Five-dimensional N = 1 AdS superspace: Geometry, off-shell multiplets
  and dynamics
arXiv:0704.1185 [hep-th]
April, 2007
Five-dimensional N = 1 AdS superspace:
Geometry, off-shell multiplets and dynamics
Sergei M. Kuzenko1 and Gabriele Tartaglino-Mazzucchelli2
School of Physics M013, The University of Western Australia
35 Stirling Highway, Crawley W.A. 6009, Australia
Abstract
As a step towards formulating projective superspace techniques for supergrav-
ity theories with eight supercharges, this work is devoted to field theory in five-
dimensional N = 1 anti-de Sitter superspace AdS5|8 = SU(2,2|1)/SO(4, 1) × U(1)
which is a maximally symmetric curved background. We develop the differential ge-
ometry of AdS5|8 and describe its isometries in terms of Killing supervectors. Vari-
ous off-shell supermultiplets in AdS5|8×S2 are defined, and supersymmetric actions
are constructed both in harmonic and projective superspace approaches. Several
families of supersymmetric theories are presented including nonlinear sigma-models,
Chern-Simons theories and vector-tensor dynamical systems. Using a suitable coset
representative, we make use of the coset construction to develop an explicit real-
ization for one half of the superspace AdS5|8 as a trivial fiber bundle with fibers
isomorophic to four-dimensional Minkowski superspace.
1kuzenko@cyllene.uwa.edu.au
2gtm@cyllene.uwa.edu.au
http://arxiv.org/abs/0704.1185v3
http://arxiv.org/abs/0704.1185
Contents
1 Introduction 2
2 Covariant derivatives 3
3 Killing supervectors 8
4 Harmonic superspace approach 10
4.1 Analytic multiplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2 Harmonic action principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5 Projective superspace approach 16
5.1 Projective multiplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2 Projective action principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6 Dynamical systems in projective superspace 25
6.1 Projective multiplets revisited . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.2 Projective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.3 Nonlinear sigma-models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.4 Vector multiplet and Chern-Simons couplings . . . . . . . . . . . . . . . . 32
6.5 Tensor multiplet and vector-tensor couplings . . . . . . . . . . . . . . . . . 34
7 Coset space realization 35
7.1 Coset representative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
7.2 SO(4,1)×U(1) covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.3 Representation of covariant derivatives . . . . . . . . . . . . . . . . . . . . 40
7.4 Torsion and curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
A 5D Conventions 43
1 Introduction
In four-dimensional N = 2 Poincaré supersymmetry, there exist two powerful for-
malisms to construct off-shell manifestly supersymmetic actions: harmonic superspace
[1, 2] and projective superspace [3, 4, 5, 6]. Both approaches make use of the superspace
R4|8 × S2 and its supersymmetric subspaces, which were introduced for the first time by
Rosly [7] who built on earlier ideas due to Witten [8]. Both approaches can naturally be
extended to the case of d-dimensional supersymmetry with eight supercharges, for d ≤ 6,
where the appropriate flat superspace with auxiliary bosonic dimensions is Rd|8 × S2.
Specifically, the harmonic superspace formulations were developed in [9, 10] for d = 5,
and in [11] for d = 6. The projective superspace formulations were developed in [10] for
d = 5, and in [12, 13] for d = 6.
In projective superspace, off-shell multiplets are reasonably short and can readily be
expressed in terms of 4D N = 1 superfields. The latter property is very appealing from
the point of view of brane(-world) models. It is also expected [14, 15] that projective
superspace should be relevant in the context of hybrid string theory [16]. For these
and similar possible applications, one actually needs projective superspace techniques for
supergravity. So far, to the best of our knowledge, the projective superspace approach
has been mastered only in the flat case.
In harmonic superspace, the prepotential structure of 4D N = 2 supergravity is well
understood [17, 2], and similar constructions are clearly applicable in five and six dimen-
sions, see [18] for the six-dimensional case. What is still missing here, in our opinion, is
a properly incorporated covariant formalism of differential geometry for superfield super-
gravity, which should be similar in spirit to the famous Wess-Zumino approach to (the old
minimal formulation for) 4D N = 1 supergravity reviewed in [19]. In four-dimensional
N = 1 supergravity, it has been recognized for a long time that the most efficient ap-
proach to superfield supergravity occurs if one merges together and uses, depending on a
concrete application, both the covariant and prepotential techniques [20, 21].
Unlike the purely prepotential approach pursued in [17, 2], this paper is targeted at
(making the first step towards) developing covariant superfield techniques for supergravity
theories with eight supercharges. Our point of departure is as follows. It is known that all
information about off-shell supergravity formulations (including the structure of possible
matter multiplets) is encoded in the corresponding algebra of covariant derivatives. We
would like to use only this input and try to develop techniques to construct supersymmet-
ric actions both in the harmonic and projective settings. In this paper we consider one
particular supergravity background – five-dimensional N = 1 anti-de Sitter superspace,
AdS5|8, and explicitly develop harmonic and projective formulations in a covariant fashion
using only the language of differential geometry. We believe that similar ideas should be
applicable for a general supergravity background, as well as in four and six space-time
dimensions. In particular, the case of 4D N = 2 anti-de Sitter superspace1 can be treated
similarly.
This paper is organized as follows. In section 2 we derive the algebra of the covariant
derivatives for 5D N = 1 anti-de Sitter superspace by solving the Bianchi identities. In
section 3 the isometries of AdS5|8 are realized in terms of Killing supervectors. In section 4
we introduce analytic multiplets over the harmonic superspace AdS5|8×S2 and formulate
the harmonic superspace action. Various projective multiplets are defined in section 5, as
well as the projective superspace action is formulated. A remarkable feature of this super-
symmetric action is that it is uniquely determined by two independent requirements: (i)
projective invariance; (ii) invariance under the isometry group SU(2, 2|1). Some impor-
tant examples of dynamical systems in the AdS projective superspace are given in section
6. An explicit coset construction for one half of AdS5|8 (Poincaré chart) is elaborated in
section 7. Our 5D notation and conventions are collected in Appendix A.
2 Covariant derivatives
In this section, we develop the differential geometry of five-dimensional N = 1 anti-de
Sitter superspace, AdS5|8. This is a supersymmetric version of spaces of constant curvature
and, similar to all symmetric spaces, it can be realized as a coset space, specifically
AdS5|8 = SU(2,2|1)/SO(4,1)×U(1). Group-theoretical aspects of AdS5|8 will be discussed
in section 7.
Let zM̂ = (xm̂, θ
i ) be local bosonic (x) and fermionic (θ) coordinates parametrizing
AdS5|8, where m̂ = 0, 1, · · · , 4, µ̂ = 1, · · · , 4, and i = 1, 2. The Grassmann variables θµ̂i
are assumed to obey a standard pseudo-Majorana reality condition. Since the holonomy
group of AdS5|8 is SO(4, 1) × U(1), the superspace covariant derivative D
= (Dâ,Diα̂)
can be chosen to have the form
b̂ĉM
J + Ω
β̂γ̂ M
. (2.1)
1The 4D N = 2 anti-de Sitter superspace was studied in detail in [22] where a manifestly supersym-
metric formulation for the off-shell 4D N = 2 anti-de Sitter higher spin supermultiplets [23] was given.
A few years later, some formal aspects of this superspace were also discussed in [24].
Here E
M̂(z)∂
is the supervielbein, with ∂
= ∂/∂zM̂ , J the Hermitian gen-
erator of the group U(1), M
the generators of the Lorentz group SO(4, 1), and Φ
and Ω
b̂ĉ(z) the corresponding connections. The Lorentz generators with vector indices
) and spinor indices (M
) are related to each other by the rule:
)α̂β̂M
, see Appendix A for more details regarding our 5D notation and
conventions. The generators of the holonomy group act on the covariant derivatives as
follows:
[J,Diα̂] = J ijD
, (2.2)
,Diγ̂] =
εγ̂α̂Diβ̂ + εγ̂β̂D
. (2.3)
The Hermitian matrix J ij should be traceless, J
i = 0, in order to preserve the pseudo-
Majorana condition enjoyed by the covariant derivatives. The latter condition is equiva-
lent to the fact that the isotensors2 J ij = εjkJ ik and Jij = εikJ
j are symmetric, J
ij = J ji,
Jij = Jji. The fact that J
j is Hermitian, can be seen to be equivalent to (J
ij)∗ = −Jij .
The algebra of covariant derivatives can be reconstructed if we impose the following
two requirements: (i) the torsion tensor is covariantly constant;3 (ii) the group SO(4, 1)×
U(1) belongs to the automorphism group. These requirements lead, in particular, to the
ansätze:
{Diα̂,D
} = −2iεijD
+ xεijε
J + f ijM
, (2.4)
[Dâ,Diα̂] = C ij(Γâ)
, (2.5)
where
= (Γâ)
Dâ , (2.6)
and x is a constant parameters, f ij = f ji, C ij is a 2×2 matrix. Eq. (2.5) can be rewritten
in the equivalent form
,Diγ̂] = −2C ij
εγ̂α̂Dj
, (2.7)
Note that setting x = m = C ij = 0 gives the flat supersymmetry algebra, see. e.g., [10].
The (covariantly) constant parameters x, f ij and C ij in (2.4) and (2.5) turn out to
be considerably constrained on general grounds. Firstly, the tensor f ij must be invariant
under the action of J ,
Jf ij = (J ikf
kj + J
f ik) = 0 ⇐⇒ f ij = mJ ij , (2.8)
2Two-component indices i, j are raised and lowered using the SL(2,C)-invariant antisymmetric tensors
εij and εij normalized by ε
ikεkj = δ
j and ε
12 = ε21 = 1.
3Then, in accordance with Dragon’s theorem [26], the curvature tensor is covariantly constant.
with m a constant parameter. Secondly, we should take care of reality conditions such as
(Diα̂F )∗ = −(−1)ǫ(F )Dα̂i F ∗ , (2.9)
where ǫ(F ) is the Grassmann parity of F . They imply that
x = x , m = m . (2.10)
and C ij is anti-Hermitian,
C† = −C , C = (C ij) . (2.11)
Of course, we should also guarantee the fulfillment of the Bianchi identities, and this
proves to lead to additional restrictions on the parameters. In particular, the dimension-
3/2 Bianchi identity
[Diα̂, {D
,Dkγ̂}] + [D
, {Dkγ̂ ,Diα̂}] + [Dkγ̂ , {Diα̂,D
}] = 0 (2.12)
can be shown to imply
C ij =
ωJ ij , ω =
. (2.13)
Imposing the dimension-2 Bianchi identity
[Dâ, {Diα̂,D
}] + {Diα̂, [D
,Dâ]} − {Dj
, [Dâ,Diα̂]} = 0 (2.14)
leads, in particular, to
[Dâ,Db̂] = −
εij(Γb̂)
α̂β̂ [Dâ, {Diα̂,D
εij(Γb̂)
{Diα̂, [D
,Dâ]} − {Dj
, [Dâ,Diα̂]}
, (2.15)
and then
[Dâ,Db̂] =
mωJ2M
, (2.16)
where
J2 ≡ − 1
J ijJij . (2.17)
Another consequence of the dimension-2 Bianchi identity (2.14) is
ω = − 1
m . (2.18)
As a result, all the parameters in (2.4) and (2.5) have been expressed in terms of ω. With
the above conditions taken into account, the remaining dimension-5
Bianchi identity
[Dâ, [Db̂,D
α̂]] + [Db̂, [D
α̂,Dâ]] + [Diα̂, [Dâ,Db̂]] = 0 , (2.19)
and dimension-3 Bianchi identity
[Dâ, [Db̂,Dĉ]] + [Db̂, [Dĉ,Dâ]] + [Dĉ, [Dâ,Db̂]] = 0 , (2.20)
are satisfied identically.
