{"title":"论拓扑递归中相关器的 x-y 对称性--通过循环插入操作符","authors":"Alexander Hock","doi":"10.1007/s00220-024-05043-1","DOIUrl":null,"url":null,"abstract":"<p>Topological Recursion generates a family of symmetric differential forms (correlators) from some initial data <span>\\((\\Sigma ,x,y,B)\\)</span>. We give a functional relation between the correlators of genus <span>\\(g=0\\)</span> generated by the initial data <span>\\((\\Sigma ,x,y,B)\\)</span> and by the initial data <span>\\((\\Sigma ,y,x,B)\\)</span>, where <i>x</i> and <i>y</i> are interchanged. The functional relation is derived with the loop insertion operator by computing a functional relation for some intermediate correlators. Additionally, we show that our result is equivalent to the recent result of Borot et al. (2021) in case of <span>\\(g=0\\)</span>. Consequently, we are providing a simplified functional relation between generating series of higher order free cumulants and moments in higher order free probability.</p>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the x-y Symmetry of Correlators in Topological Recursion via Loop Insertion Operator\",\"authors\":\"Alexander Hock\",\"doi\":\"10.1007/s00220-024-05043-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Topological Recursion generates a family of symmetric differential forms (correlators) from some initial data <span>\\\\((\\\\Sigma ,x,y,B)\\\\)</span>. We give a functional relation between the correlators of genus <span>\\\\(g=0\\\\)</span> generated by the initial data <span>\\\\((\\\\Sigma ,x,y,B)\\\\)</span> and by the initial data <span>\\\\((\\\\Sigma ,y,x,B)\\\\)</span>, where <i>x</i> and <i>y</i> are interchanged. The functional relation is derived with the loop insertion operator by computing a functional relation for some intermediate correlators. Additionally, we show that our result is equivalent to the recent result of Borot et al. (2021) in case of <span>\\\\(g=0\\\\)</span>. Consequently, we are providing a simplified functional relation between generating series of higher order free cumulants and moments in higher order free probability.</p>\",\"PeriodicalId\":522,\"journal\":{\"name\":\"Communications in Mathematical Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2024-07-01\",\"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/s00220-024-05043-1\",\"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/s00220-024-05043-1","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
拓扑递归会从一些初始数据((\Sigma ,x,y,B ))生成对称微分形式(关联形式)族。我们给出了由初始数据 \((\Sigma ,x,y,B)\) 和由初始数据 \((\Sigma ,y,x,B)\) 生成的属(g=0\)相关器之间的函数关系,其中 x 和 y 是互换的。通过计算一些中间相关器的函数关系,可以得出循环插入算子的函数关系。此外,我们还证明,在 \(g=0\) 的情况下,我们的结果等同于博罗特等人(2021 年)的最新结果。因此,我们提供了高阶自由积的生成序列与高阶自由概率矩之间的简化函数关系。
On the x-y Symmetry of Correlators in Topological Recursion via Loop Insertion Operator
Topological Recursion generates a family of symmetric differential forms (correlators) from some initial data \((\Sigma ,x,y,B)\). We give a functional relation between the correlators of genus \(g=0\) generated by the initial data \((\Sigma ,x,y,B)\) and by the initial data \((\Sigma ,y,x,B)\), where x and y are interchanged. The functional relation is derived with the loop insertion operator by computing a functional relation for some intermediate correlators. Additionally, we show that our result is equivalent to the recent result of Borot et al. (2021) in case of \(g=0\). Consequently, we are providing a simplified functional relation between generating series of higher order free cumulants and moments in higher order free probability.
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.