2025/02/04 by Tewodros Amdeberhan, Amdeberhan, Tewodros, W. Michael Griffin +3 · 2 citations
Mathematics · #11F50 #58J20 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2502.02432
openalex publication_date 2025/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We revisit and elucidate the \widehatA-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 \widehatA-genus and L-genus arise directly from Jacobi's theta function. For every k≥ 0, we obtain exact formulas for the quasimodular expressions of \widehatAk and Lk as ``traces'' of partition Eisenstein series \widehatAk(τ)= Trk(ϕ_\widehatA;τ) \text and Lk(τ)= Trk(ϕL;τ), which are easily converted to the original topological expressions. Surprisingly, Ramanujan defined twists of the \widehatAk(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 G2⋆-completion of the characteristic series of the Witten genus is the Jacobi theta function avatar of the \widehatA-genus.