Classical and quantum field theories may be thought of as appropriate functors from (some version of) the cobordism category. In this talk (based on joint work with Mnev and Reshetikhin) I will show how this can be achieved in the context of BF theories by the perturbative expansion of the BV action in the presence of a boundary. The theory associate a complex to the boundary and a cocycle (the state or partition function) to the bulk. I will show how the procedure is compatible with gluing along boundary components. This is a quantization of the classical BV-BFV theory. An outlook to the general case will be presented, time permitting.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/158