I'm reading "Heat kernels and Dirac operators" by Berline, Getzler and Vergne. I have some trouble to understand a identity on the bottom of page 146 which is essential for the proof of the Atiyah-Singer theorem:

If $a \in \Gamma(M, C(M))$ and $b \in \Gamma(M, \operatorname{End}_{C(M)}(\mathcal{E}))$, then the point-wise supertrace of the section $a \otimes b \in \Gamma(M, C(M) \otimes \operatorname{End}_{C(M)}(\mathcal{E})) \cong \Gamma(M, \operatorname{End}(\mathcal{E}))$ was shown in (3.21) to equal the Berezin integral

$$\operatorname{Str}_{\mathcal{E}}(a(x) \otimes b(x))= (-2i)^{n/2} \sigma_n(a(x)) \operatorname{Str}_{\mathcal{E}/S}(b(x))$$

I would like try to lose shortly some words on some used notations (not all eg super trace and construction of spinor bundles are well explaned in the book):

Let $M$ be a compact oriented Riemannian manifold of even dimension $n$. The Clifford bundle $C(M)$ of $M$ is the bundle of Clifford algebras over $M$ whose fibre at $x \in M$ is the Clifford algebra $C(T^*_xM)$ of the Euclidean spaces $T^*_x M$. The Clifford bundle $C(M)$ is an associated bundle to the orthonormal frame bundle,

$$C(M)=O(M) \times_{O(n)} C(\mathbb{R}^n).$$

Let $S$ be the spin group accociated to Clifford algebra $C(\mathbb{R}^n)$. Assume, that associated spinor bundle

$$\mathcal{S}= \operatorname{Spin}(M) \times_{\operatorname{Spin}(n)} S $$

on $M$ exists globally and let $\mathcal{W}$ another arbitrary finite dimensional vector bundle. Clearly $\mathcal{S}$ is a Clifford module, since the action of $C(\mathbb{R}^n)$ on $S$ leads to an action of the associated bundle $C(M)$.

Let $\mathcal{W}$ another arbitrary finite dimensional vector bundle on $M$ and define $\mathcal{E}= \mathcal{W} \otimes \mathcal{S}$. Observe that since $\operatorname{End}(\mathcal{S})= C(M)$ we obtain $\operatorname{End}_{C(M)}(\mathcal{E})= \operatorname{End}(\mathcal{E})$.

If we now come back to indenity

$$\operatorname{Str}_{\mathcal{E}}(a(x) \otimes b(x))= (-2i)^{n/2} \sigma_n(a(x)) \operatorname{Str}_{\mathcal{E}/S}(b(x))$$

I not understand, let take a look into proposition 3.21 which is proposed to imply it:

Proposition 3.21. let $V$ is an even-dimensional oriented Euclidean vector and $Q$ the induced bilinear form on $V \otimes \mathbb{C}$. Then if the quadratic form $Q$ is non-degenerate, then there is, up to a constant factor, a unique supertrace on $C(V)$, equal to $T \circ \sigma$. The supertrace $\operatorname{Str}(a)$ defined in (3.8) equals

$$\operatorname{Str}(a)=(-2i)^{n/2} T \circ \sigma(a).$$

Here $\sigma$ is the symbol map $\sigma: C(V) \otimes \mathbb{C} \to \wedge(V \otimes \mathbb{C})$ (page 104) ater complexification of $V$ and $T$ Berezin integral (page 42).

Now back to $\operatorname{Str}_{\mathcal{E}}(a(x) \otimes b(x))= (-2i)^{n/2} \sigma_n(a(x)) \operatorname{Str}_{\mathcal{E}/S}(b(x))$. What happens with Berezin on the right side?



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