Mock Modularity of Calabi-Yau Threefolds - Part 2 of 3

Explore the modular properties of rank 0 generalized Donaldson-Thomas invariants in this advanced mathematics lecture focusing on how these properties enable computation of the invariants and their implications for other topological invariants. Delve into the derivation of precise modular properties for generating functions of rank 0 DT invariants, which emerge as higher depth mock modular forms. Examine the modular anomaly equation that encodes these properties, along with its extensions and solution methods for both compact and non-compact Calabi-Yau threefolds. Learn how the modular framework provides a powerful computational tool for understanding the deep mathematical structures underlying string theory and algebraic geometry. This presentation forms the second part of a comprehensive three-part series delivered at the M-Seminar at Kansas State University, building upon foundational concepts of modular forms, mock modular forms, Jacobi forms, and indefinite theta series while preparing for advanced applications in wall-crossing phenomena and topological string integration methods.