Boundary conditions for General Relativity in three-dimensional spacetimes, integrable systems and the KdV/mKdV hierarchies

We present a new set of boundary conditions for General Relativity on AdS$_3$, where the dynamics of the boundary degrees of freedom are described by two independent left and right members of the Gardner hierarchy of integrable equations, also known as the "mixed KdV-mKdV" hierarchy. This integrable system has the very special property that simultaneously combines both, the Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) hierarchies in a single integrable structure. This relationship between gravitation in three-dimensional spacetimes and two-dimensional integrable systems is based on an extension of the recently introduced "soft hairy boundary conditions" on AdS$_3$, where the chemical potentials are now allowed to depend locally on the dynamical fields and their spatial derivatives. The complete integrable structure of the Gardner system, i.e., the phase space, the Poisson brackets and the infinite number of commuting conserved charges, are directly obtained from the asymptotic analysis and the conserved surface integrals in the gravitational theory. These boundary conditions have the particular property that they can also be interpreted as being defined in the near horizon region of spacetimes with event horizons. Black hole solutions are then naturally accommodated within our boundary conditions, and are described by static configurations associated to the corresponding member of the Gardner hierarchy. The thermodynamic properties of the black holes in the ensembles defined by our boundary conditions are also discussed. Finally, we show that our results can be naturally extended to the case of a vanishing cosmological constant, and the integrable system turns out to be precisely the same as in the case of AdS$_3$.