Group representations and the Euler characteristic of elliptically fibered Calabi–Yau threefolds
Authors:
Antonella Grassi and David R. Morrison
Journal:
J. Algebraic Geom. 12 (2003), 321356
DOI:
https://doi.org/10.1090/S1056391102003375
Published electronically:
December 17, 2002
MathSciNet review:
1949647
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Abstract  References  Additional Information
Abstract: To every elliptic Calabi–Yau threefold with a section $X$ there can be associated a Lie group $G$ and a representation $\rho$ of that group, determined from the Weierstrass model and the types of singular fibers. We explain this construction, which first arose in physics. The requirement of anomaly cancellation in the associated physical theory makes some surprising predictions about the connection between $X$ and $\rho$, including an explicit formula (in terms of $\rho$) for the Euler characteristic of $X$. We give a purely mathematical proof of that formula in this paper, introducing along the way a new invariant of elliptic Calabi–Yau threefolds. We also verify the other geometric predictions which are consequences of anomaly cancellation, under some mild hypotheses. As a byproduct we discover a novel relation between the Coxeter number and the rank in the case of the simply laced groups in the “exceptional series” studied by Deligne.

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Additional Information
Antonella Grassi
Affiliation:
Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
Email:
grassi@math.upenn.edu
David R. Morrison
Affiliation:
School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540;
Department of Mathematics, Duke University, Durham, North Carolina 277080320
MR Author ID:
189764
Email:
drm@math.duke.edu
Received by editor(s):
November 1, 2000
Published electronically:
December 17, 2002
Additional Notes:
Research partially supported by the Harmon Duncombe Foundation, by the Institute for Advanced Study, and by National Science Foundation grants DMS9401447, DMS9401495, DMS9627351 and DMS9706707. We thank the Institute for Advanced Study, the Mathematisches Forschunginstitut Oberwolfach, and the Institute for Theoretical Physics, Santa Barbara, for hospitality during various stages of this project.