Spinning strings in the $\eta$-deformed Neumann-Rosochatius system

TL;DR

This paper constructs explicit elliptic function solutions for spinning strings in an $ ext{AdS}_5 imes S^5$ $ ext{eta}$-deformed background, revealing how deformation affects string energies and dynamics.

Contribution

It provides the first general elliptic solutions for the $ ext{eta}$-deformed Neumann-Rosochatius system, extending understanding of integrable string models in deformed geometries.

Findings

Explicit elliptic solutions for spinning strings are derived.

The energy-angular momentum relation is obtained as an expansion in the coupling.

Limiting cases of the deformation parameter are analyzed.

Abstract

The sigma-model of closed strings spinning in the $\eta$-deformation of $AdS_{5} \times S^{5}$ leads to an integrable deformation of the one-dimensional Neumann-Rosochatius mechanical system. In this article we construct general solutions to this system that can be written in terms of elliptic functions. The solutions correspond to closed strings with non-constant radii rotating with two different angular momenta in an $\eta$-deformed three-sphere. We analyse the reduction of the elliptic solutions for some limiting values of the deformation parameter. For the case of solutions with constant radii we find the dependence of the classical energy of the string on the angular momenta as an expansion in the 't Hooft coupling.