INSTANTON EXPANSIONS FOR MASS DEFORMED N=4 SUPER YANG-MILLS THEORIES

We derive modular anomaly equations from the Seiberg-Witten-Donagi cur ves for softly broken N = 4 SU(n) gauge theories. From these equations we can derive recursion relations for the pre-potential in powers of m(2), where m is the mass of the adjoint hypermultiplet. Given the per turbative contribution of the pre-potential and the presence of ''gaps '', we can easily generate the m(2) expansion in terms of polynomials of Eisenstein series, at least for relatively low rank groups. This en ables us to determine efficiently the instanton expansion up to fairly high order for these gauge groups, e.g. eighth order for SU(3). We fi nd that after taking a derivative, the instanton expansion of the pre- potential has integer coefficients. We also postulate the form of the modular anomaly equations, the recursion relations and the form of the instanton expansions for the SO(2n) and E-n gauge groups, even though the corresponding Seiberg-Witten-Donagi curves are unknown at this ti me. (C) 1998 Elsevier Science B.V.