Complexity for superconformal primaries from BCH techniques

We extend existing results for the Nielsen complexity of scalar primaries and spinning primaries in four dimensions by including supersymmetry. Specifically, we study the Nielsen complexity of circuits that transform a superconformal primary with definite scaling dimension, spin and R-charge by means of continuous unitary gates from the $\mathbf{\mathfrak{su}}(2,2|\mathcal{N})$ group. Our analysis makes profitable use of Baker-Campbell-Hausdorff formulas including a special class of BCH formulas we conjecture and motivate. With this approach we are able to determine the super-K\"{a}hler potential characterizing the circuit complexity geometry and obtain explicit expressions in the case of $\mathcal{N}=1$ and $\mathcal{N}=2$ supersymmetry.