sufficient in order for the theory to remain conformally invariant at order $\lambda$. However, for the scalar fields this is not so since their mass terms are gauge invariant and contribute to the partition function at order $\lambda$. As noted in the introduction we want our theories to remain conformally invariant to leading order in $\lambda$. Therefore, we need some renormalization condition that keeps the scalars “massless” on $S^3$. In practice, we set the counterterms $Z^{ab}_{0s}$ by requiring that the state-operator mapping works (which is why we wanted our theories to be conformally invariant in the first place). It turns out that the knowledge of the sum of the anomalous dimensions of dimension 2 operators in flat space, is sufficient to determine $Z^{ab}_{0s}$ completely.

Regularization and counterterms {#sec:RegCount}
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In this section we present the computation of the various counterterms in (\[eq:CountertermsAction\]). The counterterms $Z_1$ and $Z_2$ of all fields can be computed in flat space, since from dimensional analysis they cannot depend on $R_{S^3}$. The $Z_0$ counterterms on the other hand, can depend on the global properties of $S^3$. For instance the scalar terms may include a dependence on the Ricci scalar, $Z_0^{ab}\phi_a\phi_b\supset Z'_0\cal{R}\phi_a\phi_b$, and vector terms may also 