Elliptic genera in mathematics and physics and a generalization to G-manifolds

Abstract

Elliptic genera are a special type of manifold invariants that only depend on the bordism class of the manifolds. They can be constructed from elliptic curves and take values in modular forms. Important examples include the signature for 4n-manifolds and the so-called Â-genus which is related to the Dirac operator. A more general perspective is provided by complex genera which can be studied using formal group laws. This is the right setting for a generalization of elliptic genera to G-manifolds. For this, equivariant bordism and equivariant formal group laws are considered. In physics, elliptic genera arise as partition functions of supersymmetric string theories which has been originally described by Witten. Since then, this perspective has been further considered and gave rise to several developments in the mathematical theory. In this thesis, we will give an introduction to elliptic genera and compare the mathematical and physical perspective. Furthermore, we will explain their relations to formal group laws and outline the generalization of elliptic genera to the equivariant setting.