Algebraic Curve for the SO(6) sector of AdS/CFT

TL;DR

The paper presents a four-sheeted algebraic-curve construction for the classical sigma-model on $\mathbb{R}\times S^5$, corresponding to the $\mathfrak{so}(6)\simeq \mathfrak{su}(4)$ sector of AdS/CFT. It shows that, up to two loops, this string curve matches the one-loop gauge-theory curve for scalar operators, and provides explicit one-loop string solutions (e.g., Frolov–Tseytlin circular and pulsating strings) as illustrations. The authors develop a parallel Bethe-ansatz framework for the sigma-model in the thermodynamic limit, reformulating the problem in terms of resolvents, cuts, and fillings that define a finite-gap curve, and they discuss higher-loop extensions and the symmetry structures (inversion, SU(4) vs SO(6)). The work lays the groundwork for a full quantum-integrable completion of the AdS/CFT correspondence in this sector by promoting the classical curve to a quantized object and matching more of the perturbative gauge data.

Abstract

We construct the general algebraic curve of degree four solving the classical sigma model on RxS5. Up to two loops it coincides with the algebraic curve for the dual sector of scalar operators in N=4 SYM, also constructed here. We explicitly reproduce some particular solutions.

Figures (9)

• Figure 1: Transfer matrix in ${\mathbf{4}}$ and ${\mathbf{\bar{4}}}$ representation.
• Figure 2: Transfer matrix in ${\mathbf{6}}$ representation.
• Figure 3: Frolov-Tseytlin string. The diagram on the left depicts the cuts as described in Engquist:2003rn. The cut along the imaginary line ($i\mathbb{R}$) interchanges the two involved sheets $(p_1,p_2)$. When we remove this inessential branch cut, we obtain the diagram on the right. Here the cut $\mathcal{C}_-$ goes directly from $p_1$ to $p_3$ right through $p_2$ effectively screening half of it from $\mathcal{C}_+$.
• Figure 4: Pulsating string solution. The diagram on the left depicts the cuts as described in Engquist:2003rn. The cuts along the imaginary line ($i\mathbb{R}$) interchange the two involved sheets $(p_1,p_2)$ and $(p_3,p_4)$. When we remove these inessential branch cuts, we obtain the diagram on the right. Here the cut $\mathcal{C}_-$ goes directly from $p_1$ to $p_4$ right through $p_2$ and $p_3$ effectively screening it from $\mathcal{C}_+$.
• Figure 5: Six-sheeted version of the pulsating string. Note that on the outer two sheets $p_1+p_2$ and $p_3+p_4$, the physical sheets, both cuts $\mathcal{C}_+$ and $\mathcal{C}_-$ can be seen. In particular, these sheets do not change if we "expand" $\mathcal{C}_+$ instead of $\mathcal{C}_-$ in Fig. \ref{['fig:Sol.Sym ']}. Here the screening works because on the two sheets which $\mathcal{C}_+$ connects, a cut $\mathcal{C}_-$ starts in the same direction, this effectively cancels the forces on $\mathcal{C}_+$. The middle two sheets are free.