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