AdS superprojectors
Published for SISSA by
Springer
Received: February 9, 2021
Accepted: March 7, 2021
Published: April 9, 2021
E.I. Buchbinder, D. Hutchings, S.M. Kuzenko and M. Ponds
Department of Physics M013, The University of Western Australia,
35 Stirling Highway, Perth W.A. 6009, Australia
E-mail: ,
, ,
Abstract: Within the framework of N = 1 anti-de Sitter (AdS) supersymmetry in four
dimensions, we derive superspin projection operators (or superprojectors). For a tensor
superfield Vα(m)α̇(n) := V(α1 ...αm )(α̇1 ...α̇n ) on AdS superspace, with m and n non-negative
integers, the corresponding superprojector turns Vα(m)α̇(n) into a multiplet with the properties of a conserved conformal supercurrent. It is demonstrated that the poles of such
superprojectors correspond to (partially) massless multiplets, and the associated gauge
transformations are derived. We give a systematic discussion of how to realise the unitary
and the partially massless representations of the N = 1 AdS4 superalgebra osp(1|4) in terms
of on-shell superfields. As an example, we present an off-shell model for the massive gravitino multiplet in AdS4 . We also prove that the gauge-invariant actions for superconformal
higher-spin multiplets factorise into products of minimal second-order differential operators.
Keywords: Supergravity Models, Superspaces
ArXiv ePrint: 2101.05524
Open Access, c The Authors.
Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP04(2021)074
JHEP04(2021)074
AdS superprojectors
Contents
1
2 Representations of the AdS (super)algebra
2.1 Unitary representations of so(3, 2) and osp(1|4)
2.2 On-shell fields in AdS
2.3 Partially massless and massive fields
3
3
5
6
3 AdS superprojectors
3.1 AdS superspace
3.2 Construction of superspin projection operators
3.3 Decomposing unconstrained superfields into irreducible parts
7
7
10
14
4 On-shell supermultiplets in AdS
4.1 Massless supermultiplets
4.2 Partially massless supermultiplets
4.3 Massive supermultiplets
4.4 Equivalent representations and reality conditions
15
16
17
20
20
5 Component analysis
5.1 Massive supermultiplets
5.2 Partially massless supermultiplets
5.3 Massless supermultiplets
5.4 Wess-Zumino supermultiplet
5.5 Massive vector multiplet
22
22
23
25
25
27
6 Massive gravitino supermultiplet
30
7 Factorisation of superconformal higher-spin actions
32
8 Conclusion
35
A AdS superspace toolkit
35
B Partially massless gauge symmetry
B.1 The non-supersymmetric case
B.2 The supersymmetric case
37
38
39
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JHEP04(2021)074
1 Introduction
1
Introduction
∂ β β̇ ϕT
= 0.
βα(m−1)β̇ α̇(n−1)
(1.1)
This constraint is the characteristic feature of a conserved current jα(m)α̇(n) .
The four-dimensional results of [1–6] correspond to the Poincaré supersymmetry. Not
much is known about the structure of superprojectors corresponding to the AdS 4 supersymmetry OSp(1|4). More precisely, forty years ago Ivanov and Sorin [15] introduced the
so-called transverse linear and longitudinal linear superfields on N = 1 AdS superspace
AdS4|4 and presented the projectors which single out such supermultiplets. We recall that
a complex tensor superfield Γα(m)α̇(n) on AdS4|4 is said to be transverse linear if it satisfies
the constraint
D̄β̇ Γα(m)β̇ α̇(n−1) = 0
⇐⇒
(D̄2 − 2(n + 2)µ)Γα(m)α̇(n) = 0 ,
n > 0,
(1.2)
where µ 6= 0 is a constant parameter which determines the curvature of AdS4|4 , see section 3.1 below. A complex tensor superfield Gα(m)α̇(n) is said to be longitudinal linear if it
satisfies the constraint
D̄(α̇1 Gα(m)α̇2 ...α̇n+1 ) = 0
⇐⇒
(D̄2 + 2nµ)Gα(m)α̇(n) = 0 .
