the 2-loop diagrams which contribute to the partition function. Finally, we present our conclusions in section \[sec:Conclusions\]. Additional information regarding the computation is collected in the appendices.

Theoretical background {#ssec:Overview}
----------------------

In this subsection we summarize briefly some results related to gauge theories on a small $S^3$ that will be used in this work. A more thorough account of these matters can be found in [@Sundborg:1999ue],[@Aharony:2003sx]. As stated in the introduction we are interested in calculating pertubatively the partition function of large $N$ $(S)U(N)$ **YM** theory on $S^3$ with additional matter fields in the adjoint representation. This can be done by solving the Euclidean path integral on $S^3\times S^1$ with the inverse temperature $\beta$ as the period of $S^1$, in the weak coupling regime[^2].

In order to use perturbation theory we must fix the gauge. It is convenient to use the Coulomb gauge, $$\begin{aligned}
\label{eq:CoulombGauge}
\partial^iA_i=0,\end{aligned}$$ where $i$ runs over the directions of $S^3$ and the derivative is assumed to be covariant. This gauge condition fixes spatial gauge transformations but leaves temporal transformations unfixed. In order to fix those we impose the following additional condition, $$\begin{aligned}
\partial_t\alpha \equiv \partial_t\left(\frac{1}{V_{S^3}}\int_{S^3}A_0\right)=0, \label{eq:ZeroModeFix}\end{aligned}$$ where we have defined the zero mode of the gauge 