for a certain class of generalized sine-Gordon models we have found that the expression for the one-loop S-matrix gotten from unitarity techniques agrees with that found from standard perturbation theory (including the one-loop correction coming from the procedure of integrating out the unphysical fields ) up to a scheme-dependent shift in the coupling.

Integrable theories with fermions
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The models discussed in the previous section have an interesting origin in string theory. They appear as the Pohlmeyer reduction of strings moving on an $n+1$-sphere [@Pohlmeyer:1975nb; @Eichenherr:1979yw]. In the string interpretation the reduction uses a non-local change of variables to solve the Virasoro constraints giving a classically equivalent theory [@Tseytlin:2003ii; @Barbashov:1980kz; @DeVega:1992xc]. The reduction can be extended to the Green-Schwarz action for the Type IIB superstring on AdS$_5 \times S^5$ [@Metsaev:1998it] giving a gauged WZW model for $\text{USp}(2,2)/\text{SU}(2)^2 \times \text{USp}(4)/\text{SU}(2)^2$ plus a potential and coupled to fermions [@Grigoriev:2007bu; @Mikhailov:2007xr]. There are two truncations of this model that we will also study corresponding to the reduction of the superstring on AdS$_3 \times S^3$ [@Grigoriev:2008jq] and AdS$_2 \times S^2$ [@Grigoriev:2007bu], by which we mean the formal supercoset truncations of the full 10-d superstring theories on AdS$_3 \times S^3 \times T^4$ and on AdS$_2 