2023/08/09 by Wenyuan Li, Li, Wenyuan · 2 citations
Mathematics · Physics and Astronomy · #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2308.05089
openalex publication_date 2023/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For an exact Lagrangian cobordism L between Legendrians in J1(M) from Λ- to Λ+ whose Legendrian lift is \widetildeL, we prove that sheaves in Sh_\widetildeL(M × ℝ × ℝ>0) are equivalent to sheaves at the negative end ShΛ-(M × ℝ) together with the data of local systems Loc(L) by studying sheaf quantizations for general noncompact Lagrangians. Thus we interpret the Lagrangian cobordism functor between ShΛ_±(M × ℝ) as a correspondence parametrized by Loc(L). This enables one to consider generalizations to immersed Lagrangian cobordisms. We also prove that the Lagrangian cobordism functor is action decreasing and recover results on the lengths of embedded Lagrangian cobordisms. Finally, using the construction of Courte-Ekholm, we obtain a family of Legendrians with sheaf categories Morita equivalent to chains of based loop spaces of the Lagrangian fillings.