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Two-loop Integrability of Planar N=6 Superconformal Chern-Simons Theory

Benjamin I. Zwiebel

Abstract

Bethe ansatz equations have been proposed for the asymptotic spectral problem of AdS_4/CFT_3. This proposal assumes integrability, but the previous verification of weak-coupling integrability covered only the su(4) sector of the ABJM gauge theory. Here we derive the complete planar two-loop dilatation generator of N=6 superconformal Chern-Simons theory from osp(6|4) superconformal symmetry. For the osp(4|2) sector, we prove integrability through a Yangian construction. We argue that integrability extends to the full planar two-loop dilatation generator, confirming the applicability of the Bethe equations at weak coupling. Further confirmation follows from an analytic computation of the two-loop twist-one spectrum.

Two-loop Integrability of Planar N=6 Superconformal Chern-Simons Theory

Abstract

Bethe ansatz equations have been proposed for the asymptotic spectral problem of AdS_4/CFT_3. This proposal assumes integrability, but the previous verification of weak-coupling integrability covered only the su(4) sector of the ABJM gauge theory. Here we derive the complete planar two-loop dilatation generator of N=6 superconformal Chern-Simons theory from osp(6|4) superconformal symmetry. For the osp(4|2) sector, we prove integrability through a Yangian construction. We argue that integrability extends to the full planar two-loop dilatation generator, confirming the applicability of the Bethe equations at weak coupling. Further confirmation follows from an analytic computation of the two-loop twist-one spectrum.

Paper Structure

This paper contains 31 sections, 155 equations.

Table of Contents

  1. Introduction
  2. The $\mathfrak{osp}(6|4)$ spin chain and its $\mathfrak{osp}(4|2)$ subsector
  3. The complete algebra and spin module
  4. Restriction to the $\mathfrak{osp}(4|2)$ sector
  5. The $\mathfrak{osp}(4|2)$ module and leading-order representation
  6. Light-cone superspace basis
  7. Leading corrections in the $\mathfrak{osp}(4|2)$ sector
  8. Structure of $\hat{\mathfrak{Q}}$ and $\hat{\mathfrak{S}}$ interactions
  9. Constraints from commutator with $\mathfrak{J}^{11}$
  10. Anticommutator with $\mathfrak{Q}^{a1\mathfrak{b}}$ and solution for $\hat{\mathfrak{Q}}$ and $\hat{\mathfrak{S}}$
  11. Solution for $\hat{\mathfrak{Q}}$ in light-cone superspace basis
  12. Hamiltonian, wrapping interactions, and twist-one spectrum
  13. Proof of $\mathfrak{osp}(4|2)$ sector integrability
  14. Leading order Yangian
  15. Vanishing commutator with $\hat{\mathfrak{Q}}$ and $\hat{\mathfrak{S}}$
  16. ...and 16 more sections