Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds
Antonella Grassi, David R. Morrison
TL;DR
This work links elliptic Calabi–Yau threefold geometry to six-dimensional anomaly cancellation in F-theory by introducing the Tate cycle, a computable, geometry-driven descriptor of matter representations, and a virtual matter cycle that encodes the physical content in the Chow group. Anomalies are reframed as relations among codimension-two cycles and Casimir invariants, reducible to local checks along discriminant components via Tate’s data and branching rules. The authors prove that anomaly freedom follows when the virtual cycle is Casimir-equivalent to the Tate cycle for all components and when representation-multiplicity matches intersection numbers, connecting Euler characteristics and base geometry to the gauge content. They develop extensive local tests using Weierstrass data and Katz–Vafa-type reasoning, showing anomaly cancellation across a wide array of singular fibers, and discuss extensions to abelian factors, higher dimensions, and non-CY settings. The results provide a practical, computable dictionary between elliptic fibration geometry and the low-energy 6D (and by extension 4D) physics, with explicit constraints on the Euler characteristic and matter content.
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.