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arxiv: 2604.20674 · v1 · submitted 2026-04-22 · ✦ hep-th · math-ph· math.AG· math.MP

Recognition: unknown

Wall-crossing of Instantons on the Blow-up

Baptiste Filoche, Stefan Hohenegger, Taro Kimura

Authors on Pith no claims yet

Pith reviewed 2026-05-09 23:37 UTC · model grok-4.3

classification ✦ hep-th math-phmath.AGmath.MP
keywords instanton countingblow-upquiver varietysuper-partitionswall-crossingJeffrey-Kirwan residuestability conditionsNakajima-Yoshioka formula
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The pith

Super-partitions select the contributions to the instanton partition function on the blow-up according to stability conditions in each chamber of a quiver variety.

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

This paper develops a method to count instantons in four-dimensional supersymmetric gauge theories on the blow-up of the complex plane by modeling the moduli space as a quiver variety. Two stability parameters divide the space into chambers separated by walls, and within each chamber the partition function is expressed as a contour integral evaluated via the Jeffrey-Kirwan residue prescription. Only certain super-partitions contribute in each chamber, and their selection criteria prove equivalent to stability conditions proposed in earlier literature. The formalism yields explicit expressions for the partition functions in neighboring chambers and recovers the Nakajima-Yoshioka blow-up formula in a limiting chamber.

Core claim

The authors establish that the instanton moduli space on the blow-up of C^2 can be realized as a quiver variety regularized by two stability parameters. This endows the space with infinitely many chambers. Within a given chamber the partition function is a contour integral whose relevant residues correspond to super-partitions obeying chamber-specific selection rules. These rules are equivalent to previously known stability conditions. Crossing walls between chambers produces explicit changes in the counting, and the construction reproduces the Nakajima-Yoshioka formula in a suitable limit.

What carries the argument

The quiver variety with two stability parameters that defines chambers and walls, together with the super-partitions whose selection rules classify the Jeffrey-Kirwan residues and match stability conditions.

If this is right

  • Explicit partition functions for each chamber follow directly from the super-partition classification.
  • Crossing a wall changes the instanton counting by the addition or removal of specific super-partition terms.
  • The construction reproduces the Nakajima-Yoshioka blow-up formula as a limiting case.
  • The same super-partition rules implement the stability conditions previously proposed in the literature.

Where Pith is reading between the lines

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

  • The combinatorial language of super-partitions may supply a systematic way to track wall-crossing in other instanton moduli spaces.
  • The equivalence between residue selection and stability conditions suggests the quiver-variety approach could be tested on related geometries or gauge theories.

Load-bearing premise

The contributions selected by applying the Jeffrey-Kirwan residue prescription to the contour integral of the quiver variety with chosen stability parameters are exactly the physically relevant terms in the instanton partition function.

What would settle it

A direct calculation of the instanton partition function in one specific chamber that includes nonzero contributions from super-partitions violating the stability selection criteria would show the equivalence fails.

Figures

Figures reproduced from arXiv: 2604.20674 by Baptiste Filoche, Stefan Hohenegger, Taro Kimura.

