The two-variable elliptic genus in odd dimensions

TL;DR

This paper introduces new two-variable elliptic genera for odd-dimensional spin and almost-complex manifolds, linking them to modular forms and deriving anomaly cancellation and divisibility results.

Contribution

It defines novel two-variable elliptic genera for odd-dimensional spin manifolds and explores their modular properties and applications.

Findings

Defined elliptic genera as indices of Toeplitz operators.

Established connections to $SL(2,\mathbb{Z})$ and $\,\Gamma^0(2)$ modular forms.

Derived anomaly cancellation formulas and divisibility results.

Abstract

A kind of two-variable elliptic genus for almost-complex manifolds was introduced by Ping Li and its various properties were established by him. In this paper, we define a two-variable elliptic genus for odd dimensional spin manifolds which is the index for some Toeplitz operator and a holomorphic $SL(2,Z)$-Jacobi form. We also define some two-variable elliptic genera for almost-complex manifolds and odd dimensional spin manifolds which are holomorphic $\Gamma_0(2)$, $\Gamma^0(2)$, $\Gamma _{\theta}$-Jacobi forms. By these Jacobi forms, we can get some $SL(2,{\bf Z})$ and $\Gamma^0(2)$ modular forms. By these $SL(2,{\bf Z})$ and $\Gamma^0(2)$ modular forms, we get some interesting anomaly cancellation formulas for almost complex manifolds and odd spin manifolds. As corollaries, we get some divisibility results of the holomorphic Euler characteristic number and the index of Toeplitz operators. In addition, we also define some another two-variable elliptic genera for even (rep. odd ) dimensional manifolds which are meromorphic $\Gamma_0(2)$, $\Gamma^0(2)$, $\Gamma _{\theta}$-Jacobi forms.