1231

Superspace models for S3

D.G.C. McKeon

Abstract: The simplest supersymmetric extension of the group SO(4) is discussed. Thesuperalgebra is realized in a superspace whose Bosonic subspace is the surface of a sphereS3 embedded in four-dimensional Euclidean space. By using Fermionic coordinates inthis superspace, which are chiral symplectic Majorana spinors, it proves possible to devisesuperfield models involving a complex scalar, a pair of chiral symplectic Majorana spinors,and a complex auxiliary scalar. Kinetic terms involve operators that are isometry generatorson S3.

PACS No.: 11.30.Pb

Résumé : Nous étudions la plus simple extension supersymétrique du groupe SO(4). Nousréalisons la superalgèbre dans un superespace dont le sous-espace barionique est la surfaced’une sphère S3 enchâssée dans un espace euclidien à quatre dimensions. Utilisant dans cesuperespace des coordonnées fermioniques qui sont des spineurs chiraux symplectiques deMajorana, il est possible de générer des modèles de superchamps impliquant un scalairecomplexe, une paire de spineurs chiraux symplectiques de Majorana et un scalaire auxiliairecomplexe. Les termes cinétiques impliquent des opérateurs qui sont les générateursd’isométrie sur S3.

[Traduit par la Rédaction]

1. Introduction

Supersymmetry on spaces of constant curvature in various dimensions has recently been investigated[1–5]. This is a natural extension of the analysis of models on spherical surfaces [6–10]. Superspacemodels have been formulated for the supersymmetric extensions of the groups SO(2, 1) [2,3], SO(2, 2)[4], and SO(3) [2,5]. The models all employ generators of the isometry group of the Bosonic subspacewhose curvature is constant; this is different from the analysis in which a supergravity model is spe-cialized by fixing the background metric, though it is likely that the two approaches are equivalent [3].Indeed, it follows more closely an approach used by Dirac in conjunction with spinors in AdS4 [11].

In this paper, we consider how to construct a superspace model associated with the supersymmetricextension of the group SO(4). We provide a realization of the simplest supersymmetric extension ofSO(4) in a superspace, then formulate a supersymmetric model using superfields in this superspace. Todo this, we use a pair of chiral symplectic Majorana spinors as Grassmann coordinates in superspace. Thisensures that the superfields depend only on two Fermionic degrees of freedom, so that the componentfields consist of a pair of complex scalars and a pair of chiral symplectic Majorana spinors, each witha realistic kinetic term.

Received 7 May 2003.Accepted 18 July 2003. Published on the NRC Research Press Web site at http://cjp.nrc.ca/on 31 October 2003.

D.G.C. McKeon. Department of Applied Mathematics, University of Western Ontario, London, ON N6A 5B7,Canada (e-mail: dgmckeo2@uwo.ca).

Can. J. Phys. 81: 1231–1237 (2003) doi: 10.1139/P03-095 © 2003 NRC Canada

1232 Can. J. Phys. Vol. 81, 2003

2. Supersymmetric model on S3The surface S3 is the surface of a sphere defined in four-dimensional Euclidean space

xµxµ = a2 (1)

The isometries on this sphere are generated by the operator

Lµν = −xµ∂ν + xν∂µ (2)

whose algebra is

[Lµν, Lλσ

] = δµλLνσ − δνλLµσ + δνσLµλ − δµσLνλ (3)

A set of four Dirac matrices defined in this space are

γ µ =(

0 σµ

σµ 0

)(4)

where σµ = (1, iτ i), σµ = (1,−iτ i) where the τ i are the usual Pauli spin matrices. With this

convention, if C = iγ 4γ 2 =(τ 2 00 −τ 2

), then

CγµC−1 = +γ µT (5)

Furthermore, we define

γ 5 = γ 1γ 2γ 3γ 4 =( −1 0

0 1

)(6)

and

�αβ = −1

4

[γ α, γ β

] =(σαβ 0

0 σαβ

)(7)

where

σαβ = −1

4

(σασβ − σβσα

) = +1

2εαβγ δσ γ δ (8a)

σαβ = −1

4

(σασβ − σβσα

) = −1

2εαβγ δσ γ δ (8b)

with ε1234 = +1. It is evident that we have the (anti) commutation relations

{γ µ, γ ν

} = 2δµν (9a)[�µν,�λσ

] = δµλ�νσ − δνλ�µσ + δνσ�µλ − δµσ�νλ (9b)

and also the Fierz identities

�µνij �

µνk� = −1

2�µνi� �

µνkj − 3

4

(δi�δkj + γ 5

i�γ5kj

)(10a)

