There are connections to string theory, as discussed in this 2011 <A HREF="https://www.birs.ca/events/2011/5-day-workshops/11w5090/press">conference</A>. The correspondence is explained in <A HREF="http://www.math.mcgill.ca/walcher/papers/mfsdwalcher.pdf">Open Strings and Extended Mirror Symmetry</A>, Johannes Walcher (2007): The Gromov-Witten theory of a Calabi-Yau manifold $X$ is solved by the Hodge theory of its mirror symmetric manifold $Y$.