In this talk, we revisit DT4 theory via derived algebraic geometry. Conceptually, the Borisov-Joyce/Oh-Thomas virtual cycles for (-2)-shifted symplectic derived schemes are the fundamental cycles of quasi-smooth Lagrangians. I will first introduce a virtual pullback formula for Lagrangian correspondences. I will then explain deformation invariance in terms of the exactness of the symplectic forms. I will also explain Kiem-Li's cosection localization via (-1)-shifted closed 1-forms and their connection to DT4 theory through the twisted (-2)-shifted cotangent bundles. Finally, I will explain how to construct (-1)-shifted closed 1-forms by generalizing Pantev-Toen-Vaquie-Vezzosi's integration map, which provides the reduced virtual cycles for counting surfaces.
