We apply the recently proposed quantum spectral curve technique to the study of twist operators in planar N = 4 SYM theory. We focus on the small spin expansion of anomalous dimensions in the sl(2) sector and compute its first two orders exactly for any value of the 't Hooft coupling. At leading order in the spin S we reproduced Basso's slope function. The next term of order S 2 structurally resembles the Beisert-Eden-Staudacher dressing phase and takes into account wrapping contributions. This expansion contains rich information about the spectrum of local operators at strong coupling. In particular, we found a new coefficient in the strong coupling expansion of the Konishi operator dimension and confirmed several previously known terms. We also obtained several new orders of the strong coupling expansion of the BFKL pomeron intercept. As a by-product we formulated a prescription for the correct analytical continuation in S which opens a way for deriving the BFKL regime of twist two anomalous dimensions from AdS/CFT integrability.

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We consider the classical limit of the recently obtained exact near-BPS result for the anomalous dimension of a cusped Wilson line with the insertion of an operator with L units of R-charge at the cusp in planar N = 4 SYM. The classical limit requires taking both the 't Hooft coupling and L to infinity. Since the formula for the cusp anomalous dimension involves determinants of size proportional to L, the classical limit requires a matrix model reformulation of the result. Building on results of Gromov and Sever, we construct such a matrix model-like representation and find the corresponding classical algebraic curve. Using this we find the classical value of the cusp anomalous dimension and the 1-loop correction to it. We check our results against the energy of the classical solution and numerically by extrapolating from the quantum regime of finite L.

In this paper we generalize the results of arXiv:0806.4758 to non-zero value J of angular momentum in S 5 . We compute the 1-loop correction to the energy of the folded spinning string in AdS 5 × S 5 in the particular limit of slow short string approximation. In this limit the string is moving in a near-flat central region of AdS 5 slowly rotating in both AdS 5 and S 5 . The one-loop correction should represent the first subleading correction to strong coupling expansion of the anomalous dimension of short gauge theory operators of the form Tr D S Φ J in the SL(2) sector.

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