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Strong-coupling expansion of cusp anomaly from quantum superstring

R. Roiban, A. A. Tseytlin

TL;DR

This work performs a detailed two-loop computation of the cusp anomaly function f(λ) in the AdS5×S5 context by quantizing fluctuations around the null cusp string background. Using a T-dual Green–Schwarz action and a 2D regularization that preserves κ-symmetry, the authors find a2 = -K, in exact agreement with BES-equation predictions and earlier string-theory results, while confirming UV finiteness and quantum consistency of the AdS5×S5 superstring. The paper also interprets f(λ) within a 2D quantum field theory framework on the worldsheet and outlines how higher-loop diagrams contribute (notably a likely ζ(3) term at 3 loops), reinforcing the connection between string theory and integrability-based BES results. Finally, it discusses implications for the summability of the strong-coupling expansion and suggests directions for generalizations, including non-zero angular momentum in S^5, and the issue of background stability.

Abstract

We consider the world surface in AdS_5 that ends on two intersecting null lines at the boundary. The corresponding string partition function describes the expectation value of the Wilson line with a null cusp in dual large N maximally supersymmetric gauge theory and thus determines the cusp anomaly function f(λ) of the gauge coupling. The first two coefficients in its strong-coupling or string inverse tension expansion were determined in hep-th/0210115 (a_1=1) and in arXiv:0707.4254 (a_2=- 3 log 2). Here we find that the 2-loop coefficient is a_2 = - K where K is the Catalan's constant. This is in agreement (expected on general grounds) with the previous results for f(λ) as the coefficient of log(S) term in the energy of closed spinning string in AdS_5. The string theory value for a_2 is in agreement with the numerical result in hep-th/0611135 and the recent analytic result in arXiv:0708.3933 for the solution of the BES equation following from the asymptotic Bethe ansatz for the spectrum of the theory. We explicitly verify the cancellation of 2-loop 2d logarithmic divergences thus demonstrating the quantum consistency of the AdS_5 x S^5 superstring. We also discuss the structure of higher loop string corrections to the cusp anomaly giving a 2d QFT diagrammatic interpretation to the strong-coupling expansion of the cusp anomaly function as solution of the BES equation found in arXiv:0708.3933.

Strong-coupling expansion of cusp anomaly from quantum superstring

TL;DR

This work performs a detailed two-loop computation of the cusp anomaly function f(λ) in the AdS5×S5 context by quantizing fluctuations around the null cusp string background. Using a T-dual Green–Schwarz action and a 2D regularization that preserves κ-symmetry, the authors find a2 = -K, in exact agreement with BES-equation predictions and earlier string-theory results, while confirming UV finiteness and quantum consistency of the AdS5×S5 superstring. The paper also interprets f(λ) within a 2D quantum field theory framework on the worldsheet and outlines how higher-loop diagrams contribute (notably a likely ζ(3) term at 3 loops), reinforcing the connection between string theory and integrability-based BES results. Finally, it discusses implications for the summability of the strong-coupling expansion and suggests directions for generalizations, including non-zero angular momentum in S^5, and the issue of background stability.

Abstract

We consider the world surface in AdS_5 that ends on two intersecting null lines at the boundary. The corresponding string partition function describes the expectation value of the Wilson line with a null cusp in dual large N maximally supersymmetric gauge theory and thus determines the cusp anomaly function f(λ) of the gauge coupling. The first two coefficients in its strong-coupling or string inverse tension expansion were determined in hep-th/0210115 (a_1=1) and in arXiv:0707.4254 (a_2=- 3 log 2). Here we find that the 2-loop coefficient is a_2 = - K where K is the Catalan's constant. This is in agreement (expected on general grounds) with the previous results for f(λ) as the coefficient of log(S) term in the energy of closed spinning string in AdS_5. The string theory value for a_2 is in agreement with the numerical result in hep-th/0611135 and the recent analytic result in arXiv:0708.3933 for the solution of the BES equation following from the asymptotic Bethe ansatz for the spectrum of the theory. We explicitly verify the cancellation of 2-loop 2d logarithmic divergences thus demonstrating the quantum consistency of the AdS_5 x S^5 superstring. We also discuss the structure of higher loop string corrections to the cusp anomaly giving a 2d QFT diagrammatic interpretation to the strong-coupling expansion of the cusp anomaly function as solution of the BES equation found in arXiv:0708.3933.

Paper Structure

This paper contains 9 sections, 43 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Superstring action, classical string background and fluctuations
  3. "Null cusp" solution
  4. Fluctuation Lagrangian
  5. Quantum corrections
  6. One-loop contribution
  7. Two-loop contribution
  8. Higher-loop contributions
  9. Concluding remarks

Figures (3)

  • Figure 1: Two-loop diagrams (bosonic propagators are denoted by solid lines and fermionic ones are denoted by dashed lines).
  • Figure 2: Topologies bosonic three-loop diagrams (diagrams with fermionic lines have similar topology).
  • Figure 3: $L$-loop "maximally irreducible" sunset-type graph