Z-critical connections and Bridgeland stability conditions

Journal article, 2024

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.