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

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 $\mathcal{O}$. We determine these operators and show that they are particular composites of q-deformed hierarchical vertex operators $$, 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 .