Let us summarise the results obtained. The covariant derivatives for AdS5|8 obey the
algebra
{Diα̂,D
} = −2iεijD
− 3ωεijε
J − 4ωJ ijM
, (2.21a)
[Dâ,Diβ̂] =
ωJ ij(Γâ)
, (2.21b)
[Dâ,Db̂] = −ω
. (2.21c)
It is useful to rewrite (2.21b) in the equivalent form
,Diγ̂] = −iωJ ij
εγ̂α̂Dj
. (2.22)
As follows from (2.21c), the bosonic body of the superspace is characterised by a constant
negative curvature, and therefore it is AdS5. Indeed, since J
j is Hermitian and traceless,
we have
J ij = J
I(σI)
j =⇒ J2 = −
J ijJij =
JIJJ tr(σIσJ) = J
IJI > 0 , (2.23)
where JI is a real tree-vector, with I = 1, 2, 3, and σI are the Pauli matrices. In section
7, we will give an explicit (coset space) realization of the geometry described.
Up to an isomorphism, one can always choose J ij ∝ (σ3)ij , and hence J11 = J22 = 0.
Then, it follows from (2.21a–2.21c) that each of the two subsets of covariant derivatives
(Dâ,D1α̂) and (Dâ,D
α̂) forms a closed algebra, in particular
{D1α̂,D
} = 0 , (2.24a)
[Dâ,D1
ωJ11(Γâ)β̂
γ̂D1γ̂ , (2.24b)
[Dâ,Db̂] = −ω
. (2.24c)
Therefore, one can consistently define covariantly chiral superfields obeying the constraint
Φ = 0. Unlike the case of 4D N = 1 anti-de Sitter superspace [27], such multiplets
can transform in arbitrary representations of the Lorentz group.
In what follows, it will be useful to deal with a different basis for the spinor covariant
derivatives. Let us introduce two linearly independent isospinors u+i and u
u+iu−i ≡ (u+u−) 6= 0 =⇒ δij =
(u+u−)
(u+iu−j − u−iu+j ) , (2.25)
which do not transform under the action of J , that is J u+i = J u
i = 0. Then, defining
D±α̂ ≡ Diα̂u±i , (2.26)
J++ ≡ J iju+i u+j , J+− ≡ J iju+i u−j , J−− ≡ J iju−i u−j . (2.27)
the relations (2.21a) and (2.21b) become
} = −4ωJ++M
, (2.28a)
{D+α̂ ,D−β̂ } = 2(u
+u−)iD
+ 3(u+u−)ωε
J − 4ωJ+−M
, (2.28b)
} = −4ωJ−−M
, (2.28c)
[Dâ,D+α̂ ] = −
2(u+u−)
(Γâ)
(J++D−
− J+−D+
) , (2.28d)
[Dâ,D−α̂ ] =
2(u+u−)
(Γâ)
(J−−D+
− J+−D−
) . (2.28e)
Eqs. (2.28d) and (2.28e) are equivalent to
(u+u−)
εγ̂α̂D−
(u+u−)
εγ̂α̂D+
, (2.29a)
,D−γ̂ ] =
(u+u−)
εγ̂α̂D−
D−α̂ +
(u+u−)
εγ̂α̂D+
. (2.29b)
Under general coordinate and local SO(4,1)×U(1) transfomations, the covariant deriva-
tives change as
= eτ D
e−τ , τ = τ B̂(z)D
+ i τ(z)J + τ β̂γ̂(z)M
. (2.30)
This gauge freedom can be used to impose a suitable Wess-Zumino gauge. The latter can
be chosen such that
Dâ| = ∇â = eâm̂(x) ∂m̂ +
b̂ĉ(x)M
, (2.31)
where U | means the θ independent part of a superfield U(x, θ),
U = U(z) = U(x, θ) , U | = U(x, θ = 0) . (2.32)
and ∇â stands for the covariant derivatives of anti-de Sitter space,
[∇â,∇b̂] = −ω
. (2.33)
3 Killing supervectors
Similar to the 4D N = 1 case [21], the isometry group SU(2, 2|1) of AdS5|8 is generated
by those supervector fields ξÂ(z)E
which enjoy the property
δξDÂ = −[(ξ + iρJ + Λβ̂γ̂Mβ̂γ̂),DÂ] = 0 , (3.1)
where
ξ ≡ ξÂD
= ξâDâ + ξα̂i Diα̂ = −
ξα̂β̂D
+ ξα̂i Diα̂ , (3.2)
for some real scalar ρ(z) and symmetric tensor Λβ̂γ̂(z) = Λγ̂β̂(z). The ξÂ(z)E
is called a
Killing supervector. The set of all Killing supervectors forms a Lie algebra with respect
to the Lie bracket. Given a Killing supervector, it generates a symmetry transformation
of matter superfields, which live on AdS5|8, defined as
δξχ = −(ξ + i ρJ + Λα̂β̂Mα̂β̂)χ . (3.3)
Using the (anti) commutation relations (2.21a) – (2.21c), eq. (3.1) can be seen to be
equivalent to
Diα̂ξβ̂γ̂ + 2iξiβ̂δ
j −Diα̂ξ
j + i ρJ
α̂ + δ
3ω ξiα̂ + iDiα̂ρ
2ωJ ij(ξ
α̂ + ξ
α̂) +Diα̂Λβ̂γ̂
, (3.4)
and from here we deduce the set of Killing supervector equations
Diα̂ξâ = −2i(Γâ)α̂β̂ ξ
iβ̂ , (3.5a)
J ij −Diα̂ξjβ̂ − i ρJ
jεα̂β̂ + δ
jΛα̂β̂ , (3.5b)
iDiα̂ρ = −3ω ξiα̂ , (3.5c)
Diα̂Λβ̂γ̂ = −2ωJ ij(ξ
α̂ + ξ
α̂) . (3.5d)
Note that (3.5b) is equivalent to the following equations
Diα̂ξiβ̂ = 2Λα̂β̂ , (3.6a)
Diα̂ξj
+Djα̂ξiα̂ = 8i J ijρ , (3.6b)
(Γâ)
α̂β̂(Diα̂ξ
) = −4iω J ijξâ . (3.6c)
It is seen that the parameters of U(1) and Lorentz transformations, ρ and Λ
, are uniquely
expressed in terms of the spinor components of the Killing supervector. As to the vector
components ξâ of ξ,which is also uniquely determined in terms of the spinor components
of ξ, it obeys the standard Killing equation
D(âξ b̂) = 0 . (3.7)
To prove (3.7), it suffices to represent Dâ in (3.7) in the form
Dâ = i
(Γâ)α̂β̂Diα̂Diβ̂ , (3.8)
and then make use of relations (3.5a) and (3.6a).
As is seen from eqs. (3.5c), (3.6a) and (3.6c), the components of ξ (hence, the Lorentz
parameter Λ
as well) can be expressed in terms of the scalar parameter ρ as follows:
ξiα̂ = −
Diα̂ρ , (3.9a)
ξâ = −
Jij(Γâ)
α̂β̂ Diα̂ξ
= − 1
3ω2J2
Jij(Γâ)
α̂β̂ Diα̂D
ρ (3.9b)
Diα̂ξiβ̂ = −
Diα̂Diβ̂ ρ . (3.9c)
This is similar to the situation in 4D N = 2 AdS supersymmetry [23].
We should point out that equation (3.9c) implies
0 = Diα̂ξiα̂ = Diα̂Diα̂ ρ . (3.10)
Furthermore, equations (3.5a), (3.5d) and (3.6b) imply
Jjk(Γ
â)β̂γ̂Diα̂D
ξkγ̂ − 2(Γâ)α̂β̂ξ
iβ̂ , (3.11a)
Diα̂Djβ̂ξ
j + 2ωJ
) , (3.11b)
(Diα̂Dj
+Djα̂Diα̂)ρ+ 8J ijρ . (3.11c)
From (3.11b) we also deduce
Diα̂Djβ̂ξ
j −Diα̂Djγ̂ξ
j = 0 =⇒ Diα̂Dβ̂γ̂ρ = 0 , (3.12)
and hence
ρ = 0 . (3.13)
We conclude that ρ is annihilated by the vector covariant derivatives.
For later applications, we also observe that the relation Diα̂ξα̂i = 0 and eq. (3.5a) imply
ξiβ̂ =
iωJ ijξ
. (3.14)
4 Harmonic superspace approach
In the previous two sections, we have described the differential geometry of five-
dimensional N = 1 AdS superspace and its isometries. From now on, we turn to con-
structing off-shell supersymmetric theories in AdS5|8. This section is devoted to developing
a harmonic superspace approach. To comply with the conventions generally accepted by
the harmonic superspace practitioners [1, 2], the isospinors u+ and u− in (2.28a–2.28e)
will be chosen to obey the following constraints:
− , ui
+) ∈ SU(2) , (u+i)∗ = u−i , (u+u−) = 1 . (4.1)
As a first step, it is natural to introduce analytic supermultiplets living on harmonic
superspace.
4.1 Analytic multiplets
We start our analysis with the introduction of O(n) supermultiplets living in AdS5|8.
Such a multiplet is described by a completely symmetric superfieldH i1···in(z) = H(i1···in)(z)
(with the symmetrization involving a factor of 1/n!) constrained to enjoy the analyticity
condition4
Dα̂(i1H i2···in+1)(z) = 0 . (4.2)
4In 4D N = 2 supersymmetry, off-shell superfields H(i1···in)(z) obeying the constrains
(i1Hi2···in+1)(z) = D̄α̇
(i1Hi2···in+1)(z) = 0 have a long history. In the presence of an intrinsic cen-
tral charge, the cases n = 1 and n = 2 correspond to the Fayet-Sohnius hypermultiplet [28] and the linear
multiplet [29] respectively. In the absence of central charge, the case n = 2 corresponds to the tensor
multiplet [30]. The case n = 4 was discussed in [31]. The multiplets with n > 2 were introduced in [32],
in the projective superspace approach, and then re-discovered in [5]. They were called “O(n) multiplets”
in [33]. Their harmonic superspace description was given in [34].
It follows from the algebra of covariant derivatives, that this constraint is consistent
provided the superfield is scalar with respect to SO(4,1). If one associates with H i1···in(z)
a superfield H(n)(z, u) of harmonic charge n,
H(n)(z, u) = u+i1 · · ·u
H i1···in(z) , (4.3)
the analyticity condition (4.2) can be seen to be equivalent to
H(n)(z, u) = 0 , D++H(n)(z, u) = 0 . (4.4)
Here D++ is one of the harmonic derivatives (D++, D−−, D0),
D++ = u+i
, D−− = u−i
, D0 = u+i
− u−i ∂
[D0, D±±] = ±2D±± , [D++, D−−] = D0 , (4.5)
which form a basis in the space of left-invariant vector fields for SU(2).
Without imposing the analyticity condition, eq. (4.2), one can consistently define
an isotensor superfield F i1···in(z) = F (i1···in)(z) that transforms under the action of the
isometry group as follows:
i1···in = −
ξ + i ρJ
F i1···in = −ξF i1···in − i q n ρJ (i1
F i2···in)k , (4.6)
where ξ is the Killing supervector, and q is the J-charge of F i1···in. One can associate with
F i1···in(z) the harmonic superfield F (n)(z, u) = u+i1 · · ·u
F i1···in(z). The latter obeys the
algebraic constraint D++F (n) = 0, and its isometry transformation is
(n) = −ξF (n) + i qρ(J++D−−F (n) − nJ+−F (n)) , (4.7)
where it has been used the fact that u±i are inert under the action of J . It is also worth
noting that the Killing supervector can be rewritten as
ξ = ξâDâ − ξ+α̂D−α̂ + ξ−α̂D+α̂ . (4.8)
It is easy to see that the constraint D++F (n) = 0 is preserved under the isometry trans-
formations D++δξF
(n) = 0.
If the superfield F (n) is constrained to be analytic, D+
F (n) = 0, then the value of
its J-charge turns out to be uniquely fixed, and namely q = 1. Therefore, the isometry
transformation of the O(n) multiplet is
(n) = −
ξ + i ρJ
H(n) = −
ξâDâ − ξ+α̂D−α̂ − i ρ
J++D−− − nJ+−
H(n) . (4.9)
It is not difficult to extend the above consideration to include more general multiplets.
