Positive microlocal holonomies are globally regular

TL;DR

This paper provides a geometric criterion ensuring that local microlocal holonomies become globally regular on moduli spaces of Lagrangian fillings, facilitating the study of cluster structures in Legendrian link theory.

Contribution

It introduces a new local-to-global regularity criterion for microlocal holonomies on moduli spaces of Lagrangian fillings, applicable to arbitrary Legendrian links.

Findings

Established a geometric criterion for global regularity of microlocal holonomies.

Constructed regular functions on derived moduli stacks of sheaves with Legendrian microsupport.

Proved local microlocal merodromies yield global Hochschild 0-cycles.

Abstract

We establish a geometric criterion for local microlocal holonomies to be globally regular on the moduli space of Lagrangian fillings. This local-to-global regularity result holds for arbitrary Legendrian links and it is a key input for the study of cluster structures on such moduli spaces. Specifically, we construct regular functions on derived moduli stacks of sheaves with Legendrian microsupport by studying the Hochschild homology of the associated dg-categories via relative Lagrangian skeleta. In this construction, a key geometric result is that local microlocal merodromies along positive relative cycles in Lagrangian fillings yield global Hochschild 0-cycles for these dg-categories.