The AdS(5)xS(5) superstring quantum spectrum from the algebraic curve

TL;DR

This work develops a universal, algebraic-curve based method for the semi-classical quantization of the $AdS_5\times S^5$ superstring, enabling computation of the fluctuation spectrum around classical solutions by adding microscopic cuts to the classical Riemann surface and reading energy shifts from the perturbed quasi-momenta. By treating bosonic and fermionic modes on equal footing and enforcing a consistent labeling of fluctuation frequencies, the authors derive explicit one-loop shifts for circular string configurations in both the $SU(2)$ and $SL(2)$ sectors, and connect the results to the BMN limit. The approach provides explicit expressions for the fluctuation frequencies in the $AdS_5$ and $S^5$ sectors as well as for fermions, and clarifies the origin of constant shifts and mode-number relabeling due to frame choices. Overall, the paper offers a robust, general framework for computing the quasi-classical spectrum of the $AdS_5\times S^5$ superstring and verifies consistency with known results while extending to broader classes of circular-string solutions.

Abstract

We propose a method for computing the energy level spacing around classical string solutions in AdS(5)xS(5). This method is based on the integrable structure of the string and can be applied to an arbitrary classical configuration. Our approach treats in equal footing the bosonic and fermionic excitations and provides an unambiguous prescription for the labeling of the fluctuation frequencies. Finally we revisit the computation of these frequencies for the SU(2) and SL(2) circular strings and compare our results to the existing ones.

Figures (4)

• Figure 1: Analytical structure of a quasi-momenta $p(x)$ of a one dimensional system. Left: for low lying states $p(x)$ is a collection of poles. Right: for high energy states the poles condense into a square root branch cut.
• Figure 2: A possible analytical stricture of the quasi-momenta of an integrable sigma model. Many types of cuts are now possible. Cuts can join different sheets and each cut is marked by its "mode number" $n_{ij}$. In flat space limit they become numbers of fourier modes. The number of microscopical poles constituting the given cut is called a "filling fraction" and can be calculated as a contour integral (\ref{['bohr']}).
• Figure 3: Some configuration of poles on the algebraic curve corresponding to the $S^5$ excitations (red) and $AdS_5$ excitations (blue). Black line denotes poles at $\pm 1$, connecting 4 sheets with equal residues. The crosses correspond to the residue $+\alpha(x)$, while circles to residue $-\alpha(x)$. Physical domain of the surface lies outside the unit circle.
• Figure 4: Some configuration of poles on the algebraic curve corresponding to the 8 fermionic excitations. Black line denotes poles at $\pm 1$, connecting 4 sheets with equal residues. The crosses correspond to the residue $\alpha(x)$, while circles to residue $-\alpha(x)$. Physical domain of the surface lies outside the unit circle.