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Quantum corrections to short folded superstring in AdS_3 x S^3 x M^4

Matteo Beccaria, Guido Macorini

TL;DR

The work advances understanding of quantum corrections to a short folded string in AdS3×S3×M4 by performing both world-sheet perturbation theory and algebraic-curve quantization, and by comparing with all-loop Bethe Ansatz approaches. A key result is that the dressing-energy component E1^{dressing} is independent of the radii ratio α to leading order, though a regularization mismatch in AdS3 complicates a clean matching with ABA predictions, especially in the T^4 limit. The analysis shows that, while one- and two-loop weak-coupling terms align with string theory under an interpolating coupling h(λ), the non-analytic dressing terms require a proper dressing phase and potentially refinements of the discrete Bethe Ansatz in AdS3. Overall, the paper clarifies how integrability-based methods relate to semiclassical string spectra in less supersymmetric backgrounds and highlights where further refinements are needed to achieve a complete match between approaches.

Abstract

We consider integrable superstring theory on AdS_3 x S^3 x M^4 where M^4=T^4 or M^4=S^3 x S^1 with generic ratio of the radii of the two 3-spheres. We compute the one-loop energy of a short folded string spinning in AdS_3 and rotating in S^3. The computation is performed by world-sheet small spin perturbation theory as well as by quantizing the classical algebraic curve characterizing the finite-gap equations. The two methods give equal results up to regularization contributions that are under control. One important byproduct of the calculation is the part of the energy which is due to the dressing phase in the Bethe Ansatz. Remarkably, this contribution E_1^{dressing} turns out to be independent on the radii ratio. In the M^4=T^4 limit, we discuss how E_1^{dressing} relates to a recent proposal for the dressing phase tested in the su(2) sector. We point out some difficulties suggesting that quantization of the AdS_3 classical finite-gap equations could be subtler than the easier AdS_5 x S^5 case.

Quantum corrections to short folded superstring in AdS_3 x S^3 x M^4

TL;DR

The work advances understanding of quantum corrections to a short folded string in AdS3×S3×M4 by performing both world-sheet perturbation theory and algebraic-curve quantization, and by comparing with all-loop Bethe Ansatz approaches. A key result is that the dressing-energy component E1^{dressing} is independent of the radii ratio α to leading order, though a regularization mismatch in AdS3 complicates a clean matching with ABA predictions, especially in the T^4 limit. The analysis shows that, while one- and two-loop weak-coupling terms align with string theory under an interpolating coupling h(λ), the non-analytic dressing terms require a proper dressing phase and potentially refinements of the discrete Bethe Ansatz in AdS3. Overall, the paper clarifies how integrability-based methods relate to semiclassical string spectra in less supersymmetric backgrounds and highlights where further refinements are needed to achieve a complete match between approaches.

Abstract

We consider integrable superstring theory on AdS_3 x S^3 x M^4 where M^4=T^4 or M^4=S^3 x S^1 with generic ratio of the radii of the two 3-spheres. We compute the one-loop energy of a short folded string spinning in AdS_3 and rotating in S^3. The computation is performed by world-sheet small spin perturbation theory as well as by quantizing the classical algebraic curve characterizing the finite-gap equations. The two methods give equal results up to regularization contributions that are under control. One important byproduct of the calculation is the part of the energy which is due to the dressing phase in the Bethe Ansatz. Remarkably, this contribution E_1^{dressing} turns out to be independent on the radii ratio. In the M^4=T^4 limit, we discuss how E_1^{dressing} relates to a recent proposal for the dressing phase tested in the su(2) sector. We point out some difficulties suggesting that quantization of the AdS_3 classical finite-gap equations could be subtler than the easier AdS_5 x S^5 case.

Paper Structure

This paper contains 39 sections, 159 equations.

Table of Contents

  1. Introduction and summary
  2. Folded string in $AdS_{3}\times S^{3}\times S^{1}\times S^{1}$
  3. Classical solution
  4. One-loop correction to the energy: World-sheet computation
  5. Bosonic quadratic fluctuations
  6. Fermionic quadratic fluctuations
  7. Short string expansion of frequencies in the flat space limit
  8. $\mathcal{O}({\cal S})$ frequencies and one-loop energy
  9. $\mathcal{O}({\cal S}^{2})$ frequencies and one-loop energy
  10. Large ${\cal J}$ expansion and non-analytic contributions
  11. Algebraic curve quantization in $AdS_{3}\times S^{3}\times T^{4}$
  12. Classical data
  13. Algebraic curve one-loop quantization and comparison with world-sheet
  14. Dressing from the algebraic curve
  15. Matching of AC and WS regularizations
  16. ...and 24 more sections