Nicolai maps for supersymmetric sigma models

TL;DR

This work advances the construction of Nicolai maps for nonlinear supersymmetric sigma models by applying the flow-operator framework to the four-dimensional ${ m mathbb{C}P}^1$ model. It develops a chiral Nicolai map and computes both the classical and quantum parts to specified orders, demonstrating that the free-action condition fixes only the one-edge part of the map and that higher-edge (branched) trees carry nontrivial structure tied to spin and fermion loops. To address loop decorations, the authors introduce an auxiliary vector field via a gauged sigma-model reformulation, yielding an enhanced, effectively classical map in an extended $(oldsymbol{ta},A_m)$ space that encodes fermion-loop effects through $A$-connected trees. The results illuminate the organization of Nicolai maps beyond quadratic fermions and point toward broader applications, such as gauge theories and potential supergravity formulations, by providing explicit perturbative constructions and a pathway to classicalization of loop effects.

Abstract

Supersymmetric field theories can be characterized by their Nicolai map, which is a nonlinear and nonlocal field transformation to their free-field limit. The systematic construction of such maps has recently been outlined for actions with power more than two in the fermions, which produces a perturbative expansion in loop-decorated fermionic tree diagrams. We thoroughly investigate the nonlinear $\mathbb C P^1$ sigma model in ($3{+}1$)-dimensional Minkowski space as a paradigmatical example. We construct and test a chiral form of the Nicolai map, to third order in the coupling, including all (regularized) quantum parts. In addition, all trees with one or two edges are summed up. The free-action condition determines only the one-edge part of the map. We resolve the fermion loop decoration of the Nicolai trees by injecting an auxiliary vector field and present the ensuing classical Nicolai map to second order in a dimensionful coupling.