Mirror symmetry of variations of Hodge structures for Fano varieties

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Let XX be a Fano variety, and let Y→CY\to\mathbb{C} be a one-parameter family. The Picard--Fuchs D\mathcal D-module of YY is the differential D\mathcal D-module governing the periods of the family, while the regularized quantum D\mathcal D-module is the regularized differential system associated with the quantum cohomology of XX.

Mirror symmetry of variations of Hodge structures. For any Fano variety XX there exists a one-parameter family

Y→CY\to\mathbb{C}

whose Picard--Fuchs D\mathcal D-module is isomorphic to a regularized quantum D\mathcal D-module for XX.

This conjecture proposes a mirror-symmetric realization of the quantum differential system of every Fano variety as the Picard--Fuchs system of a one-parameter family. The source provides no evidence of resolution, so the conjecture remains open.

References

Primary source

Nathan Owen Ilten, Jacob Lewis and Victor Przyjalkowski, “Toric Degenerations of Fano Threefolds Giving Weak Landau-Ginzburg Models”, arXiv:1102.4664 (2012).

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