K3 fibrations and softly broken N = 4 supersymmetric gauge theories

Global geometry of K3 fibration Calabi-Yau threefolds, with Hodge number h_2,1 = r + 1, is used to define N = 4 softly broken SU( r + 1) gauge theories, with the bare coupling constant given by the dual heterotic dilaton, and the mass of the adjoint hypermultiplet given by the heterotic string tension. The U( r + 1) Donagi-Witten integrable model is also derived from the K3 fibration structure, with the extra U(1) associated to the heterotic dilaton. The case of the SU(2) gauge group is analyzed in detail. String physics beyond the heterotic point particle limit is partially described by the N = 4 softly broken theory.