δij δk� = 1

4

[δi�δkj + γ 5

i�γ5kj + γ

µi�γ

µkj −

(γ µγ 5

)i�

(γ µγ 5

)kj

− 2�µνi� �µνkj

](10b)

©2003 NRC Canada

McKeon 1233

If Q is a spinor that transforms by the equation

Q → exp(ωµν�µν

)Q (11)

then it can be shown that QC transforms in the same way where

QC = CQ†T (12)

We note that as (QC)C = −Q, Q cannot satisfy the Majorana condition Q = QC . However, it ispossible to define a pair of symplectic Majorana spinors

Q1 = Q+QC√2

= +(Q2)C (13a)

Q2 = Q−QC√2

= −(Q1)C (13b)

If Q is taken to be a spinorial generator, then the identities of (9) and (10) can be used to show that thesimplest supersymmetric generalization of the SO(4) algebra of (3) is

{Q,Q†

}= −�µνJµν + Z + Z5γ5 (14a)[

Jµν,Q] = −�µνQ (14b)

[Z,Q] = 1

2Q (14c)

[Z5,Q] = 1

2γ5Q (14d)

with all other (anti) commutators being zero, save for[Jµν, J λσ

]whose form is that of (3).

It is apparent that if the four-component spinor Q is decomposed into two two-component spinorsq and r(q

0

)=

(1 − γ5

2

)Q,

(0r

)=

(1 + γ5

2

)Q (15)

then from (14), we can obtain the closed subalgebra

{q, q†

}= −σµνJµν + ζ (ζ = Z − Z5) (16a)[

Jµν, q] = −σµνq (16b)

[ζ, q] = −q (16c)

In fact though, for the algebra of (16) to be consistent with the duality relation of (8a), it is necessaryto replace (16a) and (16b) by

{q, q†

}= −σµνKµν + ζ (17a)[

Kµν, q] = −σµνq (17b)

where

Kµν = 1

2

(Jµν + 1

2εµνλσ J λσ

)= 1

2εµνλσKλσ (18)

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1234 Can. J. Phys. Vol. 81, 2003

with

[Kµν,Kλσ

] = δµλKνσ − δνλKµσ + δνσKµλ − δµσKνλ (19)

We now note that

(1 − γ5

2

)QC =

(qC0

)has

qC = τ 2q†T = τ 2q∗ (20a)

so that

q†C = −q†τ2 = −q̃ (20b)

We now consider

q1 = q + qC√2

= +(q2)C (21a)

q2 = q − qC√2

= −(q1)C (21b)

In terms of q1 and q2, the superalgebra of (16c) and (17) becomes

{q1, q̃1} = σµνKµν = − {q2, q̃2} (22a)

{q1, q̃2} = ζ = − {q2, q̃1} (22b)[Kµν, qa

] = −σµνqa (a = 1, 2) (22c)

[ζ, q1] = q2 (22d)

[ζ, q2] = q1 (22e)

The consistency of (22) can be proven directly from the relations

σµ† = σµ (23a)

τ 2σµτ 2 = σµT (23b)

σµν† = −σµν (23c)

τ 2σµντ 2 = −σµνT (23d)

σµij σ

µk� = 2δi�δkj (23e)

σµνij σ

µνk� = −2δi�δkj + δij δk� (23f)

It is now possible to realize the algebra of (22) in a superspace. A pair of chiral symplectic Majoranaspinors θ1 and θ2 constitute the Grassmann portion of this superspace; the remaining components arethe coordinates xµ on the surface of the sphere S3 defined by x2 = a2. It is possible to realize the

©2003 NRC Canada

McKeon 1235

superalgebra of (22) with√

2q1 = σµxµ∂

∂θ̃2+ σµ∂µθ2 (24a)

√2q2 = −σµxµ ∂

∂θ̃1+ σµ∂µθ1 (24b)

√2q̃1 = − ∂

∂θ2σµxµ + θ̃2σ

µ∂µ (24c)

√2q̃2 = ∂

∂θ1σµxµ + θ̃1σ

µ∂µ (24d)

ζ = θ̃2∂

∂θ̃1+ θ̃1

∂

∂θ̃2(24e)

Jµν = −xµ∂ν + xν∂µ (24f)

It is interesting that Jµν in (24f) contains no “spin part” dependent on θa . We also note that verifying(22c) involves using the relation

σµσνσλ = δµνσλ − δµλσ ν + δνλσµ − εµνλσ σσ (25)

We note now that if

R21 = x2 − θ̃2θ2 = R

2†2 (26a)

R22 = x2 + θ̃1θ1 = R

2†1 (26b)

then[q1, R

21

]= 0 =

[q2, R

22

](27)

