Convergence of Nekrasov instanton sum for unitary quivers

Abstract

The convergence radius of Nekrasov partition functions (as a function of instanton counting parameters) is shown to be positive for 4d $\mathcal{N}=2$ quiver gauge theories with unitary gauge groups in an open dense subset of parameters. For $U(N)$ SQCD this is established if the ratio of equivariant parameters $b^2=\epsilon_1/\epsilon_2$ belongs to $\mathbb{C}\setminus[0,+\infty)$ and Coulomb parameters or masses are away from a lattice of hyperplanes. For general quivers it is only established for $b^2\in\mathbb{C}\setminus\mathbb{R}$. When gauge multiplets are asymptotically free, the radius is infinite, whereas in the (mass-deformed) conformal case the radius admits a positive lower bound that only depends on $b^2$. The proof relies on the expression of the partition function as a sum over tuples of partitions, and a proof of absolute convergence based on combinatorial inequalities on products of (co)hook lengths. Through the AGT correspondence this implies that large classes of Virasoro and W-algebra conformal blocks on the sphere or torus have positive convergence radius, for generic dimensions and complex central charges.