Within the harmonic superspace approach [2], one has to deal with superfields of the
form Q(n)(z, u), with n an integer, such that (i) Q(n)(z, u) is a smooth function over
the group manifold SU(2) parametrized by u = (ui
−, ui
+); (ii) under harmonic phase
transformations u± → exp(±iϕ)u±, the charge of Q(n)(z, u) is equal to n,
Q(n)(z, eiϕu+, e−iϕu−) = eniϕQ(n)(z, u+, u−) , ⇐⇒ D0Q(n)(z, u) = nQ(n)(z, u) .
Such a superfield can be represented by a convergent Fourier series (for definiteness, we
choose n ≥ 0)
Q(n)(z, u) =
Q(i1···ik+nj1···jk)(z) u+i1 · · ·u
u−j1 · · ·u
. (4.10)
To realise an action of the U(1) generator J on Q(n), we define the component superfields
in (4.10) to transform by the law:
JQ(i1···ik+nj1···jk)q = q(2k + n)J
i2···ik+nj1···jk)r
q , (4.11)
with the same charge q for all the component superfields. This leads to
J Q(n)q (z, u) = q
J−−D++ − J++D−− + nJ+−
Q(n)q (z, u) . (4.12)
The J-charge turns out to be uniquely fixed, q = 1, if Q
q is covariantly analytic,
q = 0.
To summarise, given a covariantly analytic superfield Q(n)(z, u),
D+α̂Q(n) = 0 , (4.13)
the infinitesimal isometry transformation acts on it as follows:
(n) = −
ξ + i ρJ
ξâDâ − ξ+α̂D−α̂ + i ρ
J−−D++ − J++D−− + nJ+−
Q(n) . (4.14)
Given two covariantly analytic superfields Q(n) and Q(m), their product Q(n)Q(m) is co-
variantly analytic and transforms as Q(n+m). In addition, the superfield D++Q(n) can be
seen to be covariantly analytic and transform as Q(n+2).
4.2 Harmonic action principle
After having introduced various analytic multiplets in AdS5|8 × S2, let us turn to
constructing a supersymmetric action. It is worth recalling that in the flat global case
(ω = 0), the action principle in 5D harmonic superspace naturally generalizes the original
4D action rule [1, 2] and is given by [10]
du (D̂−)4 L(+4)
∣∣∣ , (D̂−)4 = −
εα̂β̂γ̂δ̂D−
, (4.15)
where D−
= Diα̂u
i , D
α̂ are the flat covariant derivatives, and L
(+4) is a real analytic
Lagrangian of harmonic charge +4, D+
L(+4) = 0.
We would like to generalize the flat action to the case of AdS5|8 using the following
ansatz:
S = S0 + a1S1 + a2S2
d5x e
(D̂−)4 + a1 ωJ−−(D̂−)2 + a2 (ωJ−−)2
L(+4)
∣∣ , (4.16)
where L(+4) is now covariantly analytic, D+
L(+4) = 0,
(D̂−)2 = D−α̂D−
, (D̂−)4 = − 1
εα̂β̂γ̂δ̂D−
, (4.17)
and a1, a2 are two constants to be determined. It is assumed that the above action is
evaluated in Wess-Zumino gauge (2.31), using the bar projection (2.32), and as usual e
stands for the determinant of the vielbein, e = det(em̂
â), with em̂
âeâ
n̂ = δm̂
In accordance with the definition of S, there are several rules for integration by parts
which one can use in practice:
d5x e
duDâQâ| = 0 , (4.18)
d5x e
duD++Q−−| =
d5x e
duD−−Q++| = 0 , (4.19)
d5x e
du J Q(0)| = 0 . (4.20)
Here Q(0) is a covariantly analytic superfield of harmonic charge 0.
Our aim is to find the constants a1, a2 for which S is invariant under the isometry
transformations of AdS5|8. Let us first compute the variation of S0 under infinitesimal
isometry transformations. Due to (3.1), we have
(D̂−)4L(+4)
= (D̂−)4δL(+4) = −(D̂−)4
ξ + i ρJ
L(+4)
ξ + i ρJ + Λα̂β̂M
(D̂−)4L(+4) = −
ξ + i ρJ
(D̂−)4L(+4) . (4.21)
Since L(+4) is covariantly analytic, we obtain
δξS0 = −
d5x e
ξ + i ρJ
(D̂−)4L(+4)
d5x e
ξ+α̂D−
(D̂−)4 − ξ−α̂[D+
, (D̂−)4]
L(+4)
∣∣∣ . (4.22)
Here we have also used eqs. (4.18) and (4.20).
To compute D−
(D̂−)4 in (4.22), we observe that D−
= 0, and then
0 = 5εβ̂γ̂δ̂ρ̂D−
D−γ̂ D−δ̂ D
= εβ̂γ̂δ̂ρ̂
. (4.23)
Moving D−
in each term to the left gives
(D̂−)4 = ωJ
εβ̂γ̂δ̂ρ̂
+ 3D−
+ 2D−
. (4.24)
This can be further transformed by moving all the Lorentz generators to the right and
factors of D−
to the left using iteratively the algebra of covariant derivatives. We end up
D−α̂ (D̂−)4 = D−α̂
ωJ−−(D̂−)2 + 3 (ωJ−−)2
ωJ−−D−β̂(D̂−)2 − 8
(ωJ−−)2D−β̂ + 1
(ωJ−−)2D−γ̂ M γ̂β̂
. (4.25)
The expression in the second line does not contribute when acting on a Lorentz scalar
such as L(+4).
To compute [D+
, (D̂−)4] in (4.22), we should iteratively use the algebra of covariant
derivatives. This is an obvious but tedious procedure. The result is:
[D+α̂ , (D̂−)4] =
D−β̂(D̂−)2 + 3
ωJD−α̂ (D̂−)2 −
ωJ−−iD
ω2J−−JD−
ωJ+−D−
(D̂−)2 − 5ω2J−−J+−D−
ωJ−−(D̂−)2D+
ωJ−−D−
D+β̂ − 3
ωJ−−D−
(ωJ−−)2D+
+ ωJ−−iD
M β̂γ̂ − 3
ω2J−−JD−β̂M
+2ω2J−−J+−D−β̂M
ωJ+−D−β̂(D̂−)2M
−ω2J−−J+−D−γ̂ M β̂γ̂Mβ̂α̂ . (4.26)
Using the relations (4.25) and (4.26), and also the integration by parts identities (4.18)
and (4.20), variation (4.22) turns into
δξS0 =
d5x e
du ξ+α̂
ωJ−−D−
(D̂−)2 + 3 (ωJ−−)2D−
L(+4)
d5x e
ξ−β̂)D−α̂(D̂−)2 + 3
ω(Jξ−α̂)D−
(D̂−)2
ωJ−−(iD
ξ−β̂)D−α̂ − 13
ω2J−−(Jξ−α̂)D−α̂
ωJ+−ξ−α̂D−α̂ (D̂−)2 + 5ω2J−−J+−ξ−α̂D−α̂
L(+4)
∣∣∣ . (4.27)
Finally, it remains to note Jξ−
= J−−ξ+
− J+−ξ−
, and also make use of eq. (3.14)
projected to the minus-harmonics
ξ−β̂ = −5
ω(J−−ξ+
− J+−ξ−
) . (4.28)
As a result, the variation of S0 under the isometry transformations takes the final form:
δξS0 =
d5x e
ωJ−−ξ+α̂D−
(D̂−)2 − 8
(ωJ−−)2ξ+α̂D−
ω2J−−J+−ξ−α̂D−
L(+4)
∣∣∣ . (4.29)
The next step is to compute the variation of the functional S1 appearing in our action
(4.16). Here the procedure is the same as for S0. Varying
(D̂−)2L(+4)
= −(D̂−)2
ξ + i ρJ
L(+4) = −
ξ + i ρJ
(D̂−)2L(+4) , (4.30)
we get
δξS1 =
d5x e
ωJ−−ξ+α̂D−
(D̂−)2 − ωJ−−ξ+α̂[D+
, (D̂−)2]
L(+4)
∣∣∣ . (4.31)
Using the algebra of covariant derivatives gives
[D+α̂ , (D̂−)2] = −4iDα̂β̂D
−β̂ − 6ωJD−α̂ + 12ωJ+−D−α̂
−2ωJ−−D+
+ 8ωJ+−D−β̂M
. (4.32)
As a result, the variation of S1 is
δξS1 =
d5x e
ωJ−−ξ+α̂D−
(D̂−)2 + 4(ωJ−−)2ξ+α̂D−
− 16ω2J−−J+−ξ−α̂D−α̂
L(+4)
∣∣∣ . (4.33)
It is seen that (4.33) is proportional to (4.29). Therefore, our ansatz (4.16) leads to the
unique supersymmetric action: a1 = 2/3 and a2 = 0.
The supersymmetric action is
d5x e
(D̂−)4 + 2
ω J−−(D̂−)2
L(+4)
∣∣∣ , D+α̂L(+4) = 0 . (4.34)
This is the main result of this section.
By construction, the Lagrangian in (4.34) is a covariantly analytic superfield of har-
monic charge +4. It should be also chosen to be real with respect to analyticity preserving
conjugation [1] (see also subsection 5.1), and then action (4.34) can be seen to be real.
Otherwise, L(+4) is completely arbitrary. Therefore, a great many flat superspace actions
[2] can be lifted to the AdS superspace. For instance, an off-shell hypermultiplet can be
realized in terms of a covariantly analytic superfield q+(z, u) and its conjugate q̃+(z, u),
with respect to the anlyticity preserving conjugation. To describe its dynamics, one can
choose
L(+4) = −q̃+D++q+ + λ (q̃+q+)2 , (4.35)
with λ a coupling constant.
5 Projective superspace approach
In the projective superspace approach to d-dimensional theories with eight super-
charges, one deals with superfields that live in Md|8 × S2, where Md|8 denotes the
conventional superspace, d ≤ 6, and S2 the two-sphere. Such superfields are required
to (i) be Grassmann analytic, i.e. to be annihilated by one half of the supercharges;
(ii) be holomorphic on an open domain of S2. The latter requirement is equivalently
achieved by considering superfields Ψ(n)(z, u+) which are holomorphic functions of a sin-
gle isotwsitor u+i ∈ C2−{0}, and have definite degree of homogeneity with respect to u+,
Ψ(n)(z, c u+) = cn Ψ(n)(z, u+). The variables u+i can be viewed as homogeneous coordi-
nates for CP 1. A second linearly independent isotwistor, u−i, is only required (as a purely
auxiliary means, without any intrinsic significance) for constructing a supersymmetric ac-
tion which was proposed originally in four dimensions in [3] and then reformulated in [4]
in terms of the projective isotwistor u+i. The terminology “isotwistor” is due to [35, 36].
In the flat global case, the 5D N = 1 extension of the 4D N = 2 supersymmetric
action [4] is as follows5 [25]:
u+i du
(u+u−)4
d5x (D̂−)4L++(z, u+)
∣∣ , (5.1)
where
L++(z, u+) = 0 , L++(z, c u+) = c2 L++(z, u+) , c ∈ C∗ . (5.2)
The action is invariant under arbitrary projective transformations of the form
− , ui
+) → (ui− , ui+)R , R =
∈ GL(2,C) . (5.3)
This gauge-like symmetry implies that the action is actually independent of u−i . It can
be fixed by imposing, for instance, the gauge
u+i ∼ (1, ζ) = ζ i −→ u+i ∼ (−ζ, 1) = ζi ,
u−i ∼ (0,−1) −→ u−i ∼ (1, 0) , (5.4)
in which the action (5.1) reduces to the standard 5D N = 1 projective superspace action
[10, 25].
5.1 Projective multiplets
Here we introduce several off-shell projective multiplets that are most interesting from
the point of view of model building. By definition, a projective superfield Q(n)(z, u+) lives
on the anti-de Sitter superspace and depends parametrically on a non-vanishing isotwistor
u+i 6= 0. It is defined to be analytic,
Q(n) = 0 , (5.5)
and transform by the rule
δQ(n) = −
ξ + i ρJ
Q(n) (5.6)
under the isometry group. We specify J to act on Q(n) as follows
J Q(n) = −J
++D−−Q(n) − nJ+−Q(n)
(u+u−)
. (5.7)
5Note that the action given in eq. (B.1) of [25] contains a wrong overal factor of
This definition involves an external isotwistor u−i subject to the only requirement
(u+u−) 6= 0 . (5.8)
Since Q(n) is independent of u−, it is natural to require J Q(n) to be independent of u− as
well, that is
Q(n) − nJiju+jQ(n) = u+i J Q(n) . (5.9)
Contracting this with u+i gives
J++u+i
Q(n) = nJ++Q(n) . (5.10)
Therefore, Q(n) is a homogeneous function of u+ of degree n,
Q(n)(z, c u+) = cnQ(n)(z, u+) , c ∈ C∗ . (5.11)
The Q(n) will be called a projective superfield of weight n.
