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Y-system and Quasi-Classical Strings

Nikolay Gromov

TL;DR

This paper tests the AdS/CFT Y-system conjecture by solving it in the strong-coupling scaling limit and directly comparing to the quasi-classical string spectrum in AdS$_3$×S$^1$ within the $ rak{sl}(2)$ subsector. It derives explicit asymptotic $Y$-functions, energy and momentum expressions, and corrected Bethe equations that incorporate all wrapping effects, showing a precise match with the one-loop string energies. This provides strong evidence that the Y-system captures the full planar spectrum beyond the asymptotic Bethe Ansatz. By connecting the algebraic-curve method with the Y-system, including finite-gap and back-reaction analyses, the work lays groundwork for extending these exact techniques to broader sectors and finite coupling. The results highlight the potential of the Y-system as a complete, non-perturbative framework for AdS/CFT spectra and finite-size corrections.

Abstract

Recently Kazakov, Vieira and the author conjectured the Y-system set of equations describing the planar spectrum of AdS/CFT. In this paper we solve the Y-system equations in the strong coupling scaling limit. We show that the quasi-classical spectrum of string moving inside AdS3 x S1 matches precisely with the prediction of the Y-system. Thus the Y-system, unlike the asymptotic Bethe ansatz, describes correctly the spectrum of one-loop string energies including all exponential finite size corrections. This gives a very non-trivial further support in favor of the conjecture.

Y-system and Quasi-Classical Strings

TL;DR

This paper tests the AdS/CFT Y-system conjecture by solving it in the strong-coupling scaling limit and directly comparing to the quasi-classical string spectrum in AdS×S within the subsector. It derives explicit asymptotic -functions, energy and momentum expressions, and corrected Bethe equations that incorporate all wrapping effects, showing a precise match with the one-loop string energies. This provides strong evidence that the Y-system captures the full planar spectrum beyond the asymptotic Bethe Ansatz. By connecting the algebraic-curve method with the Y-system, including finite-gap and back-reaction analyses, the work lays groundwork for extending these exact techniques to broader sectors and finite coupling. The results highlight the potential of the Y-system as a complete, non-perturbative framework for AdS/CFT spectra and finite-size corrections.

Abstract

Recently Kazakov, Vieira and the author conjectured the Y-system set of equations describing the planar spectrum of AdS/CFT. In this paper we solve the Y-system equations in the strong coupling scaling limit. We show that the quasi-classical spectrum of string moving inside AdS3 x S1 matches precisely with the prediction of the Y-system. Thus the Y-system, unlike the asymptotic Bethe ansatz, describes correctly the spectrum of one-loop string energies including all exponential finite size corrections. This gives a very non-trivial further support in favor of the conjecture.

Paper Structure

This paper contains 18 sections, 105 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $Y$-system in the scaling limit
  3. Infinite length solution at strong coupling
  4. Boundary conditions
  5. Y-system in $T$-hook
  6. Solution of Hirota equation in the vertical strip
  7. Matching wings
  8. Energy and Momentum
  9. Exact Bethe ansatz equations
  10. Finite gap solutions
  11. Quasi-classical string quantization
  12. Classical algebraic curve
  13. Algebraic curve for $\frak{sl}(2)$ subsector
  14. Quasi-classical corrections from algebraic curve
  15. Back-reaction
  16. ...and 3 more sections

Figures (2)

  • Figure 1: T-shaped "fat hook". It defines the interactions between $Y$-functions in the AdS/CFT $Y$-system equations.
  • Figure 2: T-shaped "fat hook". Small black filled circles correspond to the finite $T_{as}$ function. Big circles show where solution for $Y$-functions for the strips is applicable.