Communications in Number Theory and Physics
Volume 6 (2012)
Anomalies and the Euler characteristic of elliptic Calabi–Yau threefolds
Pages: 51 – 127
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” that 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.