# From the signature theorem to anomaly cancellation

## TL;DR

This paper explores the connection between the signature theorem and anomaly cancellation by applying the Atiyah-Singer index theorem and the RRGQ formula to elliptic curves, revealing insights into anomaly measures and cancellation methods.

## Contribution

It demonstrates how the family index theorem and the RRGQ formula can be used to analyze and cancel anomalies in elliptic fibrations, linking mathematical theorems to physical anomaly concepts.

## Key findings

- Derived a generalized cohomology class related to anomalies.
- Showed how to cancel local anomalies on elliptic surfaces.
- Connected mathematical index theorems with physical anomaly cancellation.

## Abstract

We survey the Hirzebruch signature theorem as a special case of the Atiyah-Singer index theorem. The family version of the Atiyah-Singer index theorem in the form of the Riemann-Roch-Grothendieck-Quillen (RRGQ) formula is then applied to the complexified signature operators varying along the universal family of elliptic curves. The RRGQ formula allows us to determine a generalized cohomology class on the base of the elliptic fibration that is known in physics as (a measure of) the local and global anomaly. Combining several anomalous operators allows us to cancel the local anomaly on a Jacobian elliptic surface, a construction that is based on the construction of the Poincar\'e line bundle over an elliptic surface.