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Notes on Wall Crossing and Instanton in Compactified Gauge Theory with Matter
High Energy Physics - Theory · Submitted 30 Jun 2010
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We verify the predicted metric by computing the leading weak coupling instanton contribution to the four fermion correlation using standard field theory techniques, and demonstrate perfect agreement.
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The abstract, sentence by sentence
We study the quantum effects on the Coulomb branch of N=2 SU(2) supersymmetric Yang-Mills with fundamental matters compactified on R^3 x S^1, and extract the explicit perturbative and leading non-perturbative corrections to the moduli space metric predicted from the recent work of Gaiotto, Moore and Neitzke on wall-crossing [1]. We verify the predicted metric by computing the leading weak coupling instanton contribution to the four fermion correlation using standard field theory techniques, and demonstrate perfect agreement. We also demonstrate how previously known three dimensional quantities can be recovered in appropriate small radius limit, and provide a simple geometric picture from brane construction.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the quantum effects on the Coulomb branch of N=2 SU(2) supersymmetric Yang-Mills with fundamental matters compactified on R^3 x S^1, and extract the explicit perturbative and leading non-perturbative corrections to the moduli space metric predicted from the recent work of Gaiotto, Moore and Neitzke on wall-crossing [1]. | contribution 0.99 | 0.92 | 0.24 | asserted 1.00 |
| s1 | We verify the predicted metric by computing the leading weak coupling instanton contribution to the four fermion correlation using standard field theory techniques, and demonstrate perfect agreement. | result 0.79 | 0.96 | 0.59 | asserted 1.00 |
| s2 | We also demonstrate how previously known three dimensional quantities can be recovered in appropriate small radius limit, and provide a simple geometric picture from brane construction. | contribution 0.67 | 0.93 | 0.08 | asserted 1.00 |
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The paper
Original abstract
We study the quantum effects on the Coulomb branch of N=2 SU(2) supersymmetric Yang-Mills with fundamental matters compactified on R^3 x S^1, and extract the explicit perturbative and leading non-perturbative corrections to the moduli space metric predicted from the recent work of Gaiotto, Moore and Neitzke on wall-crossing [1]. We verify the predicted metric by computing the leading weak coupling instanton contribution to the four fermion correlation using standard field theory techniques, and demonstrate perfect agreement. We also demonstrate how previously known three dimensional quantities can be recovered in appropriate small radius limit, and provide a simple geometric picture from brane construction.
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How the literature uses it
3 arXiv papers cite this in 3 sentences
What it cites · 75 sentences without a confident label
Builds on 1 paper
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To extract the explicit prediction for the quantum corrected moduli space metric from ([ref]), here we generalize the iterative weak coupling expansion, first done in [ref] for the pure $SU(2)$ case, to the current case with massive flavors.
Wall Crossing and Instantons in Compactified Gauge Theory · Semiclassical Limit of the Wall-Crossing Formula with Matters · 0.99
Uses the method or tool of 3 papers
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We followed the conversion procedures in [ref] between $(X^i, Ω^i)$, the bosonic $(a,z)$, and the fermionic $(λ,ψ)$ fields in the dimensionally reduced actions ([ref]) and ([ref]) [Note that the $g_eff$ in [ref] differs from ours by $√(2)$, due to the different normalization for the complex gauge coupling $τ_eff$ ([ref]) for the flavor case.].
Wall Crossing and Instantons in Compactified Gauge Theory · Semiclassical Limit of the Wall-Crossing Formula with Matters · 0.98
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To do so, we follow [ref] to define the following four dimensional fluctuation operators for the massive fundamental hypermultiplets in the monopole background: [equation] Here the three dimensional covariant derivative $D⃗=∂⃗+ i A⃗^Mon.$ is with respect to background static monopole in $A_0=0$ gauge.
Instanton Effects in Three-Dimensional Supersymmetric Gauge Theories with Matter · Semiclassical Instanton Calculation with Matters · 0.96
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The trace in ([ref]) can be evaluated analogously following the steps in [ref] for monopole of charge $n_m$, the result is: [equation] where in writing out $_H(m_i)$ we have also used the fact $Im(a/m_i)=0$.
Magnetic Monopole Dynamics, Supersymmetry, and Duality · Semiclassical Instanton Calculation with Matters · 0.87
Compares with 4 papers
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The resultant four-fermion correlator corresponds to the appearance of the following four-fermion interaction vertex in the low-energy effective action: [equation] This exactly matches the prediction obtained from the integral equations of [ref] given in ([ref]).
Four-dimensional wall-crossing via three-dimensional field theory · Semiclassical Instanton Calculation with Matters · 0.98
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This, after recalling the gauge coupling and electric charge for the W-boson, precisely matches with the four fermion vertex computed from first principles in [ref] and [ref] .
Instantons, Three-Dimensional Gauge Theory, and the Atiyah-Hitchin Manifold · Recovering Three Dimensional Quantities · 0.99
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This, after recalling the gauge coupling and electric charge for the W-boson, precisely matches with the four fermion vertex computed from first principles in [ref] and [ref] .
Instanton Effects in Three-Dimensional Supersymmetric Gauge Theories with Matter · Recovering Three Dimensional Quantities · 0.99
Cites as background 8 papers
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Recently significant progress has been made in the work of Gaiotto, Moore and Neitzke towards a systematic determination of the BPS spectra for four dimensional $=2$ supersymmetric gauge theories [ref] .
Four-dimensional wall-crossing via three-dimensional field theory · Wall Crossing Formula, Moduli Space and BPS Spectrum with Matter · 0.96
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Recently significant progress has been made in the work of Gaiotto, Moore and Neitzke towards a systematic determination of the BPS spectra for four dimensional $=2$ supersymmetric gauge theories [ref] .
Wall-crossing, Hitchin Systems, and the WKB Approximation · Wall Crossing Formula, Moduli Space and BPS Spectrum with Matter · 0.96
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A major triumph in this area is the understanding of exact Coulomb branch metrics for four dimensional $=2$ supersymmetric Yang-Mills, provided by the famous work of Seiberg and Witten [ref] .
Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD · Introduction · 0.98
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