Matching the Hagedorn temperature in AdS/CFT correspondence

We match the Hagedorn/deconfinement temperature of planar N=4 super Yang-Mills (SYM) on RxS(3) to the Hagedorn temperature of string theory on AdS(5)xS(5). The match is done in a near-critical region where both gauge theory and string theory are weakly coupled. The near-critical region is near a point with zero temperature and critical chemical potential. On the gauge-theory side we are taking a decoupling limit found in Ref. 7 in which the physics of planar N=4 SYM is given exactly by the ferromagnetic XXX1/2 Heisenberg spin chain. We find moreover a general relation between the Hagedorn/deconfinement temperature and the thermodynamics of the Heisenberg spin chain and we use this to compute it in two distinct regimes. On the string-theory side, we identify the dual limit for which the string tension and string coupling go to zero. This limit is taken of string theory on a maximally supersymmetric pp-wave background with a flat direction, obtained from a Penrose limit of AdS(5)xS(5). We compute the Hagedorn temperature of the string theory and find agreement with the Hagedorn/deconfinement temperature computed on the gauge-theory side.