92
V. Hussin et al.
For the non-susy case, a general approach for constructing holomorphic maps of
2-sphere S 2 of constant curvature into G(M, N ) has been realized in [7, 8] and the
cases G(2, 4) and G(2, 5) have been treated in detail [9, 10].
The natural question is to extend those results to susy G(M, N ) σ -models for
M > 1. In the susy case, some results have already been known when M = 1, i.e.
the CP N −1 σ -model [5, 6, 11, 12]. In particular, all the susy invariant solutions
with constant curvature holomorphic (CCH) solutions of this model have been
thoroughly discussed.
In a recent article [13], the present authors, together with W. J. Zakrzewski,
introduced a general method for characterizing the constant curvature surfaces for
the holomorphic solutions of the susy Grassmannian sigma models. The main tool
there was to use the gauge invariance of these models and to our knowledge this
was the first time in literature that this invariance is explicitly and effectively used
in such a context. In this paper, first we give some criteria for having CCH solutions
of the susy Grassmannian G(M, N ) σ -model by the help of gauge invariance and
then apply this method on a specific example, namely the G(2, 4) sigma model
thoroughly. The problem splits into four cases and we separately investigate all of
them. Whenever possible we give the canonical form of such constant curvature
surfaces. Among these four types of solutions with different curvatures, only two of
them produce the susy invariant solutions as the unique ones.
The structure of this paper is as follows; in Sect. 2, we discuss the necessary
and sufficient conditions to get the CCH solutions of the general susy G(M, N ) σ -
model. In Sect. 3 we give a detailed analysis of the susy G(2, 4) σ-model. Taking
into account the susy gauge invariance we present all the CCH solutions of this
model. Finally, we end the article by giving some comments in Sect. 4.
2 CCH Solutions of the Susy G(M, N ) σ-Model
For the susy G(M, N ) σ -model [5], a general bosonic superfield has the following
expansion Φ(x ± , θ ± ) = Φ 0 (x ± ) + iθ + Φ 1 (x ± ) + iθ − Φ 2 (x ± ) − θ + θ − Φ 3 (x ± ), where
Φ 0 and Φ 3 are N ×M bosonic complex matrices and Φ 1 and Φ 2 are N ×M fermionic
complex matrices. This bosonic superfield must satisfy Φ † Φ = I M . The energy
action functional of the model is given by S(Φ) =
S 2 dx + dx − dθ + dθ − L(Φ),
where L(Φ) = 2 T r
| ˇ
D + Φ| 2 − | ˇ
D − Φ| 2
and the supercovariant derivatives are
defined by ˇ
D ± ˜
Λ = ˇ
∂ ± ˜
Λ − ˜
Λ(Φ † ˇ
∂ ± Φ), with ˇ
∂ ± = −i∂ θ ± + θ ± ∂ ± and ∂ ± ≡ ∂ x ± .
Using the principle of least action, it is found that the superfield Φ satisfies the
Euler–Lagrange equations ˇ
D + ˇ
D − Φ + Φ| ˇ
D − Φ| 2 = 0 . As in the non-susy case,
holomorphic solutions of the susy G(M, N ) σ -model are trivial solutions of the
model [5, 13]. It has been shown that they take the form Φ = W L, where W is
an N × M matrix depending only on the coordinates (x + , θ + ), while L is a nonsingular M × M matrix that depends on the coordinates (x ± , θ ± ). It means that the
holomorphic superfield W takes the explicit form
V. Hussin et al.
For the non-susy case, a general approach for constructing holomorphic maps of
2-sphere S 2 of constant curvature into G(M, N ) has been realized in [7, 8] and the
cases G(2, 4) and G(2, 5) have been treated in detail [9, 10].
The natural question is to extend those results to susy G(M, N ) σ -models for
M > 1. In the susy case, some results have already been known when M = 1, i.e.
the CP N −1 σ -model [5, 6, 11, 12]. In particular, all the susy invariant solutions
with constant curvature holomorphic (CCH) solutions of this model have been
thoroughly discussed.
In a recent article [13], the present authors, together with W. J. Zakrzewski,
introduced a general method for characterizing the constant curvature surfaces for
the holomorphic solutions of the susy Grassmannian sigma models. The main tool
there was to use the gauge invariance of these models and to our knowledge this
was the first time in literature that this invariance is explicitly and effectively used
in such a context. In this paper, first we give some criteria for having CCH solutions
of the susy Grassmannian G(M, N ) σ -model by the help of gauge invariance and
then apply this method on a specific example, namely the G(2, 4) sigma model
thoroughly. The problem splits into four cases and we separately investigate all of
them. Whenever possible we give the canonical form of such constant curvature
surfaces. Among these four types of solutions with different curvatures, only two of
them produce the susy invariant solutions as the unique ones.
The structure of this paper is as follows; in Sect. 2, we discuss the necessary
and sufficient conditions to get the CCH solutions of the general susy G(M, N ) σ -
model. In Sect. 3 we give a detailed analysis of the susy G(2, 4) σ-model. Taking
into account the susy gauge invariance we present all the CCH solutions of this
model. Finally, we end the article by giving some comments in Sect. 4.
2 CCH Solutions of the Susy G(M, N ) σ-Model
For the susy G(M, N ) σ -model [5], a general bosonic superfield has the following
expansion Φ(x ± , θ ± ) = Φ 0 (x ± ) + iθ + Φ 1 (x ± ) + iθ − Φ 2 (x ± ) − θ + θ − Φ 3 (x ± ), where
Φ 0 and Φ 3 are N ×M bosonic complex matrices and Φ 1 and Φ 2 are N ×M fermionic
complex matrices. This bosonic superfield must satisfy Φ † Φ = I M . The energy
action functional of the model is given by S(Φ) =
S 2 dx + dx − dθ + dθ − L(Φ),
where L(Φ) = 2 T r
| ˇ
D + Φ| 2 − | ˇ
D − Φ| 2
and the supercovariant derivatives are
defined by ˇ
D ± ˜
Λ = ˇ
∂ ± ˜
Λ − ˜
Λ(Φ † ˇ
∂ ± Φ), with ˇ
∂ ± = −i∂ θ ± + θ ± ∂ ± and ∂ ± ≡ ∂ x ± .
Using the principle of least action, it is found that the superfield Φ satisfies the
Euler–Lagrange equations ˇ
D + ˇ
D − Φ + Φ| ˇ
D − Φ| 2 = 0 . As in the non-susy case,
holomorphic solutions of the susy G(M, N ) σ -model are trivial solutions of the
model [5, 13]. It has been shown that they take the form Φ = W L, where W is
an N × M matrix depending only on the coordinates (x + , θ + ), while L is a nonsingular M × M matrix that depends on the coordinates (x ± , θ ± ). It means that the
holomorphic superfield W takes the explicit form