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Volume of Moduli Space of Vortex Equations and Localization
High Energy Physics - Theory · Submitted 11 May 2011
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We apply this technique to Abelian (ANO) vortex and show that the volume of moduli space agrees with the previous results obtained by integrating over the moduli space metric.
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The abstract, sentence by sentence
We evaluate volume of moduli space of BPS vortices on a compact Riemann surface by using topological field theory and localization technique developed by Moore, Nekrasov and Shatashvili. We apply this technique to Abelian (ANO) vortex and show that the volume of moduli space agrees with the previous results obtained by integrating over the moduli space metric. We extend the evaluation to non-Abelian gauge groups and multi-flavors. We also compare our results with the volume of the Kahler quotient space inspired by the brane configuration.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We evaluate volume of moduli space of BPS vortices on a compact Riemann surface by using topological field theory and localization technique developed by Moore, Nekrasov and Shatashvili. | contribution 0.72 | 0.93 | 0.21 | asserted 1.00 |
| s1 | We apply this technique to Abelian (ANO) vortex and show that the volume of moduli space agrees with the previous results obtained by integrating over the moduli space metric. | result 0.91 | 0.88 | 0.30 | asserted 1.00 |
| s2 | We extend the evaluation to non-Abelian gauge groups and multi-flavors. | contribution 0.99 | 0.92 | 0.08 | asserted 0.95 |
| s3 | We also compare our results with the volume of the Kahler quotient space inspired by the brane configuration. | contribution 0.66 | 0.83 | 0.08 | asserted 0.82 |
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The paper
Original abstract
We evaluate volume of moduli space of BPS vortices on a compact Riemann surface by using topological field theory and localization technique developed by Moore, Nekrasov and Shatashvili. We apply this technique to Abelian (ANO) vortex and show that the volume of moduli space agrees with the previous results obtained by integrating over the moduli space metric. We extend the evaluation to non-Abelian gauge groups and multi-flavors. We also compare our results with the volume of the Kahler quotient space inspired by the brane configuration.
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References
How the literature uses it
13 arXiv papers cite this in 35 sentences
Papers citing it · 20 sentences without a confident label
Contests it 1 paper
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(4.52) of [ref] by inference from the concrete evaluation for the $N=2,3$ cases, but the conjecture turns out to be in slight disagreement with our present result ([ref]), which shows a slightly different coefficient.
Higgs and Coulomb Branch Descriptions of the Volume of the Vortex Moduli Space · Generating Function · contest 0.74
Builds on it 1 paper
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The generating function allows us to show the reduction for arbitrary values of $N$ and $k$, improving our previous result [ref] .
Higgs and Coulomb Branch Descriptions of the Volume of the Vortex Moduli Space · Introduction · 0.95
Uses its method or tool 2 papers
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We can evaluate the integral ([ref]) directly by using a usual field theoretical procedure as performed in [ref] .
Localization Method for Volume of Domain-Wall Moduli Spaces · Volume of Moduli Space · 0.96
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The purpose of this section is to investigate various relations between the $G/G$ gauged WZW-matter model and the $q$-boson model by utilizing the cohomological localization method in a similar way with [ref] .
G/G gauged WZW-matter model, Bethe Ansatz for q-boson model and Commutative Frobenius algebra · $G/G$ gauged Wess-Zumino-Witten-matter model · 0.98
Compares with it 3 papers
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Surprisingly, they completely agree with the volume of the moduli space of the local vortices on the sphere $S^2$, derived in [ref] , up to an overall normalization and a rescaling to define the moduli space.
Localization Method for Volume of Domain-Wall Moduli Spaces · T-duality to Vortex on Cylinder · 0.99
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This agrees with our previous result [ref] , and the power of $β$, $d(k)$ represents the dimension of the moduli space as we expected in the Higgs branch analysis.
Higgs and Coulomb Branch Descriptions of the Volume of the Vortex Moduli Space · Generating Function · 0.97
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is nothing but the integral expression for the volume of the vortex moduli space in $U(1)$ gauge theory with $N_f$ flavors [ref] up to a redefinition of the parameter $β$.
