) (3.48) F = X s ≥ , | l |≤ s a (+) s,l c s,l tan w (+) l (cid:18) θ (cid:19) sin s − θ sin ( l φ + s ( 2 r t − w (+) ψ ) + α (+) s,l ) (3.49)where c s,l is given by (B.26). Other components of the field strength are determined by(3.29).Let us enphasize that it is possible to turn on BPS excitation for the gauge field inde-pendently from the matter sector in the abelian sector. This behavior is different from thenon-abelian sector [27]. We also note that it is not required to use the vector sphericalharmonics to solve the BPS equation for the gauge field.The moduli space of the 1 /
16 BPS solution is given by x s,l , x s,l , x s,l ∈ C , a (+) s,l ≥ , ≤ α (+) s,l < π (3.50)for s ≥ , | l | ≤ s for x s,l A , s ≥ , | l | ≤ s for a (+) s,l . Note that we do not take into accountthe flux quantization condition here and hereafter, which would make the moduli spacequantized in a certain manner.Let us compute conserved charges (2.8), (2.9) and (2.7). We first simply them by usingthe BPS solution (3.29), (3.30). As a result the conserved charges are written in terms of X A , ∂ X A , ∂ X A , F , F . We present results by separating the matter sector and gaugesector. The conserved charges in the matter sector are simplified as H | F µν =0 = Z d Ω 4 g Tr (cid:18) r | X A | − w (+) ir X A ∂ X A + | ∂ X A | + | ∂ X A | (cid:19) , (3.51) J θ | F µν =0 =0 , (3.52) J φ | F µν =0 = Z d Ω 2 g Tr (cid:18) − ( ir X A − w (+) ∂ X A ) ∂ φ X A + ( c.c. ) (cid:19) , (3.53) J ψ | F µν =0 = Z d Ω 2 g Tr (cid:18) − ( ir X A − w (+) ∂ X A ) ∂ ψ X A + ( c.c. ) (cid:19) , (3.54) R | F µν =0 = i Z d Ω 2 g Tr (cid:18) ir | X B | − w (+) X B ∂ X B (cid:19) , (3.55) Here we exclude the mode of s = 0, which formally satisfies the 1 /
16 BPS equations but turns out tobe non-normalizable. The author would like to thank S.Kim for pointing this out. c.c. ) means the complex conjugation of the first term. The gauge field part is thefollowing. H | X AB =0 = Z d Ω 1 g Tr (cid:0) ( F ) + ( F ) (cid:1) , (3.56) J θ | X AB =0 =0 , (3.57) J φ | X AB =0 = w (+) r g Z d Ω cos θ Tr (cid:0) ( F ) + ( F ) (cid:1) , (3.58) J ψ | X AB =0 = w (+) r g Z d Ω Tr (cid:0) ( F ) + ( F ) (cid:1) , (3.59) R | X AB =0 =0 . (3.60)Note that at this stage we see the BPS relation of charges given by (3.22) in the gaugesector.Let us compute conserved charges using the BPS solution (3.37), (3.48), (3.49). Thematter part is the following. H | F µν =0 = 8 r g X s ≥ , | l |≤ s (2 s + 1) Tr | x s,l A | , (3.64) J φ | F µν =0 = − rg X s ≥ , | l |≤ s (2 s + 1) l Tr | x s,l A | , (3.65) J ψ | F µν =0 = 4 rg w (+) X s ≥ , | l |≤ s (2 s + 1) s Tr | x s,l A | , (3.66) R | F µν =0 = − rg X s ≥ , | l |≤ s (2 s + 1)Tr | x s,l A | . (3.67)In the matter sector we also find the BPS relation given by (3.22). For this computation we used formulas such that Z d Ω 1sin θ | Y s,l ,w (+) s | = s (1 + 2 s )2 ( − l + s ) , (3.61) Z d Ω cot θ sin θ | Y s,l ,w (+) s | = w (+) l (1 + 2 s )2( l − s ) , (3.62) Z d Ω cot θ | Y s,l ,w (+) s | = − l + s l − s ) , (3.63)where | l | < s . In the main text, we also used these formulas formally at | l | = s . H | X AB =0 = w (+) r J ψ | X AB =0 = 1 g X s ≥ , | l |≤ s s (2 s + 1)2 ( s − l ) Tr( a (+) s,l ) , (3.68) J φ | X AB =0 = r g X s ≥ , | l |≤ s l (2 s + 1)2( l − s ) Tr( a (+) s,l ) . (3.69)Note that the energy and momenta become divergent when the modes with | l | = s arenonzero. ǫ ( − ) Next we study BPS states preserving a Killing spinor ǫ ( − ) whose constant spinor η ( − ) isprojected in such a way that γ η ( − ) = iw ( − ) η ( − ) (3.70)where w ( − ) = ±
1. Under this projection the Killing spinor ǫ ( − ) given by (2.16) is evaluatedas ǫ ( − ) = e − i r t e − iw ( − ) φ ( c θ − w ( − ) γ s θ ) η ( − ) (3.71)where we used notation such that c θ = cos θ , s θ = sin θ . (3.72)We determine charges of this Killing spinor ( ǫ ( − ) ) ∗ to find out the BPS relation ofcharges in Table 3. Therefore the BPS relation of charges associated with the Killing
H J φ J ψ R R R R ( ǫ ( − ) ) ∗ − r w ( − ) −
34 14 14 14
Table 3: Charges of the Killing spinor ( ǫ ( − ) ) ∗ are shown.spinor ǫ ( − ) is rH = 2 R + 2 w ( − ) J φ . (3.73) The reason why we determine charges of ( ǫ ( − ) ) ∗ instead of those of ǫ ( − ) is because ( ǫ ( − ) ) ∗ has thenegative 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 particularsupersymmetry charge.
