Resumen:
This thesis extends a previously found relation between the integrable KdV
hierarchy and the boundary dynamics of pure gravity on AdS3 described in the
highest weight gauge, to a more general class of integrable systems associated
to three-dimensional gravity on AdS3 and higher spin gravity with gauge group
SL(N, R) × SL(N, R) in the diagonal gauge. We present new sets of boundary conditions for the (higher spin) gravitational theories on AdS3, where the
dynamics of the boundary degrees of freedom is described by two independent
left and right members of a hierarchy of integrable equations. For the pure
gravity case, the associated hierarchy corresponds to the Gardner hierarchy,
also known as the “mixed KdV-mKdV” one, while for the case of higher spin
gravity, they are identified with the “modified Gelfand-Dickey” hierarchies.
The complete integrable structure of the hierarchies, i.e., the phase space, the
Poisson brackets and the infinite number of commuting conserved charges, are
directly obtained from the asymptotic structure and the conserved surface in tegrals in the gravitational theories. Consequently, the corresponding Miura
transformation is recovered from a purely geometric construction in the bulk.
Black hole solutions that fit within our boundary conditions, the Hamiltonian
reduction at the boundary and more general thermodynamic ensembles called
“Generalized Gibbs Ensemble” (GGE) are also discussed.