Convergence of Nekrasov instanton sum with adjoint matter

The Nekrasov instanton partition function of the 4d

U(N) U ( N )

gauge theory (a mass deformation of 4d

\mathcal{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=\epsilon_1/\epsilon_2 b 2 = ϵ 1 / ϵ 2

of equivariant parameters is in

{\mathbb{C}\setminus[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 p , q ≠ 0

; the series diverges if

b^2 b 2

is super-exponentially well approximable by rationals; and if

b^2 b 2

is rational some terms are singular. The AGT correspondence translates these results to convergence of torus one-point conformal blocks of the Virasoro and

W_N W N

algebras with non-real

b b

, within the unit disk. For the Virasoro algebra this corresponds to a central charge in

\mathbb{C}\setminus[25,+∞) ℂ \ [ 25 , + ∞ )

.