Supersymmetric Grassmannian Sigma Model
93
W (x + , θ + ) = Z(x + ) + iθ + ηA(x + ) ,
(1)
where η is a fermionic constant and Z(x + ) and A(x + ) are usual N × M matrices
that depend on x + .
We now assume, as for the special case M = 1, that the susy Gaussian curvature
of the surface corresponding to the susy holomorphic solution W is given by the
formula
˜
κ = −
1
˜
g
∂ + ∂ − ln ˜
g,
(2)
where the susy expression of the metric is ˜
g = ∂ + ∂ − ln
det
W † W
.
Thus asking for a CCH solution is equivalent to assuming that ˜
κ = κ where κ is
a purely bosonic constant (a strictly positive real number) and must be the curvature
associated with the non-susy G(M, N ) solution Z involved in W = (1).
Let us write explicitly the condition (2) using the expression of W in (1) and
taking into account that ˜
κ = κ. In order to simplify the calculations, we take T 1 =
θ + η and T 2 = θ − η † . Notice that since T 1 and T 2 are both product of two fermionic
quantities, we have T 2
1 = 0 and T 2
2 = 0. Moreover, they are bosonic and hence
commute with all the other quantities.
We thus easily get
det
W
† W
= (det M 0 ) det
I M + iT 1 M
−1
0 M 1 + iT 2 M
−1
0 M 2 − T 1 T 2 M
−1
0 M 3
= (det M 0 ) (1 + iT 1 X 1 + iT 2 X 2 − T 1 T 2 X 3 ) ,
(3)
with M 0 = Z † Z, M 1 = Z † A, M 2 = A † Z and M 3 = A † A. The expressions of
X 1 , X 2 and X 3 remain to be explicitly computed.
The metric ˜
g = ∂ + ∂ − ln
det
W † W
takes the form
˜
g = g + ∂ + ∂ − ln (1 + iT 1 X 1 + iT 2 X 2 − T 1 T 2 X 3 ) ,
(4)
with g = ∂ + ∂ − ln(det M 0 ). Using the Taylor expansion of the logarithmic function
ln (1 + x) = x −
x 2
2 + O(x 3 ), we get
˜
g = g + ∂ + ∂ − [iT 1 X 1 + iT 2 X 2 − T 1 T 2 (X 3 − X 1 X 2 )] .
(5)
By a similar procedure we can express the quantity ∂ + ∂ − ln ˜
g as
∂ + ∂ − ln ˜
g = ∂ + ∂ − ln g + iT 1 ∂ + ∂ − Y 1 + iT 2 ∂ + ∂ − Y 2
−T 1 T 2 ∂ + ∂ −
Y 3 − Y 1 Y 2
,
(6)