(1.3)
For n = 0 the first constraint in (1.2) is not defined, while the second condition
(D̄2 − 4µ)Γα(m) = 0
(1.4)
defines a linear superfield. For n = 0 the constraint (1.3) defines a chiral superfield. The
constraints (1.2), (1.3) and (1.4) are the only differential constraints in AdS4|4 which define off-shell supermultiplets with unconstrained component fields [15]. Certain transverse
1
Ref. [7] was a natural extension of the work [12], where the spin projection operators in three dimensions
were constructed and used to obtain simple expressions for the higher-spin Cotton tensors.
2
Refs. [13, 14] made use of the four-vector notation in conjunction with the four-component spinor
formalism, which resulted in rather complicated expressions for the spin projection operators. However,
switching to the two-component spinor formalism leads to remarkably simple and compact expressions for
these projectors [3, 6].
–1–
JHEP04(2021)074
Superprojectors [1–7] are superspace projection operators which single out irreducible representations of supersymmetry. Various applications of such operators have appeared in the
literature, including the following constructions: (i) superfield equations of motion [8, 9];
(ii) gauge-invariant actions in four dimensions [10, 11]; and (iii) off-shell N -extended superconformal actions in three dimensions [7].1 Of special interest are those superprojectors which single out the highest superspin of tensor superfields Vα(m)α̇(n) (x, θ, θ̄) :=
Vα1 ...αm α̇1 ...α̇n (x, θ, θ̄) = V(α1 ...αm )(α̇1 ...α̇n ) (x, θ, θ̄), since they may be viewed as supersymmetric extensions of the Behrends-Fronsdal spin projection operators [13, 14]. We recall
that a tensor field ϕα(m)α̇(n) (x) of Lorentz type (m/2, n/2) is mapped by the corresponding
Behrends-Fronsdal projector2 to a transverse field ϕT
α(m)α̇(n) (x) such that
Dβ Jβα(m−1)α̇(n) = 0
D̄β̇ Jα(m)β̇ α̇(n−1) = 0
⇐⇒
D2 − 2(m + 2)µ̄ Jα(m)α̇(n) = 0 ,
(1.5a)
⇐⇒
D̄2 − 2(n + 2)µ Jα(m)α̇(n) = 0 .
(1.5b)
If m = n, it is consistent to restrict Jα(n)α̇(n) to be real, J¯α(n)α̇(n) = Jα(n)α̇(n) . The
m = n = 1 case corresponds to the ordinary conformal supercurrent [21]. The case
m = n > 1 was first described in Minkowski superspace in [22] (see also [23]) and extended
to AdS4|4 in [24]. The case m = n + 1 > 1 was first described in Minkowski superspace
in [23] and extended to AdS4|4 in [24].
If m > n = 0, the constraints (1.5) should be replaced with
Dβ Jβα(m−1) = 0
⇐⇒
D2 − 2(m + 2)µ̄ Jα(m) = 0 ,
(D̄ − 4µ)Jα(m) = 0 .
(1.6a)
(1.6b)
2
The m = 1 case was first considered in [25], where it was shown that the spinor supercurrent
Jα naturally originates from the reduction of the conformal N = 2 supercurrent [26] to
N = 1 superspace.
Finally, for m = n = 0 the constraints (1.6) should be replaced with
(D2 − 4µ̄)J = 0 ,
(1.7a)
(D̄2 − 4µ)J = 0 .
(1.7b)
These constraints describe a flavour current supermultiplet [27] in AdS4 . Irreducible supermultiplets of the types (1.5), (1.6) and (1.7) have been used in [28] for the covariant
quantisation of the massless supersymmetric higher-spin gauge theories in AdS 4 [16].
The Behrends-Fronsdal projectors have been generalised to the case of AdS4 only
recently in [29]. One of the important outcomes of [29] was a new understanding of the
so-called partially massless fields in AdS4 . Such fields in diverse dimensions were studied
earlier in [30–43 (...truncated)
