Counting surfaces on Calabi-Yau 4-folds II: $\mathrm{DT}$-${\mathrm{PT}}_0$ correspondence

Abstract

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 $\mathrm{DT},{\mathrm{PT}}_0,{\mathrm{PT}}_1$ invariants and conjecture a $\mathrm{DT}$-${\mathrm{PT}}_0$ correspondence. For certain tautological insertions, we derive Lefschetz principles in both the compact and toric case allowing reductions to 3-dimensional $\mathrm{DT},\mathrm{PT}$ invariants. We also develop a topological vertex and conjecture a $\mathrm{DT}$-${\mathrm{PT}}_0$ vertex correspondence. These methods enable us to verify our conjectures in several examples.

97 pages