A solution to the non-linear equations of D=10 super Yang-Mills theory

TL;DR

This work presents a formal solution to the nonlinear equations of ten-dimensional super Yang-Mills by resumming linearized, multiparticle superfields into a Lie-algebra valued generating series. Grounded in the pure spinor string framework and Berends-Giele currents, the method captures nonlinearities through off-shell contact terms and defines higher-mass-dimension superfields with explicit equations of motion. The generating-series formalism recasts the nonlinear SYM dynamics into compact, gauge-covariant equations and reveals BCJ-type color-kinematics structures, offering a direct route to amplitude-generating functions for tree and higher-loop contexts. Potential applications include dimensional reduction to lower dimensions, extensions to theories with reduced supersymmetry, and insights into higher-derivative operators such as D^6R^4 in string amplitudes, bridging field theory and string-theory perturbation theory.

Abstract

In this letter, we present a formal solution to the non-linear field equations of ten-dimensional super Yang--Mills theory. It is assembled from products of linearized superfields which have been introduced as multiparticle superfields in the context of superstring perturbation theory. Their explicit form follows recursively from the conformal field theory description of the gluon multiplet in the pure spinor superstring. Furthermore, superfields of higher mass dimensions are defined and their equations of motion spelled out.