Exact computation of one-loop correction to energy of spinning folded string in AdS_5 x S^5

TL;DR

The paper resolves the exact one-loop correction $E_1(S)$ to the energy of the folded spinning string in the $AdS_3$ sector of $AdS_5\times S^5$ by casting all quadratic fluctuations into the single-gap Lamé form. It demonstrates the UV finiteness of the energy and confirms the equivalence of conformal-gauge and static-gauge results, providing a compact determinant-based expression for $E_1$ that can be evaluated numerically. Through detailed large-spin and small-spin analyses, the authors verify reciprocity constraints at strong coupling and connect the semiclassical spectrum to finite-gap integrability, highlighting the role of Lamé operators in this context. This work both substantiates the integrable structure underlying quantum string corrections and offers a practical framework for exact quantum computations in AdS/CFT settings.

Abstract

We consider the 1-loop correction to the energy of folded spinning string solution in the AdS_3 part of AdS_5 x S^5. The classical string solution is expressed in terms of elliptic functions so an explicit computation of the corresponding fluctuation determinants for generic values of the spin appears to be a non-trivial problem. We show how it can be solved exactly by using the static gauge expression for the string partition function (which we demonstrate to be equivalent to the conformal gauge one) and observing that all the corresponding second order fluctuation operators can be put into the standard (single-gap) Lamé form. We systematically derive the small spin and large spin expansions of the resulting expression for the string energy and comment on some of their applications.

Figures (12)

• Figure 1: Plot (blue, solid curve) of the classical energy ${\mathcal{E}}_0$ as a function of the spin ${\mathcal{S}}$, compared with the large spin expansion (red, dotted curve) in (\ref{['Ecl']}), and the small spin expansion (gold, dashed curve) in (\ref{['Ec']}).
• Figure 2: Potential $V_\beta$ in (\ref{['bebi']}), for $k=0.5, 0.9, 0.99$ and $0.999$, from bottom to top.
• Figure 3: Potential $V_\phi$ in (\ref{['phb']}), for $k=0.5, 0.9, 0.99$ and $0.999$, from bottom to top. Note that singularities appear at the turning points where $\sigma=\left(n+\frac{1}{2}\right)\pi$.
• Figure 4: Potentials $V_{\psi_+}$ (red) and $V_{\psi_-}$ (blue) in (\ref{['pss']}) , for $k=0.5, 0.9, 0.99$ and $0.999$, from bottom to top. Note that $V_\pm(\sigma)$ are identical in from, but are displaced from one another by a half-period $\pi$; they are self-isospectralDunne:1997ia.
• Figure 5: The discriminant $\Delta(\Lambda)$ for the Lamé potential $V(x)=2\,k^2\,{\rm sn}^2(x\,|\,k^2)$ with $k = 0.9$. The three band edges occur at the points ${\Lambda}_1, {\Lambda}_2, {\Lambda}_3$ where $\Delta(\Lambda)$ cuts the lines $\pm 2$, while the remainder of the periodic/antiperiodic spectrum consists of points where $\Delta(\Lambda)$ touches the lines $\pm 2$.