Quartic Gauge Couplings fromK3 Geometry

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 hyperkähler 4-fold fibered over a IP base. The natural candidate for this fiber is the symmetric product Sym(K3). We are lead to this structure by analyzing the implications of higher powers of E2 in the relevant Borcherds counting functions, and in particular the appropriate generalizations of the Picard-Fuchs equations for the K3. CERN-TH/98-378 November 1998 ∗ On leave from Physics Department, U.S.C., University Park, Los Angeles, CA 90089-0484