Nicolai maps with four-fermion interactions

TL;DR

This work generalizes Nicolai maps to supersymmetric theories with four-fermion interactions, showing that the determinant-matching condition expands into an infinite hierarchy of fermion-loop constraints and that the map acquires quantum corrections organized by a coupling-flow operator $R_g[\phi]$. It provides a perturbative construction for four-dimensional SUSY nonlinear sigma models and explicitly constructs the Nicolai map for $\mathbb{C}P^N$, including leading classical and one-loop contributions, while revealing how four-fermion terms induce fermion-loop decorations in the map. The authors also explore an auxiliary-vector-field (Hubbard–Stratonovich) reformulation for $\mathbb{C}P^N$ and discuss its limitations, notably that the $g\to0$ limit becomes singular and the reformulation does not simplify the Nicolai-map expansion. They discuss the implications for regularization, potential extensions to gauge theories, and the broader goal of applying Nicolai maps to theories with richer fermionic interactions, including prospects for supergravity.

Abstract

Nicolai maps offer an alternative description of supersymmetric theories via nonlinear and nonlocal transformations characterized by the so-called `free-action' and `determinant-matching' conditions. The latter expresses the equality of the Jacobian determinant of the transformation with the one obtained by integrating out the fermions, which so far have been considered only to quadratic terms. We argue that such a restriction is not substantial, as Nicolai maps can be constructed for arbitrary nonlinear sigma models, which feature four-fermion interactions. The fermionic effective one-loop action then gets generalized to higher loops and the perturbative tree expansion of such Nicolai maps receives quantum corrections in the form of fermion loop decorations. The `free-action condition' continues to hold for the classical map, but the `determinant-matching condition' is extended to an infinite hierarchy in fermion loop order. After general considerations for sigma models in four dimensions, we specialize to the case of $\mathbb{C}\mathrm{P}^N$ symmetric spaces and construct the associated Nicolai map. These sigma models admit a formulation with only quadratic fermions via an auxiliary vector field, which does not simplify our analysis.