Embedding topological quantum field theories functorially in the UV

Renormalization group (RG) flow describes how the behavior of quantum field theories changes from high ultra-violet (UV) energies to low infra-red (IR) energies. In this thesis, I describe a new method which allows the calculation of topological IR correlators directly within the UV. This method is based on codimension-one domain-wall defects and provides a complete representation of the topological IR theory in terms of the UV theory. From this perspective, RG flow of bulk theories can be regarded as RG flow of the codimension-one identity defect in the fixed UV bulk theory. The procedure is not restricted to RG flow but generalizes to topological quantum field theories projected to subtheories by projection defects. It is furthermore shown that projection defects in triangulated defect categories (such as defects in 2d topologically twisted N = (2, 2) theories) always come with complementary projection defects. The unprojected theory then decomposes into the theories associated to the two projection defects. The new method is applied to phases of gauged linear sigma models and RG flows between 2d supersymmetric Landau-Ginzburg orbifold models, for which the respective defects can be described in terms of matrix factorizations