## Abstract

Recently, Nekrasov discovered a new “genus” for Hilbert schemes of points on C4

. 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 TotP1(O(−1)⊕O(−1)⊕O)

. Upon dimensional reduction to the resolved conifold, we recover a formula which was recently proved by Kononov–Okounkov–Osinenko.

. 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 TotP1(O(−1)⊕O(−1)⊕O)

. Upon dimensional reduction to the resolved conifold, we recover a formula which was recently proved by Kononov–Okounkov–Osinenko.

Original language | English |
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Pages (from-to) | 225-264 |

Journal | Communications in Mathematical Physics |

Volume | 396 |

DOIs | |

Publication status | Published - Nov 2022 |