  We present a renormalization group construction of a weakly interacting Bose
gas at zero temperature in the two-dimensional continuum, both in the quantum
critical regime and in the presence of a condensate fraction. The construction
is performed within a rigorous renormalization group scheme, borrowed from the
methods of constructive field theory, which allows us to derive explicit bounds
on all the orders of renormalized perturbation theory. Our scheme allows us to
construct the theory of the quantum critical point completely, both in the
ultraviolet and in the infrared regimes, thus extending previous heuristic
approaches to this phase. For the condensate phase, we solve completely the
ultraviolet problem and we investigate in detail the infrared region, up to
length scales of the order $(\lambda^3 \rho_0)^{-1/2}$ (here $\lambda$ is the
interaction strength and $\rho_0$ the condensate density), which is the largest
length scale at which the problem is perturbative in nature. We exhibit
violations to the formal Ward Identities, due to the momentum cutoff used to
regularize the theory, which suggest that previous proposals about the
existence of a non-perturbative non-trivial fixed point for the infrared flow
should be reconsidered.
