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Wall-Crossing in Supersymmetric Gauge Theories
High Energy Physics - Theory · Submitted 11 Jun 2012
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We find perfect agreement between the GMN prediction and the first-principles result.
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We study $\mathcal{N}=2$ supersymmetric Yang--Mills theory in four dimensions and then compactify it on $\mathbb{R}^{3}\times S^{1}$. The gauge symmetry of the theory is broken by a vacuum expectation value of the scalar field, which parametrises the moduli space. The spectrum of BPS states, carrying electric and magnetic charges, is piece-wise constant, changing only when the vacuum expectation value crosses the so-called walls of marginal stability. Kontsevich and Soibelman proposed an algebraic construction relating BPS spectra on both sides of a wall of marginal stability. These formulae are known to correctly relate the strong- and weak-coupling spectra in theories with gauge group SU(2) with and without fundamental flavours; we generalise this result to gauge group SU(n) without flavours in the weak-coupling regime. In addition, we find the walls of marginal stability in the SU(n) theory at the root of the Higgs branch and, employing the wall-crossing formula, determine the BPS spectrum in all regions of the moduli space. Gaiotto, Moore, and Neitzke (GMN) proposed an ansatz expressing the moduli space metric of $\mathcal{N}=2$ theory on $\mathbb{R}^{3}\times S^{1}$ in terms of a set of integral equations. Using the GMN ansatz, we find perturbative and instanton corrections in $\mathbb{R}^{3}\times S^{1}$ for gauge group SU(2) with or without flavours and for gauge group SU(n) without flavours. For gauge group SU(n), we also demonstrate that the predicted two-instanton metric is continuous across the walls. Then, we calculate instanton corrections from first principles. We find perfect agreement between the GMN prediction and the first-principles result. We also take the limit of small radius of the compactified dimension finding one- and two-instanton corrections in three dimensions, recovering some of the previously known semiclassical results.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study $\mathcal{N}=2$ supersymmetric Yang--Mills theory in four dimensions and then compactify it on $\mathbb{R}^{3}\times S^{1}$. | contribution 0.62 | 0.82 | 0.02 | asserted 0.94 |
| s1 | The gauge symmetry of the theory is broken by a vacuum expectation value of the scalar field, which parametrises the moduli space. | background 1.00 | 0.10 | 0.02 | asserted 1.00 |
| s2 | The spectrum of BPS states, carrying electric and magnetic charges, is piece-wise constant, changing only when the vacuum expectation value crosses the so-called walls of marginal stability. | background 0.99 | 0.10 | 0.07 | asserted 1.00 |
| s3 | Kontsevich and Soibelman proposed an algebraic construction relating BPS spectra on both sides of a wall of marginal stability. | background 1.00 | 0.03 | 0.05 | asserted 0.98 |
| s4 | These formulae are known to correctly relate the strong- and weak-coupling spectra in theories with gauge group SU(2) with and without fundamental flavours; we generalise this result to gauge group SU(n) without flavours in the weak-coupling regime. | contribution 0.88 | 0.78 | 0.28 | asserted 1.00 |
| s5 | In addition, we find the walls of marginal stability in the SU(n) theory at the root of the Higgs branch and, employing the wall-crossing formula, determine the BPS spectrum in all regions of the moduli space. | result 0.54 | 0.97 | 0.25 | asserted 1.00 |
| s6 | Gaiotto, Moore, and Neitzke (GMN) proposed an ansatz expressing the moduli space metric of $\mathcal{N}=2$ theory on $\mathbb{R}^{3}\times S^{1}$ in terms of a set of integral equations. | background 0.92 | 0.03 | 0.04 | asserted 0.99 |
| s7 | Using the GMN ansatz, we find perturbative and instanton corrections in $\mathbb{R}^{3}\times S^{1}$ for gauge group SU(2) with or without flavours and for gauge group SU(n) without flavours. | contribution 0.56 | 0.97 | 0.11 | asserted 1.00 |
| s8 | For gauge group SU(n), we also demonstrate that the predicted two-instanton metric is continuous across the walls. | result 0.99 | 0.94 | 0.11 | asserted 1.00 |
| s9 | Then, we calculate instanton corrections from first principles. | method 0.51 | 0.96 | 0.05 | asserted 0.98 |
| s10 | We find perfect agreement between the GMN prediction and the first-principles result. | result 1.00 | 0.95 | 0.27 | asserted 1.00 |
| s11 | We also take the limit of small radius of the compactified dimension finding one- and two-instanton corrections in three dimensions, recovering some of the previously known semiclassical results. | result 0.75 | 0.96 | 0.07 | asserted 1.00 |
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Original abstract
We study $\mathcal{N}=2$ supersymmetric Yang--Mills theory in four dimensions and then compactify it on $\mathbb{R}^{3}\times S^{1}$. The gauge symmetry of the theory is broken by a vacuum expectation value of the scalar field, which parametrises the moduli space. The spectrum of BPS states, carrying electric and magnetic charges, is piece-wise constant, changing only when the vacuum expectation value crosses the so-called walls of marginal stability. Kontsevich and Soibelman proposed an algebraic construction relating BPS spectra on both sides of a wall of marginal stability. These formulae are known to correctly relate the strong- and weak-coupling spectra in theories with gauge group SU(2) with and without fundamental flavours; we generalise this result to gauge group SU(n) without flavours in the weak-coupling regime. In addition, we find the walls of marginal stability in the SU(n) theory at the root of the Higgs branch and, employing the wall-crossing formula, determine the BPS spectrum in all regions of the moduli space. Gaiotto, Moore, and Neitzke (GMN) proposed an ansatz expressing the moduli space metric of $\mathcal{N}=2$ theory on $\mathbb{R}^{3}\times S^{1}$ in terms of a set of integral equations. Using the GMN ansatz, we find perturbative and instanton corrections in $\mathbb{R}^{3}\times S^{1}$ for gauge group SU(2) with or without flavours and for gauge group SU(n) without flavours. For gauge group SU(n), we also demonstrate that the predicted two-instanton metric is continuous across the walls. Then, we calculate instanton corrections from first principles. We find perfect agreement between the GMN prediction and the first-principles result. We also take the limit of small radius of the compactified dimension finding one- and two-instanton corrections in three dimensions, recovering some of the previously known semiclassical results.
