Matching the Hagedorn temperature in AdS/CFT
Troels Harmark, Marta Orselli
TL;DR
This paper identifies a controlled decoupled sector of AdS/CFT in which planar N=4 SYM on $\mathbb{R}\times S^3$ reduces to the ferromagnetic XXX$_{1/2}$ Heisenberg spin chain and strongly mirrors string theory in a dual pp-wave background. A general relation is derived linking the Hagedorn temperature to the spin-chain thermodynamics, enabling explicit results in both the weakly coupled small-$\tilde{\lambda}$ and large-$\tilde{\lambda}$ regimes: $\tilde{T}_H \sim 1/\log 2$ plus corrections at small $\tilde{\lambda}$, and $\tilde{T}_H \sim c\tilde{\lambda}^{1/3}$ at large $\tilde{\lambda}$. The same decoupling limit on the IIB string theory side yields a pp-wave with a flat direction, and the Hagedorn temperature computed there matches the gauge theory result, providing strong evidence for a spin-chain/gauge/string triality within a tractable weak-coupling AdS/CFT setup. The work offers a concrete bridge between confinement/deconfinement in weakly coupled SYM and Hagedorn phenomena in string theory, with potential to illuminate finite-size and non-planar corrections in AdS/CFT.
Abstract
We match the Hagedorn/deconfinement temperature of planar N=4 super Yang-Mills (SYM) on R x S^3 to the Hagedorn temperature of string theory on AdS_5 x 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 hep-th/0605234 in which the physics of planar N=4 SYM is given exactly by the ferromagnetic XXX_{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 AdS_5 x 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.