The geometry of the spinning string

The development of quantum electrodynamics is one of the major achievements of theoretical physics and mathematics of the 20th century, called the "Jewel of physics" by Richard Feynman. This talk is not about that. Instead, I explain two of its basic ingredients - Feynman diagrams, and Spinor bundles - and then describe how these can be adapted to "electron-like" strings. This will lead us naturally to the Spinor bundle on loop space, which I will describe in some detail. An element of loop space, i.e. a smooth function from the circle into some fixed manifold, is supposed to represent a string at a fixed moment in time. I will then explain the notion of a fusion product (on this bundle), and argue that this is a manifestation of the principle of locality, which is ubiquitous in physics. If time permits, I will discuss some ongoing work, in collaboration with Matthias Ludewig, Darvin Mertsch, and Konrad Waldorf, where we describe how this fusive spinor bundle on loop space fits beautifully in the higher categorical framework of 2-vector bundles.