Supersymmetry in the Seiberg–Witten theory a window into quantum field theory

We take supersymmetry in the Seiberg–Witten theory as a case study of the uses of (super)symmetry arguments in studying the ontology of four-dimensional interacting quantum field theories. Together with a double expansion, supersymmetry is a via media that helps to bridge the gap between the ontologies of an exact quantum field theory and its semi-classical limit. We discuss a class of states that exist at any value of the coupling, and whose properties such as mass, electric and magnetic charges, and spin quantum numbers can be precisely characterised at low energies. The low-energy theory is best presented as a one-dimensional complex manifold, equipped with metric and other structures: namely, the space of low-energy vacua, covered by three open regions that are interpreted as macroscopic phases. We discuss two cases of emergence: the emergence of the low-energy regime and the emergence between models at low energies, thereby highlighting the significance of the topology of the space of vacua for such cases of emergence.