Figure 1
Figure 1. Figure 1: The lines correspond to walls separating the different stability chambers, in the grey [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Graphical depiction of C 2 = C1 × C2 and Cb2 with the marked fixed points of the T-action and C ∼= P 1 is an exceptional divisor of Cb2 given by z1 = z2 = 0. with F the Lie(G)-valued 2-form field strength. The blow-up Cb2 admits an exceptional divisor C ∼= P 1 given by z1 = z2 = 0 depicted in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: On the left, the ADHM quiver for k U(N) instantons on C 2 , with K ∼= C k and N ∼= C N . On the right, the ADHM quiver for k0 +k1 U(N) instantons on Cb2 , with K0,1 ∼= C k0,1 . The real parameter ζ is introduced in (2.7) as a regularisation parameter. This definition does not hinge on the exact value of ζ, but is only sensitive to its sign. This leads to two distinct realisations, namely ζ > 0 and ζ < 0. A… view at source ↗
Figure 4
Figure 4. Figure 4: The space of stability conditions with its chamber structure, the gray area is a region [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Poles picked by the JK prescription for k0, k1 = 0, 1, 2 as functions of the stability parameters ζ0 and ζ1. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Configurations cancelled by the complex moment map constraint are indicated by [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: An example of a super-partition and its graphical representation as a super-Young [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The green dots denote a pair (s0, s1) of dimensions associated with a sub-super￾partition of Ξ2 which satisfies (3.13) strictly, the orange dots denote a similar pair which is an equality case of (3.13), while the red dots violate the inequality; the blue line corresponds to the condition (3.13) for different m. ◦ • ◦ ◦ • • ◦ ◦ • • [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A tree graph in the P-chamber, all nodes can be grouped in pair of one black node and one white node. This property implies that k0 = k1 and we can indeed verify that all graphs with k0 ̸= k1 are unstable in the P-chamber. Indeed, each branch of such tree can be associated with a sequence of maps of the form (d, B1,2, d, B1,2, . . . , d, B1,2). Graphically this guarantees that each white triangle is paired… view at source ↗
Figure 10
Figure 10. Figure 10: A bipartite graph in the SP-chamber differing from a graph in the P-chamber at the branches edges. the generating function of Young diagrams, interpreted as the partition function of 4d U(1) N = 4 theory on C 2 , is given by: ZP(Qτ ) = X λ∈P Q |λ| τ = (Qτ ; Qτ ) −1 ∞ , (3.15) where Qτ is a formal counting parameter with |Qτ | < 1 and (a; q)n the Pochhammer symbol is defined as: (a; q)n := nY−1 k=0 (1 − aq… view at source ↗
Figure 11
Figure 11. Figure 11: The three pieces (k1 − k0, λ+, λ−) of a separable super-partition where the original super-partition is obtained by gluing together the red dots and the blue dots. (k0, k1) (k0, k1) (0, 1) (2, 3) (1, 2) (2, 4) (1, 3) (3, 4) + tr [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: A 1-stable super-partition where red boxes are irrelevant and green boxes are rele￾vant. function on C 2 . The character of the tangent space to the moduli space near the fixed point labelled by Λ can be decomposed as: ch TΛMcN,k0,k1 = − X N α,β=1 e 2iπ(aβ−aα)VΛαΛβ (q1, q2), (4.12) with VΛαΛβ given by: VΛαΛβ := c λ (0) α + q −1 12 c ∨ λ (1) β − c λ (0) α c ∨ λ (0) β − c λ (1) α c ∨ λ (1) β + (q −1 1 + q −… view at source ↗
read the original abstract

We study the instanton counting in four dimensional $\mathcal{N}=2$ supersymmetric gauge theories on the blow-up of $\mathbb{C}^2$: we start by formulating the instanton moduli space as a quiver variety, which we regularise by introducing two stability parameters, thus endowing it with a structure of infinitely many chambers separated by walls. Within a given chamber, we formulate the instanton partition function as a contour integral, which can be evaluated using the Jeffrey-Kirwan residue prescription. We characterise the physically relevant contributions in terms of bipartite oriented graphs and show that they can more efficiently be classified in terms of combinatorial objects called super-partitions. Within a given chamber, only certain types of super-partitions contribute and we show that the corresponding selection criteria are equivalent to stability conditions that have previously been proposed in the literature. We use this formalism to compare how the instanton counting changes when moving across walls between neighbouring chambers and provide explicit expressions for the corresponding partition functions. In a limiting chamber and using our approach, we show how to reproduce the Nakajima-Yoshioka blow-up formula.

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 / 3 minor

Summary. The paper formulates the instanton moduli space on the blow-up of C² as a quiver variety regularized by two stability parameters, which partitions the parameter space into chambers separated by walls. Within each chamber the instanton partition function is expressed as a contour integral evaluated by the Jeffrey-Kirwan residue prescription. Contributions are classified first via bipartite oriented graphs and then more efficiently via super-partitions; explicit selection rules on these super-partitions are derived and shown to be equivalent to previously proposed stability conditions. Wall-crossing formulas between adjacent chambers are obtained by comparing the sets of contributing super-partitions, and the construction recovers the Nakajima-Yoshioka blow-up formula in a limiting chamber.

Significance. If the derivations hold, the work supplies a systematic combinatorial framework that makes the chamber structure and wall-crossing of instanton counting on the blow-up fully explicit. The equivalence between the Jeffrey-Kirwan selection rules and earlier stability conditions, together with the explicit wall-crossing expressions and the reproduction of the Nakajima-Yoshioka formula, constitute concrete technical advances that can be used to test or extend other approaches to BPS counting and moduli-space geometry in four-dimensional N=2 theories.

minor comments (3)
  1. §3.2: the definition of a super-partition is given combinatorially but an explicit low-rank example (e.g., rank 2 with small instanton number) would clarify how the bipartite-graph and super-partition descriptions map onto each other.
  2. §4.1, Eq. (4.3): the contour-integral representation is stated without a brief reminder of the precise Jeffrey-Kirwin residue prescription used; adding one sentence would make the subsequent selection-rule derivations easier to follow for readers outside the immediate subfield.
  3. §5.3: the wall-crossing formula between chambers C and C' is written as a difference of two sums over super-partitions; it would be helpful to include a small numerical check (e.g., first few instanton numbers) that the difference reproduces the expected change in the partition function.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their thorough and positive summary of our manuscript. We are pleased that the combinatorial framework based on super-partitions, the explicit wall-crossing formulas, and the recovery of the Nakajima-Yoshioka blow-up formula have been recognized as concrete technical advances. As no specific major comments or criticisms were raised, we see no need for revisions.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on explicit calculations from standard tools

full rationale

The paper constructs the instanton moduli space on the blow-up as a quiver variety regularized by two stability parameters, yielding chambers and walls. Within each chamber the partition function is expressed as a contour integral evaluated by the Jeffrey-Kirwan residue prescription. Contributions are classified via bipartite graphs and then super-partitions; selection rules are derived combinatorially and shown equivalent to prior stability conditions through direct comparison rather than by definition or fitting. Wall-crossing formulas follow from explicit set differences between adjacent chambers, and the Nakajima-Yoshioka formula is recovered in a limiting case by the same residue evaluation. All load-bearing steps use independent mathematical machinery (quiver varieties, JK residues) whose validity does not depend on the present results, with no self-referential definitions, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the claims to tautologies.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on the standard identification of instanton moduli spaces with quiver varieties, the applicability of the Jeffrey-Kirwan residue theorem to the resulting contour integrals, and the introduction of two stability parameters whose values define the chambers. Super-partitions are presented as a new combinatorial classification tool whose selection rules are shown to match prior stability conditions.

free parameters (1)
  • two stability parameters
    Introduced to regularize the moduli space and partition it into chambers separated by walls; their specific values determine which super-partitions contribute.
axioms (2)
  • domain assumption The instanton moduli space on the blow-up can be formulated as a quiver variety.
    Standard construction in the literature for instanton counting in N=2 theories.
  • domain assumption The Jeffrey-Kirwin residue prescription correctly extracts the physically relevant contributions from the contour integral in each chamber.
    Invoked to evaluate the partition function within a fixed chamber.
invented entities (1)
  • super-partitions no independent evidence
    purpose: Combinatorial objects used to classify and select the contributing terms in the instanton partition function more efficiently than bipartite oriented graphs.
    Introduced or adapted in the paper as the key classification device whose selection criteria are shown equivalent to stability conditions.

pith-pipeline@v0.9.0 · 5497 in / 1842 out tokens · 44192 ms · 2026-05-09T23:37:57.120787+00:00 · methodology

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