Superintegrability of Geodesic Motion on the Sausage Model

TL;DR

This paper demonstrates that geodesic motion on the sausage model is superintegrable, with explicit integrals of motion, and relates it to geodesics on a standard two-sphere via a canonical transformation.

Contribution

It constructs explicit integrals of motion for geodesic flow on the sausage model and links it to the well-understood two-sphere case through a canonical transformation.

Findings

Geodesic motion is described by a superintegrable system with four-dimensional phase space.

Constructed three integrals of motion satisfying an $rak{sl}(2)$ Poisson algebra.

Mapped geodesics on the sausage model to those on a standard two-sphere.

Abstract

Reduction of the $\eta$-deformed sigma model on ${\rm AdS}_5 \times {\rm S}^5$ to the two-dimensional squashed sphere $({\rm S}^2)_{\eta}$ can be viewed as a special case of the Fateev sausage model where the coupling constant $\nu$ is imaginary. We show that geodesic motion in this model is described by a certain superintegrable mechanical system with four-dimensional phase space. This is done by means of explicitly constructing three integrals of motion which satisfy the $\mathfrak{sl}(2)$ Poisson algebra relations, albeit being non-polynomial in momenta. Further, we find a canonical transformation which transforms the Hamiltonian of this mechanical system to the one describing the geodesic motion on the usual two-sphere. By inverting this transformation we map geodesics on this auxiliary two-sphere back to the sausage model. This paper is a tribute to the memory of Prof. Petr Kulish.