Convergence of Nekrasov instanton sum with adjoint matter

Abstract

The Nekrasov instanton partition function of the 4d N=2^* U(N) gauge theory (a mass deformation of 4d 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. This note proves convergence of the series in the unit disk |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=ϵ₁/ϵ₂ of equivariant parameters is in ℂ∖[0,+∞), the radius is 1, as expected. If b² is non-negative, three cases arise: the radius is finite if b² has finite exponential type (a generalization of Brjuno numbers), namely there exists C>0 such that |b²−p/q|>exp(−Cq) for all integers p,q≠ 0; the series diverges if b² is super-exponentially well approximable by rationals; and if b² 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 algebras with non-real b, within the unit disk. For the Virasoro algebra this corresponds to a central charge in ℂ∖[25,+∞).

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BibTeX RIS

BibTeX

@article{floch2026convergence,
  title = {{Convergence of Nekrasov instanton sum with adjoint matter}},
  author = {Bruno Le Floch},
  journal = {SciPost Physics Core},
  year = {2026},
  month = jun,
  doi = {10.21468/scipostphyscore.9.2.038},
  eprint = {2602.19425},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05848469v1},
  note = {CC BY 4.0},
}

RIS

TY  - JOUR
TI  - Convergence of Nekrasov instanton sum with adjoint matter
AU  - Bruno Le Floch
PY  - 2026
DA  - 2026/06/25
JO  - SciPost Physics Core
DO  - 10.21468/scipostphyscore.9.2.038
UR  - https://hal.science/hal-05848469v1
AB  - The Nekrasov instanton partition function of the 4d N=2^* U(N) gauge theory (a mass deformation of 4d 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. This note proves convergence of the series in the unit disk |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 of equivariant parameters is in ℂ∖[0,+∞), the radius is 1, as expected. If b^2 is non-negative, three cases arise: the radius is finite if b^2 has finite exponential type (a generalization of Brjuno numbers), namely there exists C>0 such that |b^2−p/q|>exp(−Cq) for all integers p,q≠ 0; the series diverges if b^2 is super-exponentially well approximable by rationals; and if 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 algebras with non-real b, within the unit disk. For the Virasoro algebra this corresponds to a central charge in ℂ∖[25,+∞).
SN  - 2666-9366
ER  -

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