The all loop AdS4/CFT3 Bethe ansatz (0807.0777v2)

Abstract: We propose a set of Bethe equations yielding the full asymptotic spectrum of the AdS4/CFT3 duality proposed in arXiv:0806.1218 to all orders in the t'Hooft coupling. These equations interpolate between the 2-loop Bethe ansatz of Minahan and Zarembo arXiv:0806.3951 and the string algebraic curve of arXiv:0807.0437. The several SU(2|2) symmetries of the theory seem to highly constrain the form of the Bethe equations up to a dressing factor whose form we also conjecture.

• The paper generalizes Bethe equations to capture the complete asymptotic spectrum of the AdS4/CFT3 duality.
• It details how SU(2|2) symmetry and integrability extend from two-loop analyses to strong coupling regimes.
• The work lays the groundwork for future research into finite-size wrapping corrections and complete quantum spectra.

Overview of the All Loop $AdS_4/CFT_3$ Bethe Ansatz

This paper proposes a set of Bethe equations which describe the full asymptotic spectrum of the $AdS_4/CFT_3$ duality. The authors, Nikolay Gromov and Pedro Vieira, present these equations as an interpolation between the two-loop Bethe ansatz and the string algebraic curve formulations, underlining the interplay between different facets of integrability in string and gauge theories. This proposal is pertinent given the significance of the $AdS/CFT$ correspondence in theoretical physics, particularly in the paper of superconformal Chern-Simons theories.

Key Contributions

1. Bethe Ansatz for $AdS_4/CFT_3$: The authors generalize the Bethe equations to describe the $AdS_4/CFT_3$ duality across all values of the t'Hooft coupling $\lambda$. The proposed equations extend from the two-loop level as studied by Minahan and Zarembo to encompass strong coupling regimes.
2. Integrability Paradigm:
   • The paper elaborates on the integrability within the $SU(N)\times SU(N)$ Chern-Simons theory with a clear association to superstring theory on $AdS_4 \times CP^3$. Integrability, which initially appears at the two-loop order, is posited to persist at higher orders due to underlying $SU(2|2)$ symmetries.
3. Symmetry Constraints: The equations adhere to the $OSp(2,2|6)$ symmetry, which strongly constrains their formulation. There's conjecture regarding a dressing factor influenced by these symmetries, similar in form to the kernel proposed in the $AdS_5/CFT_4$ field.
4. Finite-Size and Wrapping Effects: While these are asymptotic equations, the wrapping corrections that appear at finite sizes remain unaccounted for. The authors suggest further exploration into these corrections as an intriguing avenue, connecting to recent advancements in $AdS_5\times CFT_4$ duality.

Theoretical and Practical Implications

• Theoretical Consistency with Dualities: This proposal ensures a degree of coherence and compatibility with existing conjectures and results within string theory and supersymmetric gauge theory paradigms.
• Pathways for Future Research: Extending to non-asymptotic limits involving finite-size corrections would mark a future landmark in understanding the full quantum spectrum, heavily influencing research in quantum field theories and integrable models.
• Strong Coupling Checks: The approach substantiates its propositions by matching with known semiclassical results and analyses based on algebraic curves, reassured by the $SU(2|2)$ invariant structure.

Speculation and Conclusion

Given their conjectural nature, these Bethe equations remain to be cross-verified through additional calculations to ascertain completeness, such as wrapping corrections, which are complex yet pivotal. Additionally, the determination of the function governing these corrections, denoted as $h(\lambda)$, and its characteristics across weak to strong coupling regimes, is crucial.

Beside the specific results, this paper emphasizes the necessity for meticulous exploration of dualities and symmetry considerations that reinforce the significance of exactly solvable models within theoretical physics, contributing both to the toolkit used to approach complex field theories and the potential construction of new theories that may arise from such foundational exploration.

While pushing the boundaries of current knowledge, this paper invites more exhaustive investigations into the predictive capacities and conditions under which these Bethe equations hold true across the spectrum of theoretical models used in high-energy physics.