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

TL;DR

This paper extends $SL(2,Z)$ modular forms to fiber bundle families using index theory, deriving new anomaly cancellation formulas for determinant line bundles, index gerbes, and eta invariants, with applications to higher degree cases.

Contribution

It introduces a generalization of classical modular forms to fiber bundle families and establishes novel anomaly cancellation formulas in this broader context.

Findings

Derived new anomaly cancellation formulas for determinant line bundles.

Extended $SL(2,Z)$ modular forms to family cases using index theory.

Obtained results on eta invariants and residue Chern forms.

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.