Some topological genera and Jacobi forms
成果类型：
Article
署名作者：
Amdeberhan, Tewodros; Griffin, Michael J.; Ono, Ken
署名单位：
Tulane University; Vanderbilt University; University of Virginia
刊物名称：
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
ISSN/ISSBN：
0027-8695
DOI：
10.1073/pnas.2502678122
发表日期：
2025-08-12
关键词：
摘要：
We revisit and elucidate the 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 A-genus and L-genus arise directly from Jacobi's theta function. For every k >= 0, we obtain exact formulas for the quasimodular expressions of A(k) and L-k as traces of partition Eisenstein series A(K)(tau ) = Tr-k(Phi(A):tau) and L-k(tau) = Tr-k (Phi(L):tau) which are easily converted to the original topological expressions. Surprisingly, Ramanujan defined twists of the 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)*-completion of the characteristic series of the Witten genus is the Jacobi theta function avatar of the A-genus.
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