Wall-crossing for zero-dimensional sheaves and Hilbert schemes of points on Calabi--Yau 4-folds

Arkadij Bojko

Published 2021-02-01Version 1

We define a vertex algebra on the moduli stack of the auxiliary abelian category of pairs to give a precise formulation of the wall-crossing conjecture for Calabi--Yau 4-folds proposed by Gross--Joyce--Tanaka arXiv:2005.05637. Assuming that it holds, we prove the conjecture of Cao--Kool arXiv:1712.07347 for 0-dimensional DT4 invariants on projective Calabi--Yau 4-folds. We also give formulae for generating series of topological virtual fundamental classes of Hilbert schemes of points and 0-dimensional sheaves. Using these, we prove a version of Nekrasov's conjecture arXiv:1712.08128 for compacts and its generalization to K-theory classes of higher rank. We compute higher rank Segre series. After defining the correct generalization of virtual Verlinde series for 4-folds, we obtain a very simple Segre--Verlinde correspondence. Finally, we relate universal series of general invariants for elliptic surfaces and elliptic curves to those on CY 4-folds via a universal transformation.