13o determine a BPS condition preserving ǫ ( − ) , we extract the terms containing η ( − ) from (3.24), (3.25) as done previously. The results are as follows.∆ ( − ) λ A = e − iµ t e − iw ( − ) φ × (cid:20)(cid:8) ( − iw ( − ) c θ )( F − iF ) − c θ ( F + iF ) + s θ ( F − iF ) − i [ X AB , X B ]( − w ( − ) s θ ) (cid:9) γ η ( − ) + (cid:8) − is θ ( F − iF ) + s θ w ( − ) ( F + iF ) + c θ w ( − ) ( F − iF ) − i [ X AB , X B ] c θ (cid:9) η ( − ) (cid:21) + · · · ¯∆ ( − ) λ A = e iµ t e iw ( − ) φ × (cid:20)(cid:8) ( − D X A − X A ir )( iw ( − ) c θ ) − D X A ( − w ( − ) s θ ) − D X A ( is θ ) − D X A ( ic θ ) (cid:9) ( η ( − ) ) ∗ + (cid:8) ( − D X A − X A ir )( is θ ) − D X A ( c θ ) − D X A ( iw ( − ) c θ ) − D X A ( − iw ( − ) s θ ) (cid:9) γ ( η ( − ) ) ∗ (cid:21) + · · · where the ellipses describe the other terms than η ( − ) . In order for these to vanish with η ( − ) nonzero, the fields are required to satisfy (cid:8) − is θ ( F − iF ) + s θ w ( − ) ( F + iF ) + c θ w ( − ) ( F − iF ) (cid:9) δ A − i [ X AB , X B ] c θ = 0 , (cid:8) ( − iw ( − ) c θ )( F − iF ) − c θ ( F + iF ) + s θ ( F − iF ) (cid:9) δ A − i [ X AB , X B ]( − w ( − ) s θ ) = 0 , ( − D X A − X A ir )( iw ( − ) c θ ) − D X A ( − w ( − ) s θ ) − D X A ( is θ ) − D X A ( ic θ ) = 0 , ( − D X A − X A ir )( is θ ) − D X A ( c θ ) − D X A ( iw ( − ) c θ ) − D X A ( − iw ( − ) s θ ) = 0 , for all A = , , , , since η ( − ) , γ η ( − ) , ( η ( − ) ) ∗ , γ ( η ( − ) ) ∗ are linearly independent under theprojection (3.70). Using the fact that all the components of F µν are real, we can simplifythese BPS equations as follows. F = − w ( − ) F sec θ + F tan θ, F = − w ( − ) F sec θ − F tan θ, [ X AB , X B ] = 12 i (cid:8) w ( − ) sec θ ( F − iF ) − ( F + iF ) tan θ (cid:9) δ A ,D X A = − ir X A − w ( − ) r D φ X A ,D X A = iw ( − ) ( − cos θD X A + sin θD X A ) . (3.74)These are the other set of 1 /
16 BPS equations, which is a new result in this paper.14et us solve these BPS equations when the fields take values in Cartan subalgebra of agauge group. In this case these BPS equations are further simplified as F = − w ( − ) cos θF , F = w ( − ) sin θF , (3.75) F = − w ( − ) sin θF , F = − w ( − ) cos θF , (3.76)( ∂ + ir + 2 r w ( − ) ∂ φ ) X A = 0 , (3.77)( ∂ − iw ( − ) ( − cos θ∂ + sin θ∂ )) X A = 0 . (3.78)Let us first solve the matter BPS equation. By noticing that (3.78) can be written asˆ L w ( − ) X A = 0 , where ˆ L w ( − ) = ˆ L ± when w ( − ) = ±
1, the BPS solution of the matterfields are expanded by the spherical harmonics of highest (or lowest) weight of su (2) L for w ( − ) = +1 (or − X A = X s ≥ , | r |≤ s x s,r A ( t ) Y s,w ( − ) s,r ( θ, φ, ψ ) (3.79)where x s,r A ( t ) represents time-dependence of the scalar field, which is determined from(3.77). ∂ x s,r A ( t ) = − ir x s,r A ( t ) − r w ( − ) ( iw ( − ) s ) x s,r A ( t ) . (3.80)This equation is easily solved as x s,r A ( t ) = x s,r A e − i s +1 r t (3.81)where x s,r A is an integral constant. As a result we obtain BPS solutions X A = X s ≥ , | r |≤ s x s,r A e − i s +1 r t Y s,w ( − ) s,r ( θ, φ, ψ ) . (3.82)This is a general 1 /
16 BPS solution which preserves ǫ ( − ) projected by (3.20). It is notdifficult 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 theequations of motion (2.5) and Bianchi identity. In the current situation they are given by ∇ µ F µν = 0 and ∇ F + ∇ F + ∇ F = 0 , respectively. Under the BPS conditions(3.75),(3.76) the equations of motion reduce to the following three equations ∂ t F = − w ( − ) r ∂ φ F , (3.83) ∂ t F = − w ( − ) r ∂ φ F , (3.84) ∂ F = − w ( − ) ( ∂ t (cos θF ) + ∂ (sin θF )) , (3.85)and the Bianchi identity becomes( − θ ∂ ψ + cot θ∂ φ )(sin θF ) = − w ( − ) ∂ θ (sin θF ) . (3.86)15e can solve (3.85) and (3.86) by noticing the fact that they can be rewritten as ˆ L w ( − ) G =0, where G = sin θF + i sin θF . Therefore this solution is given by the spherical har-monics of the highest (or lowest) weight of su (2) L algebra. G = X s ≥ , | l |≤ s C ( − ) s,r ( t ) Y s,w ( − ) s,r ( θ, φ, ψ ) (3.87)where C ( − ) s,r ( t ) is an unknown function of time, which can be easily determined from theother BPS equations (3.83), (3.84). ∂ t C ( − ) s,r ( t ) = − i r sC ( − ) s,r ( t ) (3.88)which is solved as C ( − ) s,r ( t ) = C ( − ) s,r e − i r st = a ( − ) s,r e − i r st + iα ( − ) s,r (3.89)where a ( − ) s,r , α ( − ) s,r are integral constants with the range a ( − ) s,r ≥ , ≤ α ( − ) s,r < π . Pluggingthis back into the above gives G = X s ≥ , | l |≤ s a ( − ) s,r e − i r st + iα ( − ) s,r Y s,w ( − ) s,r ( θ, φ, ψ ) . (3.90)Plugging the polar coordinates expression of the spherical harmonics given by (B.28) wefind the 1 /
