Strong-coupling expansion of cusp anomaly from quantum superstring

We consider the world surface in AdS₅ that ends on two intersecting null lines at the boundary. The corresponding superstring 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(ł) of the gauge coupling ł or the string tension √λ/2π. The first two coefficients in its strong-coupling or string inverse tension expansion were determined in hep-th/0210115 (a₀ = 1) and in arXiv:0707.4254 (a₁ = − 3 ln 2). Here we find that the 2-loop coefficient is a₂ = −K where K is the Catalan's constant. This is in agreement (expected on the general grounds) with the previous results for f(ł) as the coefficient of ln S term in the energy of the closed spinning string in AdS₅. The string theory value for a₂ is in agreement with the numerical result in hep-th/0611135 and the recent analytic result in arXiv:0708.3933 for the coefficients in strong-coupling Ssolution of the BES equation. We explicitly verify the cancellation of all 2-loop logarithmic divergences thus demonstrating the quantum consistency of the AdS₅×S⁵ superstring action at this order. We also discuss the structure of the three and higher string loop corrections to the cusp anomaly function giving a 2d QFT diagrammatic interpretation to the result of arXiv:0708.3933 for the solution of the BES equation following from the Bethe ansatz prescription for the spectrum of the theory.