Talks

The Euler characteristic of a compact complex manifold M is a classical cohomological invariant. Depending on the viewpoint, it is most natural to interpret it as an index of an elliptic differential operator on M, or as a supersymmetric index in superconformal field theories “on M”. Refining the Euler characteristic but keeping with both index theoretic interpretations, one arrives at the notion of complex elliptic genera. We argue that superconformal field theory motivates further refinements of these elliptic genera which result in a choice of several new invariants, all of which have lost their interpretation in terms of index theory. However, at least if M is a K3 surface, then superconformal field theory and higher algebra select the same new invariant as a natural refinement of the complex elliptic genus.

Topological field theories in the sense of Atiyah–Segal are symmetric monoidal functors from a bordism category to the category of complex (super) vector spaces. A field theory E of dimension d associates vector spaces to closed (d-1)-manifolds and linear maps to manifolds of dimension d. It turns out that if E is invertible, i.e., if the vector spaces associated to (d-1)-manifolds have dimension one, then the complex number E(M) that E associates to a closed d-manifold M, is an SKK manifold invariant. Here these letters stand for schneiden=cut, kleben=glue and kontrolliert=controlled, meaning that E(M) does not change when modifying the manifold by cutting and gluing along hypersurfaces in a controlled way. The main result of this joint work with Matthias Kreck and Peter Teichner is that the map described above gives a bijection between topological field theories and SKK manifold invariants.