Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds

By A. Grassi, D. Morrison. 2011

163 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/9b070757f3f2a34c631d8d57286488b8dd8d9621

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Citations in the corpus

On the classification of 6D SCFTs and generalized ADE orbifolds (2013, unknown): “For further discussion see for example [51].”
Matter and singularities (2011, background): “In six dimensions, the space of allowed supergravity theories is strongly constrained by anomalies and other simple features of the low-energy theory [13, 14, 16, 17, 18, 19, 20, 21].”
F-theory on all toric hypersurface fibrations and its Higgs branches (2014, background): “We have shown that these effective theories are anomaly-free in 6D, which proves in turn the completeness of our analysis of codimension one and two singularities of all these models [93].”
Classifying bases for 6D F-theory models (2012, methodology): “the precise gauge algebra; monodromies can give rise to non-simply-laced algebras in some situations [32, 9].”
F-theory compactifications with multiple U(1)-factors: constructing elliptic fibrations with rational sections (2013, background): “Mathematically this translates into the Euler number of the resolution X̂ [62].”
Atomic classification of 6D SCFTs (2015, methodology): “In the language of [34], the points of div(βΣ) are all contained in the cycle div(γΣ), and the corresponding line of Table 9 could have been written (17)These are half-fundamentals because a branch point is a zero in the cycle βΣ, whereas the representations listed in Table 9 are associated to the c”
Anomaly equations and intersection theory (2011, background): “There is an elegant picture of how the geometry of the internal Calabi-Yau threefold is encoded in the non-abelian sector of the low-energy theory [11, 12, 30, 34].”