The Green-Schwarz Mechanism and Geometric Anomaly Relations in 2d (0,2) F-theory Vacua

TL;DR

The authors derive closed-form expressions for gauge and gravitational anomalies in 2d $N=(0,2)$ theories arising from F-theory on elliptically fibered Calabi–Yau 5-folds and show that abelian anomalies are canceled by a two-dimensional Green–Schwarz mechanism implemented via RR axions. They separate anomaly cancellation into flux-independent geometric identities and flux-dependent conditions tied to $G_4$ backgrounds, validating them in an explicit $SU(5) imes U(1)$ example. The work reveals a rich geometric structure linking elliptic fibration topology with anomaly inflow and GS terms, and places the 2d results in the broader context of 6d/4d anomaly relations, while highlighting open mathematical questions about proofs on CY5-fibrations. Overall, it provides a concrete framework for understanding anomaly cancellation in 2d F-theory vacua and exposes deeper connections between flux, geometry, and low-dimensional effective theories.

Abstract

We study the structure of gauge and gravitational anomalies in 2d N=(0,2) theories obtained by compactification of F-theory on elliptically fibered Calabi-Yau 5-folds. Abelian gauge anomalies, induced at 1-loop in perturbation theory, are cancelled by a generalized Green-Schwarz mechanism operating at the level of chiral scalar fields in the 2d supergravity theory. We derive closed expressions for the gravitational and the non-abelian and abelian gauge anomalies including the Green-Schwarz counterterms. These expressions involve topological invariants of the underlying elliptic fibration and the gauge background thereon. Cancellation of anomalies in the effective theory predicts intricate topological identities which must hold on every elliptically fibered Calabi-Yau 5-fold. We verify these relations in a non-trivial example, but their proof from a purely mathematical perspective remains as an interesting open problem. Some of the identities we find on elliptic 5-folds are related in an intriguing way to previously studied topological identities governing the structure of anomalies in 6d N=(1,0) and 4d N=1 theories obtained from F-theory.