Spiky strings in the SL(2) Bethe Ansatz

TL;DR

This paper analyzes spiky string solutions within the SL(2) Bethe ansatz framework, deriving energy formulas at strong coupling and exploring root distributions, connecting classical string solutions with quantum Bethe ansatz results.

Contribution

It provides the all-loop anomalous dimension for spiky strings and explores a novel two-cut Bethe root distribution at one-loop, linking classical and quantum descriptions.

Findings

Energy of spiky strings matches classical solutions at strong coupling.

Asymmetric Bethe root distribution determines anomalous dimensions.

A new limit connects Bethe ansatz results to string solutions in AdS pp-wave background.

Abstract

We study spiky strings in the context of the SL(2) Bethe ansatz equations. We find an asymmetric distribution of Bethe roots along one cut that determines the all loop anomalous dimension at leading and subleading orders in a large S expansion. At leading order in strong coupling (large lambda) we obtain that the energy of such states is given, in terms of the spin S and the number of spikes n by E-S=n sqrt{lambda}/(2 pi) (ln 16 pi S/(n sqrt{lambda})+ ln sin (pi/n) - 1)+ O(ln S/S). This result matches perfectly the same expansion obtained from the known spiky string classical solution. We then discuss a two cut spiky string Bethe root distribution at one-loop in the SL(2) Bethe ansatz. In this case we find a limit where n goes to infinity, keeping (E+S)/n^2, (E-S)/n, J/n fixed. This is the one loop version of a limit previously considered in the context of the string classical solutions in AdS5 x S5. In that case it was related to a string solution in the AdS pp-wave background.