UC Santa Barbara

Motivated by the Calabi-Yau/Landau-Ginzburg correspondence, we study BCOV-type torsion for Landau-Ginzburg B-models. To pave the way for understanding the Landau-Ginzburg model from the index theoretic point of view, we study Witten deformation, which was introduced in one of Witten's extremely influential papers in 1987, on noncompact manifolds. In Part I, we explore the analysis of Witten deformation on noncompact manifolds: we show the asymptotic growth of eigenvalues and the decay of eigenfunctions near infinity as well as the expansion and the estimate of the heat kernel for Schr\"odinger-type operators on noncompact manifolds. With heat kernel expansions of Sch\"ordinger-type operators, we define the Ray-Singer metric (analytic torsion) for the Witten deformation associated with some flat vector bundle and explore several nice properties of it (independence of metrics, Cheeger-Muller/Bismut-Zhang theorem, e.t.c.). Next, we move on to study Landau Ginzburg B-models in Part II. In chapter 6, we investigate the genus-zero theory for Landau-Ginzburg B-models, and establish Calabi-Yau/Landau-Ginzburg correspondence for the $tt^*$ structure and the Weil-Peterson type metric.