As is obvious, the complex conjugate of an analytic superfield is not analytic. However,
one can introduce a generalized, analyticity-preserving conjugation [7, 1, 3], u+i → ũ+i,
which is obtained by composing the complex conjugation, u+i → u+i, with the antipodal
map u+i → −u+i . In what follows, it is called “smile-conjugation.” It is thus defined to
act on the isotwistor u+ = (u+i) by the rule6
u+ → ũ+ = i σ2 u+ ,
(u+i) = −u+i , (5.12)
with σ2 the second Pauli matrix. Its action on the projective superfields is defined to be
Q(n)(u+) → Q̃(n)(u+) ≡ Q̄(n)
Q(n) = (−1)nQ(n) . (5.13)
It is clear that Q̃(n)(u+) is a homogeneous function of u+ of degree n, that is Q̃(n)(c u+) =
cn Q̃(n)(u+), with c ∈ C∗. Due to the identity
Q(n) = (−1)ǫ(Q(n)) D+α̂Q̃(n) , (5.14)
the smile-conjugation indeed preserves analyticity.
It is important to note that, in accordance with (5.13), for an even integer weight,
n = 2p, one can consistently define real projective superfields R(2p) with respect to the
smile-conjugation: R̃(2p) = R(2p).
6Due to projective invariance, u+i ∼ c u+i, the smile-conjugation could be also defined as u+ → ũ+ =
−iσ2 u+, instead of (5.12).
Now, let us show that the smile-conjugation is compatible with the superfield transfor-
mation law (5.6). To evaluate the smile-conjugate of J Q(n), eq. (5.7), we conventionally
define the operation of smile-conjugate for u− = (u−i) to be identical to that we have
already chosen for the isotwistor u+, that is
u− → ũ− = i σ2 u− ,
(u−i) = −u−i , (5.15)
We should emphasize that such a definition is completely conventional in the sense that
the projective superfields are independent of the isotwistor u−. Then it holds
D̃−− = D−− , J̃±± = −J±± , J̃+− = −J+− , ˜(u+u−) = (u+u−) . (5.16)
This implies
J̃ Q(n) = − J Q̃(n) , (5.17)
and the smile-conjugate of the transformation law (5.6) is
δ̃Q(n) = −(ξ + i ρJ) Q̃(n) = δ Q̃(n) . (5.18)
Therefore, the smile-conjugation preserves the superfield transformation laws under the
isometry group.
As is known, the space CP 1 can be covered by two charts that are defined in terms
of u+ = (u+1, u+2) as follows: (i) the north chart on which u+1 6= 0; (ii) the south chart
on which u+2 6= 0. As will be described below, the projective action involves the line
integral over a closed contour in CP 1, and this contour can be chosen to lie inside one of
the coordinate charts. The latter can be chosen to be the north chart, and that is why
our local considerations will be mainly restricted to that chart. In the north chart, we can
introduce a projective invariant complex coordinate ζ defined as u+i = u+1 (1, ζ), with
ζ = u+2/u+1. Since ũ+i = (u+2,−u+1), the smile-conjugation acts as follows:
ζ → − 1
. (5.19)
The simplest solution to eq. (5.11) is the O(n) multiplet defined by eqs. (4.2) and (4.3).
This multiplet is globally defined on CP 1. Allowing for singularities at some points in CP 1
offers the possibility to generate many more interesting supermultiplets. For example, a
charged hypermultiplet is described by a weight-one projective superfield Υ+(u+) being
holomorphic on CP 1 −{N}, where the North pole is identified with u+i ∼ (0, 1). We can
represent Υ+(u+) as
Υ+(u+) = u+1Υ+(u+i/u+1) ≡ u+1Υ(ζ) , Υ(z, ζ) =
Υk(z)ζ
k . (5.20)
Its smile-conjugate Υ̃+(u+) is holomorphic onCP 1−{S}, where the South pole is identified
with u+i ∼ (1, 0). We can represent Υ̃+(u+) as
Υ̃+(u+) = u+2 Υ̃+(u+i/u+2) ≡ u+2 Υ̃(ζ) , Υ̃(z, ζ) =
Ῡk(z)
(−1)k
, (5.21)
with Ῡk(z) the complex conjugate of Υk(z). To describe an off-shell vector multiplet, one
should use a real weight-zero projective superfield V (u+) being holomorphic on CP 1 −
{N ∪ S}. It can be represented as
V (z, ζ) =
Vk(z)ζ
k , V̄k = (−1)kV−k . (5.22)
5.2 Projective action principle
Our aim here is to find a generalization of the flat superspace action (5.1) to the case
of AdS5|8 superspace. We start with the following ansatz7
S = S0 + β1 S1 + β2 S2
= − 1
u+i du
(u+u−)4
d5x e
(D̂−)4 + β1ωJ−−(D̂−)2 + β2(ωJ−−)2
L++(z, u+)
∣∣∣ . (5.23)
Here L++(z, u+) is a covariantly analytic superfield, D+
L++ = 0, which is homogeneous
in u+i of degree +2. The line integral in (5.23) is carried out over a closed contour,
γ = {u+i (t)}, in the space of u+ variables. The integrand in (5.23) involves a constant
(i.e. time-independent) isotwistor u−i subject to the only condition that u
+(t) and u−
form a linearly independent basis at each point of the contour γ, that is (u+u−) 6= 0.
Our first requirement is that the action (5.23) be invariant under the projective gauge
transformations (5.3). First of all, it is obvious that (5.23) is invariant under arbitrary
7An alternative approach to introduce the projective action consists in using a proper generalization
of the procedure given in [37]. The latter allows one to derive the projective action as a singular limit of
the harmonic action.
scale transformations u+i (t) → c(t) u+i (t), with c(t) 6= 0. It is thus only necessary to
analyse projective transformations of u− of the form
u−i → ũ−i = a(t) u−i + b(t) u+i (t) , a(t) 6= 0 . (5.24)
Since both u− and ũ− should be time independent, the coefficients should obey the equa-
tions (using the notation
f ≡ df(t)/dt, for a function f(t)):
a = b
(u+u−)
b = −b (
(u+u−)
. (5.25)
As is obvious, the action (5.23) is invariant under arbitrary scale transformations u−i →
a(t) u−i , with a 6= 0. Therefore, it only remains to analyse infinitesimal transformations
of the form δu−i = b(t)u
i , with b(t) obeying the differential equation (5.25). This trans-
formation induces the following variations:
δD−α̂ = bD+α̂ , δJ−− = 2b J+− . (5.26)
Let us start by evaluating the variation of S0. Using the condensed notation
dµ++ ≡ − 1
u+i du
(u+u−)4
= − 1
(u+u−)4
dt , (5.27)
we obtain
δS0 =
d5x e
δ(D̂−)4
α̂β̂γ̂δ̂
dµ++ b
d5x e
}+ 2D−
+ {D+
∣∣∣ . (5.28)
Now, making use of the covariant derivatives algebra (2.28a–2.29b) and the identities
[J,D+
(u+u−)
J+−D+
− J++D−
, (5.29a)
[J,D−
(u+u−)
J−−D+
− J+−D−
, (5.29b)
we can systematically move in (5.28) all space-time derivatives to the left (neglecting total
space-time derivatives) and the J operator to the right. This gives
δS0 =
dµ++ b
d5x e
ωJ+−(D̂−)2 − 3
ω(u+u−)(D̂−)2J
(u+u−)ω2J−−J
∣∣∣ . (5.30)
To transform the second and third terms in the square brackets, we should first recall how
J acts on the Lagrangian,
J L++ = 1
(u+u−)
−J++D−−L++ + 2J+−L++
. (5.31)
Since L++ is a homogeneous function of degree two, we also have
L++ = (
(u+u−)
L++ − (
(u+u−)
D−−L++
(u+u−)
L++ − (
(u+u−)
D−−L++ . (5.32)
The latter results leads to
u+) J L++ = J++ d
L++ − 2(
(u+u−)
J++L++ + 2(
(u+u−)
J+−L++ . (5.33)
One more technical observation,
J++ = 2
(u+u−)
J++ − 2(
(u+u−)
J+− , (5.34)
allows us to obtain the following identity:
(u+u−)3
J L++ = d
b J++
(u+u−)3
(u+u−)4
J+−L++ . (5.35)
Then (5.30) becomes
δS0 =
dµ++ b
d5x e
ωJ+−(D̂−)2 + 22ω2J−−J+−
∣∣∣ . (5.36)
Using the same procedure, for δS1 and δS2 we find
δS1 =
dµ++ b
d5x e
2ωJ+−(D̂−)2 + 48ω2J−−J+−
∣∣∣ , (5.37a)
δS2 =
dµ++ b
d5x e 4ω2J−−J+−L++
∣∣∣ . (5.37b)
The relations obtained show that the requirement of projective invariance, δS = δS0 +
β1 δS1 + β2 δS2 = 0, uniquely fixes the coefficients in in (5.23) as follows: β1 = 25/24 and
β2 = −18. We end up with the projective-invariant action
S = − 1
u+i du
(u+u−)4
d5x e
(D̂−)4 + 25
ωJ−−(D̂−)2 − 18 (ωJ−−)2
∣∣∣ . (5.38)
Now, we are going to demonstrate that (5.38) is supersymmetric, that is this action is
invariant under the isometry group of AdS5|8. This requires us to carry out calculations
that are very similar to those presented in section 4 for the harmonic case. But there
are two technical features being specific for the projective case: (i) unlike the harmonic
case, we have (u+u−) 6= 1 in general, and therefore it is necessary to keep track of the
factors of (u+u−); (ii) unlike the harmonic superspace identity (4.20), in general we have∮
dµ++J Q−− 6= 0. In all variations involving the U(1) generator J , we will systematically
move J ’s to the right to hit the Lagrangian L++, so that eqs. (5.31) and (5.33) can be
applied.
We start by computing the variation of S0 under the infinitesimal isometry transfor-
mation. Making use of
(D̂−)4L++
= −(D̂−)4
ξ + i ρJ
L++ = −
ξ + i ρJ
(D̂−)4L++ (5.39)
gives
δξS0 =
d5x e
(u+u−)
ξ+α̂D−
(D̂−)4 − ξ−α̂[D+
, (D̂−)4]
− i ρ
[J, (D̂−)4] + (D̂−)4J
∣∣∣ . (5.40)
To evaluate D−
(D̂−)4L++, we note that eq. (4.25) holds even if (u+u−) 6= 1, since in the
derivation of (4.25) we only used eq. (2.28c) and the commutation relations of the Lorentz
generator M
with the covariant derivatives, and both results are clearly not affected by
the normalization of (u+u−). Therefore, for the first term on the right of (5.40) we have
D−α̂ (D̂−)4L++ =
ωJ−−D−α̂ (D̂−)2 + 3 (ωJ−−)2D−α̂
L++ . (5.41)
For the operator [D+
, (D̂−)4], which appears in (5.40), we have derived eq. (4.26) in
the harmonic case. Now, in evaluating the second term on the right of (5.40), we should
take care of the factors of (u+u−), as well as to move the U(1) generator J to the right.
