Two studies of topological quantum field theory in two dimensions

Take powers of the determinant line bundles on the relative moduli spaces (or stacks) of principal G-bundles over relative curves C over a base space B, and then push them down to the base space B --- the resulting sheaves over B, which are in fact vector bundles, are known as the Verlinde bundles. They satisfy certain ``gluing'' properties and yield a structure known as (K-theoretic) cohomological field theory, which is a type of families 2-dimensional topological quantum field theory. In the first part of this thesis, we carry out an investigation of the higher twisted Verlinde bundles, as defined by Teleman--Woodward, for the case when G is C^*, the multiplicative group of complex numbers. In particular, we show that their Chern characters can be written in terms of tautological classes. This generalizes the known fact that the ordinary Verlinde bundles have tautological Chern characters; indeed we also include in our study an explicit computation of the Chern characters of the C^*-Verlinde bundles. From this computation we are able to explicitly demonstrate the gluing properties in action.By results of Costello and of Konstevich and collaborators, cohomological field theories can also arise from categories that are homologically smooth, proper, and Calabi--Yau. The state space of such a theory is given by the Hochschild (co)homology of the category. In the second part of this thesis, then, we study categories of matrix factorizations for Landau--Ginzburg models (X,W), where X is a complex variety. We show that when X is smooth and Calabi--Yau as a variety and W has a proper critical locus, the corresponding category of matrix factorizations is homologically smooth, proper, and Calabi--Yau as a category. Furthermore, we compute the Hochschild cohomology of these matrix factorization categories, and we get the result predicted by Kontsevich. These results, which are joint with Daniel Pomerleano, generalize the results of Dyckerhoff for the case when X is affine local and W has an isolated singularity.We conclude with some brief remarks on forthcoming work, also joint with Daniel Pomerleano, in which we propose mirror partners to certain Fano non-toric 3-folds which can be degenerated to nodal toric varieties. Using our results on matrix factorization categories, we are able to prove some homological mirror symmetry results for these proposed mirror pairs