Some topological genera and Jacobi forms

TL;DR

This paper explores the deep connections between topological genera like the -genus, L-genus, and Witten genus with Jacobi theta functions, providing explicit formulas and revealing historical links to Ramanujan's work.

Contribution

It establishes new explicit formulas linking topological invariants to Jacobi theta functions and uncovers historical connections to Ramanujan's work on derivatives of theta functions.

Findings

-genus and L-genus derive directly from Jacobi theta functions.

Explicit quasimodular formulas for _k and L_k as traces of Eisenstein series.

The nonholomorphic G_2^* completion relates to the -genus via Jacobi theta functions.

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.