Towards a Nicolai map for supergravity

TL;DR

This paper investigates the existence of a Nicolai map for minimal four-dimensional supergravity around Minkowski space. It develops the coupling-flow framework, incorporating off-shell global supersymmetry, BRST gauge fixing, and a rescaled flow operator, to test whether a map $T_g$ exists that expresses quantum gravity correlators in terms of a flat-space bosonic theory. The analysis reveals three concrete obstacles: the Lagrangian is only almost a supervariation (density issues), gauge-fixing introduces multiplictive flow terms via BRST, and the rescaling fails to cancel problematic trace terms, preventing a clean perturbative off-shell construction. A partial, four-parameter first-order Nicolai map that passes the free-action test is obtained, but determinant-matching requires going to second order in the gravitational coupling and to the quantum level, leaving the full construction unresolved. The authors suggest unimodular supergravity as a potential route to bypass the trace obstacle and call for further work on higher-order and quantum consistency tests, as well as implications of partial maps.

Abstract

We investigate the possibility of a Nicolai map for minimal supergravity in four dimensions. Such a map would allow for the computation of quantum supergravity correlation functions in terms of flat-space correlators in an effective nonlocal bosonic theory with the help of a nonlinear field transformation, the inverse Nicolai map. Such a map is guaranteed for off-shell global supersymmetry, but local supersymmetry presents at least three obstacles for the construction. Their effects are analyzed in detail, in an attempt to set up a Nicolai map to leading order in the gravitational coupling. We find indications that the conformal factor of the metric obstructs the off-shell construction, suggesting that the unimodular variant of supergravity may do better. The on-shell supersymmetry approach, successful for super-Yang-Mills theory in its critical dimensions, also fails, because the graviton self-interaction cannot be written as a supervariation. Nevertheless, by brute force we obtain a four-parameter first-order Nicolai map fulfilling the free-action condition. For the acid test of determinant matching, however, one needs to push the general ansatz and the perturbative expansion to the second order and to the quantum level.