On certain geometrical aspects of surfaces associated with CP^N−1 sigma models

In this paper, the Weierstrass technique for harmonic maps S² → CPN−1 is employed in order to obtain surfaces immersed in multidimensional Euclidean spaces. It is proved that if the CPN−1 model equations are defined on the sphere S² and the associated action functional of this model is finite, then a specific holomorphic function (corresponding to a component of the energy-momentum tensor of a CPN−1 sigma model) vanishes. In particular it is shown that for any holomorphic or antiholomorphic solutions of this model, the Weierstrass formula for immersion X of a surface lies in the su(N) algebra and can be expressed in terms of an orthogonal projector of rank (N−1). The implementation of this method is presented for two-dimensional conformally parametrized surfaces immersed in the su(3) algebra. The usefulness of the proposed approach is illustrated with examples, including the dilation-invariant meron-type solutions and the Veronese solutions for the CP² model. Depending on the location of the critical points (zeros and poles) of the first fundamental form associated with the meron solution, it is shown that the associated surfaces are semi-infinite cylinders. It is also demonstrated that surfaces related to holomorphic and mixed Veronese solutions are immersed in R⁸ and R³, respectively