Classical/quantum integrability in AdS/CFT

TL;DR

This paper develops an integrability-driven bridge between perturbative gauge theory and semiclassical string theory in AdS5×S5 by treating the SU(2) sector of N=4 SYM as an XXX spin chain and the dual string sigma-model as a classical finite-gap system. A unified long-wavelength Bethe Ansatz framework is constructed, linking Bethe roots to densities on cuts, a Riemann-Hilbert problem for the quasi-momentum on hyperelliptic curves, and a differential that encodes all conserved charges. The authors show that one- and two-loop anomalous dimensions in the gauge theory agree with corresponding string-sigma-model predictions for a broad class of long operators, via explicit finite-gap data and spectral-parameter mappings. They illustrate this with BMN, rational (one-cut), and elliptic (two-cut) solutions, and discuss the prospects and caveats for higher-loop and all-orders extensions. The results highlight the power of integrability in unifying gauge/string duality and provide concrete computational tools to explore more general operator sectors.

Abstract

We discuss the AdS/CFT duality from the perspective of integrable systems and establish a direct relationship between the dimension of single trace local operators composed of two types of scalar fields in N=4 super Yang-Mills and the energy of their dual semiclassical string states in AdS(5) X S(5). The anomalous dimensions can be computed using a set of Bethe equations, which for ``long'' operators reduces to a Riemann-Hilbert problem. We develop a unified approach to the long wavelength Bethe equations, the classical ferromagnet and the classical string solutions in the SU(2) sector and present a general solution, governed by complex curves endowed with meromorphic differentials with integer periods. Using this solution we compute the anomalous dimensions of these long operators up to two loops and demonstrate that they agree with string-theory predictions.

Figures (5)

• Figure 1: Contours of integration in eq. (\ref{['der']}): on the physical sheet (a) and after we move the contour into the second sheet with simultaneous flip of orientation (b).
• Figure 2: Hyperelliptic Riemann surface $\Sigma$, defined by equation (\ref{['sigmaxxx']}); zig-zag line represents the condensate cut. There is a double pole of differential $dG$ at the image of the origin $x=0$ on the second sheet, while the differential $dp$ has double poles at $x=0$ on both sheets. The sheet where $dG$ has no singularities is called physical.
• Figure 3: The contour of integration in (\ref{['ZEROMODE']}). The marked points on two sheets correspond to infinities $\infty_+$ and $\infty_-$ where $x=\infty$.
• Figure 4: When an A-cycle crosses the condensate, $\oint_{A_i}dp=\,{\rm Disc}\, p=2\pi m_i$.
• Figure 5: The trace of the monodromy matrix as a function of the spectral parameter along the curve on which $\cos p(x)$ is real. The crosses correspond to degenerate periodic solutions, which represent forbidden zones shrunk to points.