Witten Deformation on Non-compact Manifolds

Witten deformation is a deformation of the de Rham complex introduced in an extremely influential paper by Witten. Witten deformation on closed manifolds has found many beautiful applications, including that of Bismut-Zhang on the Cheeger-M\"uller Theorem/Ray-Singer Conjecture. Development in mirror symmetry, in particular the Calabi-Yau/Landau-Ginzburg correspondence has highlighted the importance of mathematical study of Landau-Ginzburg models. This leads to a whole range of questions on the Witten deformation on non-compact manifolds. In this talk we will discuss our recent work, joint with Junrong Yan, on the $L^2$-cohomology, the heat asymptotic expansion, the local index theorem, and the Ray-Singer analytic torsion in this setting.