Large spin limit of AdS_5 x S^5 string theory and low energy expansion of ferromagnetic spin chains

TL;DR

This paper develops a systematic large-J expansion connecting AdS5×S5 string theory with the low-energy dynamics of the ferromagnetic SU(2) spin chain. By employing a non-diagonal uniform gauge on the string side and coherent-state methods on the spin-chain side, it derives a Landau-Lifshitz–type action that matches the string sigma model up to next-to-leading order in ${\tilde{\lambda}}$ after incorporating quantum corrections from integrating out short-wavelength modes. The work demonstrates, via folded-string checks and explicit coefficient matching, that the duality extends beyond leading order and clarifies how gauge choices and field redefinitions enable the string–gauge correspondence. It lays out a clear program to push the matching to higher orders and to better understand the underlying integrable structures linking the two descriptions.

Abstract

By considering AdS_5 x S^5 string states with large angular momenta in S^5 one is able to provide non-trivial quantitative checks of the AdS/CFT duality. A string rotating in S^5 with two angular momenta J_1,J_2 is dual to an operator in N=4 SYM theory whose conformal dimension can be computed by diagonalizing a (generalization of) spin 1/2 Heisenberg chain Hamiltonian. It was recently argued and verified to lowest order in a large J=J_1+J_2 expansion, that the Heisenberg chain can be described using a non-relativistic low energy effective 2-d action for a unit vector field n_i which exactly matches the corresponding large J limit of the classical AdS_5 x S^5 string action. In this paper we show that this agreement extends to the next order and develop a systematic procedure to compute higher orders in such large angular momentum expansion. This involves several non-trivial steps. On the string side, we need to choose a special gauge with a non-diagonal world-sheet metric which insures that the angular momentum is uniformly distributed along the string, as indeed is the case on the spin chain side. We need also to implement an order by order redefinition of the field n_i to get an action linear in the time derivative. On the spin chain side, it turns out to be crucial to include the effects of integrating out short wave-length modes. In this way we gain a better understanding of how (a subsector of) the string sigma model emerges from the dual gauge theory, allowing us to demonstrate the duality beyond comparing particular examples of states with large J.