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Elliptic genera and $SL(2,Z)$ modular forms for fibre bundles

Yong Wang

Abstract

By the family index theory, we generalize some well-known $SL(2,Z)$ modular forms to the family case and obtain some new anomaly cancellation formulas for the determinant line bundle and index gerbes, and certain results about eta invariants. Moreover, for the higher degree case, we give some anomaly cancellation formulas of residue Chern forms.

Elliptic genera and $SL(2,Z)$ modular forms for fibre bundles

Abstract

By the family index theory, we generalize some well-known modular forms to the family case and obtain some new anomaly cancellation formulas for the determinant line bundle and index gerbes, and certain results about eta invariants. Moreover, for the higher degree case, we give some anomaly cancellation formulas of residue Chern forms.
Paper Structure (7 sections, 51 theorems, 172 equations)

This paper contains 7 sections, 51 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Anomaly cancellation formulas and $SL(2,Z)$ modular forms for fibre bundles
  3. The two-variable elliptic genus for fibre bundles
  4. The higher degree case
  5. Acknowledgements
  6. Data availability
  7. Conflict of interest

Key Result

Theorem 2.2

$Q(TZ,\tau)$ is a modular form over $SL_2({\bf Z})$ with the weight $2k$.

Theorems & Definitions (55)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • ...and 45 more