The two-variable elliptic genus in odd dimensions

TL;DR

The work extends the notion of the two-variable elliptic genus to odd-dimensional spin manifolds by identifying it as the index of a Toeplitz operator and as a holomorphic $SL(2,\mathbb{Z})$-Jacobi form, and simultaneously develops analogous two-variable genera for almost-complex and odd spin manifolds that are holomorphic Jacobi forms for $\Gamma_0(2)$, $\Gamma^0(2)$, and $\Gamma_{\theta}$. Under natural topological constraints (e.g., $c_1(W)=0$, $p_1(M)=p_1(W)$, $H^3(M)=0$), these genera exhibit modularity and yield anomaly cancellation formulas that imply divisibility of holomorphic Euler characteristics and Toeplitz indices. The paper also introduces meromorphic two-variable genera for both even and odd dimensions, enriching the link between spin/complex geometry, index theory, and the theory of modular and Jacobi forms. Overall, the results deepen our understanding of how geometric invariants of manifolds encode modular-analytic structures with potential applications to divisibility phenomena and anomaly considerations in higher dimensions.

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 $Γ_0(2)$, $Γ^0(2)$, $Γ_θ$-Jacobi forms. By these Jacobi forms, we can get some $SL(2,{\bf Z})$ and $Γ^0(2)$ modular forms. By these $SL(2,{\bf Z})$ and $Γ^0(2)$ 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 $Γ_0(2)$, $Γ^0(2)$, $Γ_θ$-Jacobi forms.

Theorems & Definitions (26)

• Definition 3.1
• Lemma 3.2
• Theorem 3.3
• Proposition 3.4
• Proposition 3.5
• proof
• Proposition 3.6
• Theorem 3.7
• proof
• Definition 4.1