Communications in Number Theory and Physics

Volume 6 (2012)

Number 1

Anomalies and the Euler characteristic of elliptic Calabi–Yau threefolds

Pages: 51 – 127



Antonella Grassi (Department of Mathematics, University of Pennsylvania)

David R. Morrison (Departments of Mathematics and Physics, University of California at Santa Barbara)


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.

Full Text (PDF format)