\times S^2 \times T^6$ – see, for example, [@Babichenko:2009dk] and [@Sorokin:2011rr] and references therein.

These reduced theories are all classically integrable, demonstrated by the existence of a Lax connection, and conjectured to be UV-finite [@Roiban:2009vh]. Indeed, in [@Roiban:2009vh] finiteness at one loop and at two loops in the dimensional reduction scheme was demonstrated. Furthermore, the reduced AdS$_2 \times S^2$ theory is in fact given by the $\mathcal{N} = 2$ supersymmetric sine-Gordon model and hence is supersymmetric. The reduced AdS$_3 \times S^3$ and AdS$_5 \times S^5$ theories have a non-local $\mathcal{N}=4$ and $\mathcal{N}=8$ supersymmetry respectively [@Schmidtt:2010bi; @Hollowood:2011fq; @Goykhman:2011mq; @Schmidtt:2011nr], which manifests as a $q$-deformation of the S-matrix symmetry algebra.

The tree-level and one-loop S-matrices for these theories were computed in [@Hoare:2009fs; @Hoare:2011fj], while the exact S-matrices have been conjectured using integrability techniques in [@Kobayashi:1991rh] for the reduced AdS$_2 \times S^2$ model, [@Hoare:2011fj] for the reduced AdS$_3 \times S^3$ model and [@Hoare:2013ysa] for the reduced AdS$_5 \times S^5$ model.

In each of the reduced superstring theories the asymptotic excitations (both bosonic and fermionic) can be packaged into a single field $$\label{superfield}
\Phi_{A\dot A} \,, \qquad A=(a|\alpha) \,, \qquad [a]=0 \,, \ [\alpha]=1 \ .$$ The particular configurations relevant for the individual theories 