Smooth functorial field theories from B-fields and D-branes

In the Lagrangian approach to 2-dimensional sigma models, B-fields and D-branes contribute topological terms action of worldsheets both open closed strings. We show that these naturally fit into a 2-dimensional, smooth open-closed functorial field theory (FFT) in sense Atiyah, Segal, Stolz-Teichner. give detailed construction this FFT, based on definition suitable bordism category. category, all manifolds are equipped with map spacetime target manifold. Further, object allowed have boundaries; endpoints strings stretched between D-branes. The values our FFT obtained from B-field its via transgression. Our generalises work Bunke-Turner-Willerton include At same time, it Moore-Segal about TQFTs spaces. provide number further features FFT: we depends functorially D-branes, is thin homotopy invariant, comes positive reflection structure Freed-Hopkins. Finally, describe how related classification by Lauda-Pfeiffer.