16 BPS solutions of the field strength F = X s ≥ , | r |≤ s a ( − ) s,r c s,r tan w ( − ) r (cid:18) θ (cid:19) sin s − θ cos ( − r ψ + s ( w ( − ) φ − r t ) + α ( − ) s,r ) , (3.91) F = X s ≥ , | r |≤ s a ( − ) s,r c s,r tan w ( − ) r (cid:18) θ (cid:19) sin s − θ sin( − r ψ + s ( w ( − ) φ − r t ) + α ( − ) s,r ) , (3.92)where c s,r is given by (B.29). Other components can be obtained from (3.75), (3.76).The moduli space of the 1 /
16 BPS solution is given by x s,r , x s,r , x s,r ∈ C , a ( − ) s,r ≥ , ≤ α ( − ) s,r < π (3.93)for s ≥ , | r | ≤ s for x s,r A , s ≥ , | r | ≤ s for a (+) s,r . Here we do not take into account theflux quantization condition.Let us compute conserved charges (2.8), (2.9), (2.7) of the BPS solution. For this endwe first simplify them by using the BPS conditions (3.75), (3.76), (3.77), (3.78). The We exclude the mode with s = 0. H | F µν =0 = 4 g Z d Ω Tr (cid:18) r | X A | + | ∂ X A | + | ∂ X A | + 2 ir w ( − ) X A ∂ φ X A (cid:19) , (3.94) J θ | F µν =0 =0 , (3.95) J φ | F µν =0 = Z d Ω 1 g Tr (cid:18) − − ir X A − w ( − ) r D φ X A )) D φ X A + ( c.c. ) (cid:19) ,J ψ | F µν =0 = Z d Ω 1 g Tr (cid:18) − − ir X A − w ( − ) r D φ X A ) D ψ X A + ( c.c. ) (cid:19) , (3.96) R | F µν =0 = i Z d Ω 2 g Tr (cid:18) − ir | X A | − w ( − ) r X A D φ X A (cid:19) . (3.97)The gauge sector is as follows. H | X AB =0 = Z d Ω 1 g Tr (cid:0) ( F ) + ( F ) (cid:1) , (3.98) J θ | X AB =0 =0 , (3.99) J φ | X AB =0 = w ( − ) r g Z d Ω Tr (cid:0) ( F ) + ( F ) (cid:1) , (3.100) J ψ | X AB =0 = w ( − ) r g Z d Ω cos θ Tr (cid:0) ( F ) + ( F ) (cid:1) , (3.101)with R | X AB =0 = 0. Note that at this stage we see the BPS relation of charges given by(3.73) in the gauge sector.Let us compute these conserved charges under the BPS solution (3.82). (3.91), (3.92).17he matter parts are H | F µν =0 = 8 r g X s ≥ , | r |≤ s (2 s + 1) Tr | x s,r A | , (3.105) J φ | F µν =0 = 4 rg w ( − ) X s ≥ , | r |≤ s (2 s + 1) s Tr | x s,r A | , (3.106) J ψ | F µν =0 = 4 rg X s ≥ , | r |≤ s − (2 s + 1) r Tr | x s,r A | , (3.107) R | F µν =0 = 2 rg X s ≥ , | r |≤ s (2 s + 1)Tr | x s,r A | , (3.108)which satisfies the BPS relation of charges given by (3.73). The gauge field part is thefollowing. H | X AB =0 = w ( − ) r J φ | X AB =0 = 1 g X s ≥ , | r |≤ s s (1 + 2 s )2 ( s − r ) Tr( a ( − ) s,r ) , (3.109) J ψ | X AB =0 = r g X s ≥ , | r |≤ s r (1 + 2 s )2( r − s ) Tr( a ( − ) s,r ) . (3.110)Note that the modes with | r | = s give divergent contribution to the energy and momenta. In this subsection we count number of supersymmetries preserved by BPS solutions con-structed in the previous subsections. Let us count the number of supersymmetries of theBPS solutions given by (3.37), (3.48), (3.49), which preserves at least η (+) projected by We note useful formulas Z d Ω 1sin θ | Y s,w ( − ) s,r | = s (1 + 2 s )2 ( − r + s ) , (3.102) Z d Ω cot θ sin θ | Y s,w ( − ) s,r | = w ( − ) r (1 + 2 s )2( r − s ) , (3.103) Z d Ω cot θ | Y s,w ( − ) s,r | = − r + s r − s ) , (3.104)where | r | < s . We however used these formulas formally at | r | = s in the main text. X A = X s ≥ , | l |≤ s x s,l A e i s +1 r t Y s,l ,w (+) s ( θ, φ, ψ ) , (3.111) F = X s ≥ , | l |≤ s a (+) s,l c s,l tan w (+) l (cid:18) θ (cid:19) sin s − θ cos ( l φ + s ( 2 r t − w (+) ψ ) + α (+) s,l ) , (3.112) F = X s ≥ , | l |≤ s a (+) s,l c s,l tan w (+) l (cid:18) θ (cid:19) sin s − θ sin ( l φ + s ( 2 r t − w (+) ψ ) + α (+) s,l ) , (3.113)for arbitrary A = , , . To exclude the case of trivial angular momenta, we consider thecase whether x s,l C = 0 or a (+) s,l = 0 for some positive s and l .First let us consider the case where a (+) s,l = 0 for some positive s and l . Under this situ-ation we consider the BPS solution which also preserves η (+) satisfying γ η (+) = iw (+) η (+) .Then the form of the BPS solution gets another constraint such that X A = X s ≥ , | l |≤ s x s,l A e i s +1 r t Y s,l ,w (+) s ( θ, φ, ψ ) , (3.114) F = X s ≥ , | l |≤ s a (+) s,l c s,l tan w (+) l (cid:18) θ (cid:19) sin s − θ cos ( l φ + s ( 2 r t − w (+) ψ ) + α (+) s,l ) , (3.115) F = X s ≥ , | l |≤ s a (+) s,l c s,l tan w (+) l (cid:18) θ (cid:19) sin s − θ sin ( l φ + s ( 2 r t − w (+) ψ ) + α (+) s,l ) , (3.116)for A = , , . Under the assumption, matching of (3.112), (3.113) with (3.115), (3.116)requires the signature of the projection for η (+) , η (+) to be the same, w (+) = w (+) . Matchingof the matter solution (3.111), (3.114) demands X = X = 0 since X † = − X , X † = X from (2.1). Under this situation supersymmetries specified by η (+) and η (+) are brokenunless X † = 0. In other words, the matter becomes trivial if we keep three supersymmetriesof the form η (+) A . Can the BPS solution preserve η ( − ) A supersymmetry concurrently? Letus study this possibility. Keeping also η ( − ) makes the matter fields completely trivial andgives constraint for the field strength so that