{"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}
引用次数: 0
Abstract
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.