On instanton contributions to anomalous dimensions in N=4 supersymmetric Yang-Mills theory

TL;DR

This paper investigates how instantons affect the anomalous dimensions of gauge-invariant operators in N=4 supersymmetric Yang-Mills theory, identifying specific operators with non-perturbative corrections and exploring their implications.

Contribution

It provides a detailed analysis of instanton effects on operator dimensions, constructing operators in various representations and identifying sectors with non-perturbative contributions.

Findings

Instanton corrections occur only for specific operators of dimensions 4 and 5.

Many operators exhibit non-renormalisation properties, unaffected by instantons.

Results suggest certain sectors have universal non-perturbative behavior regardless of dimension.

Abstract

Instanton contributions to the anomalous dimensions of gauge-invariant composite operators in the N=4 supersymmetric SU(N) Yang-Mills theory are studied in the one-instanton sector. Independent sets of scalar operators of bare dimension 2, 3, 4 and 5 are constructed in all the allowed representations of the SU(4) R-symmetry group and their two-point functions are computed in the semiclassical approximation. Analysing the moduli space integrals the sectors in which the scaling dimensions receive non-perturbative contributions are identified. The requirement that the integrations over the fermionic collective coordinates which arise in the instanton background are saturated leads to non-renormalisation properties for a large class of operators. Instanton-induced corrections to the scaling dimensions are found only for dimension 4 SU(4) singlets and for dimension 5 operators in the representation [0,1,0] of SU(4). In many cases the non-renormalisation results are argued to be specific to operators of small dimension, but for some special sectors it is shown that they are valid for arbitrary dimension. Comments are also made on the implications of the results on the form of the instanton contributions to the dilation operator of the theory and on the possibility of realising its action on the instanton moduli space.