the highest or lowest weight modes in termsof su (2) L remain: a (+) s,l = 0 for ∀ s > , l = − w ( − ) s for fixed w ( − ) . To preserve η ( − ) in steadof η ( − ) , the matter field X has to vanish and the other matter fields have only the modeswhich are also the highest or lowest weight states of su (2) L . In this case projected η ( − ) is preserved. The condition to keep η ( − ) , η ( − ) are the same with that of η ( − ) by replacing by , respectively. When all scalar fields become trivial, all Killing spinors η (+) A arepreserved for all A . In this case, supersymmetry is enhanced when the gauge field strengthis expanded by the mode of the highest or lowest weight representation of su (2) L as wellso that a (+) s,l = 0 for ∀ s > , l = − w ( − ) s for fixed w ( − ) . In this case the Killing spinors ofthe form η ( − ) A are also conserved. We summarize this result in Table 4.19arameter region. Preserved Killing spinors. Number of SUSY. x s,l , x s,l , x s,l , a (+) s,l : generic. η (+) γ η (+) = iw (+) η (+) . ( BPS) x s,l = x s,l = 0 for ∀ s, l . η (+) , η (+) γ η (+) A = iw (+) η (+) A . ( BPS) x s,l , x s,l , a (+) s,l = 0 for ∀ s > , l = − w ( − ) s , η ( − ) , η (+) x s,l = 0 for ∀ s, l . with γ η ( ± ) A = iw ( ± ) η ( ± ) A . ( BPS) x s,l , a (+) s,l = 0 for ∀ s > , l = − w ( − ) s , η ( − ) , η ( − ) , η (+) , η (+) x s,l = x s,l = 0 for ∀ s, l . with γ η ( ± ) A = iw ( ± ) η ( ± ) A . ( BPS) x s,l = x s,l = x s,l = 0 for ∀ s, l . η (+) , η (+) , η (+) , η (+) γ η (+) A = iw (+) η (+) A . ( BPS) a (+) s,l = 0 for ∀ s > , l = − w ( − ) s , η ( ± ) , η ( ± ) , η ( ± ) , η ( ± ) x s,l = x s,l = x s,l = 0. for ∀ s, l with γ η ( ± ) A = iw ( ± ) η ( ± ) A . ( BPS)Table 4: We list the number of supersymmetry of BPS solutions (3.37), (3.48), (3.49)to preserve η (+) with γ η (+) = iw (+) η (+) at each point of moduli space with a (+) s,l = 0 for ∃ s > , l .Next we consider a case where x s,l = 0 for some positive s and l . This assumptionrestricts us to two case to study BPS solutions to preserve another Killing spinor as (i) η (+) , (ii) η ( − ) . They are all projected by γ . Note that the assumption excludes the cases η (+) (or η (+) ), η ( − ) , η ( − ) , which result in X = 0. (i) Let us first study BPS solutions topreserves another supersymmetry specified by η (+) which satisfies γ η (+) = iw (+) η (+) . Westudy the BPS solution of the matter fields since the constraint of the field strength canbe discussed in the same way as above. As in the above case, the matter BPS solutionshould 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 (+) , η (+) to be the same, w (+) = w (+) ,and X = X = 0 from (2.1). It is not possible to preserve η (+) or η (+) , which wouldmake all matter fields trivial, though it is possible to conserve η ( − ) projected by γ , inwhich case the non-trivial scalar field X and the field strength have to also be expandedby the highest weight states of su (2) L : x s,l , a (+) s,l = 0 for l = − w ( − ) s . This solutionautomatically has supersymmetry specified by η ( − ) as well. (ii) Let us consider the case forthe BPS solution to preserves η ( − ) which satisfies γ η ( − ) = iw ( − ) η ( − ) . This demands that X = 0 and x s,l , x s,l , a (+) s,l = 0 for l = − w ( − ) s . Notice that supersymmetry is enhancedwhen x s,l , a (+) s,l = 0 for all s >
0, in which case η ( − ) is preserved without any projection.Furthermore in order to preserve η ( − ) in addition it is necessary for X to vanish. In thiscase supersymmetry is enhanced so that η (+) is also preserved. We summarize this result20n Table 5. Parameter region. Preserved Killing spinors. Number of SUSY. x s,l , x s,l , x s,l , a (+) s,l : generic. η (+) γ η (+) = iw (+) η (+) . ( BPS) x s,l = x s,l = 0 for ∀ s, l . η (+) , η (+) γ η (+) A = iw (+) η (+) A . ( BPS) x s,l , x s,l , a (+) s,l = 0 for ∀ s > , l = − w ( − ) s , η ( − ) , η (+) x s,l = 0 for ∀ s, l . with γ η ( ± ) A = iw ( ± ) η ( ± ) A . ( BPS) x s,l = 0 for ∀ s > , l = − w ( − ) s η ( − ) , η (+) x s,l , a (+) s,l = 0 for ∀ s > , l ; x s,l = 0 for ∀ s, l . with γ η (+) = iw (+) η (+) . ( BPS) x s,l , a (+) s,l = 0 for ∀ s > , l = − w ( − ) s , η ( − ) , η ( − ) , η (+) , η (+) x s,l = x s,l = 0 for ∀ s, l . with γ η ( ± ) A = iw ( ± ) η ( ± ) A . ( BPS)Table 5: We list the number of supersymmetries of BPS solutions (3.37), (3.48), (3.49)to preserve η (+) with γ η (+) = iw (+) η (+) at each point of moduli space with x s,l = 0 for ∃ s > , l .Similarly we can count the number of supersymmetries of the BPS solution whichpreserves at least η ( − ) with projection (3.70). They are given by (3.82), (3.91), (3.92),which we write down again for convenience. X A = X s ≥ , | r |≤ s x s,r A e − i s +1 r t Y s,w ( − ) s,r ( θ, φ, ψ ) , (3.117) F = X s ≥ , | r |≤ s a ( − ) s,r c s,r tan w ( − ) r (cid:18) θ (cid:19) sin s − θ cos ( − r ψ + s ( w ( − ) φ − r t ) + α ( − ) s,r ) , (3.118) F = X s ≥ , | r |≤ s a ( − ) s,r c s,r tan w ( − ) r (cid:18) θ (cid:19) sin s − θ sin( − r ψ + s ( w ( − ) φ − r t ) + α ( − ) s,r ) . (3.119)Discussion how to count the number of supersymmetries can be done in a parallel way, sowe only present results of the table corresponding to Table 1, Table 4, Table 5 by Table 6. N = 8 SYM on R × S N = 8 SYM on R × S by carryingout dimensional reduction of N = 4 SYM on R × S so that the Hopf fiber direction ψ is degenerated. We present basic results obtained by this dimensional reduction for thispaper to be self-contained. From the metric of R × S we obtain that of R × S as ds R × S = − dt + µ − ( dθ + sin θdφ ) (4.1)21arameter region. Preserved Killing spinors. Number of SUSY. x s,r , x s,r , x s,r , a ( − ) s,r : generic. η ( − ) γ η ( − ) = iw ( − ) η ( − ) . ( BPS) x s,r = x s,r = 0 for ∀ s, r . η ( − ) , η ( − ) γ η ( − ) A = iw ( − ) η ( − ) A . ( BPS) x s,r , x s,r , a ( − ) s,r = 0 for ∀ s, r = − w (+) s , η ( − ) , η (+) x s,r = 0 for ∀ s, r . with γ η ( ± ) A = iw ( ± ) η ( ± ) A . ( BPS) x s,r , x s,r , x s,r = 0 for ∀ s > , r , η ( − ) . a ( − ) s,r = 0 for ∀ s, r . ( BPS) x s,r = 0 for ∀ s, r = − w (+) s η ( − ) , η (+) , 6 x s,r , a ( − ) s,r = 0 for ∀ s > , l ; x s,r = 0 for ∀ s, l . with γ η ( − ) = iw ( − ) η ( − ) . ( BPS) x s,r , a ( − ) s,r = 0 for ∀ s, r = − w (+) s , η ( − ) , η ( − ) , η (+) , η (+) x s,r = x s,r = 0 for ∀ s, r . with γ η ( ± ) A = iw ( ± ) η ( ± ) A . ( BPS) x s,r = x s,r = x s,r = 0 for ∀ s, r . η ( − ) , η ( − ) , η ( − ) , η ( − ) γ η ( − ) A = iw ( − ) η ( − ) A . ( BPS) x s,r = x s,r = 0 for ∀ s > , r , η ( − ) , η (+) . 8 x s,r = a ( − ) s,r = 0 for ∀ s, r . ( BPS) a ( − ) s,r = 0 for ∀ s, r = − w (+) s , η ( ± ) , η ( ± ) , η ( ± ) , η ( ± ) x s,r = x s,r = x s,r = 0 for ∀ s, r . with γ η ( ± ) A = iw ( ± ) η ( ± ) A . ( BPS) x s,r = 0 for ∀ s > , r , η ( − ) , η ( − ) , η (+) , η (+) . 16 x s,r = x s,r = a ( − ) s,r = 0 for ∀ s, r . ( BPS) x s,r = x s,r = x s,r = a ( − ) s,r = 0 for ∀ s, r . η ( ± ) , η ( ± ) , η ( ± ) , η ( ± ) . 32(Unique vacuum)Table 6: We list the number of supersymmetry of BPS solutions (3.82), (3.91), (3.92)which preserve η ( − ) with γ η ( − ) = iw ( − ) η ( − ) at each point of moduli space.22here we set µ = r , which is the inverse radius of S . For more detail, see Appendix B.2.Dimensional reduction for fields can be achieved by truncating the fields to leave the zeromodes of the Hopf fiber direction. ∂ ψ Φ = 0 , or ∂ Φ = 0 , where Φ is any four dimensionalfield. Accordingly the four dimensional gauge field is separated into the three dimensionalone and a scalar field in a way that A i = a i , A = φ, (4.2)where i = 0 , ,
2. Thus the gauge field strength become F = f , F = f , F µ = D µ φ, F = f − µφ, (4.3)where f µν = ∂ µ a ν − ∂ ν a µ + i [ a µ , a ν ] , D µ φ = ∂ µ φ + i [ a µ , φ ] with µ, ν = t, θ, φ . Note that thelocal Lorentz indices 0 , , t, θ, φ are now transformed to each otherby using the transition function of R × S given by (B.34).Performing this dimensional reduction to the N = 4 SYM action given by (2.2) weobtain the action of N = 8 SYM on R × S S = 1 g Z dt d Ω µ Tr (cid:20) − f µν f µν − D µ φD µ φ + µφf − µ φ − D µ X AB D µ X AB − µ X AB X AB + 12 [ X AB , φ ][ X AB , φ ] + 14 [ X AB , X CD ][ X AB , X CD ]+ i ( ψ A ) † γ µ D µ ψ A + µ ψ A ) † ρ t ψ A + i ( ψ A ) † [ φ, ψ A ] + ( ψ A ) † [ X AB , ( ψ B ) † ] + ψ A [ X AB , ψ B ] (cid:21) (4.4)where we set d Ω = dθ sin θdφ , λ A = ψ A , g = g πµ , and D µ is the covariant derivative interms 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 D µ f µν − i [ φ, D ν φ ] − i [ X AB , D ν X AB ] + µγ µν D µ φ = 0 , (4.5) D µ D µ φ + µf − µ φ + [ X AB , [ φ, X AB ]] = 0 , (4.6) D µ D µ X AB − µ X AB + [ φ, [ X AB , φ ]] + [ X CD , [ X AB , X CD ]] = 0 . (4.7)Conserved charges of this theory can be obtained by dimensional reduction. SU (4) R R-symmetry charge of the bosonic part is R C A = i Z d Ω 1 g Tr (cid:0) − X CB D X AB + D X CB X AB (cid:1) . (4.8)The energy is H = Z d Ω 1 g Tr (cid:18)
