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.

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