Speaker: Henry Liu (IPMU)

Abstract: On smooth quasi-projective toric 3- and 4-folds, vertices are the contributions from an affine toric chart to the enumerative invariants of Donaldson-Thomas (DT) or Pandharipande-Thomas (PT) moduli spaces. Unlike partition functions, vertices are fundamentally torus-equivariant objects, and they carry a great deal of combinatorial complexity, particularly in equivariant K-theory. In joint work with Nick Kuhn and Felix Thimm, we give two different proofs of the K-theoretic 3-fold DT/PT vertex correspondence. Both proofs use equivariant wall-crossing in a setup originally due to Toda; one uses a Mochizuki-style master space, while the other uses ideas from Joyce's recent universal wall-crossing machine. A crucial new ingredient is the construction of *symmetrized* pullbacks of symmetric obstruction theories on moduli stacks, using Kiem-Savvas'étale-local notion of almost-perfect obstruction theory. I believe our techniques, particularly the Joyce-style approach, can also be applied to related questions such as DT/PT descendent transformations, the DT crepant resolution conjecture, and the 4-fold DT/PT vertex correspondence.