form of the effective action (\[eq:Z\]) is highly restricted by gauge invariance. To 2 loops order, writing $S_{eff}(U,\beta)=S_{eff}^{1-loop}+S_{eff}^{2-loop}$, it turns out that

$$\begin{aligned}
S_{eff}^{1-loop} &= -\sum_{n=1}^{\infty}\frac{1}{n}z_n(x)\mathrm{tr}(U^n)\mathrm{tr}(U^{\dag n}),\label{eq:1LoopEffAction}\\
S_{eff}^{2-loop} &= \beta g^2\left[N\sum_{n=1}^\infty f_n(x)\left(\mathrm{tr}(U^n)\mathrm{tr}(U^{\dag n})-1\right) +\right. \nonumber\\
&\left.+\sum_{n, m =1}^{\infty}f_{n m}(x)\left(\mathrm{tr}(U^n)\mathrm{tr}(U^m)\mathrm{tr}(U^{-n-m}) + c.c.-2N\right)\right]\label{eq:2LoopEffAction},\end{aligned}$$

where $x\equiv e^{-\beta/R_{S^3}}$ and the function $z_n(x)$ is defined in terms of the “1-particle partition sums” for $N_s$ scalars, 1 vector and $N_f$ chiral fermions on $S^3$,

$$\begin{aligned}
\begin{array}{c}\label{eq:zBzF}
z_n(x)=z_B(x^n)+(-1)^{n+1}z_F(x^n),~~z_B(x)=N_sz_{s}(x)+z_V(x),~~  z_F(x)=N_fz_f(x)\\
z_s(x)=\frac{x+x^2}{(1-x)^3},~~  z_V(x)=\frac{6x^2-2x^3}{(1-x)^3},~~  z_F(x)=\frac{4x^{3/2}}{(1-x)^3}.
\end{array}\end{aligned}$$

The two and three trace terms in (\[eq:2LoopEffAction\]) arise from planar diagrams with 3 index loops, while the $U$ independent terms come from non-planar diagrams with a single index loop[^4]. In the low temperature phase, the three trace terms in the matrix integral (\[eq:Z\]) contribute only through perturbations around the gaussian model, and can be neglected to the order in the coupling that we are interested in.

The matrix model (\[eq:1LoopEffAction\])-(\[eq:2LoopEffAction\]) can be solved in the large $N$ limit by using the saddle point approximation. Below the critical temperature the minimizing eigenvalue distribution of $U$ is uniform. The matrix integral (\[eq:Z\]) can be evaluated at this saddle point to give[^5],

$$\begin{aligned}
\label{eq:UnZ}
\cal{Z}_{U(N)}=e^{-\lambda\beta \tilde{F}_2^{np}}\prod_{n=1}^{\infty}\frac{e^{\lambda\beta f_n(x)}}{1-z_n(x)+\lambda n\beta f_n(x)},\end{aligned}$$

where $\widetilde{F}_2^{np}$ is the non-planar piece given 