D.Freed has formulated and proved an index theorem on odd dimensional spin manifolds with boundary. The proof is based on analysis by Calderon and Seeley. In this note we are going to give a proof of this theorem using the heat kernels methods for boundary conditions of Dirichlet and Von Neumann type. Moreover we consider also the Atiyah-Patodi-Singer spectral boundary condition which is not considered in Freed's paper. As a direct consequence of the method, we will obtain some information about isospectral invariants of the boundary conditions. This proof does not uses the cobordism invariance of index and are easily generalized to family case.
In this paper, we establish an equivariant version of Dai-Zhang's Toeplitz index theorem for compact odd-dimensional spin manifolds with even-dimensional boundary.
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 …
We generalize the transgression formula for the eta form of Bismut, Cheeger and Berline, Getzler, Vergne for vertical Dirac operators on a fibre bundle with odd dimensional fibres where the Dirac operators have locally at most one eigenvalue of multiplicity one crossing zero transversally.
We study an index of a transversal Dirac operator on an odd-dimensional manifold $X$ with locally free $\mathbb{S}^1$-action. One difficulty of using heat kernel method lies in the understanding of the asymptotic expansion as $t\to 0^+$. By a probabilistic approach via the Feynman-Kac formula, the transversal heat kernel on $X$ can be linked to the ordinary heat kernel for functions on the orbifold $M=X/\mathbb{S}^1$ which is more tractable. After some technical results for a uniform bound estimate as $t\to 0^+$, we are reduced from the transversal, orbifold situation to the classical situatio…
A renormalized index theorem for some complete asymptotically regular metrics: the Gauss-Bonnet theorem
研究了Gauss-Bonnet定理作为重正化指标定理的边度量。这些度量包括AdS/CFT对应的庞加莱-爱因斯坦度量。重整化用于解释$L^2$ -上同调空间的曲率积分和维数，以及进行指标定理的热方程证明。对于共形紧度量even mod $x^m$，证明了共形紧流形上热核的有限时间超迹的重整化与特殊边界定义函数的选择无关.
Among those transversally elliptic operators initiated by Atiyah and Singer, Kohn's $\Box_b$ operator on CR manifolds with $S^1$ action is a natural one of geometric significance for complex analysts. Our first main result establishes an asymptotic expansion for the heat kernel of such an operator with values in its Fourier components, which involves an unprecedented contribution in terms of a distance function from lower dimensional strata of the $S^1$-action. Our second main result computes a local index density, in terms of \emph{tangential} characteristic forms, on such manifolds including…
By studying modular invariance properties of some characteristic forms, we get some new anomaly cancellation formulas on $(4r-1)$ dimensional manifolds. As an application, we derive some results on divisibilities of the index of Toeplitz operators on $(4r-1)$ dimensional spin manifolds and some congruent formulas on characteristic number for $(4r-1)$ dimensional spin$^c$ manifolds.
We compute explicitly, and without any extra regularity assumptions, the large time limit of the fibrewise heat operator for Bismut-Lott type superconnections in the L^2-setting. This is motivated by index theory on certain non-compact spaces (families of manifolds with cocompact group action) where the convergence of the heat operator at large time implies refined L^2-index formulas. As applications, we prove a local L^2-index theorem for families of signature operators and an L^2-Bismut-Lott theorem, expressing the Becker-Gottlieb transfer of flat bundles in terms of Kamber-Tandeur classes. …
