204
N. Aizawa et al.
Proposition 1 Let H be Hamiltonian of Akulov–Kudinov model: {Q +
a , Q
−
b } =
δ ab H and A =
0 Γ n
Γ n 0
. The complex vector space spanned by the matrix
differential operators
Q
±
a = I 2 ⊗ Q
±
a ,
˜
Q
±
a = iQ
±
a A, H = I 2 ⊗ H,
˜
H = HA
(19)
forms a Z 2
2 -graded superalgebra having the following non-vanishing relations:
{Q
+
a , Q
−
b } = { ˜
Q
+
a , ˜
Q
−
b } = δ ab H,
[Q
±
a , ˜
Q
∓
b ] = iδ ab ˜
H.
(20)
The assignment of Z 2
2 -degree is
deg(H) = (0, 0), deg(Q
±
a ) = (0, 1), deg( ˜
Q
±
a ) = (1, 0), deg( ˜
H) = (1, 1).
(21)
Thus (19) gives a N -extended version of Bruce–Duplij model. The Bruce–Duplij
model is recovered from (19) by setting n = 1 (N = 2) and the identification
Q 01 =
1
√
2
(Q
+
a + Q
−
a ),
Q 10 =
1
√
2
( ˜
Q
+
a + ˜
Q
−
a ).
(22)
More detailed analysis of N = 2 case is found in [7].
3.3 Z 2
2
-Graded SCM
Now we consider Z 2
2 -graded version of SCM by using Theorem 1. Many models of
SCM have been obtained so far (see, for instance, [2]). Some of the models, e.g. the
ones in [9, 10], satisfy the condition (6) so that we may have models of Z 2
2 -graded
SCM of N = 2, 4, 8 and so on.
As an example, we here present N = 1 model with osp(1|2) symmetry:
Q =
1
√
2
σ 1 p − σ 2
β
x
,
S =
x
√
2
σ 1 ,
H =
1
2
p
2
+
β 2
x 2
I 2 +
β
2x 2 σ 3 , D = −
1
4
{x, p} I 2 , K =
x 2
2
I 2 , (23)
where σ i is Pauli matrices and β ∈ R is a coupling constant. Conformal subalgebra
so(1, 2) is given by H, D, K For this realization of osp(1|2) one may
immediately see that Γ = σ 3 commute with Q and S. Thus Theorem 1 gives us
the following operators which is a model of Z 2
2 -graded SCM: