The Wilson Loop Observables of Chern-Simons Theory on ℝ3 in Axial Gauge

Hahn, Atle

doi:10.1007/s00220-004-1097-4

Cited by 20 publications

(57 citation statements)

References 28 publications

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“…It is pretty clear that such a partial gauge fixing will lead to a version of the vertex model, although it may be tricky to compute the precise vertex weights. The necessary computations are likely to be somewhat similar to those involved in studying Chern-Simons theory in an ordinary axial gauge -for example, see [52,53] -or in a certain almost axial gauge [54]. Somewhat analogous is the use of a complex version of axial gauge to derive the Knizhnik-Zamolodchikov equations [55].…”

Section: Davide Gaiotto and Edward Wittenmentioning

confidence: 99%

Knot invariants from four-dimensional gauge theory

Gaiotto

Witten

2012

Advances in Theoretical and Mathematical Physics

109

264

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It has been argued based on electric-magnetic duality and other ingredients that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory equations in four dimensions. Here, we attempt to verify this directly by analyzing the equations and counting their solutions, without reference to any quantum dualities. After suitably perturbing the equations to make their behavior more generic, we are able to get a fairly clear understanding of how the Jones polynomial emerges. The main ingredient in the argument is a link between the four-dimensional gauge theory equations in question and conformal blocks for degenerate representations of the Virasoro algebra in two dimensions. Along the way we get a better understanding of how our subject is related to a variety of new and old topics in mathematical physics, ranging from the Bethe ansatz for the Gaudin spin chain to the M -theory description of Bogomol'nyi-Prasad-Sommerfield (BPS) monopoles and the relation between Chern-Simons gauge theory and Virasoro conformal blocks. e-print archive: http://lanl.arXiv.org/abs/1106.4789v1

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Section: Davide Gaiotto and Edward Wittenmentioning

confidence: 99%

Knot invariants from four-dimensional gauge theory

Gaiotto

Witten

2012

Advances in Theoretical and Mathematical Physics

109

264

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“…The same results were obtained in [60] in the framework of white noise calculus. These result have been extended to the case where G is not abelian and M = R 3 in [29,45] by means of white noise analysis (see also [14] for a detailed exposition of this topic). The case M = S 1 × S 2 has been recently handled in [46].…”

Section: The Chern-simons Modelmentioning

confidence: 98%

Theory and Applications of Infinite Dimensional Oscillatory Integrals

Albeverio

2007

Stochastic Analysis and Applications

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“…It is also a necessary preparation for the non-Abelian case, for it turns out that the four lemmas which are used in the proof of Theorem 1 also play a role in the treatment of the non-Abelian case, cf. [16].…”

Section: Introductionmentioning

confidence: 95%

“…Of course, G ¼ Uð1Þ is Abelian but as we will show in a subsequent paper (cf. [16]), it is also possible to define and to compute the WLOs explicitly within our approach if the group G is non-Abelian.…”

Section: Introductionmentioning

confidence: 99%

Chern–Simons theory on R3 in axial gauge: a rigorous approach

Hahn¹

2004

Journal of Functional Analysis

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We study pure Chern-Simons models on M ¼ R 3 using a functional integral quantization approach which is based on axial gauge fixing. It is well-known (see, e.g., Comm. Math. Phys. 126 (1989) 167; Comm. Math. Phys. 186 (1997) 563) that in axial gauge the Chern-Simons action function is quadratic and that the Faddeev-Popov determinant of this gauge fixing procedure is a constant function. This means that the Wilson loop observables (WLOs) of the model considered can be obtained heuristically by integrating certain quantities against a functional measure of ''Gaussian type''. We demonstrate that although these heuristic integral expressions look rather singular it is possible to give a rigorous meaning to them by combining constructions from White noise analysis with certain regularization techniques like ''loop smearing'' and ''framing''. For the special case G ¼ Uð1Þ; G being the group of the model, we carry out the details of this approach, which is also applicable to the case of non-Abelian G; and find well-known linking number expressions for the WLOs. r 2004 Elsevier Inc. All rights reserved.

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The Wilson Loop Observables of Chern-Simons Theory on ℝ3 in Axial Gauge

Cited by 20 publications

References 28 publications

Knot invariants from four-dimensional gauge theory

Knot invariants from four-dimensional gauge theory

Theory and Applications of Infinite Dimensional Oscillatory Integrals

Chern–Simons theory on R3 in axial gauge: a rigorous approach

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