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SL(2,Z) modular forms and Witten genus in odd dimensions

Jianyun Guan, Yong Wang, Haiming Liu

Abstract

By some SL(2, Z) modular forms introduced in [4] and [10], we construct some modular forms over SL2(Z) and some modular forms over Γ^0(2) and Γ_0(2) in odd dimensions. In parallel, we obtain some new cancellation formulas for odd dimensional spin manifolds and odd dimensional spin^c manifolds respectively. As corollaries, we get some divisibility results of index of the Toeplitz operators on spin manifolds and spin^c manifolds .

SL(2,Z) modular forms and Witten genus in odd dimensions

Abstract

By some SL(2, Z) modular forms introduced in [4] and [10], we construct some modular forms over SL2(Z) and some modular forms over Γ^0(2) and Γ_0(2) in odd dimensions. In parallel, we obtain some new cancellation formulas for odd dimensional spin manifolds and odd dimensional spin^c manifolds respectively. As corollaries, we get some divisibility results of index of the Toeplitz operators on spin manifolds and spin^c manifolds .
Paper Structure (7 sections, 70 theorems, 223 equations)

This paper contains 7 sections, 70 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Characteristic Forms and Modular Forms
  3. Some modular forms and Witten genus over $SL_2({\bf Z})$ in odd dimensions
  4. Some modular forms and Witten genus over $\Gamma^0(2),\Gamma_0(2),\Gamma_{\theta}$
  5. Acknowledgements
  6. Data availability
  7. Conflict of interest

Key Result

Theorem 3.1

Let ${\rm dim}M=4k-1$. If $c_3(E,g,d)=0$, then for any integer $p,r\geq 1.$$Q(\nabla^{TM},g,d,\tau)$ is a modular form over $SL_2({\bf Z})$ with the weight $2p+2r=2k$.

Theorems & Definitions (77)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • Theorem 3.6
  • proof
  • ...and 67 more