K-Theoretic Donaldson-Thomas Theory of $[\mathbb{C}^2/\mu_r]\times\mathbb{C}$ and Factorization

TL;DR

This paper computes equivariant K-theoretic Donaldson-Thomas invariants for a specific orbifold product space using advanced factorization and rigidity techniques, extending existing methods to orbifold Hilbert schemes.

Contribution

It generalizes Okounkov's factorization technique to orbifold Hilbert schemes and applies it to compute invariants for $[C^2/mu_r] imes C$ using a rigidity argument.

Findings

Established factorization property for orbifold Hilbert schemes

Generated explicit formulas for invariants as plethystic exponentials

Demonstrated the effectiveness of rigidity techniques in orbifold DT theory

Abstract

We compute the equivariant K-theoretic Donaldson--Thomas invariants of $[\mathbb{C}^2/\mu_r]\times \mathbb{C}$ using factorization and rigidity techniques. For this, we develop a generalization of Okounkov's factorization technique that applies to Hilbert schemes of points on orbifolds. We show that the (twisted) virtual structure sheaves of Hilbert schemes of points on orbifolds satisfy the desired factorization property. We prove that the generating series of Euler characteristics of such factorizable systems are the plethystic exponential of a simpler generating series. For $[\mathbb{C}^2/\mu_r]\times \mathbb{C}$, the computation is then completed by a rigidity argument, involving an equivariant modification of Young's combinatorial computation of the corresponding numerical Donaldson-Thomas invariants.