The Nekrasov instanton partition function of the 4d N=2^* 𝒩 = 2 * U (N) U (N) gauge theory (a mass deformation of 4d N=4 𝒩 = 4 super-Yang-Mills theory), which is a generating series of equivariant integrals over instanton moduli spaces, is given by a sum over colored partitions weighted by a counting parameter q q. This note proves convergence of the series in the unit disk |q| | q | 1 for generic parameters. Specifically, the absolute convergence radius of this sum is determined, assuming that mass and Coulomb branch parameters avoid some lattice. If the ratio b²=₁/₂ b 2 = ϵ 1 / ϵ 2 of equivariant parameters is in C[0, +∞) ℂ \ [ 0, + ∞), the radius is 1 1, as expected. If b² b 2 is non-negative, three cases arise: the radius is finite if b² b 2 has finite exponential type (a generalization of Brjuno numbers), namely there exists C>0 C > 0 such that |b²-p/q|> (-Cq) | b 2 − p / q | > exp (− C q) for all integers p, q≠ 0 <mml: math xmlns: mml="http: /
Bruno Le Floch (Thu,) studied this question.