Alternative Derivation of the Non-Abelian Stokes Theorem in Two Dimensions

IF 2.2 3区 综合性期刊 Q2 MULTIDISCIPLINARY SCIENCES Symmetry-Basel Pub Date : 2023-10-30 DOI:10.3390/sym15112000
Seramika Ariwahjoedi, Freddy Permana Zen
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Abstract

The relation between the holonomy along a loop with the curvature form is a well-known fact, where the small square loop approximation of aholonomy Hγ,O is proportional to Rσ. In an attempt to generalize the relation for arbitrary loops, we encounter the following ambiguity. For a given loop γ embedded in a manifold M, Hγ,O is an element of a Lie group G; the curvature Rσ∈g is an element of the Lie algebra of G. However, it turns out that the curvature form Rσ obtained from the small loop approximation is ambiguous, as the information of γ and Hγ,O are insufficient for determining a specific plane σ responsible for Rσ. To resolve this ambiguity, it is necessary to specify the surface S enclosed by the loop γ; hence, σ is defined as the limit of S when γ shrinks to a point. In this article, we try to understand this problem more clearly. As a result, we obtain an exact relation between the holonomy along a loop with the integral of the curvature form over a surface that it encloses. The derivation of this result can be viewed as an alternative proof of the non-Abelian Stokes theorem in two dimensions with some generalizations.
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二维非阿贝尔斯托克斯定理的另一种推导
沿环的完整度与曲率形式之间的关系是一个众所周知的事实,其中完整度的小方环近似Hγ,O与Rσ成正比。在尝试推广任意循环的关系时,我们遇到了以下的模糊性。对于嵌入流形M中的给定环γ, Hγ,O是李群G的元素;曲率Rσ∈g是g的李代数的一个元素。然而,由小环近似得到的曲率形式Rσ是模糊的,因为γ和Hγ,O的信息不足以确定一个特定的平面σ负责Rσ。为了解决这种歧义,有必要指定由环γ包围的表面S;因此,σ被定义为当γ收缩到一点时S的极限。在本文中,我们试图更清楚地理解这个问题。因此,我们得到了沿环的完整度与其所包围的曲面上曲率形式的积分之间的精确关系。这个结果的推导可以看作是二维非阿贝尔斯托克斯定理的另一种证明,并作了一些推广。
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来源期刊
Symmetry-Basel
Symmetry-Basel MULTIDISCIPLINARY SCIENCES-
CiteScore
5.40
自引率
11.10%
发文量
2276
审稿时长
14.88 days
期刊介绍: Symmetry (ISSN 2073-8994), an international and interdisciplinary scientific journal, publishes reviews, regular research papers and short notes. Our aim is to encourage scientists to publish their experimental and theoretical research in as much detail as possible. There is no restriction on the length of the papers. Full experimental and/or methodical details must be provided, so that results can be reproduced.
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