Exact solutions of SO(3) non-linear sigma model in a conic space background

TL;DR

This paper derives exact solutions for the nonlinear sigma model in a conic space background, revealing how the conic geometry influences solutions, actions, and topological charges, with implications for instanton and meron configurations.

Contribution

It introduces a method to obtain exact solutions of the sigma model in conic space by transforming equations to Minkowski spacetime and analyzes the effects of the conic geometry on solutions.

Findings

Exact solutions depend on two, three, and four coordinates.

Differences in actions and topological charges due to the conic deficit angle.

Identification of instanton and meron solutions in conic space.

Abstract

We consider a nonlinear sigma model coupled to the metric of a conic space. We obtain restrictions for a nonlinear sigma model to be a source of the conic space. We then study nonlinear sigma model in the conic space background. We find coordinate transformations which reduce the chiral fields equations in the conic space background to field equations in Minkowski spacetime. This enables us to apply the same methods for obtaining exact solutions in Minkowski spacetime to the case of a conic spacetime. In the case the solutions depend on two spatial coordinates we employ Ivanov's geometrical ansatz. We give a general analysis and also present classes of solutions in which there is dependence on three and four coordinates. We discuss with special attention the intermediate instanton and meron solutions and their analogous in the conic space. We find differences in the total actions and topological charges of these solutions and discuss the role of the deficit angle.