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

## TL;DR

This paper proposes a new $K$-theoretic DT/PT correspondence for toric Calabi-Yau 4-folds, verifies it in specific cases, and explores its limits to connect with known conjectures and invariants in lower dimensions.

## Contribution

It introduces a conjectural $K$-theoretic DT/PT correspondence for toric Calabi-Yau 4-folds and verifies it using vertex formalism, linking it to existing conjectures in lower dimensions.

## Key findings

- Verification of the conjecture in several cases using vertex formalism
- Recovery of the cohomological DT/PT correspondence in a certain limit
- Derivation of a $K$-theoretic formula for the local resolved conifold invariants

## 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.