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

Abstract

Recently, Nekrasov discovered a new “genus” for Hilbert schemes of points on {mathbb {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}}). Upon dimensional reduction to the resolved conifold, we recover a formula which was recently proved by Kononov–Okounkov–Osinenko.