SusyCP^N−1model and surfaces in

Abstract

We describe surfaces in R N 2 −1 generated by the holomorphic solutions of the supersymmetric CP N−1 model. We show that these surfaces are described by the fundamental projector constructed out of the solutions of this model and that in the CP N−1 case the corresponding surface is a sphere.Although the coordinates of the sphere are superfields the sphere's curvature is constant. We show that for N > 2 the corresponding surfaces can also be constructed from the similar projector.

Constant curvature surfaces of the supersymmetric ℂP N−1 sigma model

“…Note that these results are more general than the one obtained in [32], where only the holomorphic case was studied. Here our procedure applies also to non-holomorphic immersions.…”

“…The explicit form of the constraint (16) may be found by direct calculations (see for example [6,32]). The action is defined in this case as [6,29,30,31,32]…”

“…A good review is given in [6]. The constraints which appear in the usual formulation of this model have limited some authors to the study of the constant curvature surfaces obtained from only the holomorphic solutions of the SUSY CP 1 model [32]. In this paper, we consider the SUSY CP N −1 model and we show that, up to gauge symmetry, the holomorphic solutions corresponding to the constant curvature surfaces are related to a generalization of the Veronese curve (13).…”

General solutions of the supersymmetric ℂP2 sigma model and its generalisation to ℂP N−1

“…These boundary conditions have the effect of compactifying the bosonic part of the superspace (x + , x − ; θ + , θ − ) into the 2-sphere S 2 via the stereographic projection. A convenient reformulation of the two-dimensional CP N −1 supersymmetric sigma model involves considering it in a gauge-invariant way in terms of orthogonal projectors [7,20,24]. Indeed, let Φ be a solution of the model and define P as…”

“…In the particular case of CP N −1 , a complete classification was given in terms of the Veronese curve [14]. In the supersymmetric case, we have a similar classification but only for the holomorphic case [20,24]. The challenge here resides in having to define the surfaces which correspond to the bosonic model but which are elements of the su(N ) Lie algebra.…”