Supersymmetric Proximity

Abstract

I argue that a certain perturbative proximity exists between some supersymmetric and non-supersymmetric theories (namely, pure Yang-Mills and adjoint QCD with two flavors, adjQCD$_{N_f=2}$). I start with ${\mathcal N}=2$ super-Yang-Mills theory built of two ${\mathcal N}=1$ superfields: vector and chiral. In ${\mathcal N}=1$ language the latter presents matter in the adjoint representation of SU$(N). $ Then I convert the matter superfield into a "{\em phantom}" one (in analogy with ghosts), breaking ${\mathcal N}=2$ down to ${\mathcal N}=1$. The global SU(2) acting between two gluinos in the original theory becomes graded. Exact results in thus deformed theory allows one to obtain insights in certain aspects of non-supersymmetric gluodynamics. In particular, it becomes clear how the splitting of the $β$ function coefficients in pure gluodynamics, $β_1 =(4 -\frac 13 )N$ and $β_2= (6-\frac 13)N^2$, occurs. Here the first terms in the braces (4 and 6, always integers) are geometry-related while the second terms ($-\frac 13$ in both cases) are {\it bona fide} quantum effects. In the same sense adjQCD$_{N_f=2}$ is close to ${\mathcal N}=2$ SYM. Thus, I establish a certain proximity between pure gluodynamics and adjQCD$_{N_f=2}$ with supersymmetric theories. (Of course, in both cases we loose all features related to flat directions and Higgs/Coulomb branches in ${\mathcal N}=2$.) As a warmup exercise I use this idea in 2D CP(1) sigma model with ${\mathcal N}=(2,2)$ supersymmetry, through the minimal heterotic ${\mathcal N}=(0,2) \to$ bosonic CP(1).