Green’s functions ofN=1super Yang-Mills theory and the radius/energy relation

Abstract

We study counterterms of one- and two-point Green's functions of some special operators in $\mathcal{N}=1$ super Yang-Mills (SYM) theory from their supergravity (SUGRA) duals from the consideration of AdS conformal field theory or gauge-gravity correspondence. We consider both the Maldacena-Nunez solution and the Klebanov-Strassler-Tseytlin solution which are proposed as SUGRA duals of $\mathcal{N}=1$ SYM theory. We obtain a radius/energy relation for each solution by comparing the SUGRA calculations with the field theory results. Using these relations we evaluate the $\ensuremath{\beta}$ function of $\mathcal{N}=1$ SYM theory. We find that the leading order term can be accurately obtained for both solutions and the higher order terms exhibit some ambiguities. We discuss the origin of these ambiguities and conclude that more studies are needed to check whether these SUGRA solutions are exactly dual to $\mathcal{N}=1$ SYM theory.