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Non-Abelian Localization For Chern-Simons Theory
High Energy Physics - Theory · Submitted 16 Mar 2005
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In the process, we show that the partition function of Chern-Simons theory on M admits a topological interpretation in terms of the equivariant cohomology of the moduli space of flat connections on M.
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The abstract, sentence by sentence
We reconsider Chern-Simons gauge theory on a Seifert manifold M (the total space of a nontrivial circle bundle over a Riemann surface). When M is a Seifert manifold, Lawrence and Rozansky have shown from the exact solution of Chern-Simons theory that the partition function has a remarkably simple structure and can be rewritten entirely as a sum of local contributions from the flat connections on M. We explain how this empirical fact follows from the technique of non-abelian localization as applied to the Chern-Simons path integral. In the process, we show that the partition function of Chern-Simons theory on M admits a topological interpretation in terms of the equivariant cohomology of the moduli space of flat connections on M.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We reconsider Chern-Simons gauge theory on a Seifert manifold M (the total space of a nontrivial circle bundle over a Riemann surface). | background 0.85 | 0.60 | 0.03 | asserted 0.67 |
| s1 | When M is a Seifert manifold, Lawrence and Rozansky have shown from the exact solution of Chern-Simons theory that the partition function has a remarkably simple structure and can be rewritten entirely as a sum of local contributions from the flat connections on M. | background 1.00 | 0.03 | 0.11 | asserted 1.00 |
| s2 | We explain how this empirical fact follows from the technique of non-abelian localization as applied to the Chern-Simons path integral. | contribution 0.62 | 0.90 | 0.38 | asserted 1.00 |
| s3 | In the process, we show that the partition function of Chern-Simons theory on M admits a topological interpretation in terms of the equivariant cohomology of the moduli space of flat connections on M. | interpretation 0.86 | 0.88 | 0.43 | asserted 1.00 |
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The paper
Original abstract
We reconsider Chern-Simons gauge theory on a Seifert manifold M (the total space of a nontrivial circle bundle over a Riemann surface). When M is a Seifert manifold, Lawrence and Rozansky have shown from the exact solution of Chern-Simons theory that the partition function has a remarkably simple structure and can be rewritten entirely as a sum of local contributions from the flat connections on M. We explain how this empirical fact follows from the technique of non-abelian localization as applied to the Chern-Simons path integral. In the process, we show that the partition function of Chern-Simons theory on M admits a topological interpretation in terms of the equivariant cohomology of the moduli space of flat connections on M.
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References
How the literature uses it
45 arXiv papers cite this in 103 sentences
Papers citing it · 66 sentences without a confident label
Builds on it 2 papers
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In [ref] , the techniques of [ref] were extended to compute the expectation values of Wilson loops along the $U(1)$ fibers.
Localization and Non-Renormalization in Chern-Simons Theory · Introduction · 0.95
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Repeating the transformations in ([ref]) and identifying $a^TB^-1a$ with the CS functional of a connected component of an abelian flat connection [equation] one obtains a generalization of ([ref]) to the CS thimbles $_a$ corresponding to abelian flat connections in the case of non-trivial torsion in homology: [equation] where $γ̃_a$ is the Lefschetz thimble with respect to $S̃(u)+u^Ta$ (cf. [ref] for the case of Seifert manifolds).
On categorification of Stokes coefficients in Chern-Simons theory · Non-trivial torsion in homology · 0.97
Uses its method or tool 3 papers
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Following the discussion in [ref] , we can Fourier expand horizontal forms into eigenmodes of the operator $ℒ_v$: ξ=∑_t=-∞^∞ξ_t, ξ∈Ω_H^∙(M,𝔤), ℒ_vξ_t=2πi t ·ξ_t.
Cohomological localization of Chern-Simons theory · Calculation of the partition function · 0.97
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We follow [ref] closely and proceed formally.
A generalized 4d Chern-Simons theory · Moment map and the quadratic action · 0.87
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The computation of the 1-loop determinant goes along the same lines as the computations in [ref] .
Twisted supersymmetric 5D Yang-Mills theory and contact geometry · Computation of the 1-loop determinant · 0.98
Compares with it 6 papers
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It was later derived in a path integral computation using non-abelian localization in [ref] , and here we have derived it starting from the $𝒩=2$ supersymmetric version of Chern-Simons theory.
Cohomological localization of Chern-Simons theory · Calculation of the partition function · 0.86
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The condition that [ref] vanishes is absent in [ref] , because of the cocycle used there is, roughly speaking, recovered from [ref] by setting $Ω→1$ and $𝕄→M$.
A generalized 4d Chern-Simons theory · Moment map and its properties · 0.97
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If one had first integrated over $B$, one would have recovered the result of [ref] obtained by nonabelian localization, which involves not only an integral over the Cartan, but also over the moduli space of flat connections on $M^3$ to which the Chern-Simons path integral localizes.
Localization and Non-Renormalization in Chern-Simons Theory · Introduction · 0.97
Cites it as background 16 papers
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Then $H_1(M,ℝ)=0$ if g =0 d/P =n+∑_j=1^Nb_j /a_j gcd(a_i,a_j) =1, i≠j. h10 For an explanation of how these conditions occur, we refer to [ref] .
Cohomological localization of Chern-Simons theory · Seifert homology spheres · 0.92
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Here is a schematic account of the history of those exact formulae: $∙$ For $𝔤 = su(2)$ or $𝔰𝔬(3)$ and $M$ a Seifert integer homology sphere, Lawrence and Rozansky [ref] have used the Reshetikhin-Turaev construction to rewrite $Z_CS^𝔤$ as a $1$-dimensional integral [ref], including contributions of all flat connections. $∙$ Mariño generalized their derivation to any simply laced-Lie algebra $𝔤$ and Seifert rational homology spheres $M$ [ref] . $∙$ Bar-Natan [ref] has computed the LMO invariant of Seifert rational homology spheres, via the Kontsevich integral. $∙$ Beasley and Witten [ref] have developed a non-abelian localization method, allowing the computation of the contribution of isolated flat connections[The trivial flat connection in a Seifert fibered spaces is isolated iff the $a_i$ are pairwise coprime.
Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces · Exact evaluations · 0.85
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In this paper we are concerned with the $q$-deformation of two-dimensional Yang-Mills theory, studied originally in [ref] and further developed in [ref] .
q-deformations of two-dimensional Yang-Mills theory: Classification, categorification and refinement · Introduction and summary · 0.87
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