Moduli spaces of 4d N=2 quantum field theories

Supersymmetry endows quantum field theories with several rich algebraic and geometric structures associated to their moduli space of vacua, providing powerful tools to study such theories non-perturbatively. For example, in four-dimensional theories with eight supercharges, the low energy dynamics of the theory is captured by an algebraic completely integrable system whose base is the Coulomb branch– a particular distinguished submanifold of the moduli space. This structure is so tightly constrained, that there is an ongoing program to classify such theories purely by understanding their Coulomb branch geometry. In this talk, I will give a gentle introduction to the geometry of the moduli spaces of 4d N=2 theories and, time permitting, discuss some recent results showcasing how the geometry of the Coulomb branch can be used to constrain certain physical quantities of the theory.