field $\alpha$. The additional condition implies that $\alpha$ is constant on $S^3\times S^1$. The mode $\alpha$ plays a special role in our computation since it is the only zero mode of the theory[^3].

As shown in [@Aharony:2003sx] the Fadeev-Popov determinant conjugate to (\[eq:ZeroModeFix\]) can be absorbed in the integration measure of $\alpha$, changing it to integration over $U=e^{i\beta\alpha}$. Furthermore, it was also shown that the effective action arising from integrating out the massive modes is only a function of $U$ and $\beta$,

$$\begin{aligned}
\label{eq:Z}
\cal{Z}(\beta) = \int[dU]e^{-\mathbf{S}_{eff}(U;\beta)} = \int[dU]\left(\int[dA_0][dA_i][d\phi^i]e^{-\mathbf{S}_E[\alpha,A_0,A_i,\phi^i;\beta]}\right),\end{aligned}$$

where by $A_0$ we mean the part of the zeroth component of the gauge field its zero mode $\alpha$, $\mathbf{S}_E$ is the gauge fixed Euclidean action on the sphere, and all the other massive fields are collectively denoted as $\phi^i$.

The new integration variable $U$ is related to the Wilson line over the time cycle, also known as the Polyakov loop $\cal{P}=\frac{1}{N}\mathrm{tr}P\mathrm{exp}(\int_0^{\beta}A_0dt)$ [@Polyakov:1978vu]. The Polyakov loop measures the free energy $F_q(T)$ of **YM** theory in the presence of an external quark through the relation $\langle\cal{P}\rangle=\mathrm{exp}(-F_q(T))$, and is a standard order parameter for deconfinement. More precisely, $\frac{1}{N}\mathrm{tr}(U)$ is the Wilson line of the zero mode of the gauge field around $S^1$.

The 