Many different non-perturbative large N equivalences, relating theories with different gauge groups and matter content, are now known to hold provided appropriate symmetry realization conditions are satisfied. These include many examples associated with orbifold (or orientifold) projections, some of which relate supersymmetric theories to non-supersymmetric QCD-like theories. A variety of novel predictions for real QCD have been obtained in this fashion. Large N volume independence is another type of large N equivalence generated by an orbifold projection. Volume independence exactly relates large N properties of a wide class of QCD-like theories on R4 to the same theory on toriodal compactifications of R4 , provided appropriate center symmetries are not spontaneously broken.
"Eguchi-Kawai" reduction, relating a large N lattice gauge theory to a single-site matrix model, is a special case of such volume independence. The original version of Eguchi-Kawai reduction (as well as "twisted" and "quenched" variants) fails for sufficiently weak coupling due to unwanted symmetry breaking. But examples of QCD-like theories which do satisfy complete volume independence are now known. Large N orientifold equivalence and volume independence may be combined to produce a fully reduced version of a large N generalization of QCD with light fermions. Lattice studies of the resulting reduced models should be possible and will allow one to extract infinite volume physics from simulations with only a single site.
Instead of relying on dynamical fermions to provide stabilization of center symmetry, it has also been recently understood how to add double trace deformations to the Yang-Mills action in a manner which preserves large N equivalence with the original undeformed theory (on R4 ) while at the same time preventing center symmetry breaking in small volume, thereby ensuring complete validity of large N volume independence. When studied on sufficiently small S1 × R3 , these deformed theories also provide an analytically tractable example of confinement which does not involve elementary scalars or supersymmetry. There are strong reasons to believe that the properties of the deformed Yang-Mills theory (or a one flavor deformed QCD) on S1 × R3 are completely smooth as one decompactifies to R4 . These theories possess a rich assortment of topological excitations and it will be quite interesting to compare the small S1 semi-classical results with lattice simulations.