Spin chains and string theory

TL;DR

The paper shows that the one-loop dilatation operator for a class of ${ m cal N}=4$ SYM operators is equivalent to a ferromagnetic Heisenberg spin chain and that its long-wavelength limit reproduces a sigma model identical to the string worldsheet action for rotating strings in $AdS_5\times S^5$ in the large-$J$ regime. It provides explicit matching between Bethe-ansatz results and semiclassical string solutions, and demonstrates a coherent-state mapping whereby the mean spin at a site corresponds to the bulk position of the string. The work argues for a more precise formulation of the AdS/CFT correspondence in the large-$N$ limit, suggesting that the all-loop spin-chain dynamics encode the classical string action and potentially offering a route to string duals of other gauge theories. It also discusses extensions to higher loops and the interpretation of the center-of-mass motion within this framework.

Abstract

Recently, an impressive agreement was found between anomalous dimensions of certain operators in N=4 SYM and rotating strings with two angular momenta in the bulk of AdS5xS5. A one-loop field theory computation, which involves solving a Heisenberg chain by means of the Bethe ansatz agrees with the large angular momentum limit of a rotating string. We point out that the Heisenberg chain can be equally well solved by using a sigma model approach. Moreover we also show that a certain limit, akin to the BMN limit, leads exactly to the same sigma model for a string rotating with large angular momentum. The agreement is then at the level of the action. As an upshot we propose that the rotating string should be identified with a coherent, semi-classical state built out of the eigenstates of the spin chain. The agreement is then complete. For example we show that the mean value of the spin <S> gives, precisely, the position of the string in the bulk. This suggests a more precise formulation of the AdS/CFT correspondence in the large-N limit and also indicates a way to obtain string theory duals of other gauge theories.