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We present elements of a general method for the quantization of a certain class of field theories on manifolds with boundary, namely the class of field theories which are topological normal to the boundary. The techniques we discuss are generalizations of methods of Kevin Costello and Owen Gwilliam, which apply to closed manifolds. As an example, we will show how Kontsevich’s star product on the space of functions on a Poisson vector space arises from the quantization of a field theory on the (two-dimensional) upper half-plane. Joint work with Ben Albert.