$AdS_5\times S^5$ mirror TBA equations from Y-system and discontinuity relations

Using the recently proposed set of discontinuity relations we translate the AdS/CFT Y-system to TBA integral equations and quantization conditions for a large subset of excited states from the sl(2) sector of the $AdS_5 \times S^5$ string sigma-model. Our derivation provides an analytic proof of the fact that the exact Bethe equations reduce to the Beisert-Staudacher equations in the asymptotic limit. We also construct the corresponding T-system and show that in the language of T-functions the energy formula reduces to a single term which depends on a single T-function.