Double boxes and double dimers

TL;DR

This paper provides a combinatorial proof linking double-box configurations in Donaldson-Thomas theory to MacMahon's plane partition generating function via double-dimer configurations and graphical condensation techniques.

Contribution

It introduces a novel combinatorial approach connecting double-box configurations to double-dimer models, proving a key generating function identity in rank 2 Donaldson-Thomas theory.

Findings

Established a correspondence between double-box and double-dimer configurations.

Applied graphical and double-dimer condensation to prove the generating function equality.

Connected combinatorial models to algebraic geometry results.

Abstract

We give a combinatorial proof of a result in rank 2 Donaldson-Thomas theory, which states that the generating function for certain plane-partition-like objects, called double-box configurations, is equal to a product of MacMahon's generating function for (boxed) plane partitions. In our proof, we first give the correspondence between double-box configurations and double-dimer configurations on the hexagon lattice with a particular tripartite node pairing. Using this correspondence, we apply graphical condensation and double-dimer condensation to prove the result.