The Volume of the Quiver Vortex Moduli Space · Two Abelian vertices · 0.96
Cites it as background 3 papers
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The vortex partition function (or the index) has been studied to certain extent in the literature, and especially the partition function of 2 dimensional vortices as instantons was investigated in [ref] .
Vortices and 3 dimensional dualities · Introduction · 0.97
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In particular, in the last decade, the localization theorem has been utilized in physics for the evaluation of the volume of the symplectic moduli space [ref] , or the non-perturbative corrections to the supersymmetric gauge theories [ref] .
Non-Abelian Localization for Supersymmetric Yang-Mills-Chern-Simons Theories on Seifert Manifold · Introduction · 0.90
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The dynamics of vortices is rather rich [ref] and deserves a separate discussion.
Toroidal and Elliptic Quiver BPS Algebras and Beyond · On vortices and 4d crystals · 0.97
What it cites · 27 sentences without a confident label
Uses the method or tool of 9 papers
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[ref] The $Ω$-background deforms the BRST transformations as [ Q_Z = iλ, Q_λ= -[Φ,Z]-Z,; Q_Z^†= -iλ^†, Q_λ^†= [Φ,Z^†]-Z^†,; Q_I = iψ_I, Q_ψ_I = ΦI-IM,; Q_I^†= -iψ_I^†, Q_ψ_I^†= I^†Φ- MI^†,; Q_J = iψ_J, Q_ψ_J = J Φ- M̃J+J,; Q_J^†= -iψ_J^†, Q_ψ_J^†= ΦJ^†- J^†M̃+J^†,; Q_Φ= 0, ; ] where $$ is a parameter of the $Ω$-background.
Integrating Over Higgs Branches · Comparison with Hanany-Tong Moduli Space · 0.82
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[ref] The $Ω$-background deforms the BRST transformations as [ Q_Z = iλ, Q_λ= -[Φ,Z]-Z,; Q_Z^†= -iλ^†, Q_λ^†= [Φ,Z^†]-Z^†,; Q_I = iψ_I, Q_ψ_I = ΦI-IM,; Q_I^†= -iψ_I^†, Q_ψ_I^†= I^†Φ- MI^†,; Q_J = iψ_J, Q_ψ_J = J Φ- M̃J+J,; Q_J^†= -iψ_J^†, Q_ψ_J^†= ΦJ^†- J^†M̃+J^†,; Q_Φ= 0, ; ] where $$ is a parameter of the $Ω$-background.
Seiberg-Witten Prepotential From Instanton Counting · Comparison with Hanany-Tong Moduli Space · 0.82
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To evaluate the integral, it is useful to introduce the “character” [ref]
Multi-Instanton Calculus and Equivariant Cohomology · Comparison with Hanany-Tong Moduli Space · 0.94
Compares with 2 papers
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We can compare our result to the previous result of the volume of the moduli space obtained from the moduli space metric by Manton and Nasir [ref] (_k) =π^k ∑_j=0^min(h,k) (4π)^j (g^2cA- 4πk )^k-j h!/j!(k-j)!(h-j)!.
Volume of Vortex Moduli Spaces · ANO vortices ($N_f=N_c=1$) · 1.00
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For the ANO vortices ($N_c=N_f=1$) where volumes were obtained from the metric before [ref] , we find that our results by the localization technique completely agree with the previous results for any topology of the Riemann surface.
Thermodynamics of Vortices in the Plane · Introduction · 1.00
Cites as background 16 papers
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Most of the previous works [ref] based on effective Lagrangian considered Abelian gauge theory with single charged scalar field at the critical coupling, allowing the BPS vortices.
Volume of Vortex Moduli Spaces · Introduction · 0.92
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Most of the previous works [ref] based on effective Lagrangian considered Abelian gauge theory with single charged scalar field at the critical coupling, allowing the BPS vortices.
Vortices and Flat Connections · Introduction · 0.92
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Most of the previous works [ref] based on effective Lagrangian considered Abelian gauge theory with single charged scalar field at the critical coupling, allowing the BPS vortices.
The K\"ahler Potential of Abelian Higgs Vortices · Introduction · 0.92
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