I. Florakis et al. / Nuclear Physics B 890 (2015) 279–301 283
The algebra (2.5), (2.11) and (2.12) is the hyperspace counterpart of the conventional super-
Poincaré algebra enlarged by dilatations. That this is so can be most easily seen by considering,
e.g. n = 2(i.e. μ = 1, 2), in which case this algebra is recognized as the D = 3 super-Poincaré al-
gebra with L
μ
ν
−
1
2
δ
ν
μ
L
ρ
ρ
= M
m
(γ
m
)
μ
ν
generating the SL(2, R) ∼ SO(1, 2) Lorentz rotations
(note that m = 0, 1, 2) and D =
1
2
L
ρ
ρ
being the dilatation generator. Note that the factor
1
2
in
the definition of the dilatation generator is required in order to have the canonical scaling of the
momentum generator P
μν
with weight 1 and the supercharge Q
μ
with weight
1
2
, as follows from
Eq. (2.11).
This algebra may be further extended to the OSp(1|2n) algebra, generating generalized su-
perconformal transformations of the flat hyper-superspace, by adding the additional set of super-
symmetry generators
S
μ
=−
X
μν
+
i
2
θ
μ
θ
ν
Q
ν
, (2.13)
together with the generalized conformal boosts
K
μν
= i
X
μρ
+
i
2
θ
μ
θ
ρ
X
νλ
+
i
2
θ
ν
θ
λ
∂
ρλ
− iθ
(μ
S
ν)
. (2.14)
The generators S
μ
and K
μν
form a superalgebra similar to (2.5)
S
μ
,S
ν
=−2K
μν
,
S
μ
,K
νρ
= 0,
K
μν
,K
ρλ
= 0, (2.15)
while the non-zero (anti)commutators of S
μ
and K
μν
with Q
μ
, P
μν
and L
μ
ν
read
Q
μ
,S
ν
=−L
μ
ν
,
S
μ
,P
νρ
= iδ
μ
(ν
Q
ρ)
,
Q
μ
,K
νρ
=−iδ
(ν
μ
S
ρ)
,
S
μ
,L
ν
ρ
= iδ
μ
ν
S
ρ
. (2.16)
2.2. Generalized superconformal algebra OSp(1|2n)
We now collect together all the non-zero (anti)commutation relations among the generators
of the OSp(1|2n) algebra
{Q
μ
,Q
ν
}=2P
μν
, [Q
μ
,P
νρ
]=0, [P
μν
,P
ρλ
]=0,
S
μ
,S
ν
=−2K
μν
,
S
μ
,K
νρ
= 0,
K
μν
,K
ρλ
= 0,
Q
μ
,S
ν
=−L
μ
ν
,
S
μ
,P
νρ
= iδ
μ
(ν
Q
ρ)
,
Q
μ
,K
νρ
=−iδ
(ν
μ
S
ρ)
,
P
μν
,L
λ
ρ
=−i
δ
ρ
μ
P
νλ
+ δ
ρ
ν
P
μλ
,
Q
μ
,L
ν
ρ
=−iδ
ρ
μ
Q
ν
S
μ
,L
ν
ρ
= iδ
μ
ν
S
ρ
,
L
ν
μ
,L
λ
ρ
= i
δ
μ
λ
L
ν
ρ
− δ
ρ
ν
L
λ
μ
,
K
μν
,L
λ
ρ
= i
δ
μ
λ
K
νρ
+ δ
ν
λ
K
μρ
,
P
μν
,K
λρ
=
i
4
δ
ρ
μ
L
ν
λ
+ δ
ρ
ν
L
μ
λ
+ δ
λ
μ
L
ν
ρ
+ δ
λ
ν
L
μ
ρ
. (2.17)
Let us note that in the case n = 4, in which the physical space–time is four-dimensional (see
Eq. (2.1)) the generalized superconformal group OSp(1|8) contains the D = 4 conformal sym-
metry group SO(2, 4) ∼ SU(2, 2) as a subgroup, but not the superconformal group SU(2, 2|1).
The reason being that, although OSp(1|8) and SU(2, 2|1) contain the same number of (eight)
generators, the anticommutators of the former close on the generators of the whole Sp(8), while