Superintegrability of Geodesic Motion on the Sausage Model

Abstract

Reduction of the $\eta$-deformed sigma model on ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ to the two-dimensional squashed sphere $({\mathrm{S}}^2{)}_{\eta }$ 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.

30 pages, 4 figures; v2: added results for geodesic motion on the mirror geometry of S^2, matches published version

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