12 ( f i ) + 14 ( f ij ) + 12 | D φ | + 12 | ~Dφ | + 12 | D X AB | + 12 | ~DX AB | − µφf + 12 µ φ + 12 | [ X AB , φ ] | + 14 | [ X AB , X CD ] | + µ | X AB | (cid:19) . (4.9)23he angular momentums are P i = Z d Ω 1 g Tr (cid:18) f µ f iµ + D φD i φ + 12 ( D X AB D i X AB + D i X AB D X AB ) (cid:19) (4.10)where i = θ, φ .Under the truncation supersymmetries specified by ǫ (+) A are all broken, since they aredependent on the Hopf fiber direction ψ . The other supersymmetries specified by ǫ ( − ) A areall preserved, which we denote by ξ A . The determining equation of ξ A comes from (2.14),which is now given by ∂ t ξ A = − iµ ξ A , ∇ S θ ξ A = − iµ ρ θt ξ A , ∇ S φ ξ A = − iµ ρ φt ξ A , (4.11)and the solution is ξ A = e − i µ t e − i ρ θ e − ρ φ η A (4.12)where η A is a constant spinor. By using this the supersymmetry transformation rule of N = 8 SYM on R × S is obtained as∆ ξ A µ = i (( ξ A ) ∗ ρ µ ψ A − ( ψ A ) † ρ µ ξ A ) , ∆ ξ φ =( ξ A ) ∗ ψ A − ( ψ A ) † ξ A , ∆ ξ X AB = i ( − ε ABCD ξ C ψ D − ( ξ A ) ∗ ( ψ B ) † + ( ξ B ) ∗ ( ψ A ) † ) , ∆ ξ ψ A =( − iD µ φρ µ + 12 f µν ρ µν − µφρ ) ξ A − D µ X AB ρ µ − [ φ, X AB ])( ξ B ) ∗ − i [ X AB , X BC ] ξ C − X AB ( ρ µ ∇ S µ − iµ ρ )( ξ B ) ∗ . (4.13)Note that due to this reduction the global symmetry reduces from P SU (2 , |
4) to
P SU (2 | N = 8 SYM on R × S can be studied in the same manner as thecase of N = 4 SYM on R × S in § § N = 8 SYM on R × S . For this purpose we study a BPSstate which preserves a Killing spinor ǫ whose constant spinor η is projected in such away that γ η = iwη (4.14)where w = ± = 1 , , ,
4. The BPS relation of charges associated with the Killingspinor ǫ is µ − H = R + wJ φ . (4.15)24y performing the dimensional reduction to (3.74), we find f = − wD φ sec θ + D φ tan θ, D φ = − wf sec θ − ( f − µφ ) tan θ, [ X AB , X B ] = 12 i (cid:8) w sec θ ( D φ − i ( f − µφ )) − ( D φ + if ) tan θ (cid:9) δ A ,D X A = − iµ X A − wµD φ X A ,D X A = iw ( − cos θD X A + sin θ [ iφ, X A ]) , (4.16)where A = , , , . These are the most general 1 / N = 8 SYM on R × S with the fermionic fields trivial, which preserves ǫ with = 1 , , , / f = − w cos θ∂ φ, ∂ t φ = − wµ∂ φ φ,f = µφ − w sin θf , ∂ φ = − w cos θf , (4.17)( ∂ + iµ µw∂ φ ) X A = 0 , ( ∂ + iw cos θ∂ ) X A = 0 . (4.18)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 µν ∂ µ φ = 0. Under the BPS condition (4.17) thisbecomes ∂ t f = − wµ∂ φ f , µ∂ φ φ = tan θ∂ θ (sin θf ) . (4.19)By carrying out dimensional reduction for (3.82), (3.91), (3.92), we find X A = X s ≥ x sA e − iµ ( s + ) t Y s,ws ( θ, φ ) , (4.20) f = X s ≥ a s c s sin s − θ cos ( s ( wφ − µt ) + α s ) , (4.21) ∂ φ = X s ≥ a s c s sin s − θ sin( s ( wφ − µt ) + α s ) , (4.22)where x sA , a s , α s are integral constants with the range x sA ∈ C , a s ≥ , ≤ α s < π forall A = , , ; s ≥ , and Y s,ws ( θ, φ ) , c s are given in Appendix B.2. Thus the real adjointscalar field φ is determined as φ = φ − wµ X s ≥ a s s c s sin s θ cos( s ( wφ − µt ) + α s ) , (4.23)where φ is another integral constant taking real values, which parametrizes the vacua ofthe theory. The other components of field strength can be obtained from (4.17). This is Different from the four dimensional case, the Bianchi identity becomes redundant in the three dimen-sion. If the flux quantization condition is taken into account, not only x sA , a s but also φ are quantized ina suitable way.
25 set of general 1 / N = 8 SYM on R × S . Note that this result isconsistent with the BPS solutions obtained in [32], where BPS solutions of the N = 8 SYMwere obtained from those of N = 6 Chern-Simons (or ABJM) theory defined on R × S 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). H = 1 g µ X s ≥ (2 s + 1) Tr | x sA | + X s ≥ (1 + 2 s )2 s Tr( a s ) , (4.26) J θ =0 , (4.27) J φ = wg X s ≥ µ (2 s + 1) s Tr | x sA | + µ − X s ≥ (1 + 2 s )2 s Tr( a s ) , (4.28) R = µg X s ≥ (2 s + 1)Tr | x sA | , (4.29)which satisfies the BPS relation of charges given by (4.15).The number of supersymmetries preserved by the 1 / φ . We have done a systematic analysis of supersymmetric states in N = 4 SYM on R × S by setting the gaugino fields to zero. As a result we have found two sets of 1 /
16 BPS con-ditions and we have solved them completely when the bosonic fields are valued in Cartansubalgebra of a gauge group. We have precisely counted the number of supersymmetriespreserved by the BPS solutions varying the parameters of the solution. We have also ob-tained the most general 1 / N = 8 SYM on R × S with the precise numberof supersymmetries under the same assumptions by performing dimensional reduction.In this paper we have solved the 1 /
16 BPS solutions assuming that they are valuedin Cartan subalgebra of a gauge group. We pointed out that in the Cartan part the If we include the non-normalizable mode, the general solution (4.21), (4.23) becomes f = a ′ sin θ + X s ≥ a s c s sin s − θ cos ( s ( wφ − µt ) + α s ) , (4.24) φ = φ − wµ a ′ log sin θ + X s ≥ a s s c s sin s θ cos( s ( wφ − µt ) + α s ) . (4.25) The degeneracy of vacua in N = 8 SYM on R × S can be encoded in non-trivial holonomy in theHopf fiber direction in N = 4 SYM on R × S / Z k , where the Hopf fiber direction is orbifolded [33, 34]. x s , x s , x s , a s , φ : generic. η γ η = iwη . ( BPS) x s = x s = 0 for ∀ s . η , η γ η A = iwη A . ( BPS) x s , x s , x s = 0 for ∀ s > η . a s = 0 for ∀ s, r . ( BPS) x s = x s = x s = 0 for ∀ s . η , η , η , η γ η A = iwη A . ( BPS) x s = 0 for ∀ s > η , η . 8 x s = x s = a s = 0 for ∀ s . ( BPS) φ : generic, η , η , η , η . 16 x s = x s = x s = a s = 0 for ∀ s . (Vacua)Table 7: We list the number of supersymmetry of BPS solutions (4.20), (4.21), (4.23)which preserve η with γ η = iwη at each point of moduli space.gauge field and the matter ones decouple and they can be independently excited preservingsupersymmetry. It would be interesting to study more general BPS solutions by relaxingthis 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 restrictingattention to a special subsector as performed in [27].An important problem is to clarify a relation between 1 /
