To understand the genus one term of the LG B-model, we develop the notion of analytic torsion for Witten deformation on noncompact manifolds. Analytic torsion on noncompact manifolds turns out to have its own significance beyond only studying the LG B-model: We discover an interesting relationship between the analytic torsion of the Witten deformation on a noncompact manifold and the analytic torsion of its compact core (with absolute/relative boundary conditions). Based on this observation, we develop a new analytic proof of gluing formulas for analytic torsions. This is partly joint work with Xianzhe Dai.