Witten deformation on non-compact manifolds: heat kernel expansion and local index theorem

Abstract

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 $${\mathbb {C}}^n$$ 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.

Singular Weyl’s law with Ricci curvature bounded below

“…In another direction, if one considers Schrödinger operators on complete noncompact manifolds, then Weyl's law could also involve log term, see [26, Theorem 1.1 and Remark 1.2]. For related results on the small time heat kernel expansion of Witten Laplacian, see [25,Theorem 1.1]. Such expansion implies a Weyl law by the Karamata-Tauberian type theorem [26,Theorem 2.4] Crucial to our construction we further develop the examples constructed by the last two authors in [61].…”

“…For related results on the small time heat kernel expansion of Witten Laplacian, see [25,Theorem 1.1]. Such expansion implies a Weyl law by the Karamata-Tauberian type theorem [26,Theorem 2.4] Crucial to our construction we further develop the examples constructed by the last two authors in [61]. These examples are the first Ricci limit spaces whose Hausdorff dimensions may not be integer and moreover, the Hausdorff dimension of the singular set is bigger than that of the regular set, see Theorem 3.1.…”