  We show how certain F^4 couplings in eight dimensions can be computed using
the mirror map and K3 data. They perfectly match with the corresponding
heterotic one-loop couplings, and therefore this amounts to a successful test
of the conjectured duality between the heterotic string on T^2 and F-theory on
K3. The underlying quantum geometry appears to be a 5-fold, consisting of a
hyperk"ahler 4-fold fibered over a IP^1 base. The natural candidate for this
fiber is the symmetric product Sym^2(K3). We are lead to this structure by
analyzing the implications of higher powers of E_2 in the relevant Borcherds
counting functions, and in particular the appropriate generalizations of the
Picard-Fuchs equations for the K3.
