Some topological genera and Jacobi forms

Abstract

We revisit and elucidate the $\hat{A}$-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 $\hat{A}$-genus and $L$-genus arise directly from Jacobi's theta function. For every $k\ge 0,$ we obtain exact formulas for the quasimodular expressions of ${\hat{A}}_k$ and $L_k$ as ``traces'' of partition Eisenstein series \[ \widehat{\mathcal{A}}_k(τ)= \operatorname{Tr}_k(ϕ_{\widehat{A}};τ)\ \ \ \ \ \ {\text {and}}\ \ \ \ \ \ \mathcal{L}_k(τ)= \operatorname{Tr}_k(ϕ_L;τ), \] which are easily converted to the original topological expressions. Surprisingly, Ramanujan defined twists of the ${\hat{\mathcal{A}}}_k(\ddot{I}„)$ 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 $\hat{A}$-genus.

We have corrected a few minor typos and updated two references. This paper will appear in the Proceedings of the National Academy of Sciences