Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds

TL;DR

This paper explores the relationship between singular fibers in elliptic Calabi-Yau threefolds and anomaly cancellation in F-theory, introducing a Tate cycle to relate geometry and physical consistency conditions.

Contribution

It introduces the Tate cycle as a new tool to relate geometric cycles to anomaly cancellation in elliptic Calabi-Yau threefolds, applicable across dimensions.

Findings

The anomaly formula is geometrically represented by relations among codimension two cycles.

The Tate cycle can be explicitly calculated from the Weierstrass equation.

The anomaly cancellation constrains the Euler characteristic of the Calabi-Yau threefold.

Abstract

We investigate the delicate interplay between the types of singular fibers in elliptic fibrations of Calabi-Yau threefolds (used to formulate F-theory) and the "matter" representation of the associated Lie algebra. The main tool is the analysis and the appropriate interpretation of the anomaly formula for six-dimensional supersymmetric theories. We find that this anomaly formula is geometrically captured by a relation among codimension two cycles on the base of the elliptic fibration, and that this relation holds for elliptic fibrations of any dimension. We introduce a "Tate cycle" which efficiently describes this relationship, and which is remarkably easy to calculate explicitly from the Weierstrass equation of the fibration. We check the anomaly cancellation formula in a number of situations and show how this formula constrains the geometry (and in particular the Euler characteristic) of the Calabi-Yau threefold.