Supersymmetric Grassmannian Sigma Model
99
for arbitrary functions f and g of given variables. Requiring it to be satisfied when
x + = 0 and x − = 0 separately we obtain β 21 (x + ) = γ 1 x + , where γ 1 is an arbitrary
constant. Upon introducing it into (32) we get f (x + ) + g(x − ) =
4|x| 2 γ 2
1
(1+|x| 2 ) 2 , which
immediately implies that γ 1 = 0 and hence β 21 = 0.
The necessary and sufficient conditions (9) and (10) are thus satisfied and finally
the CCH solution W 3 is given by the form
W 3 = Z 3 + iθ +
√
3ηβ 22 (x + )∂ + Z 3 .
(33)
Hence in this case we have obtained the susy invariant solution as the unique CCH
solution.
3.4 The Case of Z 4
In this case we have det Z
†
4 Z 4 =
1 + |x| 2 4 , i.e. r = 4 and κ =
1
2 . Again the
condition (9) becomes a third degree polynomial in x − after introducing the solution
W 4 given in (14). Similarly as what we did with W 3 , we equate the coefficients of
different powers of x − to zero and now get
β 11 (x + ) = 3x
2
+ β 22 (x + ),
β 21 (x + ) = −β 12 (x + ) + 2
√
3x + β 22 (x + ). (34)
In order to solve the last condition (10) we introduce (34) into it and find that
β 21 (x + ) = β 12 (x + ). Finally, the CCH solution W 4 is given as
W 4 = Z 4 + iθ +
1
2
ηβ 22 (x + )∂ + Z 4 .
(35)
Thus, in the case of Z 4 once more we have obtained the susy invariant solution as
the unique CCH solution.
4 Conclusions and Final Comments
In this article we give some criteria for having CCH solutions of the susy Grassmannian G(M, N ) σ -model. With the help of the susy gauge invariance of the model we
first show that the susy holomorphic solution given in (11) (i.e., generalization of
non-susy holomorphic solution) leads to a constant curvature surface. This kind of a
solution is called a susy invariant one, in analogy with the discussion given in [11].
Then we restrict ourselves to the susy G(2, N) σ -model and give the necessary and
sufficient conditions to get such solutions. The case of G(2, 4) is studied in detail
taking into account the classification of non-susy solutions [9].
99
for arbitrary functions f and g of given variables. Requiring it to be satisfied when
x + = 0 and x − = 0 separately we obtain β 21 (x + ) = γ 1 x + , where γ 1 is an arbitrary
constant. Upon introducing it into (32) we get f (x + ) + g(x − ) =
4|x| 2 γ 2
1
(1+|x| 2 ) 2 , which
immediately implies that γ 1 = 0 and hence β 21 = 0.
The necessary and sufficient conditions (9) and (10) are thus satisfied and finally
the CCH solution W 3 is given by the form
W 3 = Z 3 + iθ +
√
3ηβ 22 (x + )∂ + Z 3 .
(33)
Hence in this case we have obtained the susy invariant solution as the unique CCH
solution.
3.4 The Case of Z 4
In this case we have det Z
†
4 Z 4 =
1 + |x| 2 4 , i.e. r = 4 and κ =
1
2 . Again the
condition (9) becomes a third degree polynomial in x − after introducing the solution
W 4 given in (14). Similarly as what we did with W 3 , we equate the coefficients of
different powers of x − to zero and now get
β 11 (x + ) = 3x
2
+ β 22 (x + ),
β 21 (x + ) = −β 12 (x + ) + 2
√
3x + β 22 (x + ). (34)
In order to solve the last condition (10) we introduce (34) into it and find that
β 21 (x + ) = β 12 (x + ). Finally, the CCH solution W 4 is given as
W 4 = Z 4 + iθ +
1
2
ηβ 22 (x + )∂ + Z 4 .
(35)
Thus, in the case of Z 4 once more we have obtained the susy invariant solution as
the unique CCH solution.
4 Conclusions and Final Comments
In this article we give some criteria for having CCH solutions of the susy Grassmannian G(M, N ) σ -model. With the help of the susy gauge invariance of the model we
first show that the susy holomorphic solution given in (11) (i.e., generalization of
non-susy holomorphic solution) leads to a constant curvature surface. This kind of a
solution is called a susy invariant one, in analogy with the discussion given in [11].
Then we restrict ourselves to the susy G(2, N) σ -model and give the necessary and
sufficient conditions to get such solutions. The case of G(2, 4) is studied in detail
taking into account the classification of non-susy solutions [9].