This gives
, (D̂−)4] = 1
(u+u−)iD
D−β̂(D̂−)2 + 3
(u+u−)ωD−
(D̂−)2J − 7
ωJ+−D−
(D̂−)2
(u+u−)ωJ−−iD
D−β̂ − 17
(u+u−)ω2J−−D−α̂ J +
ω2J−−J+−D−α̂ + · · · , (5.42)
where the dots denote those terms which do not contribute when acting on Lorentz scalar
and analytic superfields such as the Lagrangian L++. Inserting (5.42) into δS0, one can get
read of the terms with vector covariant derivatives by taking into account the integration
by parts rule (4.18) and
ξ−β̂ = − 5ω
2(u+u−)
(J−−ξ+
− J+−ξ−
) . (5.43)
To evaluate the contributions to δS0 which contain JL++, we note that eqs. (5.33)
and (5.34) imply
(u+u−)4
J L++ = d
(J++L++
(u+u−)4
(u+u−)5
J+−L++ . (5.44)
The latter observation tells us
dµ++ O(u−) J L++
∣∣∣ = 4
(u+u−)
O(u−)L++
∣∣∣ , (5.45)
for any operator O(u−) independent of u+. It follows that
d5x e [D+
, (D̂−)4]L++
∣∣∣ =
d5x e
(u+u−)
ξ+α̂D−
(D̂−)2
ωJ−−ξ+α̂D−α̂ +
ωJ+− ξ−α̂D−α̂
∣∣∣ . (5.46)
Completely similar considerations, using also D
ρ = 0 (3.13), give
d5x e ρ[J, (D̂−)4]L++
d5x e
ρ J+−
(u+u−)
4(D̂−)4
ωJ−−(D̂−)2 − 22 (ωJ−−)2
∣∣∣ .(5.47)
As a result, the variation δξS0 can be represented in the form
δξS0 =
d5x e
(u+u−)
ξ+α̂D−
(D̂−)2 + 91
(ωJ−−)2
(u+u−)
ξ+α̂D−
ω2J−−J+−
(u+u−)
ξ−α̂D−
ωJ−−J+−
(u+u−)
ρ(D̂−)2 − 22 (ωJ
−−)2J+−
(u+u−)
∣∣∣ . (5.48)
The variations δξS1 and δξS2 can be computed by similar means. The results are:
δξS1 =
d5x e
[ ωJ−−
(u+u−)
ξ+α̂D−α̂ (D̂−)2 + 10
(ωJ−−)2
(u+u−)
ξ+α̂D−α̂
− 4 ω
2J−−J+−
(u+u−)
ξ−α̂D−
− 2 ωJ
−−J+−
(u+u−)
ρ(D̂−)2 − 48 (ωJ
−−)2J+−
(u+u−)
∣∣∣ ,(5.49a)
δξS2 =
d5x e
[ (ωJ−−)2
(u+u−)
ξ+α̂D−
− 4 (ωJ
−−)2J+−
(u+u−)
∣∣∣ . (5.49b)
Collecting all the results obtained, we conclude
δξS = 0 , (5.50)
and therefore the action (5.38) is supersymmetric. Actually, it proves to be the only su-
persymmetric action in the family (5.23). It is quite remarkable that projective invariance
implies supersymmetry and vice versa.
6 Dynamical systems in projective superspace
In this section we study in more detail the projective multiplets and then consider sev-
eral important supersymmetric theories. To simplify the analysis, it is useful to choose the
projective gauge u−2 = 0. Without loss of generality, one can also work in a representation
of the algebra in which J11 = J22 = 0, and hence J−− = 0.
6.1 Projective multiplets revisited
In each of the two coordinate charts for CP 1, one can describe the projective multiplets
by superfields invariant under the projective transformation (5.11). Let us restrict our
consideration to the north chart. Given a complex projective multiplet of weight n,
Q(n)(u+), it can be equivalently described by a holomorphic function Q[n](ζ) defined as
follows:
Q(n)(u+i) = (u+1)nQ[n](ζ) , Q[n](ζ) ≡ Q(n)(1, ζ) . (6.1)
Here Q[n](ζ) is clearly invariant under (5.11). For the smile-conjugate of Q(n)(u+), we get
Q̃(n)(u+i) = (u+2)nQ̃[n](ζ) , Q̃[n](ζ) = Q̄[n](−1/ζ) . (6.2)
Given a real projective multiplet R(2p)(u+), with respect to the smile-coinjugation, it can
be represented
R(2p)(u+) = (iu+1u+2)pR[2p](ζ) , R̃[2p](ζ) ≡ R̃[2p](−1/ζ) = R[2p](ζ) . (6.3)
The most general form for Q[n](z, ζ) is
Q[n](z, ζ) =
(z)ζk . (6.4)
In the projective gauge chosen (u−2 = 0, J
11 = J22 = 0), the action of the operator J on
our superfield becomes
J Q[n](u+) = − 1
(u+u−)
J++D−− − nJ+−
Q[n](u+) = J12
n− 2u+2 ∂
Q[n](u+) .
Then, since the isotwistor u+i is neutral under the action of J , it holds
(u+1)n J Q[n](ζ) = J Q(n)(u+) = J12
n(u+1)nQ[n](ζ)− 2u+2 ∂
(u+1)nQ[n](ζ)
= (u+1)nJ12
nQ[n](ζ)− 2ζ ∂
Q[n](ζ)
and therefore
J Q[n](z, ζ) = J11
Q[n](z, ζ) , J Q
(z) = (2k − n)J11Q[n]k (z) . (6.5)
In the case of a real superfield R(2p)(z, u+) = (iu+1u+2)pR[2p](z, ζ), we have for R[2p]
R[2p](z, ζ) =
(z)ζk , R̄
= (−1)kR[2p]−k . (6.6)
The operator J is represented as follows:
J R[2p](z, ζ) = 2J11ζ
R[2p](z, ζ) , J R
k (z) = 2kJ
k (z) . (6.7)
Let us analyse the implications of the analyticity condition, D+
Q(n)(u+) = 0. It is
useful to change the representation for the projective superfields, Q(n)(u+) → Q[n](ζ). We
then have D+
Q[n](ζ) = −u+1(ζD1
)Q[n](ζ), and therefore the analyticity condition
is equivalent to
Q[n](ζ) = ζD1
Q[n](ζ) . (6.8)
For the component superfields Q
(z), this implies
D2α̂Q
k = D
k−1 . (6.9)
It is natural to think of D1
and D2
as the covariant derivatives associated with two 5D
Dirac spinor coordinates, θα̂1 and their conjugates θ
2 . It then follows from (6.8) that the
dependence of Q[n](ζ) on θα̂2 is completely determined by the dependence of Q
[n](ζ) on θα̂1 .
Suppose that the expansion of Q[n](ζ) in powers of ζ terminates from below
Q[n](z, ζ) =
(z)ζk . (6.10)
Then, eq. (6.9) tells us that the two lowest components of Q[n] are constrained as follows:
L = 0 ,
(D̂2)2Q[n]L+1 = 12ω J Q
L , (6.11)
where
(D̂i)2 = Diα̂Diα̂ , i = 1, 2 . (6.12)
Therefore, Q
L is a five-dimensional chiral superfield, while Q
L+1 a complex linear super-
field. The union of Q
L and Q
L+1 forms a 5D analogue of the famous chiral-nonminimal
doublet in 4D supersymmetry [38].
Given a real O(2) multiplet H(2)(z, u+), we can represent H [2](z, ζ) in the form
H [2](z, ζ) =
Φ(z) +G(z)− ζΦ̄(z) , (6.13)
where Φ is a five-dimensional chiral superfield, and G a real linear superfield,
Φ = 0 ,
(D̂2)2G = 0 , Ḡ = G . (6.14)
If the expansion of Q[n](ζ) in powers of ζ terminates from above,
Q[n](z, ζ) =
(z)ζk . (6.15)
then eq. (6.9) implies that the two highest components of Q[n] are constrained as follows:
D1α̂Q
L = 0 ,
(D̂1)2Q[n]L−1 = −12ω J Q
L , (6.16)
Therefore, Q
L is a five-dimensional antichiral superfield, while Q
L−1 a complex antilinear
superfield.
For further analysis, it is useful to switch from the 5D four-component spinor notation
to the 4D two-component one by representing
Diα̂ =
D̄α̇i
εijD̄α̇j
In such a notation, the algebra of covariant derivatives (2.21a–2.21c) takes the form
{Diα,D
} = 2εijεαβD5 − 3ωεijεαβJ − 4ωJ ijMαβ , (6.17a)
{Diα, D̄
j } = −2iδij(σa)αβ̇Da − 4ωJ ijM β̇α , (6.17b)
{D̄α̇i , D̄
j } = −2εijεα̇β̇D5 − 3ωεijεα̇β̇J − 4ωJijM α̇β̇ , (6.17c)
[Da,Diα] = − i2ωJ
ij(σa)αβ̇D̄
j , [Da, D̄α̇i ] = i2ωJij(σ̃a)
α̇βDjβ , (6.17d)
[D5,Diα] = 12ωJ
jDjα , [D5, D̄α̇i ] = 12ωJ
j , (6.17e)
[Da,Db] = −ω2J2Mab , [Da,D5] = −ω2J2Ma5 . (6.17f)
In the two-component spinor notation, the analyticity condition D+
Q[n](ζ) = 0 is
equivalent to
D2αQ[n](ζ) = ζ D1αQ[n](ζ) , D̄α̇2Q[n](ζ) = −
D̄α̇1Q[n](ζ) . (6.18)
For the component superfields Q
k (z), this implies
= D1αQ
k−1 , D̄α̇2Q
= −D̄α̇1Q
k+1 . (6.19)
By analogy with the flat case, these constraints indicate an interesting interpretation. Let
us introduce two sets of spinor derivatives, (D1α, D̄
1 ) and (D2α, D̄
2 ) which can be viewed
as the covariant derivatives corresponding to two different sets of Grassmann variables
Θ1 and Θ2. Then, the above constraints imply that the dependence of the projective
superfields on Θ2 is uniquely determined in terms of their dependence on Θ1. Unlike the
flat case, such an interpretation is somewhat limited in the sense that one can not con-
sistently switch off the variables Θ2 (what would be necessary for reducing the multiplets
to 4D N = 1 superfields). It follows from the algebra of covariant derivatives, specifically
from eq. (6.17d), that [Da,D1α] = − i2ωJ
12(σa)αβ̇D̄
2 and [Da, D̄α̇1 ] = − i2ωJ
12(σ̃a)
α̇βD2β, and
therefore the commutation relations mix all the spinor derivatives. This is an important
difference between the flat and curved cases.
The constraints (6.19) simplify if the series in (6.4) or (6.6) is bounded from below
(above). Consider a real O(2n) multiplet H(2n)(z, u+). In accordance with the above
general consideration, it can be described by the superfield H [2n](z, ζ) which is defined by
H(2n)(z, u+) = (iu+1u+2)nH [2n](z, ζ), and can be represented in the form
H [2n](z, ζ) =
k (z)ζ
k , H̄
k = (−1)kH
−k . (6.20)
The analyticity constraints (6.19) imply that the two lowest component superfields are
constrained by
D̄α̇1H
−n = 0 ,
(D̄1)2H [2n]−n+1 = −(4D5 + 6ωJ)H
−n = −(4D5 − 12nωJ11)H
−n , (6.21)
where we have defined
(Di)2 ≡ DiαDiα , (D̄i)2 ≡ D̄iα̇D̄α̇i . (6.22)
Consider an arctic multiplet of weight n ≥ 0, Υ(n)(u+), defined to be holomorphic on
CP 1 − {N}. It can be represented as
Υ(n)(z, u+) = (u+1)n Υ[n](z, ζ) , Υ[n](z, ζ) =
k (z)ζ
k . (6.23)
Then the constraints on the two lowest components superfields are
D̄α̇1Υ
0 = 0 ,
(D̄1)2Υ[n]1 = −(4D5 + 6ωJ)Υ
0 = −(4D5 − 6nωJ11)Υ
0 . (6.24)
In the flat superspace limit, ω → 0, the constraints (6.21) and (6.24) reduce to those given
in [10].
