The Bethe ansatz for superconformal Chern-Simons

TL;DR

Problem: compute planar anomalous dimensions for scalar operators in the ABJM $\mathcal{N}=6$ superconformal Chern-Simons theory. Approach: map the two-loop dilatation operator to an integrable $SU(4)$ spin chain with alternating sites and derive Bethe equations, then extend to the full $OSp(2,2|6)$ superconformal symmetry via a nested Bethe ansatz. Contributions: explicit two-loop Hamiltonian $\Gamma$ and its spectrum, SU(4) Bethe equations with momentum-carrying outer roots, and a proposal for $OSp(2,2|6)$ Bethe equations including fermionic roots; discussion of subsectors and consistency checks. Significance: provides strong evidence for planar integrability in a three-dimensional gauge theory and offers a route to all-loop results aligned with the $AdS_4 \times CP^3$ dual.

Abstract

We study the anomalous dimensions for scalar operators for a three-dimensional Chern-Simons theory recently proposed in arXiv:0806.1218. We show that the mixing matrix at two-loop order is that for an integrable Hamiltonian of an SU(4) spin chain with sites alternating between the fundamental and the anti-fundamental representations. We find a set of Bethe equations from which the anomalous dimensions can be determined and give a proposal for the Bethe equations to the full superconformal group of OSp(2,2|6).

Figures (8)

• Figure 1: The alternating spin chain.
• Figure 2: The planar diagrams that contribute to operator mixing at two loops. The horizontal bar denotes the operator. The directions of the arrows refer to the flow of the $SU(4)$ flavor. Since the superpartners of the scalars are in the conjugate representation of $SU(4)$, the fermion arrows in (b) and (c) have the opposite orientation. The gauge propagators in (d), (e), (f) and (g) do not have arrows since they do not carry $SU(4)$ charges. It turns out that only (a), (b) and (d) contribute to the anomalous dimension.
• Figure 3: The two-loop Hamiltonian. The arrows denote $SU(4)$ index contractions.
• Figure 4: The permutation and trace operators.
• Figure 5: The $SU(4)$ Dynkin diagram where the numbers indicate the Dynkin labels of the representation. The roots $u_j$ are associated with one outer root, $v_j$ with the other outer root, and $r_j$ with the middle root.