The data of a compact Lie group G and a degree 4 cohomology class on its classifying space leads to invariants in low-dimensional topology as well as important representations of the infinite dimensional group of loops in G. The former are the famous quantum invariants of Reshetikhin-Turaev-Witten; the latter are the positive energy representations. The important algebraic data is encoded in a braided tensor category, constructed in special cases (G finite, G simply connected) by various methods. Previous joint work with Mike Hopkins and Constantin Teleman gave a unified description of the Grothendieck ring of that category in terms of topological K-theory. After reviewing these ideas I will describe joint work with Teleman which gives a geometric construction of the category, though not of its tensor structure.