6.2 Projective action
Here we turn to a more detailed analysis of the projective action (5.38). In the pro-
jective gauge (u−2 = 0, J
11 = J22 = 0) used throughout this section, we have J−− = 0,
and therefore the projective action simplifies
S = − 1
u+i du
(u+u−)4
d5x e (D̂−)4 L++
∣∣∣ . (6.25)
Of course, the Lagrangian L++ should be real with respect to the smile conjugation, and
can be represented as
L++(z, u+) = iu+1u+2L(z, ζ) . (6.26)
Then, the action turns into
S = − 1
d5x e ζ (D̂1)2(D̂1)2L(z, ζ)
∣∣∣ , (6.27)
where we have taken into account the fact that {D1
} = 0 in the projective gauge, and
also made use of the identity εα̂β̂γ̂δ̂ =
εα̂β̂εγ̂δ̂ + εα̂γ̂εδ̂β̂ + εα̂δ̂εβ̂γ̂
. Using the relation
(D̂2)2Q[n] = ζ2(D̂1)2Q[n] + 12ωζ JQ[n] , (6.28)
we can express action (6.27) in the equivalent forms
S = − 1
d5x e (D̂1)2
(D̂2)2 − ζ 12ωJ
L(z, ζ)
∣∣∣ , (6.29a)
S = − 1
d5x e (D̂2)2
(D̂1)2 + 1
L(z, ζ)
∣∣∣ , (6.29b)
where we have used the identities
, (D̂2)2] = −18ωJ11D
+ 4iD
D2β̂ + 6ωD2
J − 8ωJ11D2β̂Mα̂β̂ , (6.30)
[(D̂1)2, ( ˆ̄D2)2] = 4iDα̂β̂[D1
] + (6ω[D1
] + 96ω2J11)J − 8ωJ11[D
]M α̂β̂ . (6.31)
Then we can represent the action in the form
S = − 1
d5x e
(D̂1)2, (D̂2)2
+ 24ωJ
ζ(D̂1)2 + 1
(D̂2)2
L(z, ζ)
∣∣∣ (6.32)
which makes manifest the reality of S with respect to the smile-conjugation.
It can be seen from the above relations that there exists a natural “gauge freedom” in
the choice of L++. It occurs in the three incarnations:
L++ → L++ + Λ++ + Λ̃++ , (6.33)
L++ → L++ + i J++
Λ+ Λ̃
, (6.34)
L++ → L++ +H++ , (6.35)
with Λ++ and Λ arctic multiplets (6.23) of weight +2 and 0, respectively, and H++ a real
O(2) multiplet.
It is also instructive to express the action in a 4D N = 1 form by switching to the
two-component spinor notation
D1α̂ =
D̄1α̇
D̄α̇2
, D2α̂ =
D̄2α̇
−D̄α̇1
. (6.36)
Using the analyticity conditions (6.18) we can express D̄α̇2 via D̄α̇1 . As a result our action
(6.25) becomes
d5x e
(D1)2(D̄1)2 − ζωJ11(D1)2
L(z, ζ)
∣∣∣ . (6.37)
Using the identities
[D̄α̇1 , (D1)2] = 8ωJ11D̄α̇2 − 4i(σ̃a)α̇αDaD1α + 8ωJ11D1αMαα̇ , (6.38)
[(D1)2, (D̄1)2] = − 16ωJ11D̄1α̇D̄α̇2 + 16ωJ11D1αD2α + 96ω2J11J
+4i(σa) α̇α Da[D1α, D̄1α̇] + 8ωJ11[D1α, D̄1α̇]Mαα̇ , (6.39)
the action can also be rewritten in the following form
d5x e
(D̄1)2(D1)2 +
1(D̄1)2
L(z, ζ)
∣∣∣ , (6.40)
or in the manifestly real form
d5x e
(D1)2, (D̄1)2
1(D1)2 +
1(D̄1)2
L(z, ζ)
∣∣∣ . (6.41)
As compared with the flat superspace action [10], the second and third terms on the right
of (6.41) are due to the non-vanishing curvature.
6.3 Nonlinear sigma-models
We consider a system of interacting artic weight-one multiplets Υ+(z, u+) and their
smile-conjugates Υ̃+ described by the Lagrangian
L++ = iK(Υ+, Υ̃+) , (6.42)
with K(ΦI , Φ̄J̄) a real analytic function. Since L++ = L++(z, u+) is required to be a
weight-two projective superfield, the potential K has to respect the following homogeneity
condition (
+ Φ̄Ī
∂Φ̄Ī
K(Φ, Φ̄) = 2K(Φ, Φ̄) . (6.43)
For L++ to be real, it is sufficient to require a stronger condition
K(Φ, Φ̄) = K(Φ, Φ̄) . (6.44)
Such a Lagrangian corresponds to the superconformal sigma-model introduced in [25].
Then, representing Υ+(z, u+) = u+1Υ(z, ζ) and Υ̃+(z, u+) = u+2 Υ̃(z, ζ), we can rewrite
the Lagrangian in the form
L++(z, u+) = i u+1u+2L(z, ζ) , L = K(Υ, Υ̃) . (6.45)
Because of freedom (6.33) in the choice of Lagrangian, we can generalize the above
construction by replacing K(ΦI , Φ̄J̄) in (6.42) with
K ′(ΦI , Φ̄J̄) = K(ΦI , Φ̄J̄) + Λ(ΦI)− Λ̄(Φ̄J̄) , ΦI ∂
Λ(Φ) = 2Λ(Φ) , (6.46)
with Λ(Φ) a holomorphic homogeneous function of degree +2. Then, the homogeneity
condition (6.44) turns into
K ′(Φ, Φ̄) = K ′(Φ, Φ̄) + Λ(Φ) + Λ̄(Φ̄) . (6.47)
We can also consider a system of interacting arctic weight-zero multiplets Υ(z, u+)
and their smile-conjugates Υ̃ described by the Lagrangian
L++ = i
J++K(Υ, Υ̃) , (6.48)
with K(ΦI , Φ̄J̄) a real function which is not required to obey any homogeneity condition.
Due to the gauge freedom (6.34), the action is invariant under Kähler transformations of
the form
K(Υ, Υ̃) → K(Υ, Υ̃) +Λ(Υ) + Λ̄(Υ̃) , (6.49)
withΛ a holomorphic function. Such dynamical systems generalize the hyperkähler sigma-
models on cotangent bundles of Kähler manifolds [39, 40, 41].
6.4 Vector multiplet and Chern-Simons couplings
An Abelian vector mulitplet can be described by a weight-zero real projective superfield
V (z, u+) which is required to be holomorphic on CP 1 − {N ∪ S}.
V (z, u+) = 0 , V (z, c u+) = V (z, u+) , c ∈ C∗ . (6.50)
In the North chart, it is characterized by the series (5.22). It is defined to possess the
gauge freedom
V → V + λ+ λ̃ , λ(z, ζ) =
λk(z)ζ
k . (6.51)
with λ(z, u+) an arctic multiplet of weight 0. Using considerations similar to those given
in subsection 5.2, the field strength (compare with the flat superspace expression [25])
W (z) = − 1
u+i du
(u+u−)2
(D̂−)2 − 12ω J−−
V (z, u+) (6.52)
can be shown to be invariant under the projective transformations (5.3). The field strength
turns out to be invariant under the gauge transformations (6.51). In the projective gauge
(u−2 = 0, J
11 = J22 = 0), the field strength takes the form
W (z) = − 1
dζ (D̂1)2V (z, ζ) , (6.53)
compare with the flat superspace result [10].
The AdS transformation law of V ,
δV = −
ξ + i ρJ
V , (6.54)
can be shown to imply that W transforms as
δW = −ξ W (6.55)
under the isometry group.
The field strength can be shown to obey the Bianchi identity
Dγ̂(iDj)
W , (6.56)
and therefore
W = 2ω ε
J (ijDk)
W , (6.57)
compare with the flat superspace case [9, 10]. The Bianchi identity implies that
G++(z, u+) = i
D+α̂W D+
W (D̂+)2W − 2ω J++W 2
(6.58)
is a composite O(2) multiplet,
G++ = 0 , G++(z, u+) = Gij(z) u+i u
j . (6.59)
Let H++(z, u+) be a real O(2) multiplet. Then, similarly to the flat superspace case
[10, 25], the supersymmetric action associated with the Lagrangian
L++ = V (z, u+)H++(z, u+) (6.60)
can be shown to be invariant under the gauge transformations (6.51).
Given several Abelian vector multiplets VI(z, u
+), where I = 1, . . . , n, the composite
superfield (6.58) is generalised to the form:
G++IJ = G
D+α̂WI D+α̂WJ +
W(I (D̂+)2WJ) − 2ω J++WIWJ
G++IJ = 0 , G
IJ (z, u
+) = G
IJ(z) u
j . (6.61)
We then can construct a supersymmetric Chern-Simons action associated with the La-
grangian
L++CS =
cI,JK VI(z, u
+)G++JK(z, u
+) , cI,JK = cI,KJ , (6.62)
for some constant parameters cI,JK (compare with the flat superspace case [10, 25]). In
accordance with the above result, the Chern-Simons action is gauge invariant.
6.5 Tensor multiplet and vector-tensor couplings
Given several O(2) (or, equivalently, tensor) multiplets H++I (z, u
+), a supersymmetric
action is generated by the Lagrangian
L++ = F(H++I ) , I = 1, . . . , n (6.63)
where F(H) is a weakly homogeneous function of first degree in the variables H ,
∂F(H)
−F(H) = αI HI , (6.64)
for some constants α’s.8 Such a Lagrangian occurs in the models for superconformal
tensor multiplets in four [42] and five dimensions [25].
One can also consider systems of coupled vector and tensor multiplets described by a
Lagrangian of the form
L++ = F(H++I ) + VI
cI,JK G
, (6.65)
for some coupling constants κI and cI,JK.
8The projective action principle formulated in subsection 5.2 requires the Lagrangian to be a projective
weight-two multiplet. With αI 6= 0 in (6.64), the Lagrangian (6.63) does not have any definite weight,
and hence the results of subsection 5.2 are not applicable directly. We plan to discuss the case with
αI 6= 0 in more detail somewhere else.
7 Coset space realization
In this section we would like to give an explicit realization for the N = 1 AdS5 superge-
ometry which we have studied in section 2 using the representation-independent approach.
From the group-theoretical point of view, it is known that the N = 1 AdS5 superspace
(or simply AdS5|8) can be identified with the coset space SU(2,2|1)/SO(4,1)×U(1). Us-
ing the formalism of nonlinear realizations9 [44] (or Cartan’s coset construction), here we
introduce a suitable coset representative that makes possible to realize one half of AdS5|8
as a trivial fiber bundle with fibers isomorophic to four-dimensional Minskowski super-
space. This realization should be useful if one is interested in having the 4D N = 1 super
Poincaré symmetry manifest. However, since it corresponds to one half of AdS5|8 (known
as the Poincaré patch [45]), it is not suitable to describe the supersymmetric actions.
The analysis of this section builds on the construction given in [46], see also [47] for
related issues. Note that we use the superform convenctions of [19].
7.1 Coset representative
As is well known, the supergroup SU(2,2|1) is the four-dimensional N = 1 supercon-
formal group. It is generated by Lie-algebra elements of the form (parametrization (7.1)
was used in [48, 49])
β −∆δαβ −ibαβ̇ 2ρα
−iaα̇β −w̄α̇β̇ + ∆̄δα̇β̇ 2ǭα̇
2ǫβ 2ρ̄
2(∆̄−∆)
 , (7.1)
which satisfy the conditions
str X = 0 , BX†B = −X , B =
0 1 0
1 0 0
0 0 −1
 . (7.2)
The matrix elements in (7.1) correspond to a 4D Lorentz transformation (wα
β , w̄α̇
a translation aα̇α, a special conformal transformation bαα̇, a Q–supersymmetry (ǫ
α, ǭα̇),
an S–supersymmetry (ρα, ρ̄α̇), and a combined scale and U(1)–chiral transformation
∆ = 1
9Many years ago, this formalism was also applied to introduce the 4D N = 1 AdS superspace [43, 27].
The explicit parametrization for the algebra su(2, 2|1), which is given in (7.1), is ideally
suited to describe the compactified Minkowski space SU(2,2|1)/(P×C∗), where P denotes
the N = 1 super Poincaré group (generated by the parameters (w, w̄, b, ρ, ρ̄) in (7.1)),
and C∗ denotes the group of scale and chiral transformations generated by the parameters
∆ and ∆̄ in (7.1). In the case of the coset space SU(2,2|1)/SO(4,1)×U(1), however, this
parametrization should be slightly modified. In addition, a re-scaling of some matrix
elements is needed in order to incorporate the AdS curvature ω2 into the formalism.
