Double-dimer condensation and the PT-DT correspondence

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.

The orbifold DT/PT vertex correspondence

“…When I had obtained PT invariants of toric CY 3-orbifolds with transverse A n−1 singularities (see Theorem 3.22) through exploring the orbifold PT topological vertex as in [6], I was informed by Patrick Lei last October during the email communication on the progress of PT theory of [C 2 /Z n+1 ] × P 1 that Zhang [45] had also studied the orbifold PT topological vertex and obtained the 1-leg orbifold PT vertex formula in terms of Schur functions. Thanks to the recent proof of [35,Conjecture 2] by Jenne, Webb and Young in [15], Zhang and I had independently obtained the same formula of PT partition function for toric CY 3-orbifolds with transverse A n−1 singularities, see also [45,Theorem/Conjecture 4.19]. However, there are several differences between ours as follows.…”

“…where the conjectured DT/PT vertex correspondence is resolved recently in [15] for the Calabi-Yau case. There is an alternative approach to the DT/PT correspondence in [5,43].…”

“…In [45], the author has proved the 1-leg orbifold DT/PT vertex correspondence by obtaining the explicit formula for the 1-leg orbifold PT topological vertex. In this paper, we will extend this result to the full 3-leg case, i.e., prove the 3-leg orbifold DT/PT vertex correspondence by generalizing the DT/PT Calabi-Yau topological vertex correspondence in [15] to the orbifold case.…”

“…In [15], the DT topological vertex V λµν (see Definition 3.2) is described as some dimer model via the well-known correspondence between 3D partitions and dimer configurations on the honeycomb graph while the PT topological vertex W λµν (see Definition 3.7) has certain double-dimer model description with the established correspondence between labelled box configurations and tripartite double-dimer configurations on the honeycomb graph. With these descriptions, the so-called graphical condensation discovered by Kuo [17,Theorem 5.1] is applied to derive a graphical condensation recurrence for the DT vertex while the analogue technique developed in [14, Theorem 1.0.2] is employed to obtain a graphical condensation recurrence for the PT vertex.…”

“…The above graphical condensation recurrence (1) generalizes [15,Equation (2)] if we set q l := q for all l (or equivalently n = 1). However, it is not enough to determine the 3-leg orbifold DT/PT vertex correspondence since we don't have the 2-leg correspondence Y 1 (λ, ∅, ν) = Y 2 (λ, ∅, ν) and Y 1 (∅, µ, ν) = Y 2 (∅, µ, ν) as the base case by Remark 4.8.…”