The AdS4/CFT3 algebraic curve

TL;DR

The paper builds the OSp(2,2|6) symmetric algebraic curve for the AdS$_4$/CFT$_3$ duality, unifying classical string dynamics at strong coupling with the weak-coupling two-loop spectrum via a ten-sheeted Riemann surface of quasi-momenta. It first constructs the bosonic curve from the flat connection and then elevates it to a full superstring curve by incorporating fermionic poles, enabling semi-classical quantization around any classical solution. The BMN point-like string is worked out in detail, with explicit fluctuation energies for CP$^3$, AdS$_4$, and fermionic modes that reproduce the BMN spectrum. A Chern-Simons curve in the ABJM scaling limit connects the weak-coupling Bethe ansatz to a ten-sheeted curve and links to a Landau-Lifshitz formulation, highlighting the role of integrability across couplings and space-time backgrounds. Overall, the work provides a geometric, algebraic framework to study non-perturbative spectra in AdS$_4$/CFT$_3$, analogous to the AdS$_5$/CFT$_4$ curve, and enables semi-classical analyses around arbitrary classical solutions.

Abstract

We present the OSp(2,2|6) symmetric algebraic curve for the AdS4/CFT3 duality recently proposed in arXiv:0806.1218. It encodes all classical string solutions at strong t'Hooft coupling and the full two loop spectrum of long single trace gauge invariant operators in the weak coupling regime. This construction can also be used to compute the complete superstring semi-classical spectrum around any classical solution. We exemplify our method on the BMN point-like string.

Figures (2)

• Figure 1: Full $AdS_4 \times CP^3$ algebraic curve in the $\textbf{10}$ representation. Poles uniting $AdS_4$ and $CP^3$ quasimomenta are fermionic excitations. Regions which are trivially related are painted with the same colour. Poles at $x=\pm 1$ are marked by black filled circles. The $OSP(2,2|6)$ emerges naturally and notice that the black dots disappear as we jump though the dynkin nodes whose Dynkin labels are non-zero, precisely as expected -- Bethe equations should be difference of quasimomenta and only therefore this pattern reflects the $SU(N) \times SU(N)$ staggered spin chain of Minahan and Zarembo MZ.
• Figure 2: The several states in the Hilbert space can be constructed in the usual oscillator representation. There is one oscillator per Dynkin node of the $OSp(2,2|6)$ super Dynkin diagram. A light (dark) gray shaded node corresponds to an oscillator excited once (twice). From the Chern-Simons Bethe ansatz point of view, the number of times each oscillator is excited is the same as the number of Bethe roots of the corresponding type. Thus, for example, in the notation of MZ, the last fermionic excitation corresponds to a bound state of one root of each type $u,v,w,s$ and two Bethe roots $r$. From the string point of view fluctuations correspond to poles uniting the several sheets of the algebraic curve. Close to each fluctuation we represented some numbers like $45/67$ for the first fluctuation. They indicate which momenta are being united by this pole. In this case it is momenta $q_4$ and $q_5$. Since (\ref{['qs2']}) automatically $q_6$ and $q_7$ also share a pole.