The Painlev\'e property of $\mathbb{C}P^{N-1}$ sigma models

TL;DR

This paper investigates whether the $ ext{CP}^{N-1}$ sigma models possess the Painlevé property, analyzing their solutions' singularities and identifying conditions under which solutions are meromorphic, thus contributing to understanding their integrability.

Contribution

The study provides a comprehensive test of the Painlevé property for $ ext{CP}^{N-1}$ sigma models, revealing the structure of their solution space and singularities without restrictions on dimensionality or initial exponents.

Findings

Solutions have only pole singularities, indicating Painlevé property.

The solution space has a $(4N-5)$-parameter family.

Existence of solutions with branching essential singularities.

Abstract

We test the $\mathbb{C}P^{N-1}$ sigma models for the Painlev\'e property. While the construction of finite action solutions ensures their meromorphicity, the general case requires testing. The test is performed for the equations in the homogeneous variables, with their first component normalised to one. No constraints are imposed on the dimensionality of the model or the values of the initial exponents. This makes the test nontrivial, as the number of equations and dependent variables are indefinite. A $\mathbb{C}P^{N-1}$ system proves to have a $(4N-5)$-parameter family of solutions whose movable singularities are only poles, while the order of the investigated system is $4N-4$. The remaining degree of freedom, connected with an extra negative resonance, may correspond to a branching movable essential singularity. An example of such a solution is provided.