Advances in Theoretical and Mathematical Physics

Volume 22 (2018)

Number 5

An analytic formula for numbers of restricted partitions from conformal field theory

Pages: 1271 – 1288



Dimitri Polyakov (Center for Theoretical Physics, Sichuan University, Chengdu, China; Max-Planck-Instutut fuhr Gravitationsphysik (Albert-Einstein-Institut), Potsdam, Germany; and Institute of Information Transmission Problems (IITP), Moscow, Russia)


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 $\lambda (N \vert k)$ of length $k$ partitions of $N : N = n_1 + \dotsc + n_k ; 0 \lt n_1 \leq n_2 \leq \dotsc \leq n_k$.

2) finding the total number $\lambda (N) = \sum^N_{k=1} \lambda (N\vert k)$ of partitions of a natural number $N$.

We propose an exact analytic expression for $\lambda (N\vert k)$ 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 $f(z) = h(z)e^{-\frac{i}{z}}$ where $h(z)$ is regular and non-vanishing at $z = 0$.

The final formula for $\lambda (N\vert k)$ is given in terms of regularized ($\epsilon$-ordered) finite series in the higher-derivative Schwarzians and incomplete Bell polynomials of the above conformal transformation at $z = i \epsilon (\epsilon \to 0)$.

Published 2 May 2019