scalar fields to be conformally coupled by adding $\frac{1}{2}\phi^2_a$ to their kinetic terms[^7].

Without loss of generallity we take the quartic scalar coupling $\mathbf{Q}^{abcd}$ to be cyclic in its indices, and the Yukawa coupling $\rho^a_{IJ}$ to be antisymmetric in $I,J$. This is the most general action that is classically conformally invariant, up to the assumption of having only commutator interactions. The reader is referred to appendix \[app:Notations\] for further conventions regarding spinors and the restriction of the above action to the $\cal N = 4$ **SYM** case.

Next we impose the gauge fixing conditions (\[eq:CoulombGauge\]) and (\[eq:ZeroModeFix\]). The Fadeev-Popov determinant conjugate to (\[eq:CoulombGauge\]) leads to the addition of the ghost action,

$$\begin{aligned}
\label{eq:GhostAction}
\mathbf{S}_{FP} = -\int_0^{\beta}dt\int d^3x \mathrm{tr}(c^{\dag}\partial_iD^ic).\end{aligned}$$

The resulting gauge fixed action can be found in appendix \[sapp:action\]. Next we expand all the fields in Kaluza-Klein modes on the sphere[^8],

$$\begin{gathered}
\label{eq:HarmonicsExpansions}
A_0(t,\Omega)  = \sum_{\alpha}a^{\alpha}(t)\cal{S}^{\alpha}(\Omega), ~~ c(t,\Omega)  = \sum_{\alpha}c^{\alpha}(t)\cal{S}^{\alpha}(\Omega), ~~\vec{A}(t,\Omega)  = \sum_{\alpha}A^\alpha(t)\vec{\cal{V}}^{\alpha}(\Omega),\nonumber\\
\Psi_I(t,\Omega)  = \sum_{\alpha}\psi_I^{\alpha}(t)\cal{Y}^{\alpha}(\Omega),~~\Phi_a(t,\Omega)  = \sum_{\alpha}\phi_a^{\alpha}(t)\cal{S}^\alpha(\Omega),\end{gathered}$$

where $\cal{S}^{\alpha}(\Omega)$, $\vec{\cal{V}}^{\alpha}(\Omega)$ and $\cal{Y}^{\alpha}(\Omega)$ are scalar, vector and Weyl spinor spherical harmonics. The Greek letter indices collect within them all the relevant angular momenta quantum numbers associated with those functions and summation over all of them is implied. To write the 