Some Systematics of the Coupling Constant Dependence of N=4 Yang-Mills

TL;DR

This work develops a Ward-identity–driven framework to constrain how the coupling constant $\tau$ (and thus the gauge coupling and theta-angle) affects correlation functions in $\mathcal{N}=4$ super-Yang–Mills. By placing the coupling-variation operator ${\mathcal{O}}_\tau$ in the same current multiplet as the stress tensor and supercurrents, the authors derive a structured OPE for ${\mathcal{O}}_\tau$ with arbitrary operators, controlled by supersymmetry. Using a stepwise peeling-off procedure, they relate the ${\mathcal{O}}_\tau$ OPE to the stress-tensor and supercurrent OPEs and, crucially, show exact non-renormalization of two-point functions and many three-point functions for BPS operators, with precise constraints on the remaining OPE coefficients. The results solidify the understanding that for BPS sectors, the space-time and coupling dependences are tightly constrained, aligning with expectations from S-duality and higher-dimensional (6D) origins. The framework also clarifies under which circumstances descendents inherit protection and how instanton effects may alter these conclusions in non-BPS sectors.

Abstract

The operator, O_τ, that generates infinitesimal changes of the coupling constant in N=4 Yang-Mills sits in the same supermultiplet as the superconformal currents. We show how superconformal current Ward identities determine a class of terms in the operator product expansion of O_τwith any other operator. In certain cases, this leads to constraints on the coupling dependence of correlation functions in N=4 Yang-Mills. As an application, we demonstrate the exact non-renormalization of two and certain three-point correlation functions of BPS operators.