Legendrian-dga proof of localization for carved Weinstein subdomains

Compute the Legendrian differential graded algebras of the attaching spheres of the Weinstein subdomain X \setminus D produced by the explicit antisurgery presentation in Theorem 4.1, thereby giving a direct proof of the Ganatra–Pardon–Shende localization formula.

Background

For a Weinstein domain X and a carved Lagrangian disk D, the Ganatra–Pardon–Shende localization formula identifies the wrapped Fukaya category of the subdomain X \setminus D with the algebraic localization of the wrapped Fukaya category of X by D. The formula provides a method for computing morphism complexes using data involving X and D.

The paper’s antisurgery algorithm independently produces an explicit Weinstein presentation of X \setminus D and explicit Legendrian attaching spheres. The unresolved question is whether the Legendrian differential graded algebras of these attaching spheres can be computed directly from this presentation, yielding an alternative direct proof of the localization formula from Legendrian invariants.