Group representations and the Euler characteristic of elliptically fibered Calabi–Yau threefolds

To every elliptic Calabi–Yau threefold with a section X X there can be associated a Lie group G G and a representation ρ ρ 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 X and ρ ρ , including an explicit formula (in terms of ρ ρ ) for the Euler characteristic of X 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.