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arxiv: 2606.31953 · v1 · pith:5ZTV4XWZnew · submitted 2026-06-30 · ✦ hep-th

Convergence of Nekrasov instanton sum for unitary quivers

Bruno Le Floch This is my paper

Pith reviewed 2026-07-01 04:02 UTC · model grok-4.3

classification ✦ hep-th
keywords Nekrasov partition functioninstanton sumquiver gauge theoryconvergence radiusAGT correspondenceconformal blockshook lengthsequivariant parameters
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The pith

Nekrasov instanton sums for unitary quivers converge absolutely inside a positive-radius disk for generic parameters.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the Nekrasov partition function for 4d N=2 quiver gauge theories with unitary gauge groups can be expressed as a sum over tuples of partitions that converges absolutely when the instanton counting parameters are small enough. This holds throughout an open dense subset of the space of Coulomb parameters, masses, and equivariant parameters. For U(N) SQCD the condition is that the ratio b squared of the equivariant parameters lies outside the nonnegative reals and the other parameters avoid a lattice of hyperplanes; for general quivers the condition is simply that b squared is not real. In asymptotically free theories the radius is infinite, while in the conformal case the radius has a positive lower bound that depends only on b squared. The result also transfers through the AGT correspondence to a statement that large classes of Virasoro and W-algebra conformal blocks converge in a disk for generic dimensions and complex central charges.

Core claim

The convergence radius of Nekrasov partition functions is positive for 4d N=2 quiver gauge theories with unitary gauge groups in an open dense subset of parameters. For U(N) SQCD this holds when b squared belongs to the complement of the nonnegative reals and Coulomb parameters or masses avoid a lattice of hyperplanes; for general quivers it holds when b squared is not real. When the theories are asymptotically free the radius is infinite, whereas in the mass-deformed conformal case the radius admits a positive lower bound depending only on b squared. The proof proceeds from the explicit expression of the partition function as a sum over tuples of partitions and applies combinatorial inequal

What carries the argument

The explicit sum over tuples of partitions whose terms are bounded by combinatorial inequalities on products of hook and cohook lengths, which supplies the estimates needed to prove absolute convergence of the series.

If this is right

  • The Nekrasov partition function defines a holomorphic function of the instanton parameters inside a disk of positive radius around the origin.
  • In asymptotically free theories the sum converges for every value of the instanton parameters.
  • In the conformal case the radius of convergence is bounded below by a positive number that depends only on the ratio of equivariant parameters.
  • Via the AGT correspondence, large classes of Virasoro and W-algebra conformal blocks on the sphere or torus converge in a disk for generic dimensions and complex central charges.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same style of hook-length estimates might be adapted to obtain convergence statements for other classes of gauge theories once an analogous partition sum is available.
  • Inside the region of convergence the instanton sum can be differentiated or integrated term by term with respect to the Coulomb parameters or masses without further justification.
  • The lower bound on the radius in the conformal case supplies a uniform control that could be used to compare the instanton expansion with other perturbative expansions at the same point in parameter space.

Load-bearing premise

The individual terms in the sum over partitions can be bounded by combinatorial inequalities on products of hook and cohook lengths that are strong enough to guarantee absolute convergence inside the claimed parameter region.

What would settle it

An explicit choice of parameters with b squared not real, away from the lattice of hyperplanes, for which the ratio of successive terms in the sum over partitions fails to tend to zero for arbitrarily small values of the instanton counting parameters.

Figures

Figures reproduced from arXiv: 2606.31953 by Bruno Le Floch.

Figure 1
Figure 1. Figure 1: Linear, circular, and (an example) E-type quiver gauge theories with [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the Nekrasov instanton partition function for 4d N=2 quiver gauge theories with unitary gauge groups has positive convergence radius in the instanton counting parameters, for an open dense subset of the remaining parameters. For U(N) SQCD the result holds when b² = ε₁/ε₂ lies in C s [0, +∞) and Coulomb parameters/masses avoid a lattice of hyperplanes; for general quivers it holds when b² is non-real. The radius is infinite when the theory is asymptotically free and admits a positive lower bound depending only on b² in the (mass-deformed) conformal case. The argument expresses the partition function as an explicit sum over tuples of partitions and establishes absolute convergence via combinatorial inequalities on products of hook and cohook lengths. Via AGT this yields positive-radius convergence for large classes of Virasoro/W-algebra conformal blocks on the sphere or torus.

Significance. If the stated combinatorial bounds close the argument, the result supplies a rigorous, direct proof of convergence for Nekrasov functions in a broad, physically relevant parameter region, distinguishing the asymptotically free and conformal regimes. The AGT implication for conformal-block convergence at generic dimensions and complex central charges is a concrete payoff. The approach is self-contained and does not rely on fitted quantities or external conjectures.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'only depends on b²' for the conformal lower bound would be clearer if it explicitly noted independence from the ranks N_i and the matter content (or stated the precise dependence).
  2. The lattice of hyperplanes excluded for the Coulomb parameters and masses is described only qualitatively; a brief explicit characterization (e.g., in terms of the weight lattice) would aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments appear in the provided report, so there are no specific points requiring a point-by-point response or manuscript changes.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation expresses the Nekrasov partition function explicitly as a sum over tuples of partitions and proves absolute convergence via direct combinatorial inequalities on products of hook and cohook lengths. These bounds are established independently of the target radius result and do not rely on fitted parameters, self-referential definitions, or load-bearing self-citations. The argument is self-contained against the explicit series representation and external combinatorial facts.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two background facts: the known expression of the Nekrasov function as a sum over partitions, and the validity of the combinatorial hook-length inequalities used to bound the terms. Both are invoked directly in the abstract as the basis of the proof.

axioms (2)
  • domain assumption The Nekrasov partition function admits an explicit expression as a sum over tuples of partitions.
    Invoked in the abstract as the starting point for the convergence argument.
  • ad hoc to paper Combinatorial inequalities on products of (co)hook lengths are strong enough to produce absolute convergence inside the claimed open dense set of parameters.
    This is the load-bearing step of the proof described in the abstract.

pith-pipeline@v0.9.1-grok · 5736 in / 1455 out tokens · 55473 ms · 2026-07-01T04:02:07.475592+00:00 · methodology

discussion (0)

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