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Mirror-side deformation: corrected differential and polyvector fields on the corrected SYZ mirror

Demonstrate that, under mirror symmetry, the corrected differential δ + {µ^0,·} on the family Floer complex 𝔠 corresponds to deforming the Čech complex of polyvector fields C^*(Y^0; Λ^*TY^0) to polyvector fields (or an appropriate noncommutative analogue) on the corrected SYZ mirror Y.

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Background

Building on the conjectural Maurer–Cartan structure for the family Floer complex, the author proposes that the mirror-side manifestation is a deformation of polyvector fields on the uncorrected mirror Y0 to those on the corrected mirror Y, capturing wall-crossing and disc contributions in SYZ mirror symmetry.

References

Conjecturally, under mirror symmetry this amounts to deforming the \v{C}ech complex of polyvector fields $C(Y0;\Lambda^ TY0)$ to arrive at polyvector fields (or their appropriate noncommutative analogue) on the corrected mirror.

— Lagrangian Floer theory, from geometry to algebra and back again (2510.22476 - Auroux, 26 Oct 2025) in Section 3.1 (Floer theory for families of Lagrangians), final paragraphs