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.

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