As is known, a key role in the coset construction for M = G/H is played by a coset
representative S(p) defined to be a smooth mapping S: U → G, for some open domain
U ⊂ M, such that S(p)p0 = p, for any point p ∈ U , where p0 ∈ U is a fixed point having
H as its isotropy group. On topological grounds, it is not always possible to extend U to
the whole coset space M.
As a coset representative, S(z), for AdS5|8 = SU(2,2|1)/SO(4,1)×U(1), following
mainly [46] we choose
S(z) = g(z) · gS · gD
1 0 0
−iωx̃+ 1 2ω
2 θ 0 1
1 2ωηη̄ 2ω
0 1 0
2 η̄ 1
1 0 0
0 0 1
 (7.3)
ωyδ βα 2ωe
ωyηαη̄β̇ 2ω
−iωe− 12ωyx̃α̇β+ e
− 2iω2x̃α̇γ+ ηγ η̄β̇ + 4ωθ̄α̇η̄β̇
θ̄α̇ − iωx̃α̇γ+ ηγ
ωyθβ 2ω
+ 2ωθγηγ η̄β̇
1 + 4ωθγηγ
where xa± = x
a ± iθσaθ̄ denote ordinary 4D N = 1 (anti) chiral bosonic variables. It is
worth pointing out that the coset representative g(z) corresponds to the coset P/SO(3, 1)
and provides a matrix realization10 for 4D N = 1 Minkowski superspace, with coordinates
z = (xa, θα, θ̄α̇). Note that the isotropy group at z = 0 is H = SO(4, 1) × U(1) ∈
SU(2, 2|1) = G, and it is generated by matrices of the form
w − i
b̃ −w̄ 0
0 0 0
τ 1 0 0
0 − i
τ 1 0
0 0 −4i
trw = 0 , w̄ = w† ,
b† = b , τ̄ = τ .
(7.4)
Setting ω = 1 in (7.3) gives the parametrization used in [46].
10It is a curious historic fact that the above matirx realization for 4D N = 1 Minkowski superspace
was introduced by Akulov and Volkov [50] a year before the official discovery of superspace.
Once the coset representative S(p) is chosen, the next step in the coset construction for
M = G/H is to compute the Maurer-Cartan one-form S−1dS which proves to encode all
the information about the geometry of M. Let G and H be the Lie algebras of G and H ,
respectively, and G−H be a complement ofH in G such that [G−H,H] ⊂ G−H. Then, the
Maurer-Cartan one-form can be uniquely decomposed as S−1dS = S−1dS|G−H+S−1dS|H,
where S−1dS|G−H is identified with the vielbein, and S−1dS|H with the connection.
In our case, the vielbein E = S−1dS|G−H and the connection Ω = S−1dS|H are:
S−1 dS = E+Ω ,
ωEy δα
β − i
2 (Eη)α
ωẼα̇β 1
ωEy δ
2 (Ēθ̄)
2 (Eθ)
2 (Ēη̄)β̇ 0
 , (7.5)
β − i
ΩU(1)δα
β − i
Ω̃α̇β −Ω̄α̇
ΩU(1)δ
0 0 −4i
ΩU(1)
 . (7.6)
The components of the vielbein are given by the one-forms
Eαα̇ = eαα̇ e
−ωy(1− e2ωyω2η2η̄2) + 2ieωydηαη̄α̇ + 2ieωydη̄α̇ηα
+4iωeωydθαη
2η̄α̇ + 4iωe
ωydθ̄α̇ηαη̄
2 , (7.7a)
Ey = dy + dθ
µ(−2ηµ) + dθ̄µ̇(−2η̄µ̇) , (7.7b)
α = dθµ δαµe
ωy + em(iωe−
σ̃β̇αm ) , (7.7c)
α = dηµ δαµe
ωy + dθµ δαµ(2ωe
ωyη2) + dθ̄µ̇(2ωe
ωyη̄µ̇ηα) + em(iω2e
ωyη2η̄β̇ σ̃
m ) .(7.7d)
The components of the SO(4,1)×U(1) connection read
β = dθµ(4ωηαδ
µ − 2ωηµδβα) + em(−2iω2ηαη̄β̇σ̃β̇βm − iω2η̄γ̇σ̃γ̇γm ηγδβα) , (7.8a)
Ωαα̇ = −eαα̇ ωe−ωy(1 + ω2e2ωyη2η̄2) + dηα(2iωeωyη̄α̇) + dη̄α̇(2iωeωyηα)
+ dθα(4iω
2eωyη2η̄α̇) + dθ̄α̇(4iω
2eωyηαη̄
2) , (7.8b)
ΩU(1) = dθ
µ( 3iωηµ) + dθ̄µ̇(−3iωη̄µ̇) + em(−3ω2η̄µ̇σ̃µ̇µm ηµ) , (7.8c)
where
em = dxm − i dθµσaµµ̇θ̄µ̇ + i θµσaµµ̇dθ̄µ̇ , (7.9)
is the space-time component of the N = 1 flat superspace vielbein [19].
Note that under a group transformation g ∈ SU(2, 2|1)
g S(z) = S(g · z) ĥ(z; g) ≡ S ′ ĥ , ĥ(z; g) ∈ H , (7.10)
the vielbein E and the connection Ω transform as follows:
E′ = ĥE ĥ−1 , Ω′ = ĥΩ ĥ−1 − (dĥ) ĥ−1 . (7.11)
It is useful to introduce the inverse EA
M of the vielbein supermatrix EM
A implic-
itly used in the previous equations (EA
B = δA
B, EM
N = δM
N ). With the
definitions
em, dy, dθµ, dθ̄µ̇, dη
µ, dη̄µ̇
= EAEA
M , (7.12)
Ea,Ey, (Eθ)
α, (Ēθ̄)α̇, (Eη)
α, (Ēη̄)α̇
= εMEM
A , (7.13)
where em = −1
(σ̃m)α̇αeαα̇ and E
a = −1
(σ̃a)α̇αEαα̇, we find
eαα̇ = Eαα̇ e
ωy(1 + ω2e2ωyη2η̄2) + (Eη)α(−2ie
ωyη̄α̇) + (Ēη̄)α̇(−2ie
ωyηα)
+ (Eθ)α(2iωe
ωyη̄α̇η
2) + (Ēθ̄)α̇(2iωe
ωyηαη̄
2) , (7.14a)
dy = Ey + (Eθ)
α( 2e
ωyηα) + (Ēθ̄)α̇( 2e
ωyη̄α̇) , (7.14b)
dθµ = (Eθ)
ωyδµα(1− 2ω2e2ωyη2η̄2) + (Eη)α δµα(2ωe
ωyη̄2)
+ (Ēη̄)α̇(−2ωe
ωyη̄α̇ηµ) + Ea(−iωeωyη̄ν̇ σ̃ν̇µa ) , (7.14c)
dηµ = (Eη)
ωyδµα(1− 4ω2e2ωyη2η̄2) + (Eθ)α δµα(−2ωe
ωyη2)
+ (Ēθ̄)α̇(−2ωe
ωyη̄α̇ηµ) + Ea(2iω2eωyη̄ν̇ σ̃
2) . (7.14d)
It is also useful to decompose the connection with respect to the curved basis {EA}
Ω βα = (Eθ)
ωy(4ηαδ
γ − 2ηγδβα) + (Eη)γ ωe
ωy(4ηαη̄
2δβγ − 2ηγ η̄2δβα)
+Ea ω2eωy(−2iη̄β̇σ̃β̇βa ηα + iη̄γ̇ σ̃γ̇γa ηγδβα) , (7.15a)
Ωαα̇ = Eαα̇ ω(−1− 2ω2e2ωyη2η̄2) + (Eη)α(4iωe
ωyη̄α̇) + (Ēη̄)α̇(4iωe
ωyηα)
+ (Eθ)α(−4iω2e
ωyη̄α̇η
2) + (Ēθ̄)α̇(−4iω2e
ωyηαη̄
2) , (7.15b)
ΩU(1) = E
a( 3ω2eωyη̄
σ̃β̇βa ηβ) + (Eθ)
α( 3iωe
ωyηα) + (Ēθ̄)α̇(−3iωe
ωyη̄α̇)
+ (Eη)
α( 6iω2e
ωyηαη̄
2) + (Ēη̄)α̇(−6iω2e
ωyη̄α̇η2) . (7.15c)
7.2 SO(4,1)×U(1) covariance
To better understand the relation between the above coset construction and the AdS5|8
supergeometry of section 2, it is necessary to figure out the precise meaning of the
SO(4,1)×U(1) covariance of the vielbein and the connection. We will use several re-
sults which are collected in Appendix A and concern the reduction of 5D spinors into 4D
ones.
First of all, let us recall that choosing g = h ∈ H in relations (7.10, 7.11) gives
ĥ = h = const, and the group transformations (7.11) reduce to
E′ = hE h−1 , Ω′ = hΩ h−1 , h ∈ SO(4, 1)× U(1) . (7.16)
In particular, a 5D Lorentz transformation acts as follows:
E′ = ΛEΛ−1 , Ω′ = ΛΩΛ−1 , (7.17)
where
0 0 1
 , Λα̂β̂ =
Λĉd̂(Σ
β̂ . (7.18)
This transformation law allows us to combine components of the connection into five-
dimensional vector and spinor. Explicitly, we can write
ωEâ(Γâ)α̂
β̂ 2ω
2 Ēβ̂ 0
 , (7.19)
Ωâb̂
0 0 0
 + iΩU(1)
0 0 −4
 , (7.20)
where
Eâ = (Ea,E5) = (Ea,Ey) , (7.21a)
Eα̂ =
(Eη)α
(Ēθ̄)
, Ēα̂ =
α, (Ēη̄)α̇
, (7.21b)
Ωâb̂ = (Ωab,Ωa5) , (7.21c)
Ωab = −(σab)βαΩαβ + (σ̃ab)β̇ α̇Ω̄α̇β̇ , Ωa5 = −12(σ̃
a)α̇αΩαα̇ . (7.21d)
Note that Eâ, Ω
and ΩU(1) are real. It follows that E
â, Eα̂, Ē
α̂, Ω
and ΩU(1)
transform under the 5D Lorentz group SO(4, 1) respectively as a vector, a Dirac spinor,
its Dirac conjugate spinor, an antisymmetric two-tensor and a scalar. Due to (7.21a) we
can identify
x5 ≡ y . (7.22)
Note also that we can combine the two spinors Eα̂ and Ē
α̂ into a 5D pseudo-Majorana
spinor defined as follows:
Eα̂i = (E
i ,−Ēiα̇) , (7.23)
Eα1 = (Eθ)
α , Eα2 = (Eη)
α , Ē
α̇ = (Ēθ̄)α̇ , Ē
α̇ = (Ēη̄)α̇ . (7.24)
It remains to consider the transformation properties of the vielbein and the connection
under the U(1) part of the isotropy group. In accordance with (7.16), they transform as
E′ = ΣEΣ−1 , Ω′ = ΣΩΣ−1 ,
exp(−1
φ i δ)
0 0 e−
 . (7.25)
Clearly Ω is invariant under the U(1) transformation, while E transforms as
ωEâ(Γâ)α̂
β̂ 2ω
eφiEα̂
e−φi Ēβ̂
 , (7.26)
and hence Eâ is invariant. Note also that (7.25) induces induces the following transfor-
mation of Eiα̂:
E′α̂i = [exp(−φiJ)]i
jEα̂j , Ji
j = (σ3)i
. (7.27)
7.3 Representation of covariant derivatives
With the vielbein and the connection having been introduced, we can now construct
the covariant derivatives
b̂ĉM
bcMbc + ΩÂ
b5Mb5
= (Dâ,Diα̂) = (Da, D5, D1α, D̄1α̇, D2α, D̄2α̇) . (7.28)
The vector fields E
are defined by
Eâ, E
MDM ,
,Dµ, D̄
∂η̄µ̇
. (7.29)
Here the supermatrices E
M and EM
Â have been defined in subsection 7.1. It should be
pointed out that (∂m, Dµ, D̄
µ̇) are the 4D N = 1 flat superspace covariant derivatives,
+ iθ̄µ̇∂µµ̇ and D̄
µ̇ = ∂
∂θ̄µ̇
+ iθµ∂̃
µ̇µ. Furthermore, the connection supefields in D
are defined as
ΩU(1) = E
, Ωâb̂ = EÂ Ω
âb̂ . (7.30)
It can be shown that the explicit expressions for the covariant derivatives are as follows:
Da = eωy(1 + ω2e2ωyη2η̄2)∂a − iωeωy(η̄σ̃a)µDµ − iωeωy(ησa)µ̇D̄µ̇
+2iω2eωyη2(η̄σ̃a)
+ 2iω2eωyη̄2(ησa)µ̇
∂η̄µ̇
− 3iω2eωy(ησaη̄)J + ω2eωyηabεbcde(ησcη̄)Mde − ω(1 + 2ω2e2ωyη2η̄2)Ma5 , (7.31a)
, (7.31b)
D1α = e
ωy(1− 2ω2e2ωyη2η̄2)Dα − 2ωe
− 2ωe 12ωyηαη̄µ̇
∂η̄µ̇
+2ηαe
− iωe 52ωyη2(σmη̄)α∂m
− 3ωe 12ωyηα J + 2ωe
ωyηβ(σab)βαMab + 2iω
ωyη2(σaη̄)αMa5 , (7.31c)
D2α = e−
ωy(1− 4ω2e2ωyη2η̄2) ∂
+ 2ωe
ωyη̄2Dα − 2ωe
ωyηαη̄µ̇D̄
µ̇ + ie
ωy(σmη̄)α∂m
− 6ω2e 32ωyηαη̄2J + 2ω2e
ωyηβ η̄2(σab)βαMab − 2iωe
ωy(σaη̄)αMa5 , (7.31d)
D̄α̇1 = e
ωy(1− 2ω2e2ωyη2η̄2)D̄α̇ − 2ωe 12ωyη̄2 ∂
∂η̄α̇
− 2ωe 12ωyη̄α̇ηµ ∂
+2η̄α̇e
− iωe 52ωyη̄2(σ̃mη)α̇∂m
ωyη̄α̇ J + 2ωe
(σ̃ab)β̇α̇Mab + 2iω
ωyη̄2(σ̃aη)α̇Ma5 , (7.31e)
D̄α̇2 = e−
ωy(1− 4ω2e2ωyη2η̄2) ∂
∂η̄α̇
+ 2ωe
ωyη2D̄α̇ − 2ωe 32ωyη̄α̇ηµDµ + ie
y(σ̃mη)α̇∂m
+6ω2e
ωyη̄α̇η2 J + 2ω2e
ωyη̄β̇η
2(σ̃ab)β̇α̇Mab − 2iωe
ωy(σ̃aη)α̇Ma5 . (7.31f)
It is interesting to consider a flat superspace limit, ω → 0, for the covariant derivatives.
