becomes impossible.

In this paper, we overcome this obstacle by developing techniques for higher-loop non-compact sectors. Only using constraints from Feynman diagrammatics and superconformal symmetry, we compute the two-loop dilatation operator for the non-compact ${{\mathfrak{su}}}(1,1|2)$ sector. We also find the corrections to this sector’s symmetry algebra up to $\mathcal{O}(g^3)$. We introduce an auxiliary generator that satisfies special commutation relations with the classical and half-loop symmetry generators. This extension of the symmetry algebra enables us to find and verify solutions of the symmetry constraints at one and one-half loops only using the commutation relations of the extended algebra at zero and at one-half loops. Via our method, higher-loop computations reduce to straightforward algebraic manipulations of commutators. This can be done efficiently even without a computer.

For our computation, it is essential that the ${{\mathfrak{su}}}(1,1|2)$ sector has a hidden ${{\mathfrak{psu}}}(1|1)^2$ symmetry [@Beisert:2004ry], which adds tight restrictions. The representation for the ${{\mathfrak{psu}}}(1|1)^2$ symmetry is trivial at leading order and has an expansion in odd powers of $g$, which is proportional to $\sqrt{\lambda}$. The generators, labeled $\overrightarrow{{{\mathfrak{T}}}}$ and $\overleftarrow{{{\mathfrak{T}}}}$,[^2] change the length of the spin chain, reflecting the dynamic aspect of the full ${{\mathfrak{psu}}}(2,2|4)$ spin chain. $\overrightarrow{{{\mathfrak{T}}}}$ and $\overleftarrow{{{\mathfrak{T}}}}$ anti-commute to the dilatation operator. 