alt Hacker NewsEDITED to emphasize that the idea Kontsevich "took from string theorists" is not as big a deal as title of TFA implies, thanks sno129
On page6 is the crucial^W main idea that Kontsevich took from string theory, the Hodge Diamond
It's memorable, at least :)
En.wiki has a quick explanation
Mirror symmetry translates the dimension number of the (p, q)-th differential form h^(p,q) for the original manifold into h^(n-p,q) of that for the counter pair manifold.
(n=4 for the paper, "cubic 4-fold")
https://en.wikipedia.org/wiki/Homological_mirror_symmetry#Ho...
Don't miss the caption from the end of the previous page, what the sum means :)
I wouldn't consider the Hodge diamond the "crucial idea from string theory." It's a pretty basic/fundamental concept in geometry and really doesn't a priori have much to do with string theory. The decomposition they give on page 6 probably predates most of the development of string theory.