Spherical adjunction and Serre functor from microlocalization

TL;DR

This paper establishes that microlocalization at infinity for certain Legendrian submanifolds is a spherical functor, linking sheaf theory with Fukaya categories and providing new tools for symplectic topology.

Contribution

It proves microlocalization is a spherical functor for swappable or full Legendrian stops, connecting sheaf theory with Fukaya categories and symplectic invariants.

Findings

Microlocalization at infinity is a spherical functor under certain conditions.

The spherical cotwist corresponds to the Serre functor on a sheaf subcategory.

Constructs Guillermou doubling functor for Reeb flows.

Abstract

For a subanalytic Legendrian $\Lambda \subseteq S^{*}M$, we prove that when $\Lambda$ is either swappable or a full Legendrian stop, the microlocalization at infinity $m_\Lambda: \operatorname{Sh}_\Lambda(M) \rightarrow \operatorname{\mu sh}_\Lambda(\Lambda)$ is a spherical functor, and the spherical cotwist is the Serre functor on the subcategory $\operatorname{Sh}_\Lambda^b(M)_0$ of compactly supported sheaves with perfect stalks. This is a sheaf theory counterpart (with weaker assumptions) of the results on the cap functor and cup functors between Fukaya categories. When proving spherical adjunction, we deduce the Sato-Sabloff fiber sequence and construct the Guillermou doubling functor for any Reeb flow.