K-Theoretic DT/PT Correspondence for Toric Calabi–Yau 4-Folds

Source abstract

Abstract Recently, Nekrasov discovered a new “genus” for Hilbert schemes of points on $${\mathbb {C}}^4$$ C 4 . We extend its definition to Hilbert schemes of curves and moduli spaces of stable pairs, and conjecture a K -theoretic DT/PT correspondence for toric Calabi–Yau 4-folds. We develop a K -theoretic vertex formalism, which allows us to verify our conjecture in several cases. Taking a certain limit of the equivariant parameters, we recover the cohomological DT/PT correspondence for toric Calabi–Yau 4-folds recently conjectured by the first two authors. Another limit gives a dimensional reduction to the K -theoretic DT/PT correspondence for toric 3-folds conjectured by Nekrasov–Okounkov. As an application of our techniques, we find a conjectural formula for the generating series of K -theoretic stable pair invariants of $$\text {Tot}_{{\mathbb {P}}^1}({\mathcal {O}}(-1) \oplus {\mathcal {O}}(-1) \oplus {\mathcal {O}})$$ Tot P 1 ( O ( - 1 ) ⊕ O ( - 1 ) ⊕ O ) . Upon dimensional reduction to the resolved conifold, we recover a formula which was recently proved by Kononov–Okounkov–Osinenko.