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On the L^2-metric of vortex moduli spaces
High Energy Physics - Theory · Submitted 5 Mar 2010
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We derive general expressions for the Kaehler form of the L^2-metric in terms of standard 2-forms on vortex moduli spaces.
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The abstract, sentence by sentence
We derive general expressions for the Kaehler form of the L^2-metric in terms of standard 2-forms on vortex moduli spaces. In the case of abelian vortices in gauged linear sigma-models, this allows us to compute explicitly the Kaehler class of the L^2-metric. As an application we compute the total volume of the moduli space of abelian semi-local vortices. In the strong coupling limit, this then leads to conjectural formulae for the volume of the space of holomorphic maps from a compact Riemann surface to projective space. Finally we show that the localization results of Samols in the abelian Higgs model extend to more general models. These include linear non-abelian vortices and vortices in gauged toric sigma-models.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We derive general expressions for the Kaehler form of the L^2-metric in terms of standard 2-forms on vortex moduli spaces. | contribution 0.53 | 0.96 | 0.40 | asserted 1.00 |
| s1 | In the case of abelian vortices in gauged linear sigma-models, this allows us to compute explicitly the Kaehler class of the L^2-metric. | result 0.83 | 0.93 | 0.15 | asserted 0.97 |
| s2 | As an application we compute the total volume of the moduli space of abelian semi-local vortices. | result 0.91 | 0.96 | 0.15 | asserted 1.00 |
| s3 | In the strong coupling limit, this then leads to conjectural formulae for the volume of the space of holomorphic maps from a compact Riemann surface to projective space. | result 0.53 | 0.80 | 0.15 | speculative 0.90 |
| s4 | Finally we show that the localization results of Samols in the abelian Higgs model extend to more general models. | result 0.72 | 0.94 | 0.18 | asserted 1.00 |
| s5 | These include linear non-abelian vortices and vortices in gauged toric sigma-models. | result 0.57 | 0.76 | 0.08 | asserted 0.88 |
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The paper
Original abstract
We derive general expressions for the Kaehler form of the L^2-metric in terms of standard 2-forms on vortex moduli spaces. In the case of abelian vortices in gauged linear sigma-models, this allows us to compute explicitly the Kaehler class of the L^2-metric. As an application we compute the total volume of the moduli space of abelian semi-local vortices. In the strong coupling limit, this then leads to conjectural formulae for the volume of the space of holomorphic maps from a compact Riemann surface to projective space. Finally we show that the localization results of Samols in the abelian Higgs model extend to more general models. These include linear non-abelian vortices and vortices in gauged toric sigma-models.
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References
How the literature uses it
9 arXiv papers cite this in 14 sentences
Papers citing it · 8 sentences without a confident label
Cites it as background 4 papers
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The moduli space metric has been found recently for well-separated vortices [ref] using a generic formula for the Kähler potential on the moduli space [ref] .
Group Theory of Non-Abelian Vortices · Introduction and discussion · 0.88
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The moduli space metric has been explicitly obtained for non-Abelian vortices on $$ [ref] and on Riemann surfaces [ref] when vortices are well separated, and dynamics and scattering have been studied on $$ [ref] .
Vortices on Orbifolds · Summary and Discussion · 0.82
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The metric of the moduli space of the non-Abelian vortices with a $U(N)$ gauge group has been calculated in Ref. [ref] and for well-separated vortices in Ref. [ref] .
Fractional and semi-local non-Abelian Chern-Simons vortices · Introduction · 0.99
What it cites · 2 sentences without a confident label
Builds on 1 paper
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Since in this abelian case there exists a rather explicit description of $$ [ref] , this allows us to deduce an equally explicit formula for $[ω_]$, a formula that generalizes the result of Manton and Nasir for the abelian Higgs model [ref] .
Vortex invariants and toric manifolds · Introduction · 0.96
Compares with 2 papers
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Taking into account that in [ref] the Fubiny-Study metrics on both $ℂP^1$'s were chosen with volume $π$ – and hence for the target space this corresponds to the choice $τ= 1$ – we see that ([ref]) reduces exactly to the value $π^6 /6$ obtained there.[Since the first version of the present paper appeared, Speight has rigorously verified the volume formula ([ref]) also in the case $g=0$, $d=1$ and arbitrary $n$ [ref] .]
The volume of the space of holomorphic maps from S^2 to CP^k · Volume of spaces of holomorphic curves · 0.88
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Note: A week before this article was completed, a paper that contains some overlap with the material of section 7.1 in the case $Σ=$ appeared on the arXiv [ref] .
The Moduli Space Metric for Well-Separated Non-Abelian Vortices · Introduction · 0.86
Cites as background 2 papers
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This example is important in physics [ref] and a rather concrete description of $$ is known [ref] .
Vortex invariants and toric manifolds · The vortex Kähler class in abelian linear models · 0.92
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Other recent results on vortex metrics include [ref] .
Exact moduli space metrics for hyperbolic vortices · Introduction · 0.97
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