论文标题
非压缩歧管上的WITTEN变形：热核扩展和局部索引定理
Witten Deformation on Non-compact Manifold: Heat Kernel Expansion and Local Index Theorem
论文作者
论文摘要
很少探索Schrödinger操作员的热核和热痕迹的渐近膨胀，即使在$ \ Mathbb {C}^n $的情况下，也很简单，具有（Quasi-homosiensos）多项式电位，也很复杂。由热核的路径整体配方的动机，我们引入了抛物线距离，这也出现在Li-Yau著名的抛物线harnack估算中。在抛物线距离的帮助下，我们为Witten Laplacian的热核的渐近膨胀而得出，其剩余估计值很强。当Witten变形和时间参数的变形参数耦合时，我们将用于小型$ t $的热核痕迹的渐近膨胀，并获得局部索引定理。这是我们在非平凡空间上理解Landau-Ginzburg B模型的第二篇论文，在随后的工作中，我们将开发在非紧凑型环境中Witten变形的射线手扭转。
Asymptotic expansions of heat kernels and heat traces of Schrödinger operators on non-compact spaces are rarely explored, and even for cases as simple as $\mathbb{C}^n$ with (quasi-homogeneous) polynomials potentials, it's already very complicated. Motivated by path integral formulation of the heat kernel, we introduced a parabolic distance, which also appeared in Li-Yau's famous work on parabolic Harnack estimate. With the help of the parabolic distance, we derive a pointwise asymptotic expansion of the heat kernel for the Witten Laplacian with strong remainder estimate. 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. This is the second of our papers in understanding Landau-Ginzburg B-models on nontrivial spaces, and in subsequent work, we will develop the Ray-Singer torsion for Witten deformation in the non-compact setting.
