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Bethe ansatz and fluctuations in SU(3) Yang-Mills operators

Lisa Freyhult

TL;DR

This work uses the Bethe ansatz to compute gauge-theory anomalous dimensions for SU(3)-like operators dual to circular strings with three angular momenta in S^5, focusing on the case of two equal spins. By formulating integral Bethe equations in the J'≤J representation and introducing fluctuations through moved Bethe roots, the authors derive the 1/L and 1/L^2 corrections to the anomalous dimension and show that the resulting energy matches the quantum-corrected string energy for the corresponding semiclassical solution. A half-filling condition emerges naturally, and a detailed analysis of the fluctuation positions reveals massless and tachyonic modes depending on the winding parameters, clarifying the stability structure of the spectrum. The results constitute a nontrivial check of AdS/CFT at the quantum level and illuminate how discrete spin-chain fluctuations encode the string's quantum corrections, complementing related SU(3) spin-chain continuum descriptions.

Abstract

We consider the scalar operators corresponding to semiclassical string states in AdS_5xS^5 with the three angular momenta in S^5 non-trivial. The string states recieve quantum corrections and we study the corresponding process on the gauge theory side. The anomalous dimension of the scalar operators is computed using the Bethe ansatz and we find the correction that corresponds to the energy of the quantized string. We restrict for simplicity to the case where two of the angular momenta in S^5 are equal.

Bethe ansatz and fluctuations in SU(3) Yang-Mills operators

TL;DR

This work uses the Bethe ansatz to compute gauge-theory anomalous dimensions for SU(3)-like operators dual to circular strings with three angular momenta in S^5, focusing on the case of two equal spins. By formulating integral Bethe equations in the J'≤J representation and introducing fluctuations through moved Bethe roots, the authors derive the 1/L and 1/L^2 corrections to the anomalous dimension and show that the resulting energy matches the quantum-corrected string energy for the corresponding semiclassical solution. A half-filling condition emerges naturally, and a detailed analysis of the fluctuation positions reveals massless and tachyonic modes depending on the winding parameters, clarifying the stability structure of the spectrum. The results constitute a nontrivial check of AdS/CFT at the quantum level and illuminate how discrete spin-chain fluctuations encode the string's quantum corrections, complementing related SU(3) spin-chain continuum descriptions.

Abstract

We consider the scalar operators corresponding to semiclassical string states in AdS_5xS^5 with the three angular momenta in S^5 non-trivial. The string states recieve quantum corrections and we study the corresponding process on the gauge theory side. The anomalous dimension of the scalar operators is computed using the Bethe ansatz and we find the correction that corresponds to the energy of the quantized string. We restrict for simplicity to the case where two of the angular momenta in S^5 are equal.

Paper Structure

This paper contains 8 sections, 59 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Gauge theory computations
  3. The integral Bethe equations in the representation $J'\leq J$
  4. The positions of the moved roots
  5. The positions of the fluctuations corresponding to the first type of roots
  6. The positions of the fluctuations corresponding to the second type of roots
  7. Contribution to $\gamma$ from the modified densities
  8. Conclusions

Figures (4)

  • Figure 1: The distribution of roots in the complex plane.
  • Figure 2: Two roots of the first type is moved to the real axis. $\mu$ could be located anywhere on the real line except at the intersection with $C_+$.
  • Figure 3: Two roots of the second type is moved to the real axis.
  • Figure 4: The stability of the solution will depend on the winding number.