2025/04/10 by Roger Casals, Casals, Roger, Yoon Jae Nho +1 · 1 citation
Engineering · Mathematics · #53D12 #57K43 #Differential Geometry (math.DG) #Dynamics and Control of Mechanical Systems #Elasticity and Wave Propagation #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematics and Applications #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2504.08144
openalex publication_date 2025/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce and develop the theory of spectral networks in real contact and symplectic topology. First, we establish the existence and pseudoholomorphic characterization of spectral networks for Lagrangian fillings in the cotangent bundle of a smooth surface. These are proven via analytic results on the adiabatic degeneration of Floer trajectories and the explicit computation of continuation strips. Second, we construct a Family Floer functor for Lagrangian fillings endowed with a spectral network and prove its equivalence to the non-abelianization functor. In particular, this implies that both the framed 2d-4d BPS states and the Gaiotto-Moore-Neitzke non-abelianized parallel transport are realized as part of the A_∞-operations of the associated 4d partially wrapped Fukaya categories. To conclude, we present a new construction relating spectral networks and Lagrangian fillings using Demazure weaves, and show the precise relation between spectral networks and augmentations of the Legendrian contact dg-algebra.