Perturbative Nicolai-Map Diagrammatics: Application to Poincaré Supergravity

We develop a perturbative, diagrammatic framework for constructing Nicolai maps and apply it to four-dimensional $\mathcal{N}=1$ Poincaré supergravity expanded around flat Minkowski space. It provides an alternative to the coupling-flow-operator construction, which faces several obstructions when extended to local supersymmetry. Expanding the bosonic effective action and the Nicolai map jointly in the gravitational coupling $\kappa$ and the loop-counting parameter $\mathrm{\hslash }$, we derive the Nicolai-map defining conditions, i.e. the free-action and determinant-matching conditions, order by order. The diagrammatics enumerates all admissible local terms in the Nicolai-map ansatz from the effective-action diagrams and reduces the construction to a finite system of nonlinear polynomial equations. Carried through order ${\kappa }^2$, the resulting constraints are found to be independent of the detailed bosonic input and hierarchical, order-${\kappa }^2$ consistency further restricting the order-$\kappa$ data. A consistent Nicolai-map construction for the Einstein–Hilbert graviton sector is found to require the Rarita–Schwinger gravitino already at this order: Einstein gravity admits a Nicolai map only through its $\mathcal{N}=1$ supersymmetric completion, Poincaré supergravity, supporting Nicolai’s characterization of supersymmetry.