Anomaly Cancellation And Abelian Gauge Symmetries In F-theory

TL;DR

This work demonstrates that 4D F-theory compactifications, including non-Abelian and Abelian gauge factors, automatically implement anomaly cancellation through a generalized Green-Schwarz mechanism when analyzed via their M-theory dual on resolved Calabi-Yau fourfolds. By tracing 4D anomalies to 3D Chern-Simons terms generated on the Coulomb branch and matching them to classical flux-induced CS terms, the authors derive precise geometric conditions on G4-flux and the resolved geometry that enforce cancellation. The analysis yields explicit relations between flux data, axion gaugings, and chirality, and is validated in concrete SU(5) and SU(5)×U(1) examples, including weak-coupling Type IIB comparisons. The results provide a robust framework for constructing anomaly-free F-theory vacua with rich U(1) structures and offer geometric criteria applicable to a broad class of flux backgrounds.

Abstract

We study 4D F-theory compactifications on singular Calabi-Yau fourfolds with fluxes. The resulting N=1 effective theories can admit non-Abelian and U(1) gauge groups as well as charged chiral matter. In these setups we analyze anomaly cancellation and the generalized Green-Schwarz mechanism. This requires the study of 3D N=2 theories obtained by a circle compactification and their M-theory duals. Reducing M-theory on resolved Calabi-Yau fourfolds corresponds to considering the effective theory on the 3D Coulomb branch in which certain massive states are integrated out. Both 4D gaugings and 3D one-loop corrections of these massive states induce Chern-Simons terms. All 4D anomalies are captured by the one-loop terms. The ones corresponding to the mixed gauge-gravitational anomalies depend on the Kaluza-Klein vector and are induced by integrating out Kaluza-Klein modes of the U(1) charged matter. In M-theory all Chern-Simons terms classically arise from G_4-flux. We find that F-theory fluxes implement automatically the 4D Green-Schwarz mechanism if non-trivial geometric relations for the resolved Calabi-Yau fourfold are satisfied. We confirm these relations in various explicit examples and elucidate the general construction of U(1) symmetries in F-theory. We also compare anomaly cancellation in F-theory with its analog in Type IIB orientifold setups.

Figures (2)

• Figure 1: One-loop triangle diagram responsible for the Abelian-gravitational anomaly.
• Figure 2: One-loop diagram with KK-states in the loop generating the 3D Chern-Simons level $\Theta_{0m}$.