16 BPS states in the N = 4SYM and 1 /
16 BPS objects in type IIB supergravity on
AdS × S . It would be interestingto find out the counterparts of the BPS states constructed in this paper, especially those inwhich only electromagnetic field is turned on. Since charges of the supersymmetric statesfound in this paper can be of order N , the corresponding objects in the dual geometry canbe supersymmetric (dual) giant gravitons. General 1 /
16 BPS (dual) giant gravitons havebeen constructed in [35] by using the same technique to construct a general 1 / / E is realizedby the intersection of S and the zero locus of a polynomial of the form X n + n + n = E/R c n n n e − iEt z n z n z n (5.1)where R is the radius of AdS , z , z , z are coordinates of C into which S is embedded.(See also [37, 38, 39, 28, 40, 41, 42, 43].) Supersymmetric black holes have also been foundin [25, 24, 26], which have turned out to be 1 /
16 BPS. Reproducing behaviors of theseobjects from the N = 4 SYM side is an important issue in AdS /CFT duality.Another interesting direction is to study the N = 4 SYM on R × S / Z k by orbifoldingthe Hopf fiber direction so that the su (2) R algebra is broken and the global symmetry27lgebra of the system becomes psu (2 | psu (2 | psu (2 |
4) symmetry (calledbubbling geometries) and BPS objects in ten or eleven dimensional supergravity theorieshas extensively studied in [33, 34, 44, 45]. It would be attracting to pursue the correspon-dence of BPS spectra of both sides more. (See [13, 46, 47, 48, 49] for the study of thisdirection.)It would also be fascinating to study BPS states in other supersymmetric gauge theoriesdefined on R × S n as done in this paper, for example SYM on R × S constructed recentlyin [50]. (The case of N = 6 Chern-Simons (ABJM) theory on R × S was done in [32].)We leave these issues to future works. Acknowledgments
The author would like to thank S. Kim and S. Minwalla for useful comments for the firstversion of this paper.
A Convention
In this appendix we collect our convention used in this paper. In the flat space-time, themetric is given by g µν = diag( − , , ,
1) and SO (1 ,
3) gamma matrices are realized by γ µ = (cid:18) ρ µ ρ µ (cid:19) , γ = (cid:18) i − i (cid:19) . (A.1)Here ρ = iσ , ρ = σ , ρ = σ , where σ i is the Pauli matrices satisfying σ i σ j = δ ij + iε ijk σ k so that { γ µ , γ ν } = 2 g µν . Note that ρ µ becomes SO (1 ,
2) gamma matrices.In this paper we alway suppress spinor indices for simplicity. In a fermionic bilinear,spinor indices are contracted in the southwest-northeast manner. For example, ψ † γ µ χ = ψ † ˙ α γ µ ˙ αβ χ β = C ˙ α ˙ γ ψ † ˙ γ γ µ ˙ αβ χ β (A.2)where C ˙ α ˙ γ is the charge conjugation matrix. Note that γ αβ = − iδ αβ . We also use notationsuch that γ µν = 12 ( γ µ γ ν − γ ν γ µ ) . (A.3) B Basics on S S used in this paper. It is convenient for us to realize S by SU (2) group. Any SU (2) element denoted by g can be parametrized by using thepolar coordinates of S as g = e − i φ σ e − i θ σ e − i ψ σ = cos θ e − i ( φ + ψ )2 − sin θ e i ( − φ + ψ )2 sin θ e − i ( − φ + ψ )2 cos θ e i ( φ + ψ )2 ! (B.1)28here the parameter region is 0 ≤ θ < π, ≤ φ < π, ≤ ψ < π . On the other hand,any point on S with radius r can embedded into C by using SU (2) elements z i = rg i = r cos θ e − i ( φ + ψ )2 sin θ e − i ( − φ + ψ )2 ! (B.2)where z , z are complex coordinates of C . Therefore the metric of S with radius r isgiven by ds S = | dz | + | dz | = r dθ + sin θdφ + ( dψ + cos θdφ ) ) (B.3)where we used (B.2) to obtain the second equation. The metric of R × S is ds R × S = − dt + ds S . (B.4)Therefore a local orthonormal frame is given by e = dt, e = r dθ, e = r θdφ, e = r dψ + cos θdφ ) . (B.5)Vielbein of this geometry reads e t = 1 , e θ = r , e φ = r θ, e φ = r θ, e ψ = r . (B.6)The inverse of vielbein is e t = 1 , e θ = 2 r , e φ = 2 r sin θ , e ψ = 2 r ( − cot θ ) , e ψ = 2 r . (B.7)We interchange local Lorentz indices and global space-time ones by using these.The 1-form connection of this geometry can be determined as follows. ω = 2 r (cot θ e − e ) , ω = 1 r e , ω = − r e . (B.8)Others components are trivial. This reads ω , = 2 r cot θ, ω , = − r , ω , = 1 r , ω , = − r . (B.9)The partial derivatives with the local Lorentz indices and those with the global coordinatesare related by ∂ = 2 r ∂ θ , ∂ = 2 r ( 1sin θ ∂ φ − cot θ∂ ψ ) , ∂ = 2 r ∂ ψ , (B.10) ∂ θ = r ∂ θ , ∂ φ = r θ∂ + cos θ∂ ) , ∂ ψ = r ∂ . (B.11)29 benefit to realize S as SU (2) group is that so (4) Killing action on S is realized bythe su (2) L × su (2) R algebraic action.