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arXiv · 1702.04631

An Analytic Formula for Numbers of Restricted Partitions from Conformal Field Theory

Dimitri Polyakov
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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: 1) for given $N,k$, finding the total number $λ(N|k)$ of length $k$ partitions of $N$: $N=n_1+...+n_k;0<n_1\leq{n_2}...\leq{n_k}$. 2) finding the total number $λ(N)=\sum_{k=1}^Nλ(N|k)$ of partitions of a natural number $N$ We propose an exact analytic expression for $λ(N|k)$ 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 $λ(N|k)$ is given in terms of regularized ($ε$-ordered) finite series in the generalized higher-derivative Schwarzians and incomplete Bell polynomials of the above conformal transformation at $z=iε$ ($ε\rightarrow{0}$)

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Dimitri Polyakov. 2017-03-02. An Analytic Formula for Numbers of Restricted Partitions from Conformal Field Theory. https://arxiv.org/abs/1702.04631

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