Smoothness of the non-monotone Floer Aโˆž-algebra at isolated critical points

Establish that, in the non-monotone setting over a Novikov field, for a Lagrangian torus L equipped with a rank-1 local system corresponding to an isolated critical point of the weak bounding-cochain potential function ๐”“๐”’ on H^1(L; ฮ›_{>0}), the Aโˆž-algebra CF^*(L^, L^) is homologically smooth.

Background

The main results of the paper rely crucially on homological smoothness of the endomorphism Aโˆž-algebra CF*(L, L) for monotone Lagrangian tori at isolated critical points, deduced via a comparison with matrix factorizations (Dyckerhoff's smoothness).

To extend the theory beyond monotone tori, one must work over a Novikov field and use the potential function ๐”“๐”’ arising from weak bounding cochains. The conjecture asserts that the analogous smoothness should hold in this broader, non-monotone context at isolated critical points. Proving this would enable the extension of the automatic split-generation and quantum cohomology factorization results to the non-monotone setting.

References

We conjecture, based on the heuristic picture outlined in \cref{sscMS}, that the even in the non-monotone case the algebra $CF*(L, L)$ is smooth at isolated critical points, but one would need new techniques to establish this.

— Quantum cohomology and Fukaya summands from monotone Lagrangian tori  (2409.07922 - Smith, 2024) in Introduction, Remark on extending to non-monotone tori, item (3)