Anomaly of the Electromagnetic Duality of Maxwell Theory

TL;DR

The paper analyzes the anomaly of the $(3+1)$-D Maxwell theory under electromagnetic duality when a nontrivial $SL(2,\mathbb{Z})$ background is turned on, establishing that the duality anomaly equals $56$ times the anomaly of a Weyl fermion. It derives this result via two independent frameworks: a $(4+1)$-D bulk BdC theory capturing the duality background and a $(5+1)$-D E-string theory perspective connecting self-dual tensors to Maxwell through a $T^2$ compactification. The authors compute the anomaly on twisted backgrounds like $S^5/\mathbb{Z}_k$ using eta invariants and Arf refinements, and show consistency with stringy objects such as O3-planes. The origin of the factor $56$ is traced to the E-string tensor/Higgs branch dynamics, providing a deep higher-dimensional explanation for the Maxwell anomaly and its implications for string theory consistency.

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.

Figures (1)

• Figure 1: Maxwell to 56 fermions via E-string theory