Generalized MacMahon G(q) as q-deformed CFT Correlation Function

TL;DR

This paper derives a d-dimensional MacMahon function as a correlation function in q-deformed conformal field theory, revealing its structure through vertex operators and clarifying its limitations in generating generalized Young diagrams.

Contribution

It provides a quantum field theoretical derivation of G_d(q) as a correlation function using q-deformed vertex operators, extending the understanding of MacMahon functions in CFT.

Findings

G_d(q) expressed as a (d+1)-point correlation function

Operators are composites of q-deformed hierarchical vertex operators

G_d(q) for d≥4 does not generate all d-dimensional Young diagrams

Abstract

Using $\Gamma_{\pm}(z) $ vertex operators of the $c=1$ two dimensional conformal field theory, we give a 2d-quantum field theoretical derivation of the conjectured d- dimensional MacMahon function G$_{d}(q) $. We interpret this function G$_{d}(q) $ as a $(d+1) $- point correlation function $\mathcal{G}_{d+1}(z_{0},...,z_{d}) $ of some local vertex operators $\mathcal{O}%_{j}(z_{j}) $. We determine these operators and show that they are particular composites of q-deformed hierarchical vertex operators $% \Gamma _{\pm}^{(p)}$, with a positive integer p. In agreement with literature's results, we find that G$_{d}(q) $, $d\geq 4$, cannot be the generating functional of all \textit{d- dimensional} generalized Young diagrams .