Boundary conditions for General Relativity on AdS$_{3}$ and the KdV hierarchy

TL;DR

This work shows that 3D General Relativity with negative cosmological constant admits a family of boundary conditions labeled by $k$, under which boundary gravitons obey the $k$-th KdV equation and the boundary theory comprises two noninteracting Lifshitz movers with $z=2k+1$. The asymptotic symmetry algebra is abelian and free of central extensions for $k>0$, while the BTZ black hole and AdS spacetime fit naturally within this framework, with entropy and thermodynamics captured by a generalized Cardy formula consistent with the anisotropic scaling. The analysis extends to higher-spin cases and highlights deep links between gravitational boundary data, integrable hierarchies (via Gelfand–Dikii polynomials), and nonrelativistic holography, including an anisotropic version of modular invariance. Overall, the paper provides a geometrization of KdV dynamics in AdS$_3$, a new mechanism for Lifshitz scaling without bulk Lifshitz spacetimes, and concrete bridges to entropy counting and soft hair. The approach offers a rich platform for exploring nonrelativistic holography and integrable structures within AdS/CFT.

Abstract

It is shown that General Relativity with negative cosmological constant in three spacetime dimensions admits a new family of boundary conditions being labeled by a nonnegative integer $k$. Gravitational excitations are then described by "boundary gravitons" that fulfill the equations of the $k$-th element of the KdV hierarchy. In particular, $k=0$ corresponds to the Brown-Henneaux boundary conditions so that excitations are described by chiral movers. In the case of $k=1$, the boundary gravitons fulfill the KdV equation and the asymptotic symmetry algebra turns out to be infinite-dimensional, abelian and devoid of central extensions. The latter feature also holds for the remaining cases that describe the hierarchy ($k>1$). Our boundary conditions then provide a gravitational dual of two noninteracting left and right KdV movers, and hence, boundary gravitons possess anisotropic Lifshitz scaling with dynamical exponent $z=2k+1$. Remarkably, despite spacetimes solving the field equations are locally AdS, they possess anisotropic scaling being induced by the choice of boundary conditions. As an application, the entropy of a rotating BTZ black hole is precisely recovered from a suitable generalization of the Cardy formula that is compatible with the anisotropic scaling of the chiral KdV movers at the boundary, in which the energy of AdS spacetime with our boundary conditions depends on $z$ and plays the role of the central charge. The extension of our boundary conditions to the case of higher spin gravity and its link with different classes of integrable systems is also briefly addressed.