Hidden Symmetry and S-duality in N=4 D=4 Super Yang-Mills Theory

In this talk the Schwarz hypothesis that the duality symmetries should be pieces of the hidden gauge symmetry in a string theory is discussed. Using auxiliary linear system special dual transformations for $N=4$ SYM generalizing the Schwarz dual transformations for a principal chiral model are constructed. These transformations are related with a continuous group of hidden symmetry of a new model involving more fields as compare with $N=4$ SYM. We conjecture that the $Z_{2}$ discrete subgroup of this hidden symmetry group has a stable set of N=4 YM fields and transforms a self-dual configuration to an anti-self-dual and via versa.