Convergence of Nekrasov instanton sum with adjoint matter

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|<1 | 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 b^2=ε_1/ε_2 b 2 = ϵ 1 / ϵ 2 of equivariant parameters is in {C[0,+∞)} ℂ \ [ 0 , + ∞ ) , the radius is 1 1 , as expected. If b^2 b 2 is non-negative, three cases arise: the radius is finite if b^2 b 2 has finite exponential type (a generalization of Brjuno numbers), namely there exists C>0 C > 0 such that |b^2-p/q|>exp(-Cq) | b 2 − p / q | > exp ⁡ ( − C q ) for all integers p,q≠ 0 <mml:math xmlns:mml="http:/