Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs

Abstract: For an arbitrary semisimple Frobenius manifold we construct Hodge integrable hierarchy of Hamiltonian partial differential equations. In the particular case of quantum cohomology the tau-function of a solution to the hierarchy generates the intersection numbers of the Gromov--Witten classes and their descendents along with the characteristic classes of Hodge bundles on the moduli spaces of stable maps. For the one-dimensional Frobenius manifold the Hodge hierarchy is a deformation of the Korteweg--de Vries hierarchy depending on an infinite number of parameters. Conjecturally this hierarchy is a universal object in the class of scalar Hamiltonian integrable hierarchies possessing tau-functions.