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Generalized cusp in AdS_4 x CP^3 and more one-loop results from semiclassical strings

V. Forini, V. Giangreco M. Puletti, O. Ohlsson Sax

TL;DR

The paper develops exact one-loop quantization for macroscopic strings in AdS_4 × CP^3 and related AdS_3 backgrounds, focusing on Wilson-line-like configurations such as antiparallel lines and generalized cusps in ABJM theory. By evaluating worldsheet determinants and exploiting decoupled fluctuation sectors, it yields explicit expressions for the one-loop corrections to the quark-antiquark potential and cusp anomalous dimension, including detailed small-angle expansions and checks against known strong-coupling limits and scaling functions. It extends the analysis to folded strings in AdS_3 subspaces, deriving long- and short-string expansions and confirming reciprocity patterns, thereby connecting semiclassical string results to both AdS_4/CFT_3 and AdS_3/CFT_2 frameworks and informing potential links to integrability via algebraic-curve approaches. The work also discusses the AdS_3 × S^3 × T^4 limit and the implications for h(λ) and Bethe equations, highlighting regularization subtleties and cross-checks with known AdS_5 × S^5 results, and providing high-precision integral representations for numerical validation.

Abstract

We evaluate the exact one-loop partition function for fundamental strings whose world-surface ends on a cusp at the boundary of AdS_4 and has a "jump" in CP^3. This allows us to extract the stringy prediction for the ABJM generalized cusp anomalous dimension Gamma_{cusp}^{ABJM} (phi,theta) up to NLO in sigma-model perturbation theory. With a similar analysis, we present the exact partition functions for folded closed string solutions moving in the AdS_3 parts of AdS_4 x CP^3 and AdS_3 x S^3 x S^3 x S^1 backgrounds. Results are obtained applying to the string solutions relevant for the AdS_4/CFT_3 and AdS_3/CFT_2 correspondence the tools previously developed for their AdS_5 x S^5 counterparts.

Generalized cusp in AdS_4 x CP^3 and more one-loop results from semiclassical strings

TL;DR

The paper develops exact one-loop quantization for macroscopic strings in AdS_4 × CP^3 and related AdS_3 backgrounds, focusing on Wilson-line-like configurations such as antiparallel lines and generalized cusps in ABJM theory. By evaluating worldsheet determinants and exploiting decoupled fluctuation sectors, it yields explicit expressions for the one-loop corrections to the quark-antiquark potential and cusp anomalous dimension, including detailed small-angle expansions and checks against known strong-coupling limits and scaling functions. It extends the analysis to folded strings in AdS_3 subspaces, deriving long- and short-string expansions and confirming reciprocity patterns, thereby connecting semiclassical string results to both AdS_4/CFT_3 and AdS_3/CFT_2 frameworks and informing potential links to integrability via algebraic-curve approaches. The work also discusses the AdS_3 × S^3 × T^4 limit and the implications for h(λ) and Bethe equations, highlighting regularization subtleties and cross-checks with known AdS_5 × S^5 results, and providing high-precision integral representations for numerical validation.

Abstract

We evaluate the exact one-loop partition function for fundamental strings whose world-surface ends on a cusp at the boundary of AdS_4 and has a "jump" in CP^3. This allows us to extract the stringy prediction for the ABJM generalized cusp anomalous dimension Gamma_{cusp}^{ABJM} (phi,theta) up to NLO in sigma-model perturbation theory. With a similar analysis, we present the exact partition functions for folded closed string solutions moving in the AdS_3 parts of AdS_4 x CP^3 and AdS_3 x S^3 x S^3 x S^1 backgrounds. Results are obtained applying to the string solutions relevant for the AdS_4/CFT_3 and AdS_3/CFT_2 correspondence the tools previously developed for their AdS_5 x S^5 counterparts.

Paper Structure

This paper contains 35 sections, 199 equations, 2 figures.

Table of Contents

  1. Overview
  2. Exact one-loop results in $\textup{AdS}_4 \times\mathbb{C} \textup{P}^3$
  3. Antiparallel lines
  4. Generalized cusp
  5. The classical limit
  6. Fluctuations and one-loop correction to $\Gamma_{\textrm{cusp}}^{\textrm{ABJM}}$
  7. Folded string
  8. Long and short string limits
  9. Exact one-loop results for folded strings in $\textup{AdS}_3 \times \textup{S}^3 \times \textup{S}^3\times \textup{S}^1$
  10. Fluctuations
  11. Complete partition function, large and short string limits
  12. Long string limit.
  13. Short string limit.
  14. The $\textup{AdS}_3\times \textup{S}^3\times \textup{T}^4$ limit
  15. Notation and $\Gamma$ matrices representation
  16. ...and 20 more sections

Figures (2)

  • Figure 1: Plot of the partiton function for $\theta=0$. The straight purple line shows the small $k$ expansion \ref{['expansiontheta0']} and the red dots show the result of numerically evaluating \ref{['reg_effectiveaction_theta0']}.
  • Figure 2: Plot of the partiton function for $\phi=0$. The straight purple line shows the small $k$ expansion \ref{['expansionphi0']} and the red dots show the result of numerically evaluating \ref{['reg_effectiveaction_phi0']}.