Integrability in Asymptotic Symmetries of Spacetime: the ${\text{BMS}}_3$ scenario

Abstract

We revisit the proof of a ${\mathfrak{bms}}_3-$integrable hierarchy introduced in Fuentealba et al. (JHEP, 2018), using different structural methodologies. Specifically, from the variational complex on the ring of polynomial symbols, a ${\mathfrak{bms}}_3$ bi-Hamiltonian structure is constructed on which a Nijenhuis operator can be attached. A similar argument is made for the ${\mathrm{AdS}}_3$ case, where we check that the flat limit of the latter recovers the asymptotically flat situation. An alternative $\ddot{I}„-$scheme description is also presented. Moreover, a Lie-Poisson description suggests that such a ${\mathfrak{bms}}_3-$hierarchy is not unique. For a subclass of so-called energy-dependent Schrödinger operators, it is shown that their Lax flows are described by the coadjoint orbits of ${\mathfrak{bms}}_3$.

30 pages, 1 figure. Comments are welcome

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