4D/3D QFT and representation theory

4D/3D quantum field theory in theoretical physics is conceptually rich and gives rise to many interesting mathematical structures, even though a fully rigorous mathematical formulation of the theories themselves is still lacking. A relatively recent discovery by Beem et al. shows that to every 4D N=2 superconformal field theory one can associate a representation-theoretic object called a vertex algebra, which serves as an invariant (or observable) of the theory. Although vertex algebras are inherently algebraic, those arising as invariants of 4D QFT display striking connections with certain geometric objects that also appear as invariants of the same physical theories. Similarly, to each 3D N=4 gauge theory one can associate two vertex algebras—the A-twisted and B-twisted boundary VOAs—which may be viewed as refinements of the Higgs and Coulomb branches. In this talk, I will discuss some representation-theoretic aspects of these phenomena.