Lagrangian cobordism functor in microlocal sheaf theory I

TL;DR

This paper constructs a functor between sheaf categories induced by a Lagrangian cobordism, providing a sheaf-theoretic perspective on Legendrian contact homology and revealing new obstructions to cobordisms.

Contribution

It introduces a sheaf-theoretic functor associated with Lagrangian cobordisms, linking Legendrian contact homology and sheaf categories in a novel way.

Findings

Defines a functor between sheaf categories induced by Lagrangian cobordisms.

Establishes a sheaf-theoretic analogue of Legendrian contact homology maps.

Derives new obstructions to the existence of high-dimensional Lagrangian cobordisms.

Abstract

Given a Lagrangian cobordism $L$ of Legendrian submanifolds from $\Lambda_-$ to $\Lambda_+$, we construct a functor $\Phi_L^*: Sh^c_{\Lambda_+}(M) \rightarrow Sh^c_{\Lambda_-}(M) \otimes_{C_{-*}(\Omega_*\Lambda_-)} C_{-*}(\Omega_*L)$ between sheaf categories of compact objects with singular support on $\Lambda_\pm$ and its right adjoint on sheaf categories of proper objects, using Nadler-Shende's work. This gives a sheaf theory description analogous to the Lagrangian cobordism map on Legendrian contact homologies and the right adjoint on their unital augmentation categories. We also deduce some long exact sequences and new obstructions to Lagrangian cobordisms between high dimensional Legendrian submanifolds.