{"title":"保形场论中限制分区数的解析公式","authors":"D. Polyakov","doi":"10.4310/ATMP.2018.V22.N5.A4","DOIUrl":null,"url":null,"abstract":"We study the correlators of irregular vertex operators in two-dimensional conformal field theory (CFT) in order to propose an exact analytic formula for calculating numbers of partitions, that is: \r\n1) for given $N,k$, finding the total number $\\lambda(N|k)$ of length $k$ partitions of $N$: $N=n_1+...+n_k;0<n_1\\leq{n_2}...\\leq{n_k}$. \r\n2) finding the total number $\\lambda(N)=\\sum_{k=1}^N\\lambda(N|k)$ of partitions of a natural number $N$ \r\nWe propose an exact analytic expression for $\\lambda(N|k)$ by relating two-point short-distance correlation functions of irregular vertex operators in $c=1$ conformal field theory ( the form of the operators is established in this paper): with the first correlator counting the partitions in the upper half-plane and the second one obtained from the first correlator by conformal transformations of the form $f(z)=h(z)e^{-{i\\over{z}}}$ where $h(z)$ is regular and non-vanishing at $z=0$. The final formula for $\\lambda(N|k)$ is given in terms of regularized ($\\epsilon$-ordered) finite series in the generalized higher-derivative Schwarzians and incomplete Bell polynomials of the above conformal transformation at $z=i\\epsilon$ ($\\epsilon\\rightarrow{0}$)","PeriodicalId":187229,"journal":{"name":"String Fields, Higher Spins and Number Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2017-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"title\":\"An Analytic Formula for Numbers of Restricted Partitions from Conformal Field Theory\",\"authors\":\"D. Polyakov\",\"doi\":\"10.4310/ATMP.2018.V22.N5.A4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the correlators of irregular vertex operators in two-dimensional conformal field theory (CFT) in order to propose an exact analytic formula for calculating numbers of partitions, that is: \\r\\n1) for given $N,k$, finding the total number $\\\\lambda(N|k)$ of length $k$ partitions of $N$: $N=n_1+...+n_k;0<n_1\\\\leq{n_2}...\\\\leq{n_k}$. \\r\\n2) finding the total number $\\\\lambda(N)=\\\\sum_{k=1}^N\\\\lambda(N|k)$ of partitions of a natural number $N$ \\r\\nWe propose an exact analytic expression for $\\\\lambda(N|k)$ by relating two-point short-distance correlation functions of irregular vertex operators in $c=1$ conformal field theory ( the form of the operators is established in this paper): with the first correlator counting the partitions in the upper half-plane and the second one obtained from the first correlator by conformal transformations of the form $f(z)=h(z)e^{-{i\\\\over{z}}}$ where $h(z)$ is regular and non-vanishing at $z=0$. The final formula for $\\\\lambda(N|k)$ is given in terms of regularized ($\\\\epsilon$-ordered) finite series in the generalized higher-derivative Schwarzians and incomplete Bell polynomials of the above conformal transformation at $z=i\\\\epsilon$ ($\\\\epsilon\\\\rightarrow{0}$)\",\"PeriodicalId\":187229,\"journal\":{\"name\":\"String Fields, Higher Spins and Number Theory\",\"volume\":\"1 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2017-02-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"String Fields, Higher Spins and Number Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4310/ATMP.2018.V22.N5.A4\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"String Fields, Higher Spins and Number Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4310/ATMP.2018.V22.N5.A4","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
An Analytic Formula for Numbers of Restricted Partitions from Conformal Field Theory
We study the correlators of irregular vertex operators in two-dimensional conformal field theory (CFT) in order to propose an exact analytic formula for calculating numbers of partitions, that is:
1) for given $N,k$, finding the total number $\lambda(N|k)$ of length $k$ partitions of $N$: $N=n_1+...+n_k;0
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