Large spin limit of AdS5× S5 string theory and low energy expansion of ferromagnetic spin chains

Abstract

By considering AdS 5× S 5 string states with large S 5 angular momenta one can 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 2d action for a unit vector field n i which exactly matches the corresponding large J limit of the classical AdS 5× S 5 string action. In this paper we show that this agreement extends to the next order and develop a systematic procedure for computing 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.