Integrable Spin Chain in Superconformal Chern-Simons Theory

TL;DR

The paper demonstrates integrability of the ABJM ${ m N}=6$ superconformal Chern-Simons theory by connecting the strong-coupling spectrum to free string excitations on the AdS$_4\times$CP$^3$ supercoset and deriving an integrable alternating SU(4) spin chain that governs the dilatation operator at weak coupling. Through explicit two-loop perturbative calculations, it shows the dilatation operator matches the spin-chain Hamiltonian derived from Yang–Baxter equations, with wrapping effects carefully analyzed for short operators. Bethe ansatz analysis of the alternating chain reveals correlated excitations between ${f 4}$ and $ar{f 4}$ sectors and provides insight into the absence of meson-like bound states, while chiral primaries remain protected by supersymmetry. These results provide nontrivial, cross-regime evidence for ${ m AdS}_4/{ m CFT}_3$ integrability and set the stage for exact spectral computations in ABJM theory.

Abstract

N = 6 superconformal Chern-Simons theory was proposed as gauge theory dual to Type IIA string theory on AdS4*CP3. We study integrability of the theory from conformal dimension spectrum of single trace operators at planar limit. At strong `t Hooft coupling, the spectrum is obtained from excitation energy of free superstring on OSp(6|2,2)/SO(3,1)xSU(3)xU(1) supercoset. We recall that the worldsheet theory is integrable classically by utilizing well-known results concerning sigma model on symmetric space. With R-symmetry group SU(4), we also solve relevant Yang-Baxter equation for a spin chain system associated with the single trace operators. From the solution, we construct alternating spin chain Hamiltonian involving three-site interactions between 4 and 4*. At weak `t Hooft coupling, we study gauge theory perturbatively, and calculate action of dilatation operator to single trace operators up to two loops. To ensure consistency, we computed all relevant Feynman diagrams contributing to the dilatation opeator. We find that resulting spin chain Hamiltonian matches with the Hamiltonian derived from Yang-Baxter equation. We further study new issues arising from the shortest gauge invariant operators transforming as 15 and 1 under SU(4). We observe that `wrapping interactions' are present, compute the true spectrum and find that the spectrum agrees with prediction from supersymmetry. We also find that scaling dimension computed naively from alternating spin chain Hamiltonian coincides with the true spectrum. We solve Bethe ansatz equations for small number of excitations, and find indications of correlation between excitations of 4's and 4*'s and of nonexistence of mesonic (4 4*) bound-state.

Figures (8)

• Figure 1: Two loop contribution of scalar sextet interaction to anomalous dimension of ${\cal O}$.
• Figure 2: Two loop contribution of gauge and fermion exchange interaction to anomalous dimension of ${\cal O}$.
• Figure 3: Two loop contribution of diamagnetic gauge interactions to wave function renormalization of $Y, Y^\dagger$. They contribute to $\mathbb{I}$ operator in the dilatation operator.
• Figure 4: Two loop contribution of paramagnetic gauge interactions to wave function renormalization of $Y, Y^\dagger$. They contribute to $\mathbb{I}$ operator in the dilatation operator.
• Figure 5: Two loop contribution of Chern-Simons interaction to wave function renormalization of $Y, Y^\dagger$. They contribute to $\mathbb{I}$ operators in the dilatation operator.