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Double-dimer condensation and the PT-DT correspondence

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.