On Gammelgaard's formula for a star product with separation of variables

TL;DR

This paper demonstrates the equivalence of Gammelgaard's graph-based formula for star products with separation of variables to an inversion formula on a formal Fock space, providing new insights into the structure of the graphs involved.

Contribution

It offers a direct proof of the inversion formula, providing an alternative perspective on Gammelgaard's formula and explaining why the directed graphs are acyclic.

Findings

Equivalence between Gammelgaard's formula and an inversion formula

Direct proof of the inversion formula

Insight into the acyclic nature of directed graphs

Abstract

We show that Gammelgaard's formula expressing a star product with separation of variables on a pseudo-Kaehler manifold in terms of directed graphs without cycles is equivalent to an inversion formula for an operator on a formal Fock space. We prove this inversion formula directly and thus offer an alternative approach to Gammelgaard's formula which gives more insight into the question why the directed graphs in his formula have no cycles.