Complexity for superconformal primaries from BCH techniques

TL;DR

This paper extends the analysis of Nielsen complexity to superconformal primaries in four dimensions, incorporating supersymmetry and BCH formulas to derive explicit complexity geometries for N=1 and N=2 cases.

Contribution

It introduces a method to compute superconformal circuit complexity using BCH formulas, providing explicit results for N=1 and N=2 supersymmetry.

Findings

Derived super-Kähler potential for complexity geometry.

Explicit expressions for N=1 and N=2 supersymmetry cases.

Extended complexity analysis to superconformal primaries with supersymmetry.

Abstract

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.