We study marginal and relevant supersymmetric deformations of the N=4
super-Yang-Mills theory in four dimensions. Our primary innovation is the
interpretation of the moduli spaces of vacua of these theories as
non-commutative spaces. The construction of these spaces relies on the
representation theory of the related quantum algebras, which are obtained from
F-term constraints. These field theories are dual to superstring theories
propagating on deformations of the AdS_5xS^5 geometry. We study D-branes
propagating in these vacua and introduce the appropriate notion of algebraic
geometry for non-commutative spaces. The resulting moduli spaces of D-branes
have several novel features. In particular, they may be interpreted as
symmetric products of non-commutative spaces. We show how mirror symmetry
between these deformed geometries and orbifold theories follows from T-duality.
Many features of the dual closed string theory may be identified within the
non-commutative algebra. In particular, we make progress towards understanding
the K-theory necessary for backgrounds where the Neveu-Schwarz antisymmetric
tensor of the string is turned on, and we shed light on some aspects of
discrete anomalies based on the non-commutative geometry.Comment: 60 pages, 4 figures, JHEP format, amsfonts, amssymb, amsmat
