Modular Invariance and Characteristic Numbers

TL;DR

This paper explores how modular invariance of elliptic operators on loop spaces leads to miraculous cancellation formulas, divisibility of characteristic numbers, and other topological results, revealing deep connections between topology and modular forms.

Contribution

It demonstrates that several topological phenomena are direct consequences of the modular invariance of elliptic operators, extending previous work on rigidity to cancellation formulas and divisibility.

Findings

Proves a general miraculous cancellation formula

Shows divisibility of certain characteristic numbers

Establishes topological results from modular invariance

Abstract

We show that a general miraculous cancellation formula, the divisibility of certain characteristic numbers and some other topologiclal results are con- sequences of the modular invariance of elliptic operators on loop spaces. Previously we have shown that modular invariance also implies the rigidity of many elliptic operators on loop spaces.