主 题: 薛定谔方程，形变理论和镜像对称
报告人: 范辉军 教授 (数学学院)
时 间: 2011-04-22 14:00-15:00
地 点: 理科一号楼1114（数学所活动）
Let $(M,g)$ be a complete noncompact Kaehler manifold with bounded geometry and $f$ be a strongly tame holomorphic function defined on $M$. We can associate a twisted Laplacian operator to $(M,g,f)$. This is a Schroedinger type operator acting on forms. We can prove it has discrete spectrum in $L^2$ space. As a consequence, we can construct the Hodge theory to $(M,g,f)$. If $f_t$ is a strong deformation of $f$, then we can get the Hodge bundle over the parameter space and prove that it has $tt^*$ geometric structure. This gives a rigorous mathematical proof to Cecotti-Vafa\'s $tt^*$ geometry structure and furthermore, more interesting structures were found by us. This work can be viewed as the construction of (Landau-Ginzburg) B model theory，and has intimate relation to the deformation theory of projective varieties, toric varieties ( in the sense of CY/LG correspondence), quantum singularity theory by Fan-Jarvis-Ruan-Witten, and Gromov-Witten theory (in the sense of mirror symmetry).
