Some topological genera and Jacobi forms

Abstract

We revisit and elucidate the $\widehat{A}$-genus, Hirzebruch's $L$-genus and Witten's $W$-genus, cobordism invariants of special classes of manifolds. After slight modification, involving Hecke's trick, we find that the $\widehat{A}$-genus and $L$-genus arise directly from Jacobi's theta function. For every $k\geq 0,$ we obtain exact formulas for the quasimodular expressions of $\widehat{A}_k$ and $L_k$ as ``traces'' of partition Eisenstein series \[ \widehat{\mathcal{A}}_k(\tau)= \operatorname{Tr}_k(\phi_{\widehat{A}};\tau)\ \ \ \ \ \ {\text {and}}\ \ \ \ \ \ \mathcal{L}_k(\tau)= \operatorname{Tr}_k(\phi_L;\tau), \] which are easily converted to the original topological expressions. Surprisingly, Ramanujan defined twists of the $\widehat{\mathcal{A}}_k(\tau)$ 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 $G_2^{\star}$-completion of the characteristic series of the Witten genus is the Jacobi theta function avatar of the $\widehat{A}$-genus.