$\alpha$ must be zero.

Non-planarity and wrapping interactions \[sec:non-planar\]
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By lifting our expressions for the building blocks of ${\delta{\mathfrak{D}}}_4$ to their non-planar generalization, we will construct a candidate for the finite-$N$ ${\delta{\mathfrak{D}}}_4$. To support our conjecture that this is the correct solution, we will observe that it accurately includes wrapping interactions. The two-loop non-planar solution for the ${{\mathfrak{su}}}(2)$ sector (a subsector of the ${{\mathfrak{su}}}(1,1|2)$ sector) was found in [@Beisert:2003tq]. In that case, there is a unique lift from the planar to the non-planar theory.

The non-planar action for the one-site generators, including the $\mathcal{O}(g^0)$ terms and ${{\mathfrak{h}}}$ are straightforward to obtain. Let the gauge group of the theory have generators $\mathfrak{t_m}$ and metric $\mathfrak{g^{mn}}$. Then, for instance, using the notation of [@Beisert:2004ry] $${{\mathfrak{h}}}= \sum_{k=0}^{\infty} h(k)\, \mbox{Tr}\, (\phi^\downarrow_k \,  \check{\phi}^\downarrow_k+\phi^\uparrow_k \,  \check{\phi}^\uparrow_k)+h(k+1) \, \mbox{Tr}\, (\overleftarrow{\psi_k} \, \check{\overleftarrow{\psi_k}} + \overrightarrow{\psi_k} \, \check{\overrightarrow{\psi_k}}),$$ where for $X_i \in \{\phi_k^\updownarrow,\,\overleftrightarrow{\psi_k}\}$, we have the expansion $X_i=X^\mathfrak{m}_i\mathfrak{t_m}$, and $$\check{X_i} = \mathfrak{t}_\mathfrak{m}\mathfrak{g^{mn}}\frac{\delta}{\delta X^\mathfrak{n}_i}, \quad \frac{\delta}{\delta X^\mathfrak{m}_i}X^\mathfrak{n}_j=\delta_{ij}\delta^\mathfrak{n}_\mathfrak{m}.$$

The ${{\mathfrak{T}}}_1$ also have a natural generalization for the non-planar theory. $$\begin{aligned}
 \overrightarrow{{{\mathfrak{T}}}_1}^+=&\sum_{\substack{0 \leq m \\ 0 \leq k < m}}& \frac{1}{\sqrt{2(k+1))}}\,\mbox{Tr}\,{\left[\overrightarrow{\psi_k},\phi_{m-1-k}^\updownarrow\right]} \check{\phi}_m^\updownarrow \nonumber \\ + & \sum_{\substack{0 \leq m \\ 0 \leq k < m}} & \frac{m+1}{2 \sqrt{2(k+1)(m-k)}}\,\mbox{Tr}\,{\left\{\overrightarrow{\psi_k},\overrightarrow{\psi_m}_{-1-k}\right\}} \check{\overrightarrow{\psi_m}} \nonumber 