  We develop a method of quantization for free field theories on manifolds with
boundary where the bulk theory is topological in the direction normal to the
boundary and a local boundary condition is imposed. Our approach is within the
Batalin-Vilkovisky formalism. At the level of observables, the construction
produces a stratified factorization algebra that in the bulk recovers the
factorization algebra developed by Costello and Gwilliam. The factorization
algebra on the boundary stratum enjoys a perturbative bulk-boundary
correspondence with this bulk factorization algebra. A central example is the
factorization algebra version of the abelian Chern-Simons/Wess-Zumino-Witten
correspondence, but we examine higher dimensional generalizations that are
related to holomorphic truncations of string theory and $M$-theory and involve
intermediate Jacobians.
