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Convergence of Nekrasov instanton sum with adjoint matter

Bruno Le Floch

Abstract

The Nekrasov instanton partition function of the 4d $\mathcal{N}=2^*$ $U(N)$ gauge theory (a mass deformation of 4d $\mathcal{N}=4$ 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 $\mathfrak{q}$. This note proves convergence of the series in the unit disk $|\mathfrak{q}|<1$ 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 $\mathbb{C}\setminus[0,+\infty)$, 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 $p,q\neq 0$; 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 $\mathbb{C}\setminus[25,+\infty)$.

Convergence of Nekrasov instanton sum with adjoint matter

Abstract

The Nekrasov instanton partition function of the 4d gauge theory (a mass deformation of 4d 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 . This note proves convergence of the series in the unit disk 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 of equivariant parameters is in , the radius is , as expected. If is non-negative, three cases arise: the radius is finite if has finite exponential type (a generalization of Brjuno numbers), namely there exists such that for all integers ; the series diverges if is super-exponentially well approximable by rationals; and if is rational some terms are singular. The AGT correspondence translates these results to convergence of torus one-point conformal blocks of the Virasoro and algebras with non-real , within the unit disk. For the Virasoro algebra this corresponds to a central charge in .
Paper Structure (56 sections, 5 theorems, 130 equations)

This paper contains 56 sections, 5 theorems, 130 equations.

Table of Contents

  1. Introduction and main result
  2. Instanton partition functions
  3. Absolute convergence radius
  4. On absolute and conditional convergence.
  5. Convergence techniques.
  6. Conformal blocks
  7. Strategy and basic upper bound
  8. Formulas and notation
  9. Convergence/divergence strategy
  10. Divergence beyond the unit disk
  11. A choice that works generically.
  12. When reordering ensures the extra genericity.
  13. A different configuration.
  14. Case distinction.
  15. Optimal convergence
  16. ...and 41 more sections

Key Result

Theorem 1.1

Assume that none of the differences $a_I-a_J$ for $1\leq I<J\leq N$, nor the adjoint mass $m$, are in the closureThe lattice $\epsilon_1\mathbb{Z}+\epsilon_2\mathbb{Z}$ is discrete if $b^2\in\mathbb{C}\setminus\mathbb{R}$ or $b^2\in\mathbb{Q}$, but is dense in the line $\epsilon_1\mathbb{R}=\epsilon

Theorems & Definitions (14)

  • Theorem 1.1: On the 4d $\mathcal{N}=2^*$ instanton partition function
  • Lemma 4.1: Equidistribution of irrational orbits
  • proof
  • Lemma 4.2
  • proof
  • Definition 4.3
  • Remark 4.4
  • Proposition 4.5
  • proof
  • Proposition 4.6
  • ...and 4 more