Benjamin Young: Combinatorics of the toric Calabi-Yau topological vertex in Donaldson-Thomas theory and Pandharipande-Thomas theory

Donaldson-Thomas (DT) theory and Pandharipande-Thomas (PT) theory are, very loosely speaking, two different ways to "count" curves in a space X. In DT theory, one "counts" subschemes of X. In PT theory, one "counts" stable pairs in the derived category. Conjecturally, the two theories are equivalent, in that the generating function for "reduced" DT invariants is the same as the generating function of the PT invariants.

In the case where the space X is toric and Calabi-Yau, each of these generating functions can be computed using the enumerative combinatorics of a plane-partition-like structure (the "topological vertex" in the title), which puts both problems within my realm of expertise. I will show how to rephrase these box-counting problems in terms of the honeycomb dimer and double-dimer models, and outline a proof that they indeed do give the same answer. This greatly simplifies the current geometric proof, which is (as far as I know) not entirely written down, and which requires a lengthy detour through the relatively distant field of Gromov-Witten theory.

It should be possible to follow this talk without knowing much algebraic geometry or combinatorics.