Spring School on Field Theories and Algebraic Topology

At least since the 1980s, algebraic topology and quantum field theories have interacted in increasingly interesting manner. The purpose of the spring school was to introduce the participants to some of these interactions.

The school was held 16-20 May, 2022, in the conference center Bergse Bossen in Driebergen, close to Utrecht in the Netherlands.

The bulk of the spring school consisted of four lecture series:

Dan Berwick-Evans: Supersymmetric field theories and topological modular forms

André Henriques: Chiral conformal field theory

Peter Teichner/Luuk Steehuwer: Some computations of positive TQFTs

Katrin Wendland: The complex elliptic genus

The lecture series of Berwick-Evans had as one focus how to relate the classification of supersymmetric field theories with cohomology theories. The relevant cohomology theory for one-dimensional field theories is K-theory, while for two-dimensional field theories one considers something called topological modular forms. The subject of Henriques's lectures, chiral conformal field theories, are a particularly important class of 2-dimensional field theories. Elliptic genera, as in Wendland's lectures, are important invariants of both manifolds and quantum field theories and were a key motivation for the introduction of topological modular forms. The lecture series of Teichner and Stehouwer used topological field theories to classify topological phases. The notes/slides of the four lecture series (plus additional material) can be found below. Especially the handwritten notes should be used at the risk of the reader and mistakes in them are likely due to the notetaker, i.e. the organizer.

The school was mostly aimed at PhD students. Beyond de Rham cohomology, vector bundles and characteristic classes, some of the further assumed background knowledge can befound here. This includes in particular a short series of videos by Dan Berwick-Evans about index theory.

The schedule was the following: