$Z$-critical connections and Bridgeland stability conditions

TL;DR

This paper introduces $Z$-critical connections, a new class of geometric PDEs linked to Bridgeland stability, establishing a connection between stability and the existence of solutions, including new higher-rank examples.

Contribution

It defines $Z$-critical connections associated with Bridgeland stability and proves their existence in the large volume limit, extending solutions to higher rank cases.

Findings

$Z$-critical connections exist if and only if the bundle is asymptotically $Z$-stable.

Provides the first higher-rank solutions to deformed Hermitian Yang--Mills equations.

Connects geometric PDEs with stability conditions via infinite dimensional moment maps.

Abstract

We associate geometric partial differential equations on holomorphic vector bundles to Bridgeland stability conditions. We call solutions to these equations $Z$-critical connections, with $Z$ a central charge. Deformed Hermitian Yang--Mills connections are a special case. We explain how our equations arise naturally through infinite dimensional moment maps. Our main result shows that in the large volume limit, a sufficiently smooth holomorphic vector bundle admits a $Z$-critical connection if and only if it is asymptotically $Z$-stable. Even for the deformed Hermitian Yang--Mills equation, this provides the first examples of solutions in higher rank.