Mirror identification of the corrected differential via polyvector fields

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

Background

Building on the Maurer–Cartan conjecture for μ0, the authors propose that the resulting differential matches, under mirror symmetry, a deformation of the polyvector fields on the uncorrected SYZ mirror Y0 to those on the corrected mirror. This would align the Floer-theoretic correction mechanism with the algebraic geometry of mirror deformations and relate the Floer BV/Schouten–Nijenhuis structures across the mirror.

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,” same subsection on negative Maslov index and extended deformations