Double-dimer condensation and the PT-DT correspondence

TL;DR

This paper proves a conjecture linking two generating functions in algebraic geometry related to plane partitions and topological vertices, using condensation identities and a novel dimer model application.

Contribution

It establishes the equality of two key generating functions in Donaldson-Thomas and Pandharipande-Thomas theories, advancing understanding in algebraic geometry.

Findings

Proved the conjecture relating the two generating functions.

Connected algebraic geometry with the tripartite double-dimer model.

Used a Desnanot-Jacobi-type condensation identity in the proof.

Abstract

We resolve an open conjecture from algebraic geometry, which states that two generating functions for plane partition-like objects (the "box-counting" formulae for the Calabi-Yau topological vertices in Donaldson-Thomas theory and Pandharipande-Thomas theory) are equal up to a factor of MacMahon's generating function for plane partitions. The main tools in our proof are a Desnanot-Jacobi-type condensation identity, and a novel application of the tripartite double-dimer model of Kenyon-Wilson.