Perturbative Nicolai-Map Diagrammatics: Application to Poincaré Supergravity

Abstract

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 $\hat{I}º$ and the loop-counting parameter $\hslash$, 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 $\hat{I}{º}^2$, the resulting constraints are found to be independent of the detailed bosonic input and hierarchical, order-$\hat{I}{º}^2$ consistency further restricting the order-$\hat{I}º$ 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.

75 pages, 8 figures