Relative Calabi-Yau structure on microlocalization

TL;DR

This paper constructs canonical Calabi-Yau structures on microlocal sheaves and their adjoints on manifolds with Legendrian submanifolds, advancing the understanding of Fukaya categories without relying on local properness.

Contribution

It introduces a new approach to Calabi-Yau structures on microlocal sheaves that avoids local properness and arborealization, with applications to wrapped Fukaya categories.

Findings

Constructs a canonical strong smooth relative Calabi-Yau structure on microlocalization.

Establishes a canonical strong Calabi-Yau structure on microsheaves.

Provides a Calabi-Yau structure on the Orlov functor for wrapped Fukaya categories.

Abstract

For an oriented manifold $M$ and a compact subanalytic Legendrian $\Lambda \subseteq S^*M$, we construct a canonical strong smooth relative Calabi--Yau structure on the microlocalization at infinity and its left adjoint $m_\Lambda^l: \operatorname{\mu sh}_\Lambda(\Lambda) \rightleftharpoons \operatorname{Sh}_\Lambda(M)_0 : m_\Lambda$ between compactly supported sheaves on $M$ with singular support on $\Lambda$ and microsheaves on $\Lambda$. We also construct a canonical strong Calabi-Yau structure on microsheaves $\operatorname{\mu sh}_\Lambda(\Lambda)$. Our approach does not require local properness and hence does not depend on arborealization. We thus obtain a canonical smooth relative Calabi-Yau structure on the Orlov functor for wrapped Fukaya categories of cotangent bundles with Weinstein stops, such that the wrap-once functor is the inverse dualizing bimodule.