Finite-size corrections to the rotating string and the winding state

TL;DR

This paper calculates finite-size energy corrections for rotating strings and winding states in AdS/CFT, matching gauge theory predictions with string theory results to high order in 1/J and lambda' expansions.

Contribution

It provides explicit finite-size correction formulas for rotating strings and winding states, extending previous results and confirming gauge-string duality at higher orders.

Findings

Agreement between string and gauge theory energies up to 1/J^2 order.

Explicit solutions to Bethe equations in an expansion in m/K and all orders in J.

Finite-size corrections match effective Landau-Lifshitz model predictions.

Abstract

We compute higher order finite size corrections to the energies of the circular rotating string on AdS_5 x S^5, of its orbifolded generalization on AdS_5 x S^5/Z_M and of the winding state which is obtained as the limit of the orbifolded circular string solution when J -> infinity and J/M^2 is kept fixed. We solve, at the first order in lambda'=lambda/J^2, where lambda is the 't Hooft coupling, the Bethe equations that describe the anomalous dimensions of the corresponding gauge dual operators in an expansion in m/K, where m is the winding number and K is the "magnon number", and to all orders in the angular momentum J. The solution for the circular rotating string and for the winding state can be matched to the energy computed from an effective quantum Landau-Lifshitz model beyond the first order correction in 1/J. For the leading 1/J corrections to the circular rotating string in m^2 and m^4 and for the subleading 1/J^2 corrections to the m^2 term, we find agreement. For the winding state we match the energy completely up to, and including, the order 1/J^2 finite-size corrections. The solution of the Bethe equations corresponding to the spinning closed string is also provided in an expansion in m/K and to all orders in J.