Some topological genera and Jacobi forms
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| Format: | Preprint |
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2025
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| author | Amdeberhan, Tewodros Griffin, Michael Ono, Ken |
| author_facet | Amdeberhan, Tewodros Griffin, Michael Ono, Ken |
| contents | 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 \[ \widehat{\mathcal{A}}_k(τ)= \operatorname{Tr}_k(ϕ_{\widehat{A}};τ)\ \ \ \ \ \ {\text {and}}\ \ \ \ \ \ \mathcal{L}_k(τ)= \operatorname{Tr}_k(ϕ_L;τ), \] 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_02432 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some topological genera and Jacobi forms Amdeberhan, Tewodros Griffin, Michael Ono, Ken Number Theory Algebraic Topology 11F50, 58J20 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 \[ \widehat{\mathcal{A}}_k(τ)= \operatorname{Tr}_k(ϕ_{\widehat{A}};τ)\ \ \ \ \ \ {\text {and}}\ \ \ \ \ \ \mathcal{L}_k(τ)= \operatorname{Tr}_k(ϕ_L;τ), \] 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. |
| title | Some topological genera and Jacobi forms |
| topic | Number Theory Algebraic Topology 11F50, 58J20 |
| url | https://arxiv.org/abs/2502.02432 |