Discrete Abelian gauge symmetries

We investigate the possibility to gauge discrete Abelian symmetries. An algebraic approach to understanding general Abelian discrete groups, which govern the coupling structure of a physical theory is presented. In particular, the embedding of a general Abelian discrete group Zd1 x.. x Zdr into a general Abelian gauge group U(1)k via spontaneous symmetry breaking of the continuous group is elaborated in detail. A promising candidate for the embedding of any discrete gauge symmetry is string theory. The algebraic approach to general discrete Abelian groups establishes new possibilities of controlling the coupling structure in string derived model building. We discuss phenomenological consequences of discrete Abelian symmetries arising in string derived MSSM models. A simple discrete R-symmetry, ZR4, which contains matter parity as non-anomalous subgroup, is capable of resolving multiple issues such as dimension four and five proton decay as well as the μ-problem.