# Algebraic Cycles and Local Anomalies in F-Theory

## TL;DR

This paper establishes cohomological identities in elliptic fibrations that encode anomaly cancellation conditions in F-theory compactifications, and conjectures these relations extend to the Chow ring, supported by non-trivial examples.

## Contribution

It introduces universal cohomological relations that govern anomaly cancellation in F-theory and conjectures their extension to the Chow ring, verified in complex examples.

## Key findings

- Identities in the cohomology ring encode anomaly cancellation conditions.
- These identities are shown to govern anomalies in 4D and 6D F-theory vacua.
- Conjecture that relations hold in the Chow ring, supported by examples.

## Abstract

We introduce a set of identities in the cohomology ring of elliptic fibrations which are equivalent to the cancellation of gauge and mixed gauge-gravitational anomalies in F-theory compactifications to four and six dimensions. The identities consist in (co)homological relations between complex codimension-two cycles. The same set of relations, once evaluated on elliptic Calabi-Yau three-folds and four-folds, is shown to universally govern the structure of anomalies and their Green-Schwarz cancellation in six- and four-dimensional F-theory vacua, respectively. We furthermore conjecture that these relations hold not only within the cohomology ring, but even at the level of the Chow ring, i.e. as relations among codimension-two cycles modulo rational equivalence. We verify this conjecture in non-trivial examples with Abelian and non-Abelian gauge groups factors. Apart from governing the structure of local anomalies, the identities in the Chow ring relate different types of gauge backgrounds on elliptically fibred Calabi-Yau four-folds.