Vertex operators and character varieties

TL;DR

This paper proves combinatorial conjectures related to vertex operators and character varieties, using a gluing formula to simplify complex calculations to a basic case involving projective space with punctures.

Contribution

It introduces a novel application of vertex operators to derive a gluing formula for generating series in the context of character varieties.

Findings

Proved several combinatorial conjectures.

Developed a gluing formula for generating series.

Reduced complex computations to a basic case.

Abstract

We prove some combinatorial conjectures extending those proposed in [13, 14]. The proof uses a vertex operator due to Nekrasov, Okounkov, and the first author [4] to obtain a "gluing formula" for the relevant generating series, essentially reducing the computation to the case of complex projective space with three punctures.