Counting surfaces on Calabi-Yau 4-folds II: $\mathrm{D}\mathrm{T}$-${\mathrm{P}\mathrm{T}}_0$ 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 $\mathrm{D}\mathrm{T},{\mathrm{P}\mathrm{T}}_0,{\mathrm{P}\mathrm{T}}_1$ invariants and conjecture a $\mathrm{D}\mathrm{T}$-${\mathrm{P}\mathrm{T}}_0$ correspondence. For certain tautological insertions, we derive Lefschetz principles in both the compact and toric case allowing reductions to 3-dimensional $\mathrm{D}\mathrm{T},\mathrm{P}\mathrm{T}$ invariants. We also develop a topological vertex and conjecture a $\mathrm{D}\mathrm{T}$-${\mathrm{P}\mathrm{T}}_0$ vertex correspondence. These methods enable us to verify our conjectures in several examples.