Virtual Reductions and Intersection Theory on Moduli Space of Sheaves on Calabi-Yau 4 Folds

Explore a comprehensive lecture on virtual reductions and intersection theory in moduli spaces of sheaves on Calabi-Yau 4-folds. Delve into the speaker's collaborative work with Diaconscu and Yau, focusing on reducing virtual fundamental classes from 4-fold moduli spaces to virtual cycles of 3-fold moduli spaces. Examine specific examples of virtual reductions and their applications in studying Donaldson-Thomas (DT) invariants of torsion sheaves supported on smooth projective surfaces within non-compact Calabi-Yau 4-folds. Investigate the reduction properties of universal Atiyah classes and their implications for relating 4-fold DT theory to reduced 3-fold DT theory and moduli spaces of sheaves on base surfaces. Consider predictions about the modularity of 4-fold invariants when the base surface is an elliptic K3. Explore potential extensions to framed sheaves on cotangent bundles of surfaces and the connection between 4-fold invariants and Vafa-Witten invariants. Gain insights into generalizing these results to sheaves supported on non-algebraic surfaces and discover hidden structures in their associated partition functions.