In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, superconformal algebras are finite-dimensional and generate the superconformal group (in two Euclidean dimensions, the Lie superalgebra does not generate any Lie supergroup).
The conformal group of the
(
p
+
q
)
-dimensional space
R
p
,
q
is
S
O
(
p
+
1
,
q
+
1
)
and its Lie algebra is
s
o
(
p
+
1
,
q
+
1
)
. The superconformal algebra is a Lie superalgebra containing the bosonic factor
s
o
(
p
+
1
,
q
+
1
)
and whose odd generators transform in spinor representations of
s
o
(
p
+
1
,
q
+
1
)
. Given Kač's classification of finite-dimensional simple Lie superalgebras, this can only happen for small values of
p
and
q
. A (possibly incomplete) list is
o
s
p
∗
(
2
N
|
2
,
2
)
in 3+0D thanks to
u
s
p
(
2
,
2
)
≃
s
o
(
4
,
1
)
;
o
s
p
(
N
|
4
)
in 2+1D thanks to
s
p
(
4
,
R
)
≃
s
o
(
3
,
2
)
;
s
u
∗
(
2
N
|
4
)
in 4+0D thanks to
s
u
∗
(
4
)
≃
s
o
(
5
,
1
)
;
s
u
(
2
,
2
|
N
)
in 3+1D thanks to
s
u
(
2
,
2
)
≃
s
o
(
4
,
2
)
;
s
l
(
4
|
N
)
in 2+2D thanks to
s
l
(
4
,
R
)
≃
s
o
(
3
,
3
)
;
real forms of
F
(
4
)
in five dimensions
o
s
p
(
8
∗
|
2
N
)
in 5+1D, thanks to the fact that spinor and fundamental representations of
s
o
(
8
,
C
)
are mapped to each other by outer automorphisms.
According to the superconformal algebra with
N
supersymmetries in 3+1 dimensions is given by the bosonic generators
P
μ
,
D
,
M
μ
ν
,
K
μ
, the U(1) R-symmetry
A
, the SU(N) R-symmetry
T
j
i
and the fermionic generators
Q
α
i
,
Q
¯
i
α
˙
,
S
i
α
and
S
¯
α
˙
i
. Here,
μ
,
ν
,
ρ
,
…
denote spacetime indices;
α
,
β
,
…
left-handed Weyl spinor indices;
α
˙
,
β
˙
,
…
right-handed Weyl spinor indices; and
i
,
j
,
…
the internal R-symmetry indices.
The Lie superbrackets of the bosonic conformal algebra are given by
[
M
μ
ν
,
M
ρ
σ
]
=
η
ν
ρ
M
μ
σ
−
η
μ
ρ
M
ν
σ
+
η
ν
σ
M
ρ
μ
−
η
μ
σ
M
ρ
ν
[
M
μ
ν
,
P
ρ
]
=
η
ν
ρ
P
μ
−
η
μ
ρ
P
ν
[
M
μ
ν
,
K
ρ
]
=
η
ν
ρ
K
μ
−
η
μ
ρ
K
ν
[
M
μ
ν
,
D
]
=
0
[
D
,
P
ρ
]
=
−
P
ρ
[
D
,
K
ρ
]
=
+
K
ρ
[
P
μ
,
K
ν
]
=
−
2
M
μ
ν
+
2
η
μ
ν
D
[
K
n
,
K
m
]
=
0
[
P
n
,
P
m
]
=
0
where η is the Minkowski metric; while the ones for the fermionic generators are:
{
Q
α
i
,
Q
¯
β
˙
j
}
=
2
δ
i
j
σ
α
β
˙
μ
P
μ
{
Q
,
Q
}
=
{
Q
¯
,
Q
¯
}
=
0
{
S
α
i
,
S
¯
β
˙
j
}
=
2
δ
j
i
σ
α
β
˙
μ
K
μ
{
S
,
S
}
=
{
S
¯
,
S
¯
}
=
0
{
Q
,
S
}
=
{
Q
,
S
¯
}
=
{
Q
¯
,
S
}
=
0
The bosonic conformal generators do not carry any R-charges, as they commute with the R-symmetry generators:
[
A
,
M
]
=
[
A
,
D
]
=
[
A
,
P
]
=
[
A
,
K
]
=
0
[
T
,
M
]
=
[
T
,
D
]
=
[
T
,
P
]
=
[
T
,
K
]
=
0
But the fermionic generators do carry R-charge:
[
A
,
Q
]
=
−
1
2
Q
[
A
,
Q
¯
]
=
1
2
Q
¯
[
A
,
S
]
=
1
2
S
[
A
,
S
¯
]
=
−
1
2
S
¯
[
T
j
i
,
Q
k
]
=
−
δ
k
i
Q
j
[
T
j
i
,
Q
¯
k
]
=
δ
j
k
Q
¯
i
[
T
j
i
,
S
k
]
=
δ
j
k
S
i
[
T
j
i
,
S
¯
k
]
=
−
δ
k
i
S
¯
j
Under bosonic conformal transformations, the fermionic generators transform as:
[
D
,
Q
]
=
−
1
2
Q
[
D
,
Q
¯
]
=
−
1
2
Q
¯
[
D
,
S
]
=
1
2
S
[
D
,
S
¯
]
=
1
2
S
¯
[
P
,
Q
]
=
[
P
,
Q
¯
]
=
0
[
K
,
S
]
=
[
K
,
S
¯
]
=
0
There are two possible algebras with minimal supersymmetry in two dimensions; a Neveu–Schwarz algebra and a Ramond algebra. Additional supersymmetry is possible, for instance the N = 2 superconformal algebra.
