How do field theories detect torsion in topological modular forms?

Abstract: Since the mid 1980s, there have been hints of a connection between 2-dimensional field theories and elliptic cohomology. This led to Stolz and Teichner's conjectured geometric model for the universal elliptic cohomology theory of topological modular forms (TMF), for which cocycles are 2-dimensional (supersymmetric) field theories. Properties of these field theories lead naturally to the expected integrality and modularity properties of classes in TMF. However, the abundant torsion in TMF has always been mysterious from the field theory point of view. In this talk, we will describe a map from 2-dimensional field theories to a cohomology theory that approximates TMF. This map affords a cocycle description of certain torsion classes. In particular, we will explain how a choice of anomaly cancelation for the supersymmetric sigma model with target $S^3$ determines a cocycle representative of the generator of $\pi_3($TMF$)=\mathbb Z/24$.

Mathematics

Audience: researchers in the topic

Opening Workshop (IRP Higher Homotopy Structures 2021, CRM-Bellaterra)