We derive four dimensional gauge theories with exceptional groups F4β,
E8β, E7β, and E7β with matter, by starting from the duality between the
heterotic string on K3 and F-theory on a elliptically fibered Calabi-Yau
3-fold. This configuration is compactified to four dimensions on a torus, and
by employing toric geometry, we compute the type IIB mirrors of the Calabi-Yaus
of the type IIA string theory. We identify the Seiberg-Witten curves describing
the gauge theories as ALE spaces fibered over a P1 base.Comment: 18 pages, 5 figures, uses harvmac and eps