Higher Categories of Push-Pull Spans and Applications to Topological Quantum Field Theories

In this project we attempt a formalization of Rozansky-Witten models in the functorial field theory framework. Motivated by work of Calaque-Haugseng-Scheimbauer, we construct a family of symmetric monoidal (infinity,3)-categories PP(C; Q), parametrized by an infinity-category C with finite limits and a representable functor Q : C^op --> CAlg(Cat_infinity) with pushforwards, which contains correspondences in C with local systems in Q that compose via a push-pull formula. We apply this general construction to provide an approximation CRW to the 3-category of Rozansky-Witten models whose existence was conjectured by Kapustin-Rozansky-Saulina. This approximation behaves like a non-deformed or ``commutative'' version of the conjectured 3-category. We also prove some 2-dimensional dualizability results about CRW and explore the connections with work of Brunner-Carqueville-Fragkos-Roggenkamp on matrix factorizations, which are known to model the affine Rozansky-Witten models.