  We propose a Hodge field theory construction that captures algebraic
properties of the reduction of Zwiebach invariants to Gromov-Witten invariants.
It generalizes the Barannikov-Kontsevich construction to the case of higher
genera correlators with gravitational descendants.
  We prove the main theorem stating that algebraically defined Hodge field
theory correlators satisfy all tautological relations. From this perspective
the statement that Barannikov-Kontsevich construction provides a solution of
the WDVV equation looks as the simplest particular case of our theorem. Also it
generalizes the particular cases of other low-genera tautological relations
proven in our earlier works; we replace the old technical proofs by a novel
conceptual proof.
