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.

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