ar X iv : h ep - t h / 05 12 16 5 v 1 1 4 D ec 2 00 5 A surprise in mechanics with nonlinear chiral supermultiplet

We show that the nonlinear chiral supermultiplet allows one to construct, over given two-dimensional bosonic mechanics, the family of two-dimensional N = 4 supersymmetric mechanics parameterized with the holomorphic function λ(z). We show, that this family includes, as a particular case, the N = 4 superextensions of twodimensional mechanics with magnetic fields, which have factorizable Schroedinger equations. Introduction Since its discovery [1] supersymmetric mechanics attracts much interest as a convenient toy model for the study of dynamical consequences of supersymmetry. It is also a convenient object for developing the supersymmetry technique, particularly for the construction of supersymmetric models within the superfield approach. However, even in the latter case supersymmetric mechanics was found to have some specific properties, which have no analogs in dimensions higher than one. For instance, in [2] it was found, that in N = 4, d = 1 supersymmetry, besides the five off-shell linear finite supermultiplets [3] and the one-dimensional analog of the N = 2, d = 4 nonlinear multiplet [4], there exists some new nonlinear supermultiplet, (called in [2] nonlinear chiral supermultiplet) which seems to have no known higher-dimensional analogs. It includes, as a limiting case, the standard chiral supermultiplet and has the same components as the latter. Let us recall that the standard (linear) chiral supermultiplet corresponds to the complex superfield parameterizing the two-dimensional plane IR = I C. Opposite to that case, the nonlinear chiral supermultiplet corresponds to the complex superfield parameterizing the two-dimensional sphere (complex projective plane) S = I CP = SU(2)/U(1), but it has the same component content, as the linear one. Consequently, the standard chirality condition is modified as follows: DiZ = −αZDiZ, DiZ = αZDiZ , α = const. (1) In Ref.[5], by the use of the nonlinear chiral supermultiplet, the model of two-dimensional N = 4 supersymmetric mechanics has been suggested, with the following superfield action: S = ∫ dtdθdθ̄ K(Z,Z) + ∫ dtdθ̄ F (Z) + ∫ dtdθ F (Z) . (2) Here K(Z,Z) is an arbitrary real function playing the role of Kähler potential of the metric, while F (Z) and F (Z) are arbitrary holomorphic and antiholomorphic functions. Some interesting features of the model were observed there, e.g. the possibility to incorporate a magnetic field preserving the supersymmetry of the system. It was shown that this system includes, as a particular case, the N = 4 supersymmetric Landau problem on the sphere. Later on the nonlinear chiral multiplet has been used for the construction of N = 8 supersymmetric mechanics [6], as well as obtained by the reduction of the linear supermultiplet with four bosonic and four fermionic degrees of freedom [7]. In the present note we show that N = 4 supersymmetric mechanics with nonlinear chiral multiplet possesses a quite surprising property. When we construct the supersymmetric mechanics with linear chiral multiplet, the arbitrariness of the construction is in the choice of Kähler potential K and superpotential F (z) only, and these functions define the underlying bosonic configuration. The extension to a supersymmetric system is unique [8]. On the contrary, when dealing with nonlinear chiral supermultiplet, we have the freedom in the supersymmetric extension of the given bosonic system, encoded in the choice of the holomorphic function λ(z). When the underlying bosonic system is of the sigma-model type, the function λ(z) remains arbitrary. Otherwise it is related with the given potential and magnetic field as follows: U(z, z̄) = F (z)F ′ (z̄) (1 + λλ̄)2g , B = λ̄(z̄)F (z) + λ(z)F ′ (z̄) ( 1 + λλ̄ )2 g . (3) Here gdzdz̄ defines the Kähler metric of the underlying bosonic space, U is a potential of the underlying bosonic system, and B is the magnitude of the magnetic field. An interesting feature of supersymmetric (quantum) mechanics is the application to integrable systems of quantum mechanics. Initially, it was found that supersymmetric quantum mechanics could be naturally related with the