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with
Y 1 ≡
κ
2
(1 + |x|
2 )
2 ∂ + ∂ − X 1 , Y 2 ≡
κ
2
(1 + |x|
2 )
2 ∂ + ∂ − X 2 ,
Y 3 ≡
κ
2
(1 + |x|
2 )
2 ∂ + ∂ −
X 3 − X 1 X 2
.
(7)
Upon inserting these relations into (2) we get the following constraints
∂ + ∂ − ln g + κg = 0 ,
(8)
∂ + ∂ − (Y 1 + κX 1 ) = 0 ,
∂ + ∂ − (Y 2 + κX 2 ) = 0 ,
(9)
∂ + ∂ − ((Y 3 − Y 1 Y 2 ) + κ(X 3 − X 1 X 2 )) = 0 .
(10)
Notice that the two expressions in (9) are complex conjugate to each other and
hence we have only one independent condition, say the one involving Y 1 and X 1 .
These are necessary and sufficient conditions for the susy holomorphic solutions to
have a constant Gaussian curvature and will be the fundamental equations for our
analysis.
2.1 Susy Invariant Solutions
Here we give a sufficient condition for obtaining CCH solutions. This result
generalizes what we already proved in the case M = 1 [11]. We assume that the
susy holomorphic solution is given by
W (x + , θ + ) = Z(x + ) + iθ + η∂ + Z(x + ),
(11)
i.e. A(x + ) = ∂ + Z(x + ) in (1), where Z is a CCH solution of the non-susy model.
Using the MacFarlane parametrization [14], we can rewrite (11) as
W =
I M
K + iθ + η∂ + K
.
(12)
Then we prove that W = (12) is a CCH solution of the susy G(M, N ) model.
Remember here that, det
Z † Z
= det
I M + K † K
=
1 + |x| 2 r , for some
positive integer r and thus ˜
κ = κ =
2
r .
3 CCH Solutions of the Susy G(2, 4) σ-Model
The non-susy CCH solutions are given in [9] as