  In this article we consider the initial value problem for the m-equivariant
Chern-Simons-Schr\"odinger model in two spatial dimensions with real-valued
coupling parameter g. This is a covariant NLS type problem that is
L^2-critical. We prove that at the critical regularity, for any integer-valued
equivariance index m, the initial value problem in the defocusing case (g < 1)
is globally wellposed and the solution scatters. The problem is focusing when g
>= 1, and in this case we prove that for nonnegative integer-valued
equivariance indices m there exist constants c = c_{m, g} such that, at the
critical regularity, the initial value problem is globally wellposed and the
solution scatters when the L^2 initial data phi_0 is m-equivariant and has
L^2-norm less than the square root of c_{m, g}. We also show that c_{m,
g}^{1/2} is equal to the minimum L^2 norm of a nontrivial m-equivariant
standing wave solution. In the self-dual g = 1 case, we have the exact
numerical values c_{m, 1} = 8*pi*(m + 1).
