Microsheaf composition of Lagrangian correspondences

TL;DR

This paper explores how microsheaf quantization interacts with the composition of Lagrangian correspondences in symplectic geometry, demonstrating commutativity under certain conditions and illustrating group actions on microsheaf categories.

Contribution

It establishes the compatibility of microsheaf composition with Lagrangian correspondence composition and introduces a family version of the gappedness criterion for nearby cycles.

Findings

Operations commute when the composition is embedded.

Lie groups of symplectomorphisms act on microsheaf categories.

Develops a family version of the gappedness criterion.

Abstract

In exact symplectic manifolds whose Liouville flow is gradientlike for a proper Morse function, one can associate conic microsheaves to eventually conic exact Lagrangians. Here we study how this 'microsheaf quantization' interacts with composition of Lagrangian correspondences. In particular: these operations commute when the composition is embedded. As an illustration, we show that Lie groups of exact symplectomorphisms act on microsheaf categories. The key technical advance is a version 'in families' of the gappedness criterion for commuting nearby cycles past tensor or Hom.