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.

在拉格朗日方法中的二维sigma模型中，B场和D谱对打开和关闭弦的世界表的作用都具有拓扑作用。我们证明，这些术语自然适用于Atiyah，Segal和Stolz-Teichner的二维平滑开闭函数场论（FFT）。基于合适的平滑bordism类别的定义，我们给出了该平滑FFT的详细构造。在此bordism类别中，所有流形都配备了到时空目标流形的平滑映射。此外，允许对象歧管具有边界；例如，这些是在D形脑筋之间伸展的开放弦的端点。我们的FFT值是通过B场及其D波段通过海侵获得的。我们的构造将Bunke-Turner-Willerton的工作概括为开放弦乐。同时，它概括了Moore-Segal关于开放式TQFT的工作，以包括目标空间。我们提供了FFT的许多其他功能：我们证明它在功能上取决于B场和D谱，我们证明它是稀薄的同伦不变性，并且我们证明它配备有正反射结构。霍普金斯精神的释放。最后，我们描述了我们的构造与Lauda–Pfeiffer获得的开放式TQFT的分类之间的关系。