Superconformal Chern-Simons theories and AdS4/CFT3 correspondence

We discuss the N = 2 superspace formulation of the N = 8 superconformal Bagger-Lambert-Gustavsson theory, and of the N = 6 superconformal Aharony-Bergman-Jafferis-Maldacena U(N) x U(N) Chern-Simons theory. In particular, we prove the full SU(4)R-symmetry of the ABJM theory. We then consider orbifold projections of this theory that give non-chiral and chiral (U( N) x U(N))(n) superconformal quiver gauge theories. We argue that these theories are dual to certain AdS(4) x S-7/(Z(n) x Z (k$) over tilde) backgrounds of M-theory. We also study a SU(3) invariant mass term in the superpotential that makes the N = 8 theory flow to a N = 2 superconformal gauge theory with a sextic superpotential. We conjecture that this gauge theory is dual to the U(1)(R) x SU(3) invariant extremum of the N = 8 gauged supergravity, which was discovered by N. Warner 25 years ago and whose uplifting to 11 dimensions was found more recently.