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.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/6826