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Two loop integrability for Chern-Simons theories with N=6 supersymmetry

J. A. Minahan, W. Schulgin, K. Zarembo

TL;DR

This work establishes two-loop integrability for the ${ m ABJM}$ and ${ m ABJ}$ theories by deriving the full two-loop dilatation operator and showing its consistency with the conjectured Bethe ansatz for ${ m OSp}(6|4)$. Parity-violating differences between ${ m ABJ}$ and ${ m ABJM}$ cancel at two loops, rendering the two theories indistinguishable at this order in the planar limit. The authors construct the complete $OSp(6|4)$ spin-chain Hamiltonian by first solving the noncompact $SL(2|1)$ sector and then lifting to the full symmetry, verifying agreement with both fermionic and bosonic sectors. Their method employs an oscillator realization to build the $R$-matrix, followed by a lifting procedure that preserves the integrable structure across subsectors, including the ${ m SU}(4)$ and mixed fermion–scalar sectors. The results bolster the expectation of planar integrability in ${ m ABJM}$ and raise interesting questions about parity and integrability in the ${ m ABJ}$ theory at higher loops.

Abstract

We consider two-loop anomalous dimensions for fermionic operators in the ABJM model and the ABJ model. We find the appropriate Hamiltonian and show that it is consistent with a previously predicted Bethe ansatz for the ABJM model. The difference between the ABJ and ABJM models is invisible at the two-loop level by cancelation of parity violating diagrams. We then construct a Hamiltonian for the full two-loop OSp(6|4) spin chain by first constructing the Hamiltonian for an SL(2|1) subgroup, and then lifting to OSp(6|4). We show that this Hamiltonian is consistent with the Hamiltonian found for the fermionic operators.

Two loop integrability for Chern-Simons theories with N=6 supersymmetry

TL;DR

This work establishes two-loop integrability for the and theories by deriving the full two-loop dilatation operator and showing its consistency with the conjectured Bethe ansatz for . Parity-violating differences between and cancel at two loops, rendering the two theories indistinguishable at this order in the planar limit. The authors construct the complete spin-chain Hamiltonian by first solving the noncompact sector and then lifting to the full symmetry, verifying agreement with both fermionic and bosonic sectors. Their method employs an oscillator realization to build the -matrix, followed by a lifting procedure that preserves the integrable structure across subsectors, including the and mixed fermion–scalar sectors. The results bolster the expectation of planar integrability in and raise interesting questions about parity and integrability in the theory at higher loops.

Abstract

We consider two-loop anomalous dimensions for fermionic operators in the ABJM model and the ABJ model. We find the appropriate Hamiltonian and show that it is consistent with a previously predicted Bethe ansatz for the ABJM model. The difference between the ABJ and ABJM models is invisible at the two-loop level by cancelation of parity violating diagrams. We then construct a Hamiltonian for the full two-loop OSp(6|4) spin chain by first constructing the Hamiltonian for an SL(2|1) subgroup, and then lifting to OSp(6|4). We show that this Hamiltonian is consistent with the Hamiltonian found for the fermionic operators.

Paper Structure

This paper contains 14 sections, 89 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The ${\cal N}=6$ theory and its Bethe equations
  3. Fermionic part of the dilatation operator
  4. Length-$4$ operators
  5. Spectrum from the mixing matrix
  6. Spectrum from the Bethe equations
  7. The Complete $R$-matrix and Integrability
  8. The oscillator algebra and the singleton representations
  9. The $SL(2|1)$ sector
  10. The lift to $OSp(6|4)$
  11. Subsectors
  12. $SU(4)$
  13. Scalars and one fermion
  14. Discussion

Figures (4)

  • Figure 1: The Dynkin diagram of $OSp(6|4)$. Shown are the Dynkin labels of the state with $\{K_u,K_v,K_r,K_s,K_w\}$ Bethe roots.
  • Figure 2: The CP-breaking contributions to the dilatation operator mutually cancel.
  • Figure 3: The diagrams that contribute to the mixing of operators with single fermion insertion.
  • Figure 4: Super-Dynkin diagrams for (a) $OSp(6|4)$ (b) $SL(2|1)$.