I have a question concerning proposition $1.38$ in Heat Kernels and Dirac operators:
Let \begin{equation} A:\Gamma(M,\Lambda(TM^*)\otimes E)\to\Gamma(M,\Lambda(TM^*)\otimes E) \end{equation} be a super-connection on the super-bundle $E$, then $A^2$ is a local operator and hence is given by the action of a differential form $F\in\Gamma(M,\Lambda(TM^*)\otimes\mathrm{End}(E))$.
($F$ is called the curvature of $A$ in proposition $3.43$.) I think that a few more words could/should have been said here and I would like to make sure that my understanding is correct:
- We are using the fact that a local operator $P:\Gamma(M,E)\to\Gamma(M,F)$ is given by the action of a section $\Gamma(M,L(E,F))$.
- Applying 1. to the case $P=A^2$ yields a section of $\mathrm{End}(\Lambda(TM^*)\otimes E)$ instead of $\Lambda(TM^*)\otimes\mathrm{End}(E)$.
- To obtain $F$ we actually apply 1. to the restriction of $A^2$ to $\Gamma(M,E)$. This yields a section \begin{equation} M\to L(E,\Lambda(TM^*)\otimes E) \end{equation} and the composition with the obvious vector bundle isomorphism \begin{equation} L(E,\Lambda(TM^*)\otimes E)\to \Lambda(TM^*)\otimes\mathrm{End}(E) \end{equation} is the desired $F$.