  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.
