Yang-Mills measure on the two-dimensional torus as a random distribution
type: Item
instance of: scholarly article
title: Yang-Mills measure on the two-dimensional torus as a random distribution
zbMATH Open document ID: 1434.35134
author: Ilya Chevyrev
published in: Communications in Mathematical Physics
publication date: 2019-11-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1808.09196
review text: The author aims at giving a rigorous meaning to the (Euclidean) Yang-Mills measure on the two-dimensional Torus \(\mathbb{T}^2\). Formally, this measure is given by \[ d\mu(A) = Z^{-1} e^{-S_{\mathrm{YM}(A)}}\,dA\,, \] where \(dA\) denotes a formal Lebesgue measure on the affine space \(\mathcal{A}\) of connections on a principle \(G\)-bundle \(P\) over \(\mathbb{T}^2\), where \(G\) is a compact, connected Lie group with Lie algebra \(\mathfrak{g}\). In this article, it is assumed that \(P\) is trivial, so that \(\mathcal{A}\) can be identified with the space \(\Omega^1(\mathbb{T}^2,\mathfrak{g})\) of \(\mathfrak{g}\)-valued \(1\)-forms on \(\mathbb{T}^2\). The constant \(Z\) is chosen such that \(\mu\) becomes a probability measure. Finally, \[ S_{\mathrm{YM}}(A) := \int_{\mathbb{T}^2} |F_A(x)|^2\,dx \] is the Yang-Mills action where \(F_A\) is the curvature two-form of \(A\). One possibility to understand \(\mu\) is to investigate the distributions of certain gauge invariant observables, such as Wilson loops, which are defined via holonomies. The main goal is to use a characterization of the Yang-Mills measure by \textit{T. Lévy} [Yang-Mills measures on compact surfaces. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1036.58009)] to realize \(\mu\) as a random distribution with certain regularity properties for small scales. For this purpose, a new space of distributional \(1\)-forms \(\Omega_\alpha^1\) on \(\mathbb{T}^2\) is introduced for which ``holonomies along axis paths are well-defined and induce Hölder continuous functions on line segments''. Basic properties of \(\Omega_\alpha^1\) are discussed in Section 3. The main result of this article (Theorem 1.1) is the existence of a ``\(\Omega_\alpha^1\)-valued random variable \(A\) for which Wilson loop observables of axis paths coincide in law with the corresponding observables under the Yang-Mills measure in the sense of Lévy''. In particular, it is shown that \(A\) has regularity properties (w…
reviewed by: Konstantin Merz
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 81T13
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 7136924
zbMATH Keywords: Yang-Mills measure
zbMATH Keywords: holonomy
zbMATH Keywords: Wilson loops
zbMATH Keywords: Landau gauge
MaRDI profile type: Publication
OpenAlex ID: W3102760502
arXiv ID: 1808.09196
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Wikidata QID: Q127253460
DOI: 10.1007/S00220-019-03567-5
description: scientific article