In this limit, one finds
= e−UD
eU , U = ηθ + η̄θ̄ , (7.32)
where D
= (∂â, D
α̂) are 5D flat global covariant derivatives,
Diα̂ =
∂θα̂i
− i (Γb̂)
θβ̂i ∂
, (7.33)
with θα̂i = (θ
i ,−θ̄α̇i) and θαi = (θα, ηα).
7.4 Torsion and curvature
Now, we are prepared to demonstrate that the geometry described in the present
section reproduces the geometry of AdS5|8 constructed in section 2.
We proceed by recalling that, in accordance with the coset construction, the torsion
T and curvature R two-forms are defined as follows:
T = dE − Ω ∧E − E ∧Ω , R = dΩ − Ω ∧Ω . (7.34)
Under group transformations (7.10) they transform covariantly
T′ = ĥT ĥ−1 , R′ = ĥR ĥ−1 . (7.35)
Keeping in mind the definition E+Ω = G−1dG, we get
dE+ dΩ = G−1dG ∧G−1dG = E ∧E + E ∧Ω + Ω ∧ E + Ω ∧Ω , (7.36)
from which we obtain
dE = (E ∧E)|G−H + E ∧Ω + Ω ∧ E , dΩ = (E ∧E)|H + Ω ∧Ω , (7.37)
since (E∧Ω+Ω∧E) ∈ G −H and Ω∧Ω ∈ H. Using the previous formulae we are able
to see that the torsion and curvature two-forms are given by simple expressions
T = (E ∧E)|G−H , R = (E ∧E)|H . (7.38)
Therefore, it remains to compute E ∧E.
Direct calculations give
E ∧ E =
ω2Eâ ∧ Eb̂(Σ
β̂ + 4ωE2α̂ ∧Eβ̂1 −iω
2 Eâ ∧ E2γ̂ (Γâ)α̂γ̂
−iω 32 Eγ̂1 ∧ Eb̂ (Γb̂)γ̂ β̂ 4ωE
1 ∧ E2γ̂
 , (7.39)
and this we should represent as E ∧ E = (E ∧ E)|G−H + (E ∧ E)|H. We end up with
ωTâ(Γâ)α̂
β̂ 2ω
2 T2α̂
 , (7.40)
Râb̂
0 0 0
 + iRU(1)
0 0 −4
 , (7.41)
where
Tâ =
∧Eβ̂j
2i εjk(Γâ)
, (7.42)
Tα̂i =
Eĉ ∧ Eβ̂j
ω(σ3)
i (Γĉ)β̂
ω(σ3)
i (Γb̂)γ̂
, (7.43)
Râb̂ =
Ed̂ ∧Eĉ ω2(−δâĉ δb̂d̂ + δ
) + Eδ̂l ∧E
− 4ωεki(σ3) li (Σâb̂)γ̂δ̂
, (7.44)
RU(1) =
Eδ̂l ∧ E
3iωεklε
. (7.45)
Using standard superform definitions [19], we define the components of the torsion and
curvature as follows:
TÂ = 1
EĈ ∧ EB̂ T
Â , (7.46)
Râb̂ = 1
ED̂ ∧EĈ R
âb̂ , RU(1) =
ED̂ ∧ EĈ (RU(1))ĈD̂ . (7.47)
Now, let us return to the covariant derivatives described in section 2. Their algebra
given by eqs. (2.21a–2.21c) can be represented concisely as
+ i (RU(1))ÂB̂ J +
ĉd̂M
. (7.48)
Comparing (2.21a–2.21c) with eqs. (7.42–7.45), we find that all the components of the
torsion and curvature coincide provided
J ij = (σ3)
j . (7.49)
This completes our analysis of the coset construction.
Acknowledgements:
This work is supported in part by the Australian Research Council and by a UWA research
grant.
A 5D Conventions
Our 5D notation and conventions correspond to [10]. The 5D gamma-matrices Γm̂ =
(Γm,Γ5), with m = 0, 1, 2, 3, are defined by
{Γm̂ , Γn̂} = −2ηm̂n̂ 1 , (Γm̂)† = Γ0 Γm̂ Γ0 (A.1)
are chosen in accordance with
(Γm)α̂
0 (σm)αβ̇
(σ̃m)
α̇β 0
, (Γ5)α̂
−i δαβ 0
0 i δα̇
, (A.2)
such that Γ0Γ1Γ2Γ3Γ5 = 1. The charge conjugation matrix, C = (ε
α̂β̂), and its inverse,
C−1 = C† = (ε
) are defined by
C Γm̂C
−1 = (Γm̂)
T , εα̂β̂ =
εαβ 0
0 −εα̇β̇
εαβ 0
0 −εα̇β̇
. (A.3)
The antisymmetric matrices εα̂β̂ and ε
are used to raise and lower the four-component
spinor indices.
A Dirac spinor, Ψ = (Ψα̂), and its Dirac conjugate, Ψ̄ = (Ψ̄
α̂) = Ψ† Γ0, look like
Ψα̂ =
, Ψ̄α̂ = (φα , ψ̄α̇) . (A.4)
One can now combine Ψ̄α̂ = (φα, ψ̄α̇) and Ψ
α̂ = εα̂β̂Ψ
= (ψα,−φ̄α̇) into a SU(2) doublet,
Ψα̂i = (Ψ
i ,−Ψ̄α̇i) , (Ψαi )∗ = Ψ̄α̇i , i = 1, 2 , (A.5)
with Ψα1 = φ
α and Ψα2 = ψ
α. It is understood that the SU(2) indices are raised and
lowered by εij and εij , ε
12 = ε21 = 1, in the standard fashion: Ψ
α̂i = εijΨα̂j . The Dirac
spinor Ψi = (Ψiα̂) satisfies the pseudo-Majorana condition Ψ̄i
T = C Ψi. This will be
concisely represented as
(Ψiα̂)
∗ = Ψα̂i . (A.6)
With the definition Σm̂n̂ = −Σn̂m̂ = −14 [Γm̂,Γn̂], the matrices {1,Γm̂,Σm̂n̂} form a
basis in the space of 4 × 4 matrices. The matrices ε
and (Γm̂)α̂β̂ are antisymmetric,
εα̂β̂ (Γm̂)α̂β̂ = 0, while the matrices (Σm̂n̂)α̂β̂ are symmetric.
It is useful to write explicitly the 4D reduction of these matrices
(Γm)α̂β̂ =
0 −(σm)αβ̇
(σm)β
, (Γ5)α̂β̂ =
i εαβ 0
0 i εα̇β̇
, (A.7)
(Σmn)α̂
(σmn)α
0 (σ̃mn)
, (Σm5)α̂
0 − i
(σm)αβ̇
(σ̃m)
α̇β 0
, (A.8)
(Σmn)α̂β̂ =
(σmn)αβ 0
0 −(σ̃mn)α̇β̇
, (Σm5)α̂β̂ =
(σm)α
(σm)β
, (A.9)
where (σmn)α
β = −1
(σmσ̃n − σnσ̃m)αβ and (σ̃mn)α̇β̇ = −14(σ̃mσn − σ̃nσm)
Given a 5-vector V m̂ and an antisymmetric tensor F m̂n̂ = −F n̂m̂, we can equivalently
represent them as the bi-spinors V = V m̂ Γm̂ and F =
F m̂n̂ Σm̂n̂ with the following
symmetry properties
, εα̂β̂ V
= 0 , F
. (A.10)
The two equivalent descriptions Vm̂ ↔ Vα̂β̂ and and Fm̂n̂ ↔ Fα̂β̂ are explicitly described
as follows:
= V m̂ (Γm̂)α̂β̂ , Vm̂ = −
(Γm̂)
α̂β̂ V
F m̂n̂(Σm̂n̂)α̂β̂ , Fm̂n̂ = (Σm̂n̂)
α̂β̂ F
. (A.11)
These results can be easily checked using the identities
α̂β̂γ̂δ̂
+ εα̂γ̂ εδ̂β̂ + εα̂δ̂ εβ̂γ̂ ,
εα̂γ̂ εβ̂δ̂ − εα̂δ̂ εβ̂γ̂ = −
(Γm̂)
(Γm̂)γ̂δ̂ +
, (A.12)
and therefore
α̂β̂γ̂δ̂
(Γm̂)
(Γm̂)γ̂δ̂ +
, (A.13)
with ε
α̂β̂γ̂δ̂
the completely antisymmetric fourth-rank tensor.
Complex conjugation gives
)∗ = −εα̂β̂ , (V
)∗ = V α̂β̂ , (F
)∗ = F α̂β̂ , (A.14)
provided V m̂ and F m̂n̂ are real.
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	Introduction
	Covariant derivatives
	Killing supervectors
	Harmonic superspace approach
	Analytic multiplets
	Harmonic action principle
	Projective superspace approach
	Projective multiplets
	Projective action principle
	Dynamical systems in projective superspace
	Projective multiplets revisited
	Projective action
	Nonlinear sigma-models
	Vector multiplet and Chern-Simons couplings
	Tensor multiplet and vector-tensor couplings
	Coset space realization
	Coset representative
	SO(4,1)U(1) covariance
	Representation of covariant derivatives
	Torsion and curvature
	5D Conventions
