Phases of supersymmetric gauge theories from M-theory on G₂ manifolds

We consider M-theory on compact spaces of G₂ holonomy constructed as orbifolds of the form (CY×S¹)/ℤ with fixed point set ∑ on the CY. This describes N = 1 SU(2) gauge theories with b₁ (∑) chiral multiplets in the adjoint. For b₁ = 0, it generalizes to compact manifolds the study of the phase transition from the non-abelian to the confining phase through geometrical S³ flops. For b₁ = 1, the non-abelian and Coulomb phases are realized, where the latter arises by desingularization of the fixed point set, while an S² x S¹ flop occurs. In addition, an extremal transition between G₂ spaces can take place at conifold points of the CY moduli space where unoriented membranes wrapped on ℂℙ¹ and ℝℙ² become massless.