Superintegrability of Geodesic Motion on the Sausage Model

Authors: Gleb Arutyunov, Martin Heinze and Daniel Medina-Rincon

Preprint number: UUITP-16/16

Reduction of the η-deformed sigma model on AdS₅×S⁵ to the two-dimensional squashed sphere (S²)_η can be viewed as a special case of the Fateev sausage model where the coupling constant ν 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 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.