Circular Semiclassical String Solutions on AdS_5 x S_5

TL;DR

This work constructs two highly symmetric semiclassical string solutions in AdS_5 × S^5, a circular pulsating string in AdS_5 and another on S^5, to extract operator dimensions in the dual N=4 SYM via AdS/CFT. In AdS_5, a Bohr-Sommerfeld analysis yields an anomalous dimension scaling as $\lambda^{1/4}\sqrt{mn}$ atop a bare dimension $2n$, linking the level to $mn$; in S^5, perturbative methods give an operator dimension with an analytic correction in $m^2\lambda/(2n)^2$ and a leading term of order $2n$, aligning with BMN-like expansion ideas. The results illustrate how curvature effects render positive, controllable anomalous dimensions and reveal distinct analytic structures across AdS_5 and S^5 pulsations, with potential perturbative verification and broader applicability to less symmetric motions. The work also discusses candidate dual operators with phase structures mirroring string orientation, enriching the bridge between semiclassical string dynamics and gauge-theory operator spectra.

Abstract

We discuss two semiclassical string solutions on AdS_5\times S_5. In the first case, we consider a multiwrapped circular string pulsating in the radial direction of AdS_5, but fixed to a point on the S_5. We compute the energy of this motion as a function of a large quantum number $n$. We identify the string level with $mn$, where $m$ is the number of string wrappings. Using the AdS/CFT correspondence, we argue that the bare dimension of the corresponding gauge invariant operator is $2n$ and that its anomalous dimension scales as λ^{1/4}\sqrt{mn}, for large $n$. Next we consider a multiwrapped circular string pulsating about two opposite poles of the $S_5$. We compute the energy of this motion as a function of a large quantum number, $n$ where again the string level is given as $mn$. We find that the dimension of the corresponding operator is 2n(1+f(m^2λ/(2n)^2)), where f(x) is computible as a series about x=0 and where it is analytic. We also compare this result to the BMN result for large J operators.