Some topological genera and Jacobi forms.

Abstract

We revisit and elucidate the [Formula: see text]-genus, Hirzebruch's [Formula: see text]-genus, and Witten's [Formula: see text]-genus, cobordism invariants of special classes of manifolds. After slight modification, involving Hecke's trick, we find that the [Formula: see text]-genus and [Formula: see text]-genus arise directly from Jacobi's theta function. For every [Formula: see text] we obtain exact formulas for the quasimodular expressions of [Formula: see text] and [Formula: see text] as "traces" of partition Eisenstein series [Formula: see text] which are easily converted to the original topological expressions. Surprisingly, Ramanujan defined twists of the [Formula: see text] in his "lost notebook" in his study of derivatives of theta functions, decades before Borel and Hirzebruch rediscovered them in the context of spin manifolds. In addition, we show that the nonholomorphic [Formula: see text]-completion of the characteristic series of the Witten genus is the Jacobi theta function avatar of the [Formula: see text]-genus.