Sheaf Quantization of Legendrian Isotopy

TL;DR

This paper proves the invariance of sheaf categories associated with Legendrian isotopies embedded in Weinstein hypersurfaces, advancing the understanding of sheaf-theoretic invariants in symplectic topology.

Contribution

It establishes the invariance of sheaf categories under Legendrian isotopies embedded in Weinstein hypersurfaces, linking Legendrian isotopy to sheaf-theoretic invariants.

Findings

Sheaf categories are invariant under Legendrian isotopies embedded in Weinstein hypersurfaces.

Provides a new criterion for invariance based on embedding into Weinstein hypersurfaces.

Advances the understanding of the relationship between Legendrian isotopy and sheaf theory.

Abstract

Let $\{\Lambda^\infty_t\}$ be an isotopy of Legendrians (possibly singular) in a unit cosphere bundle $S^*M$. Let $Sh(M, \Lambda^\infty_t)$ be the differential graded (dg) derived category of constructible sheaves on $M$ with singular support at infinity contained in $\Lambda^\infty_t$. We prove that if the isotopy of Legendrians embeds into an isotopy of Weinstein hypersurfaces, then the categories $Sh(M, \Lambda^\infty_t)$ are invariant.