Sign Convention for $A_{\infty}$-Operations in Bott-Morse Case
Kaoru Ono
TL;DR
This work tackles the sign and orientation challenges that arise in the filtered $A_{ abla\infty}$-formulae within Bott--Morse Lagrangian Floer theory by employing a de Rham model. It extends the FOOO framework to a finite collection of relatively spin Lagrangian submanifolds and constructs the filtered $A_{ abla\infty}$-operations ${\mathfrak m}_{k,B}$ via de Rham pushforwards along evaluation maps from bordered polygon moduli spaces, incorporating relative spin local systems ${\Theta}_{R_\alpha}$ and degree shifts. The paper provides a rigorous verification of the filtered $A_{ abla\infty}$-relations by detailed sign tracking across boundary strata of moduli spaces using tree-like Kuranishi structures and CF-perturbations, leveraging base-change formulas for integration along fibers. The results solidify the coherence of Bott--Morse Lagrangian Fukaya categories with multiple Lagrangians and de Rham models, enabling precise handling of orientation and sign in degenerate settings with nontrivial local systems.
Abstract
We describe the sign and orientation issue appearing the filtered $A_{\infty}$-formulae in Lagrangian Floer theory using de Rham model in Bott-Morse setting. After giving the definition of filtered $A_{\infty}$-operations in a Fukaya category, we verify the filtered $A_{\infty}$-formulae.