Furthermore, if

'1 = xµ∂µ + θ̃2∂

∂θ̃2(28a)

'2 = xµ∂µ + θ̃1∂

∂θ̃1(28b)

then

[q,'1] = 0 = [q2,'2] (29)

If now

(1 (x, θ2) = φ(x)+ iψ̃2(x)θ2 + i

2F(x)θ̃2θ2 (30)

(φ, F — complex scalars and ψ2 — spinor) then

S1 =∫

d4x

∫d2θ1δ(θ1)

∫d2θ2δ

(R2

1 − a2)(1 (x, θ2) q̃1q1(1 (x, θ2) (31)

is invariant under supersymmetry transformations generated by q1. (We take δ(θ1) = θ̃1θ and∫d2θ1θ̃1θ1 = 1.) Using the equations

q̃1q1 = −x2 ∂

∂θ2

∂

∂θ̃2+ θ̃2θ2

[1

x2

(1

2JµνJµν + 2x · ∂ + (x · ∂)2

)]

− 2x · ∂ + 2θ̃2(2x · ∂ + 4 − 2σµνJµν

) ∂

∂θ̃2(32)

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1236 Can. J. Phys. Vol. 81, 2003

and

δ(R2

1 − a2)

= δ(x2 − a2

)− θ̃2θ2δ

′ (x2 − a2)

= δ(x2 − a2

) [1 + θ̃2θ2

1

x2

(1

2x · ∂ + 1

)](33)

we arrive at

S1 = 1

2

∫d4xδ

(x2 − a2

) [1

x2 φ

(1

2JµνJµν − ω2

)φ + 1

2x2F 2

+ψ̃2(−σµνJµν + 2

)ψ2 + i(1 − ω)φF

](34)

To define (1 off of the surface x2 = a2, we have imposed the condition

'1(1 (x, θ2) = ω(1 (x, θ2) (35)

so that

(x · ∂ − ω)φ = (x · ∂ − ω + 1)ψ2 ≡ (x · ∂ − ω + 2)F = 0 (36)

By (29), the condition (35) is invariant under supersymmetry transformations generated by q1.The action S1 by itself is not Hermitian; we see that

S†1 =

∫d4x

∫d2θ2δ (θ2)

∫d2θ1δ

(R2

2 − a2)(2 (x, θ1) q̃2q2(2 (x, θ1) ≡ S2 (37)

where

(2 (x, θ1) = φ∗(x)+ iψ̃1θ1 + i

2F ∗θ̃1θ1 = (

†1 (x, θ2) (38)

In terms of component fields, one can show that

S2 = 1

2

∫d4xδ

(x2 − a2

) [1

x2 φ∗(

1

2JµνJµν − ω2

)φ∗ + 1

2x2F ∗2

+ψ̃1(σµνJµν − 2

)ψ1 − i(1 − ω)φ∗F ∗] (39)

An action for the free fields φ, F , ψ1, and ψ2 = − (ψ1)C that is both Hermitian and supersymmetric is

S = i (S1 − S2) (40)

Interactions of the form SN = S1N + S2N with

S1N = λN

∫d4x

∫d2θ1δ(θ1)

∫d4θ2δ

(R2

1 − a2)(N1 (x, θ2) (N = 2, 3, 4 . . .) (41a)

S2N = S†1N = λ∗

N

∫d4x

∫d2θ2δ (θ2)

∫d2θ1δ

(R2

2 − a2)(N2 (x, θ1) (41b)

can be defined. It is curious that on the surface S3, the couplings λN in (41) are dimensionless for allN = 2, 3 . . ..

©2003 NRC Canada

McKeon 1237

3. Discussion

Using features employed in generating superfield models on AdS2 [2,3], S2 [2,5], and AdS3 [4],we have been able to formulate a superfield model on the surface S3. The space S3 is treated as beinga surface embedded in four-dimensional Euclidean space and kinetic operators involve the isometrygenerators on this surface.

Extending this construction to formulate superfield models on S4 is complicated by the absence of

chiral spinors in five-dimensional Euclidean space. Possibly, the projection operators 12

(1 ± γ · x/√a2

)can be used on the surface S4 to reduce the number of independent components of spinors in the em-bedding space. (These projection operators have many of the properties of chiral projection operatorson S4 [12,13].)

Computing radiative effects and analyzing their renormalizability using the interactions SN is ob-viously of interest, especially since each of the couplings λN is dimensionless.

Acknowledgements

The author would like to thank T.N. Sherry and C. Schubert for helpful discussions and the NaturalSciences and Engineering Research Council of Canada for financial support. R. and D. MacKenzie werehelpful as well.

References

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©2003 NRC Canada