Some results on the moduli spaces of quiver bundles

This article is concerned with the study of gauge theory, stability and moduli for twisted quiver bundles in algebraic geometry. We review natural vortex equations for twisted quiver bundles and their link with a stability condition. Then we provide a brief overview of their relevance to other geometric problems and explain how quiver bundles can be viewed as sheaves of modules over a sheaf of associative algebras and why this view point is useful, e.g., in their deformation theory. Next we explain the main steps of an algebro-geometric construction of their moduli spaces. Finally, we focus on the special case of holomorphic chains over Riemann surfaces, providing some basic links with quiver representation theory. Combined with the analysis of the homological algebra of quiver sheaves and modules, these links provide a criterion for smoothness of the moduli spaces and tools to study their variation with respect to stability.