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- A natural reﬁnement of the Euler characteristic

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18080079

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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