EDITED 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 :)

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sno129 2 days ago | parent [-]

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.

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gsf_emergency_6 2 days ago | parent [-]

I "blame" Quantamagazine for this.. upselling string theory via Kontsevich, because I don't think there's anything in this work related to string theory other than the Hodge diamond + related "elementary" symmetries (see my other unedited comment in response to a geometer)

It was probably not intentional, though, and might trigger noone besides snobs like us :)

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calf 2 days ago | parent [-]

There's different ways to define "related", here what does Quanta explicitly claim is "related", plausibly without looking maybe they meant historically related but not conceptually related.

	▲	gsf_emergency_6 2 days ago \| parent [-]
		Tbf I was maybe a bit indignant over this sentence from TFA: >The proof _relies_ on ideas imported from the world of string theory. Its techniques are wholly unfamiliar to the mathematicians who have dedicated their careers to classifying polynomials. They should have said "differential geometry", unless you count Kontsevich himself as a string theorist (maybe he does. I don't know) From the paper, sec3.1.2: While historically prevalent in the mirror symmetry and Gromov-Witten literature, the complex analytic or formal analogues of an F-bundle will not be useful for constructing birational invariants directly Later on, however: One largely unexplored aspect of Gromov-Witten theory is its algebraic flexibility.. I guess we can't really not credit the string theorists if Kontsevich can be so inspired by them :)