Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds

2011/08/31 by Grassi, Antonella, Morrison, David R. · 3 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th)

paper · doi:10.48550/arxiv.1109.0042

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.

Cited by

• Reduction of topological invariants on null hypersurfaces
• Explicit Bounds on the Spectrum of 6d N=(1,0) Supergravity
• FTheoryTools: Advancing Computational Capabilities for F-Theory Research

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