The coupling flow for supergravity

Abstract

Quantum correlation functions in supersymmetric field theories can be encoded in the Nicolai map. The infinitesimal inverse map provides their response to a change in a coupling constant through a ``flow equation''. We investigate the flow in the gravitational constant $κ$ for four-dimensional Poincaré supergravity via its superconformal formulation. By expressing the full superconformal off-shell action as a supervariation we derive a ``flow operator'' in the effective vierbein theory, whose exponentiation yields the gauge-invariant part of an all-order (inverse) Nicolai map for supergravity. Alas, the BRST gauge fixing adds to this functional differential flow operator a multiplicative term, which ruins its derivational property and therefore jeopardizes its connection with the Nicolai map. We argue and present an example, however, that such multiplicative flow contributions might be rewritable as derivational ones with the help of Wick's theorem in the free effective theory where the Nicolai-transformed correlators are ultimately evaluated. At least in the Landau gauge, the supergravity flow equation is shown to be regular at $κ{=}0$, allowing for a perturbative expansion around Minkowski geometry. Therewith, we have overcome two of the three obstacles met in an earlier attempt towards a Nicolai map for Poincaré supergravity. Finally, we compare with the direct on-shell construction of a Nicolai map to leading order in $κ$.