  We propose a new approach to solve conformal field theories and apply it to
Chern-Simons Matter theories and three-dimensional bosonization duality. All
three-point correlation functions of single-trace operators are obtained in the
large-$N$ as a simple application. The idea is to construct, as an effective
weakly-coupled theory, a nonlinear realization of the conformal algebra in
terms of physical, gauge-invariant, operators. The efficiency of the method is
also in the use of an analog of the light-cone gauge and of the momentum-space
on the CFT side. AdS/CFT is used as a convenient regulator and as a source of
the canonical bracket. The uniqueness of the nonlinear realization manifests
the three-dimensional bosonization duality at this order. We also find two more
non-unitary solutions which should be analogous to the fishnet theories. The
results can also be viewed as an explicit realization of the slightly-broken
higher spin symmetry.
  As a by-product, the cubic action of the Higher Spin Gravity in $AdS_4$ is
constructed. While generic Higher Spin Gravities are obstructed at higher
orders by nonlocality, we point out the existence of two especially simple and
well-defined theories: chiral and anti-chiral whose three-point functions
correspond to the two new solutions. These two theories are supposed to give a
quantum complete and local example of gravitational bulk duals.
