computation contains divergences. We use the regularization scheme of [@Aharony:2005bq], which involves cutoff regulators which break gauge invariance. This scheme involves multiplying each vector, scalar and fermion line which carries momentum $\vec{p}$, with smooth regulating functions $R_{g,s,f}\left(\frac{|\vec{p}|}{M}\right)$[^12]. In the argument of the regulators, $M$ denotes the cutoff regularization scale. We demand that all the regulators have Heaviside-like properties: $R(0)=1$, $R'(0)=0$, $R(k\rightarrow\infty)=0$, but other than that they are arbitrary.

These regulators preserve rotational symmetry on $S^3$, but break Lorentz and gauge invariance. It was shown by ’t Hooft [@'tHooft:1971fh], that gauge invariance can be restored by adding the appropriate set of renormalizable local counterterms which may break Lorentz and gauge symmetries, while preserving rotational symmetry. To first order in $\lambda = g^2N$, adding the following counterterms is necessary[^13],

$$\begin{aligned}
\label{eq:CountertermsAction}
S_{ct} &= \lambda\int_0^{\beta}dt\int_{S^3}\mathrm{tr}^{SU(N)}\bigg\{ A_i(Z_{0g}-Z_{1g}\partial^2-Z_{2g}D_t^2)A_i+ \nonumber\\
&~~~ +\Phi^a(Z^{ab}_{0s}-Z^{ab}_{1s}\partial^2-Z^{ab}_{2s}D_t^2)\Phi^b+\nonumber\\
&~~~ +i\Psi^{\dag I}(Z^{IJ}_{1f}\sigma^i\partial_i+Z^{IJ}_{2f}D_t)\Psi_J\bigg\}=\\
&=\lambda\int_0^{\beta}dt\mathrm{ tr}^{SU(N)}\Big\{A^{\bar{\alpha}}(Z_{0g}+Z_{1g}{(j_{\alpha}+1)}-Z_{2g}D_t^2)A^{\alpha}+\nonumber\\
&~~~+\phi^{\bar{\alpha}}_a(Z^{ab}_{0s}+Z^{ab}_{1s}{j_{\alpha}(j_{\alpha}+2)}-Z^{ab}_{2s}D_t^2)\phi^{\alpha}_b+\nonumber\\
&~~~+i\psi^{\dag\alpha I}(Z^{IJ}_{1f}{\epsilon_{\alpha}(j_{\alpha}+\frac{1}{2})}+Z^{IJ}_{2f}D_t)\psi^{\alpha}_J\Big\}.\nonumber\end{aligned}$$

The notation in (\[eq:CountertermsAction\]) suggests that we take the trace only over the $SU(N)$ part of the fields, (e.g. $A_i^{SU(N)}=A_i-\frac{1}{N}\mathrm{tr}(A_i)$). Since we assume only commutator interactions, the $U(1)$ part of the theory is free and only the $SU(N)$ part of the fields contributes to the counterterms.

For the gauge boson and fermions, imposing gauge invariance is 