The Geometry of Quivers ^†

Quivers are oriented graphs that have profound connections to various areas of mathematics, including representation theory and geometry. Quiver representations correspond to a vast generalization of classical linear algebra problems. The geometry of these representations can be described in the framework of Hamiltonian reduction and geometric invariant theory, giving rise to the concept of quiver variety. In parallel to these developments, quivers have appeared to naturally encode certain supersymmetric quantum field theories. The associated quiver variety then corresponds to a part of the moduli space of vacua of the theory. However, physics tells us that another natural geometric object associated with quivers exists, which can be seen as a magnetic analog of the (electric) quiver variety. When viewed from that angle, magnetic quivers are a new tool, developed in the past decade, that help mathematicians and physicists alike to understand geometric spaces. This note is the writeup of a talk in which I review these developments from both the mathematical and physical perspective, emphasizing the dialogue between the two communities.