Some topological genera and Jacobi forms

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 $k\ge 0,$ we obtain exact formulas for the quasimodular expressions of ${\widehat{A}}_k$ and $L_k$ as traces'' of partition Eisenstein series \[ \widehat{\mathcal{A}}_k(τ)= \operatorname{Tr}_k(φ_{\widehat{A}};τ)\ \ \ \ \ \ {\text {and}}\ \ \ \ \ \ \mathcal{L}_k(τ)= \operatorname{Tr}_k(φ_L;τ), \] which are easily converted to the original topological expressions. Surprisingly, Ramanujan defined twists of the $\widehat{\mathcal{A}}_k(τ)$ 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.