# Curve counting and DT/PT correspondence for Calabi-Yau 4-folds

## TL;DR

This paper explores curve counting invariants for Calabi-Yau 4-folds, proposing a conjectural DT/PT correspondence, providing evidence, and developing a vertex formalism for toric cases with localization techniques.

## Contribution

It introduces a conjectural DT/PT correspondence for Calabi-Yau 4-folds, extending stable pair invariants and formulating a vertex formalism for toric cases.

## Key findings

- Evidence for the conjecture in the compact case
- Development of a vertex formalism for toric Calabi-Yau 4-folds
- Verification of the DT/PT vertex relation in several examples

## Abstract

Recently, Cao-Maulik-Toda defined stable pair invariants of a compact Calabi-Yau 4-fold $X$. Their invariants are conjecturally related to the Gopakumar-Vafa type invariants of $X$ defined using Gromov-Witten theory by Klemm-Pandharipande. In this paper, we consider curve counting invariants of $X$ using Hilbert schemes of curves and conjecture a DT/PT correspondence which relates these to stable pair invariants of $X$. After providing evidence in the compact case, we define analogous invariants for toric Calabi-Yau 4-folds using a localization formula. We formulate a vertex formalism for both theories and conjecture a relation between the (fully equivariant) DT/PT vertex, which we check in several cases. This relation implies a DT/PT correspondence for toric Calabi-Yau 4-folds with primary insertions.