Matching the Hagedorn temperature in AdS/CFT correspondence

Abstract

We match the Hagedorn/deconfinement temperature of planar $\mathcal{N}=4$ super Yang-Mills (SYM) on $\mathbb{R}\ifmmode\times\else\texttimes\fi{}{S}^{3}$ to the Hagedorn temperature of string theory on ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$. 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 $\mathcal{N}=4$ SYM is given exactly by the ferromagnetic $XX{X}_{1/2}$ 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 ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$. We compute the Hagedorn temperature of the string theory and find agreement with the Hagedorn/deconfinement temperature computed on the gauge-theory side.