G2 holonomy, mirror symmetry and phases of N = 1 SYM

We study the phase structure of four-dimensional N = 1 super Yang-Mills theories realized on D6-branes wrapping the RP3 of a Z2 orbifold of the deformed conifold. The non-trivial fundamental group of RP3 allows for the gauge group to be broken to various product groups by Z2 Wilson lines. We study the classical moduli space of theories in various pictures related by dualities including an M-theory lift. The quantum moduli space is analyzed in a dual IIB theory, where a complex curve contained in the target space plays a key role. We find that the quantum moduli space is made up of several branches, characterized by the presence or absence of a low energy U(1) gauge symmetry, which are connected at points of monopole condensation. The resulting picture of the quantum moduli space shows how the various gauge theories with different product gauge groups are connected to one another