the perturbative corrections to the ${{\mathfrak{psu}}}(1,1|2)$ generators. They are needed for constraining the one and one-half loop ${{\mathfrak{psu}}}(1|1)^2$ generators, which anti-commute to the two-loop dilatation generator. Moreover, the solution we present below for the ${{\mathfrak{psu}}}(1,1|2)$ generators has an interesting and simple structure. Next we present this solution and discuss its possible modifications and its proof.

The solution \[sec:g2solution\]
-------------------------------

We define two auxiliary generators that play central parts in our solution. ${{\mathfrak{h}}}$ is a one-site generator of harmonic numbers. Its action is $$\begin{aligned}
 {{\mathfrak{h}}}{|\phi^\updownarrow_k\rangle}&={{{\textstyle\frac{1}{2}}}}h(k){|\phi^\updownarrow_k\rangle},&  {{\mathfrak{h}}}{|\overleftrightarrow{\psi_k}\rangle}&= {{{\textstyle\frac{1}{2}}}}h(k+1){|\overleftrightarrow{\psi_k}\rangle}. \end{aligned}$$ ${{\mathfrak{x}}}$ is a two-site to two-site generator that we can write in two equivalent ways, $$\begin{aligned}
 \label{eq:definex} {{\mathfrak{x}}}&=& {\left\{\overleftarrow{{{\mathfrak{T}}}_1}^-,{\left[\overrightarrow{{{\mathfrak{T}}}_1}^+,{{\mathfrak{h}}}\right]}\right\}}-{\left\{\overleftarrow{{{\mathfrak{T}}}_1}^+,{\left[\overrightarrow{{{\mathfrak{T}}}_1}^-,{{\mathfrak{h}}}\right]}\right\}} \nonumber \\
&=& {\left\{\overrightarrow{{{\mathfrak{T}}}}^-_1,{\left[\overleftarrow{{{\mathfrak{T}}}}^+_1,{{\mathfrak{h}}}\right]}\right\}}-{\left\{\overrightarrow{{{\mathfrak{T}}}}^+_1,{\left[\overleftarrow{{{\mathfrak{T}}}}^-_1,{{\mathfrak{h}}}\right]}\right\}}. \end{aligned}$$ The equality in (\[eq:definex\]) follows from (\[eq:psU11\^2Alg\]), since $${\left\{\overleftarrow{{{\mathfrak{T}}}_1}^-,{\left[\overrightarrow{{{\mathfrak{T}}}_1}^+,{{\mathfrak{h}}}\right]}\right\}}+{\left\{\overrightarrow{{{\mathfrak{T}}}_1}^+,{\left[\overleftarrow{{{\mathfrak{T}}}_1}^-,{{\mathfrak{h}}}\right]}\right\}} = {{{\textstyle\frac{1}{2}}}}{\left[{\delta{\mathfrak{D}}}_2,{{\mathfrak{h}}}\right]},$$ and the analogous equation for $\overrightarrow{{{\mathfrak{T}}}_1}^-$ and its conjugate is satisfied. Because of this equality in (\[eq:definex\]), ${{\mathfrak{x}}}$ is hermitian.

Let ${{\mathfrak{X}}}^\pm$ represent ${{\mathfrak{J}}}^{++}$, ${{\mathfrak{J}}}^{--}$, or the eight ${{\mathfrak{Q}}}$’s, where we retain only the ${{\mathfrak{su}}}(1,1)$ charge. Then the solution of the symmetry and Feynman diagram constraints is $$\label{eq:g2solution} {{\mathfrak{X}}}^\pm_2=\pm {\left[{{\mathfrak{X}}}^\pm_0,{{\mathfrak{x}}}\right]} +{\left[{{\mathfrak{X}}}^\pm_0,{{\mathfrak{y}}}\right]}.$$ We present an outline of the proof that this satisfies the algebra relations in section \[sec:g2discussion\] and more details in Appendix \[sec:g2proof\].

${{\mathfrak{y}}}$ is a two-site to 