Chiral anomalies on a circle and their cancellation in F-theory

TL;DR

The paper addresses how four-dimensional local gauge anomalies manifest when the theory is compactified on a circle, showing that anomaly information is retained in three dimensions through field-dependent Chern-Simons terms generated at one loop by KK modes. Using a regulator that preserves four-dimensional Lorentz invariance, the authors connect the 3D CS coefficients to the underlying 4D anomaly structure, and demonstrate that anomaly cancellation is equivalent to the gauge invariance of the resulting 3D effective action, up to anomaly inflow. They extend the framework to include gravity and Green-Schwarz mechanisms, deriving the corresponding CS coefficients and inflow relations, and then apply these results to F-theory by comparing circle reductions with M-theory on Calabi-Yau fourfolds with $G_4$-flux; in this regime, local anomalies cancel automatically. The work provides a robust bridge between 4D anomaly conditions and 3D topological terms, offering a practical criterion for anomaly consistency in F-theory compactifications and a method to extract higher-order corrections from the 3D effective action.

Abstract

We study in detail how four-dimensional local anomalies manifest themselves when the theory is compactified on a circle. By integrating out the Kaluza-Klein modes in a way that preserves the four-dimensional symmetries in the UV, we show that the three-dimensional theory contains field-dependent Chern-Simons terms that appear at one-loop. These vanish if and only if the four-dimensional anomaly is canceled, so the anomaly is not lost upon compactification. We extend this analysis to situations where anomalies are canceled through a Green-Schwarz mechanism. We then use these results to show automatic cancellation of local anomalies in F-theory compactifications that can be obtained as a limit of M-theory on a smooth Calabi-Yau fourfold with background flux.

Figures (3)

• Figure 1: One-loop diagrams contributing to the field independent (a) and field dependent (b) Chern- Simons terms in three dimensions.
• Figure 2: Plot of the value of the CS coefficient $\Theta^{\rm \textsc{af}}$, given in \ref{['theta_af']}, with charges being the hypercharges of the Standard Model, as a function of the Coulomb branch parameter $\langle \zeta \rangle$, over the fundamental period $\langle \zeta \rangle \in \left [-\frac{1}{2} , \frac{1}{2} \right ]$. The red points denote the values of $\langle \zeta \rangle$ for which some $q_a \, \langle \zeta \rangle \in \mathbb Z$, where at least a fermion is massless.
• Figure 3: One-loop diagrams contributing to the field independent CS terms involving the KK photon $A^0$ in three dimensions.