  Infinite-dimensional conformal symmetry in two dimensions leads to
integrability of 2d conformal field theories through an infinite tower of local
conserved qKdV charges in involution. We discuss the role this integrable
structure plays in equilibration of 2d CFTs. We show that in the thermodynamic
limit large central charge 2d CFTs satisfy generalized eigenstate
thermalization, with the values of qKdV charges forming a complete set of
thermodynamically relevant quantities, which unambiguously determine
expectation values of all local observables from the vacuum family. Our work
settles the question if non-local or quasi-local charges are necessary to
describe equilibrium of large central charge 2d CFTs by showing that upon
equilibration local physics can be described by the Generalized Gibbs Ensemble
that only includes qKdV charges. In the case of a general initial state, upon
equilibration, emerging Generalized Gibbs Ensemble will include negative
chemical potentials and holographically will be described by a quasi-classical
black hole with quantum soft hair.
