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Anomalous dimensions in the symmetric orbifold

Matthias R. Gaberdiel, Felix Lichtner, Beat Nairz

TL;DR

The paper provides an independent validation of the large-$w$ anomalous-dimension spectrum for quarter-BPS states in the symmetric orbifold by performing direct second-order perturbation theory via both four-point and three-point function methods. Using covering-map techniques and a detailed account of intermediate states, the authors reproduce the Gaberdiel et al. predictions and clarify the role of higher-magnon intermediates, demonstrating consistency with the integrable structures associated with the AdS$_3$/CFT$_2$ dual. A key outcome is that including higher-magnon sectors, and the associated tails, is essential to obtain the correct spectrum, while a reduced two- and four-magnon subspace already captures the leading large-$w$ behavior. The work also furnishes a practical Mathematica toolkit for automating these calculations for arbitrary quarter-BPS states, enabling broad application to similar perturbative analyses in symmetric orbifolds.

Abstract

Recently, the anomalous conformal dimensions of the symmetric orbifold under the $2$-cycle twisted sector deformation were calculated using the perturbed action of the supercharges. In particular, explicit and simple formulae for the dispersion relations of the torus magnons in the $w$-cycle twisted sector were derived for large $w$. In this paper we reproduce these results from a direct perturbed $2$-point function calculation. In the process we also develop techniques (and a Mathematica code) that allows one to do these calculations for arbitrary quarter BPS states at finite $w$.

Anomalous dimensions in the symmetric orbifold

TL;DR

The paper provides an independent validation of the large- anomalous-dimension spectrum for quarter-BPS states in the symmetric orbifold by performing direct second-order perturbation theory via both four-point and three-point function methods. Using covering-map techniques and a detailed account of intermediate states, the authors reproduce the Gaberdiel et al. predictions and clarify the role of higher-magnon intermediates, demonstrating consistency with the integrable structures associated with the AdS/CFT dual. A key outcome is that including higher-magnon sectors, and the associated tails, is essential to obtain the correct spectrum, while a reduced two- and four-magnon subspace already captures the leading large- behavior. The work also furnishes a practical Mathematica toolkit for automating these calculations for arbitrary quarter-BPS states, enabling broad application to similar perturbative analyses in symmetric orbifolds.

Abstract

Recently, the anomalous conformal dimensions of the symmetric orbifold under the -cycle twisted sector deformation were calculated using the perturbed action of the supercharges. In particular, explicit and simple formulae for the dispersion relations of the torus magnons in the -cycle twisted sector were derived for large . In this paper we reproduce these results from a direct perturbed -point function calculation. In the process we also develop techniques (and a Mathematica code) that allows one to do these calculations for arbitrary quarter BPS states at finite .

Paper Structure

This paper contains 22 sections, 67 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The approach of Gaberdiel:2023lco
  3. The direct calculation
  4. The four-point function calculation
  5. The method of calculation
  6. The covering map
  7. The anomalous dimension expressed through the covering map
  8. Two-magnon external states
  9. The full anomalous dimension matrix
  10. The three-point function calculation
  11. The method of calculation
  12. Intermediate states
  13. Consistency checks
  14. Algebraic structure of the eigenstates
  15. Conclusions
  16. ...and 7 more sections

Figures (2)

  • Figure 1: The anomalous dimension associated to the two-magnon states as a function of $m$ for $w=16$. Shown are the diagonal elements $\bra{\Psi_m}\tilde{\gamma}\ket{\Psi_m}$ (blue diamonds), the eigenvalues of the full eigenstates with the largest overlap with the respective two-magnon state $\ket{\Psi_m}$ (orange dots), and the (scaled) predicted spectrum for large $w$, see eq. (\ref{['eq:int_anomalous_dimension']}) (dashed green line).
  • Figure 2: The coefficients $\mathcal{N}_{8,m}^\text{old}\,c^\text{old}(k;m)$ (blue diamonds) and $\mathcal{N}_{8,m}\,c(k;m)$ (orange circles) for the eigenstates associated to $\alpha^2_{-m/8}\alpha^2_{-1+m/8}\ket{8}_-$ and $m=1,2,3,4$ from top left to bottom right.