Finite ε2ε2-corrections to the N=2 SYMN=2 SYM prepotential

We derive the first ε2ε2-correction of the instanton partition functions in 4D N=2N=2 Super Yang–Mills (SYM) to the Nekrasov–Shatashvili limit ε2→0ε2→0. In the latter we recall the emergence of the famous Thermodynamic Bethe Ansatz-like equation which has been found by Mayer expansion techniques. Here we combine efficiently these to field theory arguments. In a nutshell, we find natural and resolutive the introduction of a new operator ∇ that distinguishes the singularities within and outside the integration contour of the partition function.