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Some topological genera and Jacobi forms.

Tewodros Amdeberhan1, Michael J Griffin2, Ken Ono3

  • 1Department of Mathematics, Tulane University, New Orleans, LA 70118.

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Summary
This summary is machine-generated.

This study connects Jacobi theta functions to cobordism invariants like the Hirzebruch [Formula: see text]-genus and Witten [Formula: see text]-genus. It reveals surprising links to Ramanujan

Keywords:
Jacobi formgenuspartition Eisenstein series

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Area of Science:

  • Mathematics
  • Topology
  • Number Theory

Background:

  • The study revisits Hirzebruch's [Formula: see text]-genus and Witten's [Formula: see text]-genus, known cobordism invariants for specific manifolds.
  • These invariants are crucial in understanding the topological properties of manifolds.

Purpose of the Study:

  • To elucidate the mathematical origins and interconnections of the [Formula: see text]-genus, Hirzebruch [Formula: see text]-genus, and Witten [Formula: see text]-genus.
  • To establish a direct link between these genera and Jacobi's theta function.

Main Methods:

  • Employing Hecke's trick to modify the [Formula: see text]-genus and [Formula: see text]-genus.
  • Utilizing Jacobi's theta function and partition Eisenstein series [Formula: see text].
  • Analyzing Ramanujan's "lost notebook" for connections to theta function derivatives.

Main Results:

  • The [Formula: see text]-genus and [Formula: see text]-genus are shown to arise directly from Jacobi's theta function after modification.
  • Exact formulas for quasimodular expressions of [Formula: see text] and [Formula: see text] are derived as "traces" of partition Eisenstein series.
  • A connection is established between Ramanujan's work on theta function twists and the later discovery of these genera in spin manifolds.
  • The nonholomorphic [Formula: see text]-completion of the Witten genus characteristic series is identified as a Jacobi theta function avatar of the [Formula: see text]-genus.

Conclusions:

  • Jacobi's theta function provides a unifying framework for understanding the [Formula: see text]-genus and [Formula: see text]-genus.
  • The research highlights unexpected historical links between number theory (Ramanujan) and topology (Borel, Hirzebruch).
  • This work offers new perspectives on the relationship between modular forms and topological invariants.