Felix Thimm (UiO) - K-Theoretic Orbifold Donaldson-Thomas Invariants and Factorization

Donaldson-Thomas invariants "virtually" count curves in a given threefold. They factor into two parts: a part which only counts curves, and a degree 0 part, which counts 0-dimensional subschemes. The degree 0 part can be fully computed with a closed formula by relating them to combinatorial counting of plane partitions, which are certain configurations of boxes in 3D space. DT theory comes in various refinements. Nekrasov's formula refines the relation to counts of plane partitions to equivariant K-theoretic DT theory and gives a closed formula for refined degree 0 DT invariants.

 

Degree 0 DT invariants of orbifolds are related to counts of colored plane partitions, where the boxes are colored in a way determined by the orbifold structure. This allows the computation of closed formulas for some orbifolds. We refine these closed formulas to equivariant K-theoretic DT theory by modifying the techniques used in Okounkov's proof of Nekrasov's formula to work for orbifolds. We will explain these techniques in the case of schemes and describe some of the modifications to make them work for orbifolds.