| Witten-Morse view the stable submanifold from a critical point of a Morse function ƒ as semi-classical limit of an eigenform with respect to Witten Laplacian Deltaƒ. We prove an enhancement of this correspondence in Chapter 1 by showing operations on eigenforms, involving taking wedge product and Witten's twisted Green's operator G ƒ successively, has semi-classical limit counting gradient trees between critical points. The main tool is the semi-classical analysis for the Witten's twisted Green's operator.;Mirror symmetry is a duality relating A-model on a Calabi-Yau manifold X to B-model on its mirror Xˇ . Limiting to their large structure limits we obtain (noncompact) semi- at mirror pairs which are torus bundle X0 and Xˇ 0 over a base B0. Strominger-Yau-Zaslow's approach suggests that fiberwise Fourier transform over the base, given in Chapter 2, is responsible for the mirror correspondence. In particular, the deformation theory from the large structure limits which is responsible for capturing X and Xˇ should be identified via semi- at transform.;We propose a differential graded Lie algebra (dgLa)L* X0 on fiberwise loop space of X 0, which is transformed to Kodaira-Spencer dgLa on Xˇ 0, to capture the quantum deformation of X0 that is necessary to recover X which is believed to have informations from holomorphic disk instantons. In Chapter 3, we interpret holomorphic disk instantons as semi-classical limit of 1-form on loop space, motivated from Witten-Morse theory. We prove that solving Maurer-Cartan equation inL*X 0 has semi-classical limit as the scattering process introduced by Kontsevich- Soibelman in [24], which is known to govern the deformation from Xˇ0 to Xˇ . |
