Bijectivizing the PT-DT Correspondence

TL;DR

This paper establishes a bijective correspondence between PT and DT generating functions in enumerative geometry, using vertex operators and toggle involutions to provide a combinatorial proof of their relationship.

Contribution

It introduces a bijective proof of the PT-DT correspondence in specific cases by employing vertex operators and toggle involutions, offering a combinatorial perspective.

Findings

Demonstrates a bijective proof of the PT-DT relationship

Utilizes vertex operators and toggle involutions in the proof

Provides a combinatorial interpretation of the generating functions

Abstract

Pandharipande-Thomas theory and Donaldson-Thomas theory (PT and DT) are two branches of enumerative geometry in which particular generating functions arise that count plane-partition-like objects. That these generating functions differ only by a factor of MacMahon's function was proven recursively by Jenne, Webb, and Young using the double dimer model. We bijectivize two special cases of the result by formulating these generating functions using vertex operators and applying a particular type of local involution known as a toggle, first introduced in the form we use by Pak.