Nicolai maps for super Yang-Mills on the light cone

TL;DR

This paper constructs explicit Nicolai maps for supersymmetric Yang-Mills theories in four and ten dimensions within the light-cone gauge, overcoming previous obstructions caused by four-fermi interactions and providing detailed second-order formulations.

Contribution

It provides the first explicit second-order Nicolai maps for super Yang-Mills theories in four and ten dimensions, addressing nonlocal interactions and systematic cancellations.

Findings

Explicit second-order Nicolai maps in 4D and 10D

Polynomial form possible for certain gauge choices in 4D

Infinite series with systematic cancellations in 10D

Abstract

We construct Nicolai maps for supersymmetric Yang-Mills theory in four and ten spacetime dimensions in the light-cone gauge, where the elimination of non-propagating degrees of freedom causes nonlocal and four-fermi interactions in the Lagrangian. The presence of the latter used to be an obstruction to the Nicolai map, which has recently been overcome at the price of quantum corrections to the map. No gauge-fixing or ghost terms arise in this formulation, since only physical transverse degrees of freedom occur. We present an explicit form of the Nicolai map to second order in the gauge coupling. In four dimensions, a `chiral' choice of the map leaves one of the two transverse gauge-field modes invariant, which forces the classical part of the map (on the other mode) to become a polynomial (quadratic in the gauge coupling, cubic in the gauge field)! In the power series expansion for the ten-dimensional map however, cancellations at each order in the coupling are systematic but incomplete, still leaving an infinite power series for the Nicolai map (on all eight transverse modes). Nevertheless, the existence of a polynomial variant is conceivable, also for the maximal ${\cal N}{=}\,4$ theory in four dimensions.