equations of motion, namely a zero of the kinetic operator at non-zero momenta. As in the case of the Brazovskii model, it generates a $1/M$ singularity in the two-point function. Thus, we expect that the homogeneous phase will remain meta-stable for $T < T_c$. If quantum corrections are parametrically suppressed (e.g., by $1/N$), the spatially modulated phase will eventually acquire lower energy and the first order phase transition will take place at that point.

Linear response {#sec:GreensFunction}
===============

Finally, let us examine linear response of our solution when we couple a gauge field to the current $J$ at the boundary. Because of the Chern-Simons term, the current is anomalous. We therefore treat the boundary gauge field as an external and non-dynamical source as in recent papers, for example [@Son; @D'Hoker].

Note that the background solution we consider is inhomogeneous, carrying the momentum $k$. In the homogeneous setup, it is natural to choose a translationally invariant source. In our case, we consider a small perturbation of the form $\tilde A=\tilde A^{(0)}+a$, where $\tilde A^{(0)}$ is our non-linear solution and $a$ is a small perturbation with non-zero components, $$\label{E:aAnsatz}
a_3(x_2,r,t)+i a_4(x_2,r,t)= - i a_T(r,t) e^{-i k x_2},~~~ a_2(r,t)=a_L(r,t)\;.$$ Here, $a_T(r,t)$ and $a_L(r,t)$ 