Matching gauge theory and string theory in a decoupling limit of AdS/CFT

We identify a regime of the AdS/CFT correspondence in which we can quantitatively match N=4 super Yang-Mills (SYM) for small 't Hooft coupling with weakly coupled type IIB string theory on AdS_5 x S^5. We approach this regime by taking the same decoupling limit on both sides of the correspondence. On the gauge theory side only the states in the SU(2) sector survive, and in the planar limit the Hamiltonian is given by the XXX_{1/2} Heisenberg spin chain. On the string theory side we show that the decoupling limit corresponds to a non-relativistic limit. In this limit some of the bosonic modes and all of the fermionic modes of the string become infinitely heavy and decouple. We first take the decoupling limit of the string sigma-model classically. This enables us to identify a semi-classical regime with semi-classical string states even though we are in a regime corresponding to small 't Hooft coupling. We furthermore analyze the quantum corrections that enter in taking the limit. From this we infer that gauge theory and string theory match, both in terms of the action and the spectrum, for the leading part and the first correction away from the semi-classical regime. Finally we consider the implications for the hitherto unexplained matching of the one-loop contribution to the energy of certain gauge theory and string theory states, and we explain how our results give a firm basis for the matching of the Hagedorn temperature in hep-th/0608115.