3d gravity as a source of integrable systems and hierarchies

In 1988 Witten showed the theory of (pure) General Relativity in 3 dimensions is exactly solvable, using an “equivalent” Chern-Simons formulation. In this talk we will describe the theory using the Newman- Penrose formalism and explicitly show how the equations of motion reduce to an AKNS system, proving in this way its integrability in a more direct way. Then we will see how boundary conditions for higher spin 3d gravity naturally provide hierarchies of integrable systems. Explicitly, we will see how to obtain the mBoussinesq hierarchy from boundary conditions in spin-3 gravity on AdS_3 and how the boundary dynamics of the theory could be completely described in terms of this family of integrable systems.