Asymptotic expansion of perturbative Chern-Simons theory via Wiener space
type: Item
instance of: scholarly article
title: Asymptotic expansion of perturbative Chern-Simons theory via Wiener space
zbMATH Open document ID: 1169.57030
author: Itaru Mitoma
author: S. Albeverio
published in: Bulletin des Sciences Mathématiques
publication date: 2009-05-12T00:00:00Z
review text: After Witten's seminal work on relations between quantum link invariants and heuristic Chern-Simons path integrals there has been a sustained effort to make rigorous mathematical sense of the latter. In the context of white noise analysis the non-existence of an infinite-dimensional Lebesgue measure on the space of connections is circumvented by using a suitable Gaussian measure instead. Integration is then replaced by pairing functionals of distributional connections with regularized loop holonomies. An additional difficulty in the Chern-Simons case is that the Gaussian kernel is degenerate and a gauge is usually fixed before making sense of the integrals. Unfortunately, this defeats the original purpose of Witten's work, which was to find an intrinsic 3D representation for quantum link invariants. However, no alternative approach has emerged so far in the non-Abelian case. This paper is one of the first attempts to define integrals appearing in the formal expansion of Chern-Simons integrals in powers of the level \(k\) along similar lines. Heuristically, these integrals correspond to Vassiliev (finite-type) invariants rather than link polynomials. Unitary bundles of connections over general closed 3-manifolds are considered, and the expansion is centered at a flat reference connection assumed isolated and irreducible. The Lorentz gauge is chosen, and the usual trick with anti-commuting variables is used to integrate out the degeneracy and obtain a formal expression for the Faddeev-Popov determinant. The reduced kernel then becomes elliptic and a regularized determinant can be defined. Since the Gaussian measure is not supported by smooth connections the usual loop holonomies (Wilson loops) are not integrable with respect to it. Hence regularizations are defined by smearing Wilson loops into smooth currents via convolution. Under additional technical assumtions on the reference connection the authors show that partial sums of the regularized series make sense as Ga…
reviewed by: Sergiy Koshkin
Mathematics Subject Classification ID: 57R56
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 58J28
Mathematics Subject Classification ID: 81T13
zbMATH DE Number: 5552986
zbMATH Keywords: Chern-Simons integral
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: Gaussian measure
zbMATH Keywords: white noise
zbMATH Keywords: Malliavin-Taniguchi formula
zbMATH Keywords: linking number
MaRDI profile type: Publication
full work available at URL: j.bulsci.2007.07.003
OpenAlex ID: W2078029375
cites work: Abelian Chern–Simons theory and linking numbers via oscillatory integrals
cites work: A mathematical construction of the non-Abelian Chern-Simons functional integral
cites work: Some new developments in the theory of path integrals, with applications to quantum theory
cites work: Vassiliev knot invariants and Chern-Simons perturbation theory to all orders
cites work: Q3998730
cites work: Non-abelian Stokes formula
cites work: Q4320925
cites work: Chern-Simons perturbation theory. II
cites work: PERTURBATIVE CHERN-SIMONS THEORY
cites work: On the Vassiliev knot invariants
cites work: Perturbative expansion of Chern-Simons theory with non-compact Gauge group
cites work: On the self-linking of knots
cites work: Q3961493
cites work: Iterated path integrals
cites work: Q4002453
cites work: Q3317770
cites work: Q3857069
cites work: The Chern-Simons theory and knot polynomials
cites work: Q5509277
cites work: Q3679959
cites work: Recent developments in Chern-Simons theory and link invariants
cites work: The Wilson loop observables of Chern-Simons theory on \({\mathbb{R}}^3\) in axial gauge
cites work: Chern-Simons models on \(S^2\times S^1\), torus gauge fixing, and link invariants. I
cites work: Gaussian measures in Banach spaces
cites work: Q3995493
cites work: Invariant Sobolev calculus on the free loop space
cites work: Analytic functions, Cauchy formula, and stationary phase on a real abstract Wiener space
cites work: One loop approximation of the Chern-Simons integral
cites work: Wiener space approach to a perturbative Chern-Simons integral.
cites work: Asymptotic expansion of the one-loop approximation of the Chern-Simons integral in an abstract Wiener space setting
cites work: Q5518064
cites work: Q3909512
cites work: The partition function of a degenerate functional
cites work: Chen integrals, generalized loops and loop calculus
cites work: A METHOD OF CONSTRUCTION OF MEASURES ON FUNCTION SPACES AND ITS APPLICATIONS TO STOCHASTIC PROCESSES
cites work: Quantum field theory and the Jones polynomial
cites work: Q5620934
DOI: 10.1016/J.BULSCI.2007.07.003
description: scientific article; zbMATH DE number 5552986