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Non-planar ABJM Theory and Integrability

Charlotte Kristjansen, Marta Orselli, Konstantinos Zoubos

TL;DR

The authors derive the complete two-loop dilatation generator for ABJM in the SU(2)×SU(2) sector, including non-planar corrections, and apply it to short and BMN-type operators to test planar integrability beyond the planar limit. Using effective vertices, they show D = :V_F^{bos}: with cancellations of D-terms and a planar reduction to two coupled SU(2) Heisenberg magnets; non-planar terms introduce trace-number changing interactions, expanding as D = λ^2(D_0 + N^{-1}D_+ + N^{-1}D_- + N^{-2}D_{00} + N^{-2}D_{++} + N^{-2}D_{--}). Planar parity degeneracies between opposite-parity operators persist at planar level and reflect an extra conserved charge, but non-planar corrections lift these degeneracies, signaling a breakdown of integrability beyond planar; BMN analyses show that in the BMN limit the theory resembles N=4 SYM at leading order, while away from the limit extra non-planar terms appear. Overall, the work clarifies structural differences between planar and non-planar ABJM dynamics and lays groundwork for connecting non-planar spectral data to the dual string theory on AdS4×CP3.

Abstract

Using an effective vertex method we explicitly derive the two-loop dilatation generator of ABJM theory in its SU(2)xSU(2) sector, including all non-planar corrections. Subsequently, we apply this generator to a series of finite length operators as well as to two different types of BMN operators. As in N=4 SYM, at the planar level the finite length operators are found to exhibit a degeneracy between certain pairs of operators with opposite parity - a degeneracy which can be attributed to the existence of an extra conserved charge and thus to the integrability of the planar theory. When non-planar corrections are taken into account the degeneracies between parity pairs disappear hinting the absence of higher conserved charges. The analysis of the BMN operators resembles that of N=4 SYM. Additional non-planar terms appear for BMN operators of finite length but once the strict BMN limit is taken these terms disappear.

Non-planar ABJM Theory and Integrability

TL;DR

The authors derive the complete two-loop dilatation generator for ABJM in the SU(2)×SU(2) sector, including non-planar corrections, and apply it to short and BMN-type operators to test planar integrability beyond the planar limit. Using effective vertices, they show D = :V_F^{bos}: with cancellations of D-terms and a planar reduction to two coupled SU(2) Heisenberg magnets; non-planar terms introduce trace-number changing interactions, expanding as D = λ^2(D_0 + N^{-1}D_+ + N^{-1}D_- + N^{-2}D_{00} + N^{-2}D_{++} + N^{-2}D_{--}). Planar parity degeneracies between opposite-parity operators persist at planar level and reflect an extra conserved charge, but non-planar corrections lift these degeneracies, signaling a breakdown of integrability beyond planar; BMN analyses show that in the BMN limit the theory resembles N=4 SYM at leading order, while away from the limit extra non-planar terms appear. Overall, the work clarifies structural differences between planar and non-planar ABJM dynamics and lays groundwork for connecting non-planar spectral data to the dual string theory on AdS4×CP3.

Abstract

Using an effective vertex method we explicitly derive the two-loop dilatation generator of ABJM theory in its SU(2)xSU(2) sector, including all non-planar corrections. Subsequently, we apply this generator to a series of finite length operators as well as to two different types of BMN operators. As in N=4 SYM, at the planar level the finite length operators are found to exhibit a degeneracy between certain pairs of operators with opposite parity - a degeneracy which can be attributed to the existence of an extra conserved charge and thus to the integrability of the planar theory. When non-planar corrections are taken into account the degeneracies between parity pairs disappear hinting the absence of higher conserved charges. The analysis of the BMN operators resembles that of N=4 SYM. Additional non-planar terms appear for BMN operators of finite length but once the strict BMN limit is taken these terms disappear.

Paper Structure

This paper contains 19 sections, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. ABJM theory
  3. The derivation of the full dilatation generator
  4. The structure of the dilatation generator
  5. Planar parity pairs, conserved charges and integrability
  6. Short Operators
  7. Operators with only one type of excitation
  8. Operators with two types of excitations
  9. Length 8 with 2 excitations
  10. Length 8 with 3 excitations
  11. Length 8 with 4 excitations
  12. BMN operators
  13. BMN operators with only one type of excitation
  14. BMN operators with two different types of excitations
  15. Conclusion
  16. ...and 4 more sections

Figures (1)

  • Figure 1: The four types of two--loop diagrams contributing to anomalous dimensions. For operators in the $SU(2)\times SU(2)$ sector diagrams in class (d) do not contribute.