We compute the equivariant K-theoretic DT invariants of certain CY3 orbifolds using factorization and rigidity techniques. This proves a conjecture by Cirafici and refines earlier work by Young to the equivariant K-theoretic setting. Our proof follows the outline of Okounkov's proof of Nekrasov's formula, which this project generalizes to some orbifolds. The main technical tool is an extension Okounkov's factorization technique to the orbifold setting, which allows us to simplify the DT generating series by splitting 0-dimensional substacks into smaller parts. The computation is finished by a limit-equivariant version of Young's combinatorial computation, using the rigidity principle. We are currently working on extending this to analogous CY4 orbifolds.

This is joint work with Henry Liu and Nick Kuhn. We prove the equivariant K-theoretic DT/PT vertex correspondence using wall-crossing. The more classical DT/PT correspondence, conjectured by Pandharipande and Thomas was proven by Bridgeland and Toda using Joyce-Song's wall-crossing framework for DT invariants defined by the Behrend function. We prove the equivariant K-theoretic refinement, conjectured by Nekrasov and Okounkov, using Liu's equivariant K-theoretic version of Joyce's more recent wall-crossing framework for virtual classes. An important technical tool is the construction of a symmetized pullback of a symmetric perfect obstruction theory on the orginial DT and PT moduli stacks to a symmetric almost perfect obstruction theory on auxiliary moduli stacks.