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Comments on knotted 1/2 BPS Wilson loops
High Energy Physics - Theory · Submitted 26 Apr 2012
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this paper, we show that the localization of three-dimensional N = 2 supersymmetric Chern-Simons theory on the ellipsoid-like squashed sphere is related to a nontrivial knot structure called torus knot.
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The abstract, sentence by sentence
In this paper, we show that the localization of three-dimensional N = 2 supersymmetric Chern-Simons theory on the ellipsoid-like squashed sphere is related to a nontrivial knot structure called torus knot. More precisely, we can capture the three sphere as the nontrivial so-called Seifert fibrations by regarding 1/2 BPS Wilson loops as U(1) fibers. The topology of knotted 1/2 BPS Wilson loops is controlled by squashing parameters. We calculate the 1/2 BPS condition of the Wilson loop and find perfect agreement with known results. We also remark on the level shift and framing anomaly.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper, we show that the localization of three-dimensional N = 2 supersymmetric Chern-Simons theory on the ellipsoid-like squashed sphere is related to a nontrivial knot structure called torus knot. | contribution 0.54 | 0.92 | 0.81 | asserted 1.00 |
| s1 | More precisely, we can capture the three sphere as the nontrivial so-called Seifert fibrations by regarding 1/2 BPS Wilson loops as U(1) fibers. | method 0.38 | 0.77 | 0.14 | asserted 0.76 |
| s2 | The topology of knotted 1/2 BPS Wilson loops is controlled by squashing parameters. | result 0.90 | 0.57 | 0.24 | asserted 1.00 |
| s3 | We calculate the 1/2 BPS condition of the Wilson loop and find perfect agreement with known results. | result 0.98 | 0.95 | 0.10 | asserted 1.00 |
| s4 | We also remark on the level shift and framing anomaly. | contribution 0.48 | 0.36 | 0.03 | unclear 0.78 |
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The paper
Original abstract
In this paper, we show that the localization of three-dimensional N = 2 supersymmetric Chern-Simons theory on the ellipsoid-like squashed sphere is related to a nontrivial knot structure called torus knot. More precisely, we can capture the three sphere as the nontrivial so-called Seifert fibrations by regarding 1/2 BPS Wilson loops as U(1) fibers. The topology of knotted 1/2 BPS Wilson loops is controlled by squashing parameters. We calculate the 1/2 BPS condition of the Wilson loop and find perfect agreement with known results. We also remark on the level shift and framing anomaly.
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References
How the literature uses it
18 arXiv papers cite this in 26 sentences
Papers citing it · 21 sentences without a confident label
Cites it as background 5 papers
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Several works have shown that Coulomb branch localization reduces a class of BPS observables in the $𝒩= 2$ theories to certain matrix integrals [ref] (see also [ref] ).
Higgs branch localization of 3d N=2 theories · Introduction · 0.86
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These have been studied in the present context in [ref] .
Wilson loops on three-manifolds and their M2-brane duals · Localization in the matrix model · 0.85
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One can also squash the $S^3$ to obtain the Berger sphere with $SU(2)×U(1)$ isometry (where the Killing vector has closed orbits and points along the Seifert fiber) [ref] or the ellipsoid with $U(1)×U(1)$ isometry (where the Killing vector does not, generically, point along the fiber) [ref] ; on the latter background, one can compute expectation values of nontrivial torus knots [ref] .
Localization and Non-Renormalization in Chern-Simons Theory · Overview · 0.80
What it cites · 10 sentences without a confident label
Uses the method or tool of 1 paper
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In this paper, we find that one of modification can be accomplished by using Hama, Hosomichi and Lee's localization on ellipsoid-like squashed three spheres [ref] .
SUSY Gauge Theories on Squashed Three-Spheres · Introduction · 0.97
Compares with 2 papers
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Our result matches with known results [ref] , and of course matches with Jones polynomials gained by Witten.
Localization for Wilson Loops in Chern-Simons Theory · Introduction · 0.97
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Our result matches with known results [ref] , and of course matches with Jones polynomials gained by Witten.
Cohomological localization of Chern-Simons theory · Introduction · 0.97
Cites as background 7 papers
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On the other hand, in these days, so called localization techniques were developed by Pestun [ref] in four-dimension, Kapustin, Willett, Yaakov [ref] in three-dimension.
Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter · Introduction · 0.87
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On the other hand, in these days, so called localization techniques were developed by Pestun [ref] in four-dimension, Kapustin, Willett, Yaakov [ref] in three-dimension.
Localization of gauge theory on a four-sphere and supersymmetric Wilson loops · Introduction · 0.87
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Another squashing is discussed in [ref] and localization on other three-dimensional manifolds is discussed by [ref] on lens spaces, [ref] on general Seifert manifolds by performing topological twist on $𝒩=2$ vector multiplet.
Cohomological localization of Chern-Simons theory · Discussion · 0.93
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