Foundations of supermathematics with applications to N=1 supersymmetric field theory

We use the term "supermathematics" to encompass all the various extensions of Alice Roger's original work on G infinity supermanifolds. Background on how numbers, functions, linear algebra, matrix calculations, real analysis, complex analysis, manifold theory and Lie theory generalize to the context of supermathematics is provided. We use countably many Grassmann generators so this work is within the realm of infinite dimensional Banach space theory.; We find that Lie's Third Theorem holds for G infinity super Lie groups. We also prove that the exponential mapping and other standard constructions in Lie theory apply equally well in the Ginfinity setting. Portions of this work are similar to existing research, but our proofs are distinct and we have focused on the Ginfinity category with infinitely many Grassmann generators. Other workers typically either use finitely many Grassmann generators or focus attention to the superanalytic category.; We provide a supersmooth principle fiber bundle framework for super gauge theory. Special sections are constructed and provide pure gauge solutions on zero curvature submanifolds. Quotient spaces and bundles are used to implement certain physical constraints. We apply these general geometric constructions to recover the superfield transformation laws of N = 1 super Yang-Mills theory.; We develop a gauged Wess-Zumino model in noncommutative Minkowski superspace. This is a natural extension of the work of Carlson and Nazaryan, who extended N = 1/2 supersymmetry over deformed Euclidean superspace to Minkowski superspace. Noncommutativity is implemented by replacing products with star products. As in the N = 1/2 theory, a reparameterization of the gauge parameter, vector superfield and chiral superfield are necessary to write standard C-independent gauge theory. However, our choice of parametrization differs from that used in the N = 1/2 supersymmetry, which leads to some unexpected new terms.