Yes, this story is heavily expanded upon :-)

As far as I understand it, the genus-zero Gromov-Witten invariants of the A-side and the Hodge theory of the B-side can be arranged into a gadget called 'Variations of semi-infinite Hodge structures" introduced by Barannikov. Mirror symmetry predicts that if $X$ and $Y$ are mirrors, then there should be an isomorphism between their respective VSHS's. Kontsevich proposed the Homological Mirror Symmetry conjecture which sees mirror symmetry as an equivalence between two non-commutative spaces - the (derived) Fukaya category of the A-side, and the derived category of coherent sheaves on the B-side. The question is how to get from this very sophisticated statement to the classical one.

This expectation was made precise by Barannikov and Katzarkov-Kontsevich-Pantev: the idea is that the cyclic homology of a 'nice' $A_\infty$-category (proper, smooth, Hodge-to-deRham degeneration conjecture holds) carries a VSHS. That's what you need to make the connection, i.e., taking cyclic homology of the Fukaya category should recover the A-side VSHS and taking cyclic homology of the bounded derived category should recover the B-side VSHS.

Proving this is exactly the content of Gantara-Perutz-Sheridan work ... see: Mirror symmetry: from categories to curve counts and Formulae in noncommutative Hodge theory.

Yes, this story is heavily expanded upon :-)

As far as I understand it, the genus-zero Gromov-Witten invariants of the A-side and the Hodge theory of the B-side can be arranged into a gadget called 'Variations of semi-infinite Hodge structures" introduced by Barannikov. Mirror symmetry predicts that if $X$ and $Y$ are mirrors, then there should be an isomorphism between their respective VSHS's. Kontsevich proposed the Homological Mirror Symmetry conjecture which sees mirror symmetry as an equivalence between two non-commutative spaces - the (derived) Fukaya category of the A-side, and the derived category of coherent sheaves on the B-side. The question is how to get from this very sophisticated statement to the classical one.

This expectation was made precise by Barannikov and Katzarkov-Kontsevich-Pantev: the idea is that the cyclic homology of a 'nice' $A_\infty$-category (proper, smooth, Hodge-to-deRham degeneration conjecture holds) carries a VSHS. That's the bridge you need to make the connection, i.e., taking cyclic homology of the Fukaya category should recover the A-side VSHS and taking cyclic homology of the bounded derived category should recover the B-side VSHS.

Proving this is exactly the content of Gantara-Perutz-Sheridan work ... see: Mirror symmetry: from categories to curve counts and Formulae in noncommutative Hodge theory.