Thermodynamics and scattering in 2D for 4D theories

Considering classical open string solutions with a null polygonal contour at the boundary of AdS3, the geometric problem of minimal surfaces related to gluon scattering amplitudes at strong coupling is developed and solved in a set of TBA-like integral equations. Their form resembles that of a TBA system whose free energy yields the dynamic contribution to the area. The corresponding set of functional relations, the so called Y-system, is derived and extensively analyzed in the cases of octagon and decagon together with the set of integral equations. Useful rewrites of the latter establish the connection with a particular class of HSG models and allow us to identify their Y-system (related to a universal TBA) with the one derived in AdS3 when the algebra level k is set to 2. A useful insight that clarifies how to properly reduce to the An series is offered. Thanks to a well defined change of reference frame, whose mathematical structure has been studied in detail, the dynamic part of the remainder function coincides with the extremum of the Yang-Yang functional for the modified TBA equations. Explicit examples have been treated in terms of Y-functions and pseudo-energies, extrapolating useful links with several works concerning null WLs, Hitchin systems and integrable perturbations of CFTs corresponding to Gk-parafermions.