A natural refinement of the Euler characteristic

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.