Noncommutativeo⋆(N)andusp⋆(2N)algebras and the corresponding gauge field theories

Abstract

The extension of the noncommutative ${\mathrm{u}}_{\ensuremath{\star}}(N)$ Lie algebra to noncommutative orthogonal and symplectic Lie algebras is studied. Using an antiautomorphism of the star-matrix algebra, we show that the ${\mathrm{u}}_{\ensuremath{\star}}(N)$ can consistently be restricted to ${\mathrm{o}}_{\ensuremath{\star}}(N)$ and ${\mathrm{usp}}_{\ensuremath{\star}}(N)$ algebras that have new mathematical structures. We give explicit fundamental matrix representations of these algebras, through which the formulation for the corresponding noncommutative gauge field theories are obtained. In addition, we present a D-brane configuration with an orientifold that realizes geometrically our algebraic construction, thus embedding the new noncommutative gauge theories in a superstring theory in the presence of a constant background magnetic field. Some algebraic generalizations that may have applications in other areas of physics are also discussed.