Invariants counting sheaves on Calabi--Yau 4-folds are obtained by virtual integrals over moduli spaces. These are expressed in terms of virtual fundamental classes, which conjecturally fit into
a wall-crossing framework proposed by Joyce. I will review the construction of vertex algebras in terms of which one can express the WCF. I describe how to use them to obtain explicit results for Hilbert schemes of points. As a consequence, I reduce multiple conjectures to a technical proof of the WCF. Surprisingly, one gets a complete correspondence between invariants of Hilbert schemes of CY 4-folds and elliptic surfaces.
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- Junior Geometry and Topology Seminar