Elliptic genus of Calabi-Yau manifolds and Jacobi and Siegel modular forms

TL;DR

This paper explores the mathematical relationships between elliptic genera of Calabi-Yau manifolds, Jacobi and Siegel modular forms, and their connections to Lorentzian Kac-Moody algebras, providing new structural insights.

Contribution

It establishes new links between elliptic genera, modular forms, and algebraic structures, and characterizes the graded ring of weak Jacobi forms with integral Fourier coefficients.

Findings

Relations between elliptic genus and Jacobi modular forms

Connections between second quantized elliptic genus and Siegel modular forms

Structural description of the graded ring of weak Jacobi forms

Abstract

In the paper we study two types of relations: a one is between the elliptic genus of Calabi-Yau manifolds and Jacobi modular forms, another one is between the second quantized elliptic genus, Siegel modular forms and Lorentzian Kac-Moody Lie algebras. We also determine the structure of the graded ring of the weak Jacobi forms with integral Fourier coefficients. It gives us a number of applications to the theory of elliptic genus and of the second quantized elliptic genus.