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arXiv · 1906.07856

$K$-theoretic DT/PT correspondence for toric Calabi-Yau 4-folds

Yalong Cao·Martijn Kool·Sergej Monavari
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Abstract

Recently, Nekrasov discovered a new "genus" for Hilbert schemes of points on $\mathbb{C}^4$. We conjecture a DT/PT correspondence for Nekrasov genera for toric Calabi-Yau 4-folds. We verify our conjecture in several cases using a vertex formalism. 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 the local resolved conifold. Upon dimensional reduction to the resolved conifold, we recover a formula which was recently proved by Kononov-Okounkov-Osinenko.

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Yalong Cao, Martijn Kool, Sergej Monavari. 2019-06-19. $K$-theoretic DT/PT correspondence for toric Calabi-Yau 4-folds. https://doi.org/10.1007/s00220-022-04472-0

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