are then as follows: $$\begin{array}{lll}
\textbf{Reduced AdS}\mathbf{_2 \times S^2:}  \qquad a=1\,, \ \alpha=2 &
\\\textbf{Reduced AdS}\mathbf{_3 \times S^3:}  \qquad a=1,2\,, \ \alpha=3,4 \quad & \text{and} \quad \Phi_{A\dot A\vphantom{\dot B}} = \Omega_{AB\vphantom{\dot B}}\Omega_{\dot A \dot B} \Phi_{B\dot B}
\\\textbf{Reduced AdS}\mathbf{_5 \times S^5:}  \qquad a=1,2\,, \ \alpha=3,4 & \end{array}$$ where $\Omega_{ab} = \epsilon_{ab} \,, \ \Omega_{\alpha\beta} =\epsilon_{\alpha\beta}$ and $\Omega_{a\beta} = \Omega_{\alpha b} = 0$. In the reduced AdS$_5 \times S^5$ model the indices $a,\alpha,\dot a,\dot \alpha$ are $\text{SU}(2)$ fundamental indices, while in the reduced AdS$_3 \times S^3$ model they are $\text{SO}(2)$ vector indices.

In each case the global symmetry [@Hoare:2011fj] is such that the S-matrix should factorize under its structure [^7] $$\label{s2smatfact}
S_{A\dot A,B\dot B}^{C \dot C, D \dot D}(\theta) = (-1)^{[\dot A][B]+[\dot C][D]} S_{AB\vphantom{\dot B}}^{CD \vphantom{\dot D}}(\theta) S_{\dot A \dot B}^{\dot C \dot D}(\theta)$$ and indeed the tree-level results have this structure. Let us now present the tree-level S-matrices (we write only the undotted factor – the dotted factor is given by the same expression) for each of the three reduced theories [^8] where $$\begin{aligned}
\no
&  M^{(0)}_1 (\theta) = - M^{(0)}_2 (\theta) = \frac{i\pi}k {\operatorname{csch}}\theta
&& M^{(0)}_3 (\theta) = M^{(0)}_4 (\theta) = -\frac{i\pi}{2k} {\operatorname{sech}}\frac\theta 2
\\
&  M^{(0)}_5 (\theta) = 