  The partition function of N=6 supersymmetric Chern-Simons-matter theory
(known as ABJM theory) on S^3, as well as certain Wilson loop observables, are
captured by a zero dimensional super-matrix model. This super-matrix model is
closely related to a matrix model describing topological Chern-Simons theory on
a lens space. We explore further these recent observations and extract more
exact results in ABJM theory from the matrix model. In particular we calculate
the planar free energy, which matches at strong coupling the classical IIA
supergravity action on AdS_4 x CP^3 and gives the correct N^{3/2} scaling for
the number of degrees of freedom of the M2 brane theory. Furthermore we find
contributions coming from world-sheet instanton corrections in CP^3. We also
calculate non-planar corrections, both to the free energy and to the Wilson
loop expectation values. This matrix model appears also in the study of
topological strings on a toric Calabi-Yau manifold, and an intriguing
connection arises between the space of couplings of the planar ABJM theory and
the moduli space of this Calabi-Yau. In particular it suggests that, in
addition to the usual perturbative and strong coupling (AdS) expansions, a
third natural expansion locus is the line where one of the two 't Hooft
couplings vanishes and the other is finite. This is the conifold locus of the
Calabi-Yau, and leads to an expansion around topological Chern-Simons theory.
We present some explicit results for the partition function and Wilson loop
observables around this locus.
