There are connections to string theory as explained in <A HREF="http://www.math.mcgill.ca/walcher/papers/mfsdwalcher.pdf">Open Strings and Extended Mirror Symmetry</A>, by Johannes Walcher (2007). In one sentence: The Gromov-Witten theory of a Calabi-Yau manifold is solved by the Hodge theory of its mirror symmetric manifold.

There are some talks you might find interesting from this 2011 <A HREF="https://www.birs.ca/events/2011/5-day-workshops/11w5090/press">conference</A>:

> The mathematical theory that describes how integrals and differential
> equations control the shape of algebraic spaces in various dimensions
> is known as Hodge theory. The most important conjecture in algebraic
> geometry -- the Hodge Conjecture -- can be thought of as "a metaphor
> for transforming transcendental computations into algebraic ones." The
> physical theory able to describe the universe at both micro- (quantum
> mechanics) and macro- (general relativity) scales, and at the same
> time thought to be a suitable candidate for unifying all known forces
> of nature, is string theory. There are several variants of this
> "theory of everything," linked by dualities which can radically alter
> mathematical formulations while preserving physical predictions.
> String dualities thus imply conjectures: seemingly unrelated pieces of
> mathematics must be related since they offer different descriptions of
> the same physical world. Although a role for Hodge theory in string
> theory has been hinted at for some time, only very recently has the
> depth and precision of this relationship begun to emerge. Recent
> results suggest that a mathematical "grand unification" relating
> arithmetic geometry and symplectic geometry is taking shape.