Color Extension of SUSY QM
203
condition of Theorem 1. The model is formulated by using matrix representations
of the Clifford algebra. For a given positive integer n we introduce the N = 2n
Hermitian block-antidiagonal matrices subject to the relations:
{γ I , γ J } = 2δ I J I 2 n ,
γ
†
I = γ I ,
(10)
where I, J run from 1 to N . We mainly work on an alternative choice of the basis
of γ -matrices:
γ
±
a =
1
2
(γ 2a−1 ± iγ 2a ), a = 1, 2, . . . , n.
(11)
In this basis the relation (10) reads as follows:
{γ
±
a , γ
±
b } = 0,
{γ
+
a , γ
−
b } = δ ab I 2 n .
(12)
We also consider n Hermitian block-diagonal matrices given by a product of γ I ’s:
Γ a = i
a γ 1 γ 2 . . . γ 2a , a = 1, 2, . . . , n.
(13)
It is then immediate to verify that
Γ
2
a = I 2 n ,
[Γ a , Γ b ] = 0
(14)
and
[γ
±
k , Γ a ] = 0 (k > a),
{γ
±
k , Γ a } = 0 (k ≤ a).
(15)
The N supercharges of Akulov–Kudinov model are defined by the matrices γ ±
a
and Γ a as follows:
Q
+
a =
1
√
2
γ
+
a (p + iW
(n)
a (x, Γ 1 , . . . , Γ n )),
Q
−
a = (Q
+
a )
† .
(16)
The superpotentials W
(n)
a are defined recursively. For instance, for n = 1 W (1) is
chosen to be W (1) = w 0 (x) and for n = 2
W
(2)
1 = w 0 (x) + Γ 2 w 1 (x), w 1 (x) =
∂ x w 0 (x)
2w 0 (x)
(17)
and so on. It is seen from (15) that Γ n anticommutes with all the supercharges:
{Q
±
a , Γ n } = 0, ∀a.
(18)
Thus one may apply Theorem 1 to Akulov–Kudinov model.