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What this paper does with the arXiv papers it cites
What it cites · 151 sentences without a confident label
Builds on 4 papers
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The recursion relation ([ref]) should be generalised to [ref]
Four-dimensional wall-crossing via three-dimensional field theory · Wall-crossing in SU(2) theories · 0.96
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Let us generalise our results for gauge group $SU(2)$ to any gauge group $SU(n)$, whose weak-coupling spectrum was found in [ref] .
On the weak coupling spectrum of N=2 supersymmetric SU(n) gauge theory · SU(n) theory · 0.86
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The same effect leads to the non-cancelling one-loop factor $ℛ$ in the three-dimensional instanton calculation of [ref] , and we find a similar effect in the present case of $ℝ^3×S^1$.
Instantons, Three-Dimensional Gauge Theory, and the Atiyah-Hitchin Manifold · Semiclassical instanton calculation · 0.88
Uses the method or tool of 13 papers
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In [ref] , it was shown that this condition is satisfied by the following choice of coordinates in ([ref]):
Four-dimensional wall-crossing via three-dimensional field theory · Hyper-Kähler description · 0.84
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Our goal is to express the spectra and decays discussed above in terms of Kontsevich–Soibelman operators [ref] and show that the wall-crossing formulae are satisfied.
Stability structures, motivic Donaldson-Thomas invariants and cluster transformations · Wall-crossing in higher-rank theories · 0.91
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Let us now show how to dualise [ref] the low-energy action ([ref]), following [ref] .
Duality in N=2 SUSY SU(2) Yang-Mills Theory: A pedagogical introduction to the work of Seiberg and Witten · Moduli space and duality · 0.89
Uses data or values from 2 papers
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Let us begin by writing out the wall-crossing formula implied by the known spectra of the $SU(3)$ theory [ref] on either side of the walls of marginal stability (this equation will be proven shortly):
On the weak coupling spectrum of N=2 supersymmetric SU(n) gauge theory · SU(3) theory · 0.84
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This result allows one to recover the wall of marginal stability: the curve in the complex $u$ plane is approximately (up to $10^-2$), although not exactly, an ellipse [ref] with axes $[-1,1]$ and $≈[-0.86,0.86]$ (figure [ref]).
The Strong-Coupling Spectrum of the Seiberg-Witten Theory · Moduli space and duality · 0.82
Compares with 8 papers
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Our result confirms that the weak-coupling BPS spectrum for the $SU(n)$ theory found in [ref] is correct.
On the weak coupling spectrum of N=2 supersymmetric SU(n) gauge theory · SU(n) theory · 0.85
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Having matched the one-loop factor derived semiclassically ([ref]) and the corresponding expression in the GMN expansion ([ref], [ref]), we conclude that the one-instanton metric ([ref]) reproduces the first-principles result in [ref] .
Instantons, Three Dimensional Gauge Theories and Monopole Moduli Spaces · Interpolating to three dimensions · 0.89
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(29) and (34) in [ref] obtained from the direct semiclassical analysis of three-dimensional instantons, we see that we have correctly reproduced the four-fermion vertex.
Instantons, Three-Dimensional Gauge Theory, and the Atiyah-Hitchin Manifold · Interpolating to three dimensions · 1.00
Cites as support 1 paper
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As has been shown in [ref] , the resulting four-fermion action implies that the moduli space of the theory is, in fact, the Atiyah–Hitchin manifold [ref] .
Instantons, Three-Dimensional Gauge Theory, and the Atiyah-Hitchin Manifold · Interpolating to three dimensions · 0.97
Cites as background 16 papers
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The special case of the $𝒩=2$ theory with gauge group $SU(2)$ was solved by Seiberg and Witten [ref] .
Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD · Gauge symmetry breaking in N=2 theory · 0.92
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The special case of the $𝒩=2$ theory with gauge group $SU(2)$ was solved by Seiberg and Witten [ref] .
Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory · Gauge symmetry breaking in N=2 theory · 0.92
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Kontsevich and Soibelman conjectured an exact formula relating the BPS spectra on both sides of any wall of marginal stability [ref] .
Stability structures, motivic Donaldson-Thomas invariants and cluster transformations · Hyper-Kähler description · 0.85
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