In 1 + 1 dimensions the N = 1 supersymmetry algebra (also known as
N
=
(
1
,
1
)
because we have one left-moving SUSY generator and one right moving one) has the following generators:
supersymmetric charges:
Q
,
Q
¯
supersymmetric central charge:
Z
time translation generator:
H
space translation generator:
P
boost generator:
N
fermionic parity:
Γ
unit element:
I
The following relations are satisfied by the generators:
{
Γ
,
Γ
}
=
2
I
{
Γ
,
Q
}
=
0
{
Γ
,
Q
¯
}
=
0
{
Q
,
Q
¯
}
=
2
Z
{
Q
,
Q
}
=
2
(
H
+
P
)
{
Q
¯
,
Q
¯
}
=
2
(
H
−
P
)
[
N
,
Q
]
=
1
2
Q
[
N
,
Q
¯
]
=
−
1
2
Q
¯
[
N
,
Γ
]
=
0
[
N
,
H
+
P
]
=
H
+
P
[
N
,
H
−
P
]
=
−
(
H
−
P
)
Z
is a central element.
The supersymmetry algebra admits a
Z
2
-grading. The generators
H
,
P
,
N
,
Z
,
I
are even (degree 0), the generators
Q
,
Q
¯
,
Γ
are odd (degree 1).
2(H − P) gives the left-moving momentum and 2(H + P) the right-moving momentum.
Basic representations of this algebra are the vacuum, kink and boson-fermion representations, which are relevant e.g. to the supersymmetric (quantum) sine-Gordon model.
