2025/04/22 by Sushmita Venugopalan, Venugopalan, Sushmita, Chris Woodward +1 · 1 voice · 1 citation
Mathematics · #53D37 #53D40 #53D42 #FOS: Mathematics #Symplectic Geometry (math.SG) #math.SG
paper · pdf · doi:10.48550/arxiv.2504.15583
In previous work, we introduced a version of the Fukaya algebra associated to a degeneration of a symplectic manifold, whose structure maps count collections of maps in the components of the degeneration satisfying matching conditions. In this paper, we introduce a further degeneration of the matching conditions (similar in spirit to Bourgeois' version of symplectic field theory) which results in a "split Fukaya algebra" whose structure maps are, in good cases, sums of products over vertices of tropical graphs. In the case of toric Lagrangians contained in a toric component of the degeneration, an invariance argument implies the existence of projective Maurer-Cartan solutions, which gives an alternate proof of the unobstructedness result of Fukaya-Oh-Ohta-Ono for toric manifolds. Our result also proves unobstructedness in more general cases, such as for toric Lagrangians in almost toric four-manifolds.