Softly-Broken {\cal N} = 4 Supersymmetry in the Large-N Limit

TL;DR

This work computes exact holomorphic observables in softly broken ${ m Cal N}=4$ SU(N) SYM with ${ m Cal N}=1$-preserving masses, including chiral condensates $u_{2k}$ and the central charge that fixes BPS domain-wall tensions. The analysis uses the exact superpotential and holomorphy to show covariant transformation under the duality group ${ m SL}(2,Z)$ acting on the complexified coupling $ ho=( au+m)/N$, with notable large-$N$ phenomena such as anomalous modular behaviour and a vacuum structure featuring a Higgs vacuum and multiple confining vacua. In the large-$N$ limit at fixed $ ho$ (via $ ho=rac{ au+m}{N}$ and $ au$ related to the 't Hooft coupling $ ho=i/ ilde{ ho}$), chiral condensates scale differently in Higgs versus confining vacua, and fractional instanton effects $q^{1/N}$ arise in confining sectors, indicating intrinsically strong-coupling physics. The results provide concrete tests and implications for AdS/CFT, suggesting how holomorphic observables map to IIB string theory data, while highlighting regimes where supergravity may be insufficient and stringy corrections or alternative dual descriptions become essential.

Abstract

We calculate the exact values of the holomorphic observables of {\cal N}=4 supersymmetric SU(N) Yang-Mills theory deformed by mass terms which preserve {\cal N}=1 SUSY. These include the chiral condensates in each massive vacuum of the theory as well as the central charge which determines the tension of BPS saturated domain walls interpolating between these vacua. Several unexpected features emerge in the large-N limit, including anomalous modular properties under an SL(2,Z) duality group which acts on a complexification of the 't Hooft coupling λ=g^{2}N/4π. We discuss our results in the context of the AdS/CFT correspondence.