We introduce quantum Hodge correlators. They have the following format: Take a family $X/B$ of compact Kahler manifolds. Given an oriented topological surface $S$ with {special points} on the boundary, we assign to each interval between special points an irreducible local system on $X$, and to each special point an Ext between the neighboring local systems. A quantum Hodge correlator is assigned to this data and lives on the base $B$. It is a sum of finite dimensional convergent Feynman type integrals. The simplest Hodge correlators are given by the Rankin-Selberg integrals for $L$-functions.

Quantum Hodge correlators can be perceived as Hodge-theoretic analogs of the invariants of knots and threefolds provided by the perturbative Chern-Simons theory. Here is an example: Hodge theory suggests to view a Riemann surface as a threefold, and its points as knots in the threefold. Then the Green function $G(x,y)$, the basic Hodge correlator, is an analog of the linking number, the simplest Chern- Simons type invariant. What do the quantum Hodge correlators do? Let $B$ be a point; consider trivial local systems.

- Hodge correlators ($S$ is a disc) describe an action of the Hodge Galois group by $A_\infty$ automorphisms of the cohomology algebra $H(X)$ preserving the PoincarĂ© pairing.
- Quantum Hodge correlators ($S$ is any surface) describe an action of the Hodge Galois group by quantum $A_\infty$ automorphisms of the algebra $H(X)$ with the PoincarĂ© pairing.