Counting surfaces on Calabi-Yau 4-folds II: DT-PT₀ correspondence

This is the second part in a series of papers on counting surfaces on Calabi-Yau 4-folds. In this paper, we introduce K-theoretic DT, PT₀, PT₁ invariants and conjecture a DT-PT₀ correspondence. For certain tautological insertions, we derive Lefschetz principles in both the compact and toric case allowing reductions to 3-dimensional DT, PT invariants. We also develop a topological vertex and conjecture a DT-PT₀ vertex correspondence. These methods enable us to verify our conjectures in several examples.