Lagrangian cobordism functor in microlocal sheaf theory II

TL;DR

This paper develops a sheaf-theoretic framework for Lagrangian cobordisms, establishing equivalences of sheaf categories, interpreting cobordism functors as correspondences, and exploring implications for immersed Lagrangians and Legendrian invariants.

Contribution

It introduces a sheaf-theoretic approach to Lagrangian cobordisms, generalizes to immersed cases, and links sheaf categories to loop space chains of Lagrangian fillings.

Findings

Sheaves on Lagrangian cobordisms are equivalent to sheaves at the negative end plus local systems.

The Lagrangian cobordism functor is action decreasing.

Constructs Legendrians with sheaf categories Morita equivalent to loop space chains.

Abstract

For an exact Lagrangian cobordism $L$ between Legendrians in $J^1(M)$ from $\Lambda_-$ to $\Lambda_+$ whose Legendrian lift is $\widetilde{L}$, we prove that sheaves in $Sh_{\widetilde{L}}(M \times \mathbb{R} \times \mathbb{R}_{>0})$ are equivalent to sheaves at the negative end $Sh_{\Lambda_-}(M \times \mathbb{R})$ 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 $Sh_{\Lambda_\pm}(M \times \mathbb{R})$ 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.