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The authors of Heat kernels and Dirac operators chose to consider a generalized Laplacian $H$ on a bundle $E\otimes|\Lambda|^{1/2}$ and heat kernels of $H$ are defined to be sections of $(E\otimes|\Lambda|^{1/2})\boxtimes(E^*\otimes|\Lambda|^{1/2})$. I have always wondered why

  1. $H$ is not assumed to be a Laplacian on $E$ and
  2. heat kernels are not defined to be sections of $E\boxtimes E^*$

as in other literature: The most important case studied in the book is the square of a Dirac operator $D$, which by definition 3.36 is an operator on the bundle $E$. In addition, the kernel of $D^2$ is clearly assumed to be a section of $E\boxtimes E^*$ in the the McKean-Singer formula (theorem 3.50). Hence I am really curious to know the motivation of the authors.

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  • $\begingroup$ I am also quite puzzled by this. My Riemannian geometry education told me integration is by default associated with the metric, in locally coordinate with respect to $\sqrt g dx_1...dx_n$. At this moment I understand this as a kind of Bourbaki style thing, want to do integration without explicitly mentioning the metric on base space - although usually there is one. My view could change when I get further into the BGV book. $\endgroup$
    – Yuval
    Commented Dec 16, 2023 at 17:45
  • $\begingroup$ @Yuval Yeah I also thought that they wanted to consider a more general setup, i.e. a manifold $M$ instead of a Riemannian manifold. But the heat kernel is only defined for generalized Laplacians and whether a differential operator is a generalized Laplacian depends on the metric, right? So we always have to assume a metric, right? $\endgroup$
    – Filippo
    Commented Dec 17, 2023 at 4:29

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