# Anomaly of the Electromagnetic Duality of Maxwell Theory

## TL;DR

This paper investigates the electromagnetic duality anomaly in 3+1D Maxwell theory, revealing it is 56 times that of a Weyl fermion, with implications for string theory consistency.

## Contribution

It provides two independent derivations of the Maxwell duality anomaly magnitude, linking it to topological phases and superconformal field theories.

## Key findings

- Anomaly is 56 times that of a Weyl fermion.
- Derived using bulk SPT phases in 4+1D.
- Connected to properties of 5+1D E-string theory.

## Abstract

We consider the ($3{+}1$)-dimensional Maxwell theory in the situation where going around nontrivial paths in the spacetime involves the action of the duality transformation exchanging the electric field and the magnetic field, as well as its $\mathrm{SL}(2,\mathbb{Z})$ generalizations. We find that the anomaly of this system in a particular formulation is 56 times that of a Weyl fermion. This result is derived in two independent ways: one is by using the bulk symmetry protected topological phase in $4{+}1$ dimensions characterizing the anomaly, and the other is by considering the properties of a ($5{+}1$)-dimensional superconformal field theory known as the E-string theory. This anomaly of the Maxwell theory plays an important role in the consistency of string theory.