The Algebraic Curve of Classical Superstrings on AdS_5xS^5

TL;DR

Beisert, Kazakov, Sakai, and Zarembo develop a gauge-independent algebro-geometric description of classical IIB superstrings on $AdS_5\times S^5$ by constructing the monodromy-based spectral curve of degree $4+4$. They show that the curve encodes all conserved quantities, including local and global charges, with fermions contributing poles that couple the two bosonic sectors; the Virasoro constraint links the sectors and yields a consistent moduli space matching flat-space intuition. By reformulating the curve as a Riemann-Hilbert problem, they derive integral equations that resemble thermodynamic Bethe equations, and demonstrate agreement with planar one-loop $\mathcal{N}=4$ SYM results, supporting the AdS/CFT spectral correspondence at this order. The framework preserves gauge and kappa symmetry and paves the way toward a quantum spectral solution via a discrete Bethe ansatz or related methods, highlighting integrability as a guiding principle for string theory in $AdS_5\times S^5$.

Abstract

We investigate the monodromy of the Lax connection for classical IIB superstrings on AdS_5xS^5. For any solution of the equations of motion we derive a spectral curve of degree 4+4. The curve consists purely of conserved quantities, all gauge degrees of freedom have been eliminated in this form. The most relevant quantities of the solution, such as its energy, can be expressed through certain holomorphic integrals on the curve. This allows for a classification of finite gap solutions analogous to the general solution of strings in flat space. The role of fermions in the context of the algebraic curve is clarified. Finally, we derive a set of integral equations which reformulates the algebraic curve as a Riemann-Hilbert problem. They agree with the planar, one-loop N=4 supersymmetric gauge theory proving the complete agreement of spectra in this approximation.

Figures (6)

• Figure 1: The monodromy $\mathnormal{\Omega}(z)$ is the open Wilson loop of the Lax connection $A(z)$ around the string.
• Figure 2: Special points of the quasi-momenta. The expansion around $z=0,\infty$ yields one sequence of local charges each, see Sec. \ref{['sec:Super.Local ']}. At $z=\pm 1,\pm i$ one finds the Noether charges, discussed in Sec. \ref{['sec:Super.Global ']}, and multi-local charges. All other points are related to non-local charges.
• Figure 3: Some configuration of cuts and poles for the sigma model. Cuts $\tilde{\mathcal{C}}_a$ between the sheets $\tilde{p}_k$ correspond to $S^5$ excitations and likewise cuts $\hat{\mathcal{C}}_a$ between the sheets $\hat{p}_k$ correspond to $AdS^5$ excitations. Poles $x^\ast_a$ on sheets $\tilde{p}_k$ and $\hat{p}_l$ correspond to fermionic excitations. The dashed line in the middle is related to physical excitations, cuts and poles which cross it contribute to the total momentum, energy shift and local charges.
• Figure 4: Cycles for $S^5$-cuts (top), fermionic poles (middle) and $AdS_5$-cuts (bottom). Generically, $S^5$-cuts are along aligned in the imaginary direction while $AdS_5$-cuts are along the real axis.
• Figure 5: The Frolov-Tseytlin limit of the configuration of cuts and poles in Fig. \ref{['fig:sheetssigma ']}. All inverse cuts and poles as well as the poles at $x=\pm 1$ have been scaled to $u=0$ and absorbed into an effective pole.