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

Abstract

Abstract Recently, Nekrasov discovered a new “genus” for Hilbert schemes of points on $${\mathbb {C}}^4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msup> </mml:math> . 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 $$\text {Tot}_{{\mathbb {P}}^1}({\mathcal {O}}(-1) \oplus {\mathcal {O}}(-1) \oplus {\mathcal {O}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mtext>Tot</mml:mtext> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⊕</mml:mo> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⊕</mml:mo> <mml:mi>O</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . Upon dimensional reduction to the resolved conifold, we recover a formula which was recently proved by Kononov–Okounkov–Osinenko.