ˆ L a g = − σ a g, ˆ R a g = 12 gσ a . (B.12)Thanks to this definition we can easily show that h ˆ L a , ˆ L b i = iǫ abc ˆ L c , [ ˆ R a , ˆ R b ] = iǫ abc ˆ R c , [ ˆ L a , ˆ R b ] = 0 . (B.13)The explicit coordinate expressions of these operators are given byˆ L = i (sin φ∂ θ − csc θ cos φ∂ ψ + cot θ cos φ∂ φ ) , ˆ L = i ( − cos φ∂ θ − csc θ sin φ∂ ψ + cot θ sin φ∂ φ ) , ˆ L = − i∂ φ , (B.14)ˆ R = i (sin ψ∂ θ + cot θ cos ψ∂ ψ − csc θ cos ψ∂ φ ) , ˆ R = i (cos ψ∂ θ − cot θ sin ψ∂ ψ + csc θ sin ψ∂ φ ) , ˆ R = i∂ ψ . (B.15)For later convenience to construct the spherical harmonics, we defineˆ L ± = ˆ L ± i ˆ L , ˆ R ± = ˆ R ± i ˆ R . (B.16)Then we can show that h ˆ L , ˆ L ± i = ± ˆ L ± , [ ˆ L + , ˆ L − ] = 2 ˆ L , (B.17) h ˆ R , ˆ R ± i = ± ˆ R ± , [ ˆ R + , ˆ R − ] = 2 ˆ R , (B.18)and ˆ L ± = e ± iφ ( i cot θ∂ φ ± ∂ θ − i θ ∂ ψ ) , (B.19)ˆ R ± = e ∓ iψ ( i cot θ∂ ψ ∓ ∂ θ − i θ ∂ φ ) . (B.20)Note that ˆ L = ˆ R = ∂ θ + cot θ∂ θ + 1sin θ ( ∂ φ + ∂ ψ − θ∂ φ ∂ ψ ) (B.21)which is the same as the Laplacian of S with radius r = 2 acting on a scalar field.30 .1 The scalar spherical harmonics on S In this subsection we construct the scalar spherical harmonics on S . Since the sphericalharmonics are roots of the su (2) L × su (2) R algebra, we can construct them by the standardalgebraic method. ˆ L Y s,l ,r = ˆ R Y s,l ,r = s ( s + 1) Y s,l ,r , (B.22)ˆ L Y s,l ,r = l Y s,l ,r , ˆ R Y s,l ,r = r Y s,l ,r , (B.23)where s is non-negative half integer and | l | , | r | ≤ s . Firstly we determine the sphericalharmonics of the highest or lowest weight of su (2) R by solving R ± Y s,l , ± s = 0 as well, whichis given in the polar coordinates by (cid:18) ∂ θ ∓ θ l − s cot θ (cid:19) y s,l , ± s ( θ ) = 0 (B.24)where Y s,l , ± s = y s,l , ± s ( θ ) e i ( l φ ∓ sψ ) . The solution is given by Y s,l , ± s = c s,l tan ± l (cid:18) θ (cid:19) sin s θe i ( l φ ∓ sψ ) , (B.25)where c s,l is a normalization constant given by c s,l = 1 π s Γ(2 s + 2)2 s +1 Γ( l + s + 1)Γ( − l + s + 1) (B.26)which we determine by demanding R d Ω | Y s,l , ± s | = 1.In the same way, we can construct the spherical harmonics of the highest or lowestweight of su (2) L by solving L ± Y s, ± s,r = 0, which is (cid:18) ∂ θ ∓ θ r − s cot θ (cid:19) y s, ± s,r ( θ ) = 0 (B.27)where Y s, ± s,r = y s, ± s,r ( θ ) e i ( − r ψ ± sφ ) . The solution thereof is given by Y s, ± s,r = c s,r tan ± r (cid:18) θ (cid:19) sin s θe i ( − r ψ ± sφ ) , (B.28)where c s,r is determined to satisfy R d Ω | Y s, ± s,r | = 1 so that c s,r = 1 π s Γ(2 s + 2)2 s +1 Γ( r + s + 1)Γ( − r + s + 1) . (B.29)A general scalar spherical harmonics, Y s,l ,r , can be obtained by acting lowering oper-ators suitably. Y s,l ,r = s Y n = l +1 ˆ L − p s ( s + 1) − n ( n − ! Y s,s,r (B.30)where we normalized the spherical harmonics to satisfy the orthonormal relation Z d Ω( Y s ′ ,l ′ ,r ′ ) ∗ Y s,l ,r = δ ss ′ δ l l ′ δ r r ′ . (B.31)31 .2 Reduction to S In this subsection we extract information on S from the results obtained in the previoussubsections by degenerating the Hopf fiber direction ψ of S . The metric of R × S is ds R × S = − dt + 1 µ ( dθ + sin θdφ ) (B.32)where µ is the inverse radius of S , which is related to the S radius by µ = r . A localorthonormal frame is e = dt, e = µ − dθ, e = µ − sin θdφ. (B.33)From this we find vielbein of R × S as e t = 1 , e θ = µ − , e φ = µ − sin θ,e t = 1 , e θ = µ, e φ = µ sin θ . (B.34)One-form connection is ω S = µ cot θe . (B.35)Other components are zero. so (3) Killing actions on S are given by ˆ L i with ∂ ψ eliminated. It is automatic that ˆ L i restricted on S form su (2) algebra. By using the operators ˆ L i the S spherical harmonicsare defined as root vectors of the su (2) algebra.ˆ L Y s,s = s ( s + 1) Y s,s , ˆ L Y s,s = l Y s,s , (B.36)where s is non-negative half integer and | s | ≤ s . The S spherical harmonics can beobtained from S spherical harmonics by taking a zero mode of ψ direction. Y s,s = r π Y s,s ,r =0 . (B.37)For instance, the S spherical harmonics of the highest or lowest weight are given by Y s, ± s = c s sin s θe ± isφ , (B.38)where c s = p π c s, = q Γ(2 s +2) π s +2 Γ( s +1) . A general scalar spherical harmonics is given by Y s,s = s Y n = s +1 ˆ L − p s ( s + 1) − n ( n − ! Y s,s . 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