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

TL;DR

This paper analyzes gauge and gravitational anomalies in 2d (0,2) theories from F-theory compactifications on elliptic Calabi-Yau 5-folds, deriving anomaly expressions and topological identities crucial for consistency.

Contribution

It provides explicit formulas for anomalies and Green-Schwarz counterterms in 2d F-theory vacua, linking topological invariants to anomaly cancellation conditions.

Findings

Derived closed-form expressions for gauge and gravitational anomalies.

Identified topological identities necessary for anomaly cancellation.

Verified these identities in a non-trivial example.

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.