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**1 - 4**of**4**### PREPARED FOR SUBMISSION TO JHEP On a discrete symmetry of the Bremsstrahlung function in N = 4 SYM

"... ABSTRACT: We consider the quark anti-quark potential on the three sphere in planar N = 4 SYM and the associated vacuum potential in the near BPS limit with L units of R-charge. The associated Bremsstrahlung function B L has been recently computed ana-lytically by means of the Thermodynamical Bethe A ..."

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ABSTRACT: We consider the quark anti-quark potential on the three sphere in planar N = 4 SYM and the associated vacuum potential in the near BPS limit with L units of R-charge. The associated Bremsstrahlung function B L has been recently computed ana-lytically by means of the Thermodynamical Bethe Ansatz. We discuss it at strong coupling by computing it at large but finite L. We provide strong support to a special symmetry of the Bremsstrahlung function under the formal discrete Z 2 symmetry L! 1 L. In this context, it is the counterpart of the reciprocity invariance discovered in the past in the spectrum of various gauge invariant composite operators. The Z 2 symmetry has remark-able consequences in the scaling limit where L is taken to be large with fixed ratio to the ’t Hooft coupling. This limit organizes in inverse powers of the coupling and resembles the semiclassical expansion of the dual string theory which is indeed known to capture the leading classical term. We show that the various higher-order contributions to the Bremsstrahlung function obey several constraints and, in particular, the next-to-leading term, formally associated with the string one-loop correction, is completely determined by the classical contribution. The large L limit at strong coupling is also discussed.ar X iv

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"... Abstract: We give a derivation of quantum spectral curve (QSC)- a finite set of Riemann-Hilbert equations for exact spectrum of planar N = 4 SYM theory proposed in our recent paper Phys.Rev.Lett.112 (2014). We also generalize this construction to all local single trace operators of the theory, in co ..."

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Abstract: We give a derivation of quantum spectral curve (QSC)- a finite set of Riemann-Hilbert equations for exact spectrum of planar N = 4 SYM theory proposed in our recent paper Phys.Rev.Lett.112 (2014). We also generalize this construction to all local single trace operators of the theory, in contrast to the TBA-like approaches worked out only for a limited class of states. We reveal a rich algebraic and analytic structure of the QSC in terms of a so called Q-system – a finite set of Baxter-like Q-functions. This new point of view on the finite size spectral problem is shown to be completely compatible, though in a far from trivial way, with already known exact equations (analytic Y-system/TBA, or FiNLIE). We use the knowledge of this underlying Q-system to demonstrate how the classical finite gap solutions and the asymptotic Bethe ansatz emerge from our formalism in appropriate limits.a rX iv:1

### PREPARED FOR SUBMISSION TO JHEP On the one-loop curvature function in the sl(2) sector of N = 4 SYM

"... ABSTRACT: We consider twist J operators with spin S in the sl(2) sector of N = 4 SYM. The small spin expansion of their anomalous dimension defines the so-called slope func-tions. Much is known about the linear term, but the study of the quadratic correction, the curvature function, started only ver ..."

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ABSTRACT: We consider twist J operators with spin S in the sl(2) sector of N = 4 SYM. The small spin expansion of their anomalous dimension defines the so-called slope func-tions. Much is known about the linear term, but the study of the quadratic correction, the curvature function, started only very recently. At any fixed J, the curvature function can be extracted at all loops from the Pµ-system formulation of the Thermodynamical Bethe Ansatz. Here, we work at the one-loop level and follow a different approach. We present a systematic double expansion of the Bethe Ansatz equations at large J and small wind-ing number. We succeed in fully resumming this expansion and obtain a closed explicit simple formula for the one-loop curvature function. The formula is parametric in J and can be evaluated with minor effort for any fixed J. The result is an explicit series in odd-index ζ values. Our approach provides a complete reconciliation between the Pµ-system predictions and the large J approach. ar X iv