Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds

Details Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds

2011-08-31

arxiv.org/abs/1109.0042v2

Physics

High Energy Physics

High Energy Physics - Theory

63 pages, 15 tables. v2: minor changes

Scientific paper

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.

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