On the Convergence of Nekrasov Functions

In this note, we present some results on the convergence of Nekrasov partition functions as power series in the instanton counting parameter. We focus on U(N) N = 2 gauge theories in four dimensions with matter in the adjoint and in the fundamental representations of the gauge group, respectively, and find rigorous lower bounds for the conver gence radius in the two cases: if the theory is conformal, then the series has at least a finite radius of convergence, while if it is asymptotically free it has infinite radius of convergence. Via AGT correspondence, this im plies that the related irregular conformal blocks of WN algebrae admit a power expansion in the modulus converging in the whole plane. By spec ifying to the SU(2) case, we apply our results to analyze the convergence properties of the corresponding Painlev´e τ -functions