{
  "id": "2502.02432",
  "version": "v2",
  "published": "2025-02-04T15:56:57.000Z",
  "updated": "2025-02-05T02:15:46.000Z",
  "title": "Some topological genera and Jacobi forms",
  "authors": [
    "Tewodros Amdeberhan",
    "Michael Griffin",
    "Ken Ono"
  ],
  "categories": [
    "math.NT",
    "math.AT"
  ],
  "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. In this way, 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 {\\rm and}}\\ \\ \\ \\ \\ \\ \\mathcal{L}_k(\\tau)= \\operatorname{Tr}_k(\\phi_L;\\tau). \\] 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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-02-05T02:15:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F50",
      "58J20"
    ],
    "keywords": [
      "jacobi forms",
      "topological genera",
      "jacobi theta function avatar",
      "jacobis theta function",
      "partition eisenstein series"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}