states with energy $\frac{n}{2R_{S^3}}$ with degeneracy $d\left(\frac{n}{2}\right)$, and $\delta E_{n,i}$ is the order $\lambda$ correction to the state $i$ with energy $\frac{n}{2R_{S^3}}$.

By using the state-operator mapping of conformal field theories, the perturbative energy corrections of the theory on $S^3\times\mathds{R}$ are equivalent to the set of 1-loop anomalous dimensions of the theory on $\mathds{R}^4$. All the theories that we are considering are classically conformally invariant and remain so to leading order in $\lambda$. Therefore we can use the partition function on $S^3$ to compute the sums of anomalous dimensions in flat space.

Setup of the calculation {#sec:Setup}
========================

In this section we present some notational conventions and the basic ingredients used throughout the computation.

The action on $S^3$. {#ssec:3SphereAction}
--------------------

We write the Euclidean action of **YM** theory on $S^3\times S^1$ with the inverse temperature $\beta$ as the period of $S^1$. We include scalars $\Phi_a$ ($a,b,\ldots = 1,\ldots,N_s$ ) and $SO(4)$ Weyl spinors $\Psi^I$ ($I,J,\ldots=1,\ldots,N_f$), all in the adjoint representation of the gauge group $U(N)$,

$$\begin{aligned}
\label{eq:FlatAction}
\mathbf{S}_E &= \int_0^{\beta}dt\int_{S^3}d\Omega \textrm{tr}\Big\{\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}\Phi^a\left(-D^2+1\right)\Phi^a+i\Psi^{\dag I}\sigma^{\mu}D_{\mu}\Psi_I-\nonumber\\
&-\frac{1}{4}g^2\mathbf{Q}^{abcd}\Phi_a\Phi_b\Phi_c\Phi_d+ \frac{1}{2}g(\rho^{a\dag})^{IJ}\Psi_{I}\varepsilon[\Phi^a,\Psi_J]+\frac{1}{2}g\rho^a_{IJ}\Psi^{\dag I}\varepsilon[\Phi^a,\Psi^{\dag J}]\Big\},\\
&F_{\mu\nu}\equiv \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-ig[A_{\mu},A_{\nu}]~,~  D_{\mu}\equiv \partial_{\mu}-ig[A_{\mu},*].\end{aligned}$$

The appropriate veilbein and connection terms for the vector and fermion covariant derivatives should be understood. We also take the 