  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. The group
is determined from the Weierstrass model, which has singularities that are
generically rational double points; these double points lead to local factors
of $G$ which are either the corresponding A-D-E groups or some associated
non-simply laced groups.
  The representation $\rho$ is a sum of representations coming from the local
factors of $G$, and of other representations which can be associated to the
points at which the singularities are worse than generic.
  This construction first arose in physics, and the requirement of anomaly
cancellation in the associated physical theory makes some surprising
predictions about the connection between $X$ and $\rho$. In particular, an
explicit formula (in terms of $\rho$) for the Euler characteristic of $X$ is
predicted. 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 about the types of singularities
which occur.
  As a byproduct we also 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.
