Is the Nicolai map unique?

TL;DR

This paper investigates the uniqueness of the Nicolai map in supersymmetric theories, revealing conditions under which it is unique and exploring the effects of topological couplings, with explicit computations in supersymmetric quantum mechanics.

Contribution

It demonstrates how the Nicolai map's ambiguity depends on the coupling space contour and identifies special theta values where the map becomes unique and polynomial.

Findings

The Nicolai map depends on the integration contour in coupling space.

At certain theta values, the map is unique and polynomial in fields.

Explicit one-loop computations confirm theta independence of correlation functions.

Abstract

The Nicolai map is a field transformation that relates supersymmetric theories at finite couplings $g$ with the free theory at $g=0$. It is obtained via an ordered exponential of the coupling flow operator integrated from $0$ to $g$. Allowing multiple couplings, we find that the map in general depends on the chosen integration contour in coupling space. This induces a large functional freedom in the construction of the Nicolai map, which cancels in all correlator computations. Under a certain condition on the coupling flow operator the ambiguity disappears, and the power-series expansion for the map collapses to a linear function in the coupling. A special role is played by topological (theta) couplings, which do not affect perturbative correlation functions but also alter the Nicolai map. We demonstate that for certain 'magical' theta values the uniqueness condition holds, providing an exact map polynomial in the fields and independent of the integration contour. This feature is related to critical points of the Nicolai map and the existence of 'instantons'. As a toy model, we work with $\mathcal{N}=1$ supersymmetric quantum mechanics. For a cubic superpotential and a theta term, we explicitly compute the one-, two- and three-point correlation function to one-loop order employing a graphical representation of the (inverse) Nicolai map in terms of tree diagrams, confirming the cancellation of theta dependence. Comparison of Nicolai and conventional Feynman perturbation theory nontrivially yields complete agreement, but only after adding all (1PI and 1PR) contributions.