{"title":"特征变量的交映几何与 SU(2) 格态量子理论 I","authors":"T. R. Ramadas","doi":"10.1007/s00220-024-04968-x","DOIUrl":null,"url":null,"abstract":"<p>Associated to any finite graph <span>\\(\\Lambda \\)</span> is a closed surface <span>\\({\\textbf{S}}={\\textbf{S}}_\\Lambda \\)</span>, the boundary of a regular neighbourhood of an embedding of <span>\\(\\Lambda \\)</span> in any three manifold. The surface retracts to the graph, mapping loops on the surface to loops on the graph. The (<i>SU</i>(2)) character variety <span>\\({{\\mathcal {M}}}\\)</span> of <span>\\({\\textbf{S}}\\)</span> has a symplectic structure and associated Liouville measure; on the other hand, the character variety <span>\\({\\textbf{M}}\\)</span> of <span>\\(\\Lambda \\)</span> carries a natural measure inherited from the Haar measure. Loops on <span>\\({\\textbf{S}}\\)</span> define functions on the character varieties, the <i>Wilson loops</i>. By the works of W. Goldman, L. Jeffrey and J. Weitsman, the formalism of Duistermaat-Heckman applies to the relevant integrals over <span>\\({{\\mathcal {M}}}\\)</span>. We develop a calculus for calculating correlations of Wilson loops on <span>\\({{\\mathcal {M}}}\\)</span> w.r.to the normalised Liouville measure, and present evidence that they approximate—for large graphs—the corresponding integrals over <span>\\({\\textbf{M}}\\)</span>. Lattice field theory involves integrals over <span>\\({\\textbf{M}}\\)</span>; we present “symplectic” analogues of expressions for partition functions, Wilson loop expectations, etc., in two and three space-time dimensions.</p>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Symplectic Geometry of Character Varieties and SU(2) Lattice Gauge Theory I\",\"authors\":\"T. R. Ramadas\",\"doi\":\"10.1007/s00220-024-04968-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Associated to any finite graph <span>\\\\(\\\\Lambda \\\\)</span> is a closed surface <span>\\\\({\\\\textbf{S}}={\\\\textbf{S}}_\\\\Lambda \\\\)</span>, the boundary of a regular neighbourhood of an embedding of <span>\\\\(\\\\Lambda \\\\)</span> in any three manifold. The surface retracts to the graph, mapping loops on the surface to loops on the graph. The (<i>SU</i>(2)) character variety <span>\\\\({{\\\\mathcal {M}}}\\\\)</span> of <span>\\\\({\\\\textbf{S}}\\\\)</span> has a symplectic structure and associated Liouville measure; on the other hand, the character variety <span>\\\\({\\\\textbf{M}}\\\\)</span> of <span>\\\\(\\\\Lambda \\\\)</span> carries a natural measure inherited from the Haar measure. Loops on <span>\\\\({\\\\textbf{S}}\\\\)</span> define functions on the character varieties, the <i>Wilson loops</i>. By the works of W. Goldman, L. Jeffrey and J. Weitsman, the formalism of Duistermaat-Heckman applies to the relevant integrals over <span>\\\\({{\\\\mathcal {M}}}\\\\)</span>. We develop a calculus for calculating correlations of Wilson loops on <span>\\\\({{\\\\mathcal {M}}}\\\\)</span> w.r.to the normalised Liouville measure, and present evidence that they approximate—for large graphs—the corresponding integrals over <span>\\\\({\\\\textbf{M}}\\\\)</span>. Lattice field theory involves integrals over <span>\\\\({\\\\textbf{M}}\\\\)</span>; we present “symplectic” analogues of expressions for partition functions, Wilson loop expectations, etc., in two and three space-time dimensions.</p>\",\"PeriodicalId\":522,\"journal\":{\"name\":\"Communications in Mathematical Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2024-04-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications in Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1007/s00220-024-04968-x\",\"RegionNum\":1,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s00220-024-04968-x","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
摘要
与任意有限图(\Lambda \)相关的是一个封闭曲面({\textbf{S}}={\textbf{S}}_\Lambda \),它是\(\Lambda \)在任意三流形中嵌入的规则邻域的边界。曲面向图形收缩,将曲面上的循环映射到图形上的循环。({\textbf{S}}\)的(SU(2))特征集({\mathcal {M}})具有交映结构和相关的Liouville度量;另一方面,\(\Lambda \)的特征集({\textbf{M}}\)带有从Haar度量继承而来的自然度量。\({\textbf{S}}\)上的循环定义了特征集上的函数,即威尔逊循环。根据 W. Goldman、L. Jeffrey 和 J. Weitsman 的著作,杜斯特马特-赫克曼的形式主义适用于 \({{\mathcal {M}}\) 上的相关积分。)我们开发了一种计算方法,用于计算\({\mathcal {M}})上的威尔逊环与归一化柳维尔量度的相关性,并提出证据表明,对于大型图,它们近似于\({\textbf{M}})上的相应积分。晶格场论涉及对\({textbf{M}}\)的积分;我们提出了分割函数、威尔逊环期望等在二维和三维时空中的 "交映 "类似表达式。
Symplectic Geometry of Character Varieties and SU(2) Lattice Gauge Theory I
Associated to any finite graph \(\Lambda \) is a closed surface \({\textbf{S}}={\textbf{S}}_\Lambda \), the boundary of a regular neighbourhood of an embedding of \(\Lambda \) in any three manifold. The surface retracts to the graph, mapping loops on the surface to loops on the graph. The (SU(2)) character variety \({{\mathcal {M}}}\) of \({\textbf{S}}\) has a symplectic structure and associated Liouville measure; on the other hand, the character variety \({\textbf{M}}\) of \(\Lambda \) carries a natural measure inherited from the Haar measure. Loops on \({\textbf{S}}\) define functions on the character varieties, the Wilson loops. By the works of W. Goldman, L. Jeffrey and J. Weitsman, the formalism of Duistermaat-Heckman applies to the relevant integrals over \({{\mathcal {M}}}\). We develop a calculus for calculating correlations of Wilson loops on \({{\mathcal {M}}}\) w.r.to the normalised Liouville measure, and present evidence that they approximate—for large graphs—the corresponding integrals over \({\textbf{M}}\). Lattice field theory involves integrals over \({\textbf{M}}\); we present “symplectic” analogues of expressions for partition functions, Wilson loop expectations, etc., in two and three space-time dimensions.
期刊介绍:
The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.