Quantum Hodge symmetries of Kahler manifolds

The product of two harmonic forms on a Riemannian manifold X is usually not harmonic. Instead,
there is an A-infinity algebra structure on the space of harmonic forms, which is quasiisomorphic
to the cohomology algebra of  X.

On a Kahler manifold Y the higher products above vanish.  We show that instead
a new phenomenon appears: the Tannakain Galois group of the category real mixed Hodge structures
acts by A-infinity automorphisms of the cohomology algebra of X.

Both constructions have a quantum = higher genus generalization:
For a Riemannian X, the A-infinity product extends to a quantum A-infinity algebra structure, and
for a Kahler manifold Y the real Hodge Galois group acts by quantum A-infinity automorphisms
of the cohomology algebra of Y.

There are similar further generalizations of these constructions when  for the Ext's between local
systems on X / semisimple local systems on Y.