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In this paper, we study rank-1 projector solutions to the completelyintegrable Euclidean $CP^{N-1}$ sigma model in two dimension and theirassociated surfaces immersed in the $su(N)$ Lie algebra. We reinterpret andgeneralize the proof of A.M. Din and W.J. Zakzrewski [1980] that any solutionfor the $CP^{N-1}$$ sigma model defined on the Riemann sphere with finiteaction can be written as a raising operator acting on a holomorphic one, or alowering operator acting on a antiholomorphic one. Our proof is formulated interms of rank-1 Hermitian projectors so it is explicitly gauge invariant andgives new results on the structure of the corresponding sequence of rank-1projectors. Next, we analyze surfaces associated with the $CP^{N-1}$ modelsdefined using the Generalized Weierstrass Formula for immersion, introduced byB. Konopelchenko [1996]. We show that the surfaces are conformallyparameterized by the Lagrangian density with finite area equal to the action ofthe model and express several other geometrical characteristics of the surfacein terms of the Lagrangian density and topological charge density of the model.We demonstrate that any such surface must be orthogonal to the sequence ofprojectors defined by repeated application of the raising and loweringoperators. Finally, we provide necessary and sufficient conditions that asurface be related to a $CP^{N-1}$ sigma model.

Analysis of $CP^{N-1}$ sigma models via projective structure