Abstract:
To understand the genus 1 term of the Landau-Ginzburg B model, we study heat kernel expansions of some Schordinger type operators on noncompact spaces. Motivated by the path integral formulation of the heat kernel, we introduced the parabolic distance, which also appeared in Li-Yau's famous work on Harnack inequality. With the help of parabolic distance, we could derive a nice point-wise asymptotic expansion of the heat kernel with a strong remainder estimate. In particular, we get an asymptotic expansion of the heat kernel of Witten Laplacian $\Box_{Tf}$ induced by $d+Tdf\wedge$, where $T>0$ is the deformation parameter. When the deformation parameter of Witten deformation and time parameter are coupled, we derive an asymptotic expansion of trace of heat kernel for small-time $t$ and obtain a local index theorem. Also, we invented a new rescaling technique to write down the local index density explicitly. If time permits, I will also explain how we define Ray-Singer torsion on noncompact spaces. This is joint work with Xianzhe Dai.
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