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Exact AdS/CFT spectrum: Konishi dimension at any coupling

Nikolay Gromov, Vladimir Kazakov, Pedro Vieira

TL;DR

This paper establishes a numerical framework to compute the exact planar N=4 SYM spectrum by solving the AdS/CFT Y-system in its integral form for excited states, focusing on the Konishi operator in the SL(2) sector. By coupling the Y-system with exact Bethe equations and performing iterative updates of the Bethe roots, the authors obtain the Konishi dimension across a wide range of the 't Hooft coupling, validating perturbative results and revealing strong-coupling behavior that matches theoretical predictions. The work confirms Δ_K(λ) behaves as $2λ^{1/4}$ at large λ with computable subleading corrections and demonstrates a robust numerical approach capable of exploring the full finite-size spectrum at any coupling. These results solidify the Y-system as a practical tool for non-perturbative spectral analysis in planar N=4 SYM and point toward potential simplifications via the Hirota framework.

Abstract

We compute the full dimension of Konishi operator in planar N=4 SYM theory it for a wide range of couplings, from weak to strong coupling regime, and predict the subleading terms in its strong coupling asymptotics. For this purpose we solve numerically the integral form of the AdS/CFT Y-system equations for the exact energies of excited states proposed by us and A.Kozak.

Exact AdS/CFT spectrum: Konishi dimension at any coupling

TL;DR

This paper establishes a numerical framework to compute the exact planar N=4 SYM spectrum by solving the AdS/CFT Y-system in its integral form for excited states, focusing on the Konishi operator in the SL(2) sector. By coupling the Y-system with exact Bethe equations and performing iterative updates of the Bethe roots, the authors obtain the Konishi dimension across a wide range of the 't Hooft coupling, validating perturbative results and revealing strong-coupling behavior that matches theoretical predictions. The work confirms Δ_K(λ) behaves as at large λ with computable subleading corrections and demonstrates a robust numerical approach capable of exploring the full finite-size spectrum at any coupling. These results solidify the Y-system as a practical tool for non-perturbative spectral analysis in planar N=4 SYM and point toward potential simplifications via the Hirota framework.

Abstract

We compute the full dimension of Konishi operator in planar N=4 SYM theory it for a wide range of couplings, from weak to strong coupling regime, and predict the subleading terms in its strong coupling asymptotics. For this purpose we solve numerically the integral form of the AdS/CFT Y-system equations for the exact energies of excited states proposed by us and A.Kozak.

Paper Structure

This paper contains 6 sections, 11 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Y-system functional and integral equations for AdS/CFT
  3. Exact Bethe equations
  4. Numerics and its interpretation
  5. Conclusions
  6. Appendix

Figures (3)

  • Figure 1: Numerical solution of exact finite size integral Y-system equations for the Konishi dimension $\Delta_K(\lambda)$ in a wide range of 't Hooft couplings $\lambda$, compared to the asymptotic Bethe ansatz curve and to the predicted large $\lambda$ asymptotics $\Delta_K(\lambda)\simeq2\lambda^{1/4}+2/\lambda^{1/4}$ obtained by fit.
  • Figure 2: Plot of $\Delta_K(\lambda)-2\lambda^{1/4}$ from the numerical data compared with the Bethe ansatz prediction and some fits. The fits in this plot are done assuming the asymptotics $\Delta_K(\lambda)=2\lambda^{1/4}+2/\lambda^{1/4}+\dots$.
  • Figure 3: T-shaped domain (T-hook) Kazakov:2007fy. It defines the interactions between $Y$'s in the AdS/CFT $Y$-system equations.