Matching the Hagedorn temperature in AdS/CFT

TL;DR

This paper establishes a precise match between the Hagedorn temperatures of planar N=4 SYM and string theory on AdS_5 x S^5 in a near-critical, weakly coupled regime, linking gauge theory thermodynamics to spin chain models.

Contribution

It provides the first exact comparison of Hagedorn temperatures in gauge and string theories in a near-critical limit, using a decoupling limit and spin chain correspondence.

Findings

Hagedorn temperatures match in both theories within the specified regime.

Derived a relation between Hagedorn temperature and Heisenberg spin chain thermodynamics.

Computed the string theory Hagedorn temperature in a pp-wave background and confirmed agreement.

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.