The Nicolai map for super Yang-Mills theory and application to the supermembrane

TL;DR

This paper develops a universal formula for the Nicolai map in supersymmetric gauge theories, enabling explicit perturbative construction and application to supermembrane regularization and vertex-operator computations.

Contribution

It introduces a canonical, perturbative construction of the Nicolai map applicable to supersymmetric gauge theories and demonstrates its use in supermembrane regularization.

Findings

Universal formula for the Nicolai map using ordered exponential

Explicit perturbative construction in any gauge for off-shell supersymmetry

Application to supermembrane linearization and vertex-operator calculations

Abstract

The nonlocal bosonic theory obtained from integrating out all anticommuting and auxiliary variables in a globally supersymmetric theory is characterized by the Nicolai map. We present a universal formula for the latter in terms of an ordered exponential of the integrated coupling flow operator, which can be canonically constructed. Also for supersymmetric gauge theories, this allows us to perturbatively construct the Nicolai map explicitly in terms of tree diagrams. For off-shell supersymmetry this works in any gauge, in the on-shell case the Landau gauge is required. The dimensional reduction of $D{=}10$ super Yang-Mills to maximally supersymmetric SU($N$) matrix mechanics (the BFSS model) is known to provide a regularization of the $D{=}11$ supermembrane in light-cone gauge via its incarnation as a one-dimensional gauge theory of area-preserving membrane diffeomorphisms. We show how a well-defined corresponding Nicolai map perturbatively linearizes the supermembrane in the small-tension regime and points the way for a computation of vertex-operator correlators.