What are Elliptic Genera?

In the early 1950's, F. Hirzebruch found a very beautiful way of associating formal power series to multiplicative cobordism invariants.  For example, under this correspondence, the Taylor series of the hyperbolic tangent is associated with the signature of oriented manifolds.  Similarly, the Taylor series of the hyperbolic sine function is associated with the so-called Â-genus.  It turns out that the analytic properties of these hyperbolic (simply-periodic) functions result in some very striking properties of the corresponding invariants.

Elliptic genera correspond to doubly-periodic (elliptic) functions and retain many of the properties of the signature and the Â-genus which appear as limit cases of elliptic genera.  In particular, elliptic genera behave nicely in the presence of a Lie group action on a Spin manifold (Rigidity Theorem of Bott and Taubes).