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Quantum corrections to spinning superstrings in AdS_3 x S^3 x M^4: determining the dressing phase

M. Beccaria, F. Levkovich-Maslyuk, G. Macorini, A. A. Tseytlin

TL;DR

This work demonstrates that the leading quantum correction to the dressing phase for strings in AdS_3 × S^3 × M^4 with RR flux deviates from the BES phase known from AdS_5 × S^5. By combining a concrete SU(2) circular-string calculation with an algebraic-curve (finite-gap) analysis, the authors derive antisymmetric 1-loop dressing-phase coefficients that reproduce the correct non-analytic part of the 1-loop energy and reconcile the ABA with string theory in the AdS_3 setting. They show a regularization ambiguity in the algebraic-curve approach can be resolved by enforcing antisymmetry, effectively selecting the standard regularization; this yields consistent results for both SU(2) circular strings and long folded strings, and extends to the alpha-dependent AdS_3 × S^3 × S^3 × S^1 background. A notable outcome is that the 1-loop shift in the effective string tension h(λ) decouples from the non-analytic dressing contributions in the AdS_3 × S^3 × T^4 limit, while remaining nontrivial for the S^3 × S^3 × S^1 theory. Overall, the results refine the integrability-based S-matrix structure for low-dimensional AdS/CFT and provide explicit coefficients guiding ABA-based spectral equations in these backgrounds.

Abstract

We study the leading quantum string correction to the dressing phase in the asymptotic Bethe Ansatz system for superstring in AdS_3 x S^3 x T^4 supported by RR flux. We find that the phase should be different from the BES phase appearing in the AdS_5 x S^5 case. We use the simplest example of a rigid circular string with two equal spins in S^3 and also consider the general approach based on the algebraic curve description. We also discuss the case of the AdS_3 x S^3 x S^3 x S^1 theory and find the dependence of the 1-loop correction to the effective string tension function h(λ) (expected to enter the magnon dispersion relation) on the parameters alpha related to the ratio of the two 3-sphere radii. This correction vanishes in the AdS_3 x S^3 x T^4 case.

Quantum corrections to spinning superstrings in AdS_3 x S^3 x M^4: determining the dressing phase

TL;DR

This work demonstrates that the leading quantum correction to the dressing phase for strings in AdS_3 × S^3 × M^4 with RR flux deviates from the BES phase known from AdS_5 × S^5. By combining a concrete SU(2) circular-string calculation with an algebraic-curve (finite-gap) analysis, the authors derive antisymmetric 1-loop dressing-phase coefficients that reproduce the correct non-analytic part of the 1-loop energy and reconcile the ABA with string theory in the AdS_3 setting. They show a regularization ambiguity in the algebraic-curve approach can be resolved by enforcing antisymmetry, effectively selecting the standard regularization; this yields consistent results for both SU(2) circular strings and long folded strings, and extends to the alpha-dependent AdS_3 × S^3 × S^3 × S^1 background. A notable outcome is that the 1-loop shift in the effective string tension h(λ) decouples from the non-analytic dressing contributions in the AdS_3 × S^3 × T^4 limit, while remaining nontrivial for the S^3 × S^3 × S^1 theory. Overall, the results refine the integrability-based S-matrix structure for low-dimensional AdS/CFT and provide explicit coefficients guiding ABA-based spectral equations in these backgrounds.

Abstract

We study the leading quantum string correction to the dressing phase in the asymptotic Bethe Ansatz system for superstring in AdS_3 x S^3 x T^4 supported by RR flux. We find that the phase should be different from the BES phase appearing in the AdS_5 x S^5 case. We use the simplest example of a rigid circular string with two equal spins in S^3 and also consider the general approach based on the algebraic curve description. We also discuss the case of the AdS_3 x S^3 x S^3 x S^1 theory and find the dependence of the 1-loop correction to the effective string tension function h(λ) (expected to enter the magnon dispersion relation) on the parameters alpha related to the ratio of the two 3-sphere radii. This correction vanishes in the AdS_3 x S^3 x T^4 case.

Paper Structure

This paper contains 24 sections, 148 equations, 2 figures.

Table of Contents

  1. Introduction and summary
  2. Asymptotic Bethe Ansatz equations for $AdS_{3} \times S^{3} \times T^{4}$ model
  3. One-loop correction to energy of circular string with two equal spins in $S^{3}$: fixing leading quantum term in the dressing phase
  4. Non-analytic term in one-loop string energy
  5. Constraining the dressing phase in $SU(2)$ sector
  6. Semiclassical dressing phase from the algebraic curve approach
  7. Scaling limit of the Bethe equations and finite-gap equations
  8. Quasi-momenta and algebraic curve
  9. Semiclassical one-loop dressing factor
  10. On consistency condition on the phase
  11. Algebraic curve approach applied to $SU(2)$ circular string case: regularization ambiguity
  12. Non-analytic part of one-loop energy
  13. Regularization origin of the mismatch
  14. Proposal for the 1-loop dressing phase coefficients from the algebraic curve approach
  15. Non-analytic term in the 1-loop energy of $SU(2)$ circular string in $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$
  16. ...and 9 more sections

Figures (2)

  • Figure 1: Setup for computation of $\mathcal{V}$.
  • Figure 2: Contour defining the AC regularization