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Superintegrability of geodesic motion on the sausage \(\mathrm{model}^{*}\). (English) Zbl 1369.81074

Summary: 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.

MSC:

81T20 Quantum field theory on curved space or space-time backgrounds
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
14D15 Formal methods and deformations in algebraic geometry
81R12 Groups and algebras in quantum theory and relations with integrable systems
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
70H05 Hamilton’s equations
53C22 Geodesics in global differential geometry
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