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Generalized scaling function from light-cone gauge AdS_5 x S^5 superstring

S. Giombi, R. Ricci, R. Roiban, A. A. Tseytlin, C. Vergu

TL;DR

The paper computes the first nontrivial two-loop correction to the generalized scaling function $f(\ell,\lambda)$ for fast-spinning strings in $AdS_5\times S^5$ using the AdS light-cone gauge around a generalized null cusp background. By performing a careful 2-loop fluctuation analysis, including bosonic and fermionic sectors and tadpoles, the authors obtain explicit expressions for $f_2(\ell)$ and its leading logarithms, showing exact agreement with the asymptotic Bethe Ansatz (BES phase) for $\hat{m}=0$ and with related $O(6)$ model data. This provides a stringent confirmation of the BA proposal at strong coupling and demonstrates the ultraviolet finiteness and quantum integrability of the AdS5xS5 superstring in this gauge. The results yield the leading and subleading $\ell$-dependent terms in the $1/\sqrt{\lambda}$ expansion, including all-order leading-log structure, and pave the way for further higher-loop and finite-size investigations in the light-cone framework.

Abstract

We revisit the computation of the 2-loop correction to the energy of a folded spinning string in AdS_5 with an angular momentum J in S^5 in the scaling limit log S, J >>1 with J / log S fixed. This correction gives the third term in the strong-coupling expansion of the generalized scaling function. The computation, using the AdS light-cone gauge approach developed in our previous paper, is done by expanding the AdS_5 x S^5 superstring partition function near the generalized null cusp world surface associated to the spinning string solution. The result corrects and extends the previous conformal gauge result of arXiv:0712.2479 and is found to be in complete agreement with the corresponding terms in the generalized scaling function as obtained from the asymptotic Bethe ansatz in arXiv:0805.4615 (and also partially from the quantum O(6) model and the Bethe ansatz data in arXiv:0809.4952). This provides a highly nontrivial strong coupling comparison of the Bethe ansatz proposal with the quantum AdS_5 x S^5 superstring theory, which goes beyond the leading semiclassical term effectively controlled by the underlying algebraic curve. The 2-loop computation we perform involves all the structures in the AdS light-cone gauge superstring action of hep-th/0009171 and thus tests its ultraviolet finiteness and, through the agreement with the Bethe ansatz, its quantum integrability. We do most of the computations for a generalized spinning string solution or the corresponding null cusp surface that involves both the orbital momentum and the winding in a large circle of S^5.

Generalized scaling function from light-cone gauge AdS_5 x S^5 superstring

TL;DR

The paper computes the first nontrivial two-loop correction to the generalized scaling function for fast-spinning strings in using the AdS light-cone gauge around a generalized null cusp background. By performing a careful 2-loop fluctuation analysis, including bosonic and fermionic sectors and tadpoles, the authors obtain explicit expressions for and its leading logarithms, showing exact agreement with the asymptotic Bethe Ansatz (BES phase) for and with related model data. This provides a stringent confirmation of the BA proposal at strong coupling and demonstrates the ultraviolet finiteness and quantum integrability of the AdS5xS5 superstring in this gauge. The results yield the leading and subleading -dependent terms in the expansion, including all-order leading-log structure, and pave the way for further higher-loop and finite-size investigations in the light-cone framework.

Abstract

We revisit the computation of the 2-loop correction to the energy of a folded spinning string in AdS_5 with an angular momentum J in S^5 in the scaling limit log S, J >>1 with J / log S fixed. This correction gives the third term in the strong-coupling expansion of the generalized scaling function. The computation, using the AdS light-cone gauge approach developed in our previous paper, is done by expanding the AdS_5 x S^5 superstring partition function near the generalized null cusp world surface associated to the spinning string solution. The result corrects and extends the previous conformal gauge result of arXiv:0712.2479 and is found to be in complete agreement with the corresponding terms in the generalized scaling function as obtained from the asymptotic Bethe ansatz in arXiv:0805.4615 (and also partially from the quantum O(6) model and the Bethe ansatz data in arXiv:0809.4952). This provides a highly nontrivial strong coupling comparison of the Bethe ansatz proposal with the quantum AdS_5 x S^5 superstring theory, which goes beyond the leading semiclassical term effectively controlled by the underlying algebraic curve. The 2-loop computation we perform involves all the structures in the AdS light-cone gauge superstring action of hep-th/0009171 and thus tests its ultraviolet finiteness and, through the agreement with the Bethe ansatz, its quantum integrability. We do most of the computations for a generalized spinning string solution or the corresponding null cusp surface that involves both the orbital momentum and the winding in a large circle of S^5.

Paper Structure

This paper contains 28 sections, 197 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Generalized scaling function from string partition function
  3. String action in AdS light-cone gauge, generalized null cusp solution and closed spinning string with winding in $S^5$
  4. Action
  5. Generalized null cusp solution
  6. Closed string spinning in $AdS_3$ with momentum and winding in $S^1 \subset S^5$
  7. Equivalence between the generalized null cusp and the large spin limit of the closed string solution
  8. Expansion of the string action near the classical solution and the 1-loop partition function
  9. Expansion of the action
  10. Quadratic part of the action
  11. One-loop partition function
  12. Two-loop partition function
  13. Bosonic Sunset
  14. Bosonic double-bubble
  15. Fermionic sunset
  16. ...and 13 more sections

Figures (4)

  • Figure 1: Profile of an arc of fast-spinning string in $AdS_3$ wound around a big circle of $S^5$.
  • Figure 2: The 2-loop 1PI topologies: "sunset" and "double-bubble". The propagators here are either bosonic or fermionic.
  • Figure 3: The 2-loop tadpole topology. The non-vanishing graphs have the internal line corresponding to a $\tilde{\phi}$-propagator while the propagators in the loops can be either bosonic or fermionic.
  • Figure 4: The two contributions to the 1-loop expectation value of $J^{56}$.